Upload
others
View
7
Download
0
Embed Size (px)
Citation preview
Research ArticleBifurcation Analysis for Phage Lambda withBinding Energy Uncertainty
Ning Xu12 Xue Lei3 Ping Ao3 and Jun Zhang12
1 Joint Institute of UMich-SJTU Shanghai Jiao Tong University Shanghai 200240 China2 Key Laboratory of System Control and Information Processing Ministry of Education Shanghai 200240 China3 Shanghai Center for Systems Biomedicine Shanghai Jiao Tong University Shanghai 200240 China
Correspondence should be addressed to Jun Zhang zhangjun12sjtueducn
Received 30 October 2013 Accepted 25 December 2013 Published 3 February 2014
Academic Editor Jose Nacher
Copyright copy 2014 Ning Xu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In a phage 120582 genetic switch model bistable dynamical behavior can be destroyed due to the bifurcation caused by inappropriatelychosen model parameters Since the values of many parameters with biological significance often cannot be accurately acquired itis thus of fundamental importance to analyze how and to which extent the system dynamics is influenced by model parametersespecially those parameters pertaining to binding energies In this paper we apply a Jacobian method to investigate the relationbetween bifurcation and parameter uncertainties on a phage 120582 OR model By introducing bistable range as a measure of systemrobustness we find that RNApolymerase binding energies have theminimumbistable ranges among all the binding energies whichimplies that the uncertainties on these parameters tend to demolish the bistability more easily Moreover parameters describingmutual prohibition between proteins Cro and CI have finite bistable ranges whereas those describing self-prohibition have infinitybistable ranges Hence the former are more sensitive to parameter uncertainties than the latter These results help to understandthe influence of parameter uncertainties on the bifurcation and thus bistability
1 Introduction
Bistability is a salient feature of phage 120582 genetic switchMathematical models with a number of parameters havebeen established to describe its bistable behavior Howeverparameter variation in phage 120582 model may considerablychange the system dynamics Therefore in the case whenthe model parameters are not precisely known it is usefulto study the influence of parameter uncertainty on systemdynamics
Since 1980s researchers have developed several modelsfor phage 120582 genetic switch Reference [1] proposed a quan-titative model for phage 120582 according to the principles ofstatistical thermodynamics The model by [2] included twovariables that is concentrations of proteins CI and CroReference [3] extended this model and pointed out thediscrepancy between experimental and theoretical resultsMoreover models with four variables were built to includemore ingredients such as mRNA concentration in phage 120582[4] Other useful models include Markov chain model [5]
delayed reaction stochastic model [6] fuzzy logic model [7]decision making [8] and qualitative model [9] Based onthesemodels significant research progresses on phage 120582 havebeen made [10ndash15]
Most of these models contain parameters that cannot beprecisely determined For example themodel by [2] hasmorethan 10 binding energy related parameters each of whichmayvary up to plusmn006 kcalmol and thus affects system dynamicsNonspecific binding also introduces additional factors thatinfluence the system models [16] Reference [17] developeda model that has 13 binding energy related parameterswith uncertainty ranges around 05 kcalmol Moreover [18]pointed out that some parameters such as protein synthesisrates can be changed as surrounding environment changesReference [19] reported that single point mutation of DNAleads to plusmn2 kcalmol change for binding free energy Due tothese parameter uncertainties it is often desired to analyzethe effect of these parameters on system dynamics
Researchers have been using sensitivity and robustnessto study the influence of parameter uncertainty Sensitivity
Hindawi Publishing CorporationComputational Biology JournalVolume 2014 Article ID 465216 11 pageshttpdxdoiorg1011552014465216
2 Computational Biology Journal
and robustness are closely related because a robust systemis not sensitive to parameter variations [20] By using lysisfrequency as a sensitivity measure [21] simulated a mutationmodel introduced by [22] and concluded that spontaneousswitching from lysogen to lysis depends sensitively on modelparameters By systematically perturbing binding energieswith plusmn1 kcalmol [23] concluded that lysogenic activity atpromoter 119875
119877119872remains stable under this perturbation
In this paper we investigate bifurcation by using bistablerange as a system robustness measure In a typical case phage120582 genetic switch model has three equilibria Two of them arestable corresponding to the lysogenic and lytic states andthe third one is a saddle point When some parameters areperturbed it is possible that one stable equilibrium and thesaddle point coalesce then both of them disappear resultingin the loss of bistable behavior Hence bistable region can bedefined as the parameter region in which bistable behaviorexists [18] The bistable region for a parameter is large if thesystem dynamics is not sensitive to that parameter
We are interested in analyzing bistable ranges for bindingenergy related parameters Some other parameters such asprotein degradation rate and protein synthesis rate withrespect to gene transcription have already been proved thattheir variations may lead to the loss of bistability [3 24]Our calculations are based on a phage 120582 OR model [3]Although DNA looping formed by cooperations betweenthe left operators and the right operators affects the stabilityof the genetic switch [25] the OR model only includesthe effects of the right operators which are believed toplay a critical role for lysogeny maintenance [26] We willcalculate bistable ranges for all binding energies which arecrucial in determining the affinity that proteins bind tophage 120582 genes Furthermore we define Z-shaped bifurcationand calculate Jacobian matrix to study robustness We findout that different binding energy affects lysogeny stabilitydifferently some of them may destroy bistability whereasothers have little influence on system dynamics Thereforewe need to precisely determine the values of those sensitiveparameters in the model because otherwise we may not getan effective bistable switch
2 Phage 120582 Modeling
We first present phage 120582 gene regulatory networks and thenintroduce their mathematical model Lysogeny and lysis aretwo possible states for phage 120582 It is believed that wild-typephage 120582 has quite a stable lysogenic state since lysogeny lossrate is even lower than 10minus8 per cell generation [27] Howevercertain conditions such as the exposure to UV light [28 29]the presence of mitomycin C [30] or the starvation of thehost cells [31] can induce phage 120582 from lysogeny to lysisMathematical models have been built to successfully explainstability and high efficiency of genetic switch
The concentrations of two proteins CI and Cro can serveas an indication of which state phage 120582 is in lysogeny orlysis In lysogeny the concentration of protein CI in the cell issignificantly higher than that of protein Cro whereas in lysis
Gene for CI expression Gene for Cro expression
Cro dimerCI dimer
OR1OR3 OR2middot middot middot
PRM PR
Figure 1 Three binding sites on a DNA segment The sites 1198741198771
1198741198772 and 119874
1198773 can be bound by CI dimer Cro dimer or RNA
polymeraseThis DNA segment consists of promoters119875119877119872
and119875119877 If
the RNA polymerase binds to 1198741198773 cI gene is expressed if the RNA
polymerase binds to1198741198771 and119874
1198772 cro gene is expressedThe amount
of Cro and CI in the cell directly determines which state phage 120582 isin
Table 1 Parameter values for our model
Δ1198661119888= minus120 kcalmoldagger Δ119866
2119888= minus108 kcalmoldagger
Δ1198663119888= minus134 kcalmoldagger Δ119866
1119903= minus125 kcalmolDagger
Δ1198662119903= minus105 kcalmolDagger Δ119866
3119903= minus95 kcalmolDagger
Δ11986612119903
= minus27 kcalmolDagger Δ11986623119903
= minus29 kcalmolDagger
Δ11986611987712
= minus125 kcalmoldagger Δ1198661198773= minus115 kcalmoldagger
119870119903= 556 times 10
minus9Mdagger 119870119888= 326 times 10
minus7Mdagger
119873RNAP = 30 times 10minus8Msect
1205831= 20 times 10
minus2minminus1dagger
1205832= 36 times 10
minus2minminus1dagger 119877 = 199 cal(molsdotK)119879 = 310K 119896 = 111
para
daggerdenotes the parameter values from [14] Daggerfrom [32] sectfrom [21] and parafrom[3]
protein Cro dominates the cell The amount of CI and Crothus implies which state phage 120582 is in
Figure 1 illustrates the gene segment regulating cro andcI gene expression Three binding sites 119874
1198771 1198741198772 and 119874
1198773
control the expression of two kinds of regulatory proteinsThe 119875
119877119872promoter covers the whole binding site 119874
1198773 and
part of 1198741198772 controlling cI gene expression When RNA
polymerase (RNAP) binds to the site1198741198773 the gene expression
is on resulting in the production of protein CI The 119875119877
promoter covers all of 1198741198771 and part of 119874
1198772 controlling
cro expression When RNAP binds to sites 1198741198771 and 119874
1198772
simultaneously the 119888119903119900 gene expression is on The bindingsites may be occupied by either Cro or CI protein UsuallyCro and CI first form their dimers so that they are able tobind to 119874
1198771 1198741198772 and 119874
1198773 prohibiting the gene expressions
on these sitesSeveral phage 120582 models have been established based on
this regulating mechanism In this paper we adopt the well-known two-dimensionalORmodel stated in [3]with updatedparameter values given in Table 1 This model involves three119874119877gene binding sites which are believed to be the most
critical factors for the stability of genetic switch [33] Thistwo-dimensional model is in line with the general biologicalswitch model for two-gene networks [34] To keep it simplewe eliminate the effects from other activities in the cell suchas nonspecific binding [16 17] stochastic gene expression
Computational Biology Journal 3
[35] and intrinsic and extrinsic noises [36] The model isdescribed by the following system equations
119889119873119903
119889119905= 1198601119891119903minus 1205831119873119903
119889119873119888
119889119905= 1198602119891119888minus 1205832119873119888
(1)
where
119891119903= 11987528+ 11987529+ 11987530+ 11987540
119891119888=
33
sum
119894=31
119875119894+ 119896(
40
sum
119895=34
119875119895)
(2)
119875119904=
exp (minusΔ119866119904119877119879)119873
119894(119904)
1199032119873119895(119904)
1198882119873119896(119904)
RNAP
sum40
119898=1exp (minusΔ119866
119898119877119879)119873
119894(119898)
1199032119873119895(119898)
1198882119873119896(119898)
RNAP
(119904 = 1 2 40)
(3)
The subscript 119904 corresponds to 40 binding configurationsshown in Table 2 The conventions between CI dimer (119873
1199032)
Cro dimer (1198731198882) and their monomers (119873
119903and119873
119888) are
1198731199032=1
2119873119903+1
8119870119903minus1
8radic1198702119903+ 8119873119903119870119903
1198731198882=1
2119873119888+1
8119870119888minus1
8radic1198702119888+ 8119873119888119870119888
(4)
From (1)ndash(4)119873119903119873119888represent concentrations of proteins
CI and Cro The parameters 1198601and 119860
2are synthesis rates of
CI and Cro respectively 1205831and 120583
2are the overall rates that
decrease protein concentration which are caused by normalprotein degradation and cell growth [37]The function 119891
119903(or
119891119888) represents the overall probability that the 119875
119877(or 119875119877119872
)promoter is occupied by RNAP 119875
119904is the probability that
119904th configuration occurs and Δ119866119904is the total binding energy
needed for this configuration The exponents 119894(119904) 119895(119904) and119896(119904) represent the numbers of CI dimers Cro dimers andRNAP at 119904th configuration respectively Δ119866
119904is calculated
by summing up the needed single binding energies listed inTable 1 including cooperation energy For example Δ119866
5=
Δ11986612119903
+Δ1198661119903+Δ1198662119903 The left right arrows in Table 2 indicate
the cooperations 119896 is the coefficient for positive feedbackWhen CI dimer binds to 119874
1198772 positive feedback makes the
transcription of cI gene more efficient hence promotes itsexpression 119870
119903 119870119888are dissociation constants for CI and Cro
proteins respectively 119877 represents the universal gas constantand 119879 the absolute temperature
3 Bifurcation in Phage 120582
The model used in this paper is two-dimensional nonlinearordinary differential equations To get all the equilibria we setthe left hand side of (1) to zero and solve the transcendentalequations It can be found that in general the model has threeequilibria However the number of equilibria may vary dueto bifurcation under parameter variation In phage 120582 model
Table 2 40 Binding configurations for the three sites
119904 1198741198773 119874
1198772 119874
1198771
12 CI23 CI24 CI25 CI2harr CI26 CI2 CI27 CI2harr CI28 CI2harr CI2harr CI29 Cro210 Cro211 Cro212 Cro2 Cro213 Cro2 Cro214 Cro2 Cro215 Cro2 Cro2 Cro216 Cro2 CI217 Cro2 CI218 Cro2 Cro2 CI219 Cro2 CI220 CI2 Cro221 Cro2 CI2 Cro222 CI2 Cro223 CI2 Cro224 CI2 Cro2 Cro225 Cro2 CI2harr CI226 CI2 Cro2 CI227 CI2harr CI2 Cro228 RNAP29 CI2 RNAP30 Cro2 RNAP31 RNAP CI232 RNAP CI2harr CI233 RNAP CI2 Cro234 RNAP35 RNAP CI236 RNAP Cro237 RNAP Cro238 RNAP Cro2 CI239 RNAP Cro2 Cro240 RNAP RNAP
frequently occurring is the saddle node bifurcation whoseprecise mathematical definition is given as follows [38]
Definition 1 (saddle node bifurcation) A saddle node bifur-cation in dynamical system takes place when two equilibriumpoints coalesce and disappear
In (1) two parameters 1198601and 119860
2are not specified
in Table 1 They represent proportional constants relatingthe transcription initiation rate to the absolute rate of CI and
4 Computational Biology Journal
CA CBAB
Nr at equilibrium Nc at equilibrium
A2 A2
Figure 2 Saddle node bifurcation with the variation of Crosynthesis rate 119860
2 This is a schematic plot and not drawn to scale
Cro protein synthesis [3] We use Figure 2 to illustrate Z-shaped bifurcation with the variation of 119860
2 Set 119860
1= 8 times
10minus9Mmin and divide the whole plane into three regions
as shown in Figure 2 labeled by A B and C Initially whenthe value of 119860
2is small in region A for example 119860
2asymp
5times10minus9Mmin there exists only one stable point and thus one
single curve segment for both 119873119903and 119873
119888 At the boundary
of regions A and B saddle node bifurcation occurs 1198602
increases to region B for example a typical value 1198602asymp
5 times 10minus8Mmin and two new equilibria (one saddle and
one stable equilibrium) emerge resulting in three equilibriaNote that this is a saddle node bifurcation with the reverseddirection in comparison to Definition 1
At the boundary of regions B and C saddle nodebifurcation takes place again leading to the vanishing oftwo equilibria (one saddle point and one stable equilibrium)In region C say 119860
2asymp 5 times 10
minus7Mmin only one stableequilibrium remains We notice the motion of the saddlepoint When the saddle point shows up at the boundary ofregions A and B the lysis stable equilibrium appears as wellAs parameter119860
2increases the saddle pointmoves toward the
lysogen stable point and finallymergeswith it at the boundaryof regions B and C Two successive bifurcations form a Z-shaped curve are shown in Figure 2
We have not yet specified the boundaries between theseregions quantitatively Next we solve the values of the bound-aries by using the Jacobian method Let us first introduce thefollowing two theorems [38]
Theorem 2 (Hartman-Grobman Theorem) If the Jacobianof the nonlinear system = 119892(119909) at equilibrium namely119869(1199090) has no zero or purely imaginary eigenvalues then the
phase trajectory of the system = 119892(119909) resembles that of thelinearized system = 119869(119909
0)119911 near 119909
0if none of the eigenvalues
of 119869(1199090) are on the 119895120596 axis
For a given equilibrium 1199090 in most cases we can
determine its dynamical behavior qualitatively by calculatingits Jacobian matrix 119869(119909
0) For a two-dimensional system its
Jacobian matrix can be calculated as
119869 =
[[[[
[
1205971198921
120597119873119903
1205971198921
120597119873119888
1205971198922
120597119873119903
1205971198922
120597119873119888
]]]]
]
(5)
The following theorem gives the necessary conditions forthe critical point that bifurcation occurs [38]
Theorem 3 For any nonlinear system = 119892(119911 120583) the systemgoes through a saddle node bifurcation at 119911 = 119911
dagger 120583 = 120583dagger only
if the following relations are satisfied
119892 (119911dagger 120583dagger) = 0 (6)
10038161003816100381610038161003816100381610038161003816
120597119892
120597119911(119911dagger 120583dagger)
10038161003816100381610038161003816100381610038161003816= 0 (7)
From (6) 119911dagger is an equilibrium point In a planar system(7) is the determinant of system Jacobian When the deter-minant of Jacobian matrix vanishes at least one eigenvalueis zero Jacobian is a characterization of planar systemdynamics For variation of parameter 119860
2 the system in (1)
can be rewritten as
119889119873119903
119889119905= 1198921(119873119888 119873119903 1198602)
119889119873119888
119889119905= 1198922(119873119888 119873119903 1198602)
(8)
From Theorems 2 and 3 for a given 1198602 the Jacobian matrix
is evaluated at equilibrium points 119873dagger119888and 119873
dagger
119903 These two
equilibrium points can be represented as a function of 1198602as
shown in Figure 2ByTheorem 3we know that the bifurcation appearswhen
Jacobian has a zero eigenvalue From Figure 3 we can obtainthe approximate values of critical points by checking thepoints of the eigenvalue curves crossing zero line Thesevalues are 56 times 10minus9Mmin and 34 times 10minus7Mmin
To determine the stability of an equilibrium point wecalculate the eigenvalue of the Jacobian matrix Figure 3 plotsthe eigenvalues as a function of 119860
2 The Jacobian matrix
can be calculated analytically (see the Appendix for details)By solving the value of 119860
2that makes the determinant of
the Jacobian matrix vanish we get the critical value of 1198602
at which bifurcation occurs When 1198602is small (less than
56 times 10minus9Mmin in Figure 3) there are only two negative
eigenvalues corresponding to one equilibrium point Thisequilibrium is thus stable [38]
As 1198602exceeds 56 times 10
minus9Mmin we have the casewhen there are three equilibria and correspondingly sixeigenvalues Among these three equilibria two are stableand the third is unstable (saddle point) This is becausepositive eigenvalue appears at saddle point (see dash-dotline in Figure 3) For a two-dimensional linear system if theJacobian has one positive and one negative eigenvalues at theequilibrium the equilibrium is a saddle point If119860
2continues
increasing only one equilibrium point is left (1198602greater than
34 times 10minus7Mmin in this case) and both of its eigenvalues
are negative The curves in Figure 3 have two intersectionswith zero eigenvalue which indicates that the eigenvalues ofJacobian vanish twice forming a Z-shaped bifurcation
Computational Biology Journal 5
0
002
004
Eige
nval
ues o
f Jac
obia
n
At equilibrium for lysogenAt equilibrium for lysisAt saddle point
10minus9
10minus8
10minus7
10minus6
10minus5
10minus4
minus012
minus01
minus008
minus006
minus004
minus002
A2 (Mmin)
Figure 3 Jacobian eigenvalues at equilibria The figure shows howthe eigenvalues of Jacobian at equilibria change with 119860
1fixed at
8 times 10minus9Mmin and 119860
2varying The solid line (black) shows
the eigenvalues of lysis equilibrium The dashed line (blue) is forlysogeny equilibrium The dot-dashed line (red) is for saddle
4 Bifurcation Analysis of BindingEnergy Uncertainty
In this section we will analyze bifurcation for Δ119866rsquos byusing the Jacobian method In Table 1 the symbol Δ119866 withsubscripts such as Δ119866
1119903and Δ119866
2119888represents Gibbs free
energy required for CI or Cro dimers to bind to the ORsites Because of the difficulty in determining the precisevalues of the binding energy it is useful to study how theequilibria change as these Gibbs free binding energies vary Inthis section we set synthesis rates 119860
1and 119860
2at appropriate
values 1198601= 8 times 10
minus9Mmin and 1198602= 5 times 10
minus8Mmin Tocalculate the equilibria we set the left hand side of (1) to zeroand obtain the expression of119873
119903and119873
119888as functions of Δ119866rsquos
119873lowast
119903= 1198911(Δ1198661119888 Δ1198662119888 Δ119866
1198773)
119873lowast
119888= 1198912(Δ1198661119888 Δ1198662119888 Δ119866
1198773)
(9)
where 119873lowast119903and 119873lowast
119888are the concentrations of proteins CI and
Cro at the equilibrium point Once (9) is obtained we candetermine the quantitative behavior around equilibria whenchanging Δ119866rsquos The relative sensitivities are given by
119878119873119903
(Δ119866119904) =
1205971198911
120597Δ119866119904
119878119873119888
(Δ119866119904) =
1205971198912
120597Δ119866119904
(10)
where the subscript 119904 isin 1119903 2119903 11987712 1198773 Since the systemequilibria cannot be analytically solved we may apply the
numerical procedure to calculate (10) as a measure of param-eter sensitivity As to the study on system robustness theJacobianmethod can be used to study bifurcation introducedby parameter variation
Since there are ten differentΔ119866s we divide them into fourgroups
(1) Δ1198661119903 Δ1198662119903 and Δ119866
3119903 binding free energies of CI
(2) Δ1198661119888 Δ1198662119888 and Δ119866
3119888 binding free energies of Cro
(3) Δ11986612119903
and Δ11986623119903
the cooperation between proteinsCIrsquos if both 119874
1198771 and 119874
1198772 or 119874
1198772 and 119874
1198773 are bound
by CI dimers(4) Δ119866
11987712and Δ119866
1198773 RNAP binding energies
We then study each of these four groups
41 Δ1198661119903 Δ1198662119903 and Δ119866
3119903 We systematically perturb the
values of Δ1198661119903 Δ1198662119903 and Δ119866
3119903to investigate their impact
on the system dynamics Two proper robustness measuresare the range of parameters in which lysogeny exists (stablerange) and inwhich bistable dynamics exists (bistable range)In stable range it is possible to keep phage 120582 dormant andintegrate into the gene of the host On the other hand bistablerange defines the parameter range to keep bistability and themaximal uncertainty tolerance In this case it implies thatthe system favors dynamics with two stable equilibria and asaddle Further if the stochastic force is strong enough phage120582 can switch from the lysogenic stable point to the lytic stablepoint even without the parameter changes
Figures 4(a)ndash4(c) illustrate the impact of Δ1198661119903 Δ1198662119903 and
Δ1198663119903
variations We set parameter values as in Table 1 andfind out all the equilibria when varying oneΔ119866
119904 We find that
the energy Δ1198663119903does not show Z-shaped bifurcation while
the other two parameters Δ1198661119903and Δ119866
2119903show quite similar
behaviors since Figures 4(a) and 4(b) are almost identicallythe sameThe result is not surprising because Δ119866
1119903and Δ119866
2119903
play similar roles in system dynamics These two parametersdetermine the likelihood that CI dimer binds to sites 119874
1198771
and 1198741198772 When either of the two sites is occupied by protein
dimers the expression of 119888119903119900 gene is prohibited resulting inlysogenic stateNonetheless the parameterΔ119866
3119903indicates the
likelihood of CI dimer binding to the site1198741198773 If the absolute
value of Δ1198663119903is too large it is easy for CI dimer to bind to
1198741198773 thus cI gene expression is prohibited resulting in lysis
If the absolute value ofΔ1198663119903is small the Z-shaped bifurcation
vanishes In this case cI gene expression is not prohibited butit has no influence on cro gene expression Bistable behavioralways exists when Δ119866
3119903is less than its nominal value in
Table 1 Without stopping or weakening the expression of crogene single lysogen stable point layout cannot be achievedjust by increasingΔ119866
3119903 Comparedwith other parameters the
system dynamics is more robust to Δ1198663119903variation
Table 3 lists the stable range bistable range and the lengthof bistable range for all tenΔ119866s It can be shownbyTheorem 2that one of the three equilibria is a saddle and the other twoare stable points Jacobian matrix is also used to determinethe boundary of stable and bistable ranges according toTheorem 3 As long as Δ119866
1119903is less than the nominal value
6 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG1r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG1r
(a) Z-shaped bifurcation for Δ1198661119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG2r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG2r
(b) Z-shaped bifurcation for Δ1198662119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG3r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG3r
(c) Bifurcation for Δ1198663119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG1c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG1c
(d) Bifurcation for Δ1198661119888
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG2c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG2c
(e) Bifurcation for Δ1198662119888
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG3c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG3c
(f) Z-shaped bifurcation for Δ1198663119888
Figure 4 Movement of equilibria under parameter variation Z-shaped bifurcation can be generated by variation of Δ1198661119903 Δ1198662119903 and Δ119866
3119888
All vertical axes are plotted on a log scale
at minus1133 kcalmol or Δ1198662119903
is less than minus934 kcalmol thesystem always has lysogenic stateThey are critical bifurcationpoints For Δ119866
3119903 however lysogenic state always exists if it is
greater than minus1233 kcalmol Consequently for CI protein alarger binding energy to119874
1198771 and119874
1198772 helps to keep the DNA
of phage 120582 silent in the host The system tends to be unstableif the values of binding energies for CI to 119874
1198771 and 119874
1198772
become small thus forming Z-shaped bifurcation Parametervariations may lead to Z-shaped bifurcation if and only if thebistable range is bounded
Computational Biology Journal 7
Table 3 Robustness measures of binding energies
Variable Stable Range(kcalmol)
Bistable Range(kcalmol)
Length(kcalmol)
Δ1198661119903 (minusinfin minus1133) (minus1549 minus1133) 416
Δ1198662119903 (minusinfin minus934) (minus1304 minus934) 370
Δ1198663119903 (minus1233 +infin) (minus1233 +infin) infin
Δ1198661119888 (minusinfin +infin) (minus1514 +infin) infin
Δ1198662119888 (minusinfin +infin) (minus1512 +infin) infin
Δ1198663119888 (minus1567 +infin) (minus1567 minus1224) 343
Δ11986612119903 (minusinfin minus155) (minus603 minus155) 448
Δ11986623119903 (minus771infin) (minus771 +infin) infin
Δ11986611987712 (minus1367infin) (minus1367 minus1086) 281
Δ1198661198773 (minusinfin minus1050) (minus1266 minus1050) 216
When the stochastic noise is strong enough the noisemayinduce the system to lytic stateMoreover if free energy variesdue to gene mutation bistable behavior might be destroyedand only one stable equilibrium is left This is what themathematical model implies about the parameter influenceon the system dynamics
42 Δ1198661119888 Δ1198662119888 and Δ119866
3119888 The bifurcation results of Δ119866
1119888
Δ1198662119888 andΔ119866
3119888are shown in Figures 4(d)ndash4(f)The equilibria
behaviors with respect toΔ1198661119888andΔ119866
2119888variation are similar
to Δ1198663119903 Δ1198661119888represents the likelihood that Cro dimer binds
to site 1198741198771 If Cro dimer binds to site 119874
1198771 the expression
of cro is prohibited resulting in only one single lysogenicstable point When Δ119866
1119888or Δ119866
2119888is small the prohibition
effect is weak but this is not enough to drive the systemdynamics to only one single lysis stable point Hence bistablelayout always exists if these two parameters are small Theinfluence ofΔ119866
3119888is similar to the influence ofΔ119866
1119903andΔ119866
2119903
The bistable range of Δ1198663119888
is about 343 kcalmol which iscomparable to those of Δ119866
1119903and Δ119866
2119903
The six parameters we have shown so far are the basicbinding energies The values of Δ119866
3119903 Δ1198661119888 and Δ119866
2119888decide
the likelihood that negative feedback (ie self-prohibition)happens It is known that the sites 119874
1198771 and 119874
1198772 are used
to transcript and then translate protein Cro The site 1198741198773
contributes to the production of protein CI If protein CIbinds to 119874
1198773 or Cro binds to 119874
1198771 and 119874
1198772 they can stop
their own transcription Thus we call the phenomenon neg-ative feedback Figure 4 shows that such negative feedbackcannot take the system to Z-shaped bifurcationThe negativefeedback occurs when the amount of Cro or CI is too largeFor instance the binding between CI dimer and 119874
1198773 readily
occurs when the amount of protein CI is too large On theother hand Δ119866
1119903 Δ1198662119903 and Δ119866
3119888represent the likelihood
of positive feedback (ie mutual prohibition) For instancewith largeΔ119866
3119888 the binding between Cro and119874
1198773 is favored
stopping the expression of cI gene If the value of Δ1198663119903
issmall and the negative feedback is weakened but it is notenough to make the system dynamics change from singlelysis equilibrium to single lysogenic equilibrium Parametersassociated with positive feedback can show Z-shaped bifur-cation whereas parameters associatedwith negative feedback
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG12r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG12r
(a) Z-shaped bifurcation for Δ11986612119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG23r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG23r
(b) Bifurcation for Δ11986623119903
Figure 5 Z-shaped bifurcation generated by variation of 11986612119903
and11986623119903The variation ofΔ119866
12119903also leads to Z-shaped bifurcation while
Δ11986623119903
is not sensitive enough to create Z-shaped bifurcation
cannotHence the system ismore robust to negative feedbackparameters than to positive feedback parameters
43 Δ11986612119903
and Δ11986623119903
The parameters Δ11986612119903
and Δ11986623119903
determine the energy difference caused by the cooperationof adjacent binding CI dimers Although [32] discovered thatadjacent Cro dimers also have cooperations we ignore thiscooperation effect since it has a small cooperation energyand focus on CI cooperation in our analysis Figure 5 showsthe bifurcation results for the parameters Δ119866
12119903and Δ119866
23119903
The nominal values of Δ11986612119903
and Δ11986623119903
are close to zeroSince protein cooperation process is an exothermic reactionΔ11986612119903
and Δ11986623119903
must be less than zero We find out thatthe variation of Δ119866
12119903leads to Z-shaped bifurcation whereas
Δ11986623119903
does not The value of Δ11986623119903
affects the expression ofboth cro and cI genes whereas Δ119866
12119903only promotes mutual
prohibition from CI to Cro Δ11986623119903
with large value prohibitsthe binding of RNAP to both sites119874
1198772 and119874
1198773 thus both 119888119868
and 119888119903119900 gene expressions are closed
8 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR12
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR12
(a) Z-shaped bifurcation for Δ11986611987712
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR3
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR3
(b) Z-shaped bifurcation for Δ1198661198773
Figure 6 Z-shaped bifurcation generated by variation of 11986611987712
and1198661198773 Both of them are able to generate Z-shaped bifurcation Their
bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope
Moreover we know the length of bifurcation range forΔ11986612119903
is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866
12119903
is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866
12119903is equal to the center value at minus379 kcalmol
the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866
12119903from this point of view
44 Δ11986611987712
and Δ1198661198773 Figure 6 illustrates the influence of
Δ1198661198773
and Δ11986611987712
variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the
appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866
11987712is 281 kcalmol and forΔ119866
1198773is only 216 kcalmol
which is the minimum among all ten parametersNotice that Δ119866
1198773and Δ119866
11987712are the binding free energies
of RNAP to 119875119877119872
promoter (1198741198773) and to 119875
119877promoter (119874
1198771
and 1198741198772) Because RNAP is necessary for gene expression
the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866
11987712and Δ119866
1198773
than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866
2119903are around 4 kcalmol while those of Δ119866
11987712
and Δ1198661198773
are less than 3 kcalmol Parameter uncertaintiesof Δ119866
11987712and Δ119866
1198773most easily destroy bistable behaviorThe
concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866
11987712or Δ119866
1198773uncertainty
45 Discussion From the above analysis we find that thebinding free energies to sites119874
1198771 and119874
1198772 have similar effects
on system dynamics As we have shown in Table 3 the resultsof Δ119866
1119903 Δ1198662119903 Δ1198661119888 and Δ119866
2119888are similar This is mainly
because of the similar status for binding sites 1198741198771 and 119874
1198772
both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for
Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866
23119903 All of the above four parameters
do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866
1119888andΔ119866
2119888can be used tomaintain
lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866
1119888lower than
minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582
stabilizes at lysogenyAs to the other six parameters with limited bistable ranges
in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866
1119903
Δ1198662119903 andΔ119866
12119903indicate the likelihood of protein CI binding
to sites 1198741198771 and 119874
1198772 or the likelihood of mutual prohi-
bition occurring Similarly Δ1198663119888indicates the likelihood of
protein Cro binding to site 1198741198773 and also results in mutual
prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior
Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866
11987712and Δ119866
1198773 Since
Δ11986611987712
and Δ1198661198773
have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins
Computational Biology Journal 9
In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866
11987712and Δ119866
1198773 because otherwise the system dynamics
may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily
5 Conclusion
In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866
11987712and Δ119866
1198773 Moreover parameters
describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866
1119903 Δ1198662119903 Δ1198663119888 and Δ119866
12119903) whereas parameters
describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866
2119888) can-
not Hence system dynamics is more sensitive to mutualprohibition parameters
There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors
Appendix
Calculation of Jacobian
From (1)ndash(5) we may represent system Jacobian by
119869 =[[[
[
1198601
120597119875119903
120597119873119903
minus 1205831
1198601
120597119875119903
120597119873119888
1198602
120597119875119888
120597119873119903
1198602
120597119875119888
120597119873119888
minus 1205832
]]]
]
(A1)
Since 119875119903and 119875
119888are sums of 119875
119904 119904 = 1 2 40 as (2) shows
the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875
119904120597119873119888 Since 119875
119904is a function with respect to
Cro and CI dimers we evaluate the partial derivatives byChain Rule
120597119875119904
120597119873119903
=120597119875119904
1205971198731199032
1198891198731199032
119889119873119903
120597119875119904
120597119873119888
=120597119875119904
1205971198731198882
1198891198731198882
119889119873119888
(A2)
Table 4 Exponents for 30 binding configurations in Table 2
119904 119894(119904) 119895(119904) 119896(119904)
1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2
where
1198891198731199032
119889119873119903
=1
2(1 minus
119870119903
radic1198702119903+ 8119873119903119870119903
)
1198891198731198882
119889119873119888
=1
2(1 minus
119870119888
radic1198702119888+ 8119873119888119870119888
)
(A3)
10 Computational Biology Journal
Thenwe have to deal with other terms in (A2) Note that119875119904is
a fractional function in (3) so we write its numerator119860119904and
denominator 119861 as
119872119904= exp(minus
Δ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)
1198882119873119896(119904)
RNAP (A4)
119863 =
40
sum
119904=1
119872119904 (A5)
The partial derivatives for 119875119904with respect to119873
1198882and119873
1199032are
given by
120597119875119904
1205971198731199032
=120597119872119904
1205971198731199032
1
119863minus
120597119863
1205971198731199032
119872119904
1198632
120597119875119904
1205971198731198882
=120597119872119904
1205971198731198882
1
119863minus
120597119863
1205971198731198882
119872119904
1198632
(A6)
The partial derivative of119872119904can be obtained as
120597119872119904
1205971198731199032
= 119894 (119904) exp(minusΔ119866119904
119877119879)119873119894(119904)minus1
1199032119873119895(119904)
1198882119873119896(119904)
RNAP
120597119872119904
1205971198731198882
= 119895 (119904) exp (minusΔ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)minus1
1198882119873119896(119904)
RNAP
(A7)
The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872
119904partial
derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM
References
[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982
[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985
[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990
[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004
[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006
[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007
[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009
[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010
[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011
[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002
[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004
[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004
[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007
[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008
[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012
[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004
[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004
[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000
[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996
[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995
Computational Biology Journal 11
[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002
[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999
[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004
[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008
[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009
[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992
[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010
[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982
[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004
[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975
[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009
[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000
[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967
[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000
[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998
[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002
[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984
[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999
[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004
[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
2 Computational Biology Journal
and robustness are closely related because a robust systemis not sensitive to parameter variations [20] By using lysisfrequency as a sensitivity measure [21] simulated a mutationmodel introduced by [22] and concluded that spontaneousswitching from lysogen to lysis depends sensitively on modelparameters By systematically perturbing binding energieswith plusmn1 kcalmol [23] concluded that lysogenic activity atpromoter 119875
119877119872remains stable under this perturbation
In this paper we investigate bifurcation by using bistablerange as a system robustness measure In a typical case phage120582 genetic switch model has three equilibria Two of them arestable corresponding to the lysogenic and lytic states andthe third one is a saddle point When some parameters areperturbed it is possible that one stable equilibrium and thesaddle point coalesce then both of them disappear resultingin the loss of bistable behavior Hence bistable region can bedefined as the parameter region in which bistable behaviorexists [18] The bistable region for a parameter is large if thesystem dynamics is not sensitive to that parameter
We are interested in analyzing bistable ranges for bindingenergy related parameters Some other parameters such asprotein degradation rate and protein synthesis rate withrespect to gene transcription have already been proved thattheir variations may lead to the loss of bistability [3 24]Our calculations are based on a phage 120582 OR model [3]Although DNA looping formed by cooperations betweenthe left operators and the right operators affects the stabilityof the genetic switch [25] the OR model only includesthe effects of the right operators which are believed toplay a critical role for lysogeny maintenance [26] We willcalculate bistable ranges for all binding energies which arecrucial in determining the affinity that proteins bind tophage 120582 genes Furthermore we define Z-shaped bifurcationand calculate Jacobian matrix to study robustness We findout that different binding energy affects lysogeny stabilitydifferently some of them may destroy bistability whereasothers have little influence on system dynamics Thereforewe need to precisely determine the values of those sensitiveparameters in the model because otherwise we may not getan effective bistable switch
2 Phage 120582 Modeling
We first present phage 120582 gene regulatory networks and thenintroduce their mathematical model Lysogeny and lysis aretwo possible states for phage 120582 It is believed that wild-typephage 120582 has quite a stable lysogenic state since lysogeny lossrate is even lower than 10minus8 per cell generation [27] Howevercertain conditions such as the exposure to UV light [28 29]the presence of mitomycin C [30] or the starvation of thehost cells [31] can induce phage 120582 from lysogeny to lysisMathematical models have been built to successfully explainstability and high efficiency of genetic switch
The concentrations of two proteins CI and Cro can serveas an indication of which state phage 120582 is in lysogeny orlysis In lysogeny the concentration of protein CI in the cell issignificantly higher than that of protein Cro whereas in lysis
Gene for CI expression Gene for Cro expression
Cro dimerCI dimer
OR1OR3 OR2middot middot middot
PRM PR
Figure 1 Three binding sites on a DNA segment The sites 1198741198771
1198741198772 and 119874
1198773 can be bound by CI dimer Cro dimer or RNA
polymeraseThis DNA segment consists of promoters119875119877119872
and119875119877 If
the RNA polymerase binds to 1198741198773 cI gene is expressed if the RNA
polymerase binds to1198741198771 and119874
1198772 cro gene is expressedThe amount
of Cro and CI in the cell directly determines which state phage 120582 isin
Table 1 Parameter values for our model
Δ1198661119888= minus120 kcalmoldagger Δ119866
2119888= minus108 kcalmoldagger
Δ1198663119888= minus134 kcalmoldagger Δ119866
1119903= minus125 kcalmolDagger
Δ1198662119903= minus105 kcalmolDagger Δ119866
3119903= minus95 kcalmolDagger
Δ11986612119903
= minus27 kcalmolDagger Δ11986623119903
= minus29 kcalmolDagger
Δ11986611987712
= minus125 kcalmoldagger Δ1198661198773= minus115 kcalmoldagger
119870119903= 556 times 10
minus9Mdagger 119870119888= 326 times 10
minus7Mdagger
119873RNAP = 30 times 10minus8Msect
1205831= 20 times 10
minus2minminus1dagger
1205832= 36 times 10
minus2minminus1dagger 119877 = 199 cal(molsdotK)119879 = 310K 119896 = 111
para
daggerdenotes the parameter values from [14] Daggerfrom [32] sectfrom [21] and parafrom[3]
protein Cro dominates the cell The amount of CI and Crothus implies which state phage 120582 is in
Figure 1 illustrates the gene segment regulating cro andcI gene expression Three binding sites 119874
1198771 1198741198772 and 119874
1198773
control the expression of two kinds of regulatory proteinsThe 119875
119877119872promoter covers the whole binding site 119874
1198773 and
part of 1198741198772 controlling cI gene expression When RNA
polymerase (RNAP) binds to the site1198741198773 the gene expression
is on resulting in the production of protein CI The 119875119877
promoter covers all of 1198741198771 and part of 119874
1198772 controlling
cro expression When RNAP binds to sites 1198741198771 and 119874
1198772
simultaneously the 119888119903119900 gene expression is on The bindingsites may be occupied by either Cro or CI protein UsuallyCro and CI first form their dimers so that they are able tobind to 119874
1198771 1198741198772 and 119874
1198773 prohibiting the gene expressions
on these sitesSeveral phage 120582 models have been established based on
this regulating mechanism In this paper we adopt the well-known two-dimensionalORmodel stated in [3]with updatedparameter values given in Table 1 This model involves three119874119877gene binding sites which are believed to be the most
critical factors for the stability of genetic switch [33] Thistwo-dimensional model is in line with the general biologicalswitch model for two-gene networks [34] To keep it simplewe eliminate the effects from other activities in the cell suchas nonspecific binding [16 17] stochastic gene expression
Computational Biology Journal 3
[35] and intrinsic and extrinsic noises [36] The model isdescribed by the following system equations
119889119873119903
119889119905= 1198601119891119903minus 1205831119873119903
119889119873119888
119889119905= 1198602119891119888minus 1205832119873119888
(1)
where
119891119903= 11987528+ 11987529+ 11987530+ 11987540
119891119888=
33
sum
119894=31
119875119894+ 119896(
40
sum
119895=34
119875119895)
(2)
119875119904=
exp (minusΔ119866119904119877119879)119873
119894(119904)
1199032119873119895(119904)
1198882119873119896(119904)
RNAP
sum40
119898=1exp (minusΔ119866
119898119877119879)119873
119894(119898)
1199032119873119895(119898)
1198882119873119896(119898)
RNAP
(119904 = 1 2 40)
(3)
The subscript 119904 corresponds to 40 binding configurationsshown in Table 2 The conventions between CI dimer (119873
1199032)
Cro dimer (1198731198882) and their monomers (119873
119903and119873
119888) are
1198731199032=1
2119873119903+1
8119870119903minus1
8radic1198702119903+ 8119873119903119870119903
1198731198882=1
2119873119888+1
8119870119888minus1
8radic1198702119888+ 8119873119888119870119888
(4)
From (1)ndash(4)119873119903119873119888represent concentrations of proteins
CI and Cro The parameters 1198601and 119860
2are synthesis rates of
CI and Cro respectively 1205831and 120583
2are the overall rates that
decrease protein concentration which are caused by normalprotein degradation and cell growth [37]The function 119891
119903(or
119891119888) represents the overall probability that the 119875
119877(or 119875119877119872
)promoter is occupied by RNAP 119875
119904is the probability that
119904th configuration occurs and Δ119866119904is the total binding energy
needed for this configuration The exponents 119894(119904) 119895(119904) and119896(119904) represent the numbers of CI dimers Cro dimers andRNAP at 119904th configuration respectively Δ119866
119904is calculated
by summing up the needed single binding energies listed inTable 1 including cooperation energy For example Δ119866
5=
Δ11986612119903
+Δ1198661119903+Δ1198662119903 The left right arrows in Table 2 indicate
the cooperations 119896 is the coefficient for positive feedbackWhen CI dimer binds to 119874
1198772 positive feedback makes the
transcription of cI gene more efficient hence promotes itsexpression 119870
119903 119870119888are dissociation constants for CI and Cro
proteins respectively 119877 represents the universal gas constantand 119879 the absolute temperature
3 Bifurcation in Phage 120582
The model used in this paper is two-dimensional nonlinearordinary differential equations To get all the equilibria we setthe left hand side of (1) to zero and solve the transcendentalequations It can be found that in general the model has threeequilibria However the number of equilibria may vary dueto bifurcation under parameter variation In phage 120582 model
Table 2 40 Binding configurations for the three sites
119904 1198741198773 119874
1198772 119874
1198771
12 CI23 CI24 CI25 CI2harr CI26 CI2 CI27 CI2harr CI28 CI2harr CI2harr CI29 Cro210 Cro211 Cro212 Cro2 Cro213 Cro2 Cro214 Cro2 Cro215 Cro2 Cro2 Cro216 Cro2 CI217 Cro2 CI218 Cro2 Cro2 CI219 Cro2 CI220 CI2 Cro221 Cro2 CI2 Cro222 CI2 Cro223 CI2 Cro224 CI2 Cro2 Cro225 Cro2 CI2harr CI226 CI2 Cro2 CI227 CI2harr CI2 Cro228 RNAP29 CI2 RNAP30 Cro2 RNAP31 RNAP CI232 RNAP CI2harr CI233 RNAP CI2 Cro234 RNAP35 RNAP CI236 RNAP Cro237 RNAP Cro238 RNAP Cro2 CI239 RNAP Cro2 Cro240 RNAP RNAP
frequently occurring is the saddle node bifurcation whoseprecise mathematical definition is given as follows [38]
Definition 1 (saddle node bifurcation) A saddle node bifur-cation in dynamical system takes place when two equilibriumpoints coalesce and disappear
In (1) two parameters 1198601and 119860
2are not specified
in Table 1 They represent proportional constants relatingthe transcription initiation rate to the absolute rate of CI and
4 Computational Biology Journal
CA CBAB
Nr at equilibrium Nc at equilibrium
A2 A2
Figure 2 Saddle node bifurcation with the variation of Crosynthesis rate 119860
2 This is a schematic plot and not drawn to scale
Cro protein synthesis [3] We use Figure 2 to illustrate Z-shaped bifurcation with the variation of 119860
2 Set 119860
1= 8 times
10minus9Mmin and divide the whole plane into three regions
as shown in Figure 2 labeled by A B and C Initially whenthe value of 119860
2is small in region A for example 119860
2asymp
5times10minus9Mmin there exists only one stable point and thus one
single curve segment for both 119873119903and 119873
119888 At the boundary
of regions A and B saddle node bifurcation occurs 1198602
increases to region B for example a typical value 1198602asymp
5 times 10minus8Mmin and two new equilibria (one saddle and
one stable equilibrium) emerge resulting in three equilibriaNote that this is a saddle node bifurcation with the reverseddirection in comparison to Definition 1
At the boundary of regions B and C saddle nodebifurcation takes place again leading to the vanishing oftwo equilibria (one saddle point and one stable equilibrium)In region C say 119860
2asymp 5 times 10
minus7Mmin only one stableequilibrium remains We notice the motion of the saddlepoint When the saddle point shows up at the boundary ofregions A and B the lysis stable equilibrium appears as wellAs parameter119860
2increases the saddle pointmoves toward the
lysogen stable point and finallymergeswith it at the boundaryof regions B and C Two successive bifurcations form a Z-shaped curve are shown in Figure 2
We have not yet specified the boundaries between theseregions quantitatively Next we solve the values of the bound-aries by using the Jacobian method Let us first introduce thefollowing two theorems [38]
Theorem 2 (Hartman-Grobman Theorem) If the Jacobianof the nonlinear system = 119892(119909) at equilibrium namely119869(1199090) has no zero or purely imaginary eigenvalues then the
phase trajectory of the system = 119892(119909) resembles that of thelinearized system = 119869(119909
0)119911 near 119909
0if none of the eigenvalues
of 119869(1199090) are on the 119895120596 axis
For a given equilibrium 1199090 in most cases we can
determine its dynamical behavior qualitatively by calculatingits Jacobian matrix 119869(119909
0) For a two-dimensional system its
Jacobian matrix can be calculated as
119869 =
[[[[
[
1205971198921
120597119873119903
1205971198921
120597119873119888
1205971198922
120597119873119903
1205971198922
120597119873119888
]]]]
]
(5)
The following theorem gives the necessary conditions forthe critical point that bifurcation occurs [38]
Theorem 3 For any nonlinear system = 119892(119911 120583) the systemgoes through a saddle node bifurcation at 119911 = 119911
dagger 120583 = 120583dagger only
if the following relations are satisfied
119892 (119911dagger 120583dagger) = 0 (6)
10038161003816100381610038161003816100381610038161003816
120597119892
120597119911(119911dagger 120583dagger)
10038161003816100381610038161003816100381610038161003816= 0 (7)
From (6) 119911dagger is an equilibrium point In a planar system(7) is the determinant of system Jacobian When the deter-minant of Jacobian matrix vanishes at least one eigenvalueis zero Jacobian is a characterization of planar systemdynamics For variation of parameter 119860
2 the system in (1)
can be rewritten as
119889119873119903
119889119905= 1198921(119873119888 119873119903 1198602)
119889119873119888
119889119905= 1198922(119873119888 119873119903 1198602)
(8)
From Theorems 2 and 3 for a given 1198602 the Jacobian matrix
is evaluated at equilibrium points 119873dagger119888and 119873
dagger
119903 These two
equilibrium points can be represented as a function of 1198602as
shown in Figure 2ByTheorem 3we know that the bifurcation appearswhen
Jacobian has a zero eigenvalue From Figure 3 we can obtainthe approximate values of critical points by checking thepoints of the eigenvalue curves crossing zero line Thesevalues are 56 times 10minus9Mmin and 34 times 10minus7Mmin
To determine the stability of an equilibrium point wecalculate the eigenvalue of the Jacobian matrix Figure 3 plotsthe eigenvalues as a function of 119860
2 The Jacobian matrix
can be calculated analytically (see the Appendix for details)By solving the value of 119860
2that makes the determinant of
the Jacobian matrix vanish we get the critical value of 1198602
at which bifurcation occurs When 1198602is small (less than
56 times 10minus9Mmin in Figure 3) there are only two negative
eigenvalues corresponding to one equilibrium point Thisequilibrium is thus stable [38]
As 1198602exceeds 56 times 10
minus9Mmin we have the casewhen there are three equilibria and correspondingly sixeigenvalues Among these three equilibria two are stableand the third is unstable (saddle point) This is becausepositive eigenvalue appears at saddle point (see dash-dotline in Figure 3) For a two-dimensional linear system if theJacobian has one positive and one negative eigenvalues at theequilibrium the equilibrium is a saddle point If119860
2continues
increasing only one equilibrium point is left (1198602greater than
34 times 10minus7Mmin in this case) and both of its eigenvalues
are negative The curves in Figure 3 have two intersectionswith zero eigenvalue which indicates that the eigenvalues ofJacobian vanish twice forming a Z-shaped bifurcation
Computational Biology Journal 5
0
002
004
Eige
nval
ues o
f Jac
obia
n
At equilibrium for lysogenAt equilibrium for lysisAt saddle point
10minus9
10minus8
10minus7
10minus6
10minus5
10minus4
minus012
minus01
minus008
minus006
minus004
minus002
A2 (Mmin)
Figure 3 Jacobian eigenvalues at equilibria The figure shows howthe eigenvalues of Jacobian at equilibria change with 119860
1fixed at
8 times 10minus9Mmin and 119860
2varying The solid line (black) shows
the eigenvalues of lysis equilibrium The dashed line (blue) is forlysogeny equilibrium The dot-dashed line (red) is for saddle
4 Bifurcation Analysis of BindingEnergy Uncertainty
In this section we will analyze bifurcation for Δ119866rsquos byusing the Jacobian method In Table 1 the symbol Δ119866 withsubscripts such as Δ119866
1119903and Δ119866
2119888represents Gibbs free
energy required for CI or Cro dimers to bind to the ORsites Because of the difficulty in determining the precisevalues of the binding energy it is useful to study how theequilibria change as these Gibbs free binding energies vary Inthis section we set synthesis rates 119860
1and 119860
2at appropriate
values 1198601= 8 times 10
minus9Mmin and 1198602= 5 times 10
minus8Mmin Tocalculate the equilibria we set the left hand side of (1) to zeroand obtain the expression of119873
119903and119873
119888as functions of Δ119866rsquos
119873lowast
119903= 1198911(Δ1198661119888 Δ1198662119888 Δ119866
1198773)
119873lowast
119888= 1198912(Δ1198661119888 Δ1198662119888 Δ119866
1198773)
(9)
where 119873lowast119903and 119873lowast
119888are the concentrations of proteins CI and
Cro at the equilibrium point Once (9) is obtained we candetermine the quantitative behavior around equilibria whenchanging Δ119866rsquos The relative sensitivities are given by
119878119873119903
(Δ119866119904) =
1205971198911
120597Δ119866119904
119878119873119888
(Δ119866119904) =
1205971198912
120597Δ119866119904
(10)
where the subscript 119904 isin 1119903 2119903 11987712 1198773 Since the systemequilibria cannot be analytically solved we may apply the
numerical procedure to calculate (10) as a measure of param-eter sensitivity As to the study on system robustness theJacobianmethod can be used to study bifurcation introducedby parameter variation
Since there are ten differentΔ119866s we divide them into fourgroups
(1) Δ1198661119903 Δ1198662119903 and Δ119866
3119903 binding free energies of CI
(2) Δ1198661119888 Δ1198662119888 and Δ119866
3119888 binding free energies of Cro
(3) Δ11986612119903
and Δ11986623119903
the cooperation between proteinsCIrsquos if both 119874
1198771 and 119874
1198772 or 119874
1198772 and 119874
1198773 are bound
by CI dimers(4) Δ119866
11987712and Δ119866
1198773 RNAP binding energies
We then study each of these four groups
41 Δ1198661119903 Δ1198662119903 and Δ119866
3119903 We systematically perturb the
values of Δ1198661119903 Δ1198662119903 and Δ119866
3119903to investigate their impact
on the system dynamics Two proper robustness measuresare the range of parameters in which lysogeny exists (stablerange) and inwhich bistable dynamics exists (bistable range)In stable range it is possible to keep phage 120582 dormant andintegrate into the gene of the host On the other hand bistablerange defines the parameter range to keep bistability and themaximal uncertainty tolerance In this case it implies thatthe system favors dynamics with two stable equilibria and asaddle Further if the stochastic force is strong enough phage120582 can switch from the lysogenic stable point to the lytic stablepoint even without the parameter changes
Figures 4(a)ndash4(c) illustrate the impact of Δ1198661119903 Δ1198662119903 and
Δ1198663119903
variations We set parameter values as in Table 1 andfind out all the equilibria when varying oneΔ119866
119904 We find that
the energy Δ1198663119903does not show Z-shaped bifurcation while
the other two parameters Δ1198661119903and Δ119866
2119903show quite similar
behaviors since Figures 4(a) and 4(b) are almost identicallythe sameThe result is not surprising because Δ119866
1119903and Δ119866
2119903
play similar roles in system dynamics These two parametersdetermine the likelihood that CI dimer binds to sites 119874
1198771
and 1198741198772 When either of the two sites is occupied by protein
dimers the expression of 119888119903119900 gene is prohibited resulting inlysogenic stateNonetheless the parameterΔ119866
3119903indicates the
likelihood of CI dimer binding to the site1198741198773 If the absolute
value of Δ1198663119903is too large it is easy for CI dimer to bind to
1198741198773 thus cI gene expression is prohibited resulting in lysis
If the absolute value ofΔ1198663119903is small the Z-shaped bifurcation
vanishes In this case cI gene expression is not prohibited butit has no influence on cro gene expression Bistable behavioralways exists when Δ119866
3119903is less than its nominal value in
Table 1 Without stopping or weakening the expression of crogene single lysogen stable point layout cannot be achievedjust by increasingΔ119866
3119903 Comparedwith other parameters the
system dynamics is more robust to Δ1198663119903variation
Table 3 lists the stable range bistable range and the lengthof bistable range for all tenΔ119866s It can be shownbyTheorem 2that one of the three equilibria is a saddle and the other twoare stable points Jacobian matrix is also used to determinethe boundary of stable and bistable ranges according toTheorem 3 As long as Δ119866
1119903is less than the nominal value
6 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG1r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG1r
(a) Z-shaped bifurcation for Δ1198661119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG2r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG2r
(b) Z-shaped bifurcation for Δ1198662119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG3r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG3r
(c) Bifurcation for Δ1198663119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG1c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG1c
(d) Bifurcation for Δ1198661119888
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG2c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG2c
(e) Bifurcation for Δ1198662119888
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG3c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG3c
(f) Z-shaped bifurcation for Δ1198663119888
Figure 4 Movement of equilibria under parameter variation Z-shaped bifurcation can be generated by variation of Δ1198661119903 Δ1198662119903 and Δ119866
3119888
All vertical axes are plotted on a log scale
at minus1133 kcalmol or Δ1198662119903
is less than minus934 kcalmol thesystem always has lysogenic stateThey are critical bifurcationpoints For Δ119866
3119903 however lysogenic state always exists if it is
greater than minus1233 kcalmol Consequently for CI protein alarger binding energy to119874
1198771 and119874
1198772 helps to keep the DNA
of phage 120582 silent in the host The system tends to be unstableif the values of binding energies for CI to 119874
1198771 and 119874
1198772
become small thus forming Z-shaped bifurcation Parametervariations may lead to Z-shaped bifurcation if and only if thebistable range is bounded
Computational Biology Journal 7
Table 3 Robustness measures of binding energies
Variable Stable Range(kcalmol)
Bistable Range(kcalmol)
Length(kcalmol)
Δ1198661119903 (minusinfin minus1133) (minus1549 minus1133) 416
Δ1198662119903 (minusinfin minus934) (minus1304 minus934) 370
Δ1198663119903 (minus1233 +infin) (minus1233 +infin) infin
Δ1198661119888 (minusinfin +infin) (minus1514 +infin) infin
Δ1198662119888 (minusinfin +infin) (minus1512 +infin) infin
Δ1198663119888 (minus1567 +infin) (minus1567 minus1224) 343
Δ11986612119903 (minusinfin minus155) (minus603 minus155) 448
Δ11986623119903 (minus771infin) (minus771 +infin) infin
Δ11986611987712 (minus1367infin) (minus1367 minus1086) 281
Δ1198661198773 (minusinfin minus1050) (minus1266 minus1050) 216
When the stochastic noise is strong enough the noisemayinduce the system to lytic stateMoreover if free energy variesdue to gene mutation bistable behavior might be destroyedand only one stable equilibrium is left This is what themathematical model implies about the parameter influenceon the system dynamics
42 Δ1198661119888 Δ1198662119888 and Δ119866
3119888 The bifurcation results of Δ119866
1119888
Δ1198662119888 andΔ119866
3119888are shown in Figures 4(d)ndash4(f)The equilibria
behaviors with respect toΔ1198661119888andΔ119866
2119888variation are similar
to Δ1198663119903 Δ1198661119888represents the likelihood that Cro dimer binds
to site 1198741198771 If Cro dimer binds to site 119874
1198771 the expression
of cro is prohibited resulting in only one single lysogenicstable point When Δ119866
1119888or Δ119866
2119888is small the prohibition
effect is weak but this is not enough to drive the systemdynamics to only one single lysis stable point Hence bistablelayout always exists if these two parameters are small Theinfluence ofΔ119866
3119888is similar to the influence ofΔ119866
1119903andΔ119866
2119903
The bistable range of Δ1198663119888
is about 343 kcalmol which iscomparable to those of Δ119866
1119903and Δ119866
2119903
The six parameters we have shown so far are the basicbinding energies The values of Δ119866
3119903 Δ1198661119888 and Δ119866
2119888decide
the likelihood that negative feedback (ie self-prohibition)happens It is known that the sites 119874
1198771 and 119874
1198772 are used
to transcript and then translate protein Cro The site 1198741198773
contributes to the production of protein CI If protein CIbinds to 119874
1198773 or Cro binds to 119874
1198771 and 119874
1198772 they can stop
their own transcription Thus we call the phenomenon neg-ative feedback Figure 4 shows that such negative feedbackcannot take the system to Z-shaped bifurcationThe negativefeedback occurs when the amount of Cro or CI is too largeFor instance the binding between CI dimer and 119874
1198773 readily
occurs when the amount of protein CI is too large On theother hand Δ119866
1119903 Δ1198662119903 and Δ119866
3119888represent the likelihood
of positive feedback (ie mutual prohibition) For instancewith largeΔ119866
3119888 the binding between Cro and119874
1198773 is favored
stopping the expression of cI gene If the value of Δ1198663119903
issmall and the negative feedback is weakened but it is notenough to make the system dynamics change from singlelysis equilibrium to single lysogenic equilibrium Parametersassociated with positive feedback can show Z-shaped bifur-cation whereas parameters associatedwith negative feedback
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG12r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG12r
(a) Z-shaped bifurcation for Δ11986612119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG23r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG23r
(b) Bifurcation for Δ11986623119903
Figure 5 Z-shaped bifurcation generated by variation of 11986612119903
and11986623119903The variation ofΔ119866
12119903also leads to Z-shaped bifurcation while
Δ11986623119903
is not sensitive enough to create Z-shaped bifurcation
cannotHence the system ismore robust to negative feedbackparameters than to positive feedback parameters
43 Δ11986612119903
and Δ11986623119903
The parameters Δ11986612119903
and Δ11986623119903
determine the energy difference caused by the cooperationof adjacent binding CI dimers Although [32] discovered thatadjacent Cro dimers also have cooperations we ignore thiscooperation effect since it has a small cooperation energyand focus on CI cooperation in our analysis Figure 5 showsthe bifurcation results for the parameters Δ119866
12119903and Δ119866
23119903
The nominal values of Δ11986612119903
and Δ11986623119903
are close to zeroSince protein cooperation process is an exothermic reactionΔ11986612119903
and Δ11986623119903
must be less than zero We find out thatthe variation of Δ119866
12119903leads to Z-shaped bifurcation whereas
Δ11986623119903
does not The value of Δ11986623119903
affects the expression ofboth cro and cI genes whereas Δ119866
12119903only promotes mutual
prohibition from CI to Cro Δ11986623119903
with large value prohibitsthe binding of RNAP to both sites119874
1198772 and119874
1198773 thus both 119888119868
and 119888119903119900 gene expressions are closed
8 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR12
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR12
(a) Z-shaped bifurcation for Δ11986611987712
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR3
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR3
(b) Z-shaped bifurcation for Δ1198661198773
Figure 6 Z-shaped bifurcation generated by variation of 11986611987712
and1198661198773 Both of them are able to generate Z-shaped bifurcation Their
bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope
Moreover we know the length of bifurcation range forΔ11986612119903
is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866
12119903
is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866
12119903is equal to the center value at minus379 kcalmol
the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866
12119903from this point of view
44 Δ11986611987712
and Δ1198661198773 Figure 6 illustrates the influence of
Δ1198661198773
and Δ11986611987712
variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the
appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866
11987712is 281 kcalmol and forΔ119866
1198773is only 216 kcalmol
which is the minimum among all ten parametersNotice that Δ119866
1198773and Δ119866
11987712are the binding free energies
of RNAP to 119875119877119872
promoter (1198741198773) and to 119875
119877promoter (119874
1198771
and 1198741198772) Because RNAP is necessary for gene expression
the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866
11987712and Δ119866
1198773
than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866
2119903are around 4 kcalmol while those of Δ119866
11987712
and Δ1198661198773
are less than 3 kcalmol Parameter uncertaintiesof Δ119866
11987712and Δ119866
1198773most easily destroy bistable behaviorThe
concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866
11987712or Δ119866
1198773uncertainty
45 Discussion From the above analysis we find that thebinding free energies to sites119874
1198771 and119874
1198772 have similar effects
on system dynamics As we have shown in Table 3 the resultsof Δ119866
1119903 Δ1198662119903 Δ1198661119888 and Δ119866
2119888are similar This is mainly
because of the similar status for binding sites 1198741198771 and 119874
1198772
both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for
Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866
23119903 All of the above four parameters
do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866
1119888andΔ119866
2119888can be used tomaintain
lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866
1119888lower than
minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582
stabilizes at lysogenyAs to the other six parameters with limited bistable ranges
in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866
1119903
Δ1198662119903 andΔ119866
12119903indicate the likelihood of protein CI binding
to sites 1198741198771 and 119874
1198772 or the likelihood of mutual prohi-
bition occurring Similarly Δ1198663119888indicates the likelihood of
protein Cro binding to site 1198741198773 and also results in mutual
prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior
Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866
11987712and Δ119866
1198773 Since
Δ11986611987712
and Δ1198661198773
have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins
Computational Biology Journal 9
In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866
11987712and Δ119866
1198773 because otherwise the system dynamics
may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily
5 Conclusion
In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866
11987712and Δ119866
1198773 Moreover parameters
describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866
1119903 Δ1198662119903 Δ1198663119888 and Δ119866
12119903) whereas parameters
describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866
2119888) can-
not Hence system dynamics is more sensitive to mutualprohibition parameters
There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors
Appendix
Calculation of Jacobian
From (1)ndash(5) we may represent system Jacobian by
119869 =[[[
[
1198601
120597119875119903
120597119873119903
minus 1205831
1198601
120597119875119903
120597119873119888
1198602
120597119875119888
120597119873119903
1198602
120597119875119888
120597119873119888
minus 1205832
]]]
]
(A1)
Since 119875119903and 119875
119888are sums of 119875
119904 119904 = 1 2 40 as (2) shows
the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875
119904120597119873119888 Since 119875
119904is a function with respect to
Cro and CI dimers we evaluate the partial derivatives byChain Rule
120597119875119904
120597119873119903
=120597119875119904
1205971198731199032
1198891198731199032
119889119873119903
120597119875119904
120597119873119888
=120597119875119904
1205971198731198882
1198891198731198882
119889119873119888
(A2)
Table 4 Exponents for 30 binding configurations in Table 2
119904 119894(119904) 119895(119904) 119896(119904)
1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2
where
1198891198731199032
119889119873119903
=1
2(1 minus
119870119903
radic1198702119903+ 8119873119903119870119903
)
1198891198731198882
119889119873119888
=1
2(1 minus
119870119888
radic1198702119888+ 8119873119888119870119888
)
(A3)
10 Computational Biology Journal
Thenwe have to deal with other terms in (A2) Note that119875119904is
a fractional function in (3) so we write its numerator119860119904and
denominator 119861 as
119872119904= exp(minus
Δ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)
1198882119873119896(119904)
RNAP (A4)
119863 =
40
sum
119904=1
119872119904 (A5)
The partial derivatives for 119875119904with respect to119873
1198882and119873
1199032are
given by
120597119875119904
1205971198731199032
=120597119872119904
1205971198731199032
1
119863minus
120597119863
1205971198731199032
119872119904
1198632
120597119875119904
1205971198731198882
=120597119872119904
1205971198731198882
1
119863minus
120597119863
1205971198731198882
119872119904
1198632
(A6)
The partial derivative of119872119904can be obtained as
120597119872119904
1205971198731199032
= 119894 (119904) exp(minusΔ119866119904
119877119879)119873119894(119904)minus1
1199032119873119895(119904)
1198882119873119896(119904)
RNAP
120597119872119904
1205971198731198882
= 119895 (119904) exp (minusΔ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)minus1
1198882119873119896(119904)
RNAP
(A7)
The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872
119904partial
derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM
References
[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982
[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985
[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990
[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004
[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006
[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007
[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009
[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010
[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011
[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002
[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004
[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004
[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007
[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008
[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012
[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004
[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004
[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000
[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996
[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995
Computational Biology Journal 11
[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002
[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999
[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004
[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008
[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009
[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992
[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010
[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982
[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004
[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975
[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009
[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000
[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967
[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000
[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998
[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002
[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984
[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999
[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004
[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
Computational Biology Journal 3
[35] and intrinsic and extrinsic noises [36] The model isdescribed by the following system equations
119889119873119903
119889119905= 1198601119891119903minus 1205831119873119903
119889119873119888
119889119905= 1198602119891119888minus 1205832119873119888
(1)
where
119891119903= 11987528+ 11987529+ 11987530+ 11987540
119891119888=
33
sum
119894=31
119875119894+ 119896(
40
sum
119895=34
119875119895)
(2)
119875119904=
exp (minusΔ119866119904119877119879)119873
119894(119904)
1199032119873119895(119904)
1198882119873119896(119904)
RNAP
sum40
119898=1exp (minusΔ119866
119898119877119879)119873
119894(119898)
1199032119873119895(119898)
1198882119873119896(119898)
RNAP
(119904 = 1 2 40)
(3)
The subscript 119904 corresponds to 40 binding configurationsshown in Table 2 The conventions between CI dimer (119873
1199032)
Cro dimer (1198731198882) and their monomers (119873
119903and119873
119888) are
1198731199032=1
2119873119903+1
8119870119903minus1
8radic1198702119903+ 8119873119903119870119903
1198731198882=1
2119873119888+1
8119870119888minus1
8radic1198702119888+ 8119873119888119870119888
(4)
From (1)ndash(4)119873119903119873119888represent concentrations of proteins
CI and Cro The parameters 1198601and 119860
2are synthesis rates of
CI and Cro respectively 1205831and 120583
2are the overall rates that
decrease protein concentration which are caused by normalprotein degradation and cell growth [37]The function 119891
119903(or
119891119888) represents the overall probability that the 119875
119877(or 119875119877119872
)promoter is occupied by RNAP 119875
119904is the probability that
119904th configuration occurs and Δ119866119904is the total binding energy
needed for this configuration The exponents 119894(119904) 119895(119904) and119896(119904) represent the numbers of CI dimers Cro dimers andRNAP at 119904th configuration respectively Δ119866
119904is calculated
by summing up the needed single binding energies listed inTable 1 including cooperation energy For example Δ119866
5=
Δ11986612119903
+Δ1198661119903+Δ1198662119903 The left right arrows in Table 2 indicate
the cooperations 119896 is the coefficient for positive feedbackWhen CI dimer binds to 119874
1198772 positive feedback makes the
transcription of cI gene more efficient hence promotes itsexpression 119870
119903 119870119888are dissociation constants for CI and Cro
proteins respectively 119877 represents the universal gas constantand 119879 the absolute temperature
3 Bifurcation in Phage 120582
The model used in this paper is two-dimensional nonlinearordinary differential equations To get all the equilibria we setthe left hand side of (1) to zero and solve the transcendentalequations It can be found that in general the model has threeequilibria However the number of equilibria may vary dueto bifurcation under parameter variation In phage 120582 model
Table 2 40 Binding configurations for the three sites
119904 1198741198773 119874
1198772 119874
1198771
12 CI23 CI24 CI25 CI2harr CI26 CI2 CI27 CI2harr CI28 CI2harr CI2harr CI29 Cro210 Cro211 Cro212 Cro2 Cro213 Cro2 Cro214 Cro2 Cro215 Cro2 Cro2 Cro216 Cro2 CI217 Cro2 CI218 Cro2 Cro2 CI219 Cro2 CI220 CI2 Cro221 Cro2 CI2 Cro222 CI2 Cro223 CI2 Cro224 CI2 Cro2 Cro225 Cro2 CI2harr CI226 CI2 Cro2 CI227 CI2harr CI2 Cro228 RNAP29 CI2 RNAP30 Cro2 RNAP31 RNAP CI232 RNAP CI2harr CI233 RNAP CI2 Cro234 RNAP35 RNAP CI236 RNAP Cro237 RNAP Cro238 RNAP Cro2 CI239 RNAP Cro2 Cro240 RNAP RNAP
frequently occurring is the saddle node bifurcation whoseprecise mathematical definition is given as follows [38]
Definition 1 (saddle node bifurcation) A saddle node bifur-cation in dynamical system takes place when two equilibriumpoints coalesce and disappear
In (1) two parameters 1198601and 119860
2are not specified
in Table 1 They represent proportional constants relatingthe transcription initiation rate to the absolute rate of CI and
4 Computational Biology Journal
CA CBAB
Nr at equilibrium Nc at equilibrium
A2 A2
Figure 2 Saddle node bifurcation with the variation of Crosynthesis rate 119860
2 This is a schematic plot and not drawn to scale
Cro protein synthesis [3] We use Figure 2 to illustrate Z-shaped bifurcation with the variation of 119860
2 Set 119860
1= 8 times
10minus9Mmin and divide the whole plane into three regions
as shown in Figure 2 labeled by A B and C Initially whenthe value of 119860
2is small in region A for example 119860
2asymp
5times10minus9Mmin there exists only one stable point and thus one
single curve segment for both 119873119903and 119873
119888 At the boundary
of regions A and B saddle node bifurcation occurs 1198602
increases to region B for example a typical value 1198602asymp
5 times 10minus8Mmin and two new equilibria (one saddle and
one stable equilibrium) emerge resulting in three equilibriaNote that this is a saddle node bifurcation with the reverseddirection in comparison to Definition 1
At the boundary of regions B and C saddle nodebifurcation takes place again leading to the vanishing oftwo equilibria (one saddle point and one stable equilibrium)In region C say 119860
2asymp 5 times 10
minus7Mmin only one stableequilibrium remains We notice the motion of the saddlepoint When the saddle point shows up at the boundary ofregions A and B the lysis stable equilibrium appears as wellAs parameter119860
2increases the saddle pointmoves toward the
lysogen stable point and finallymergeswith it at the boundaryof regions B and C Two successive bifurcations form a Z-shaped curve are shown in Figure 2
We have not yet specified the boundaries between theseregions quantitatively Next we solve the values of the bound-aries by using the Jacobian method Let us first introduce thefollowing two theorems [38]
Theorem 2 (Hartman-Grobman Theorem) If the Jacobianof the nonlinear system = 119892(119909) at equilibrium namely119869(1199090) has no zero or purely imaginary eigenvalues then the
phase trajectory of the system = 119892(119909) resembles that of thelinearized system = 119869(119909
0)119911 near 119909
0if none of the eigenvalues
of 119869(1199090) are on the 119895120596 axis
For a given equilibrium 1199090 in most cases we can
determine its dynamical behavior qualitatively by calculatingits Jacobian matrix 119869(119909
0) For a two-dimensional system its
Jacobian matrix can be calculated as
119869 =
[[[[
[
1205971198921
120597119873119903
1205971198921
120597119873119888
1205971198922
120597119873119903
1205971198922
120597119873119888
]]]]
]
(5)
The following theorem gives the necessary conditions forthe critical point that bifurcation occurs [38]
Theorem 3 For any nonlinear system = 119892(119911 120583) the systemgoes through a saddle node bifurcation at 119911 = 119911
dagger 120583 = 120583dagger only
if the following relations are satisfied
119892 (119911dagger 120583dagger) = 0 (6)
10038161003816100381610038161003816100381610038161003816
120597119892
120597119911(119911dagger 120583dagger)
10038161003816100381610038161003816100381610038161003816= 0 (7)
From (6) 119911dagger is an equilibrium point In a planar system(7) is the determinant of system Jacobian When the deter-minant of Jacobian matrix vanishes at least one eigenvalueis zero Jacobian is a characterization of planar systemdynamics For variation of parameter 119860
2 the system in (1)
can be rewritten as
119889119873119903
119889119905= 1198921(119873119888 119873119903 1198602)
119889119873119888
119889119905= 1198922(119873119888 119873119903 1198602)
(8)
From Theorems 2 and 3 for a given 1198602 the Jacobian matrix
is evaluated at equilibrium points 119873dagger119888and 119873
dagger
119903 These two
equilibrium points can be represented as a function of 1198602as
shown in Figure 2ByTheorem 3we know that the bifurcation appearswhen
Jacobian has a zero eigenvalue From Figure 3 we can obtainthe approximate values of critical points by checking thepoints of the eigenvalue curves crossing zero line Thesevalues are 56 times 10minus9Mmin and 34 times 10minus7Mmin
To determine the stability of an equilibrium point wecalculate the eigenvalue of the Jacobian matrix Figure 3 plotsthe eigenvalues as a function of 119860
2 The Jacobian matrix
can be calculated analytically (see the Appendix for details)By solving the value of 119860
2that makes the determinant of
the Jacobian matrix vanish we get the critical value of 1198602
at which bifurcation occurs When 1198602is small (less than
56 times 10minus9Mmin in Figure 3) there are only two negative
eigenvalues corresponding to one equilibrium point Thisequilibrium is thus stable [38]
As 1198602exceeds 56 times 10
minus9Mmin we have the casewhen there are three equilibria and correspondingly sixeigenvalues Among these three equilibria two are stableand the third is unstable (saddle point) This is becausepositive eigenvalue appears at saddle point (see dash-dotline in Figure 3) For a two-dimensional linear system if theJacobian has one positive and one negative eigenvalues at theequilibrium the equilibrium is a saddle point If119860
2continues
increasing only one equilibrium point is left (1198602greater than
34 times 10minus7Mmin in this case) and both of its eigenvalues
are negative The curves in Figure 3 have two intersectionswith zero eigenvalue which indicates that the eigenvalues ofJacobian vanish twice forming a Z-shaped bifurcation
Computational Biology Journal 5
0
002
004
Eige
nval
ues o
f Jac
obia
n
At equilibrium for lysogenAt equilibrium for lysisAt saddle point
10minus9
10minus8
10minus7
10minus6
10minus5
10minus4
minus012
minus01
minus008
minus006
minus004
minus002
A2 (Mmin)
Figure 3 Jacobian eigenvalues at equilibria The figure shows howthe eigenvalues of Jacobian at equilibria change with 119860
1fixed at
8 times 10minus9Mmin and 119860
2varying The solid line (black) shows
the eigenvalues of lysis equilibrium The dashed line (blue) is forlysogeny equilibrium The dot-dashed line (red) is for saddle
4 Bifurcation Analysis of BindingEnergy Uncertainty
In this section we will analyze bifurcation for Δ119866rsquos byusing the Jacobian method In Table 1 the symbol Δ119866 withsubscripts such as Δ119866
1119903and Δ119866
2119888represents Gibbs free
energy required for CI or Cro dimers to bind to the ORsites Because of the difficulty in determining the precisevalues of the binding energy it is useful to study how theequilibria change as these Gibbs free binding energies vary Inthis section we set synthesis rates 119860
1and 119860
2at appropriate
values 1198601= 8 times 10
minus9Mmin and 1198602= 5 times 10
minus8Mmin Tocalculate the equilibria we set the left hand side of (1) to zeroand obtain the expression of119873
119903and119873
119888as functions of Δ119866rsquos
119873lowast
119903= 1198911(Δ1198661119888 Δ1198662119888 Δ119866
1198773)
119873lowast
119888= 1198912(Δ1198661119888 Δ1198662119888 Δ119866
1198773)
(9)
where 119873lowast119903and 119873lowast
119888are the concentrations of proteins CI and
Cro at the equilibrium point Once (9) is obtained we candetermine the quantitative behavior around equilibria whenchanging Δ119866rsquos The relative sensitivities are given by
119878119873119903
(Δ119866119904) =
1205971198911
120597Δ119866119904
119878119873119888
(Δ119866119904) =
1205971198912
120597Δ119866119904
(10)
where the subscript 119904 isin 1119903 2119903 11987712 1198773 Since the systemequilibria cannot be analytically solved we may apply the
numerical procedure to calculate (10) as a measure of param-eter sensitivity As to the study on system robustness theJacobianmethod can be used to study bifurcation introducedby parameter variation
Since there are ten differentΔ119866s we divide them into fourgroups
(1) Δ1198661119903 Δ1198662119903 and Δ119866
3119903 binding free energies of CI
(2) Δ1198661119888 Δ1198662119888 and Δ119866
3119888 binding free energies of Cro
(3) Δ11986612119903
and Δ11986623119903
the cooperation between proteinsCIrsquos if both 119874
1198771 and 119874
1198772 or 119874
1198772 and 119874
1198773 are bound
by CI dimers(4) Δ119866
11987712and Δ119866
1198773 RNAP binding energies
We then study each of these four groups
41 Δ1198661119903 Δ1198662119903 and Δ119866
3119903 We systematically perturb the
values of Δ1198661119903 Δ1198662119903 and Δ119866
3119903to investigate their impact
on the system dynamics Two proper robustness measuresare the range of parameters in which lysogeny exists (stablerange) and inwhich bistable dynamics exists (bistable range)In stable range it is possible to keep phage 120582 dormant andintegrate into the gene of the host On the other hand bistablerange defines the parameter range to keep bistability and themaximal uncertainty tolerance In this case it implies thatthe system favors dynamics with two stable equilibria and asaddle Further if the stochastic force is strong enough phage120582 can switch from the lysogenic stable point to the lytic stablepoint even without the parameter changes
Figures 4(a)ndash4(c) illustrate the impact of Δ1198661119903 Δ1198662119903 and
Δ1198663119903
variations We set parameter values as in Table 1 andfind out all the equilibria when varying oneΔ119866
119904 We find that
the energy Δ1198663119903does not show Z-shaped bifurcation while
the other two parameters Δ1198661119903and Δ119866
2119903show quite similar
behaviors since Figures 4(a) and 4(b) are almost identicallythe sameThe result is not surprising because Δ119866
1119903and Δ119866
2119903
play similar roles in system dynamics These two parametersdetermine the likelihood that CI dimer binds to sites 119874
1198771
and 1198741198772 When either of the two sites is occupied by protein
dimers the expression of 119888119903119900 gene is prohibited resulting inlysogenic stateNonetheless the parameterΔ119866
3119903indicates the
likelihood of CI dimer binding to the site1198741198773 If the absolute
value of Δ1198663119903is too large it is easy for CI dimer to bind to
1198741198773 thus cI gene expression is prohibited resulting in lysis
If the absolute value ofΔ1198663119903is small the Z-shaped bifurcation
vanishes In this case cI gene expression is not prohibited butit has no influence on cro gene expression Bistable behavioralways exists when Δ119866
3119903is less than its nominal value in
Table 1 Without stopping or weakening the expression of crogene single lysogen stable point layout cannot be achievedjust by increasingΔ119866
3119903 Comparedwith other parameters the
system dynamics is more robust to Δ1198663119903variation
Table 3 lists the stable range bistable range and the lengthof bistable range for all tenΔ119866s It can be shownbyTheorem 2that one of the three equilibria is a saddle and the other twoare stable points Jacobian matrix is also used to determinethe boundary of stable and bistable ranges according toTheorem 3 As long as Δ119866
1119903is less than the nominal value
6 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG1r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG1r
(a) Z-shaped bifurcation for Δ1198661119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG2r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG2r
(b) Z-shaped bifurcation for Δ1198662119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG3r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG3r
(c) Bifurcation for Δ1198663119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG1c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG1c
(d) Bifurcation for Δ1198661119888
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG2c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG2c
(e) Bifurcation for Δ1198662119888
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG3c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG3c
(f) Z-shaped bifurcation for Δ1198663119888
Figure 4 Movement of equilibria under parameter variation Z-shaped bifurcation can be generated by variation of Δ1198661119903 Δ1198662119903 and Δ119866
3119888
All vertical axes are plotted on a log scale
at minus1133 kcalmol or Δ1198662119903
is less than minus934 kcalmol thesystem always has lysogenic stateThey are critical bifurcationpoints For Δ119866
3119903 however lysogenic state always exists if it is
greater than minus1233 kcalmol Consequently for CI protein alarger binding energy to119874
1198771 and119874
1198772 helps to keep the DNA
of phage 120582 silent in the host The system tends to be unstableif the values of binding energies for CI to 119874
1198771 and 119874
1198772
become small thus forming Z-shaped bifurcation Parametervariations may lead to Z-shaped bifurcation if and only if thebistable range is bounded
Computational Biology Journal 7
Table 3 Robustness measures of binding energies
Variable Stable Range(kcalmol)
Bistable Range(kcalmol)
Length(kcalmol)
Δ1198661119903 (minusinfin minus1133) (minus1549 minus1133) 416
Δ1198662119903 (minusinfin minus934) (minus1304 minus934) 370
Δ1198663119903 (minus1233 +infin) (minus1233 +infin) infin
Δ1198661119888 (minusinfin +infin) (minus1514 +infin) infin
Δ1198662119888 (minusinfin +infin) (minus1512 +infin) infin
Δ1198663119888 (minus1567 +infin) (minus1567 minus1224) 343
Δ11986612119903 (minusinfin minus155) (minus603 minus155) 448
Δ11986623119903 (minus771infin) (minus771 +infin) infin
Δ11986611987712 (minus1367infin) (minus1367 minus1086) 281
Δ1198661198773 (minusinfin minus1050) (minus1266 minus1050) 216
When the stochastic noise is strong enough the noisemayinduce the system to lytic stateMoreover if free energy variesdue to gene mutation bistable behavior might be destroyedand only one stable equilibrium is left This is what themathematical model implies about the parameter influenceon the system dynamics
42 Δ1198661119888 Δ1198662119888 and Δ119866
3119888 The bifurcation results of Δ119866
1119888
Δ1198662119888 andΔ119866
3119888are shown in Figures 4(d)ndash4(f)The equilibria
behaviors with respect toΔ1198661119888andΔ119866
2119888variation are similar
to Δ1198663119903 Δ1198661119888represents the likelihood that Cro dimer binds
to site 1198741198771 If Cro dimer binds to site 119874
1198771 the expression
of cro is prohibited resulting in only one single lysogenicstable point When Δ119866
1119888or Δ119866
2119888is small the prohibition
effect is weak but this is not enough to drive the systemdynamics to only one single lysis stable point Hence bistablelayout always exists if these two parameters are small Theinfluence ofΔ119866
3119888is similar to the influence ofΔ119866
1119903andΔ119866
2119903
The bistable range of Δ1198663119888
is about 343 kcalmol which iscomparable to those of Δ119866
1119903and Δ119866
2119903
The six parameters we have shown so far are the basicbinding energies The values of Δ119866
3119903 Δ1198661119888 and Δ119866
2119888decide
the likelihood that negative feedback (ie self-prohibition)happens It is known that the sites 119874
1198771 and 119874
1198772 are used
to transcript and then translate protein Cro The site 1198741198773
contributes to the production of protein CI If protein CIbinds to 119874
1198773 or Cro binds to 119874
1198771 and 119874
1198772 they can stop
their own transcription Thus we call the phenomenon neg-ative feedback Figure 4 shows that such negative feedbackcannot take the system to Z-shaped bifurcationThe negativefeedback occurs when the amount of Cro or CI is too largeFor instance the binding between CI dimer and 119874
1198773 readily
occurs when the amount of protein CI is too large On theother hand Δ119866
1119903 Δ1198662119903 and Δ119866
3119888represent the likelihood
of positive feedback (ie mutual prohibition) For instancewith largeΔ119866
3119888 the binding between Cro and119874
1198773 is favored
stopping the expression of cI gene If the value of Δ1198663119903
issmall and the negative feedback is weakened but it is notenough to make the system dynamics change from singlelysis equilibrium to single lysogenic equilibrium Parametersassociated with positive feedback can show Z-shaped bifur-cation whereas parameters associatedwith negative feedback
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG12r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG12r
(a) Z-shaped bifurcation for Δ11986612119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG23r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG23r
(b) Bifurcation for Δ11986623119903
Figure 5 Z-shaped bifurcation generated by variation of 11986612119903
and11986623119903The variation ofΔ119866
12119903also leads to Z-shaped bifurcation while
Δ11986623119903
is not sensitive enough to create Z-shaped bifurcation
cannotHence the system ismore robust to negative feedbackparameters than to positive feedback parameters
43 Δ11986612119903
and Δ11986623119903
The parameters Δ11986612119903
and Δ11986623119903
determine the energy difference caused by the cooperationof adjacent binding CI dimers Although [32] discovered thatadjacent Cro dimers also have cooperations we ignore thiscooperation effect since it has a small cooperation energyand focus on CI cooperation in our analysis Figure 5 showsthe bifurcation results for the parameters Δ119866
12119903and Δ119866
23119903
The nominal values of Δ11986612119903
and Δ11986623119903
are close to zeroSince protein cooperation process is an exothermic reactionΔ11986612119903
and Δ11986623119903
must be less than zero We find out thatthe variation of Δ119866
12119903leads to Z-shaped bifurcation whereas
Δ11986623119903
does not The value of Δ11986623119903
affects the expression ofboth cro and cI genes whereas Δ119866
12119903only promotes mutual
prohibition from CI to Cro Δ11986623119903
with large value prohibitsthe binding of RNAP to both sites119874
1198772 and119874
1198773 thus both 119888119868
and 119888119903119900 gene expressions are closed
8 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR12
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR12
(a) Z-shaped bifurcation for Δ11986611987712
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR3
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR3
(b) Z-shaped bifurcation for Δ1198661198773
Figure 6 Z-shaped bifurcation generated by variation of 11986611987712
and1198661198773 Both of them are able to generate Z-shaped bifurcation Their
bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope
Moreover we know the length of bifurcation range forΔ11986612119903
is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866
12119903
is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866
12119903is equal to the center value at minus379 kcalmol
the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866
12119903from this point of view
44 Δ11986611987712
and Δ1198661198773 Figure 6 illustrates the influence of
Δ1198661198773
and Δ11986611987712
variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the
appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866
11987712is 281 kcalmol and forΔ119866
1198773is only 216 kcalmol
which is the minimum among all ten parametersNotice that Δ119866
1198773and Δ119866
11987712are the binding free energies
of RNAP to 119875119877119872
promoter (1198741198773) and to 119875
119877promoter (119874
1198771
and 1198741198772) Because RNAP is necessary for gene expression
the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866
11987712and Δ119866
1198773
than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866
2119903are around 4 kcalmol while those of Δ119866
11987712
and Δ1198661198773
are less than 3 kcalmol Parameter uncertaintiesof Δ119866
11987712and Δ119866
1198773most easily destroy bistable behaviorThe
concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866
11987712or Δ119866
1198773uncertainty
45 Discussion From the above analysis we find that thebinding free energies to sites119874
1198771 and119874
1198772 have similar effects
on system dynamics As we have shown in Table 3 the resultsof Δ119866
1119903 Δ1198662119903 Δ1198661119888 and Δ119866
2119888are similar This is mainly
because of the similar status for binding sites 1198741198771 and 119874
1198772
both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for
Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866
23119903 All of the above four parameters
do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866
1119888andΔ119866
2119888can be used tomaintain
lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866
1119888lower than
minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582
stabilizes at lysogenyAs to the other six parameters with limited bistable ranges
in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866
1119903
Δ1198662119903 andΔ119866
12119903indicate the likelihood of protein CI binding
to sites 1198741198771 and 119874
1198772 or the likelihood of mutual prohi-
bition occurring Similarly Δ1198663119888indicates the likelihood of
protein Cro binding to site 1198741198773 and also results in mutual
prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior
Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866
11987712and Δ119866
1198773 Since
Δ11986611987712
and Δ1198661198773
have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins
Computational Biology Journal 9
In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866
11987712and Δ119866
1198773 because otherwise the system dynamics
may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily
5 Conclusion
In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866
11987712and Δ119866
1198773 Moreover parameters
describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866
1119903 Δ1198662119903 Δ1198663119888 and Δ119866
12119903) whereas parameters
describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866
2119888) can-
not Hence system dynamics is more sensitive to mutualprohibition parameters
There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors
Appendix
Calculation of Jacobian
From (1)ndash(5) we may represent system Jacobian by
119869 =[[[
[
1198601
120597119875119903
120597119873119903
minus 1205831
1198601
120597119875119903
120597119873119888
1198602
120597119875119888
120597119873119903
1198602
120597119875119888
120597119873119888
minus 1205832
]]]
]
(A1)
Since 119875119903and 119875
119888are sums of 119875
119904 119904 = 1 2 40 as (2) shows
the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875
119904120597119873119888 Since 119875
119904is a function with respect to
Cro and CI dimers we evaluate the partial derivatives byChain Rule
120597119875119904
120597119873119903
=120597119875119904
1205971198731199032
1198891198731199032
119889119873119903
120597119875119904
120597119873119888
=120597119875119904
1205971198731198882
1198891198731198882
119889119873119888
(A2)
Table 4 Exponents for 30 binding configurations in Table 2
119904 119894(119904) 119895(119904) 119896(119904)
1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2
where
1198891198731199032
119889119873119903
=1
2(1 minus
119870119903
radic1198702119903+ 8119873119903119870119903
)
1198891198731198882
119889119873119888
=1
2(1 minus
119870119888
radic1198702119888+ 8119873119888119870119888
)
(A3)
10 Computational Biology Journal
Thenwe have to deal with other terms in (A2) Note that119875119904is
a fractional function in (3) so we write its numerator119860119904and
denominator 119861 as
119872119904= exp(minus
Δ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)
1198882119873119896(119904)
RNAP (A4)
119863 =
40
sum
119904=1
119872119904 (A5)
The partial derivatives for 119875119904with respect to119873
1198882and119873
1199032are
given by
120597119875119904
1205971198731199032
=120597119872119904
1205971198731199032
1
119863minus
120597119863
1205971198731199032
119872119904
1198632
120597119875119904
1205971198731198882
=120597119872119904
1205971198731198882
1
119863minus
120597119863
1205971198731198882
119872119904
1198632
(A6)
The partial derivative of119872119904can be obtained as
120597119872119904
1205971198731199032
= 119894 (119904) exp(minusΔ119866119904
119877119879)119873119894(119904)minus1
1199032119873119895(119904)
1198882119873119896(119904)
RNAP
120597119872119904
1205971198731198882
= 119895 (119904) exp (minusΔ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)minus1
1198882119873119896(119904)
RNAP
(A7)
The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872
119904partial
derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM
References
[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982
[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985
[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990
[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004
[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006
[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007
[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009
[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010
[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011
[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002
[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004
[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004
[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007
[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008
[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012
[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004
[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004
[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000
[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996
[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995
Computational Biology Journal 11
[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002
[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999
[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004
[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008
[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009
[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992
[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010
[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982
[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004
[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975
[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009
[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000
[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967
[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000
[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998
[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002
[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984
[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999
[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004
[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
4 Computational Biology Journal
CA CBAB
Nr at equilibrium Nc at equilibrium
A2 A2
Figure 2 Saddle node bifurcation with the variation of Crosynthesis rate 119860
2 This is a schematic plot and not drawn to scale
Cro protein synthesis [3] We use Figure 2 to illustrate Z-shaped bifurcation with the variation of 119860
2 Set 119860
1= 8 times
10minus9Mmin and divide the whole plane into three regions
as shown in Figure 2 labeled by A B and C Initially whenthe value of 119860
2is small in region A for example 119860
2asymp
5times10minus9Mmin there exists only one stable point and thus one
single curve segment for both 119873119903and 119873
119888 At the boundary
of regions A and B saddle node bifurcation occurs 1198602
increases to region B for example a typical value 1198602asymp
5 times 10minus8Mmin and two new equilibria (one saddle and
one stable equilibrium) emerge resulting in three equilibriaNote that this is a saddle node bifurcation with the reverseddirection in comparison to Definition 1
At the boundary of regions B and C saddle nodebifurcation takes place again leading to the vanishing oftwo equilibria (one saddle point and one stable equilibrium)In region C say 119860
2asymp 5 times 10
minus7Mmin only one stableequilibrium remains We notice the motion of the saddlepoint When the saddle point shows up at the boundary ofregions A and B the lysis stable equilibrium appears as wellAs parameter119860
2increases the saddle pointmoves toward the
lysogen stable point and finallymergeswith it at the boundaryof regions B and C Two successive bifurcations form a Z-shaped curve are shown in Figure 2
We have not yet specified the boundaries between theseregions quantitatively Next we solve the values of the bound-aries by using the Jacobian method Let us first introduce thefollowing two theorems [38]
Theorem 2 (Hartman-Grobman Theorem) If the Jacobianof the nonlinear system = 119892(119909) at equilibrium namely119869(1199090) has no zero or purely imaginary eigenvalues then the
phase trajectory of the system = 119892(119909) resembles that of thelinearized system = 119869(119909
0)119911 near 119909
0if none of the eigenvalues
of 119869(1199090) are on the 119895120596 axis
For a given equilibrium 1199090 in most cases we can
determine its dynamical behavior qualitatively by calculatingits Jacobian matrix 119869(119909
0) For a two-dimensional system its
Jacobian matrix can be calculated as
119869 =
[[[[
[
1205971198921
120597119873119903
1205971198921
120597119873119888
1205971198922
120597119873119903
1205971198922
120597119873119888
]]]]
]
(5)
The following theorem gives the necessary conditions forthe critical point that bifurcation occurs [38]
Theorem 3 For any nonlinear system = 119892(119911 120583) the systemgoes through a saddle node bifurcation at 119911 = 119911
dagger 120583 = 120583dagger only
if the following relations are satisfied
119892 (119911dagger 120583dagger) = 0 (6)
10038161003816100381610038161003816100381610038161003816
120597119892
120597119911(119911dagger 120583dagger)
10038161003816100381610038161003816100381610038161003816= 0 (7)
From (6) 119911dagger is an equilibrium point In a planar system(7) is the determinant of system Jacobian When the deter-minant of Jacobian matrix vanishes at least one eigenvalueis zero Jacobian is a characterization of planar systemdynamics For variation of parameter 119860
2 the system in (1)
can be rewritten as
119889119873119903
119889119905= 1198921(119873119888 119873119903 1198602)
119889119873119888
119889119905= 1198922(119873119888 119873119903 1198602)
(8)
From Theorems 2 and 3 for a given 1198602 the Jacobian matrix
is evaluated at equilibrium points 119873dagger119888and 119873
dagger
119903 These two
equilibrium points can be represented as a function of 1198602as
shown in Figure 2ByTheorem 3we know that the bifurcation appearswhen
Jacobian has a zero eigenvalue From Figure 3 we can obtainthe approximate values of critical points by checking thepoints of the eigenvalue curves crossing zero line Thesevalues are 56 times 10minus9Mmin and 34 times 10minus7Mmin
To determine the stability of an equilibrium point wecalculate the eigenvalue of the Jacobian matrix Figure 3 plotsthe eigenvalues as a function of 119860
2 The Jacobian matrix
can be calculated analytically (see the Appendix for details)By solving the value of 119860
2that makes the determinant of
the Jacobian matrix vanish we get the critical value of 1198602
at which bifurcation occurs When 1198602is small (less than
56 times 10minus9Mmin in Figure 3) there are only two negative
eigenvalues corresponding to one equilibrium point Thisequilibrium is thus stable [38]
As 1198602exceeds 56 times 10
minus9Mmin we have the casewhen there are three equilibria and correspondingly sixeigenvalues Among these three equilibria two are stableand the third is unstable (saddle point) This is becausepositive eigenvalue appears at saddle point (see dash-dotline in Figure 3) For a two-dimensional linear system if theJacobian has one positive and one negative eigenvalues at theequilibrium the equilibrium is a saddle point If119860
2continues
increasing only one equilibrium point is left (1198602greater than
34 times 10minus7Mmin in this case) and both of its eigenvalues
are negative The curves in Figure 3 have two intersectionswith zero eigenvalue which indicates that the eigenvalues ofJacobian vanish twice forming a Z-shaped bifurcation
Computational Biology Journal 5
0
002
004
Eige
nval
ues o
f Jac
obia
n
At equilibrium for lysogenAt equilibrium for lysisAt saddle point
10minus9
10minus8
10minus7
10minus6
10minus5
10minus4
minus012
minus01
minus008
minus006
minus004
minus002
A2 (Mmin)
Figure 3 Jacobian eigenvalues at equilibria The figure shows howthe eigenvalues of Jacobian at equilibria change with 119860
1fixed at
8 times 10minus9Mmin and 119860
2varying The solid line (black) shows
the eigenvalues of lysis equilibrium The dashed line (blue) is forlysogeny equilibrium The dot-dashed line (red) is for saddle
4 Bifurcation Analysis of BindingEnergy Uncertainty
In this section we will analyze bifurcation for Δ119866rsquos byusing the Jacobian method In Table 1 the symbol Δ119866 withsubscripts such as Δ119866
1119903and Δ119866
2119888represents Gibbs free
energy required for CI or Cro dimers to bind to the ORsites Because of the difficulty in determining the precisevalues of the binding energy it is useful to study how theequilibria change as these Gibbs free binding energies vary Inthis section we set synthesis rates 119860
1and 119860
2at appropriate
values 1198601= 8 times 10
minus9Mmin and 1198602= 5 times 10
minus8Mmin Tocalculate the equilibria we set the left hand side of (1) to zeroand obtain the expression of119873
119903and119873
119888as functions of Δ119866rsquos
119873lowast
119903= 1198911(Δ1198661119888 Δ1198662119888 Δ119866
1198773)
119873lowast
119888= 1198912(Δ1198661119888 Δ1198662119888 Δ119866
1198773)
(9)
where 119873lowast119903and 119873lowast
119888are the concentrations of proteins CI and
Cro at the equilibrium point Once (9) is obtained we candetermine the quantitative behavior around equilibria whenchanging Δ119866rsquos The relative sensitivities are given by
119878119873119903
(Δ119866119904) =
1205971198911
120597Δ119866119904
119878119873119888
(Δ119866119904) =
1205971198912
120597Δ119866119904
(10)
where the subscript 119904 isin 1119903 2119903 11987712 1198773 Since the systemequilibria cannot be analytically solved we may apply the
numerical procedure to calculate (10) as a measure of param-eter sensitivity As to the study on system robustness theJacobianmethod can be used to study bifurcation introducedby parameter variation
Since there are ten differentΔ119866s we divide them into fourgroups
(1) Δ1198661119903 Δ1198662119903 and Δ119866
3119903 binding free energies of CI
(2) Δ1198661119888 Δ1198662119888 and Δ119866
3119888 binding free energies of Cro
(3) Δ11986612119903
and Δ11986623119903
the cooperation between proteinsCIrsquos if both 119874
1198771 and 119874
1198772 or 119874
1198772 and 119874
1198773 are bound
by CI dimers(4) Δ119866
11987712and Δ119866
1198773 RNAP binding energies
We then study each of these four groups
41 Δ1198661119903 Δ1198662119903 and Δ119866
3119903 We systematically perturb the
values of Δ1198661119903 Δ1198662119903 and Δ119866
3119903to investigate their impact
on the system dynamics Two proper robustness measuresare the range of parameters in which lysogeny exists (stablerange) and inwhich bistable dynamics exists (bistable range)In stable range it is possible to keep phage 120582 dormant andintegrate into the gene of the host On the other hand bistablerange defines the parameter range to keep bistability and themaximal uncertainty tolerance In this case it implies thatthe system favors dynamics with two stable equilibria and asaddle Further if the stochastic force is strong enough phage120582 can switch from the lysogenic stable point to the lytic stablepoint even without the parameter changes
Figures 4(a)ndash4(c) illustrate the impact of Δ1198661119903 Δ1198662119903 and
Δ1198663119903
variations We set parameter values as in Table 1 andfind out all the equilibria when varying oneΔ119866
119904 We find that
the energy Δ1198663119903does not show Z-shaped bifurcation while
the other two parameters Δ1198661119903and Δ119866
2119903show quite similar
behaviors since Figures 4(a) and 4(b) are almost identicallythe sameThe result is not surprising because Δ119866
1119903and Δ119866
2119903
play similar roles in system dynamics These two parametersdetermine the likelihood that CI dimer binds to sites 119874
1198771
and 1198741198772 When either of the two sites is occupied by protein
dimers the expression of 119888119903119900 gene is prohibited resulting inlysogenic stateNonetheless the parameterΔ119866
3119903indicates the
likelihood of CI dimer binding to the site1198741198773 If the absolute
value of Δ1198663119903is too large it is easy for CI dimer to bind to
1198741198773 thus cI gene expression is prohibited resulting in lysis
If the absolute value ofΔ1198663119903is small the Z-shaped bifurcation
vanishes In this case cI gene expression is not prohibited butit has no influence on cro gene expression Bistable behavioralways exists when Δ119866
3119903is less than its nominal value in
Table 1 Without stopping or weakening the expression of crogene single lysogen stable point layout cannot be achievedjust by increasingΔ119866
3119903 Comparedwith other parameters the
system dynamics is more robust to Δ1198663119903variation
Table 3 lists the stable range bistable range and the lengthof bistable range for all tenΔ119866s It can be shownbyTheorem 2that one of the three equilibria is a saddle and the other twoare stable points Jacobian matrix is also used to determinethe boundary of stable and bistable ranges according toTheorem 3 As long as Δ119866
1119903is less than the nominal value
6 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG1r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG1r
(a) Z-shaped bifurcation for Δ1198661119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG2r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG2r
(b) Z-shaped bifurcation for Δ1198662119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG3r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG3r
(c) Bifurcation for Δ1198663119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG1c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG1c
(d) Bifurcation for Δ1198661119888
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG2c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG2c
(e) Bifurcation for Δ1198662119888
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG3c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG3c
(f) Z-shaped bifurcation for Δ1198663119888
Figure 4 Movement of equilibria under parameter variation Z-shaped bifurcation can be generated by variation of Δ1198661119903 Δ1198662119903 and Δ119866
3119888
All vertical axes are plotted on a log scale
at minus1133 kcalmol or Δ1198662119903
is less than minus934 kcalmol thesystem always has lysogenic stateThey are critical bifurcationpoints For Δ119866
3119903 however lysogenic state always exists if it is
greater than minus1233 kcalmol Consequently for CI protein alarger binding energy to119874
1198771 and119874
1198772 helps to keep the DNA
of phage 120582 silent in the host The system tends to be unstableif the values of binding energies for CI to 119874
1198771 and 119874
1198772
become small thus forming Z-shaped bifurcation Parametervariations may lead to Z-shaped bifurcation if and only if thebistable range is bounded
Computational Biology Journal 7
Table 3 Robustness measures of binding energies
Variable Stable Range(kcalmol)
Bistable Range(kcalmol)
Length(kcalmol)
Δ1198661119903 (minusinfin minus1133) (minus1549 minus1133) 416
Δ1198662119903 (minusinfin minus934) (minus1304 minus934) 370
Δ1198663119903 (minus1233 +infin) (minus1233 +infin) infin
Δ1198661119888 (minusinfin +infin) (minus1514 +infin) infin
Δ1198662119888 (minusinfin +infin) (minus1512 +infin) infin
Δ1198663119888 (minus1567 +infin) (minus1567 minus1224) 343
Δ11986612119903 (minusinfin minus155) (minus603 minus155) 448
Δ11986623119903 (minus771infin) (minus771 +infin) infin
Δ11986611987712 (minus1367infin) (minus1367 minus1086) 281
Δ1198661198773 (minusinfin minus1050) (minus1266 minus1050) 216
When the stochastic noise is strong enough the noisemayinduce the system to lytic stateMoreover if free energy variesdue to gene mutation bistable behavior might be destroyedand only one stable equilibrium is left This is what themathematical model implies about the parameter influenceon the system dynamics
42 Δ1198661119888 Δ1198662119888 and Δ119866
3119888 The bifurcation results of Δ119866
1119888
Δ1198662119888 andΔ119866
3119888are shown in Figures 4(d)ndash4(f)The equilibria
behaviors with respect toΔ1198661119888andΔ119866
2119888variation are similar
to Δ1198663119903 Δ1198661119888represents the likelihood that Cro dimer binds
to site 1198741198771 If Cro dimer binds to site 119874
1198771 the expression
of cro is prohibited resulting in only one single lysogenicstable point When Δ119866
1119888or Δ119866
2119888is small the prohibition
effect is weak but this is not enough to drive the systemdynamics to only one single lysis stable point Hence bistablelayout always exists if these two parameters are small Theinfluence ofΔ119866
3119888is similar to the influence ofΔ119866
1119903andΔ119866
2119903
The bistable range of Δ1198663119888
is about 343 kcalmol which iscomparable to those of Δ119866
1119903and Δ119866
2119903
The six parameters we have shown so far are the basicbinding energies The values of Δ119866
3119903 Δ1198661119888 and Δ119866
2119888decide
the likelihood that negative feedback (ie self-prohibition)happens It is known that the sites 119874
1198771 and 119874
1198772 are used
to transcript and then translate protein Cro The site 1198741198773
contributes to the production of protein CI If protein CIbinds to 119874
1198773 or Cro binds to 119874
1198771 and 119874
1198772 they can stop
their own transcription Thus we call the phenomenon neg-ative feedback Figure 4 shows that such negative feedbackcannot take the system to Z-shaped bifurcationThe negativefeedback occurs when the amount of Cro or CI is too largeFor instance the binding between CI dimer and 119874
1198773 readily
occurs when the amount of protein CI is too large On theother hand Δ119866
1119903 Δ1198662119903 and Δ119866
3119888represent the likelihood
of positive feedback (ie mutual prohibition) For instancewith largeΔ119866
3119888 the binding between Cro and119874
1198773 is favored
stopping the expression of cI gene If the value of Δ1198663119903
issmall and the negative feedback is weakened but it is notenough to make the system dynamics change from singlelysis equilibrium to single lysogenic equilibrium Parametersassociated with positive feedback can show Z-shaped bifur-cation whereas parameters associatedwith negative feedback
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG12r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG12r
(a) Z-shaped bifurcation for Δ11986612119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG23r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG23r
(b) Bifurcation for Δ11986623119903
Figure 5 Z-shaped bifurcation generated by variation of 11986612119903
and11986623119903The variation ofΔ119866
12119903also leads to Z-shaped bifurcation while
Δ11986623119903
is not sensitive enough to create Z-shaped bifurcation
cannotHence the system ismore robust to negative feedbackparameters than to positive feedback parameters
43 Δ11986612119903
and Δ11986623119903
The parameters Δ11986612119903
and Δ11986623119903
determine the energy difference caused by the cooperationof adjacent binding CI dimers Although [32] discovered thatadjacent Cro dimers also have cooperations we ignore thiscooperation effect since it has a small cooperation energyand focus on CI cooperation in our analysis Figure 5 showsthe bifurcation results for the parameters Δ119866
12119903and Δ119866
23119903
The nominal values of Δ11986612119903
and Δ11986623119903
are close to zeroSince protein cooperation process is an exothermic reactionΔ11986612119903
and Δ11986623119903
must be less than zero We find out thatthe variation of Δ119866
12119903leads to Z-shaped bifurcation whereas
Δ11986623119903
does not The value of Δ11986623119903
affects the expression ofboth cro and cI genes whereas Δ119866
12119903only promotes mutual
prohibition from CI to Cro Δ11986623119903
with large value prohibitsthe binding of RNAP to both sites119874
1198772 and119874
1198773 thus both 119888119868
and 119888119903119900 gene expressions are closed
8 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR12
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR12
(a) Z-shaped bifurcation for Δ11986611987712
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR3
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR3
(b) Z-shaped bifurcation for Δ1198661198773
Figure 6 Z-shaped bifurcation generated by variation of 11986611987712
and1198661198773 Both of them are able to generate Z-shaped bifurcation Their
bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope
Moreover we know the length of bifurcation range forΔ11986612119903
is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866
12119903
is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866
12119903is equal to the center value at minus379 kcalmol
the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866
12119903from this point of view
44 Δ11986611987712
and Δ1198661198773 Figure 6 illustrates the influence of
Δ1198661198773
and Δ11986611987712
variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the
appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866
11987712is 281 kcalmol and forΔ119866
1198773is only 216 kcalmol
which is the minimum among all ten parametersNotice that Δ119866
1198773and Δ119866
11987712are the binding free energies
of RNAP to 119875119877119872
promoter (1198741198773) and to 119875
119877promoter (119874
1198771
and 1198741198772) Because RNAP is necessary for gene expression
the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866
11987712and Δ119866
1198773
than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866
2119903are around 4 kcalmol while those of Δ119866
11987712
and Δ1198661198773
are less than 3 kcalmol Parameter uncertaintiesof Δ119866
11987712and Δ119866
1198773most easily destroy bistable behaviorThe
concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866
11987712or Δ119866
1198773uncertainty
45 Discussion From the above analysis we find that thebinding free energies to sites119874
1198771 and119874
1198772 have similar effects
on system dynamics As we have shown in Table 3 the resultsof Δ119866
1119903 Δ1198662119903 Δ1198661119888 and Δ119866
2119888are similar This is mainly
because of the similar status for binding sites 1198741198771 and 119874
1198772
both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for
Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866
23119903 All of the above four parameters
do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866
1119888andΔ119866
2119888can be used tomaintain
lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866
1119888lower than
minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582
stabilizes at lysogenyAs to the other six parameters with limited bistable ranges
in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866
1119903
Δ1198662119903 andΔ119866
12119903indicate the likelihood of protein CI binding
to sites 1198741198771 and 119874
1198772 or the likelihood of mutual prohi-
bition occurring Similarly Δ1198663119888indicates the likelihood of
protein Cro binding to site 1198741198773 and also results in mutual
prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior
Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866
11987712and Δ119866
1198773 Since
Δ11986611987712
and Δ1198661198773
have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins
Computational Biology Journal 9
In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866
11987712and Δ119866
1198773 because otherwise the system dynamics
may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily
5 Conclusion
In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866
11987712and Δ119866
1198773 Moreover parameters
describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866
1119903 Δ1198662119903 Δ1198663119888 and Δ119866
12119903) whereas parameters
describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866
2119888) can-
not Hence system dynamics is more sensitive to mutualprohibition parameters
There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors
Appendix
Calculation of Jacobian
From (1)ndash(5) we may represent system Jacobian by
119869 =[[[
[
1198601
120597119875119903
120597119873119903
minus 1205831
1198601
120597119875119903
120597119873119888
1198602
120597119875119888
120597119873119903
1198602
120597119875119888
120597119873119888
minus 1205832
]]]
]
(A1)
Since 119875119903and 119875
119888are sums of 119875
119904 119904 = 1 2 40 as (2) shows
the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875
119904120597119873119888 Since 119875
119904is a function with respect to
Cro and CI dimers we evaluate the partial derivatives byChain Rule
120597119875119904
120597119873119903
=120597119875119904
1205971198731199032
1198891198731199032
119889119873119903
120597119875119904
120597119873119888
=120597119875119904
1205971198731198882
1198891198731198882
119889119873119888
(A2)
Table 4 Exponents for 30 binding configurations in Table 2
119904 119894(119904) 119895(119904) 119896(119904)
1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2
where
1198891198731199032
119889119873119903
=1
2(1 minus
119870119903
radic1198702119903+ 8119873119903119870119903
)
1198891198731198882
119889119873119888
=1
2(1 minus
119870119888
radic1198702119888+ 8119873119888119870119888
)
(A3)
10 Computational Biology Journal
Thenwe have to deal with other terms in (A2) Note that119875119904is
a fractional function in (3) so we write its numerator119860119904and
denominator 119861 as
119872119904= exp(minus
Δ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)
1198882119873119896(119904)
RNAP (A4)
119863 =
40
sum
119904=1
119872119904 (A5)
The partial derivatives for 119875119904with respect to119873
1198882and119873
1199032are
given by
120597119875119904
1205971198731199032
=120597119872119904
1205971198731199032
1
119863minus
120597119863
1205971198731199032
119872119904
1198632
120597119875119904
1205971198731198882
=120597119872119904
1205971198731198882
1
119863minus
120597119863
1205971198731198882
119872119904
1198632
(A6)
The partial derivative of119872119904can be obtained as
120597119872119904
1205971198731199032
= 119894 (119904) exp(minusΔ119866119904
119877119879)119873119894(119904)minus1
1199032119873119895(119904)
1198882119873119896(119904)
RNAP
120597119872119904
1205971198731198882
= 119895 (119904) exp (minusΔ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)minus1
1198882119873119896(119904)
RNAP
(A7)
The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872
119904partial
derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM
References
[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982
[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985
[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990
[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004
[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006
[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007
[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009
[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010
[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011
[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002
[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004
[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004
[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007
[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008
[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012
[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004
[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004
[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000
[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996
[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995
Computational Biology Journal 11
[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002
[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999
[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004
[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008
[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009
[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992
[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010
[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982
[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004
[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975
[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009
[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000
[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967
[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000
[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998
[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002
[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984
[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999
[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004
[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
Computational Biology Journal 5
0
002
004
Eige
nval
ues o
f Jac
obia
n
At equilibrium for lysogenAt equilibrium for lysisAt saddle point
10minus9
10minus8
10minus7
10minus6
10minus5
10minus4
minus012
minus01
minus008
minus006
minus004
minus002
A2 (Mmin)
Figure 3 Jacobian eigenvalues at equilibria The figure shows howthe eigenvalues of Jacobian at equilibria change with 119860
1fixed at
8 times 10minus9Mmin and 119860
2varying The solid line (black) shows
the eigenvalues of lysis equilibrium The dashed line (blue) is forlysogeny equilibrium The dot-dashed line (red) is for saddle
4 Bifurcation Analysis of BindingEnergy Uncertainty
In this section we will analyze bifurcation for Δ119866rsquos byusing the Jacobian method In Table 1 the symbol Δ119866 withsubscripts such as Δ119866
1119903and Δ119866
2119888represents Gibbs free
energy required for CI or Cro dimers to bind to the ORsites Because of the difficulty in determining the precisevalues of the binding energy it is useful to study how theequilibria change as these Gibbs free binding energies vary Inthis section we set synthesis rates 119860
1and 119860
2at appropriate
values 1198601= 8 times 10
minus9Mmin and 1198602= 5 times 10
minus8Mmin Tocalculate the equilibria we set the left hand side of (1) to zeroand obtain the expression of119873
119903and119873
119888as functions of Δ119866rsquos
119873lowast
119903= 1198911(Δ1198661119888 Δ1198662119888 Δ119866
1198773)
119873lowast
119888= 1198912(Δ1198661119888 Δ1198662119888 Δ119866
1198773)
(9)
where 119873lowast119903and 119873lowast
119888are the concentrations of proteins CI and
Cro at the equilibrium point Once (9) is obtained we candetermine the quantitative behavior around equilibria whenchanging Δ119866rsquos The relative sensitivities are given by
119878119873119903
(Δ119866119904) =
1205971198911
120597Δ119866119904
119878119873119888
(Δ119866119904) =
1205971198912
120597Δ119866119904
(10)
where the subscript 119904 isin 1119903 2119903 11987712 1198773 Since the systemequilibria cannot be analytically solved we may apply the
numerical procedure to calculate (10) as a measure of param-eter sensitivity As to the study on system robustness theJacobianmethod can be used to study bifurcation introducedby parameter variation
Since there are ten differentΔ119866s we divide them into fourgroups
(1) Δ1198661119903 Δ1198662119903 and Δ119866
3119903 binding free energies of CI
(2) Δ1198661119888 Δ1198662119888 and Δ119866
3119888 binding free energies of Cro
(3) Δ11986612119903
and Δ11986623119903
the cooperation between proteinsCIrsquos if both 119874
1198771 and 119874
1198772 or 119874
1198772 and 119874
1198773 are bound
by CI dimers(4) Δ119866
11987712and Δ119866
1198773 RNAP binding energies
We then study each of these four groups
41 Δ1198661119903 Δ1198662119903 and Δ119866
3119903 We systematically perturb the
values of Δ1198661119903 Δ1198662119903 and Δ119866
3119903to investigate their impact
on the system dynamics Two proper robustness measuresare the range of parameters in which lysogeny exists (stablerange) and inwhich bistable dynamics exists (bistable range)In stable range it is possible to keep phage 120582 dormant andintegrate into the gene of the host On the other hand bistablerange defines the parameter range to keep bistability and themaximal uncertainty tolerance In this case it implies thatthe system favors dynamics with two stable equilibria and asaddle Further if the stochastic force is strong enough phage120582 can switch from the lysogenic stable point to the lytic stablepoint even without the parameter changes
Figures 4(a)ndash4(c) illustrate the impact of Δ1198661119903 Δ1198662119903 and
Δ1198663119903
variations We set parameter values as in Table 1 andfind out all the equilibria when varying oneΔ119866
119904 We find that
the energy Δ1198663119903does not show Z-shaped bifurcation while
the other two parameters Δ1198661119903and Δ119866
2119903show quite similar
behaviors since Figures 4(a) and 4(b) are almost identicallythe sameThe result is not surprising because Δ119866
1119903and Δ119866
2119903
play similar roles in system dynamics These two parametersdetermine the likelihood that CI dimer binds to sites 119874
1198771
and 1198741198772 When either of the two sites is occupied by protein
dimers the expression of 119888119903119900 gene is prohibited resulting inlysogenic stateNonetheless the parameterΔ119866
3119903indicates the
likelihood of CI dimer binding to the site1198741198773 If the absolute
value of Δ1198663119903is too large it is easy for CI dimer to bind to
1198741198773 thus cI gene expression is prohibited resulting in lysis
If the absolute value ofΔ1198663119903is small the Z-shaped bifurcation
vanishes In this case cI gene expression is not prohibited butit has no influence on cro gene expression Bistable behavioralways exists when Δ119866
3119903is less than its nominal value in
Table 1 Without stopping or weakening the expression of crogene single lysogen stable point layout cannot be achievedjust by increasingΔ119866
3119903 Comparedwith other parameters the
system dynamics is more robust to Δ1198663119903variation
Table 3 lists the stable range bistable range and the lengthof bistable range for all tenΔ119866s It can be shownbyTheorem 2that one of the three equilibria is a saddle and the other twoare stable points Jacobian matrix is also used to determinethe boundary of stable and bistable ranges according toTheorem 3 As long as Δ119866
1119903is less than the nominal value
6 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG1r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG1r
(a) Z-shaped bifurcation for Δ1198661119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG2r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG2r
(b) Z-shaped bifurcation for Δ1198662119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG3r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG3r
(c) Bifurcation for Δ1198663119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG1c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG1c
(d) Bifurcation for Δ1198661119888
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG2c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG2c
(e) Bifurcation for Δ1198662119888
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG3c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG3c
(f) Z-shaped bifurcation for Δ1198663119888
Figure 4 Movement of equilibria under parameter variation Z-shaped bifurcation can be generated by variation of Δ1198661119903 Δ1198662119903 and Δ119866
3119888
All vertical axes are plotted on a log scale
at minus1133 kcalmol or Δ1198662119903
is less than minus934 kcalmol thesystem always has lysogenic stateThey are critical bifurcationpoints For Δ119866
3119903 however lysogenic state always exists if it is
greater than minus1233 kcalmol Consequently for CI protein alarger binding energy to119874
1198771 and119874
1198772 helps to keep the DNA
of phage 120582 silent in the host The system tends to be unstableif the values of binding energies for CI to 119874
1198771 and 119874
1198772
become small thus forming Z-shaped bifurcation Parametervariations may lead to Z-shaped bifurcation if and only if thebistable range is bounded
Computational Biology Journal 7
Table 3 Robustness measures of binding energies
Variable Stable Range(kcalmol)
Bistable Range(kcalmol)
Length(kcalmol)
Δ1198661119903 (minusinfin minus1133) (minus1549 minus1133) 416
Δ1198662119903 (minusinfin minus934) (minus1304 minus934) 370
Δ1198663119903 (minus1233 +infin) (minus1233 +infin) infin
Δ1198661119888 (minusinfin +infin) (minus1514 +infin) infin
Δ1198662119888 (minusinfin +infin) (minus1512 +infin) infin
Δ1198663119888 (minus1567 +infin) (minus1567 minus1224) 343
Δ11986612119903 (minusinfin minus155) (minus603 minus155) 448
Δ11986623119903 (minus771infin) (minus771 +infin) infin
Δ11986611987712 (minus1367infin) (minus1367 minus1086) 281
Δ1198661198773 (minusinfin minus1050) (minus1266 minus1050) 216
When the stochastic noise is strong enough the noisemayinduce the system to lytic stateMoreover if free energy variesdue to gene mutation bistable behavior might be destroyedand only one stable equilibrium is left This is what themathematical model implies about the parameter influenceon the system dynamics
42 Δ1198661119888 Δ1198662119888 and Δ119866
3119888 The bifurcation results of Δ119866
1119888
Δ1198662119888 andΔ119866
3119888are shown in Figures 4(d)ndash4(f)The equilibria
behaviors with respect toΔ1198661119888andΔ119866
2119888variation are similar
to Δ1198663119903 Δ1198661119888represents the likelihood that Cro dimer binds
to site 1198741198771 If Cro dimer binds to site 119874
1198771 the expression
of cro is prohibited resulting in only one single lysogenicstable point When Δ119866
1119888or Δ119866
2119888is small the prohibition
effect is weak but this is not enough to drive the systemdynamics to only one single lysis stable point Hence bistablelayout always exists if these two parameters are small Theinfluence ofΔ119866
3119888is similar to the influence ofΔ119866
1119903andΔ119866
2119903
The bistable range of Δ1198663119888
is about 343 kcalmol which iscomparable to those of Δ119866
1119903and Δ119866
2119903
The six parameters we have shown so far are the basicbinding energies The values of Δ119866
3119903 Δ1198661119888 and Δ119866
2119888decide
the likelihood that negative feedback (ie self-prohibition)happens It is known that the sites 119874
1198771 and 119874
1198772 are used
to transcript and then translate protein Cro The site 1198741198773
contributes to the production of protein CI If protein CIbinds to 119874
1198773 or Cro binds to 119874
1198771 and 119874
1198772 they can stop
their own transcription Thus we call the phenomenon neg-ative feedback Figure 4 shows that such negative feedbackcannot take the system to Z-shaped bifurcationThe negativefeedback occurs when the amount of Cro or CI is too largeFor instance the binding between CI dimer and 119874
1198773 readily
occurs when the amount of protein CI is too large On theother hand Δ119866
1119903 Δ1198662119903 and Δ119866
3119888represent the likelihood
of positive feedback (ie mutual prohibition) For instancewith largeΔ119866
3119888 the binding between Cro and119874
1198773 is favored
stopping the expression of cI gene If the value of Δ1198663119903
issmall and the negative feedback is weakened but it is notenough to make the system dynamics change from singlelysis equilibrium to single lysogenic equilibrium Parametersassociated with positive feedback can show Z-shaped bifur-cation whereas parameters associatedwith negative feedback
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG12r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG12r
(a) Z-shaped bifurcation for Δ11986612119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG23r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG23r
(b) Bifurcation for Δ11986623119903
Figure 5 Z-shaped bifurcation generated by variation of 11986612119903
and11986623119903The variation ofΔ119866
12119903also leads to Z-shaped bifurcation while
Δ11986623119903
is not sensitive enough to create Z-shaped bifurcation
cannotHence the system ismore robust to negative feedbackparameters than to positive feedback parameters
43 Δ11986612119903
and Δ11986623119903
The parameters Δ11986612119903
and Δ11986623119903
determine the energy difference caused by the cooperationof adjacent binding CI dimers Although [32] discovered thatadjacent Cro dimers also have cooperations we ignore thiscooperation effect since it has a small cooperation energyand focus on CI cooperation in our analysis Figure 5 showsthe bifurcation results for the parameters Δ119866
12119903and Δ119866
23119903
The nominal values of Δ11986612119903
and Δ11986623119903
are close to zeroSince protein cooperation process is an exothermic reactionΔ11986612119903
and Δ11986623119903
must be less than zero We find out thatthe variation of Δ119866
12119903leads to Z-shaped bifurcation whereas
Δ11986623119903
does not The value of Δ11986623119903
affects the expression ofboth cro and cI genes whereas Δ119866
12119903only promotes mutual
prohibition from CI to Cro Δ11986623119903
with large value prohibitsthe binding of RNAP to both sites119874
1198772 and119874
1198773 thus both 119888119868
and 119888119903119900 gene expressions are closed
8 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR12
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR12
(a) Z-shaped bifurcation for Δ11986611987712
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR3
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR3
(b) Z-shaped bifurcation for Δ1198661198773
Figure 6 Z-shaped bifurcation generated by variation of 11986611987712
and1198661198773 Both of them are able to generate Z-shaped bifurcation Their
bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope
Moreover we know the length of bifurcation range forΔ11986612119903
is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866
12119903
is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866
12119903is equal to the center value at minus379 kcalmol
the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866
12119903from this point of view
44 Δ11986611987712
and Δ1198661198773 Figure 6 illustrates the influence of
Δ1198661198773
and Δ11986611987712
variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the
appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866
11987712is 281 kcalmol and forΔ119866
1198773is only 216 kcalmol
which is the minimum among all ten parametersNotice that Δ119866
1198773and Δ119866
11987712are the binding free energies
of RNAP to 119875119877119872
promoter (1198741198773) and to 119875
119877promoter (119874
1198771
and 1198741198772) Because RNAP is necessary for gene expression
the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866
11987712and Δ119866
1198773
than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866
2119903are around 4 kcalmol while those of Δ119866
11987712
and Δ1198661198773
are less than 3 kcalmol Parameter uncertaintiesof Δ119866
11987712and Δ119866
1198773most easily destroy bistable behaviorThe
concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866
11987712or Δ119866
1198773uncertainty
45 Discussion From the above analysis we find that thebinding free energies to sites119874
1198771 and119874
1198772 have similar effects
on system dynamics As we have shown in Table 3 the resultsof Δ119866
1119903 Δ1198662119903 Δ1198661119888 and Δ119866
2119888are similar This is mainly
because of the similar status for binding sites 1198741198771 and 119874
1198772
both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for
Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866
23119903 All of the above four parameters
do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866
1119888andΔ119866
2119888can be used tomaintain
lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866
1119888lower than
minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582
stabilizes at lysogenyAs to the other six parameters with limited bistable ranges
in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866
1119903
Δ1198662119903 andΔ119866
12119903indicate the likelihood of protein CI binding
to sites 1198741198771 and 119874
1198772 or the likelihood of mutual prohi-
bition occurring Similarly Δ1198663119888indicates the likelihood of
protein Cro binding to site 1198741198773 and also results in mutual
prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior
Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866
11987712and Δ119866
1198773 Since
Δ11986611987712
and Δ1198661198773
have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins
Computational Biology Journal 9
In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866
11987712and Δ119866
1198773 because otherwise the system dynamics
may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily
5 Conclusion
In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866
11987712and Δ119866
1198773 Moreover parameters
describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866
1119903 Δ1198662119903 Δ1198663119888 and Δ119866
12119903) whereas parameters
describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866
2119888) can-
not Hence system dynamics is more sensitive to mutualprohibition parameters
There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors
Appendix
Calculation of Jacobian
From (1)ndash(5) we may represent system Jacobian by
119869 =[[[
[
1198601
120597119875119903
120597119873119903
minus 1205831
1198601
120597119875119903
120597119873119888
1198602
120597119875119888
120597119873119903
1198602
120597119875119888
120597119873119888
minus 1205832
]]]
]
(A1)
Since 119875119903and 119875
119888are sums of 119875
119904 119904 = 1 2 40 as (2) shows
the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875
119904120597119873119888 Since 119875
119904is a function with respect to
Cro and CI dimers we evaluate the partial derivatives byChain Rule
120597119875119904
120597119873119903
=120597119875119904
1205971198731199032
1198891198731199032
119889119873119903
120597119875119904
120597119873119888
=120597119875119904
1205971198731198882
1198891198731198882
119889119873119888
(A2)
Table 4 Exponents for 30 binding configurations in Table 2
119904 119894(119904) 119895(119904) 119896(119904)
1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2
where
1198891198731199032
119889119873119903
=1
2(1 minus
119870119903
radic1198702119903+ 8119873119903119870119903
)
1198891198731198882
119889119873119888
=1
2(1 minus
119870119888
radic1198702119888+ 8119873119888119870119888
)
(A3)
10 Computational Biology Journal
Thenwe have to deal with other terms in (A2) Note that119875119904is
a fractional function in (3) so we write its numerator119860119904and
denominator 119861 as
119872119904= exp(minus
Δ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)
1198882119873119896(119904)
RNAP (A4)
119863 =
40
sum
119904=1
119872119904 (A5)
The partial derivatives for 119875119904with respect to119873
1198882and119873
1199032are
given by
120597119875119904
1205971198731199032
=120597119872119904
1205971198731199032
1
119863minus
120597119863
1205971198731199032
119872119904
1198632
120597119875119904
1205971198731198882
=120597119872119904
1205971198731198882
1
119863minus
120597119863
1205971198731198882
119872119904
1198632
(A6)
The partial derivative of119872119904can be obtained as
120597119872119904
1205971198731199032
= 119894 (119904) exp(minusΔ119866119904
119877119879)119873119894(119904)minus1
1199032119873119895(119904)
1198882119873119896(119904)
RNAP
120597119872119904
1205971198731198882
= 119895 (119904) exp (minusΔ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)minus1
1198882119873119896(119904)
RNAP
(A7)
The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872
119904partial
derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM
References
[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982
[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985
[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990
[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004
[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006
[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007
[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009
[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010
[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011
[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002
[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004
[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004
[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007
[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008
[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012
[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004
[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004
[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000
[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996
[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995
Computational Biology Journal 11
[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002
[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999
[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004
[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008
[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009
[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992
[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010
[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982
[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004
[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975
[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009
[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000
[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967
[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000
[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998
[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002
[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984
[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999
[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004
[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
6 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG1r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG1r
(a) Z-shaped bifurcation for Δ1198661119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG2r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG2r
(b) Z-shaped bifurcation for Δ1198662119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG3r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG3r
(c) Bifurcation for Δ1198663119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG1c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG1c
(d) Bifurcation for Δ1198661119888
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG2c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG2c
(e) Bifurcation for Δ1198662119888
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG3c
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG3c
(f) Z-shaped bifurcation for Δ1198663119888
Figure 4 Movement of equilibria under parameter variation Z-shaped bifurcation can be generated by variation of Δ1198661119903 Δ1198662119903 and Δ119866
3119888
All vertical axes are plotted on a log scale
at minus1133 kcalmol or Δ1198662119903
is less than minus934 kcalmol thesystem always has lysogenic stateThey are critical bifurcationpoints For Δ119866
3119903 however lysogenic state always exists if it is
greater than minus1233 kcalmol Consequently for CI protein alarger binding energy to119874
1198771 and119874
1198772 helps to keep the DNA
of phage 120582 silent in the host The system tends to be unstableif the values of binding energies for CI to 119874
1198771 and 119874
1198772
become small thus forming Z-shaped bifurcation Parametervariations may lead to Z-shaped bifurcation if and only if thebistable range is bounded
Computational Biology Journal 7
Table 3 Robustness measures of binding energies
Variable Stable Range(kcalmol)
Bistable Range(kcalmol)
Length(kcalmol)
Δ1198661119903 (minusinfin minus1133) (minus1549 minus1133) 416
Δ1198662119903 (minusinfin minus934) (minus1304 minus934) 370
Δ1198663119903 (minus1233 +infin) (minus1233 +infin) infin
Δ1198661119888 (minusinfin +infin) (minus1514 +infin) infin
Δ1198662119888 (minusinfin +infin) (minus1512 +infin) infin
Δ1198663119888 (minus1567 +infin) (minus1567 minus1224) 343
Δ11986612119903 (minusinfin minus155) (minus603 minus155) 448
Δ11986623119903 (minus771infin) (minus771 +infin) infin
Δ11986611987712 (minus1367infin) (minus1367 minus1086) 281
Δ1198661198773 (minusinfin minus1050) (minus1266 minus1050) 216
When the stochastic noise is strong enough the noisemayinduce the system to lytic stateMoreover if free energy variesdue to gene mutation bistable behavior might be destroyedand only one stable equilibrium is left This is what themathematical model implies about the parameter influenceon the system dynamics
42 Δ1198661119888 Δ1198662119888 and Δ119866
3119888 The bifurcation results of Δ119866
1119888
Δ1198662119888 andΔ119866
3119888are shown in Figures 4(d)ndash4(f)The equilibria
behaviors with respect toΔ1198661119888andΔ119866
2119888variation are similar
to Δ1198663119903 Δ1198661119888represents the likelihood that Cro dimer binds
to site 1198741198771 If Cro dimer binds to site 119874
1198771 the expression
of cro is prohibited resulting in only one single lysogenicstable point When Δ119866
1119888or Δ119866
2119888is small the prohibition
effect is weak but this is not enough to drive the systemdynamics to only one single lysis stable point Hence bistablelayout always exists if these two parameters are small Theinfluence ofΔ119866
3119888is similar to the influence ofΔ119866
1119903andΔ119866
2119903
The bistable range of Δ1198663119888
is about 343 kcalmol which iscomparable to those of Δ119866
1119903and Δ119866
2119903
The six parameters we have shown so far are the basicbinding energies The values of Δ119866
3119903 Δ1198661119888 and Δ119866
2119888decide
the likelihood that negative feedback (ie self-prohibition)happens It is known that the sites 119874
1198771 and 119874
1198772 are used
to transcript and then translate protein Cro The site 1198741198773
contributes to the production of protein CI If protein CIbinds to 119874
1198773 or Cro binds to 119874
1198771 and 119874
1198772 they can stop
their own transcription Thus we call the phenomenon neg-ative feedback Figure 4 shows that such negative feedbackcannot take the system to Z-shaped bifurcationThe negativefeedback occurs when the amount of Cro or CI is too largeFor instance the binding between CI dimer and 119874
1198773 readily
occurs when the amount of protein CI is too large On theother hand Δ119866
1119903 Δ1198662119903 and Δ119866
3119888represent the likelihood
of positive feedback (ie mutual prohibition) For instancewith largeΔ119866
3119888 the binding between Cro and119874
1198773 is favored
stopping the expression of cI gene If the value of Δ1198663119903
issmall and the negative feedback is weakened but it is notenough to make the system dynamics change from singlelysis equilibrium to single lysogenic equilibrium Parametersassociated with positive feedback can show Z-shaped bifur-cation whereas parameters associatedwith negative feedback
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG12r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG12r
(a) Z-shaped bifurcation for Δ11986612119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG23r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG23r
(b) Bifurcation for Δ11986623119903
Figure 5 Z-shaped bifurcation generated by variation of 11986612119903
and11986623119903The variation ofΔ119866
12119903also leads to Z-shaped bifurcation while
Δ11986623119903
is not sensitive enough to create Z-shaped bifurcation
cannotHence the system ismore robust to negative feedbackparameters than to positive feedback parameters
43 Δ11986612119903
and Δ11986623119903
The parameters Δ11986612119903
and Δ11986623119903
determine the energy difference caused by the cooperationof adjacent binding CI dimers Although [32] discovered thatadjacent Cro dimers also have cooperations we ignore thiscooperation effect since it has a small cooperation energyand focus on CI cooperation in our analysis Figure 5 showsthe bifurcation results for the parameters Δ119866
12119903and Δ119866
23119903
The nominal values of Δ11986612119903
and Δ11986623119903
are close to zeroSince protein cooperation process is an exothermic reactionΔ11986612119903
and Δ11986623119903
must be less than zero We find out thatthe variation of Δ119866
12119903leads to Z-shaped bifurcation whereas
Δ11986623119903
does not The value of Δ11986623119903
affects the expression ofboth cro and cI genes whereas Δ119866
12119903only promotes mutual
prohibition from CI to Cro Δ11986623119903
with large value prohibitsthe binding of RNAP to both sites119874
1198772 and119874
1198773 thus both 119888119868
and 119888119903119900 gene expressions are closed
8 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR12
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR12
(a) Z-shaped bifurcation for Δ11986611987712
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR3
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR3
(b) Z-shaped bifurcation for Δ1198661198773
Figure 6 Z-shaped bifurcation generated by variation of 11986611987712
and1198661198773 Both of them are able to generate Z-shaped bifurcation Their
bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope
Moreover we know the length of bifurcation range forΔ11986612119903
is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866
12119903
is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866
12119903is equal to the center value at minus379 kcalmol
the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866
12119903from this point of view
44 Δ11986611987712
and Δ1198661198773 Figure 6 illustrates the influence of
Δ1198661198773
and Δ11986611987712
variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the
appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866
11987712is 281 kcalmol and forΔ119866
1198773is only 216 kcalmol
which is the minimum among all ten parametersNotice that Δ119866
1198773and Δ119866
11987712are the binding free energies
of RNAP to 119875119877119872
promoter (1198741198773) and to 119875
119877promoter (119874
1198771
and 1198741198772) Because RNAP is necessary for gene expression
the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866
11987712and Δ119866
1198773
than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866
2119903are around 4 kcalmol while those of Δ119866
11987712
and Δ1198661198773
are less than 3 kcalmol Parameter uncertaintiesof Δ119866
11987712and Δ119866
1198773most easily destroy bistable behaviorThe
concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866
11987712or Δ119866
1198773uncertainty
45 Discussion From the above analysis we find that thebinding free energies to sites119874
1198771 and119874
1198772 have similar effects
on system dynamics As we have shown in Table 3 the resultsof Δ119866
1119903 Δ1198662119903 Δ1198661119888 and Δ119866
2119888are similar This is mainly
because of the similar status for binding sites 1198741198771 and 119874
1198772
both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for
Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866
23119903 All of the above four parameters
do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866
1119888andΔ119866
2119888can be used tomaintain
lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866
1119888lower than
minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582
stabilizes at lysogenyAs to the other six parameters with limited bistable ranges
in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866
1119903
Δ1198662119903 andΔ119866
12119903indicate the likelihood of protein CI binding
to sites 1198741198771 and 119874
1198772 or the likelihood of mutual prohi-
bition occurring Similarly Δ1198663119888indicates the likelihood of
protein Cro binding to site 1198741198773 and also results in mutual
prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior
Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866
11987712and Δ119866
1198773 Since
Δ11986611987712
and Δ1198661198773
have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins
Computational Biology Journal 9
In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866
11987712and Δ119866
1198773 because otherwise the system dynamics
may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily
5 Conclusion
In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866
11987712and Δ119866
1198773 Moreover parameters
describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866
1119903 Δ1198662119903 Δ1198663119888 and Δ119866
12119903) whereas parameters
describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866
2119888) can-
not Hence system dynamics is more sensitive to mutualprohibition parameters
There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors
Appendix
Calculation of Jacobian
From (1)ndash(5) we may represent system Jacobian by
119869 =[[[
[
1198601
120597119875119903
120597119873119903
minus 1205831
1198601
120597119875119903
120597119873119888
1198602
120597119875119888
120597119873119903
1198602
120597119875119888
120597119873119888
minus 1205832
]]]
]
(A1)
Since 119875119903and 119875
119888are sums of 119875
119904 119904 = 1 2 40 as (2) shows
the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875
119904120597119873119888 Since 119875
119904is a function with respect to
Cro and CI dimers we evaluate the partial derivatives byChain Rule
120597119875119904
120597119873119903
=120597119875119904
1205971198731199032
1198891198731199032
119889119873119903
120597119875119904
120597119873119888
=120597119875119904
1205971198731198882
1198891198731198882
119889119873119888
(A2)
Table 4 Exponents for 30 binding configurations in Table 2
119904 119894(119904) 119895(119904) 119896(119904)
1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2
where
1198891198731199032
119889119873119903
=1
2(1 minus
119870119903
radic1198702119903+ 8119873119903119870119903
)
1198891198731198882
119889119873119888
=1
2(1 minus
119870119888
radic1198702119888+ 8119873119888119870119888
)
(A3)
10 Computational Biology Journal
Thenwe have to deal with other terms in (A2) Note that119875119904is
a fractional function in (3) so we write its numerator119860119904and
denominator 119861 as
119872119904= exp(minus
Δ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)
1198882119873119896(119904)
RNAP (A4)
119863 =
40
sum
119904=1
119872119904 (A5)
The partial derivatives for 119875119904with respect to119873
1198882and119873
1199032are
given by
120597119875119904
1205971198731199032
=120597119872119904
1205971198731199032
1
119863minus
120597119863
1205971198731199032
119872119904
1198632
120597119875119904
1205971198731198882
=120597119872119904
1205971198731198882
1
119863minus
120597119863
1205971198731198882
119872119904
1198632
(A6)
The partial derivative of119872119904can be obtained as
120597119872119904
1205971198731199032
= 119894 (119904) exp(minusΔ119866119904
119877119879)119873119894(119904)minus1
1199032119873119895(119904)
1198882119873119896(119904)
RNAP
120597119872119904
1205971198731198882
= 119895 (119904) exp (minusΔ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)minus1
1198882119873119896(119904)
RNAP
(A7)
The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872
119904partial
derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM
References
[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982
[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985
[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990
[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004
[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006
[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007
[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009
[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010
[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011
[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002
[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004
[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004
[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007
[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008
[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012
[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004
[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004
[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000
[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996
[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995
Computational Biology Journal 11
[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002
[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999
[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004
[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008
[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009
[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992
[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010
[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982
[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004
[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975
[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009
[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000
[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967
[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000
[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998
[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002
[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984
[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999
[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004
[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
Computational Biology Journal 7
Table 3 Robustness measures of binding energies
Variable Stable Range(kcalmol)
Bistable Range(kcalmol)
Length(kcalmol)
Δ1198661119903 (minusinfin minus1133) (minus1549 minus1133) 416
Δ1198662119903 (minusinfin minus934) (minus1304 minus934) 370
Δ1198663119903 (minus1233 +infin) (minus1233 +infin) infin
Δ1198661119888 (minusinfin +infin) (minus1514 +infin) infin
Δ1198662119888 (minusinfin +infin) (minus1512 +infin) infin
Δ1198663119888 (minus1567 +infin) (minus1567 minus1224) 343
Δ11986612119903 (minusinfin minus155) (minus603 minus155) 448
Δ11986623119903 (minus771infin) (minus771 +infin) infin
Δ11986611987712 (minus1367infin) (minus1367 minus1086) 281
Δ1198661198773 (minusinfin minus1050) (minus1266 minus1050) 216
When the stochastic noise is strong enough the noisemayinduce the system to lytic stateMoreover if free energy variesdue to gene mutation bistable behavior might be destroyedand only one stable equilibrium is left This is what themathematical model implies about the parameter influenceon the system dynamics
42 Δ1198661119888 Δ1198662119888 and Δ119866
3119888 The bifurcation results of Δ119866
1119888
Δ1198662119888 andΔ119866
3119888are shown in Figures 4(d)ndash4(f)The equilibria
behaviors with respect toΔ1198661119888andΔ119866
2119888variation are similar
to Δ1198663119903 Δ1198661119888represents the likelihood that Cro dimer binds
to site 1198741198771 If Cro dimer binds to site 119874
1198771 the expression
of cro is prohibited resulting in only one single lysogenicstable point When Δ119866
1119888or Δ119866
2119888is small the prohibition
effect is weak but this is not enough to drive the systemdynamics to only one single lysis stable point Hence bistablelayout always exists if these two parameters are small Theinfluence ofΔ119866
3119888is similar to the influence ofΔ119866
1119903andΔ119866
2119903
The bistable range of Δ1198663119888
is about 343 kcalmol which iscomparable to those of Δ119866
1119903and Δ119866
2119903
The six parameters we have shown so far are the basicbinding energies The values of Δ119866
3119903 Δ1198661119888 and Δ119866
2119888decide
the likelihood that negative feedback (ie self-prohibition)happens It is known that the sites 119874
1198771 and 119874
1198772 are used
to transcript and then translate protein Cro The site 1198741198773
contributes to the production of protein CI If protein CIbinds to 119874
1198773 or Cro binds to 119874
1198771 and 119874
1198772 they can stop
their own transcription Thus we call the phenomenon neg-ative feedback Figure 4 shows that such negative feedbackcannot take the system to Z-shaped bifurcationThe negativefeedback occurs when the amount of Cro or CI is too largeFor instance the binding between CI dimer and 119874
1198773 readily
occurs when the amount of protein CI is too large On theother hand Δ119866
1119903 Δ1198662119903 and Δ119866
3119888represent the likelihood
of positive feedback (ie mutual prohibition) For instancewith largeΔ119866
3119888 the binding between Cro and119874
1198773 is favored
stopping the expression of cI gene If the value of Δ1198663119903
issmall and the negative feedback is weakened but it is notenough to make the system dynamics change from singlelysis equilibrium to single lysogenic equilibrium Parametersassociated with positive feedback can show Z-shaped bifur-cation whereas parameters associatedwith negative feedback
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG12r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG12r
(a) Z-shaped bifurcation for Δ11986612119903
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔG23r
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔG23r
(b) Bifurcation for Δ11986623119903
Figure 5 Z-shaped bifurcation generated by variation of 11986612119903
and11986623119903The variation ofΔ119866
12119903also leads to Z-shaped bifurcation while
Δ11986623119903
is not sensitive enough to create Z-shaped bifurcation
cannotHence the system ismore robust to negative feedbackparameters than to positive feedback parameters
43 Δ11986612119903
and Δ11986623119903
The parameters Δ11986612119903
and Δ11986623119903
determine the energy difference caused by the cooperationof adjacent binding CI dimers Although [32] discovered thatadjacent Cro dimers also have cooperations we ignore thiscooperation effect since it has a small cooperation energyand focus on CI cooperation in our analysis Figure 5 showsthe bifurcation results for the parameters Δ119866
12119903and Δ119866
23119903
The nominal values of Δ11986612119903
and Δ11986623119903
are close to zeroSince protein cooperation process is an exothermic reactionΔ11986612119903
and Δ11986623119903
must be less than zero We find out thatthe variation of Δ119866
12119903leads to Z-shaped bifurcation whereas
Δ11986623119903
does not The value of Δ11986623119903
affects the expression ofboth cro and cI genes whereas Δ119866
12119903only promotes mutual
prohibition from CI to Cro Δ11986623119903
with large value prohibitsthe binding of RNAP to both sites119874
1198772 and119874
1198773 thus both 119888119868
and 119888119903119900 gene expressions are closed
8 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR12
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR12
(a) Z-shaped bifurcation for Δ11986611987712
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR3
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR3
(b) Z-shaped bifurcation for Δ1198661198773
Figure 6 Z-shaped bifurcation generated by variation of 11986611987712
and1198661198773 Both of them are able to generate Z-shaped bifurcation Their
bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope
Moreover we know the length of bifurcation range forΔ11986612119903
is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866
12119903
is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866
12119903is equal to the center value at minus379 kcalmol
the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866
12119903from this point of view
44 Δ11986611987712
and Δ1198661198773 Figure 6 illustrates the influence of
Δ1198661198773
and Δ11986611987712
variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the
appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866
11987712is 281 kcalmol and forΔ119866
1198773is only 216 kcalmol
which is the minimum among all ten parametersNotice that Δ119866
1198773and Δ119866
11987712are the binding free energies
of RNAP to 119875119877119872
promoter (1198741198773) and to 119875
119877promoter (119874
1198771
and 1198741198772) Because RNAP is necessary for gene expression
the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866
11987712and Δ119866
1198773
than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866
2119903are around 4 kcalmol while those of Δ119866
11987712
and Δ1198661198773
are less than 3 kcalmol Parameter uncertaintiesof Δ119866
11987712and Δ119866
1198773most easily destroy bistable behaviorThe
concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866
11987712or Δ119866
1198773uncertainty
45 Discussion From the above analysis we find that thebinding free energies to sites119874
1198771 and119874
1198772 have similar effects
on system dynamics As we have shown in Table 3 the resultsof Δ119866
1119903 Δ1198662119903 Δ1198661119888 and Δ119866
2119888are similar This is mainly
because of the similar status for binding sites 1198741198771 and 119874
1198772
both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for
Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866
23119903 All of the above four parameters
do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866
1119888andΔ119866
2119888can be used tomaintain
lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866
1119888lower than
minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582
stabilizes at lysogenyAs to the other six parameters with limited bistable ranges
in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866
1119903
Δ1198662119903 andΔ119866
12119903indicate the likelihood of protein CI binding
to sites 1198741198771 and 119874
1198772 or the likelihood of mutual prohi-
bition occurring Similarly Δ1198663119888indicates the likelihood of
protein Cro binding to site 1198741198773 and also results in mutual
prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior
Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866
11987712and Δ119866
1198773 Since
Δ11986611987712
and Δ1198661198773
have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins
Computational Biology Journal 9
In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866
11987712and Δ119866
1198773 because otherwise the system dynamics
may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily
5 Conclusion
In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866
11987712and Δ119866
1198773 Moreover parameters
describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866
1119903 Δ1198662119903 Δ1198663119888 and Δ119866
12119903) whereas parameters
describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866
2119888) can-
not Hence system dynamics is more sensitive to mutualprohibition parameters
There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors
Appendix
Calculation of Jacobian
From (1)ndash(5) we may represent system Jacobian by
119869 =[[[
[
1198601
120597119875119903
120597119873119903
minus 1205831
1198601
120597119875119903
120597119873119888
1198602
120597119875119888
120597119873119903
1198602
120597119875119888
120597119873119888
minus 1205832
]]]
]
(A1)
Since 119875119903and 119875
119888are sums of 119875
119904 119904 = 1 2 40 as (2) shows
the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875
119904120597119873119888 Since 119875
119904is a function with respect to
Cro and CI dimers we evaluate the partial derivatives byChain Rule
120597119875119904
120597119873119903
=120597119875119904
1205971198731199032
1198891198731199032
119889119873119903
120597119875119904
120597119873119888
=120597119875119904
1205971198731198882
1198891198731198882
119889119873119888
(A2)
Table 4 Exponents for 30 binding configurations in Table 2
119904 119894(119904) 119895(119904) 119896(119904)
1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2
where
1198891198731199032
119889119873119903
=1
2(1 minus
119870119903
radic1198702119903+ 8119873119903119870119903
)
1198891198731198882
119889119873119888
=1
2(1 minus
119870119888
radic1198702119888+ 8119873119888119870119888
)
(A3)
10 Computational Biology Journal
Thenwe have to deal with other terms in (A2) Note that119875119904is
a fractional function in (3) so we write its numerator119860119904and
denominator 119861 as
119872119904= exp(minus
Δ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)
1198882119873119896(119904)
RNAP (A4)
119863 =
40
sum
119904=1
119872119904 (A5)
The partial derivatives for 119875119904with respect to119873
1198882and119873
1199032are
given by
120597119875119904
1205971198731199032
=120597119872119904
1205971198731199032
1
119863minus
120597119863
1205971198731199032
119872119904
1198632
120597119875119904
1205971198731198882
=120597119872119904
1205971198731198882
1
119863minus
120597119863
1205971198731198882
119872119904
1198632
(A6)
The partial derivative of119872119904can be obtained as
120597119872119904
1205971198731199032
= 119894 (119904) exp(minusΔ119866119904
119877119879)119873119894(119904)minus1
1199032119873119895(119904)
1198882119873119896(119904)
RNAP
120597119872119904
1205971198731198882
= 119895 (119904) exp (minusΔ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)minus1
1198882119873119896(119904)
RNAP
(A7)
The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872
119904partial
derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM
References
[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982
[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985
[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990
[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004
[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006
[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007
[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009
[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010
[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011
[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002
[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004
[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004
[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007
[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008
[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012
[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004
[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004
[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000
[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996
[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995
Computational Biology Journal 11
[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002
[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999
[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004
[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008
[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009
[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992
[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010
[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982
[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004
[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975
[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009
[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000
[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967
[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000
[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998
[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002
[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984
[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999
[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004
[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
8 Computational Biology Journal
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR12
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR12
(a) Z-shaped bifurcation for Δ11986611987712
10minus6
10minus8
10minus10
10minus12
Nr
Nc
minus20 minus15 minus10 minus5 0
ΔGR3
10minus6
10minus8
10minus10
10minus12
minus20 minus15 minus10 minus5 0
ΔGR3
(b) Z-shaped bifurcation for Δ1198661198773
Figure 6 Z-shaped bifurcation generated by variation of 11986611987712
and1198661198773 Both of them are able to generate Z-shaped bifurcation Their
bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope
Moreover we know the length of bifurcation range forΔ11986612119903
is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866
12119903
is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866
12119903is equal to the center value at minus379 kcalmol
the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866
12119903from this point of view
44 Δ11986611987712
and Δ1198661198773 Figure 6 illustrates the influence of
Δ1198661198773
and Δ11986611987712
variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the
appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866
11987712is 281 kcalmol and forΔ119866
1198773is only 216 kcalmol
which is the minimum among all ten parametersNotice that Δ119866
1198773and Δ119866
11987712are the binding free energies
of RNAP to 119875119877119872
promoter (1198741198773) and to 119875
119877promoter (119874
1198771
and 1198741198772) Because RNAP is necessary for gene expression
the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866
11987712and Δ119866
1198773
than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866
2119903are around 4 kcalmol while those of Δ119866
11987712
and Δ1198661198773
are less than 3 kcalmol Parameter uncertaintiesof Δ119866
11987712and Δ119866
1198773most easily destroy bistable behaviorThe
concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866
11987712or Δ119866
1198773uncertainty
45 Discussion From the above analysis we find that thebinding free energies to sites119874
1198771 and119874
1198772 have similar effects
on system dynamics As we have shown in Table 3 the resultsof Δ119866
1119903 Δ1198662119903 Δ1198661119888 and Δ119866
2119888are similar This is mainly
because of the similar status for binding sites 1198741198771 and 119874
1198772
both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for
Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866
23119903 All of the above four parameters
do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866
1119888andΔ119866
2119888can be used tomaintain
lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866
1119888lower than
minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582
stabilizes at lysogenyAs to the other six parameters with limited bistable ranges
in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866
1119903
Δ1198662119903 andΔ119866
12119903indicate the likelihood of protein CI binding
to sites 1198741198771 and 119874
1198772 or the likelihood of mutual prohi-
bition occurring Similarly Δ1198663119888indicates the likelihood of
protein Cro binding to site 1198741198773 and also results in mutual
prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior
Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866
11987712and Δ119866
1198773 Since
Δ11986611987712
and Δ1198661198773
have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins
Computational Biology Journal 9
In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866
11987712and Δ119866
1198773 because otherwise the system dynamics
may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily
5 Conclusion
In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866
11987712and Δ119866
1198773 Moreover parameters
describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866
1119903 Δ1198662119903 Δ1198663119888 and Δ119866
12119903) whereas parameters
describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866
2119888) can-
not Hence system dynamics is more sensitive to mutualprohibition parameters
There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors
Appendix
Calculation of Jacobian
From (1)ndash(5) we may represent system Jacobian by
119869 =[[[
[
1198601
120597119875119903
120597119873119903
minus 1205831
1198601
120597119875119903
120597119873119888
1198602
120597119875119888
120597119873119903
1198602
120597119875119888
120597119873119888
minus 1205832
]]]
]
(A1)
Since 119875119903and 119875
119888are sums of 119875
119904 119904 = 1 2 40 as (2) shows
the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875
119904120597119873119888 Since 119875
119904is a function with respect to
Cro and CI dimers we evaluate the partial derivatives byChain Rule
120597119875119904
120597119873119903
=120597119875119904
1205971198731199032
1198891198731199032
119889119873119903
120597119875119904
120597119873119888
=120597119875119904
1205971198731198882
1198891198731198882
119889119873119888
(A2)
Table 4 Exponents for 30 binding configurations in Table 2
119904 119894(119904) 119895(119904) 119896(119904)
1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2
where
1198891198731199032
119889119873119903
=1
2(1 minus
119870119903
radic1198702119903+ 8119873119903119870119903
)
1198891198731198882
119889119873119888
=1
2(1 minus
119870119888
radic1198702119888+ 8119873119888119870119888
)
(A3)
10 Computational Biology Journal
Thenwe have to deal with other terms in (A2) Note that119875119904is
a fractional function in (3) so we write its numerator119860119904and
denominator 119861 as
119872119904= exp(minus
Δ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)
1198882119873119896(119904)
RNAP (A4)
119863 =
40
sum
119904=1
119872119904 (A5)
The partial derivatives for 119875119904with respect to119873
1198882and119873
1199032are
given by
120597119875119904
1205971198731199032
=120597119872119904
1205971198731199032
1
119863minus
120597119863
1205971198731199032
119872119904
1198632
120597119875119904
1205971198731198882
=120597119872119904
1205971198731198882
1
119863minus
120597119863
1205971198731198882
119872119904
1198632
(A6)
The partial derivative of119872119904can be obtained as
120597119872119904
1205971198731199032
= 119894 (119904) exp(minusΔ119866119904
119877119879)119873119894(119904)minus1
1199032119873119895(119904)
1198882119873119896(119904)
RNAP
120597119872119904
1205971198731198882
= 119895 (119904) exp (minusΔ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)minus1
1198882119873119896(119904)
RNAP
(A7)
The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872
119904partial
derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM
References
[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982
[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985
[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990
[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004
[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006
[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007
[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009
[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010
[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011
[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002
[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004
[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004
[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007
[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008
[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012
[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004
[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004
[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000
[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996
[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995
Computational Biology Journal 11
[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002
[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999
[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004
[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008
[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009
[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992
[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010
[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982
[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004
[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975
[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009
[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000
[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967
[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000
[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998
[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002
[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984
[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999
[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004
[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
Computational Biology Journal 9
In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866
11987712and Δ119866
1198773 because otherwise the system dynamics
may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily
5 Conclusion
In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866
11987712and Δ119866
1198773 Moreover parameters
describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866
1119903 Δ1198662119903 Δ1198663119888 and Δ119866
12119903) whereas parameters
describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866
2119888) can-
not Hence system dynamics is more sensitive to mutualprohibition parameters
There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors
Appendix
Calculation of Jacobian
From (1)ndash(5) we may represent system Jacobian by
119869 =[[[
[
1198601
120597119875119903
120597119873119903
minus 1205831
1198601
120597119875119903
120597119873119888
1198602
120597119875119888
120597119873119903
1198602
120597119875119888
120597119873119888
minus 1205832
]]]
]
(A1)
Since 119875119903and 119875
119888are sums of 119875
119904 119904 = 1 2 40 as (2) shows
the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875
119904120597119873119888 Since 119875
119904is a function with respect to
Cro and CI dimers we evaluate the partial derivatives byChain Rule
120597119875119904
120597119873119903
=120597119875119904
1205971198731199032
1198891198731199032
119889119873119903
120597119875119904
120597119873119888
=120597119875119904
1205971198731198882
1198891198731198882
119889119873119888
(A2)
Table 4 Exponents for 30 binding configurations in Table 2
119904 119894(119904) 119895(119904) 119896(119904)
1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2
where
1198891198731199032
119889119873119903
=1
2(1 minus
119870119903
radic1198702119903+ 8119873119903119870119903
)
1198891198731198882
119889119873119888
=1
2(1 minus
119870119888
radic1198702119888+ 8119873119888119870119888
)
(A3)
10 Computational Biology Journal
Thenwe have to deal with other terms in (A2) Note that119875119904is
a fractional function in (3) so we write its numerator119860119904and
denominator 119861 as
119872119904= exp(minus
Δ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)
1198882119873119896(119904)
RNAP (A4)
119863 =
40
sum
119904=1
119872119904 (A5)
The partial derivatives for 119875119904with respect to119873
1198882and119873
1199032are
given by
120597119875119904
1205971198731199032
=120597119872119904
1205971198731199032
1
119863minus
120597119863
1205971198731199032
119872119904
1198632
120597119875119904
1205971198731198882
=120597119872119904
1205971198731198882
1
119863minus
120597119863
1205971198731198882
119872119904
1198632
(A6)
The partial derivative of119872119904can be obtained as
120597119872119904
1205971198731199032
= 119894 (119904) exp(minusΔ119866119904
119877119879)119873119894(119904)minus1
1199032119873119895(119904)
1198882119873119896(119904)
RNAP
120597119872119904
1205971198731198882
= 119895 (119904) exp (minusΔ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)minus1
1198882119873119896(119904)
RNAP
(A7)
The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872
119904partial
derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM
References
[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982
[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985
[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990
[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004
[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006
[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007
[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009
[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010
[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011
[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002
[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004
[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004
[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007
[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008
[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012
[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004
[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004
[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000
[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996
[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995
Computational Biology Journal 11
[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002
[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999
[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004
[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008
[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009
[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992
[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010
[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982
[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004
[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975
[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009
[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000
[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967
[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000
[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998
[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002
[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984
[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999
[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004
[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
10 Computational Biology Journal
Thenwe have to deal with other terms in (A2) Note that119875119904is
a fractional function in (3) so we write its numerator119860119904and
denominator 119861 as
119872119904= exp(minus
Δ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)
1198882119873119896(119904)
RNAP (A4)
119863 =
40
sum
119904=1
119872119904 (A5)
The partial derivatives for 119875119904with respect to119873
1198882and119873
1199032are
given by
120597119875119904
1205971198731199032
=120597119872119904
1205971198731199032
1
119863minus
120597119863
1205971198731199032
119872119904
1198632
120597119875119904
1205971198731198882
=120597119872119904
1205971198731198882
1
119863minus
120597119863
1205971198731198882
119872119904
1198632
(A6)
The partial derivative of119872119904can be obtained as
120597119872119904
1205971198731199032
= 119894 (119904) exp(minusΔ119866119904
119877119879)119873119894(119904)minus1
1199032119873119895(119904)
1198882119873119896(119904)
RNAP
120597119872119904
1205971198731198882
= 119895 (119904) exp (minusΔ119866119904
119877119879)119873119894(119904)
1199032119873119895(119904)minus1
1198882119873119896(119904)
RNAP
(A7)
The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872
119904partial
derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM
References
[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982
[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985
[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990
[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004
[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006
[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007
[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009
[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010
[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011
[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002
[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004
[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004
[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007
[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008
[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012
[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004
[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004
[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000
[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996
[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995
Computational Biology Journal 11
[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002
[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999
[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004
[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008
[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009
[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992
[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010
[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982
[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004
[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975
[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009
[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000
[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967
[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000
[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998
[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002
[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984
[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999
[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004
[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
Computational Biology Journal 11
[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002
[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999
[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004
[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008
[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009
[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992
[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010
[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982
[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004
[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975
[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009
[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000
[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967
[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000
[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998
[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002
[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984
[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999
[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004
[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Anatomy Research International
PeptidesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
International Journal of
Volume 2014
Zoology
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Molecular Biology International
GenomicsInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioinformaticsAdvances in
Marine BiologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Signal TransductionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
Evolutionary BiologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Biochemistry Research International
ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Genetics Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Virolog y
Hindawi Publishing Corporationhttpwwwhindawicom
Nucleic AcidsJournal of
Volume 2014
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Enzyme Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Microbiology