STOCHASTIC CHEMICAL KINETICS: Theory and …geza.kzoo.edu/~erdi/ma/tulec.pdfSTOCHASTIC CHEMICAL KINETICS: Theory and Systems Biological Applications I. Stochastic kinetics: why and

STOCHASTIC CHEMICAL KINETICS: Theory and SystemsBiological Applications

Péter É[email protected]

Henry R. Luce ProfessorCenter for Complex Systems Studies

Kalamazoo Collegehttp://people.kzoo.edu/ perdi/

andInstitute for Particle and Nuclear Physics, Wigner Research Centre,

Hungarian Academy of Sciences, Budapesthttp://cneuro.rmki.kfki.hu/

– Typeset by FoilTEX – 1

STOCHASTIC CHEMICAL KINETICS: Theory andSystems Biological Applications

1973 1989 2014


STOCHASTIC CHEMICAL KINETICS: Theory andSystems Biological Applications

I. Stochastic kinetics: whyand how?• Chemical kinetics: some basic concepts• Fluctuations phenomena: introductory and histor-

ical remarks

III. Systems Biological Appli-cations• Fluctuations near instabilities• Enzyme kinetics• Signal processing in biochemical networks• The beneficial role of noise: fluctuation-

dissipation theorem and stochastic resonance• Stochastic models of gene expression• Chiral symmetry

II. Continuous time dis-crete state stochasticmodels• Stochastic processes: some basic concepts• Model frameworks, master equation• Solutions of the master equation• Stationary distributions• Simulation methods• Non-Markovian approaches


Stochastic kinetics: why and how?

Chemical kinetics: some basic concepts

CHEMICAL REACTION

0 =n

∑i=1

ν j,iAi ( j = 1,2, . . . ,m) (1)

m: the number of different reactions; ν j,i: stoi-chiometric coefficient of component Ai in reac-tion step j, is positive for species that are pro-duced, negative for species that are consumedin the reaction step j.

KINETIC EQUATIONdc(t)

dt= f(c(t);k); c(0) = c0 (2)

where f is the function which governs the tem-poral evolution of the system, k is the vector ofthe parameters (rate constants or rate coef-ficients) and c0 (with elements [A1]0, [A2]0,. . . ,[An]0) is the initial value vector of the componentconcentrations.Continuous time Continuous state Deterministic(CCD model

MASS ACTION KINETICS

fi(c) =m

∑j=1

ν j,iv j (i = 1,2, . . . ,n) (3)

constituent functions of f:the sum of the rates ofthe individual stepsPower law kinetics:

v j = k j

n

∏k=1

[Ak]α j,k (4)

fi(c) =m

∑j=1

ν j,ik j

n

∏k=1

[Ak]α j,k (i = 1,2, . . . ,n) (5)

α j,k: order of reaction



Fluctuations phenomena: introductory and historicalremarks

Einstein-Smoluchowski-Langevintheory of the Brownian motion I.• gave a relationship for the time-dependence of the av-

erage of the square of the displacement X(t) of theBrownian particle,

< X(t)2 >:=< [x(t)− x(0)]2 >= Dt, (6)

where x(t) is the actual and x(o) is the initial coordinateof the Brownian particle;

• found the connection between the mobility of the parti-cle and the - macroscopic - diffusion constant

D = µkBT. (7)

• considered the motion of the Brownian particles asmemoryless and non-differentiable trajectories, andhelped prepared the pathway to the formulation of thetheory of stochastic (actually Markov) processes.

• offered a new method to determine the Avogadro con-stant

Figure 1: there is a large discrepancy between

the classical displacement and the Brownian motion

displacement

FDT: the fluctuation of the particle at rest has thesame origin as the dissipative frictional force

Figure 2: non-differentiable trajectories



Fluctuations phenomena: introductory and historicalremarks

Einstein-Smoluchowski-Langevin theory of theBrownian motion II.

Langevin equation: forcing function has a systematicand deterministic part, and a term due to the rapidlyvarying, highly irregular random effects:

dx/dt = a(x, t)+b(x, t)ξ (t) (8)

The equation (8) is not precise, since ξ (t) is of-ten non-differentiable, and therefore x(t) is also non-differentiable. (further studies: stochastic integrals:"Itoversus Stratonovich")Retrospectively, Einstein’s theory: a(x, t) = 0 andb(x, t) =

√2Dξ t and assuming Gaussian white noise

dv/dt =−γv+√

2DdW (t). (9)

White noise is considered as a stationary Gaussianprocess with E[ξt ] = 0 and E[ξtξt ′] = δtt ′|t− t ′|, where δ

is the Dirac delta function.

Figure 3: white noise and its spec-trum



Fluctuations phenomena: introductory and historicalremarks: Diffusion processes

Fokker–Planck equation

A = a∂

∂x+

12

b∂ 2

∂x2 (10)

• a(x, t): the velocity of the condi-tional expectation (called “drift”)

• b(x, t): the velocity of the condi-tional variance (called a “diffusionconstant”).

• The general form of the FP equa-tion:

dPdt

= AP (11)

b = 0: Liouville process

p(x, t|x0, t0) axis

δ(x− x0)

x axis

t = t0

t axis

x0

x(t) solution ofdxdt

= A(x, t)

with x(t0) = x0

Figure 4: Temporal evolution of a Liouville pro-

cess: drift of the PDF in time without changing its

shape. Starting from a degenerate density function

(i.e. when the initial PDF is concentrated to a point),

the point will be drifted.


a(x, t) = a, b(x, t) = b: Wiener process,X(t)≡W (t).a = 0 b = 1:b = 0: Special Wiener process

p(x, t|x0, t0) axis δ(x− x0)

x axis

[D(t− t0)]1/2

x(t) = x0 +A(t− t0)

t axis

t = t0

Fig. 6.3. Temporal evolution of special Wiener process. From [204].

Figure 5: Temporal evolution of a special

Wiener process: s a Gaussian process, but is not

stationary. It is used in modeling the Brownian pro-

cess.For finite t > t0 the PDF is Gaussian, with mean

x0 + a(t − t0), and the standard deviation |b(t − t0)|1/2

is flattening out as time tends to infinite. For a spe-

cial Wiener process the center of the spreading curve

remains unchanged.

Ornstein-Uhlenbeck process (OU):

a(x, t) =−kx,b(x, t) = b, (k > 0,D≥ 0). (12)

p(x, t|x0, t0) axis

δ(x− x0)

x axis

t = t0

t axis

[(D/2k)(1− exp[−2k(t− t0)])]1/2

x(t) = x0 exp[−k(t− t0)]

Figure 6: OU has a stationary PDF, and it is

the normal density function. For any t > 0 the PDF

is the normal density function with an exponentially i

moving mean. The temporal evolution of the standard

deviation describes spreading out, but the width of the

curve is finite even for infinite time. The OU process,

defined by a linearly state-dependent drift and a con-

stant diffusion term, proved to be a very good model

of the velocity of a Brownian particle characterized by

normal distribution.



Fluctuation-dissipation theorem• Noise in electric circuits being in thermal equi-

librium by Nyquist (theory) and Johnson (exper-iments)

• Internal voltage fluctuation: proportional to the re-sistance R, temperature T and the b bandwidth ofthe measurement:

• Same forces that cause the fluctuations also re-sult in their dissipation

• Chemical kinetics: estimation of individual rateconstants from concentration fluctuations

• Chemical fluctuations measurements: electricconductance; fluorescence correlation spec-troscopy

• Fluctuation or noise phenomena: representationin the time domain and the frequency domain

<V 2 >= 4RKBT (13)

C(t) := E[ξ (t)ξ (t− τ)]

(14)

S(ω) =1

2π

∫ +∞

−∞

C(t)cosωdτ =1

2π< ξ

2 >eq γ(ω)

γ(ω)+ω2

(15)

γ(ω) is the Fourier transform of the dissipation con-stant, and < ξ 2 >eq characterized the measure of theequilibrium fluctuations. The relationship is famouslycalled the Wiener–Khinchine theorem.



Model framework: preliminary remarks

Pn(t) := P(ξ (t) = n (16)

dPn

dt= APn(t). (17)

dPn

dt=∑n′[ann′Pn(t)−an′nPn(t)].

(18)

• chemical reaction is considered asa Markovian jump process

• Kolmogorov equation (masterequation)

• ann′: infinitesimal transition prob-ability, which gives the probability(per unit time) of the jump from n′

to n

• master equation: a gain-lossequation for the probability of eachstate n:



A model of genetic expression: first encounter

inactive geneλ+1−−

λ−1

active gene λ2−→mRNA λ3−→ protein (19)

supplemented with two degradation steps (the second also called as proteolysis):

mRNAγm−→ 0 protein

γp−→ 0 (20)

dP(n1,n2,n3)(t)dt = λ

+1 (nmax

1 −n1 +1)P(n1−1,n2,n3)−λ+1 (nmax

1 −n1)P(n1,n2,n3)+λ−1 (n1 +1)P(n1 +1,n2,n3)−λ

−1 n1P(n1,n2,n3)

+λ2n1P(n1,n2−1,n3)−λ2n1P(n1,n2,n3)

+ (n2+1)τ2

P(n1,n2 +1,n3)− n2τ2

P(n1,n2,n3)

+λ3n2P(n1,n2,n3−1)−λ3n2P(n1,n2,n3)

+ (n3+1)τ3

P(n1,n2,n3 +1)− n3τ3

P(n1,n2,n3),

(21)where nmax

1 denotes a constant number of switching genes


Continuous time discrete state stochastic models

Model frameworks, master equation

CDS model of chemical kinetics

• (a) takes into consideration– the discrete character of the quan-

tity of components– the inherently random character of

the phenomena• (b) is in accordance (more or less)

– with the theories of thermodynam-ics

– with the theory of stochastic pro-cesses

• (c) is appropriate to describe– ’small systems’– instability phenomena

Figure 7: Renyi, A. ( 1953). Treating

chemical reactions using the theory of stochastic

processes. MT A Alk. Mat. Int. Kozl., 2, 83-101

(in Hungarian). The first complete treatment of

a second-order reaction was given. This pa-

per give evidence that the differential equation

for the expectation of the stochastic model can-

not be identified with the differential equation as-

sociated with the usual deterministic model.



Model frameworks, master equation

Mathematical backgroundX(t)t∈(R) is a continuous time Markov pro-cess if the following equation holds for everyt1 < t2 < · · ·< tn+1 (n is a positive integer):

P(X(tn+1) = j|X(t1) = i1,X(t2) = i2, . . .X(tn) = in) =

= P(X(tn+1) = j|X(tn) = in)(22)

The transition probability for a Markov process isdefined as:

pi j(s, t) = P(X(t) = j|X(s) = i) . (23)

Obviously, the following equation holds for thetransition probabilities for all possible i values:

∑k∈S

pik(s, t) = 1 (24)

Chapman–Kolmogorov equation:

pi j(s, t) = ∑k

pik(s,u)pk j(u, t) (i, j = 0,1,2, . . .)

(25)The transition probability matrix P(s, t) can beconstructed from the individual transition proba-bilities:

P(s, t) =

p1,1(s, t) p1,2(s, t) · · ·p2,1(s, t) p2,2(s, t) · · ·

...... . . .

(26)

The absolute state probabilities Pi(t) :=P(X(t) = i) of a CDS Markov process, i. e. theprobabilities for the system to be in the state i,satisfy a relatively simple recursive equation withthe transition probabilities:

Pi(t) = ∑j

p ji(s, t)Pj(s) (27)

Pi(t) functions are very often the most pre-ferred characteristics of CDS Markov processesin physical and chemical applications.



Master equation

dPi(t)dt

= ∑j(q jiPj(t)−qi jPi(t)),

(28)The qi j values: infinitesimaltransition probabilities (tran-sition rates): can be obtainedfrom the transition probabilities:

qii = 0

qi j = lims→tpi j(s,t)

t−s (i 6= j)(29)

Example: a birth-and-death process

Ø k1−→ A1

A1k2−→ 2A1

A1k3−→Ø

(30)

pi,i+1(t, t +h) = λih+o(h)

pi,i−1(t, t +h) = µih+o(h)

pi,i(t, t +h,) = 1− (λi +µi)h+o(h)

pi, j(t, t +h) = o(h) if j 6= i and j 6= i±1 h→ 0.(31)

dP0(t)dt =−λ0P0(t)+µ1P1(t)

dPn(t)dt =−(λn +µn)Pn(t)+λn−1Pn−1(t)+µn+1Pn+1(t) (n1)

(32)



Chemical Master equation: general case

transition rate of reaction j starting from state (a1,a2, ...,an):combinatorial kinetics; ”kn(n−1)”

The full master equation of the process: summing all transition rates rele-vant to a given state:

dPf (a1,a2,...,an)dt =−∑

mj=1 v j(a1,a2, ...,an)Pf (a1,a2,...,an)

+∑mj=1 v j(a1−ν j,1,a2−ν j,2, ...,an−ν j,n)Pf (a1−ν j,1,a2−ν j,2,...,an−ν j,n)

(33)

dP(t)dt

= ΩP(t) (34)



Solutions of the master equation

• Direct matrix operations

• Laplace transformation

• Generating functions

• Other techniques (Q-functions,Poisson representation)

• Stationary distributions

• Simulation method



Solutions of the master equation

Generating function:

G(z1,z2, . . . ,zn, t) = ∑all states za1za2 · · ·zabPf (a1,a2,...,an)(t)

zi ∈ C i = 1,2, . . . ,n(35)

WHY?

all the individual variables of interest in the stochas-tic description can be obtained from it in a relativelystraightforward manner

The expectation of the number of Ai molecules can begenerated using first partial derivatives:

〈ai〉(t) =∂G(1,1, . . . ,1, t)

∂ zi(36)

Second order moments and correlations can be givenwith a using second and mixed partial derivatives asfollows:

⟨a2

i⟩(t) =

∂ 2G(1,1, . . . ,1, t)∂ z2

i+

∂G(1,1, . . . ,1, t)∂ zi

(37)

⟨aia j

⟩(t) =

∂ 2G(1,1, . . . ,1, t)∂ zi∂ z j

i 6= j (38)

transient solutions for compartmental systems fully knownMaster equation (differential-difference equation ->Partial DE for the generating function(can be solved if linear)



Stationary distributions

• A stationary distribution is a vector, usually denoted as π, and satisfies the following equa-tion, which is derived from equation (34) by setting the left side 0:

0 = Ωπ (39)

• In typical cases, the stationary distribution π can also be thought of as the limit of the vectorof probability functions with time approaching infinity:

π f (a1,a2,...,an) = limt→∞

Pf (a1,a2,...,an)(t) (40)

• unimodality versus multimodality

• stochastic theory of bistable reactions


Stationary distributions

Schlögl reaction of the first-order phase transition

A+2Xk1−−k2

3X (41)

Bk3−−k4

X,. (42)

deterministic model is

dx(t)/dt = k1ax2− k2x3− k4x+ k3b; x(0) = x0. (43)

three stationary states: bistability

master equation is given as

dPn(t)dt

= λn−1Pn−1 +µn+1Pn+1− (λn +µn)Pn, (44)

for n = 1 . . .∞, and

dP0

dt= µ1P1−λ0P0. (45)

Here λn = k3nB + k1nAn(n− 1) and µn = k4n+ k2n(n−1(n− 2), λn and µn are the birth and death rates, re-spectively.

volume V is explicitly taken into account: ki =ki

V m−1 .

λn =ak1n(n−1)

V NA+bk3V NA, (46)

andµn = nk4 +

k2n(n−1)(n−2)(NAV )2 (47)

The stationary distribution calculated by using the de-tailed balance assumption λn−1Pss

n−1 = µnPssn as

Pssn = Pss

0

n−1

∏i=0

λi

µi+1,Pss

0 = 1−∞

∑j=1

Pssj . (48)

Figure 8: The volume-dependence of the

modality


Continuous time discrete state stochastic models:Simulation methods

The Doob-Hanusse-Hárs-Tóth-Érdi-Gillespiealgorithm

• when will the next reaction occur?

• what kind of reaction will it be?

• Doob theorem

• P(τ,µ): reaction probabilitydensity function

• variations, accelerated, hybrid etcmethods.

P(τ,µ)dτ) means the probabilityat time t that the next reaction inV will occur in the differential timeinterval (t+τ, t+τ+dτ), and it willbe an Rµ reaction

• simulation softwares: Cain

• algorithms

• visualization methods

Figure 9: Realization

Figure 10: Histogram


CCD versus CDS models

A more satisfactory way of dealing with this problem could be calledstochastic mapping (G. Lente) , which attempts to identify the part of theparameter space of a given kinetic scheme in which only the stochastic ap-proach is viable. A convenient definition of this stochastic region is the setof parameter values for which the stochastic approach shows that the rela-tive standard error of the target variable is larger than a pre-set critical value(often 1% because of the usual precision of analytical methods used forconcentration determination). Although there is no general proof known yet,a small standard deviation of the expectation of a variable calculated basedon the stochastic approach seems to ensure that the stochastic expectationis very close to the deterministic solution.

x


Systems Biological Applications

• Fluctuations near instabilities

• Enzyme kinetics

• Signal processing in biochemical networks

• Calcium signaling

• The beneficial role of noise: fluctuation-dissipation theorem and stochasticresonance

• Stochastic models of gene expression

• (Chiral symmetry)


Fluctuations near instabilities

CCD vs CDS models

A+X λ ′→ 2X (49)

Xµ→ 0. (50)

dx(t)/dt = (λ −µ)x(t); x(0) = x0, (51)

(here λ = λ ′[A].) The solution is

x(t) = x0exp(λ −µ)t. (52)

If λ > µ, x is exponentially increasing function of time.λ = µ:

x(t) = x0. (53)

dPk(t)/dt =−k(λ +µ)Pk(t)+λ (k−1)Pk−1(t)+µ(k+1)Pk+1(t)(54)

Pk(0) = δkx0;k = 1,2, ...N.(55)

E[ξ (t)] = x0exp(λ −µ)t, (56)

D2[ξ (t)] = (λ +µ)t. (57)

For the case of λ = µ

D2[ξ (t)] = 2Dλ t,

PSfrag replacements

E[ξ(t)]±D[ξ(t)] :

E[ξ(t)] :

t

Figure 11: Amplifications of fluctuations might

imply instability. While the expectation is constant,

the variance increases in time.



Enzyme kinetics

E+Sk1−−

k−1ES

k2−→ E+P (58)

dPe,s(t)dt =− [κ1es+(κ−1+κ2)(e0− e)]Pe,s(t)+

κ1(e+1)(s+1)Pe+1,s+1(t)+

+κ−1(e0− e+1)Pe−1,s−1(t)+

κ2(e0− e+1)Pe−1,s(t)(59)

Figure 12: Stochastic map of theMichaelis-Menten mechanism withthe number of product moleculesformed as the target variable.



Enzyme kinetics

Figure 13: An overlooked very im-portant paper

Figure 14: Comparison of CDSand CCD model



Enzyme kinetics

Oxidation of molecular hydrogen by HynSL hydrogenase from Thiocapsaroseopersicina

E2 +E3κb−→ 2E3

E3κc−→ E4

E4 +H2 +2Moκd−→ E2 +2Mr

(60)

dPe2,e3,p(t)dt =− [κbe2e3 +κce3 +κd(n− e2− e3)m(m−1)h]Pe2,e3,p(t)

+κb(e2 +1)(e3−1)Pe2+1,e3−1,p(t)+κc(e3 +1)Pe2+1,e3+1,p(t)

+κd(n− e2− e3 +1)(m+2)(m−1)(h+1)Pe2−1,e3,p−2(t)(61)

three step catalytic cycle

E2, E3 and E4 are different enzyme forms in the cat-alytic cycle, H2 is hydrogen, whereas Mo and Mr arethe oxidized and reduced form of the electron accep-tor compound benzyl viologen. state: E2, E3 and Mrmolecules as e2, e3 and p. The master equation canthen be stated as using m and h for the number of Mrand H2 species, and introducing the constant n as thetotal number of all enzyme forms (n = e2+e3+e4): It ispossible to find rate constants leading to extinction.

Extinction in autocatalytic system , which occurswhen the molecule number of the autocatalyticspecies falls to zero in a system that involves apathway for the decay of the autocatalyst. Thisphenomenon is unknown in deterministic kinet-ics, as an initially nonzero concentration can atno time be exactly zero there.



Signal processing in biochemical networks

Evaluation of signal transfer by mutual informa-tion

• Biochemical networks map time-dependent in-puts to time-dependent outputs.

• The efficiency of information transmission: mu-tual information between the input signal I andoutput signal O

M(I,O) = H(O)−H(O|I) (62)

H(O)≡−∫

p(O)logp(O))dO

is the information-theoretical entropy of the out-put O having P(O) probability distribution, andH(O|I)≡−

∫p(I)dI

∫p(O|I)logp(O|I)dO is the average

(over inputs I) information-theoretical entropy ofO given I, with p(O|I) the conditional probabilitydistribution of O given I

Input species S and output species X( Tostevin andten Wolde (2009):

M(S,X)=∫

DS(t)∫

DX(t)p(S(t),X(t))logp(S(t),X(t))

p(S(t))p(X(t))(63)

.




Analytical results: small Gaussian fluctuations:Mutual information rate: R(s,x) = limT→∞ M(s,x)/T - calculated from the power spectra of the fluctuations:

R(s,x) =− 14π

∞∫−∞

dω ln

1− |Ssx(ω)|2

Sss(ω)Sxx(ω)

(64)

R(s,x): temporal correlations between the input and output signals. Equation 64 is exact for linear systems with Gaussiannoise. Detection of input signals may generate correlations between the signal and the intrinsic noise of the reactions. If there isNO correlation:spectral addition rule (65).

Sxx(ω) = N(ω)+g2(ω)Sss(ω) (65)

Here N(ω): internal fluctuation; Sss(ω): the power spectrum of the input signal, g2(ω) = |Ssx(ω)|2Sss(ω)2 : frequency-dependent gain.

The spectrum of transmitted signal: as Pω = g2(ω)Sss(ω), so equation (64):

R(s,x) =− 14π

∞∫−∞

dω ln

1+P(ω)

N(ω)

(66)




Figure 15: The enzymatic fu-tile cycle reaction mechanism.Adapted from Samoilov 2005.

The effect of external noise (on E+):addition of a correction (diffusion)term to the deterministic result (σ isthe forward enzyme noise strength)

RN(σ)(Xss,E+;E−)=E+−k−E− (X0−Xss)(K++Xss

K+Xss)(K−+X0−Xss)+

σ 2k+K+

(K++Xss)2 = 0

(67)

Figure 16: Transition betweenuni- and multistaionarity. Thebifurcation parameter p is theexponent of E+ in a relationshipconnecting the variance E+ andE+. (Samoilov 2005)



Stochastic bifurcation in a signaling pathway

Figure 17: Stochastic bifurcation plot of the fractional steady-state values ofactivated kinase, as a function of the volume V. The solid curves represent themaxima of the steady-state distribution, and the dashed curve represents theminimum of the distribution. V = 1.67 is the critical value, where the bistabilitydisappears. (Bishop-Qian 2010) .


Calcium signaling

Hierarchy of spatiotemporal events

• single channel opening (blip)• the opening of several closely

packed channels (puff)• cooperation of puffs may set off a

wave traveling through the cell• Waves may occur periodically, so

they can be seen as global oscil-lations

Figure 18: A lumped kinetic scheme of the

channel kinetics. X00: state with no Ca2+ bound; X10:

activated state; X11 and X01: inhibited states. An index

is 1 if an ion is bound and 0 if not. Transition rates are

shown at the edges of the rectangle. Falcke03.

Figure 19: Distributions of clusters and of receptors involved obtained withsimulations of intercluster dynamics with calcium accumulation. Power law dis-tribution can be rather well fitted. Based on Lopez 2012.



The beneficial role of noise: fluctuation-dissipation theoremand stochastic resonance

• Estimation of rate constantsfrom equilibrium fluctuations

• Chemical fluctuations mea-surements: electric conduc-tance; fluorescence correla-tion spectroscopy

• Membrane noise analysis

For the association-dissociation reaction of the beryl-

lium sulfate described by the reaction Xk1−−k−1

Y, the

spectrum of the electric fluctuation (which reflects theconcentration fluctuation) was found to be

Sν =const

1+(2πν(k1 + k2))2 . (68)

Since from deterministic equilibrium the ratiok1

k−1is

given, therefore, the individual rate constants can becalculated.


Fluorescence correlation spectroscopy is able to measure the fluctuationof the concentration of fluorescent particles (molecules). Temporal changesin the fluorescence emission intensity caused by single fluorophores arerecorded. The autocorrelation function C(t) := E[ξ (t)ξ (t− τ)] of the signalξ (t) is calculated, and from their time-dependent decay of the fluorescenceintensity the rate parameters can be calculated. Higher order correlationsCmn(t) := E[ξ (t)mξ (t−τ)n] were used to study the details of molecular aggre-gation. To extract more information from the available data beyond averageand variance at least two efficient methods were suggested. Fluorescence-intensity distribution analysis is able to calculate the expected distributionof the number of photon counts, and photon counting histogram gives anaccount of the spatial brightness function. Forty years after, fluorescencefluctuation spectroscopy still is a developing method.


Parameter estimation for stochastic kinetic models: beyondthe fluctuation-dissipation theorem

• time-dependent data

• maximum likelihood estima-tor for the rate constants

• kinetic parameters of bio-chemical reactions, suchas gene regulatory, signaltransduction and metabolicnetwork, generally cannotmeasured directly,

L(k) =m

∏j=1

n

∏i=1

f (o ji , ti;k), (69)

where the jth experimental replicateso j

1,oj2, . . .o j

n are taken at time points t1,t2, ... tn for j = 1, 2, ..., m (i.e. the ex-periments are done in m replicates).f (o j

i , ti;k is the likelihood function de-termined by the density function his-togram constructed from the realiza-tions of the stochastic process speci-fied by the master equation.The maximization of the likelihoodfunction (actually from numerical rea-sons the minimization of the nega-tive log-likelihood function) gives thebest estimated parameters (i.e. itgives the greatest possible probabilityto the given data set): training data.

k∗=argmink−logL(k)= argmin

k

m

∑j=1

n

∑i−1− logP(o j

i , ti),

(70)where P(o j

i , ti) is the conditional prob-ability density function reconstructedfrom the simulated realizations.


Stochastic resonance

Figure 20: A typical curve of stochastic reso-

nance shows a single maximum of the output perfor-

mance as a function of the intensity of the input noise.

Figure 21: Signal-to-noise ratio as a function

of the width of the double well. Since width can be

identified as the noise parameter, the peak is the fea-

ture of stochastic resonance. From Leonard 1994

less effect of noise in the light intensity was observed, becausethe value of the light flux at the Hopf bifurcation is about 10-fold smaller than that in the case of the low concentration ofBrMA.

From the experimental point of view, a modified Oregonatormodel used in this study is realistic enough to reproduce theexperimentally observed behavior of the photosensitive BZreaction in a batch system.35 The photochemical production ofactivator HBrO236 38 through the process (P2) is crucial to thequantitative agreement between the experiments and the model,in addition to the photochemical production of inhibitorBr 29 34 through the process (P1). The good agreementbetween the experimental and modeling results35 in the batchsystem suggests that the model can readily be applied to theflow system as well. In practice, the bifurcation diagram asshown in Figure 1b can explain the experimental behavior suchas the photoinhibition31,38,47 and the photoinduction36,38,47 ofoscillations in the flow system.

A photosensitive flow Oregonator model (eq 3) can bereduced to a simple form by replacing the three photochemical

processes with only one process that represents zero-orderkinetics for the production of Br proposed by Krug, et al.26The reduction can be made as follows: p2 0, p1 in the dz/dequation is equal to zero, and p1 in the dy/d equation is equalto 1. This reduced form of a modified Oregonator model in abatch system has been found to be very useful, particularly instudies of the effect of light on spatio-temporal behavior in theBZ system.48,49 However, if we use the simple scheme for thephotochemical reaction, the resulting flow-Oregonator modelis considered to be quasi two-parameter system, because boththe light flux term ( ) and the flow term ( fy0) control thesystem in the same way under the conditions where the otherflow terms are negligibly small: f 8.6 10 4 1, f2 10 3 1, and f 4.6 10 6 q 9.5 10 5, withkf 2 10 3. Contrarily, the model presented in this studyinvolves two independent bifurcation parameters that willrepresent the different physical roles in the real photosensitiveBZ reaction in a flow system.

V. ConclusionComputational study of a model of the photosensitive BZ

reaction in a flow system showed that SR occurs in the situationwhere a periodic signal and noise are added to the differentphysical inputs (a flow rate and light flux). The two bifurcationparameters control the investigated system differently, whichresulted in the difference in the noise amplitudes for the SRpeak. The model is based on the experimental behavior of thephotosensitive BZ reaction and is readily applied to the SRexperiments. Two-parameter SR or, in general, multiparameterSRmay widely be seen in biological7 9 and natural10 12 systems,which are always suffered from many kinds of periodic andnoisy fluctuations.

Figure 3. Interspike histogram in the case of the high BrMAconcentration (V 0.05 M). The number of adjacent spikes (NAS) inY, whose time interval was between 270 and 330 s (i.e., (1 0.1)T1and (1 0.1)T2, was plotted as a function of the noise amplitude underthe two conditions: ( ) a periodic signal in the flow rate (kf0 1.05

10 3 s 1, 2 0.05, T2 300 s, 2 0) with noise in the light flux( 0 9 10 7 M s 1, 1 0); ( ) a periodic signal in the light flux( 0 9 10 7 M s 1, 1 0.05, T1 300 s, 1 0) with noise inthe flow rate (kf0 1.05 10 3 s 1, 2 0). The NAS is divided bythe integration time, 6 104 s. Error bars are the standard deviationsof the NAS with a set of six random numbers at each noise amplitude.The solid line is to guide the reader’s eye. The parameters used are thesame as those in Figure 1a.

Figure 4. Interspike histogram in the case of the low BrMAconcentration (V 0.005 M) under the two conditions: ( ) a periodicsignal in the flow rate (kf0 1.7 10 3 s 1, 2 0.05, T2 300 s,2 0) with noise in the light flux ( 0 1.5 10 5 M s 1, 1 0);

( ) a periodic signal in the light flux ( 0 1.5 10 5 M s 1, 10.05, T1 300 s, 1 0) with noise in the flow rate (kf0 1.7 10 3s 1, 2 0). The NAS is divided by 6 104 s. Error bars are thestandard deviations of the NAS with a set of six random numbers ateach noise amplitude. The solid line is to guide the reader’s eye. Theparameters used are the same as those in Figure 1b.

Figure 5. Threshold paradigm of SR. (a) One-parameter system: athreshold is constant in time and the system responds when the sum ofa signal and additive noise crosses the threshold. (b) Two-parametersystem: a periodic modulation in one parameter changes a thresholdof the other parameter, and the system responds when noise in the latterparameter crosses the modulated threshold.

Photosensitive Belousov Zhabotinsky Reaction J. Phys. Chem. A, Vol. 102, No. 24, 1998 4541

Figure 22: (a) One-parameter system: a

threshold is constant in time and the system gener-

ates a response if the sum of the signal and additive

noise exceeds the threshold. (b) Two-parameter sys-

tem: a periodic modulation in one parameter changes

a threshold of the other parameter, and the system

generates response when noise in the latter parame-

ter exceeds the modulated threshold. From Tomo Ya-

maguchi’s lab: 1998.


Stochastic resonance of aperiodic signalsNot only periodic, but aperiodic signals might be the subject of amplification by noise, both in experimental (actually electrochemical)and model studies. Information transfer is quantified by the C0 cross correlation function

C0 = 〈(x1−〈x1〉t(x2−〈x2〉t)〉t〉, (71)

where x1 and x2 represents the time series of the aperiodic input signal, and the noise induced response of the electrochemicalsystem, respectively. 〈〉 denotes the respective time averages.

Figure 23 illustrates the existence of optimal noise level for information transfer.

Figure 23: Cross-correlation as a function of noise level. Parma-show 2005.



Stochastic models of gene expression: second encounter

A three-stage model of gene expression (say: Paulsson 2005):

inactive geneλ+1−−

λ−1

active geneλ2−→mRNA

λ3−→ protein (72)

supplemented with two degradation steps (the second also called as prote-olysis):

mRNAγm−→ 0 protein

γp−→ 0 (73)


TEMPLATE DESIGN © 2008

www.PosterPresentations.com

The circular gene hypothesis: feedback effects and protein

fluctuations

Raoul Wadhwa, Làszlò Zalànyi, Judit Szente, Làszlò Nègyessy, Pèter Èrdi

1. Canonical stochastic models of gene expression 4. Stochastic Simulation 6. Where are we now?

7. A Similar Project

8. References

• PROTEIN FLUCTUATION seems to be sensitive

• mRNA FLUCTUATION is smaller but shows similar tendencies

• MODEL DISCRIMINATION seems to be possible

• Some ANALYTICAL calculations look feasible

• The justification of the circular gene expression hypothesis

needs kinetic studies

2. Circular Gene Expression Hypothesis

3. Direct Kinetic measurements are missing

A possible test:

HOW does the feedback mechanism

influence protein fluctuation?

5. Simulation Results – Canonical Model

From Paulsson (2005)

5. Simulation Results – Feedback Model

Figure 2. The stationary distribution of the number of protein particles at

t=50,000 for n=100,000 simulated stochastic trajectories of the canonical

model with an elevated transcription rate (propensity=0.1155). The resulting

distribution is statistically well-modeled by a Poisson distribution with expected

value λ=3,270.


t=50,000 for n=100,000 simulated stochastic trajectories of the canonical

model with a normal transcription rate (propensity=0.0231). The resulting

distribution is statistically well-modeled by a Poisson distribution with expectedvalue λ=654.


t=50,000 for n=100,000 simulated stochastic trajectories of the feedback model

with a normal transcription rate (propensity=0.0231). The resulting distribution

is statistically well-modeled by an exponential distribution.


t=50,000 for n=100,000 simulated stochastic trajectories of the feedback model

with an elevated transcription rate (propensity=0.1155). The resulting

distribution is statistically well-modeled by an exponential distribution.

The skeleton network model of coordinated mRNA

degradation and synthesis

Table 1. The reaction list for the feedback model, which takes into account the

presence of the xrn1 promoter. The value of the transcript_rate parameter is

0.0231. The table above was taken from the reaction setup using the Cain:

Stochastic Simulations for Chemical Kinetics environment. Simulations were

run using this environment.

Haimovich, G., Medina, D., Causse, S., & Garber, M. Gene Expression Is

Circular: Factors for mRNA Degradation Also Foster mRNA

Synthesis. Cell, 153, 1000-1011.

Paulsson, J. Models of stochastic gene expression. Physics of Life Reviews, 2,

157-175.

Mauch, S., & Stalzer, M. Efficient Formulations for Exact Stochastic Simulation

of Chemical Systems. IEEE/ACM Transactions on Computational

Biology and Bioinformatics, 27-35.

Investigation of transmitter-receptor interactions by

analyzing postsynaptic membrane noise using stochastic

kinetics

Erdi, P., Ropolyi, L.

AbstractThe stoichiometric and kinetic details of transmitter-receptor interaction

(the number of conformations and the rate constants of conformation

changes) in synaptic transmission have been investigated analyzing

postsynaptic membrane noises by the aid of the fluctuation-dissipation

theorem of stochastic chemical kinetics. The main assumptions are the

following: (i) the transmitter-receptor model interactions are modelled by

a closed compartment system (a special complex chemical reaction) of

unknown length, (ii) the quantity of transmitter is maintained at a

constant level, (iii) the conductance is a linear function of the

conformation quantity vector. The main conclusion is the conductance

spectral density function is determined by three qualitatively different

factors, (i) the length of the compartment system, (ii) the precise form of

the conductance-conformation quantity vector, (iii) the matrix of the

reaction rate constants.

a



Chiral symmetry: An important topic: Gábor Lente

• Molecular chirality: lack of certain symme-try elements in the three dimensional struc-tures of molecules

• Homochirality or biological chirality: mir-ror image counterparts have very differentroles: typically, only one of them is abun-dant and it cannot be ex- changed with theother one.

• Racemic mixtures (R and S)

P(r,s) =(

r+ sr

)(0.5+ ε)r(0.5− ε)s (74)

Here, P(r,s) is the probability that r molecules of the Renantiomer and s molecules of the S enantiomer oc-cur in an ensemble of (r + s) molecules. Parameterε is characteristic of the degree of inherent differencebetween the two enantiomers (ε ≤ 0.5), and is con-nected to the energy difference (∆E) between the Rand S molecules as follows:

ε =e∆E/RT −1

2(e∆E/RT +1)(75)

The expectation and standard deviation for the num-ber of R enantiomers from this distribution is given bystraightforward formulae:

〈r〉= (0.5+ ε)(r+ s) (76)

σr =√

(r+ s)(0.5+ ε)(0.5− ε) (77)



Chiral symmetry

Figure 24: The classical Frank model (1953)


!!!

),,,1()1(),1,,1()1(

),,1,1()1(),1,1,()1)(1(

),1,,1()1)(1()1(

),,1,1()1)(1()1(

),,,()(2),,,(

tsraP srantsraP s

tsraP rtsraP sr

tsraP sa a

tsraP ra a

tsraP anrs as ar a dt

tsradP

ff

fd

cu

cu

fdccu

! !! !!

!! !!!!!!!

! ! !!!!

! ! !!!!

! !!!! "

(59)

Figure 25: Master equation


Figure 26: Final probability distributions in the Frank model in a closed system


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