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Tiejun Li
Quasi-potential and Two-Scale Large Deviation Theory for Gillespie Dynamics
School of Mathematical Sciences
Peking University, PRC
UQAW 2016, KAUST, 2016-01-07
Outline
Ø Introduction
Ø Landscape and quasi-potential
Ø Two-scale LDT
Ø Conclusion
Stochastic Dynamics in Systems Biology
Elowitz et al., Science 297 (2002), 391.Intrinsic and Extrinsic noise in E. Coli
The dynamics for cellular systems (such as the gene expression) is random, which can be modeled as Gillespie dynamics if the spatial inhomogeneity is ignored.
S. Xie et al, Nature 475 (2011), 308. Protein synthesis with single molecule tech (Protein bursting)
Gillespie Dynamics: Math Setup
Birth-Death process:
n molecular species, M reactions, Propensity function State change vector Living space
aj(x)⌫j Nn
A Markov jump process on Nn
Example: Central dogma (simplest case)
DNA(D)
k1�! mRNA(m)
k2�! Protein(p)
n = 3,M = 2 ⌫1 =
0
@010
1
A ,⌫2 =
0
@001
1
A
Xt : The number of molecules
D = 1, a1(x) = k1, a2(x) = k2m
Gillespie Dynamics: Equations
�tP (x, t|x0, t0) =MX
j=1
aj(x� �j)P (x� �j , t|x0, t0)�MX
j=1
aj(x)P (x, t|x0, t0).
Chemical Master Equation:
Law of mass action: (Elementary reactions)
A Markov jump process on
aj(x) = cjxk11 · · ·xkn
ncj ⇠ O(V 1�k), k =
nX
i=1
ki
Nn
D.T. Gillespie, Markov processes: an introduction to physical scientists, 1991.S.N. Ethier-T.G. Kurtz, Markov processes: characterization and convergence, 1986.
SDEs by random time change: (Kurtz)
Xt = X0 +MX
j=1
⌫jPj(
Z t
0aj(Xs)ds)
System Size V is Very Important！
Large volume limit (Kurtz limit):
Mathematically, the volume V（or equivalently the number of molecules）plays the most important role in analysis.
dx
dt=
MX
j=1
aj(x)�j
dXt =MX
j=1
aj(Xt)�jdt+MX
j=1
qaj(Xt)�jdW
jt .
Chemical Langevin Approximation:
Kurtz, J. Appl. Prob. 1970,1971;Ann. Appl. Prob. 2006, 2013
Gillespie, J. Chem. Phys. 2000
in concentration variables (small noise):
dxt =MX
j=1
aj(xt)�jdt+1pV
MX
j=1
qaj(xt)�jdW
jt
Numerical Methods
SSA for trajectory simulation VS Tau-leaping acceleration with a deterministic time step
Tau-leaping Algorithm:
Xn+1 = Xn +MX
j=1
�jP(aj(Xn)�tn)
Gillespie, J. Chem. Phys. 2001
Further developed with L. Petzold, Y. Cao et al.
Gillespie Algorithm (SSA, Gillespie, JCP 22(1976), 403): Given .
�Step 1: Generate exponentially distributed waiting time ;�Step 2: Generate the reaction label j, which will fiﾞre;�Step 3: Advance the process:
⌧
Xt
⌧
Xt+� = Xt + �j
Finding suitable stepsize is non-trivial!
Analysis of Tau-leaping
Tau-leaping is essentially an explicit Euler scheme to the jump SDEs describing the stochastic process.
Decomposition:
L., SIAM MMS, 2007; Anderson-Ganguly-Kurtz, Ann. Appl. Prob. 2011 (Large Volume Scaling).
dXt =
MX
j=1
Z A
0�jcj(x;Xt�)�(dt⇥ dx),
where
cj(x;Xt) =
⇢1, if x 2 (
Pj�1i=1 ai(Xt),
Pji=1 ai(Xt)],
0, otherwise.j = 1, 2, . . . ,M.
SDEs via Poisson random measure
dXt =MX
j=1
Z A
0�jcj(x;Xt�)m(dt⇥ dx)
+MX
j=1
Z A
0�jcj(x;Xt�)(��m)(dt⇥ dx)
= P 1 + P 2.
Theorem: Tau-leaping is of strong order ½ and weak order 1 in the scaling .
Explicit to both terms P1 and P2
�t ! 0
Further Improvement of tau-leaping
Hu-L.-Min, J. Chem. Phys, 2009, 2011; CMS, 2011; Burrage et al., BMC Syst. Biol. 2012, 2014.
Attempts for higher order tau-leaping:
Xn+1 = Xn + � · r⇤. r⇤ = r + r r : tau-leaping, : correction rChoosing statistics of to make the scheme second order.
Improvement includes the construction of higher order schemes, avoiding negative populations and more effiﾞcient methods, etc.
r
Avoiding negative populations: reducing the probability to a tolerance. Cao-Gillespie-Petzold, J. Chem. Phys, 2005, 2006; Moraes-Tempone-Vilanova, MMS 2014.
Multilevel Monte Carlo to Chemical Kinetic System. Anderson-Higham, MMS 2012, SINUM 2014; Moraes-Tempone-Vilanova, BIT 2015.Only a partial list!
Outline
Ø Introduction
Ø Landscape and quasi-potential
Ø Two-scale LDT
Ø Conclusion
Energy LandscapeConstructing the Energy Landscape of a biological system is intuitively attractive in systems biology: Understanding the dynamics and stability under noise perturbation.
Waddington potential for the cell differentiation
J. Wang’s Landscape Theory：
�P
�t+� · J(x, t) = 0
SDE:
FPE:
J(x, t) = FP �D ·⇥x
P
U(x) = � lnPss(x)Potential:
Wang et al, PNAS, 2008, 2009, 2011, 2013
F =1
PssD · �
�xPss +
Jss(x)
Pss= �D
�U
�x+
Jss(x)
Pss
Decomposition of F :
Gradient flow flux:
�U�x . “Curl flow” flux: � · Jss(x) = 0
Interpretation:
dX
dt= F (X) + ⌘, h⌘(t)⌘(s)i = 2D�(t� s)
Connection with Rare Events
SDEs with small perturbations:
When the volume size V is large, the system corresponds to stochastic dynamics with small perturbations, which exhibits metastablity.
W. E, W. Ren, E. Vanden-Eijnden, C. Schuette, F. Noe, Q. Du ……dXt = b(Xt)dt+
p✏dWt
Interested issues: Rare events
Transition state, Transition path and Transition rate
dxt =MX
j=1
aj(xt)�jdt+1pV
MX
j=1
qaj(xt)�jdW
jt
Rare Events for Diffusion Process
SDEs with small perturbations:
For diffusion process, the transition path can be quantitated by minimizing the rate functional in large deviation theory.
dXt = b(Xt)dt+p✏dWt
Action functional:
Large deviation theory:
IT ['] =
Z T
0L(', ')dt
L(', ') = |'� b(')|2
Application in polymers (Cahn-Hilliard): �L.-Zhang-Zhang, CMS, 2012; MMS 2013
Prob
sup
t2[0,T ]kXt � '(t)k �
!⇡ exp
✓� 1
2✏IT (')
◆
State Transitions Are Observed in ExperimentsThe state transitions driven by random noise in cellular biology indeed exist and have been observed in single molecule experiments (phenotype switching).
S. Xie et al, Science 322 (2008), 442. Phenotype switching triggered by stochastic events.
Fig. 1. The expression of lactose permease in E.coli. (A) The repressor LacI and permease LacY forma positive feedback loop. Expression of permeaseincreases the intracellular concentration of theinducer TMG, which causes dissociation of LacI fromthe promoter, leading to even more expression ofpermeases. Cells with a sufficient number of per-meases will quickly reach a state of full induction,whereas cells with too few permeases will stayuninduced. (B) After 24 hours in M9 mediumcontaining 30 mM TMG, strain SX700 expressing aLacY-YFP fusion exhibits all-or-none fluorescence ina fluorescence-phase contrast overlay (bottom, im-age dimensions 31 mm × 31 mm). Fluorescence im-aging with high sensitivity reveals single moleculesof permease in the uninduced cells (top, imagedimensions 8 mm × 13 mm). (C) After 1 day of con-tinuous growth in medium containing 0 to 50 mMTMG, the resulting bimodal fluorescence distribu-tions show that a fraction of the population existseither in an uninduced or induced state, with therelative fractions depending on the TMG concentra-tion. (D) The distributions of LacY-YFP molecules inthe uninduced fraction of the bimodal population atdifferent TMG concentrations, measured with single-molecule sensitivity, indicate that one permeasemolecule is not enough to induce the lac operon, aspreviously hypothesized (12). More than 100 cellswere analyzed at each concentration. Error bars areSE determined by bootstrapping.
Fig. 2. Measurement of the threshold of permeasemolecules for induction. (A) Single-cell time tracesof fluorescence intensity, normalized by cell size,starting from different initial permease numbers.The initial LacY-YFP numbers were preparedthrough dilution by cell division of fully inducedcells after removal of the inducer. Upon adding 40mM TMG at time 0, those cells with low initialpermease numbers lost fluorescence with time as aresult of dilution by cell division and photo-bleaching, whereas those cells with high initialpermease numbers exhibited an increase in fluo-rescence as a result of reinduction. Permease mole-cule numbers were estimated from cell fluorescence(28). The dashed red line indicates the determinedthreshold. (B) The probability of induction of a cellwithin 3 hours as a function of the initial permeasenumber was determined using traces from 90 cells.The probability of induction (p) was fit with a Hillequation p = y4.5/(y4.5 + 3754.5) for the initialpermease number y. The threshold of permeasenumbers for induction was thus determined to be 375molecules. Error bars are the inverse square root ofthe sample size at each point. (C) To prove that thecomplete dissociation of tetrameric repressor fromtwo operators triggers induction, we constructedstrain SX702 with auxiliary operators removed (noDNA looping). The figure shows single-cell traces ofpermease numbers in single cells grown in 40 mMTMG as a function of time. Unlike the looping strainSX700, the rapid induction of SX702 is no longerdependent on the initial number of permease molecules. This proves thatphenotype switching is the result of a complete dissociation of the tetramericrepressor, as shown in (B). (D) In the absence of DNA looping, the entire pop-
ulation of strain SX702 rapidly induces in a coordinated manner from far belowthe threshold for a concentration as low as 20 mM TMG. DNA looping is necessaryfor bistability of the lac operon under these conditions.
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Fig. 1. The expression of lactose permease in E.coli. (A) The repressor LacI and permease LacY forma positive feedback loop. Expression of permeaseincreases the intracellular concentration of theinducer TMG, which causes dissociation of LacI fromthe promoter, leading to even more expression ofpermeases. Cells with a sufficient number of per-meases will quickly reach a state of full induction,whereas cells with too few permeases will stayuninduced. (B) After 24 hours in M9 mediumcontaining 30 mM TMG, strain SX700 expressing aLacY-YFP fusion exhibits all-or-none fluorescence ina fluorescence-phase contrast overlay (bottom, im-age dimensions 31 mm × 31 mm). Fluorescence im-aging with high sensitivity reveals single moleculesof permease in the uninduced cells (top, imagedimensions 8 mm × 13 mm). (C) After 1 day of con-tinuous growth in medium containing 0 to 50 mMTMG, the resulting bimodal fluorescence distribu-tions show that a fraction of the population existseither in an uninduced or induced state, with therelative fractions depending on the TMG concentra-tion. (D) The distributions of LacY-YFP molecules inthe uninduced fraction of the bimodal population atdifferent TMG concentrations, measured with single-molecule sensitivity, indicate that one permeasemolecule is not enough to induce the lac operon, aspreviously hypothesized (12). More than 100 cellswere analyzed at each concentration. Error bars areSE determined by bootstrapping.
Fig. 2. Measurement of the threshold of permeasemolecules for induction. (A) Single-cell time tracesof fluorescence intensity, normalized by cell size,starting from different initial permease numbers.The initial LacY-YFP numbers were preparedthrough dilution by cell division of fully inducedcells after removal of the inducer. Upon adding 40mM TMG at time 0, those cells with low initialpermease numbers lost fluorescence with time as aresult of dilution by cell division and photo-bleaching, whereas those cells with high initialpermease numbers exhibited an increase in fluo-rescence as a result of reinduction. Permease mole-cule numbers were estimated from cell fluorescence(28). The dashed red line indicates the determinedthreshold. (B) The probability of induction of a cellwithin 3 hours as a function of the initial permeasenumber was determined using traces from 90 cells.The probability of induction (p) was fit with a Hillequation p = y4.5/(y4.5 + 3754.5) for the initialpermease number y. The threshold of permeasenumbers for induction was thus determined to be 375molecules. Error bars are the inverse square root ofthe sample size at each point. (C) To prove that thecomplete dissociation of tetrameric repressor fromtwo operators triggers induction, we constructedstrain SX702 with auxiliary operators removed (noDNA looping). The figure shows single-cell traces ofpermease numbers in single cells grown in 40 mMTMG as a function of time. Unlike the looping strainSX700, the rapid induction of SX702 is no longerdependent on the initial number of permease molecules. This proves thatphenotype switching is the result of a complete dissociation of the tetramericrepressor, as shown in (B). (D) In the absence of DNA looping, the entire pop-
ulation of strain SX702 rapidly induces in a coordinated manner from far belowthe threshold for a concentration as low as 20 mM TMG. DNA looping is necessaryfor bistability of the lac operon under these conditions.
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Expression of lactose permease (lacY) in E. Coli. Fig. B: Above or below a threshold of permease copy numbers exhibits all-or-none flﾟuorescence.
Classical LDT for Gillespie Dynamics
The LDT has been well established in the classical large volume limit.
Shwartz-Weiss, Large Deviations for Performance Analysis, 1995.
The LDT in the large volume limit: Under mild conditions
lim supV!1
1
VP(XV
· 2 F ) � inf'2F
IT0 [']
For closed set F, upper bound：
For open set G, lower bound：
lim infV!1
1
VP(XV
· 2 G) � � inf'2G
IT0 [']
IT0 ['] =
Z T
0L(', ')dt
Action functional： Dual Hamiltonian：
H(x, p) =
MX
j=1
aj(x)(exp(p · ⌫j)� 1)
Landscape From LDT Perspective
Quasi-potential is a perfect candidate for the energy landscape theory!
Freidlin-Wentzell, Random perturbations of dynamical systems, 1984.
J. Wang’s Landscape Theory：Depends on the system size V;Only contains the equilibrium information.
S(x;x0) = infT>0
inf�,�(0)=x0,�(T )=x
Z T
0L(�, �)dt.
Quasi-potential from large deviation theory：
peq(x) ⇥ exp{�V S(x)},Connections between steady state distribution and quasi-potential：
H(x,rS) = 0Steady Hamilton-Jacobi equation satisfiﾞed by quasi-potential S(x)：
Detailed comparisons: Zhou-L., arXiv: 1511.02088
Quasi-potential: why?
Quasi-potential is a generalization of potential landscape.
S(x;x0) = infT>0
inf�,�(0)=x0,�(T )=x
Z T
0L(�, �)dt.
Zhou-L., arXiv: 1511.02088
In case of with single stable state , we have b(x) = �rV (x)
S(x;x0) = V (x)
x0
dual of the Lagrangian [29]
H( , p) = b( )tp+ ptD( )p. (12)
Assume x0 is a stable fixed point of the deterministic dynamical system dx/dt = b(x),representing a meta-stable biological state. Then the local quasi-potential at state x withrespect to x0 is defined as
�QPloc (x;x0) = inf
T>0inf
(0)=x0, (T )=x
Z T
0LFW ( , )dt. (13)
The heuristic explanation of this definition is that the energy di↵erence between statex and x0 can be evaluated by the least action cost of moving the system from x0 to x,because only the minimum action path dominates in Eq. (9) in the limit "! 0.
To understand the intuition behind the quasi-potential, let us consider a gradientdynamics with a single-well potential V (x), i.e.
b(x) = �rV (x), D(x) = I.
We assume that V (x) � 0 and V (x0) = 0 is the unique minimum of V (x). By definition,we have
ST [ ] =1
4
Z T
0| +rV ( )|2dt. (14)
First we show that ST [ ] � V (x) for all with endpoints (0) = x0 and (T ) = x because
ST [ ] =1
4
Z T
0| +rV ( )|2dt
=1
4
Z T
0| �rV ( )|2dt+
Z T
0 trV ( )dt
�Z T
0 trV ( )dt =
Z T
0dV ( ) = V (x). (15)
On the other hand, we can choose a special and T > 0 such that ˙ = rV ( ) and (T ) = x (this T equals 1 indeed). For this special ,
ST [ ] =
Z T
0
˙ trV ( )dt = V (x). (16)
The above discussion shows that �QPloc (x;x0) = V (x) in this single-well gradient case. The
quasi-potential generalizes the potential concept in general situation.The LDT result (11) also implies that the minimizer of the variational problem (13)
gives the minimum action path or most probable path connecting two metastable statesin zero noise limit. It can be shown [7, 30] that the minimizer (t) satisfies Hamilton’scanonical equation
(t) = rpH( ,rx�QPloc ( ;x0)) = b( ) + 2D( )rx�
QPloc ( ;x0), (0) = x0. (17)
9
In this case:
Quasi-potential: how?
Global quasi-potential is derived by pruning and sticking local ones.
Construction of global quasi-potential from local ones.
S(x;x0) = infT>0
inf�,�(0)=x0,�(T )=x
Z T
0L(�, �)dt.
Zhou-L., arXiv: 1511.02088
x
V(x)
x1 x3 x2
(a)The schematics of double-well potential V (x).
xx1 x3
x2
φQPloc (x;x1)
(b)Local quasi-potential constructed from x1.
xx1 x3 x2
φQPloc (x;x2)
(c)Local quasi-potential constructed from x2.
x
V(x)
x1 x3 x2
φQPglob(x)
(d)The constructed global quasi-potential is a
shift of V (x) in the gradient case.
FIG. 1. (Color online). The original potential field and construction of local and global quasi-
potential for a gradient system. The subfigure (a) shows the original potential field V (x). The
subfigures (b) and (c) show the constructed local quasi-potential starting from metastable states
x1 and x2, respectively. In subfigure (d), the green dashed line is the original potential V (x) and
the gray solid line is the global quasi-potential φQPglob(x).
D. Force Decomposition: HJE and Orthogonality
As in Section (II B), we will investigate the decomposition of the force b(x) in terms of the
global quasi-potential. We will obtain an ε-independent decomposition of b(x), which can
13
Example Beyond Large Volume Scaling
Metastability under noise perturbation
Consider the following typical gene regulation dynamics with positive feedback, which exhibits bistability. N.B.: DNA has only 2 states!
Assaf et al, PRL, 2011
Mathematical ResultsDeveloping two-scale LDT for the CKS to obtain the Hamiltonian (combining F-W and D-V LDT), obtaining Langevin approximation via CLT.
Liptser, Prob. Theor. Rel. Fields, 1996; Lv-Li-Li-L., PLoS ONE 9(2014), e88167; L.-Lin, arXiv:1504.03781
H(x,p) = supq
(⇥q,p⇤ � L(x, q))
= supq
✓⇥q,p⇤ � inf
µ
(L1(x, q,µ) + L2(x,µ))
◆
= supq
supµ
(⇥q,p⇤ � L1(x, q,µ)� L2(x,µ))
= supµ
supq
(⇥q,p⇤ � L1(x, q,µ)� L2(x,µ))
= supµ
(H1(x,p,µ)� L2(x,µ)) .
H1(x,p,µ) = µ(0)Ho�(x,p) + µ(1)Hon(x,p)
L2(x,µ) = � infu
X
��{0,1}
(Lx
u)(�)
u(�)µ(�),
Hamiltonian：
L1(x, q, µ)
LDT Rate functional corresponds to translation：
LDT Rate functional corresponds to switching：
L2(x, µ)
Mathematical ResultsDeveloping two-scale LDT for the CKS to obtain the Hamiltonian (combining F-W and D-V LDT), obtaining Langevin approximation via CLT.
M =af(N)/[f(N) + g(N)]� �M,
N =�bM �N,
Lv-Li-Li-L., PLoS ONE 9(2014), e88167; L.-Lin, arXiv:1504.03781
Mean fiﾞeld ODE: (QSSA)
Langevin Approximation：
a ⇠ b�1, Xt ⇠ M,Yt ⇠ N
dXt =⇣ b�1f
f + g� �Xt
⌘dt+
sb�1f
f + g+
2b�2fg
(f + g)3dW 1
t
�p
�XtdW2t
Residual flﾟuctuation from switching process!
dx
dt=
@H
@p
���p=0
@2H
@p2
���p=0
Fluctuation from：
Compared With Physics Approach
Physics approach: WKB asymptotics + Hamiltonian dynamics, diffiﾞcult to be rationalized. Variational approach is more natural.
Lv-Li-Li-L., PLoS ONE, 2014; L.-Lin, arXiv:1504.03781
Physics approach:
Pm,n = g(n)Qm,n � f(n)Pm,n +APm,n,
Qm,n = �g(n)Qm,n + f(n)Pm,n + [A+ a(E�1m � 1)]Qm,n.
H = A+ g(y)�1[A+ b�1(epx � 1)][f(y)�A],
WKB asymptotics:
Hamiltonian:
min'
Z 1
0⇤0 · ⇥d�
H(⇤, �) = 0,
H✓(⇤, �) = ⇥⇤0
Chemical Master Equation:
Maupertuis principle：
s.t.Pm,n ⌘ P (x, y) ⇠ exp[�V S(x, y)]
Heymann-Vanden-Eijnden, CPAM, 2008.
This Hamiltonian H does NOT have convexity with respect to momentum variable p, which is essential for the robust minimization.
LDT and Energy Landscape
Uphill and downhill transition path（Exhibiting non-gradient nature）
Construction of Global Energy Landscape
Global QP computed by sticking local versions.
The convexity of the obtained Hamiltonian (by-product of LDT) can guarantee the robustness and effiﾞciency in the numerical computations.
�
�1k
0a
a
b
0k0�
1�
n�
m�S(x) = infT
inf�(0)=x0,�(T )=x
Z T
0L(�, �)dt
Non-gradient Nature
The non-gradientness can be shown from the analysis of the uphill and downhill paths (drift term is not parallel to the gradient of the potential).
x = b(x) +MX
j=1
aj(x)�j
�e�x
S·⌫j � 1�
e�x
S·�j � 1 +⇥xS · ⌫j
Uphill path：
Downhill path：
Degenerate to the Langevin approximation near the steady states
x = rp
H(x, 0), p = 0
0 = rx
H(x, 0)
x = b(x) +�"�(x) · w
b(x) = �a ·⇥U(x) + l(x)
l(x) ·�U = 0
x = a ·�U(x) + l(x)
Diffusion process:
Decomposition via quasi-potential U(x) :
Uphill path：
satisfying
Switching Rate Analysis
Mean switching time (MST) VS Different choices of parameters: Faster switching gives more robust gene ON state!
MST VS the strength of trigger signal to mRNA production: Slight change of trigger signal can reduce MST signifiﾞcantly.
The mean switching time can be approximated with the exponential of the energy barrier times the system size.
of off-to-on transition (red dashed lines) while fast rates lead to longMST of off-to-on transition (red dash-dotted lines). This is due tothe reason that faster promoter transition rates lead to smallermRNA and protein noise strength (see Fig. 6 for more detailedinformation). Ignoring the difference of the mechanism of initialtranscription between prokaryotes and eukaryotes, in the simplecase, the faster promoter transition rates correspond to the geneexpression process in prokaryotes, and the slower promotertransition rates correspond to the slow chromatin remodelingprocess in eukaryotic case [39]. The results suggest thatprokaryotes may have stronger cellular memory than eukaryotes.
Local Property: Fluctuation Around Stable StatesAnother quantitative information that quasi-potential energy
landscape can provide is the noise strength of stable states. Herewe use the coefficient of variation (CV, i.e. the standard deviationover the mean) to measure the strength of fluctuation instead ofthe Fano factor, for the system here has positive feedback thusdeviates far from Poisson statistics. Notice that the stationarydistribution p(x) ! expf{KS(x)g, we can expand S(x) in thevicinity of high stable state xon up to second order thus get theGaussian approximation.
p(x)^1
(2p)32jSj
12
expf{ 1
2(x{m)S{1(x{m)Tg: ð27Þ
Here, x~(x,y,z),m~(xon,yon,zon), S~(Sij)3|3, and jSj is the
determinant of matrix S. Eq. (27) holds only in the vicinity of the
on state with standard deviations sm~(K(S00yyS00zz{S002yz )=jSj)12,
sn~(K(S00xxS00zz{S002xz )=jSj)12 and sd~(K(S00xxS00yy{S002xy)=jSj)
12.
With the sm, sn and sd above, we can easily obtain the CV asshown in Fig. 6.
Figure 6 demonstrates that when the average expression levelsincrease, the noise strength of mRNA and protein decreases in ourpositive feedback model. The fluctuation of mRNA is usuallylarger than that of protein. Furthermore, the noise level with slowpromoter transition rates is almost always larger than the one withfast promoter transition rates. This is in accordance with theresults of MST that the system with long MST has small noise andvise versa. The inconsistent portion between analytical andsimulation results (the left part of the line with slow promotertransition rates in Fig. 6B) is due to the inapplicability of Eq. (27)
Figure 4. The mean switching time (MST) and quasipotential energy landscape as a function of parameters. (A) and (B): MST as afunction of transcription rate a. Promoter transition rates k0~1, c0~50, the gMAM results with numerical prefactor of off-to-on transition (red solidline) and on-to-off transition (blue dashed line), compared with MC simulations (.) and (0), respectively. (C) and (D): The gMAM results with differentpromoter transition rates of off-to-on transition (red) and on-to-off transition (blue), where solid line with k0~1, c0~50 is same as (A) and (B), thefaster transition rate in dashed line with k0~5, c0~250, the slower transition rate in dotted line with k0~0:2, c0~10. Other parameters area0~0:4, b~40, cn~1, k1~0:0002, c1~2; in (A,C), cm~10 and (B,D) a~400:doi:10.1371/journal.pone.0088167.g004
Constructing Energy Landscape for Genetic Switch
PLOS ONE | www.plosone.org 7 February 2014 | Volume 9 | Issue 2 | e88167
during the low barrier crossing process for the on state. Moredetails may be referred to Text S1:V.
Application in Transcriptional CascadesTo further illustrate the power of quasi-potential energy
landscape and the abundant quantitative information it contains,we apply our methodology to a transcriptional cascades modelbased on the previous work of S. Hooshangi et al. [40]. In theirwork, S. Hooshangi et al. synthesized transcriptional cascadescomprised of one, two, and three repression layers and analyzedthe sensitivity and noise propagation as a function of networkcomplexity. They used different concentrations of anhydrotetra-cycline (aTc) as inducer and measured the fluorescence intensitiesof protein eyfp (the last layer of each cascade) by the flowcytometer.
Here we simplify the 3-layer cascades as x?y1 a y2 a y3,where x denotes the concentration of aTc as inducing signal andy1, y2, y3 denote the output of proteins in different layersrespectively. Then we directly construct the quasi-potential energylandscape for each layer and obtained the normalized probabilitydistribution of the output to certain signal x from Eq. (15). Thedose response curves to increasing signal x are shown in Fig. 7,which are consistent well with the previous experimental results.Further more, two features of transcriptional cascades can beobserved. Firstly, the more layers the transcriptional cascadeshave, the sharper the response curves are (as the Hill coefficient ofthe 3-layer cascades is 2.00, 3.15 and 4.08 respectively). Thus thesensitivity is increased in the cascades. Secondly, the fluctuation ofoutput can be described by the spreading width of its distribution,so more layers of cascades amplify the cell-cell variability (see Fig.
S2). In short, when a cascade has more layers, its response curvegets steeper with a wider probability distribution and thus largerfluctuations. The straightforward calculation of CV based on Fig. 7has been done and it agrees well with the MC simulations (seeText S1:VI and Figure S2).
Limitations of The Study, Open Questions, and FutureWorks
We have already illustrated a general methodology based onLDT to quantitatively understand the metastability in geneexpression processes perturbed by the intrinsic noise and appliedit to a dimer auto-regulatory circuit model. It is clear that thismethodology can be extended to more general systems, provideone can explicitly write down the Hamiltonian of the system. If allof the considered species have relatively large numbers, theHamiltonian is simply the Eq. (4). For the case where the largevolume limit fails to be true, our method is also applicable underan additional assumption that the low copy number of speciesreach their stationary distribution much faster than the others.This is the situation that we treat DNA in our dimer model.However, we would like to mention the limitations of our work,which of course motivates us for future studies.
The main limitations or the corresponding open questions canbe summarized into the following three aspects:
1. The case where the large volume limit and the fast switchingmechanism are both invalid. This prevents us to construct theLDT for the considered system. Thus there is no Hamiltonianand the current methodology fails. How to quantitatively studysuch systems and define the proper Waddington energylandscape is an issue.
Figure 5. The mean switching time (MST) of off-to-on transition as a function of (A) trigger signal strength that transcribes mRNA atconstant rate and (B) degradation rate of protein cn. (C) and (D): Quasipotential energy landscape with different trigger strength. Strigger~5 in(C), and Strigger~10 in (D). Other parameters are a0~0:4, a~400, b~40, c1~1, k1~0:0002, cm~10; cn~1 in (A,C,D), and Strigger~0 in (B).doi:10.1371/journal.pone.0088167.g005
Constructing Energy Landscape for Genetic Switch
PLOS ONE | www.plosone.org 8 February 2014 | Volume 9 | Issue 2 | e88167
Continuation of DNA States VS CLT for the Occupation Measure
DNA continuation: Second quantization path integral + semiclassical approximation. Mathematically: CLT for the occupation measure.
Zhang-Sasai-Wang, PNAS, 2013
⇥ = cos �,
Continuation of DNA states
CLT for the occupation measure 1
Vd�t =
h� �tf(y) + (1� �t)g(y)
idt
+1pV
hp�tf(y)dW
1t +
p(1� �t)g(y)dW
2t
i
Application to Noise Cascading Model
Based on LDT
1y 2y 3y
� � �
xxx
The methodology is extended to a noise Cascading model: Dimensional reductions
p(x, y) =
Zp(x, y, z)dz ⇥
Zexp{�KS(x, y, z)}dz.
S(x, y) = minz
S(x, y, z)
Lv-Li-Li-L., PLoS ONE 9(2014), e88167; L.-Lin, 2014, arXiv:1504.03781.
Budding Yeast Cell Cycle: 3-node Model
Extended to a 3-node model for budding Yeast cell cycle
Lv-Li-Li-L., 2015, PLoS CB 11(2015), e1004156.
Landscape and Stability
An excitable system with single stable node: What does the energy landscape look like?
A section of the global quasi-potential landscape from S phase to early M phase
The 3-node excitable system: ODE trajectory
Lv-Li-Li-L., 2015, PLoS CB 11(2015), e1004156.
Global Quasi-potential
Global quasi-potential: Adaptivity, Checkpoints, Finite size effect
Concentration
Potential
Concentration
nutrientssufficient
checkpointactivation
finite0volumeeffectG1
S0phase0chk
M0phase0chk
�
�
�
�
Time Irreversibility and Non-Gradientness
Local quasi-potential: Time irreversibility and non-gradientness.
Potential
Action Action
Backward
Forward
Nongradientness and Time irreversibility
Outline
Ø Introduction
Ø Landscape and quasi-potential
Ø Two-scale LDT
Ø Conclusion
Different Scalings for DNA Switching
Different scalings: The considered DNA switches with fast but different magnitudes compared with translation process.
Zhang-Sasai-Wang, PNAS, 2013, Sasai-Wolynes, PNAS 2003;
Ge-Qian-Xie, PRL, 2015.
Different regimes:Case 1: k0, k1 � f, g
Case 2: k0, k1 ⇠ f, g
Case 3: k0, k1 ⌧ f, g
k0, k1 � � ⇠ 1
f, g � 1
Condition:
Reduced Models and Analysis
Utilized approach: The reduced model is considered after the limit has already been taken to superfast reactions.
Bressloff-Newby, PRE, 2014Ge-Qian-Xie, PRL, 2015.
Stochastic hybrid system if the protein is taken to be continuous ( )：n � 1
f = f/V, g = g/V, k0 = k0/V, k1 = k1/V
@
t
p1 = �f(x)p1 + g(x)p2 � @
x
h(k0 � �x)p1
i
@
t
p2 = f(x)p1 � g(x)p2 � @
x
h(k1 � �x)p2
i
Rescaled parameters:
Case 1：
Case 3：Averaged system with discrete states：
@tp(n, t) = (E�1n � 1)k(n)p(n, t) + �(E1
n � 1)np(n, t)
Analysis: HJB equation by WKB asymptotics to the reduced model or the path integral.
Motivation for More General Two-Scale LDT
Motivation: Studying the more general two-scale LDT with different scalings in the sense of double limit instead of repeated limit.
ki ⇠ "�1, f, g ⇠ "�↵, ↵ > 0Consider
Case 1: ↵ < 1
Case 2: ↵ = 1
Case 3: ↵ > 1
LDT? Done.LDT?
What will the LDT be? Mean-fiﾞeld limit, Diffusion approximation?
L.-Lin, arXiv: 1508.06907.
V = "�1
Starting From Formal Approach
We start from the formal approach: the second quantization path integral to guess the result at fiﾞrst.
CME:
Generator: Q = (qij(n))
a† |ni = |n+ 1i , a |ni = n |n� 1i | i =X1
n=0P (n, t) |ni
@
@tP (n, t) = A{P (n� 1, t)� P (n, t)}
+B{(n+ 1)P (n+ 1, t)� nP (n, t)}+QTP (n, t)
Doi-Peliti Path Integral:
@t | (t)i = ⌦ | (t)i ⌦ = A(a† � 1) +B(a� a†) +Q†
A, B diagonal matrix
Creation-Annihilation Operator State
Schrodinger-like equation Not self-adjoint in general
Doi, J. Phys A, 1976; Peliti, J. Phys, 1985
Formal Rate Functional
Obtain the formal Lagrangian via the Doi-Peliti second quantization path integral.
Doi-Peliti Path Integral:P (nf , ⌧ |ni, 0) = hnf | exp(⌦⌧) |nii
P (nf , ⌧ |ni, 0) /Z
DxD↵DcD� exp
✓�Z
dtL
◆Resolution of identity: introducing augmented variables --- Momentum
L = i↵dn
dt+
N�1X
j=1
i�jd(cN � cj)
dt�H(n, c, i↵, i�) H(n, c, i↵, i�) = H1 +H2
Formal Lagrangian:
H1(n, c, i↵) =NX
j=1
kjcj [exp(i↵)� 1] + �n[exp(�i↵)� 1], H2(c, n, i�) =NX
m,j
cmqmj(n)(e�m��j � 1)
Large Volume Scaling
Obtain the formal Lagrangian via the Doi-Peliti second quantization path integral.
Condition: kj ⇠1
✏, qij ⇠
1
✏↵, ↵ > 0
x/✏ = n, ki/✏ = ki, qij(x)/✏↵ = qij(n)Rescaling:
V = "�1
With steepest descent asymptotics:
L1(x, x, c) = supp{px�H1(x, c, p)},
L2(c, x) = sup'
{�H2(c, x,')}
Lagrangian for translation:
Lagrangian for switching:
P (xf , ⌧ |xi, 0) /Z
DxDc exp
✓�1
✏
ZdtL1(x, x, c) +
�1
✏
↵
ZdtL2(c, x)
◆
General Framework
Singular perturbation analysis yields the desired LDT result for the different scalings.
Case 3: LDT Lagrangian
" lnP ⇠ L
x
(x, x) = infc
⇢L1(x, x, c) +
1
✏
↵�1L2(c, x)
�, " ⌧ 1
L2 � 0 and L2(c, x) = 0 at c = c0(x)
Case 2: LDT Lagrangian
Case 1: LDT Lagrangian
"
↵ lnP ⇠ L
x
(x, x) = infc
✓1
✏
1�↵
ZdtL1(x, x, c) +
ZdtL2(c, x)
◆, " ⌧ 1
L1 � 0 and L1(x, x, c) = 0 at c = cx
L
x
(x, x) = L2(cx, x)
L
x
(x, x) = L1(x, x, c0(x))
" lnP ⇠ L
x
(x, x) = infc
{L1(x, x, c) + L2(c, x)}
(↵ > 1)
(↵ < 1)
Application
Application of the framework to the previous genetic-switching model.
Case 3:
Mean fiﾞeld ODE:
Langevin approximation:
L
x
(x, x) = L1(x, x, c0(x))
c0(x) =⇣
f(x)
f(x) + g(x),
g(x)
f(x) + g(x)
⌘
H(x, p) =˜k1f +
˜k0g
f + g[exp(p)� 1] + �x[exp(�p)� 1]
dxt =
k1f + k0g
f + g
� �x
!dt+
p✏
0
@s
k1f + k0g
f + g
dw
1t �
p�xdw
2t
1
A
dx
dt
=k1f + k0g
f + g
� �x
Application
Application of the framework to the previous genetic-switching model.
Case 2:
Mean fiﾞeld ODE:
Langevin approximation:
L
x
(x, x) = infc
{L1(x, x, c) + L2(c, x)}H(x, p) =
⇣˜k1s+ ˜k0(1� s)
⌘[exp(p)� 1] + �x[exp(�p)� 1]
�⇣q
(
˜f(1� s)�p
gs⌘2
dxt =
k1f + k0g
f + g
� �x
!dt+
p✏
0
@s
k1f + k0g
f + g
+2fg(k1 � k0)2
(f + g)3dw
1t �
p�xdw
2t
1
A
dx
dt
=k1f + k0g
f + g
� �x
Application
Application of the framework to the previous genetic-switching model.
Case 1:
Mean fiﾞeld ODE:
Langevin approximation:
dx
dt
=k1f + k0g
f + g
� �x
L
x
(x, x) = L2(cx, x) =⇣q
f(x)c2 �p
g(x)c1⌘2
c1 = x�k0+�x
k1�k0and c1 + c2 = 1
Hx
(x, p) = p�0 � Lx
(x,�0)
dxt =
k1f + k0g
f + g
� �x
!dt+
p"
↵
s2fg(k1 � k0)2
(f + g)3dwt
Classical Michaelis-Menten Kinetics:
Problem: What happens if we consider the diffusion approximation of the classical Michaelis-Menten system?
Michaelis-Menten kinetics:
lost. This resembles the issue of the choice of Hamiltonians for parametrized curve problem in classical mechanics(see p. 40 in [5]).Let us make some comments on the obtained mean-field limit and Langevin approximations. Recall that there are
two parts of noise in the original dynamics: one is from the translation process and the other from DNA switchingprocess. Our result tells that di↵erent di↵usion approximations arise according to the magnitudes of residual noisein di↵erent reaction channels. If ↵ > 1, the dominant part of noise is from the translation, so only the fluctuationfrom translation process survives. If 0 < ↵ < 1, the dominant part is from switching process, so only the fluctuationfrom DNA switching process survives. And if ↵ = 1, both fluctuations from protein translation and genetic switchingcontribute. Similar situation occurs in the LDT analysis, where the singular perturbation is performed for variationalminimizations. The obtained results show the validity of the procedure by taking the limit for faster process at firstand then performing the corresponding analysis for slower process. Although we only consider the two-state models,the essential structure and results hold for general cases. It is a natural extension of the classical large volume limitfor chemical reaction processes. We summarize our discussions for the three regimes in Table I.
TABLE I. Comparison of LDTs, mean-field limits and Langevin approximations for three regimes
LDT Hamiltonian Deterministic Drift Noise in Langevin Approximation
Case 1: (↵ > 1) Hx
(x, p) in (37)
k1f(x) + k0g(x)
f(x) + g(x)� �x
p✏
0
@s
k1f + k0g
f + gdw1 �
p�xdw2
1
A
Case 2: (↵ = 1) Hx
(x, p) in (50)p✏
sk1f + k0g
f + g+
2f g
(f + g)3(k1 � k0)2dw1 �
p�xdw2
!
Case 3: (↵ < 1) Hx
(x, p) in (44)p✏↵
s2f g
(f + g)3(k1 � k0)2dw1
V. APPLICATION TO THE SINGLE-MOLECULE MICHAELIS-MENTEN KINETICS
Our approach and observation have interesting implications on the single-molecule Michaelis-Menten system14, inwhich a substrate S binds reversibly with an enzyme E to form an enzyme-substrate complex ES that decomposes toform a product P. The reaction schemes can be schematically shown as
E + Sk1
GGGGGGBF GGGGGG
k�1
ESk2
GGGGGGA E+P, P�
GGGGGA ;. (55)
In case of single-molecule enzyme set-up, the reaction system (55) falls in the framework considered in this paper. Asin [14], we assume that the substrate is abundant enough and there is essentially no depletion of substrate by a singleenzyme molecule. That is, we assume the concentration of substrate is a constant, which will be denoted as [S]. It iswell-known that the rate of product formulation v has the following form in the quasi-steady state approximation12
v =k2[S]
[S] + kM
, (56)
where kM
= (k�1 + k2)/k1. In [14], the statistics of enzymatic turn-over time and dynamical disorder are considered.Here we are interested in deriving the Langevin approximations of the Michaelis-Menten system in di↵erent regimes.In [14], k�1 ranges from 0s�1 to 2000s�1, k1 is usually taken as 107M�1s�1, k2 = 250s�1, and [S] ranges from the
order 0.001mM to 0.1mM , where 1M = 1mol/L. Some specific choices of these parameters include
• Case 1: k�1 = 2000s�1, k1[S] = 107M�1s�1 ⇥ 0.30mM = 3000s�1, k2 = 250s�1;
• Case 2: k�1 = 200s�1, k1[S] = 107M�1s�1 ⇥ 0.02mM = 200s�1, k2 = 250s�1;
• Case 3: k�1 = 50s�1, k1[S] = 107M�1s�1 ⇥ 0.005mM = 50s�1, k2 = 250s�1.
10
Michaelis-Menten law (QSSA):
d[P ]
dt=
k2[S]
[S] + kMkM =
k�1 + k2k1
What happens if the flﾟuctuation is taken into account?
Implications on Enzymatic Kinetics
Application of the framework to the single-molecule Michaelis-Menten system.
Single-molecule Michaelis-Menten:
lost. This resembles the issue of the choice of Hamiltonians for parametrized curve problem in classical mechanics(see p. 40 in [5]).Let us make some comments on the obtained mean-field limit and Langevin approximations. Recall that there are
two parts of noise in the original dynamics: one is from the translation process and the other from DNA switchingprocess. Our result tells that di↵erent di↵usion approximations arise according to the magnitudes of residual noisein di↵erent reaction channels. If ↵ > 1, the dominant part of noise is from the translation, so only the fluctuationfrom translation process survives. If 0 < ↵ < 1, the dominant part is from switching process, so only the fluctuationfrom DNA switching process survives. And if ↵ = 1, both fluctuations from protein translation and genetic switchingcontribute. Similar situation occurs in the LDT analysis, where the singular perturbation is performed for variationalminimizations. The obtained results show the validity of the procedure by taking the limit for faster process at firstand then performing the corresponding analysis for slower process. Although we only consider the two-state models,the essential structure and results hold for general cases. It is a natural extension of the classical large volume limitfor chemical reaction processes. We summarize our discussions for the three regimes in Table I.
TABLE I. Comparison of LDTs, mean-field limits and Langevin approximations for three regimes
LDT Hamiltonian Deterministic Drift Noise in Langevin Approximation
Case 1: (↵ > 1) Hx
(x, p) in (37)
k1f(x) + k0g(x)
f(x) + g(x)� �x
p✏
0
@s
k1f + k0g
f + gdw1 �
p�xdw2
1
A
Case 2: (↵ = 1) Hx
(x, p) in (50)p✏
sk1f + k0g
f + g+
2f g
(f + g)3(k1 � k0)2dw1 �
p�xdw2
!
Case 3: (↵ < 1) Hx
(x, p) in (44)p✏↵
s2f g
(f + g)3(k1 � k0)2dw1
V. APPLICATION TO THE SINGLE-MOLECULE MICHAELIS-MENTEN KINETICS
Our approach and observation have interesting implications on the single-molecule Michaelis-Menten system14, inwhich a substrate S binds reversibly with an enzyme E to form an enzyme-substrate complex ES that decomposes toform a product P. The reaction schemes can be schematically shown as
E + Sk1
GGGGGGBF GGGGGG
k�1
ESk2
GGGGGGA E+P, P�
GGGGGA ;. (55)
In case of single-molecule enzyme set-up, the reaction system (55) falls in the framework considered in this paper. Asin [14], we assume that the substrate is abundant enough and there is essentially no depletion of substrate by a singleenzyme molecule. That is, we assume the concentration of substrate is a constant, which will be denoted as [S]. It iswell-known that the rate of product formulation v has the following form in the quasi-steady state approximation12
v =k2[S]
[S] + kM
, (56)
where kM
= (k�1 + k2)/k1. In [14], the statistics of enzymatic turn-over time and dynamical disorder are considered.Here we are interested in deriving the Langevin approximations of the Michaelis-Menten system in di↵erent regimes.In [14], k�1 ranges from 0s�1 to 2000s�1, k1 is usually taken as 107M�1s�1, k2 = 250s�1, and [S] ranges from the
order 0.001mM to 0.1mM , where 1M = 1mol/L. Some specific choices of these parameters include
• Case 1: k�1 = 2000s�1, k1[S] = 107M�1s�1 ⇥ 0.30mM = 3000s�1, k2 = 250s�1;
• Case 2: k�1 = 200s�1, k1[S] = 107M�1s�1 ⇥ 0.02mM = 200s�1, k2 = 250s�1;
• Case 3: k�1 = 50s�1, k1[S] = 107M�1s�1 ⇥ 0.005mM = 50s�1, k2 = 250s�1.
10
S. Xie et al, JPC B 2005.
TABLE II. The mean-field limits and Langevin approximations for single-molecule Michaelis-Menten kinetics
Deterministic Drift Noise in Langevin Approximation
Case 1: (↵ > 1)k1k2
k1 + k�1
� �xp✏
sk1k2
k1 + k�1
dw1 �p�xdw2
!
Case 2: (↵ = 1)k1k2
k1 + k�1 + k2� �x
p✏
sk1k2
k1 + k�1 + k2+
2k21 k
22
(k1 + k�1 + k2)3dw1 �
p�xdw2
!
Case 3: (↵ < 1) k1 � �x↵
p✏↵✓q
k1dw1 �p�x
↵
dw2
◆
State1
State2
Product1 ( )n1
f(n1, n2) g(n1, n2)
Product2 ( )n2
;
;
�1
�2
k11
k 21
k12
k22
FIG. 2. Schematics of a two-scale kinetic model with two kinds of products.
VI. DISCUSSIONS AND CONCLUSION
The methods and LDT results we proposed in this paper are not limited to the two-state model, single-moleculeMichaelis-Menten and single kind of product case. It is indeed general for a class of two-scale kinetic systems. Toshow this, let us consider the following extension as shown in Fig. 2.
We assume similar scaling as considered in (12):
kij
⇠ 1
✏, f, g ⇠ 1
✏↵, ↵ > 0, (i, j = 1, 2).
Define xj
= nj
✏, kij
= kij
✏ for i, j = 1, 2 and f(x1, x2) = f(n1, n2)✏↵, g(x1, x2) = g(n1, n2)✏↵. Performing the sameapproach as in Sec. II, we get the transition probability
P (nf
, ⌧ |ni
, 0) /Z
DxDc exp
✓
�1
✏
Z
dtL1(x, x, c)�1
✏↵
Z
dtL2(x, c)
◆
.
Here the Lagrangian
L1(x, x, c) = supp{p · x� H1(x, c,p)}, L2(x, c) = sup
'{�H2(x, c,')},
where x = (x1, x2), c = (c1, c2), p = (p1, p2), ' = ('1,'2) and
H1(x, c,p) = (k11c1 + k21c2)(ep1 � 1) + �1x1[e
�p1 � 1] + (k12c1 + k22c2)(ep2 � 1) + �2x2(e
�p2 � 1),
H2(x, c,') = c1g(x)(e'1�'2 � 1) + c2f(x)(e
'2�'1 � 1).
All of the analysis performed for the two-state model can be applied here to obtain the LDTs for variable x = (x1, x2)with di↵erent ↵.
One can also employ the WKB ansatz Pj
(x) ⇠ exp(�✏�1�j
(x)) for the stationary distribution of the stochastichybrid system (32), where x = n✏ and j is a state of DNA. In the asymptotics, one gets a static Hamilton-Jacobiequation for the quasi-potential �
j
and it turns out �j
does not depend on the specific choice of j. However if nothandled appropriately, the WKB approximation may lead to totally di↵erent forms of Hamiltonian15 as mentioned inthe end of Section IV. This non-uniqueness is due to the lack of variational selection in LDT, which gives a uniqueHamiltonian dual to the obtained Lagrangian in rate functional. And this Hamiltonian has the superiority that it
14
x↵ = n✏
↵
L.-Lin, arXiv: 1508.06907. All are consistent with MM.
Insights and Remarks
The time scales play an important role in the fiﾞnal LDT and related results.
Ø The LDT can be obtained by the singular perturbation analysis from the LDTs in different scales.
Ø The strength of the noise in the diffusion approximation involves different terms from different sources, depending on the time scaling.
Ø Similar results can be obtained if the time scale of switching has more general form than the algebraic power of .
Ø The rigorous analysis is more technical, but the essential details are similar for the latter part.
"
L.-Lin, arXiv: 1508.06907.
Outline
Ø Introduction
Ø Landscape and quasi-potential
Ø Two-scale LDT
Ø Conclusion
Summary
• Quasi-potential is a good candidate to rationalize the potential landscape, which could exhibit non-equilibrium nature of biological systems.
• Two scale LDT for the Gillespie dynamics with positive feedback is obtained. Different Lagrangian and diffusion approximations arise under different scalings. Application to enzymatic dynamics is insightful for future study.
• Issues: More effiﾞcient numerics? High dimensionality? Transition rate formula for non-gradient systems with caustics?
Acknowledgement
Thank you！
Collaborators:
Funding: NSFC
Prof. Fangting LiPhysics and CQB
Dr. Xiaoguang LiPKU Math
Dr. Ye ChenPKU Math
Dr. Feng LinPKU Math
Dr. Cheng LvRockefeller
Dr. Peijie ZhouPKU Math
