Nonequilibrium phase transition ina mesoscopic biochemical system:

From stochastic to nonlineardynamics and beyond

Hao Gegehao@fudan.edu.cn

School of Mathematical Sciencesand Centre for Computational Systems Biology

Fudan UniversityShanghai, China

Biological signal transduction• In biology, “signal transduction” refers to any

process by which a cell converts one kind of signal or stimulus into another;

• Most signal transduction processes involve ordered sequences of biochemical reactions inside the cell, resulting in a signal transduction pathway;

• They are increasingly understood in quantitative terms based on mathematical models;

• Nowadays, there is a rapid growing awareness of noise in biological networks, stimulating the study of stochastic models.

Signaling cascade

EGF Signal Transduction Pathway.

PdPC and GTPase

• Phosphorylation-dephosphorylation cycle (PdPC) becomes a universal module in biological signaling processes (Nobel Prize 1992, Fischer and Krebs);

• Another important component of signaling network is the GTPase (Nobel Prize 1994, Gilman and Rodbell).

We consider a simple phosphorylation-dephosphorylation

cycle, or a GTPase cycle:

E E*

K*

ATP ADP

P

Pi

a1

a-1

a2

a-2

with a positive feedback K and K* are inactive and active forms of a kinase. E* is the phosphorylated E, a signaling molecule. Usually E∗ is functionally active, i.e., “turned-on”.

Traditional theory for chemical reaction systems is based on

the law of mass-action

Simple Kinetic Model based on the Law of Mass Action

.][]][[])E([],[])[]][K[(])E([

,])E([])E([]E[

3***

2**

***

EEPwEEEv

wvdt

d

εβ

δα

+=

+=

−=

ATP ADP

Pi

K*

P

E E*

Saddle-node Bifurcation diagram

Ge, H. and Qian, H. : Phys. Rev. Lett. 103, 148103 (2009)

Beyond deterministic approach

• A deterministic model only describes the averaged behavior of a system based on large populations;

• Many biochemical regulatory networks inside a cell consist of molecular species with low concentrations, where noise is significant;

• So how to understand for a cell with only a few copies of the molecules?

The Stochastic Nature of Chemical Reactions

• Single-molecule measurements• Relevance to cellular biology: small copy #• Kramers’ theory for unimolecular reaction

rate in terms of diffusion process (1940)• Delbrück’s theory of stochastic chemical

reaction in terms of birth-death process (1940)

Bimodal distributions

Choi, P. J. et al.:Science, 322, 442 (2008)

Buckmaster, R. et al. :Analyst, 134, 1440 (2009)

Xu, L. et al. : Neoplasia, 11, 345 (2009)

Theoretical Basis: The Chemical Master Equations: A New Stochastic Mathematical

Foundation of Chemical and Biochemical Reaction Systems

E E*

K

P

2E*0E* 1E* 3E* … (N-1)E* NE*

Markov Chain Representation

v1

w1

v2

w2

v0

w0

.))2)(1((

),1)()2)(1((

2

2

nnnV

w

nNnnV

v

n

n

δε

β

δα

+−−+=

+−+−−=

A Theorem of T. Kurtz (1972)In the limit of V →∞, the stochastic solution to CME in volume V with initial condition XV(0), XV(t) will approach to x(t), the deterministic solution of the differential equations, based on the law of mass action, with initial condition x0.

.)(lim

;)()(supPrlim

0V1

V

V1

tsV

x0XV

0εsxsXV

=

=

>−

−

∞→

−

≤∞→

Beyond Kurtz’s Theorem

• Therefore, the stochastic CME model has superseded the deterministic law of mass action model during finite time intervals.

• However, the presence of noise not only leads to corrections to the deterministic analysis but may give rise to emergentbehaviors.

• W will show that it is not an alternative; It is a more general theory.

Steady distribution for large V

( )

*( ) ( )

( )exp log( )

exp ( )

E

x

p x P N xV

v yV dyw y

V xφ

=

∝ ∫

= −

＝

　　　

We could derive that

** * * [E ]( ) 0 ( ) ( ) 0dx v x w x

dtϕ = ⇔ = ⇔ = 　　

Emergent Landscape

Steady states

Maxwell construction• The most important feature is that the function Φ(x)

is independent of V, provided that V is sufficiently large;

• Even though Φ(x) exhibits stochastic bistability which corresponds closely to the deterministic model when V →∞, only one of the two stable fixed points is relevant in the thermodynamic limit. It is just the one with smaller Φ(x).

• Therefore, a corrupt transition rather than hysteresis will occur at the critical value for certain parameter k1 when Φ(x1) = Φ(x2) ，which is similar to Maxwell’s original construction for Van der Waals gas;

Maxwell construction

Ge, H. and Qian, H. : Phys. Rev. Lett. 103, 148103 (2009)

Life time of the states

.)()()(

2

,)()()(

2

*3

''*2

''*3

))()((

12

*3

''*1

''*3

))()((

21

*2

*3

*1

*3

xxxeT

xxxeT

xxV

xxV

φφω

π

φφω

π

φφ

φφ

−∝

−∝

−

→

−

→

The switching time between states:

It explains the results about CaMKII switch in Miller, P. et al: PLOS Biology, 3, e107, 2005

Exponentially increase with V

Life time of the states

There are about 600-1200 number of total molecules in a cell. This gives the switching time around 3×104 to 1012 sec. That is 10 hours to thirty thousand years!

Biological implications and discussion

Defining states

• Experimental molecular genetics defines the state(s) of a cell by their “transcription pattern” via expression level (i.e., RNA microarray).

• Biochemistry defines the state(s) of a cell via concentrations of metabolites and copy numbers of proteins.

• But biologists define the state(s) of a cell by its phenotype(s).

How does computational biology define the biological phenotype(s)

of a cell in terms of the biochemical copy numbers of proteins?

Bistability and Emergent Sates

Pk

number of R* molecules: k

Emergent Mesoscopic Complexity

Beyond deterministic dynamics.• It is generally believed that when a system’s size

increases, the stochastic behavior at a mesoscopic level averaged out, and a deterministic behavior emerges;；

• Beyond that time scale, another “macroscopic”stochastic behavior exists! This stochastic dynamics is completely non-obvious from the level of pair-wise, static, molecule interactions. It can only be understood from a mesoscopic, open driven chemical dynamic system perspective.

Emergent Mesoscopic Complexity

Biological stability and robustness are stochasticconcepts.

• A state of a biological cell, called a functional cellular attractor, should be dynamically stable against various minor perturbations which are inevitable in living systems.

• In cellular biology, it is incorrect to model biological stability and robustness in terms of deterministic trajectories or sizes of basins of attractors from a deterministic model;

• The relative stability and robustness of PdPC can not be properly inferred without an explicit consideration of the intrinsic noise in the model.

Three time scales

The mathematical analysis suggests three distinct time scales, and related mathematical descriptions. (i) molecular signaling, (ii) biochemical network dynamics (iii) cellular evolution.

The (i) and (iii) are stochastic while (ii) is deterministic.

Three time scales

If one perturbs such a multi-attractor stochastic system:

• Rapid relaxation back to local minimum following deterministic dynamics (level ii);

• Stays at the “equilibrium” for a quite long time (metastable, quasi-steady state);

• With sufficiently long waiting, exit to a next cellular state (Final steady state).

A B

discrete stochastic model among attractors

ny

nx

chemical master equation cy

cx

A

B

fast nonlinear differential equations

appropriate reaction coordinate

ABpr

obab

ility

emergent slow stochastic dynamics and landscape

(a) (b)

(c)

(d)

Three time scales

Fixed finite molecule numbers

The emergent cellular, stochastic “evolutionary” dynamics (Time scale III) follows not gradual

changes, but rather punctuated transitions between cellular

attractors.

An example: Lac operon

Quasi-steady distribution rather than the real steady distribution will depend on the initial state.

Choi, Paul J. et al.: Science, 322, 442-446 (2008)

An example: Lac operon• We proposed a hybrid model for the regulatory

mechanism of Lac operon.• Without the DNA loop, the bistability region may become

too narrow to be observed;• The time for the system to arrive at the real steady state

is very long, so it may be impratical to be observed in experiments;

• Both small and large bursts could cause a transition from the uninduced state to the induced one, but it will take an extremely long time when there are only the small bursts.

• The paper is under preparation, joint with Prof. Hong Qian and Prof. Xie.

Possible Chemical basis of epi-genetics:

• Could this be a chemical definition for epi-genetics inheritance: Exactly same environment setting and gene, different internal biochemical states (i.e., concentrations and fluxes)?

• It is likely that the code for epi-genetic inheritance is distributed.

• Could it be readily inherited during the process of cell volume change and division?

Possible Chemical basis of epi-genetics:

• As soon as cellular concentrations are deviated from their most stable state values, the dynamics will cause them to spontaneously relax back as long as the deviations are within the limit of the basin of the attractor (stable state).

• On the large landscape scale, the system is “digital”. This kind of “inheritable code” is not only distributed, it is also dynamic instead of static as in Watson-Crick basepairng.

Chemistry may be inheritable

Change V and N by a factor of 2 at the beginning

Into the Future:Toward a Computational

Elucidation of Cellular attractor(s) and inheritable epi-

genetic phenotype(s)

What do We Need?

• It requires a theory for chemical reaction networks with small numbers of molecules.

• The CME theory is an appropriate starting point.

• It requires all the rate constants under the appropriate conditions.

• One should treat the rate constants as the “force field parameters” in the computational macromolecular structures.

Thank You!