Abstract

AbstractWe consider a small driven biochemical network, the phosphorylation-

dephosphorylation cycle (or GTPase), with a positive feedback. We investigate its bistability, with fluctuations, in terms of a nonequilibrium phase transition based on ideas from large-deviation theory. We show that the nonequilibrium phase transition has many of the characteristics of classic equilibrium phase transition: Maxwell construction, discontinuous first-derivative of the “free energy function”, Lee-Yang's zero for the generating function, and a tricritical point that matches the cusp in nonlinear bifurcation theory. As for the biochemical system, we establish mathematically an emergent “landscape” for the system. The landscape suggests three different time scales in the dynamics: (i) molecular signaling, (ii) biochemical network dynamics, and (iii) cellular evolution. For finite mesoscopic systems such as a cell, motions associated with (i) and (iii) are stochastic while that with (ii) is deterministic. We suggest that the mesoscopic signature of the nonequilibrium phase transition is the biochemical basis of epi-genetic inheritance.

Nonequilibrium Phase Transition in a Biochemical System: Emerging

landscape, time scales, and a possible basis for epigenetic-inheritance

Hong Qian

Department of Applied Mathematics

University of Washington

Background• Newton-Laplace’s world view is deterministic;

• Boltzmann tried to derive the stochastic dynamics from the Newtonian view

• Darwin’s view on biological world: stochasticity plays a key part.

• Gibbs assumed the world around a system is stochastic (i.e., canonical ensemble)

• Khinchin justified Gibbs’ equilibrium theory, Kubo-Zwanzig derived the stochastic dynamics by projection operator method, both considering small subsystems in a deterministic world.

In Molecular Cellular Biology (MCB):

• Amazingly, the dominant thinking in the field of MCB, since 1950s, has been deterministic! The molecular biologists, while taking the tools from solution physical chemists, did not take their thinking to heart: Chemical reactions are stochastic in aqueous environment (Kramers, BBGKY, Marcus, etc.)

• But things are changing dramatically …

Here are some recent headlines:

The Biochemical System Inside CellsE

GF

Signal T

ransduction Pathw

ay

k1 [A*]

k-1[A*]

k2

k-2

B B*

A A*

Biologically active forms of signaling molecules

Introducing the amplitude of a switch (AOS):

[ *] [ *] 0

1 2

1 1 2 2 1 1 2 2

[ *] [ *]

[ ] [ *] [ ] [ *]

1

/ 1 / 1

1tanh

42 1

A A

B BAOS

B B B B

k k

k k k k k k k k

G

RT

Amplitude of the switch as afunction of the intracellular phosphorylation potential

No energy, no switch!

H. Qian, Phosphorylation energy hypothesis: Open chemical systems and their biological functions. Annual Review of Physical Chemistry, 58, 113-142 (2007).

The kinetic isomorphism between PdPC and GTPase

(A)K

P

(C)

P

(B)

(D) (F)(E)

activation signal

phos

phor

ylat

ed

activation signal

phos

phor

ylat

ed

activation signal

phos

phor

ylat

ed

PdPC with a Positive FeedbackF

rom

Co

ope

r an

d Q

ian

(20

08)

Bio

chem

., 4

7, 5

68

1.F

rom

Zhu

, Qia

n an

d L

i (20

09) P

LoS

ON

E. S

ubm

itted

Simple Kinetic Model based on the Law of Mass Action

NTP NDP

Pi

E

P

R R*

].][[

],)[]][[(

,][

*

*

*

RPβJ

RREαJ

JJdt

Rd

2

χ1

21

activating signal:

acti

vati

on

leve

l: f

1 4

1

Bifurcations in PdPC with Linear and Nonlinear Feedback

= 0

= 1

= 2

hyperbolic delayed onset

bistability

Biochemical reaction systems inside a small volume like a cell: dynamics based on Delbrück’s

chemical master equation (CME), whose stochastic trajectory is

defined by the Gillespie algorithm.

A Markovian Chemical Birth-Death Process

k1

X+Y Zk-1

nZ

k-1nZ k-1(nZ +1)

nx,ny

k1(nx+1)(ny+1)V2

k1nxny

V2

VV

R R*

K

P

2R*0R* 1R* 3R* … (N-1)R* NR*

Markov Chain Representation

v1

w1

v2

w2

v0

w0

Steady State Distribution for Number Fluctuations

1

1k

1k

00

1k

00

1

2k

1k

1k

k

0

k

w

v1p

w

v

p

p

p

p

p

p

p

p

,

Large V Asymptotics

)(exp

)(

)(logexp

logexp

xφV

xw

xvdxV

w

v

w

v

11

Relations between dynamics from the CME and the LMA

• Stochastic trajectory approaches to the deterministic one, with probability 1 when V→∞, for finite time, i.e., t <T .

• Lyapunov properties of (x) with respect to the deterministic dynamics based on LMA.

• However, developing a Fokker-Planck approximation of the CME to include fluctuations can not be done in general (Hänggi, Keizer, etc)

Keizer’s Paradox: bistability, multiple time scale, exponential small transitions,

non-uniform convergence

t → , V → V→ , t →

stationary solution toFokker-Planck Equation(x)

=

Using the PdPC with positive feedback system to learn more:

Simple Kinetic Model based on the Law of Mass Action

NTP NDP

Pi

E

P

R R*

].][[

],)[]][[(

,][

*

*

*

RPβJ

RREαJ

JJdt

Rd

2

χ1

21

Beautiful, or Ugly Formulae

= 0

= 0

0

nondifferential point of c()

*

(x,)

Extrema value

(A)

(B)

dc(0)d

(C)

dc(0)d = xss

(D)

*

0

0.3

0.6

0.9

1.2

1.5

3 4 5 6 7 8

e

1(e) 2(e)

the cusp

the tri-critical point

*(e)(B)

4

6

8

10

0.01 0.1 1

xss

(A)

“Landscape” and limit cycle

θd

θdUθF

dt

θd )()(

θaθF sin)(

Further insights on “Landscape” with limit cycle

When there is a rotation

Large deviation theory or WBK

...)()(exp)( xφxφ

V

1xp 10

approaching a limit cycle,constant on the limit cycle

on the limit cycle, inversely proportional to angular velocity

Our findings on this type of non-equilibrium phase transition

• In the infinite volume limit of bistable chemical reaction system

• Beyond the Kurtz’s theorem, Maxwell type construction. Metastable state has probability e-aV, and exit rate e-bV.

• There is no bistability after all! The steady state is a monotonic function of a parameter, though with discountinuity.

• Lee-Yang’s mechanism is still valid.• Landscape is an emergent property!

Now Some Biological Implications:

for systems not too big, not too small, like a cell …

Emergent Mesoscopic Complexity• It is generally believed that when systems become

large, stochasticity disappears and a deterministic dynamics rules.

• However, this simple example clearly shows that beyond the “infinite-time” in the deterministic dynamics, there is another, emerging stochastic, multi-state dynamics!

• 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.

In a cartoon: Three time scales

ny

nx

appropriate reaction coordinate

ABpr

obab

ility

A B

chemical master equation

discrete stochastic model among attractors

emergent slow stochastic dynamics and landscape

cy

cx

A

B

fast nonlinear differential equationsmolecular s

ignaing t.s.

biochemica

l netw

ork t.s

.

cellular evolution t.s.

Ch

oi, P

.J.; Ca

i, L.; F

rieda

, K. an

d X

ie, X

.S.

Scie

nce

, 322

, 44

2- 4

46 (2

008

). Bistability in E. coli lac operon

switching

Bistability during the apoptosis of human brain tumor cell (medulloblatoma) induced by topoisomerase II inhibitor (etoposide)

Buckmaster, R., Asphahani, F., Thein, M., Xu, J. and Zhang, M.-Q.Analyst, 134, 1440-1446 (2009)

Bistability in DNA damage-induced apoptosis of human osteosarcoma

(U2OS) cells

L. Xu, Y. Chen, Q. Song, Y. Wang and D. Ma, Neoplasia, 11, 345-354 (2009)

Tip6: histone acetyltransferase; PDCD5: programed cell death 5 protein

12 hours irradiation

Chemical basis of epi-genetics:

Exactly same environment setting and gene, different internal

biochemical states (i.e., concentrations and fluxes). Could

this be a chemical definition for epi-genetics inheritance?

The inheritability is straight forward: Note that (x) is independent of volume of the cell, and x is the

concentration!steady state chemical concentration distribution

concentration of regulatory molecules

c1* c2*2

c1*

2

c2*

Could it be? Epigenetics is a kind of nonequilibrium phase transition?

Thank you!