We 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
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 Cells EGF Signal Transduction Pathway
k 1 [ A* ] k -1 [ A* ] k 2 k -2 B B * A A * Biologically active forms of signaling molecules
Introducing the amplitude of a switch (AOS):
Amplitude of the switch as a function 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 phosphorylated activation signal phosphorylated activation signal phosphorylated
PdPC with a Positive Feedback From Cooper and Qian (2008) Biochem ., 47 , 5681. From Zhu, Qian and Li (2009) PLoS ONE. Submitted
Simple Kinetic Model based on the Law of Mass Action NTP NDP Pi E P R R*
Bifurcations in PdPC with Linear and Nonlinear Feedback = 0 = 1 = 2 activating signal: activation level: f 1 4 1 hyperbolic delayed onset bistability
Biochemical reaction systems inside a small volume like a cell: dynamics based on Delbr ü ck’s chemical master equation (CME), whose stochastic trajectory is defined by the Gillespie algorithm.
A Markovian Chemical Birth-Death Process k 1 X+Y Z k -1 n Z k -1 n Z k -1 ( n Z +1 ) n x ,n y k 1 (n x +1 )( n y +1 ) V 2 k 1 n x n y V 2 V V
Markov Chain Representation v 1 w 1 v 2 w 2 v 0 w 0 R R* K P 2R* 0R* 1R* 3R* … (N-1)R* NR*
Steady State Distribution for Number Fluctuations
Large V Asymptotics
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 (H ä nggi, Keizer, etc )
Keizer’s Paradox : bistability, multiple time scale, exponential small transitions, non-uniform convergence ( x ) t -> , V -> V -> , t -> stationary solution to Fokker-Planck Equation =
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*
Beautiful, or Ugly Formulae = 0 = 0
0 nondifferential point of c ( ) * ( x, ) Extrema value (A) (B) dc ( 0 ) d (C) dc ( 0 ) d = x ss (D) *
(B) 1 ( ) 2 ( ) the cusp the tri-critical point *( ) x ss (A)
“ Landscape” and limit cycle
Further insights on “Landscape” with limit cycle
When there is a rotation
Large deviation theory or WBK 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 chemical master equation discrete stochastic model among attractors emergent slow stochastic dynamics and landscape n y n x appropriate reaction coordinate A B probability A B c y c x A B fast nonlinear differential equations molecular signaing t.s. biochemical network t.s. cellular evolution t.s.
Bistability in E. coli lac operon switching Choi, P.J.; Cai, L.; Frieda, K. and Xie, X.S. Science , 322 , 442- 446 (2008).
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 c 1 * c 2 * 2 c 1 * 2 c 2 *
Could it be? Epigenetics is a kind of nonequilibrium phase transition?