Abstract

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Abstract

• 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.

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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

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Background

• Newton-Laplaces world view is deterministic
• Boltzmann tried to derive the stochastic dynamics
  from the Newtonian view
• Darwins 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.

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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

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Here are some recent headlines
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The Biochemical System Inside Cells
EGF Signal Transduction Pathway
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Introducing the amplitude of a switch (AOS)
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Amplitude of the switch as afunction of the
intracellular phosphorylation potential
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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).
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The kinetic isomorphism between PdPC and GTPase
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PdPC with a Positive Feedback
From Zhu, Qian and Li (2009) PLoS ONE. Submitted
From Cooper and Qian (2008) Biochem., 47, 5681.
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Simple Kinetic Model based on the Law of Mass
Action
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Bifurcations in PdPC with Linear and Nonlinear
Feedback
c 0
c 1
c 2
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Biochemical reaction systems inside a small
volume like a cell dynamics based on Delbrücks
chemical master equation (CME), whose stochastic
trajectory is defined by the Gillespie algorithm.
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A Markovian Chemical Birth-Death Process
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Markov Chain Representation
v1
v2
v0
w1
w2
w0
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Steady State Distribution for Number Fluctuations
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Large V Asymptotics
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Relations between dynamics from the CME and the
LMA

• Stochastic trajectory approaches to the
  deterministic one, with probability 1 when V?8,
  for finite time, i.e., t ltT .
• Lyapunov properties of f (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)

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Keizers Paradox bistability, multiple time
scale, exponential small transitions, non-uniform
convergence
f (x)
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Using the PdPC with positive feedback system to
learn more
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Simple Kinetic Model based on the Law of Mass
Action
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Beautiful, or Ugly Formulae
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(B)
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Landscape and limit cycle
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Further insights on Landscape with limit cycle
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When there is a rotation
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Large deviation theory or WBK
approaching a limit cycle, constant on the limit
cycle
on the limit cycle, inversely proportional to
angular velocity
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Our findings on this type of non-equilibrium
phase transition

• In the infinite volume limit of bistable chemical
  reaction system
• Beyond the Kurtzs 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-Yangs mechanism is still valid.
• Landscape is an emergent property!

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Now Some Biological Implicationsfor systems
not too big, not too small, like a cell
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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.

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In a cartoon Three time scales
chemical master equation
discrete stochastic model among attractors
emergent slow stochastic dynamics and landscape
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Bistability in E. coli lac operon switching
Choi, P.J. Cai, L. Frieda, K. and Xie, X.S.
Science, 322, 442- 446 (2008).
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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)
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Bistability in DNA damage-induced apoptosis of
human osteosarcoma (U2OS) cells
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Chemical basis of epi-geneticsExactly same
environment setting and gene, different internal
biochemical states (i.e., concentrations and
fluxes). Could this be a chemical definition for
epi-genetics inheritance?
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The inheritability is straight forward Note that
f (x) is independent of volume of the cell, and x
is the concentration!
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Could it be? Epigenetics is a kind of
nonequilibrium phase transition?
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Thank you!