Cellular Dynamics From A Computational Chemistry Perspective

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Cellular Dynamics From A Computational Chemistry
Perspective

• Hong Qian
• Department of Applied Mathematics
• University of Washington

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The most important lesson learned from protein
science is
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The current state of affair of cell biology(1)
Genomics A,T,G,C symbols(2) Biochemistry
molecules
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Experimental molecular genetics defines the
state(s) of a cell by their transcription
pattern via expression level (i.e., RNA
microarray).
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Biochemistry defines the state(s) of a cell via
concentrations of metabolites and copy numbers of
proteins.
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Protein Copy Numbers in Yeast
Ghaemmaghami, S. et. al. (2003) Global analysis
of protein expression in yeast, Nature, 425,
737-741.
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Metabolites Levels in Tomato
Roessner-Tunali et. al. (2003) Metabolic
profiling of transgenic tomato plants , Plant
Physiology, 133, 84-99.
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But biologists define the state(s) of a cell by
its phenotype(s)!
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How does computational biology define the
biological phenotype(s) of a cell in terms of the
biochemical copy numbers of proteins?
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Theoretical Basis The Chemical Master
Equations A New Mathematical Foundation of
Chemical and Biochemical Reaction Systems
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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)
• Delbrücks theory of stochastic chemical reaction
  in terms of birth-death process (1940)

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Single Channel Conductance
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First Concentration Fluctuation Measurements
(1972)
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Fast Forward to 1998
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Stochastic Enzyme Kinetics
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Stochastic Chemistry (1940)
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The Kramers theory and the CME clearly marked
the domains of two areas of chemical research
(1) The computation of the rate constant of a
chemical reaction based on the molecular
structures, energy landscapes, and the solvent
environment and (2) the prediction of the
dynamic behavior of a chemical reaction network,
assuming that the rate constants are known for
each and every reaction in the system.
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Kramers Theory, Markov Process Chemical
Reaction Rate
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But cellular biology has more to do with reaction
systems and networks
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Traditional theory for chemical reaction systems
is based on the law of mass-action
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Nonlinear Biochemical Reaction Systems and
Kinetic Models
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The Law of Mass Action and Differential Equations
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Nonlinear Chemical Oscillations
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A New Mathematical Theory of Chemical and
Biochemical Reaction Systems based on Birth-Death
Processes that Include Concentration Fluctuations
and Applicable to small systems.
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The Basic Markovian Assumption
The chemical reaction contain nX molecules of
type X and nY molecules of type Y. X and Y bond
to form Z. In a small time interval of Dt, any
one particular unbonded X will react with any one
particular unbonded Y with probability k1Dt
o(Dt), where k1 is the reaction rate.
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A Markovian Chemical Birth-Death Process
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Chemical Master Equation Formalism for Chemical
Reaction Systems
M. Delbrück (1940) J. Chem. Phys. 8, 120. D.A.
McQuarrie (1963) J. Chem. Phys. 38, 433. D.A.
McQuarrie, Jachimowski, C.J. M.E. Russell
(1964) Biochem. 3, 1732. I.G. Darvey P.J.
Staff (1966) J. Chem. Phys. 44, 990 45, 2145
46, 2209. D.A. McQuarrie (1967) J. Appl.
Prob. 4, 413. R. Hawkins S.A. Rice (1971) J.
Theoret. Biol. 30, 579. D. Gillespie (1976) J.
Comp. Phys. 22, 403 (1977) J. Phys. Chem. 81,
2340.
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Nonlinear Biochemical Reaction Systems
Stochastic Version
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Stochastic Markovian Stepping Algorithm (Monte
Carlo)
l q1q2q3q4 k1nA k-1n k2nB k3n(n-1)m
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Picking Two Random Variables T n derived from
uniform r1 r2
fT(t) l e -l t, T - (1/l) ln (r1)
Pn(m) km/l , (m1,2,,4)
r2
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Concentration Fluctuations
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Stochastic Oscillations Rotational Random Walks
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Defining Biochemical Noise
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An analogy to an electronic circuit in a radio
If one uses a voltage meter to measure a node in
the circuit, one would obtain a time varying
voltage. Should this time-varying behavior be
considered noise, or signal? If one is lucky and
finds the signal being correlated with the audio
broadcasting, one would conclude that the time
varying voltage is in fact the signal, not noise.
But what if there is no apparent correlation
with the audio sound?
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Continuous Diffusion Approximation of Discrete
Random Walk Model
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Stochastic Dynamics Thermal Fluctuations vs.
Temporal Complexity
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Temporal dynamics should not be treated as noise!
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A Second Example Simple Nonlinear Biochemical
Reaction System From Cell Signaling
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We consider a simple phosphorylation-dephosphoryla
tion cycle, or a GTPase cycle
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with a positive feedback
Ferrell Xiong, Chaos, 11, pp. 227-236 (2001)
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Two Examples
From Cooper and Qian (2008) Biochem., 47, 5681.
From Zhu, Qian and Li (2009) PLoS ONE. Submitted
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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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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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Beautiful, or Ugly Formulae
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Bistability and Emergent Sates
Pk
number of R molecules k
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A Theorem of T. Kurtz (1971)

• In the limit of V ?8, the stochastic solution to
  CME in volume V with initial condition XV(0),
  XV(t), approaches to x(t), the deterministic
  solution of the differential equations, based on
  the law of mass action, with initial condition x0.

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We Prove a Theorem on the CME for Closed Chemical
Reaction Systems

• We define closed chemical reaction systems via
  the chemical detailed balance. In its steady
  state, all fluxes are zero.
• For ODE with the law of mass action, it has a
  unique, globally attractive steady-state the
  equilibrium state.
• For the CME, it has a multi-Poisson distribution
  subject to all the conservation relations.

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Therefore, the stochastic CME model has
superseded the deterministic law of mass action
model. It is not an alternative It is a more
general theory.
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The Theoretical Foundations of Chemical Dynamics
and Mechanical Motion
The Semiclassical Theory.
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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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Chemistry is inheritable!
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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 fashion
(a)
(b)
(c)
(d)
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The mathematical analysis suggests three
distinct time scales, and related mathematical
descriptions, of (i) molecular signaling, (ii)
biochemical network dynamics, and (iii) cellular
evolution. The (i) and (iii) are stochastic
while (ii) is deterministic.
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The emergent cellular, stochastic evolutionary
dynamics follows not gradual changes, but rather
punctuated transitions between cellular
attractors.
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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 tme
• With sufficiently long waiting, exit to a next
  cellular state.

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Relaxation, Wating, Barrier Crossing R-W-BC of
Stochastic Dynamics
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• Elimination
• Equilibrium
• Escape

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In Summary

• There are two purposes of this talk
• On the technical side, a suggestion on
  computational cell biology, and proposing the
  idea of three time scales
• On the philosophical side, some implications to
  epi-genetics, cancer biology and evolutionary
  biology.

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Into the FutureToward a Computational
Elucidation of Cellular attractor(s) and
inheritable epi-genetic phenotype(s)
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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.

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Analogue with Computational Protein Structures
40 yr ago

• While the equation is known in principle
  (Newtons equation), the large amount of unknown
  parameters (force field) makes a realistic
  computation essentially impossible.
• It has taken 40 years of continuous development
  to gradually converge to an acceptable set of
  parameters
• But the issues are remarkably similar defining
  biological (conformational) states, extracting
  the kinetics between them, and ultimately,
  functions.

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Thank You!