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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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Ghaemmaghami, S. et. al. (2003) “Global analysis of protein expression in
yeast”, Nature, 425, 737-741.
Protein Copy Numbers in Yeast
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Roessner-Tunali et. al. (2003) “Metabolic profiling of transgenic tomato plants …”, Plant Physiology, 133, 84-99.
Metabolites Levels in Tomato
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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)
• Delbrück’s 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)
(FCS)
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Fast Forward to 1998
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Stochastic Enzyme Kinetics
0.2mM
2mM
Lu, P.H., Xun, L.-Y. & Xie, X.S. (1998) Science, 282, 1877-1882.
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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
PxF
xx
PD
tP
)(
2
2
xxE
xF )(
)(
A Bk2
k1
),(),(),(
21 tBPktAPkdt
tAdP
A B
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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
A Xk1
k-1
B Yk2
2X+Y 3Xk3
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The Law of Mass Action and Differential Equations
dtd cx(t) = k1cA - k-1 cx+k3cx
2cy
k2cB - k3cx2cy=dt
d cy(t)
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u u
a = 0.1, b = 0.1 a = 0.08, b = 0.1
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:
X+Y Zk1
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 t, any one particular unbonded X will react
with any one particular unbonded Y with probability k1t + o(t), where k1 is the
reaction rate.
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A Markovian Chemical Birth-Death Process
nZ
k1nxnyk1(nx+1)(ny+1)
k-1nZ k-1(nZ +1)
k1
X+Y Zk-1
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Chemical Master Equation Formalism for Chemical
Reaction SystemsM. 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
A Xk1
k-1
B Yk2
2X+Y 3Xk3
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(0,0)
(0,1)
(0,2)
(1,0)
(1,1)
(2,0)
(1,2)
(3,0)
(2,1)
k1nA k1nA
k1nA
k1nA
k1nA
k2 nB
k2 nB
k2 nB
k2 nB
k2 nB
2k3
k-1 2k-1 3k-1 4k-1
k-1(n+1)
(n,m)(n-1,m) (n+1,m)
(n,m+1) (n+1,m+1)
k1nAk1nA
(n,m-1)
k2 nB
k2 nB
(n-1,m+1)
k3 n (n-1)m
k3 (n-1)n(m+1)k3 (n-2)(n-1)(m+1)
k-1mk-1(m+1)
k2 nB k2 nBk2 nB
(n+1,m-1)k1nA
k3 (n-2)(n-1)n
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Stochastic Markovian Stepping Algorithm (Monte Carlo)
=q1+q2+q3+q4 = k1nA+ k-1n+ k2nB+ k3n(n-1)m
Next time T and state j? (T > 0, 1< j < 4)
q3q1
q4
q2
(n,m)(n-1,m) (n+1,m)k1nA
(n,m-1)
k2 nB
k3 n (n-1)mk-1n
(n+1,m-1)
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Picking Two Random Variables T & n derived from uniform r1 & r2 :
fT(t) = e - t, T = - (1/) ln (r1)
Pn(m) = km/, (m=1,2,…,4)
r2
0 p1 p1+p2 p1+p2+p3p1+p2+p3+p4=1
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Concentration Fluctuations
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Stochastic Oscillations: Rotational Random Walks
a = 0.1, b = 0.1 a = 0.08, b = 0.1
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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
)1,1()1)(2)(1(
),1()1(
)1,(),1(
),()1(),,(
3
1
21
3211
YXYXX
YXX
YXBYXA
YXYXXBXAYX
nnPnnnk
nnPnk
nnPnknnPnk
nnPnnnknknknkdt
tnndP
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Stochastic Dynamics: Thermal Fluctuations vs. Temporal Complexity
FPPDt
tvuP
),,(
vubvuvuvuuaD 22
22
2
vubvuua
F2
2
Stochastic Deterministic, Temporal Complexity
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Time
Num
ber
of m
olec
ules
(A)
(C)
(D)
(B) (E)
(F)
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-dephosphorylation
cycle, or a GTPase cycle:
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A A*
S
ATP ADP
I
Pi
k1
k-1
k2
k-2
Ferrell &
Xiong, C
haos, 11, pp. 227-236 (2001)with a positive feedback
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Two ExamplesF
rom
Coo
per
and
Qia
n (2
008)
Bio
chem
., 4
7, 5
681.
From
Zhu, Q
ian and Li (2009) PLoS
ON
E. S
ubmitted
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Simple Kinetic Model based on the Law of Mass Action
NTP NDP
Pi
E
P
R R*
].][[
],)[]][[(
,][
*
*
*
RPβJ
RREαJ
JJdt
Rd
2
χ1
21
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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
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R R*
K
P
2R*0R* 1R* 3R* … (N-1)R* NR*
Markov Chain Representation
v1
w1
v2
w2
v0
w0
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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
,
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Large V Asymptotics
)(exp
)(
)(logexp
logexp
xφV
xw
xvdxV
w
v
w
v
11
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Beautiful, or Ugly Formulae
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Bistability and Emergent Sates
Pk
number of R* molecules: k
defining cellular attra
ctors
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A Theorem of T. Kurtz (1971)In the limit of V →∞, 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.
.)(lim
;)()(supPrlim
0V1
V
V1
tsV
x0XV
0εsxsXV
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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.
Newton’s Law of Motion The Schrödinger’s Eqn.ħ → 0
The Law of Mass Action The Chemical Master Eqn.V →
x1(t), x2(t), …, xn(t)
c1(t), c2(t), …, cn(t)
(x1,x2, …, xn,t)
p(N1,N2, …, Nn,t)
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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?
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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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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)
In a cartoon fashion
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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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alternative attractor
localattractor
Relaxation process
abrupt transition
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 Future:Toward 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 (Newton’s 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!