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Stochastic Nonlinear Dynamics of Cellular Biochemical
Systems:
Hong Qian
Department of Applied Mathematics
University of Washington
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abstractI present the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under
a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state
with concentration fluctuations. We discuss nonlinear biochemical reaction systems such as phosphorylation-dephosphorylation cycle (PdPC) with
bistability. Emphasis is paid to the comparison between thestochastic dynamics and the prediction based on the traditional approach
based on the Law of Mass Action. We introduce the dirence between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart.
For systems with nonlinear bistability, there are three dirent time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution. For mesoscopic systems with size of a
living cell, dynamics in (a) and (c) are stochastic while that with (b) is dominantly deterministic. Both (b) and (c) are emergent properties of a
dynamic biochemical network; We suggest that the (c) is most relevant to major cellular biochemical processes such as epigenetic regulation, apoptosis,
and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b) in a "punctuated equilibrium" manner.
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An analytical theory for Darwin’s variations in mesoscopic scale?
Intrinsic variations = Stochasticity
Natural environmental selections = Bias
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Here are some recent headlines:
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Physics
Molecular Cellular Systems
EvolutionaryBiology
Chemistry
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Stochastic Physical 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 system,
assuming that the rate constants are known for each and every reaction in the
system.
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Basic Facts on Single Molecule Stochastic Transition
Time is in the waiting, the transition is instaneous!
A Bk1
k2
A
B GΔek ‡
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The Biochemical System Inside CellsE
GF
Signal T
ransduction Pathw
ay
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The kinetic isomorphism between PdPC and GTPase
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PdPC with a Positive Feedback
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Gene regulatory and
Biochemical signaling Networks
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(B)
gene state 0 gene state 1f
ho[TF]
synthesis
degradationTF
k
g0 g1
(A)E E*
K + E* K†
k3[P]
k[K ] †
k2
k-2
k[K]
Another Kinetic Isophorphism
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According to macroscopic chemical kinetics following the
Law of Mass Action
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Simple Kinetic Model based on the Law of Mass Action
NTP NDP
Pi
E
P
R R*
.
].][[
],)[]][[(
,][
*
*
*
θβ
α
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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According to mesoscopic chemical kinetics following the
Chemical Master Equation
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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.T.L. Hill & I.W. Plesner (1965) J. Chem. Phys. 43, 267; (1971) 54, 34.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. G. Nicolis & A. Babloyantz (1969) J. Chem. Phys. 51, 2632.R. Hawkins & S.A. Rice (1971) J. Theoret. Biol. 30, 579.T.G. Kurtz (1971) J. App. Prob. 8, 344; (1972) J. Chem. Phys. 57, 2976.J. Keizer (1972) J. Stat. Phys. 6, 67.D. Gillespie (1976) J. Comp. Phys. 22, 403; (1977) J. Phys. Chem. 81, 2340.
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and more recently …
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1-dimensional, 1-stable, 1-unstable fixed pts1-dimensional, 2-stable, 1-unstable fixed pts2-dimensional, 1-stable limit cycle via Hopf bifurcation
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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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Bistability and Emergent Sates
Pk
number of R* molecules: k
defining cellular attra
ctors
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Landscape and Lyapunov Property
kP )(xf )(ln1
)( xfV
x
)(x
V
kx
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*
(x,)
Extrema value
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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 critical point
*(e)(B)
4
6
8
10
0.01 0.1 1
xss
(A)
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A fundamental difference of two types of landscapes
• For a detailed balance system, such as protein folding dynamics, the energy landscape is given a priori. It directs the dynamics of the system.
• For a system without detailed balance, can be considered as an “landscape for dynamics”. However, it is a consequence of the dynamics. That is why we call it emergent. It dynamics is non-local.
• Q: which one is the “fitness landscape”?
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Biological Implications:
for 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
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.
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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
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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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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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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*
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0
25
50
75
100
0 50 100 150 200
time
nu
mb
er o
f E
*
0
40
80
120
160
0 1 2 3 4 5
0.001
0.01
0.1
1
10
100
0 0.2 0.4 0.6 0.8 1
0
50
100
150
200
0 2500 5000 7500 10000
concentration of E *
V=100
V=200
(A)
(B) (C)
(D)
Ntot=100
Ntot=200
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1000 2000500 15000
total number of molecule E
swit
chin
g t
ime
in m
sec 1030
1015
3x1022
1.0
3x107
10 hrs
1011 yrs
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Another insight
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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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The emergent cellular, stochastic “evolutionary” dynamics follows not
gradual changes, but rather punctuated transitions between
cellular attractors.
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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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An emerging “thermo”-dynamic structure in stochastic dynamics
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The Thermodynamic Structure of Stochastic Systems
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Two Origins of Irreversibility
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Summary for Systems Biol.(1) As a physical chemistry approach to
cellular biochemical dynamics, mesoscopic reaction systems can be
modeled according to the CME: A new mathematical theory.
(2) A possible chemical bases of epi-genetic inheritance is proposed;
(3) Emerging landscape is introduced;(4) Beyond deterministic physics, there is stochastic diversity in evolutionary time!
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Summary for Theoret. Physics (5) Nonlinear multi-attractors become
stochastic attractors. Infinite large systems exhibit nonequilibrium phase
transition with Maxwell construction and Lee-Yang theory;
(6) A nonequilibrium statistical “thermo- dynamics” emerges from stochastic
nonlinear dynamics; (7) Epigenetic switching is a form of
nonequilibrium phase transition?
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Thank You!