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AbstractWe 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
Background• 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 CellsE
GF
Signal T
ransduction Pathw
ay
k1 [A*]
k-1[A*]
k2
k-2
B B*
A A*
Biologically active forms of signaling molecules
Introducing the amplitude of a switch (AOS):
[ *] [ *] 0
1 2
1 1 2 2 1 1 2 2
[ *] [ *]
[ ] [ *] [ ] [ *]
1
/ 1 / 1
1tanh
42 1
A A
B BAOS
B B B B
k k
k k k k k k k k
G
RT
Amplitude of the switch as afunction 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
phos
phor
ylat
ed
activation signal
phos
phor
ylat
ed
activation signal
phos
phor
ylat
ed
PdPC with a Positive FeedbackF
rom
Co
ope
r an
d Q
ian
(20
08)
Bio
chem
., 4
7, 5
68
1.F
rom
Zhu
, Qia
n an
d L
i (20
09) P
LoS
ON
E. S
ubm
itted
Simple Kinetic Model based on the Law of Mass Action
NTP NDP
Pi
E
P
R R*
].][[
],)[]][[(
,][
*
*
*
RPβJ
RREαJ
JJdt
Rd
2
χ1
21
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
Biochemical reaction systems inside a small volume like a cell: dynamics based on Delbrück’s
chemical master equation (CME), whose stochastic trajectory is
defined by the Gillespie algorithm.
A Markovian Chemical Birth-Death Process
k1
X+Y Zk-1
nZ
k-1nZ k-1(nZ +1)
nx,ny
k1(nx+1)(ny+1)V2
k1nxny
V2
VV
R R*
K
P
2R*0R* 1R* 3R* … (N-1)R* NR*
Markov Chain Representation
v1
w1
v2
w2
v0
w0
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
,
Large V Asymptotics
)(exp
)(
)(logexp
logexp
xφV
xw
xvdxV
w
v
w
v
11
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 (Hänggi, Keizer, etc)
Keizer’s Paradox: bistability, multiple time scale, exponential small transitions,
non-uniform convergence
t → , V → V→ , t →
stationary solution toFokker-Planck Equation(x)
=
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*
].][[
],)[]][[(
,][
*
*
*
RPβJ
RREαJ
JJdt
Rd
2
χ1
21
Beautiful, or Ugly Formulae
= 0
= 0
0
nondifferential point of c()
*
(x,)
Extrema value
(A)
(B)
dc(0)d
(C)
dc(0)d = xss
(D)
*
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 tri-critical point
*(e)(B)
4
6
8
10
0.01 0.1 1
xss
(A)
“Landscape” and limit cycle
θd
θdUθF
dt
θd )()(
θaθF sin)(
Further insights on “Landscape” with limit cycle
When there is a rotation
Large deviation theory or WBK
...)()(exp)( xφxφ
V
1xp 10
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
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.
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
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
c1* c2*2
c1*
2
c2*
Could it be? Epigenetics is a kind of nonequilibrium phase transition?
Thank you!