Nonequilibrium stochastic Processes at Single-molecule and Single-cell levels Hao Ge (葛颢) [email protected]1 Beijing International Center for Mathematical Research 2 Biodynamic Optical Imaging Center Peking University, China http://bicmr.pku.edu.cn/~gehao/
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Nonequilibriumstochastic Processes at Single-molecule and Single-cell levels
Which kind of physical/chemical processes can be described by stochastic processes?
• Mesoscopic scale (time and space)
• Single-molecule and single-cell (subcellular) dynamics
• Trajectory perspective
Single-molecule experiments
Lu, et al. Science (1998)
Single-molecule enzyme kinetics
E. Neher and B. SakmanNobel Prize in 1991
Single Ion channel
Single-cell dynamics (in vivo)
Eldar, A. and Elowitz, M. Nature (2010)
Choi, et al. Science (2008)
Stochastic theory of nonequilibrium statistical
mechanics
The fundamental equation in nonequilibrium thermodynamics
Ilya Prigogine(1917-2003)Nobel Prize
in 1977
Entropy production
SdSddS ie
0 k
kki XJSdepr
Clausius inequality
T
QdS
Rudolf Clausius(1822-1888)
Second law of thermodynamics
0T
QdSepr
rewrite
More general
Carl Eckart(1902-1973)
P.W. Bridgman(1882-1961)Nobel Prize
in 1946
Lars Onsager(1903-1976)Nobel Prize
in 1968
Mathematical theory of nonequilibrium steady state
Min Qian (1927-)Recipient of Hua Loo-KengMathematics Prize (华罗庚数学奖) in 2013
Time-independent(stationary) Markov process
Ge, H.: Stochastic Theory of Nonequilibrium Statistical Physics (review). Advances in Mathematics(China) 43, 161-174 (2014)
Master equation model for the single-molecule system
No matter starting from any initial distribution, it will finally approach its stationary distribution satisfying
01
N
j
ij
ss
iji
ss
j kpkp
j
ijijiji kpkp
dt
tdp )(
Consider a motor protein with N different conformations R1,R2,…,RN. kij is the first-order or pseudo-first-order rate constants for the reaction Ri→Rj.
Self-assembly or self-organization
ij
eq
iji
eq
j kpkp
Detailed balance (equilibrium state)
NESS thermodynamic force and entropy production rate
0 ji
ss
ij
ss
ij
ness AJeprT
ji
ss
jij
ss
i
ss
ij kpkpJ
ji
ss
j
ij
ss
i
B
ss
ijkp
kplogTkA NESS thermodynamic force
NESS flux
NESS entropy production rate
Time-dependent case
j
ijijiji tkptkp
dt
tdp
Free energy
j
kTtE
BjeTktF
/)(log)(
j
kTtE
kTtEeq
i
ss
ij
i
e
etptp
/)(
/)(
)()(Boltzmann’s law
If {kij(t)} satisfys the detailed balance condition for fixing t
01
N
j
ij
ss
iji
ss
j tktptktpQuasi-stationary distribution
0 tktptktp ij
ss
iji
ss
j
Mathematical equivalence of Jarzynski and Hatano-Sasa equalities
Ge, H. and Qian, M., JMP (2007); Ge, H. and Jiang, D.Q., JSP (2008);
Jarzynski equality: local equilibrium
kTF
pp
kTW eeeq
/
)0()0(
/
.)0()0(
FW eqpp
Hatano-Sasa equality: without local equilibrium
1
)0()0(
/
ss
ex
pp
kTQe
.kT/Q
S
ex
pp ss
00
Same theorem for time-dependent Markov process
Are these inequalities already known in the Second Law of classic thermodynamics? Do they only hold for the whole transition process between two steady states?
The traditional Clausius inequality can be in a differential form.
Using Feynman-Kac formula of the time-dependent case
Decomposition of mesoscopic thermodynamic forces
tAtA
tktp
tktpTktA ij
ss
ij
jij
iji
Bij log
i j
0 ji
ijijp tAtJteT Entropy production
0 ji
ss
ijijhk tAtJtQ )()()(Housekeeping heat
Free energy dissipation 0 ji
ijijd tAtJtf )()()(
t t t dhkp fQeT Ge, H., PRE (2009);
Ge, H. and Qian, H., PRE (2010) (2013)
tktp
tktpTktA
ji
ss
j
ij
ss
i
B
ss
ij log
All the results have also been proved for multidimensional diffusion process.
0tep for any time tIn the absence of external energy input and at steady state.
0tQhk for any time t In the absence of external energy input
0tfd for any time t At steady state
Two origins of nonequilibrium
Ge, H., PRE (2009); Ge, H. and Qian, H., PRE (2010) (2013)
0 ptot e
T
Q
dt
dS
The new Clausius inequality is stronger than the traditional one.
0
d
hktotex fT
QQ
T
Q
dt
dS
Decomposition of entropy production rate
.QfeT
,Q,f
hkdp
hkd
0
00
Ge, H., PRE (2009);
Ge, H. and Qian, H., PRE (2010) (2013)
T
Qe
dt
dS totp
Mathematical problems left
Mathematical proof for some newly recent
developed finite-time fluctuation theorems of
diffusion process with time-dependent diffusion
coefficients;
Mathematical proof for the large deviation
principle of sample entropy production of
diffusion process on Rn;
Stochastic theory of nonequilibrium statistical
mechanics of second-order stochastic process;
……
A first step towards the stochastic theory of nonequilibrium statistical
mechanics for second-order stochastic system:
Time-reversibility and anomalous behavior
tX,XFdt
Xdm
.
2
2
Two different definitions of entropy production rates
dxdvlogm
xDFF
xDk
Ts:rP
Ts:Plog
TlimkEPR
ttvB
PsTts
st
TB
2
01
22
2
0
01
v,xFF
Spinney, R.E., and Ford, I.J., PRL (2012); Lee, H.K., Kwon, C., and Park, H., PRL (2013)
dxdvlog
m
xDvx
xDkEPR ttvB
2
22
2
Kim, K.H. and Qian, H., PRL (2004)
correspond to time-reversibility and Maxwell-Boltzmann distribution for thermodynamic equilibrium respectively
00 21 EPR,EPR
thermodynamic equilibrium Ge, H.: PRE (2014)
When the external forceis only dependent on position
xGvxv,xF
xxTk
xD
B
2
021 EPREPR
thermodynamic equilibrium
xUxG
;TxT
x
Thermal equilibrium
Mechanical equilibrium
Flow of kinetic energy along the spatial coordinate
t,xQt,xWJxEdt
d kinetic
xx
kinetic
t
00 qx JJEPR
x
kinetic
t
kinetic
xq vxEJxJ (measurable) Heat flux
Ge, H.: PRE (2014)
Thermodynamic equivalence between mesoscopic and macroscopic scales
The entropy production rate in the small-noise limit
Celani, et al.: PRL (2012);
Ge, H.: PRE (2014)
dxˆ
ˆ
J
xT
xEPREPR t
t
over
xoverspatial
2
dxxˆxTx
xTk
nt
xB
over
22
6
2
Entropy production rate of the overdamped-limit
Anomalous contribution of EPR
Hence the overdamped approximation only keeps the dynamics rather than the second law of thermodynamics.
spatialEPREPR
Decomposition of entropy production rate
Local reciprocal relation between linear coefficients
Ge, H.: JSTAT (2015)
qqqpqxq
qxqpxx
over
x
XLXLxJ
XLXLJ
xˆx
xTknL t
Bqq
32
6
8
xˆx
xTL txx
x
xTkLL B
xqqx
2
Always hold, even for far-from-equilibrium system.
Reciprocal relation between Soret effect (thermal diffusion) and Dufour effect
Come from the second moment of velocity along each dimension.
Two-state model of central dogma without feedback
Help to uncover the mechanism of transcriptional burst
DNA transcription
Regulation of gene expression
Induced condition
Highly expressed,
i.e. low repression
Transcriptional burst under induced condition
Golding et al. Cell (2005)
DNA topology and transcriptional burst
Levens and Larson: Cell (2014) (preview) Shasha et al. Cell (2014)
Anchored DNA segment
High-throughput in-vitro experiments
Supercoiling accumulation and gyrase activity
Gene OFF mRNA øGene ON
1k
Shasha et al. Cell (2014)
Two-state model without feedback
0 1 n…… n+1 ……
0 1 n…… n+1 ……
Gene ON
Gene OFF
1k 1k 1k
2
2
)1( n
)1( n
Gene OFF mRNA øGene ON
1k
Chemical master equation
Copy number of mRNA
Ge
ne
sta
tes
The mean-field deterministic model has only one stable fixed point!
Poisson distribution with a spike at zero
When α,β<<k1,γ, then
.1,!
)()()(
;)0()0()0(
1
21
21
1
1
nn
k
enpnpnp
eppp
n
k
k
Poisson distribution with a spike (bimodal)
Duty cycle ratio
Two-state model of central dogma with positive feedback
A rate formula for stochastic phenotype transition—in an intermediate region of gene-state switching
Central Dogma
Copy numbers in a single cell
Bacteria Eukaryotic cells
DNA 1 or 2 ~2
mRNA A few 1 - 103
Protein 1 - 104 1 - 106
Not enough attention has been paid to this fact.
Regulation of gene expression
An example of gene circuit with positive feedback:
Lac operon
Bimodal distributions in biology: multiple phenotypic states
Choi, et al., Science (2008) To, T. and Maheshri, N. Science (2010)
Ferrell, J. and Machleder, E. Science (1998)
Two-state model with positive feedback
1k
n max
large
Mean-field deterministic model with positive feedback
0 5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Bifurcation diagram for simple example
1/Keq
x*
OFF state
ON state
x
flux
Influx g(x)
Outflux γx
Stable On-state
Stable Off-state
Unstable threshold
Flux-balance plot
Bifurcation diagram
xxgdt
dx )(
Sigmoidal influx
x xg
maxn
nx
**)( xxg
Interconversion of different phenotypic states
Choi, et al., Science (2008) Gupta, et al., Cell (2011)
How to quantify the transition rates between different phenotypic states, provided their existence?
Recall Langevin dynamics and Kramers rate formula
Chemical reaction activated by diffusional fluctuations
H.A. Kramers (1894-1952)
tf
dt
dx
dx
xdU
dt
xdm
2
2
P. Langevin (1872-1946)
.,0
;2
;0
2
tssftf
Tktf
tf
B
Tk
U
a Be2
large is k
ǂǂ
x. around x,xx m2
1xU
;x around x,xxm2
1xU
22
A
2
a
2
a
ǂǂǂ
𝑞‡ = 1 𝑞A =2𝜋𝑘𝐵𝑇
𝜔𝑎ℎ
𝑘+ = 𝜅𝑘𝐵𝑇
ℎ𝑒−Δ𝐺‡
𝑘𝐵𝑇
= κ𝑘𝐵𝑇
ℎ
𝑞‡
𝑞A𝑒−Δ𝑈‡
𝑘𝐵𝑇
𝜅 =𝜔‡
𝛾
𝛾 =𝜂
𝑚
From single chemical reaction to biochemical networks (biology)
Chemical master equation (CME)
Max Delbruck(1906-1981)Nobel Prize in 1969
Physical state of atoms
Molecular copy number Phenotypic state
Conformational state
Single cell: biochemical network
Single chemical reaction
The state of system
Emergent state at a higher level
M
j
j
M
j
jjj tXPXrtXPXrdt
tXdP
11
,,,
Two-state model with positive feedback
1k
n max
largeThe analytical results introduced here can be applied to
any self-regulating module of a single gene, while the
methodology is valid for a much more general context.
Three time scales and three different scenarios
( protein ofrate synthesis:
)(switching state -gene:
)(cycle cell:
))(
)(,)(
)(
1
1
kiii
nhnfii
i
)i( )iii( )ii( Ao, et al. (2004); Huang, et al. (2010)
When stochastic gene-state switching is extremely rapid
)ii( )i( )iii(Qian, et al. (2009); Wolynes, et al. (2005)
When stochastic gene-state switching is extremely slow
When the time scales of (ii) and (iii) are comparableAssaf, et al. (2011); Li, et al. (2014)
)i( )ii( Wolynes, et al. (2005);Ge, et al. (2015)
When stochastic gene-state switching is relatively slow
)iii(
)i( )ii( )iii(
A single-molecule fluctuating-rate model is derived
Ge, H., Qian, H. and Xie, X.S., PRL (2015)
(A) ,,1 hfk (B)hfk ,,1
0 5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Bifurcation diagram for simple example
1/Keq
x*
Bifurcation diagram
xxgdt
dx )(
OFF state
ON state
Rescaled dynamics
Continuous Mean-field limit Fluctuating-rate model
maxn
nx
xn
k
dt
dx
max
1
xn
k
dt
dx
max
2
f2xh
Ge, H., Qian, H. and Xie, X.S., PRL (2015)
Stochastic dynamics of fluctuating-rate model
xn
k
dt
dx
max
1
xn
k
dt
dx
max
2
f2xh
Nonequilibrium landscape function emerges
Dynamics in the mean field limit model
xxgdt
dx )(
0
~
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-12
-10
-8
-6
-4
-2
0
2
4
6
x
Stable(OFF)
Unstable Stable(ON)
0.01 0.0105 0.011 0.0115 0.0120.69
0.695
0.7
0.705
0.71
0.715
0.72
0.725
0.73
x
xss exp 0
0
Landscape function
analog to energy function at equilibrium case
xn
k
xh
xn
k
f
dx
xd
maxmax
2
2
1
0
Ge, H., Qian, H. and Xie, X.S., PRL (2015)
As gene-state switching is much faster than the
cell cycle
Quantify the relative stability of stable fixed points
෪Φ0 = Φ0 ∕ 𝑓
alternative
attractor
2: fluctuating in local
attractor, waiting
1: relaxation
process
3: abrupt transition
via barrier-crossing
The uphill dynamics is the rare event, related to phenotype switching, punctuated transition in evolution, et al.
Dynamics of bistable systems
Intra-attractorial dynamics
Inter-attractorial dynamics
A B
discrete stochastic model
among attractors
ny
nx
chemical master equation cy
cx
A
B
fast nonlinear differential equations
appropriate reaction coordinate
ABp
robabil
ity
emergent slow stochastic dynamics
and landscape
(a) (b)
(c)
(d)
Three time scalesFixed finite molecule numbers
Stochastic
Stochastic
Deterministic
Ge and Qian: PRL (2009), JRSI (2011)
Rate formulae associated with the landscape function
Gene-state switching is extremely slow
k linearly depend on gene-state switching rates
Wolynes, et al. PNAS (2005)
Gene-state switching is relatively slow
Barrier crossing
ǂ0ekk 0
Ge, H., Qian, H. and Xie, X.S., PRL (2015)
maxn
maxn
maxn
maxn
maxn
A recent example: HIV therapy (activator + noise enhancer)
0ekk 0
off
onoff
k
kk 00
~
Activator: increasing kon , lower the barrier
Noise enhancer: Decreasing both kon and koff, further lower the barrier
Significantly increasing the transition rate
Weinberger group, Science (2014)
Gene ON Gene OFFonk
offk
Rigorous analysis: quasi potential in LDP
Local: The Donsker-Varadhan large deviation theory for Markov process
Global: The Freidlin-Wentzell large deviation theory for random perturbed dynamic system
+
LDT of Fluctuating-rate model (Switching ODE)
Two-scale LDT of Switching(Coupled) Diffusion
See Chapter 7 in Freidlin and Wentzell: Random Perturbations of Dynamical Systems (2nd Ed). Springer 1984
Compared to previous rate formulae for bursty dynamics
Eldar, A. and Elowitz, M. Nature (2010) Cai, et al. Science (2006)
Burst sizef
xMax
k
f
1b off
1
b
x x
0off
ekk
ǂ xx ,
b
1
dx
xd 0
ǂIf
Walczak,et al.,PNAS (2005);Choi, et al.,JMB (2010);Ge,H.,Qian,H.and Xie, X.S.,PRL (2015)