nonequilibrium thermodynamics of stochastic chemical ...bicmr.pku.edu.cn/~gehao/chinese...
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Nonequilibriumthermodynamics of stochastic chemical reaction systems
Hao Ge (葛颢)[email protected]
1Beijing International Center for Mathematical Research2Biodynamic Optical Imaging Center
Peking University, Chinahttp://bicmr.pku.edu.cn/~gehao/
Stochastic Biophysics(Biomath)
Stochastic theory of nonequilibrium statistical mechanics
JSP06,08PRE09,10,13,14JCP12; JSTAT15
Nonequilibrium landscape theory and rate formulas
PRL09,15JRSI11; Chaos12
Stochastic modeling of biophysical systems
Statistical machine learning of single-cell data
JPCB08,13,16; JPA12Phys. Rep. 12Science13;Cell14;MSB15
CR15
Theory Applications Providing analytical tools
Providing scientific problems
Summary of Ge group
What is life
• Erwin Schrödinger was the first one trying to apply mathematical/physical arguments to study living matters; • He explains that living matter evades the decay to thermodynamical equilibrium (death) by homeostatically maintaining negative entropy in an open system.
Nonequilibrium thermodynamics and chemical oscillationBelousov–Zhabotinsky reaction
Ilya PrigogineNobel Prize in 1977
Entropy productionNonequilibrium steady stateNonequilibrium thermodynamics
SdSddS ie
0 k
kki XJSdepr
Brusselator
The fundamental equation in nonequilibrium thermodynamics
Ilya Prigogine(1917-2003)Nobel Prize
in 1977
Entropy production
SdSddS ie
0 k
kki XJSdepr
Clausius inequalityTQdS
Rudolf Clausius(1822-1888)
Second law of thermodynamics
0TQdSepr
rewrite
More general
Carl Eckart(1902-1973)
P.W. Bridgman(1882-1961)Nobel Prize
in 1946
Lars Onsager(1903-1976)Nobel Prize
in 1968
Nonequilibrium statistical mechanics
• Needs the idea of stochastic process;• Understand where is the energy input and output
Nonequilibrium thermodynamics is not easy to be applied.
Ion pump at cell membrane T.L. Hill, Nature (1982) E. Eisenberg and T.L. Hill, Science (1985)
T.L. Hill
Muscle contraction: transduction from chemical energy to mechanical energy
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)
Second law of thermodynamics is statistical
“The truth of the second law is therefore a statistical, not mathematical, truth, for it depends on the fact the bodies we deal with consist of millions of molecules, and that we never can get hold of single molecules.”
“Hence the second law of thermodynamics is continuously violated, and that to a considerable extent, in any sufficiently small group of molecules belonging to a real body.”
Nature 17, 278 (1878) J.C. Maxwell(1831-1879)
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
Stochastic trajectory observed
McCann et al. Nature, 402, 785 (1999) Lu, et al. Science (1998)
Single-molecule enzyme kineticsSingle Brownian particle
Second law: from inequality to equality (Fluctuation theorems)
tsrP
tsPt
Eprt
eprsts
s
tt
0:0:log1lim1lim
Qian, M., et al. (1981,1984)
Jarzynski, C. (1997) FW ee
Kurchan, J. (1998); Lebowitz, J. and Spohn, H. (1999); Seifert, U. (2005); Spinney, R. and Ford, I. (2012); et al.
Jarzynski equality
1 Epre
zezEprP
zEprP
Crooks, G. (1998); Van den Broeck, C. and Esposito, M. (2010); et al.
1 Ke
zezK~P~
zKP
zzIzI )()(
FW
0K
Entropy production along the trajectory
Experimental validation and application
Gupta, et al. Nat. Phys. (2011)Wang, et al.: Phys. Rev. Lett. (2002)
Reversible Michaelis-Menten enzyme kinetics
Kinetic scheme of a simple reversible enzyme. From the perspective of a single enzyme molecule, the reaction is unimolecular and cyclic.
Ge, H., Qian, M. and Qian, H.: Phys. Rep. (2012)
Ge, H.: J. Phys. Chem. B (2008)
Two reversibleMichaelis-Menten reactions
From concentration to probability
][011 Skk ][0
33 Pkk Pseudo-first order reaction constants
The evolution of probability distribution
Ek-1
k01 [S]
ES
EPk2
k-2
k3k0
-3 [P]
[S] and [P] held constant
Reactant ProductPS
ESES
EPES
PEEP
Split it into the following three reactions:
(1)
(2)
(3)
E
ESB pS
pTk][
ln011
ES
EPB p
pTk ln022
EP
EB p
PpTk ][ln033
][][ln0
30
20
1321 SPTkB
Constant conc. [S] and [P] held constant
Cyclic kinetics of enzyme moleculesFree energy
(chemical potential)
cTkB ln0
03
02
01 Now we need to evaluate
ESES
01 eq1eqeq01 ]ES[k]E[]S[k
1
010
10
1 ln][][
][ln0
kkTk
ESES
Tk Beqeq
eqB
1
010
1 ln
kkTkB
Take for example, at equilibrium,
Therefore,
Therefore,
0321
32010
30
20
1 ln
kkkkkkTkB
][][ln 0
321
3201
PkkkSkkkTkB
3201
0321
kkkkkk
]P[]S[
3201
0321
kkkkkk
]P[]S[
When
At equilibrium
Min, et al. Nano Lett, (2005)
PS
Simulated turnover traces of a single molecule
Ge, H.: J. Phys. Chem. B (2008)
)(t
PSt :)( SPt :)( the number of occurrences of forward and backward cycles up to time t
Steady-state cycle fluxes and nonequilibrium steady state
.][][1
][][)(lim ssss
mPmS
mPP
mSS
t
ss JJ
KP
KS
KPV
KSV
ttJ
.][][1
][)(lim ;][][1
][)(lim
mPmS
mPP
t
ss
mPmS
mSS
t
ss
KP
KS
KPV
ttJ
KP
KS
KSV
ttJ
Michaelis-Menten kinetics
Chemical potential difference: )ln(][
][ln 0321
3201
ss
ss
BB JJTk
PkkkSkkkTk
Ge, H.: JPCB (2008)Ge, H., Qian, M. and Qian,H.: Phys. Rep. (2012)
00
ssJ
Waiting cycle times
ESE EP E ES EP E-2-3 -1 0 1 2 3 PS
k2k1 k3 k1 k2 k3
k-1 k-2 k-3 k-1 k-2 k-3
The kinetic scheme for computing the waiting cycle times, which also serves for molecular motors.
.1;1;1ssssssss J
TJ
TJJ
T
Generalized Haldane equality
).|()|( TTtTPTTtTP EE
Waiting cycle time T is independent of whether the enzyme completes a forward cycle or a backward cycle, although the probability weight of these two cycles might be rather different.
Carter, N. J.; Cross, R. A. Nature (2005)
Superposed distributions!
Average time course of forward and backward steps
Qian, H. and Xie, X.S.: Phys. Rev. E (2006);
Ge, H.: J. Phys. Chem. B (2008); J. Phys. A (2012)
Jia, et al. Ann. Appl. Prob. (2016): general Markovian jumping process
Ge, et al. submitted to Stoc. Proc. Appl. (2016): diffusion on a circle
Nonequilibrium steady state
Ge, H., et al. Phys. Rep. (2012)
321
321BPS kkk
kkklogTk
)|()|( TTtTPTTtTP EE
Ge, H. J. Phys. Chem. B (2008); J. Phys. A (2012)
Generalized Haldane Equality
Fluctuation theorem of fluctuating chemical work
Tkn
E
E BentWP
ntWP
))(())((
1)(
TktW
BeSecond law in terms of equality
Free energy conservation0
tWepr t
Entropy production: Free energy dissipationFree energy input
PS 0.mEquilibriu
Traditional Second law
ttW
Steady state Gallavotti-Cohen-type fluctuation theorems
ttJ t
ttJ t
ttt JJJ
tt J,J has the large deviation principle with rate function I1.
21121211 xxxxIxxI ,,
tJ has the large deviation principle with rate function I2.
xxIxI 22Jia, et al. Ann. Appl. Prob. (2016): Ge, et al. submitted to Stoc. Proc. Appl. (2016)
t,x,xIx,xJ,JPlogt tt 211211
t,xIxJPlogt t 21
Quasi-time-reversal invariance of 1d B.M. and 3d Bessel process
The distribution is the same as a random flipped 3d Bessel Process
)|()|( TTtTPTTtTP EE
For diffusion on a 1d circleGe, et al. submitted to Stoc. Proc. Appl. (2016)
translation
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
jij
ssiji
ssj kpkp
j
ijijiji kpkp
dttdp )(
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
ijeqiji
eqj kpkp
Detailed balance (equilibrium state)
Canonical ensemble
Free energy j
kTE jekTF /log
j
kTE
kTEeqi
ssi j
i
eepp /
/
Boltzmann’s law
ijssiji
ssj kpkp
Under detailed balance condition (equilibrium)
EntropyT
FEppkS
j
eqj
eqj
log
0
iiE
Time reversibilityTime-reversed process: still satisfy the master equation
j
ijijiji kpkp
dttdp
ssj
ijssi
ji pkp
k
ijssiji
ssjjiji kpkpkk
For each cycle, niiiiiiii
iiiiiiii iiickkkk
kkkk
nnnnn
n ,,, ,1 10012110
0322110
Kolmogorov cyclic condition
NESS thermodynamic force and entropy production rate
0 ji
ssij
ssij
ness AJeprT
jissjij
ssi
ssij kpkpJ
jissj
ijssi
Bssij kp
kplogTkA NESS thermodynamic force
NESS flux
NESS entropy production rate
Energy transduction efficiency at NESS
A mechanical system coupled fully reversibly to a chemical reactions, with a constant force resisting the mechanical movement driven by the chemical gradient.
l)(MechanicaOutput Energy (Chemical)Input Energy nessdh
1outputEnergy
outputEnergy inputEnergy outputEnergy
nessdh
or(Chemical)Output Energy l)(MechanicaInput Energy ness
dh
nessnessd eprTh all dissipated! Ge, H. and Qian, H.: PRE (2013)
Time-dependent case
jijijij
i tkptkpdt
tdp
Free energy j
kTtEB
jeTktF /)(log)(
j
kTtE
kTtEeqi
ssi j
i
eetptp /)(
/)(
)()(Boltzmann’s law
If {kij(t)} satisfies the detailed balance condition for fixing t
01
N
jij
ssiji
ssj tktptktpQuasi-stationary distribution
0 tktptktp ijssiji
ssj
Work and heat along the trajectory
1kT kT
)(tX
jTX k )(
iTX k )( 1
…
…
Heat
)()()()( 11)()( 11 kikikTXkTX TETETETEW
kkWork
)()()()( )()( 1 kikjkTXkTX TETETETEQkk
Jarzynski equality with detailed balance condition
Heat
11)()( )()(
11k
kTXkTX TETEWkk
Work
1)()( )()(
1k
kTXkTX TETEQkk
)0()( )0()( XTX ETEQWE
)0()( FTFF Jarzynski equality
kTF
pp
kTW ee eq
/
)0()0(
/
.)0()0(
FW eqpp
FWWd Dissipative work
Hatano-Sasa equality without detailed balance condition
)(log)( tpt ssii Entropy
kTFE /
FkTWEWQQex Hatano-Sasa equality
1)0()0(
/
ss
ex
pp
kTQe
./)0()0(
kTQS expp ss
Excess heat
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 equilibriumkTF
pp
kTW ee eq
/
)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 really need those conditions under which the fluctuations theorems is valid? 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
tktptktp
TktA ijssij
jij
ijiBij log
i j
0 ji
ijij tAtJteprT )()()(Entropy production
0 ji
ssijijhk tAtJtQ )()()(Housekeeping heat
Free energy dissipation 0 ji
ijijd tAtJtf )()()(
t t t dhk fQeprT Esposito, M. et al. PRE (2007)Ge, H., PRE (2009); Ge, H. and Qian, H., PRE (2010) (2013)
tktp
tktpTktA
jissj
ijssi
Bssij log
All the results here have also been proved for multidimensional diffusion process.
0tepr for any time t In 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)
The evolution of entropy and relative entropy
Relaxation to steady state(Time-independent system)
dd fWdt
dH
Dissipative work in Jarzynski equality
Time-dependent system
TQepr
dtdS tot ;f
dtdH
d
Generalized free energy i
ssi
iiB p
plogpTkH
TQepr
dtdS tot
i
iiB ppkS logEntropy:
0 eprT
QdtdS tot
The new Clausius inequality is stronger than the traditional one.
0 d
hktotex fT
QQT
QdtdS
Extended Clausius inequality
.QfeprT,Q,f
hkd
hkd
000
Ge, H., PRE (2009);
Ge, H. and Qian, H., PRE (2010) (2013)
TQepr
dtdS tot
0 hkin QE Generalized free energy input
0 eprTEdis
A generalization of free energy and its balance equation
.0 disin EEdt
dH
Ge, H., PRE (2009); Ge, H. and Qian, H., PRE (2010) (2013)
Generalized free energy dissipation
Approaching nonequilibriumsteady state (NESS)
Closed system
Very slowly changing [ATP], [ADP] and [Pi]
Quasi-steady-state (QSS) approaching equilibrium state
The chemical kinetics of the internal system (B+C) are the same.
Quasi-steady-state perspective versus steady-state perspective
Ge, H. and Qian, H., PRE (2013); in preparation (2016)
Thermodynamics of a closed isothermal “universe”
Closed system
.0 closep
closed
close
eTfdt
dF
Gibbs states free energy never increase in a closed, isothermal system; while Prigogine states that the entropy production is non-negative in an open system.They are equivalent.
In this case, everything is well defined from traditional chemical thermodynamics.
Ge, H. and Qian, H., PRE (2013); in preparation (2016)
The
dtdS open
dopenp
open
dtdFfee
closeclose
dclosep
openp
Thermodynamics of the open system with regenerating system
dtdFfee
closeclose
dclosep
openp )1(
A quasi-steady-state interpretation of epr decomposition
Ge, H., PRE (2009); Ge, H. and Qian, H., PRE (2010) (2013)
.0
,0,0
openhk
opend
openp
openhk
opend
QfeT
QfGe, H. and Qian, H., in preparation (2016)
dtdH
ttpttp
TktJ
tf
closeqss
jiqssij
qssji
Bij
opend
)(
log)(
)(2Quasi-steady-state distribution depending on certain parameters which is slowly changing with time
iqssi
iiB
closeqss t
tptpTkH
logHolds under quasi-steady-state assumption
Chemical reaction system at macroscopic level
⋯ ⇔ ⋯1,2,… ,Backward flux
Stoichiometric matrix
: concentration vector
=
Flux vector =
Rate equation∗ =0
Steady state
Forward flux
Entropy production rate
⇔ = , ∀
0
Equilibrium state: detailed balanceEquilibrium state is always steady state
Wegscheider-Lewis cycle condition:
For any M-dimensional vector , , … , satisfying S 0
∑ 0, ∀ .haveRight-null-space
Chemical reaction system at the mesoscopic level
Copy number of all the chemical species , , … , ·
Denote , , … , ,
→ →
Assuming Markovian dynamics
Chemical master equation
,
, ,
, ,
Mesoscopic nonequilibriumthermodynamics
.
, log,
, , log
, , log,,
Ge, H., PRE (2009); Ge, H. and Qian, H., PRE (2010) (2013)
Law of large number: from mesoscopic to macroscopic
Large deviation principle and rate function
In the limit of V tending to infinity, with the same initial state
1 · 1 · 0
/ → / →
Hu, G. Z. Physik (1982)
Shwartz, A. and Weiss, A. Large deviation for performance analysis. (1995)
Kurtz. T. JCP (1972)
Emergent macroscopic nonequilibrium thermodynamics
0.
→
→ log · 0
→ log 0
A generalization of macroscopic free energy
Entropy production rate
Free energy input
.
Ge, H. and Qian, H., arxiv:1601.03159; arxiv: 1604.07115 (2016)
→ ∞.
Strong and weak detail balance conditions
Weak detailed balance condition for each x is equivalent to
Wegscheider-Lewis cycle condition.
0 =
Strong detailed balance condition (equilibrium)
0 log · , ∀
Weak detailed balance condition
0 .
Ge, H. and Qian, H., arxiv:1601.03159; arxiv: 1604.07115 (2016)
General fluctuation-dissipation theorem for chemical reactions
1,2, … ,
Rate equation∗ =0
Stable fixed point
Ξ∗ , ∗ ∗
∗ ,
Variance matrix Fluctuation matrix Dissipation matrix
Ge, H. and Qian, H., arxiv:1601.03159; arxiv: 1604.07115 (2016)
Acknowledgement
Prof. Hong QianUniversity of Washington
Fundings: NSFC, MOE of PRC
Prof. Min QianPeking University