nonequilibrium thermodynamics of stochastic chemical ...bicmr.pku.edu.cn/~gehao/chinese...

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/

BICMR: Beijing International Center for Mathematical Research

BIOPIC: Biodynamic Optical Imaging Center

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


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


.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,,


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 .


General fluctuation-dissipation theorem for chemical reactions

1,2, … ,

Rate equation∗ =0

Stable fixed point

Ξ∗ , ∗ ∗

∗ ,

Variance matrix Fluctuation matrix Dissipation matrix


Acknowledgement

Prof. Hong QianUniversity of Washington

Fundings: NSFC, MOE of PRC

Prof. Min QianPeking University

Thanks for your attention!

nonequilibrium thermodynamics of stochastic chemical ...bicmr.pku.edu.cn/~gehao/chinese...

Documents