Nonequilibrium Thermodynamics of open driven systems

Nonequilibrium Thermodynamics of open driven systems Hao Ge 1Biodynamic

Optical Imaging Center (BIOPIC) 2Beijing International Center for Mathematical Research (BICMR) Peking University, China

Laws of thermodynamics Zeroth law: The definition of temperature First law: Energy conservation Second law: the arrow of time

dU  W  Q Q Clausius inequality dS   T

Third law: absolute zero temperature

At equilibrium

Microscopic reversibility

Detailed balance

Evolution of entropy System

dStot  dS system  dSmedium  0 T

Medium

dS system  dSi  dSe

Two different perspectives

T  epr  T  dS i  T  dS tot   J i X i  0 i

dSi, dSe and dStot, rather than Si, Se and Stot are the state functions of the internal system. Detailed balance

Ji  0  Xi  0 Generalized flux

Generalized force

I. Prigogine: Introduction to thermodynamics of irreversible processes. 3rd ed. (1967) T.L. Hill: Free energy transduction in biology. (1977)

Two major questions dS system  epr  dSe  epr  dSmedium 1. In steady state, what does the state function T·dSmedium mean? Total heat dissipation? Can it be used to perform work? It requires a “real driven” perspective and a minimum work argument. 2. In the relaxation process towards steady state, how to distinguish the two origin of nonequilibrium, i.e. nonstationary and nondetailed-balance (driven) of the steady state?

T  epr  f d  Qhk

A single biochemical reaction cycle

Spontaneous ATP hydrolysis and related ATP regenerating system.

A single biochemical reaction cycle

B  ATP B  Pi

k1

k 1 k2

k2

Equilibrium condition:

C  ADP

(1)

C

(2)

[ ATP ]eq k 1k  2  [ ADP]eq [ Pi ]eq k1k 2

Open driven system: regenerating system k1k 2 [ ATP ]ss 1 keeping the concentrations of ATP, ADP and   ss ss k 1k  2 [ ADP] [ Pi ] Pi

Heat dissipation

( 2)

(1)

After an internal clockwise cycle, the traditional heat dissipation during ATP hydrolysis 0 0   hC0  hPi0  hB0  hd  hB0  hATP  hC0  hADP 0 0  hATP  hADP  hPi0  T  S e  T  S medium .

Could not be calculated only from the dynamics of the internal system.

Heat dissipation There is an external step for the regenerating system converting ADP+Pi back to ATP after each completion of a cycle. The minimum work (non-PV) it has to do is just the free energy difference between ADP+Pi and ATP, i.e.

Wmin   ATP   ADP   Pi Driven energy of the internal system

The extra heat dissipation

0 0 hdext  Wmin  ( hATP  hADP  hPi0 )

The total heat dissipation of such a reaction cycle is hd  hdext  Wmin  k BT log   T  S e  T  S medium .

Master equation model 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.

dci (t )   (c j k ji  ci kij ) dt j No matter starting from any initial distribution, it will finally approach its stationary distribution satisfying

 c N

j 1

ss j



k ji  c k  0 ss i ij

Self-assembly or self-organization

c k ji  c k eq j

eq i ij

Detailed balance

Coupled with energy source Assume only one of the transition is involved in the energy source, i.e. ATP and ADP.

~ ~ k12  k12 [ ATP ], k 21  k 21[ ADP]

If there is no external mechanism to keep the concentrations of ATP and ADP, then

~ ~ dcT dc D    k12 cT c1  k 21cD c2 . dt dt

Thermodynamic constrains i0  k BT log cieq

Boltzmann’s law

i (cieq )   j (c eqj ), T (cTeq )   D (cDeq )

eq c i0   0j  k BT log ; T0   D0  k BT log Deq , cT k ji ~ k12 0 0 0 0 1  T   2   D  k BT log ~ . k 21

kij

Heat dissipation 

~ open hd (t )  k BT  ci (t )kij  c j (t )k ji  hi0  h 0j



i j

 k BT c1 (t )k12  c2 (t )k 21 T   D 

In an NESS, its kinetics and thermodynamics can be decomposed into different cycles (Kirchhoff’s law, Beijing school). The minimum amount of total heat dissipation for each internal cycle k i 0 i1 k i1 i 2 ... k i n i 0

c  k B T log Q min

k i 0 i n k i n i n  1 ... k i1 i 0

;

c  { i 0  i1  i 2   i n  i 0 }





kij ~ hdness  k BT  ciss kij  c ssj k ji log  T  dS e  T  dS medium . k ji i j

Energy transduction efficiency A mechanical system coupled fully reversibly to a chemical reactions, with a constant force resisting the mechanical movement driven by the chemical gradient.

Wmin  J cm

~ ness  hd  Pmech  Te ness  Pmech p

Transduction from chemical energy to mechanical energy

Wmin  0, J cm  0, e

ness p

 0, Pmech

0  

Pmech P  nessmech 1 Wmin  J cm Te p  Pmech

Transduction from mechanical energy to chemical energy

Wmin  0, J cm  0, e ness  0, Pmech  0 p

 Wmin  J cm  Wmin  J cm   ness 1  Pmech Te p  Wmin  J cm The steady-state entropy production is always the total dissipation, which is nonnegative

The evolution of entropy ~ ~ F open  H 0  TS open

S 0   si0 ci ; S open  k B  ci log ci . i

i

  0  TS open

~ open 0 open S S S

H 0   hi0 ci ,  0   i0 ci

~ open ~ open hd dS open  ep  ; T dt open h dS open d  e open  . p dt T

i

Enthalpy-entropy compensation

hdopen (t )  k BT  ci (t )kij  c j (t )k ji log i j

e

open p

(t )  k B  ci (t )kij  c j (t )k ji log i j

i

kij k ji

Operationally defined heat if we do not know the temperature dependence of 

;

ci kij c j k ji

.

ness d

h

~ ness  hd

QSS v.s. NESS

Closed system Very slow changing environment

dF close   f dclose ; dt close h dS close d  e close  ; p dt T close close Te open  Te  f  0. p p d

This reflects the different perspective of Boltzmann/Gibbs and Prigogine: 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.

Real driven: Housekeeping heat Housekeeping heat

Qhk (t )  k BT  ci (t )kij  c j (t )k ji log i j

ciss kij ss j

c k ji

.

The minimum heat dissipation for each cycle could be distributed to each i→j as kij Qij  k BT log  T si0  s 0j  k ji ciss ss The steady-state entropy difference Sij  k B log ss  s 0j  si0 cj

Qij  TSijss  k BT log Qhk (t )  Qij  TSijss  0.

ciss kij

c ssj k ji

driven (approaching equilibrium Qhk (t )  0  No state with detailed balance)

Time-independent systems Relative entropy

dF   fd ; dt dE  Qhk  hd ; dt

hd dS  ep  ; dt T

fd:  free energy dissipation rate hd:  heat dissipation/work out  Qhk:  house keeping heat/work in Qhk (t )  k BT  ci (t )kij  c j (t )k ji log i j

ss j

c k ji

.

ep: entropy production rate

e p (t )  k B  ci (t )kij  c j (t )k ji log i j

ciss kij

ci kij c j k ji

；hd (t )  k BT  ci (t )kij  c j (t )k ji log i j

kij k ji

;

Two origins of irreversibility f d  0, Qhk  0, Te p  f d  Qhk  0. ep characterizes total time irreversibility in a Markov process.  When system reaches stationary, fd = 0. When system is closed (i.e., no active energy  drive, detailed balaned) Qhk = 0.  Boltzmann: fd = T∙ep >0 but Qhk=0; Prigogine (Brussel school, NESS): Qhk=T∙ep > 0 but fd=0. fd ≥ 0 in driven systems is “self‐organization”.    

Time-dependent systems dci (t )   (c j k ji (t )  ci kij (t )) dt j Entropy in HatanoSasa equality. We would like to call it intrinsic entropy, which could be defined at individual level.

hd (t ) dS (t )  e p (t )  dt T

dF (t )  W ext (t )  f d (t ) dt dU (t )  W ext (t )  Qex (t ) dt

Dissipative work in Jarzynski equality

Qhk (t )  Te p (t )  f d (t )  hd (t )  Qex (t )

Two kinds of Second Law hd dS   e p  0  dt T

dF  W ext  f d  0  dt

In detailed-balance case, they are equivalent.

Te p  f d , Qhk  0 In non-detailed balance case, the new one is stronger than the traditional one.

dF dU ext W   Qex dt dt Qex hd dS    dt T T

Summary Regenerating system approach would distinguish quasisteady-state and nonequilibrium-steady-state, and supply an equilibrium thermodynamic foundation for the expression of heat dissipation in nonequilibrium steady state of subsystems, without the need to know “environment”; Thermodynamic superstructure would explicitly distinguish Boltzmann and Prigogine’s thesis, and further clarify the two kinds of the Second Law;  So far, a comprehensive framework for both equilibrium and nonequilibrium statistical mechanics is proposed. 

Acknowledgement

Prof. Hong Qian University of Washington Department of Applied Mathematics

Thanks for your attention!