Nonequilibrium Thermodynamics of open driven systems Hao Ge 1 - PowerPoint PPT Presentation

Nonequilibrium Thermodynamics of open driven systems Hao Ge 1 Biodynamic Optical Imaging Center (BIOPIC) 2 Beijing International Center for Mathematical Research (BICMR) Peking University, China

Laws of thermodynamics Zeroth law: The definition of temperature   dU W Q First law: Energy conservation Q   dS Second law: the arrow of time Clausius inequality T Third law: absolute zero temperature Microscopic reversibility Detailed balance At equilibrium

Evolution of entropy    0 dS dS dS tot system medium Two System different T   perspectives dS dS dS system i e Medium          0 T epr T dS T dS J X i tot i i i dS i , dS e and dS tot , rather than S i , S e and S tot are the state functions of the internal system.    0 0 J X Detailed balance i i Generalized force Generalized flux I. Prigogine: Introduction to thermodynamics of irreversible processes. 3 rd ed. (1967) T.L. Hill: Free energy transduction in biology . (1977)

Two major questions     dS epr dS epr dS system e medium 1. In steady state, what does the state function T·dS medium 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 non- detailed-balance (driven) of the steady state?    T epr f Q d hk

A single biochemical reaction cycle Spontaneous ATP hydrolysis and related ATP regenerating system.

A single biochemical reaction cycle k B  1 C  ATP ADP (1) k  1 k B  2 P C (2) i k  2 [ ] eq ATP k k    1 2 Equilibrium condition: [ ] [ ] eq eq ADP P k k 1 2 i Open driven system: regenerating system [ ] ss k k ATP    1 2 1 keeping the concentrations of ATP , ADP and [ ] [ ] ss ss k k ADP P   1 2 i P i

Heat dissipation ( 1 ) ( 2 ) After an internal clockwise cycle, the traditional heat dissipation during ATP hydrolysis            0 0 0 0 0 0 0 h h h h h h h h d B ATP C ADP C Pi B          0 0 0 . h h h T S T S ATP ADP Pi e 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+ P i 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.       W min ATP ADP Pi Driven energy of the internal system     0 0 0 ( ) ext h W h h h The extra heat dissipation min d ATP ADP Pi The total heat dissipation of such a reaction cycle is           log ext h h W k T T S T S . min d d B e medium

Master equation model Consider a motor protein with N different conformations R 1 , R 2 , … , R N . k ij is the first-order or pseudo-first-order rate constants for the reaction R i → R j . ( ) dc t    ( ) i c k c k j ji i ij dt j No matter starting from any initial distribution, it will finally approach its stationary distribution satisfying   N     eq eq 0 ss ss c k c k c k c k j ji i ij j ji i ij  1 j Self-assembly or self-organization Detailed balance

Coupled with energy source Assume only one of the transition is involved in the energy source, i.e. ATP and ADP . ~ ~   [ ], [ ] k k ATP k k ADP 12 12 21 21 If there is no external mechanism to keep the concentrations of ATP and ADP , then ~ ~ dc dc      . T D k c c k c c 12 1 21 2 T D dt dt

Thermodynamic constrains    0 log eq Boltzmann ’ s law k T c i B i       ( ) ( ), ( ) ( ) eq eq eq eq c c c c i i j j T T D D eq k c         0 0 0 0 ij log ; log , D k T k T i j B T D B eq k c ji T ~ k         0 0 0 0 12 log . k T ~ 1 2 T D B k 21

Heat dissipation     ~     0 0 ( ) ( ) ( ) open h t k T c t k c t k h h d B i ij j ji i j  i j         ( ) ( ) k T c t k c t k 1 12 2 21 B 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 k k  log i i i i i i ; c 0 1 1 2 0 Q k T n min B ... k k k i i i i i i  0 1 1 0 n n n      { }  c i i i i i 0 1 2 0 n   ~ k        log ij . ness ss ss h k T c k c k T dS T dS d B i ij j ji e medium k  i j ji

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. ~      ness ness W J h P Te P  min c m d mech p mech Transduction from chemical energy to mechanical energy P P     1     mech mech 0 , 0 , 0 , 0 ness W J e P    min ness c m p mech W J Te P  min c m p mech Transduction from mechanical energy to chemical energy     0 , 0 , 0 , 0 ness W J e P  min c m p mech     W J W J       min min 1 c m c m    ness P Te W J  min mech p c m The steady-state entropy production is always the total dissipation, which is nonnegative

The evolution of entropy   ~ ~ 0  0   ; log . open  0  S s c S k c c open open F H T S i i B i i i i    ~ 0 open TS   0 open open S S S   0  0  0   0 , H h c c ~ ~ i i i i i i open open d S h   ; open Enthalpy-entropy compensation d e p dt T Operationally defined open open dS h   heat if we do not know . open d e p dt T the temperature dependence of     k    ( ) ( ) ( ) log ij ; open h t k T c t k c t k ~ d B i ij j ji k   i j ji ness ness h h   c k  d d   ( ) ( ) ( ) log i ij . open e t k c t k c t k p B i ij j ji c k  i j j ji

QSS v.s. NESS close dF   ; close f d dt close close dS h   ; close d e p dt T    0 . Closed system open close close Te Te f p p d Very slow changing environment 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   ss c k    ( ) ( ) ( ) log i ij . Housekeeping heat Q t k T c t k c t k hk B i ij j ji ss c k  i j j ji The minimum heat dissipation for each cycle could be distributed to each i → j as   k    0 0 log ij Q k T T s s ij B i j k ji ss c     0 0 log ss i The steady-state entropy difference S k s s ij B j i ss c j ss c k    log i ij ss Q T S k T ij ij B ss c k j ji  0      ( t ) No driven (approaching equilibrium ( ) 0. ss Q hk Q t Q T S hk ij ij state with detailed balance)

； Time-independent systems Relative entropy dF   ; f d : free energy dissipation rate f d dt h d : heat dissipation/work out dE   ; Q h Q hk : house keeping heat/work in hk d dt   ss c k    ( ) ( ) ( ) log i ij . Q t k T c t k c t k hk B i ij j ji ss c k  i j j ji dS h  p  ; d e p : entropy production rate e dt T     c k k       i ij ij ( ) ( ) ( ) log ( ) ( ) ( ) log ; e t k c t k c t k h t k T c t k c t k p B i ij j ji d B i ij j ji c k k   i j i j j ji ji

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