Open Chemical Systems and Their Biological Function Hong Qian Hong - PowerPoint PPT Presentation

Open Chemical Systems and Their Biological Function Hong Qian Hong Qian Department of Applied Mathematics University of Washington

Dynamics and Thermodynamics of Stochastic Nonlinear (Mesoscopic) Systems (Mesoscopic) Systems

Mesoscopic description of physical and chemical systems: • Gibbs (1870-1890s) – complex system in equilibrium in terms of ensembles q • Einstein, Smoluchowski, Langevin – simple motions linear dynamics (1900-1910s) motions, linear dynamics (1900 1910s) • Kramers (1940) – emergent rare events in nonlinear systems nonlinear systems • Onsager (1953) – general linear dynamical th theory

Mesoscopic description of physical and chemical systems: • Gibbs (1870-1890s) – complex system in equilibrium in terms of ensembles q • Einstein, Smoluchowski, Langevin – simple motions linear dynamics (1900-1910s) motions, linear dynamics (1900 1910s) • Kramers (1940) – emergent rare events in nonlinear systems nonlinear systems • Onsager (1953) – general linear dynamical th theory

Mesoscopic description of a p p system - the mathematical tool: • Kolmogorov (1933) – mathematical theory of random variables & stochastic processes p

The Aim of Statistical Mechanics The Aim of Statistical Mechanics ``to develop a formalism from which one can p deduce the macroscopic behavior of physical systems composed of a large p y y p g number of molecules from a specification of the component molecular species, the p p laws of force which govern intermolecular interactions, and the nature of their surroundings.'‘ [Montroll and Green, Ann. Rev. Phys. Chem . (1954)] y ( )

The Aim of Statistical Physics (now half a century later) (now half a century later) T To develop a formalism from which one d l f li f hi h can deduce the macroscopic behavior of complex systems from a specification of l t f ifi ti f the components, the laws of force which govern dynamics, and the nature of their d i d th t f th i surroundings.

In Traditional Physics: y • A law of a force is an interaction between particles; particles; • However, as many physical chemists k know well, entropic force is not a force ll t i f i t f between particles; in fact it is an emergent entity on a population level. The Fick’s tit l ti l l Th Fi k’ law makes no sense on an individual B Brownian particle level; i ti l l l • Still, a force is something that causes a system to change.

What is thermodynamics? What is thermodynamics? Thermodynamics deals with energy, entropy, their balance, and inter-relationships in h i b l d i l i hi i complex systems ( (no temperature in this talk). i hi lk)

What is Kolmogorov’s Stochastic Process? • It is a mathematical description of It i th ti l d i ti f dynamics with “uncertainties”. It has both a trajectory perspective and a population t j t ti d l ti perspective. They are complementary; neither is a complete story. ith i l t t • Classical dynamics of Newton and Laplace has singular distribution, quantum dynamics has distribution but no trajectory, stochastic process requires both.

For Stochastic Process with Continuous Paths • Its trajectory can be described by a • Its trajectory can be described by a stochastic differential equation (generalized nonlinear Langevin equation) (generalized nonlinear Langevin equation) • Its distribution is described by a Fokker- Planck (Kolmogorov forward) equation. Pl k (K l f d) ti   dx t b x dt σ x dB t ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) )       2 f x t σ f x t x ( , ) ( , )     ( )        b b x x f f x x t t ( ( ) ) ( ( , ) )         t x 2 x x  

For Stochastic Process with Discrete States & Jumps • Its trajectory can be described by the Bortz-Kalos-Lebowitz-Gillespie algorithm • Its distribution is described by master equation q       ξ j ξ i q dt i j Pr | ( )  t dt t ij dp n t ( , )    p m t q p n t q ( , ) ( , ) mn nm dt dt m

A disclaimer …

“Generalized” Energy, Entropy and gy, py Free Energy in a Markov System Let us assume a Markov dynamics has a unique stationary (invariant) distribution. This means that there is a probability based “force” pushing a system from low probability to high probability:   ss E p ln n n [Haken & Graham, Kubo et al ., Nicolis & Lefevere, Ao]

“Generalized” Energy Entropy “Generalized” Energy, Entropy and Free Energy – Cont. gy Then one has the energetics of the system:         ss E t p t p S t p t p t ( ) ( ) ln , ( ) ( ) ln ( ); n n n n n n   p t ( )      n n F F t t E E t t S S t t p p t t ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) ln ln     n ss p   n n F(t) is also known relative entropy.

source sink Then we uniquely have Then we uniquely have, dF dF ( t t ( ) )   E t e t ( ) ( ); in p dt dt         p p ( q t t q ( ) )    i ij  e t p t q p t q 0   ( ) ( ) ( ) ln ; p i ij j ji p t q   ( )   i j , j ji   ss       p q  i ij   E t p p t q q p p t q q     0 ( ( ) ) ( ( ) ) ( ( ) ) ln . in in i i ij ij j j ji ji ss ss p q       i j , j ji

Non-negative energy input E in   s p q 1       i ij E p q p q ( ) ln     in i ij j j j ji j s 2 2 p p q q     i j , j ji     s s p q p q         i ij j ji    p p q q p p q q ln ln ln ln         i i ij ij i i ij ij s s p q p q     i j i j , j ji , i ij     s s p q p q             j j ji ji j j ji ji      p q p q 1 1 ln l     i ij i ij s s p q p q       i j i i j i , i ij , i ij         s s p p q q p p p p q q           j ji i j ji     p q 1 p q     i ij i ij s s p q p     i j i j , i ij , i p p        s p q i p q 0 . i ij j ji s p i j i j i

Energy balance equation for a gy q subsystem is a generalization of energy conservation of an energy conservation of an isolated system: We interpret y p this mathematical result as “the 1 st L 1 st Law of thermodynamics”. f th d i ”

Furthermore we have Furthermore, we have dF t ( )   ,   0 0 or or e e E E p in dt We interpret this mathematical result as “the 2 nd Law of Thermodynamics” Thermodynamics .

Non-positive free energy change p gy g     dp p p dF            j j j ln ln ( ( ) ) ln ln p p q q p p q q         i i ij ij j j ji ji s s dt dt p p     j i , j j j       p p             j   i ln ln ln ln p p q q p p q q         i i ij ij i i ij ij s s     p p     , i j j i     s s p p p p           j i j i ln ln ln ln p p q q p p q q         i i ij ij i i ij ij s s p p p p      , , i j i j i j i j i     s s p p p p             j i j i p p q q 1 p p q q 1         i i ij ij i i ij ij s s p p p p      , , i j i i j j i j i s p p q p               j i ij j s p q p q p q i i ij ij i i ij ij i i ij ij s s s s p p , i j j i i j j j  . 0

An alternative interpretation: An alternative interpretation: E in is also known as house- Boltzmann’s thesis keeping heat Q hk Q Prigogine’s thesis (Oono and Paniconi, 1998) (Oono and Paniconi, 1998) dF dF ( t t ( ) )     e Q 0 . p hk dt dt Two origins of irreversibility, e p is p the total entropy production.

For System with Detailed For System with Detailed Balance:   ss       p q   i ij  E t p p t q q p p t q q 0     ( ( ) ) ( ( ) ) ( ( ) ) ln . in in i i ij ij j j ji ji ss ss p q       i j j ji , Then Then, dF t ( )   e t ( ); p p dt dt