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

• f 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
• f 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

• f 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 d l f li f hi h To develop a formalism from which one can deduce the macroscopic behavior of l t f ifi ti f complex systems from a specification of the components, the laws of force which d i d th t f th i govern dynamics, and the nature of their surroundings.

In Traditional Physics: y

• A law of a force is an interaction between

particles; particles;

• However, as many physical chemists

k ll t i f i t f know well, entropic force is not a force between particles; in fact it is an emergent tit l ti l l Th Fi k’ entity on a population level. The Fick’s law makes no sense on an individual B i ti l l l Brownian particle level;

• Still, a force is something that causes a

system to change.

What is thermodynamics? What is thermodynamics?

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

What is Kolmogorov’s Stochastic Process?

It i th ti l d i ti f

• It is a mathematical description of

dynamics with “uncertainties”. It has both t j t ti d l ti a trajectory perspective and a population

• perspective. They are complementary;

ith i l t t neither is a complete story.

• 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-

Pl k (K l f d) ti Planck (Kolmogorov forward) equation. ) ( ) ( ) ( ) ( t dB x σ dt x b t dx  

 

) ( ) ( ) , ( ) ( ) , ( ) ( ) ( ) ( ) ( t x f x b t x f x σ t x f

2

           

 

) , ( ) ( t x f x b x x 2 x t           

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

 

   

 ij t dt t

j i dt q i ξ j ξ ) ( | Pr



 

nm mn

q t n p q t m p dt t n dp ) , ( ) , ( ) , (

m

dt

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 n n

p E ln  

[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:

 

   

n n ss n n

t p t p t S p t p t E ); ( ln ) ( ) ( , ln ) ( ) (

  

   

n n n

t p t p t S t E t F ) ( ln ) ( ) ( ) ( ) (



      

n ss n n n

p t p t S t E t F ln ) ( ) ( ) ( ) ( F(t) is also known relative entropy.

Then we uniquely have

source sink

Then we uniquely have, ) (t dF ); ( ) ( ) ( t e t E dt t dF

p in

 

 

) ( q t p  

dt

 

; ) ( ) ( ln ) ( ) ( ) (

,

q t p q t p q t p q t p t e

ji j ij i j i ji j ij i p

          

 

. ln ) ( ) ( ) ( q p q t p q t p t E

ss ij ss i ji j ij i in

        



) ( ) ( ) (

,

q p q p q p

ji ss j j i ji j ij i in

     



Non-negative energy input Ein

ln ) ( q p q p q p q p 2 1 E

s ij s i ji j ij i in

           ln ln

,

q p q p q p q p q p 2

ji s j ij i ij s i ij i j i ji j j j j

                  

  

l ln ln

, ,

q p 1 q p q p q p q p q p

ji s j ji s j j i ij s i ij i j i ji s j ij i

                

   

ln

, ,

q p p q p q p 1 q p q p q p

s s i j i ij s i ji j ij i i j i ij s i ji j ij i

                       

 

  , ,

p p q p p q p q p q p 1 q p

j i s i ji j i ij i j i ij s i ji j ij i

                   

 

. q p p p q p

j ji s j i s i i i j ij 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 1st L f th d i ” 1st Law of thermodynamics”.

Furthermore we have Furthermore, we have

E e

• r

t dF   ) (

in p

E e

• r

dt   ,

We interpret this mathematical result as “the 2nd Law of Thermodynamics” Thermodynamics .

Non-positive free energy change p gy g

ln ) ( ln p q p q p p dp dF

j ji j ij i j j

              

 

ln ln ln ) ( ln

,

p q p p q p p q p q p p dt dt

i ij i j ij i j i s j ji j ij i s j j

                         

  

ln ln ln ln

,

p p q p p p q p p q p p q p

s i j ij i s i j ij i j i s i ij i s j ij i

                             

  

ln ln

, ,

1 p p q p 1 p p q p p p q p p p q p

s i j ij i s i j ij i i j i i s j ij i j i i s j ij i

                       

   

 , ,

q p q p p q p q p p p p q p p p q p

ij i ij s i s j ij i s ij s i j j i i s j ij i i j i i s j ij i

           

      



.

,

p p

j ij i i j i ij i s j j i ij i s j



    

An alternative interpretation: An alternative interpretation: Ein is also known as house- Q

Boltzmann’s thesis

keeping heat Qhk

(Oono and Paniconi, 1998)

Prigogine’s thesis

(Oono and Paniconi, 1998)

) (t dF . ) ( Q dt t dF e

hk p

    dt

Two origins of irreversibility, ep is

p

the total entropy production.

For System with Detailed For System with Detailed Balance:

 

. ln ) ( ) ( ) ( q p q t p q t p t E

ss ij ss i ji j ij i in

        



) ( ) ( ) (

,

q p q p q p

ji ss j j i ji j ij i in

     



Then Then,

); ( ) ( t e dt t dF

p

  dt

p

This is known to Gibbs: While for This is known to Gibbs: While for canonical ensemble the appropriate potential function is free energy not entropy but the free energy, not entropy, but the

• rigin of 2nd Law is still entropy

production.

For System with detailed For System with detailed balance and uniform stationary distribution:

. ln ) ( ) ( const p t p t E

n ss n n

    . ) ( ) ( const t S t F

n

  

This is a microcanonical ensemble.

Entropy Balance Equation py q (de Groot and Mazur, 1962)

; ) ( ) ( ) (    

e i d p

dt S d dt S d t h t e dt t dS

  l

) ( ) ( ) (    



ij

q t t t h dt dt dt

 

. ln ) ( ) ( ) (

,

       

ji ij j i ji j ij i d

q q t p q t p t h

[Onsager (1931), Eckart & Bridgman (1940), Prigogine (1945) de Groot (1951) Prigogine (1945), de Groot (1951), Bergmann & Lebowitz (1955)]

Now Nonlinear Stochastic Dynamics Dynamics …

We consider Markov processes We consider Markov processes with continuous path:

) ( ) ( ) ( ) ( t dB x σ dt x b t dx  

 

] [ ) ( ) ( ) ( ) ( ) , ( f L x f x b x f x A t t x f        

 

) ( f t 

W ld lik t i t d We would like to introduce a symmetric-anti-symmetric symmetric anti symmetric decomposition for L[f ]. To do th t i t d i that, we introduce an inner product: product:

 

1 



  dx

x f x φ x ψ φ ψ

1 ss



 ) ( ) ( ) ( ) , (

Then we have Then we have

   

) ( ) ( ) ( x φ x b φ A φ L       

   

) ( ) ( , ) ( ) ( ) ( φ L φ L x φ x b φ A φ L

A S

        ) ( ) ( φ φ

A S

   ,

) ( ) ( ln ) ( x φ x f A φ A φ L

ss S

     

       ,

) ( ) ( ) ( ln ) ( , ) ( ) ( ln ) ( x φ x b x f A φ L x φ x f A φ A φ L

ss A S

       

     

) ( ) ( ) ( , ) ( ) ( ) ( ) ( x f x j x j dx φ f φ

ss A

   

 

. ) ( ) ( ), ( x f x j x j dt   

More importantly More importantly

For dynamics with only the symmetric part

u  . ) ( ], [ t E u L t u

in S

   

F d i i h l h For dynamics with only the Anti-symmetric part

. ) ( ], [ t dF u L u

A

  

y p

. ], [ dt u L t

A



The symmetric and anti- symmetric parts of the dynamics symmetric parts of the dynamics generalize nicely Fourier’s g y dissipative dynamics (heat eq ation) and Ne ton’s equation) and Newton’s conservative dynamics (volume y ( preserving).

• I. Prigogine
• I. Prigogine

Nobel Lecture (1977)

[L]et us emphasize that one hundred fifty years after its formulation the second law of years after its formulation, the second law of thermodynamics still appears more as a program than a well defined theory in the program than a well defined theory in the usual sense, as nothing precise (except the sign) is said about the entropy production sign) is said about the entropy production. Even the range of validity of this inequality is left unspecified left unspecified.

Our Major Claim Our Major Claim

For complex systems the thermodynamic For complex systems, the thermodynamic laws are consequences of (nonlinear) dynamical descriptions of a system with dynamical descriptions of a system with

• stochastic. The mathematical theory of

stochastic processes for mesoscopic stochastic processes for mesoscopic systems supports a (equilibrium and nonequilibrium) thermodynamic structure nonequilibrium) thermodynamic structure which consists of both 1st and 2nd Laws.

The validity of non-equilibirum The validity of non equilibirum thermodynamics, therefore, no longer relies on “local equilibrium longer relies on local equilibrium assumption” as in the past. Rather th b d i hift d t th lidit f the burden is shifted to the validity of a Markovian description of a natural process, be it from physics, biology, economics, or sociology. There is , gy absolutely no assumption on linearity! linearity!

Thermodynamic relations are y not natural laws; they are th ti l th ith mathematical theorems with applications in nature. pp Dynamics is more f d t l it i th d l f fundamental; it is the model for natural phenomena. natural phenomena.

Thermodynamic relations are y not natural laws; they are th ti l th ith mathematical theorems with applications in nature. pp Dynamics is more f d t l it i th d l f fundamental; it is the model for natural phenomena. natural phenomena. Thermodynamics, however, is absolute.

Two further developments in Two further developments in the making: (1) Temperature as a measure (1) Temperature as a measure

• f “distance” between a

mesoscopic system and its deterministic limit; zeroth law deterministic limit; zeroth law and third laws;

(2) Stochastic Partial Differential Equations with real physical space (i e reaction diffusion) space (i.e., reaction-diffusion)

P Chi f li

• Pope-Ching formalism;
• This is essentially the fluctuating

hydrodynamic formalism;

• Again, Langevin and Fokker-Planck are

g , g just two different perspectives of a same dynamic process. y p

Thank You!