Introduction to Large Deviation Principle and its applications

Hao Ge1Biodynamic Optical Imaging Center (BIOPIC) 2Beijing International Center for Mathematical

Research (BICMR)Peking University, China

Mathematical foundation of statistical mechanics

Mathematical analysis Classic mechanics

Statistical mechanics?

Mathematical structure of statistical mechanics?

Only math can be applied beyond physics!

Law of Large Number

,...,...,, 21 nXXX Independent, identical distributed random variables

nn XXXS +⋅⋅⋅++= 21

+∞→=→ nXnSn ,1µ

0,,0 >∀+∞→→

>− εεµ n

nSP n

Strong version

Weak version

Central Limit Theorem

( )122

1

var,21)(

0,,)(

2

Xey

ndyyxn

nSP

y

xn

==

>∀+∞→→

<

−

−

∞−∫

σπ

φ

εφσµ

( ) ( ) 22 2/

21 σµ

πσnnS

nne

nSp −−≈

Central Limit Theorem

Variance = nσ2

Sn

µn

P(Sn)

Cramer’s theorem

)(~ xnIn exnSP −

=

{ })(sup)( θθθ

CxxI −=

∫ == )(log)( 1 xXPeC xθθ Cumulant generating function

Rate function

{ } 0)(inf)( == xIIx

µ 0)(' =µI

21)(''

σµ =I ( ) 1,

21)( 2

2 <<−−≈ µµσ

xxxI

)(log1 xIxnSP

nn −→

=

Central limit theorem

Sanov theorem

Empirical distribution:

∑=

=n

ijXjn in

L1

,,1 δ

( ) ∑=

=M

i i

iiM p

lllllI1

21 ln,...,,

jpjXP == )( 1

( ) ),...,(,

1~,...,2,1, MllnIjjn eMjlLP −==

Relative entropy

Gibbs conditioning

{Xi | i=1,2,…,n} i.i.d. with probability density fpriori

Under the conditiong(X1) + g(X2) +…+ g(Xn) = α = na

What is the asymptotic posterior distribution?

µ≠a

( )naXgXgXgXPxf nnposterior =+++=∞→

)(...)()(|lim)( 211

Minimum relative entropy principle

Xi

Xj

Now, if n is very large, then a is essentially the expected value of the posterior distribution for each Xi!Minimum relative entropy:

dxxfxf

xfpriori

posteriorposterior )(

)(ln)(∫

subjected to: .)()( adxxfxg posterior =∫)()()( xg

prioriposterior exfxf β−∝

Varadhan theorem

Generalization of Laplace method

{ }.)()(supln1lim)( )( xIxfen

fx

Anf

nn −==

∞→λ

( ) )(~ xnIn exAP −=If

Then

Varadhan, SRS (1966). "Asymptotic probabilities and differential equations". Communications on Pure and Applied Mathematics 19 (3): 261–286

(Large deviation)

Contraction principle

( ) )(~ xnIn exAP −=If

Then ( )( ) )(~ ynJn eyAfP −=

)(inf)()(

xIyJyxf =

=

== ∑=

xillllIxIM

iiM

121 :),...,,(inf)(E.g.

Gartner-Ellis theoremHugo Touchette, Phys. Rep. 478, 1-69 (2009)Ellis, Entropy, Large Deviation and Statistical Mechanics. (1984)

nnkA

ne

nk ln1lim)(

∞→=λIf

exists and is differentiable

Then ( ) )(~ xnIn exAP −=

{ }.)(sup)( kkxxIk

λ−=

Statistical mechanics

{ }.)(sup)( kkxxIk

λ−=Legendre–Fenchel transformation

)()( ** kxkxI λ−= )(' *kx λ=

entropy Free energy/temperature energy 1/temperature

Hugo Touchette, Phys. Rep. 478, 1-69 (2009)Ellis, Entropy, Large Deviation and Statistical Mechanics. (1984)

)(~ xnIn exnhP −

=

energy

Markov processDonsker and Varadhan: Asymptotic evaluation of certain Markov process expectations for large time. I, Communications on Pure and Applied Mathematics 28 (1975), pp. 1–47; part II, 28 (1975), pp. 279–301; part III, 29 (1976), pp. 389–461; part IV, 36 (1983), pp. 183–212 Dembo and Zeitouni, Large Deviations Techniques and Applications (2009)

{ },...,...,, 21 nXXX Finite state Markov chain

( ) ijkk piXjXP ===+ |1 ∑=

=n

ikn Xf

nA

1

)(1

)(),( jfijepjiP λ

λ =( ) )(~ xnIn exAP −=

( ){ }λλ

ρλ PxxI logsup)( −= Perron-Frobenius eigenvalue

Freidlin-Wentzell theoryFreidlin and Wentzell: Random Perturbation of Dynamical System. (1983)

Ge and Qian: PRL (2009), JRSI (2011), Chaos (2012)

( ) ttt Xb

dtdX η+= Aststt εδηηη ,,0 ==

{ }( ) ( )( ) ( ) ( )( )∫=

− −−=≤≤T

ssss

Tsss dsxbxxAxbxTsxI

0

'1'21

0:

{ }( ) { }( )TssxI

ss eTsxXP≤≤−

≤≤=0:1

~0: ε

Strongly dependent on the Gaussian feature of the white noise

Freidlin-Wentzell theory

A nonequilibrium generalization of Kramer’s rate theory

{ }( ) ( )( ) ( ) ( )( )∫=

− −−=≤≤T

ssss

Tsss dsxbxxAxbxTsxI

0

'1'21

0:

O1

O2

Saddle

Φ

Transition rate

( ) { }( )TsxIx sxTxOx≤≤=Φ

==0:inf

,10

ε∆Φ

−

∝ ekk 0

( )0

)(≤

Φdt

txd( )tt xb

dtdx

=

Quasipotential/landscape

Emergent dynamic landscape

)(loglim)(0

xpx ssalso

εεεφ

→−=

Global minimum

Local minimum

Maximum: the barrier

Stable fixed points of deterministic models

Unstable fixed point of deterministic models

( )tt xb

dtdx

=

Relative stability of stable steady states

Many nonlinear dynamical systems have multiple, locally stable steady states.Is one attractor more “important” than another?

( ) .0,1exp)( →

−≈ εφ

εε xxpss

The most important steady state when V is large would be the global minimum of dynamic landscape.

Maxwell construction

),( θxbdtdx

=θ *

φ (x,θ )

θ

Steady States x*

x

Global minimum abruptly transferred.

Ge and Qian: PRL (2009), JRSI (2011)

Nonequilibrium phase transition

( ) ( ){ }kkck

φλλ −= sup ( ) ( ){ }kckxxck

−= sup*

Ge and Qian: PRL (2009), JRSI (2011)

Nonequilibrium phase transitionGe and Qian: PRL (2009), JRSI (2011)

alternative attractor

2: fluctuating in local attractor, waiting

1: relaxation process

3: abrupt transition via barrier-crossing

The uphill dynamics is the rare event, related to phenotype switching, punctuated transition in evolution, et al.

Dynamics of bistable systems

Intra-attractorial dynamics

Inter-attractorial dynamics

Local-global confliction

Global landscape: stationary distribution

Just cut and glue on the local landscapes (non-derivative point).

The emergent Markovian jumping process being nonequilibrium is equivalent to the discontinuity of the local landscapes (time symmetry breaking).

Dynamic on a ring as an example.

Local landscapes

Kramers’ rate formula

Ge and Qian: Chaos (2012)

Chemical master equation

Chemical Master Equation

Gillespie algorithm Langevin dynamics

Fokker-Planck equation

Gillespie algorithm (GA) is really an equation that describes the dynamic trajectory of chemical master equation.

Trajectory view

Probability distribution view

Discrete Markov Chain

Diffusion process

Phosphorylation cycle and GTPase cycle are isomorphic

Also isomorphic to self-regulating gene system

Self-regulating gene

Phosphorylation-dephosphorylation cycle

With positive feedback

The dimer case χ=2

E E*

K*

ATP ADP

P

Pi

a1

a-1

a2

a-2

K and K* are inactive and active forms of a kinase. E* is the phosphorylated E, a signaling molecule. Usually E∗ is functionally active, i.e., “turned-on”.

E E*

K

P

2E*0E* 1E* 3E* … (N-1)E* NE*

Chemical master equation Representation

v1

w1

v2

w2

v0

w0

.))2)(1((

),1)()2)(1((

2

2

nnnV

w

nNnnV

v

n

n

δε

β

δα

+−−+=

+−+−−=

Emergent dynamic landscape

)(uφ

Global minimum

Local minimum

Maximum: the barrier

Stable fixed points of deterministic models

Unstable fixed point of deterministic models

( )( )

.,

,exp)(

+∞→=

−∝

VVnu

uVnpssV φ

Maxwell construction

),( θubdtdu

=θ *

φ (u,θ )

θ

Steady States u*

u

Global minimum abruptly transferred.

Ge and Qian: PRL (2009), JRSI (2011)

Kramers’ theory for CME

., 12211221

→→ −→

−→ ∝∝ VHVH eTeT

21→H

The switching time between attractors:

12→H

The barrier H here may not be the same as the barrier in the global landscape φ(u) for high dimensional multistable cases.

A B

discrete stochastic model among attractors

ny

nx

chemical master equation cy

cx

A

B

fast nonlinear differential equations

appropriate reaction coordinate

A Bprob

abil

ity

emergent slow stochastic dynamics and landscape

(a) (b)

(c)(d)

Three time scalesFixed finite molecule numbers

Stochastic

Stochastic

Deterministic

Ge and Qian: PRL (2009), JRSI (2011)

Possible Chemical basis of epi-genetics:

Could this be a chemical definition for epi-genetics inheritance: Exactly same environment setting and gene, different internal biochemical states (i.e., concentrations and fluxes)?

It is likely that the code for epi-genetic inheritance is distributed.

Could it be readily inherited during the process of cell volume change and division?

Chemistry may be inheritable

Change V and N by a factor of 2 at the beginning

Acknowledgement

Prof. Hong Qian

University of WashingtonDepartment of Applied Mathematics

My collaborator:

Thanks for your attention!