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Short Course

Mathematical Molecular BiologyBob Eisenberg

Shanghai Jiao Tong University

Sponsor Zhenli Xu

u

2

How can we use mathematics to describe biological systems?

I believe some biology isPhysics as usualGuess and Check

But you have to know which biology!

u

3

All of Biology occurs in Salt Solutions of definite composition and concentration

and that matters!

Salt Water is the Liquid of LifePure H2O is toxic to cells and molecules!

Salt Water is a Complex FluidMain Ions are Hard Spheres, close enough

Sodium Na+ Potassium K+ Calcium Ca2+ Chloride Cl-

3

K+Na+ Ca++ Cl-

Page 4

Trajectories in Condensed Phases are Noisy

Note: Brownian noise looks the same on all scales! Function has unbounded variation, crossing any line an infinite number of times in any interval no matter how small.

Page 5

Instruction to BobShow Videos!!!

Page 6

Page 7

Trajectories Over Barriers in Condensed Phases are Noisy

Barcilon, Chen, Ratner, Eisenberg

9

From Trajectories to Probabilitiesin Diffusion Processes

Life Work of Zeev Schuss Department of Mathematics, Tel Aviv University

Theory and Applications of Stochastic Differential Equations. 1980 John Wiley

Theory And Applications Of Stochastic Processes: An Analytical Approach 2009 Springer

Singular perturbation methods for stochastic differential equations of mathematical physics.

SIAM Review, 1980 22: p. 116-155.

Schuss, Nadler, Singer, Eisenberg

10

Here I start from

Stochastic PDEand

Field Theory

Other methodsgive nearly identical results

MSA (Mean Spherical Approximation)SPM (Primitive Solvent Model)

Non-equil MMC (Boda, Gillespie) several forms

DFT (Density Functional Theory of fluids, not electrons)DFT-PNP (Poisson Nernst Planck)

EnVarA (Energy Variational Approach)Steric PNP (simplified EnVarA)

Poisson Fermi

MATHField

Theory

ChemistryModels

11

Solved with PNP including Correlations

Other methodsgive nearly identical results

MMC Metropolis Monte Carlo (equilibrium only)DFT (Density Functional Theory of fluids, not electrons)

DFT-PNP (Poisson Nernst Planck)MSA (Mean Spherical Approximation)

SPM (Primitive Solvent Model)EnVarA (Energy Variational Approach)

Non-equil MMC (Boda, Gillespie) several formsSteric PNP (simplified EnVarA)

Poisson Fermi

12

Always start with Trajectoriesbecause

Always start with TrajectoriesZeev Schuss

Department of Mathematics, Tel Aviv University

1) Trajectories are the equivalent of SAMPLES in probability theory

2) Trajectories satisfy PHYSICAL boundary conditions

3) Trajectories satisfy classical PHYSICAL ordinary differential equations(we hope)

13

From Trajectories to Probabilitiesin Diffusion Processes

Life Work of Zeev Schuss Department of Mathematics, Tel Aviv University

Theory and Applications of Stochastic Differential Equations1980

Theory and Applications Of Stochastic Processes: An Analytical Approach 2009

Singular perturbation methods for stochastic differential equations of mathematical physics

SIAM Review, 1980 22: 116-155

Schuss, Nadler, Singer, Eisenberg

Page 14

Trajectories in Condensed Phases are Noisy

Note: Brownian noise looks the same on all scales! Function has unbounded variation, crossing any line an infinite number of times in any interval no matter how small.

15

We start with Langevin equations of charged particlesSimplest stochastic trajectories

areBrownian Motion of Charged Particles

Einstein, Smoluchowski, and Langevin ignored chargeand therefore

do not describe Brownian motion of ions in solutions

Once we learn to count Trajectories of Brownian Motion of Charge,we can count trajectories of Molecular Dynamics

Opportunity and Need

We useTheory of Stochastic Processes

to go

from Trajectories to Probabilities

Schuss, Nadler, Singer, Eisenberg

16Schuss, Nadler, Singer, Eisenberg

Langevin Equations Bulk Solution

; 2kpk x q pp pkk k

f kTm mx x w

Positive cat ion, e.g., p = Na+

;

Newton's Law Friction & Noise

2knk x q nn nkk k

f kTm mx x w

Negative an ion, e.g., n = Cl

Global Electric Forcefrom all charges including

Permanent charge of Protein,Dielectric Boundary charges,Boundary condition charge

17

00

0( ) ( ) ( ) ( )div ( )xs k ik

i

px x ix x

ef q xe Ef ze

Electric Force in Ion Channelsnot assumed

Excess Chemical

Force

All Spheres Implicit Solvent

Primitive Model

GLOBAL Electric Forcefrom all charges including

Permanent charge of Protein,Dielectric Boundary charges,Boundary condition charge,

MOBILE IONS

Schuss, Nadler, Singer, Eisenberg

Total Force

18

From Trajectories to Probabilities

Joint probability density of position and velocity

Coordinates are positions and velocities of N particles in 12N dimensional phase space

Schuss, Nadler, Singer, Eisenberg

c cj j

c c c c c

x vj j j j j

cj cj

kTmp v p v v f m p p L

with Fokker Planck Operator

,, x vp x v Pr 2Nj = 1

Main Result of Theory of Stochastic ProcessesSum the trajectories

Sum satisfies Fokker-Planck equation

More MathMany papers

We actually performed the sum and showed it was the same as a MARGINAL PROBABILITY estimator of SINGLET CONCENTRATION defined in chemistry

We actually did a nonequilibrium BBGKY expansion with electrostatic & steric correlations

We like everyone else had to assume a closure

Page

1) Nadler, B., T. Naeh and Z. Schuss (2001). SIAM J Appl Math 62: 443-447.2) Nadler, B., T. Naeh and Z. Schuss (2003). "SIAM J Appl Math 63: 850-873.

3) Nadler, B., Z. Schuss and A. Singer (2005). "" Physical Review Letters 94(21): 218101.

4) Nadler, B., Z. Schuss, A. Singer and B. Eisenberg (2003). Nanotechnology 3: 439.

5) Nadler, B., Z. Schuss, A. Singer and R. Eisenberg (2004). Journal of Physics: Condensed Matter 16: S2153-S2165.

6) Schuss, Z., B. Nadler and R. S. Eisenberg (2001). Physical Review E 64: 036116 1-14.

7) Schuss, Z., B. Nadler and R. S. Eisenberg (2001). "Phys Rev E Stat Nonlin Soft Matter Phys 64(3 Pt 2): 036116.

8) Schuss, Z, B Nadler, A Singer, R Eisenberg (2002) Unsolved Problems Noise & Fluctuations, UPoN 2002, Washington, DC AIP

9) Singer, A, Z Schuss, B Nadler, R. Eisenberg (2004) Physical Review E Statistical Nonlinear Soft Matter Physics 70 061106.

10) Singer, A, Z Schuss, B Nadler, R Eisenberg (2004). Fluctuations & Noise in Biological Systems II V. 5467. D. Abbot, S. M.

20

Finite OPEN System Fokker Planck EquationBoltzmann Distribution

TrajectoriesConfigurations

NonequilibriumEquilibriumThermodynamics Schuss, Nadler, Singer &

Eisenberg

Thermodynamics Device Equation

lim ,N V

StatisticalMechanics

Theory of Stochastic Processes

Conditional PNP

0 |

| |

yy ye y x

y y

y

x x

P

|1 0y y xy yx

xx em

Other Forces

21

Electric Force depends on Conditional Density of Charge

Nernst-Planck gives UNconditional Density of Charge

Schuss, Nadler, Singer, Eisenberg

Closure Needed: CORRELATIONSGuess and Check

22

Probability and Conditional Probability are

Measures on DIFFERENT Sets

that may be VERY DIFFERENT

Considerall trajectories that end on the right

vs.all trajectories that end on the left

23

Conditioning and Correlations are VERY strong and GLOBAL

when Electric Fields are Involved, as in

Ionic Solutions and Channels

so cannot do the probability theorywithout variational methods

We had to guessthe conditioned sets

24

Main Biological Ions are Hard Spheres, close enough

Sodium Na+ Potassium K+ Calcium Ca2+ Chloride Cl-

General Theory of Hard Spheresis now available

Thanks to Chun Liu, more than anyone else

Took a long time, because dissipation, multiple fields, and multiple ion types had to be included

VARIATIONAL APPROACH IS NEEDED

3

K+Na+ Ca++ Cl-

Page 25

EnVarA

212 ( log log )B n n p pk T c c c c E dx

Microscopic

Finite Size EffectElectrostatic Entropy

(atomic)

Solid Spheres

212 ( )IPE t u w

Macroscopic

Hydrodynamc Potential EnergyHydrodynamcEquation of StateKinetic Energy

(hydrodynamic)

Primitive Phase;

26

EnergeticVariationalApproachEnVarA

ChunLiu,RolfRyham,andYunkyong Hyon

Mathematicians and Modelers: two different partial variationswritten in one framework, using a pullback of the action integral

12 0

E

'' Dissipative 'Force'Conservative Force

x u

Action Integral, after pullback Rayleigh Dissipation Function

FieldTheoryofIonicSolutions:Liu,Ryham,Hyon,EisenbergAllowsboundaryconditionsandflow

DealsConsistentlywithInteractionsofComponents

Composite

Variational Principle

Euler Lagrange Equations

Shorthand for Euler Lagrange processwith respect to

x

Shorthand for Euler Lagrange processwith respect to

u