Short Course Mathematical Molecular Biology - THIRD DAY... · Short Course Mathematical Molecular Biology

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  • 1

    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