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An application of Girsanovs theorem to a model of molecular dynamics - - PowerPoint PPT Presentation

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An application of Girsanovs theorem to a model of molecular dynamics Han C. Lie (joint work with C. Hartmann, Ch. Sch utte) Berlin-Padova Young Researchers Meeting Friday, 24th October 2014 Berlin, Germany Why study molecules? Molecular

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An application of Girsanov’s theorem to a model of molecular dynamics

Han C. Lie (joint work with C. Hartmann, Ch. Sch¨ utte)

Berlin-Padova Young Researchers Meeting Friday, 24th October 2014 Berlin, Germany

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Why study molecules?

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Molecular biology and drug design

Principle: structure ↔ function

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Molecular biology and drug design

Principle: structure ↔ function Protein aggregation

◮ Example: Blood clot formation

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Molecular biology and drug design

Principle: structure ↔ function Protein aggregation

◮ Example: Blood clot formation ◮ Cause: Change in spatial structure

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Molecular biology and drug design

Principle: structure ↔ function Protein aggregation

◮ Example: Blood clot formation ◮ Cause: Change in spatial structure

Features of macromolecules with N atoms

◮ High-dimensional state space: 3N degrees of freedom

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Molecular biology and drug design

Principle: structure ↔ function Protein aggregation

◮ Example: Blood clot formation ◮ Cause: Change in spatial structure

Features of macromolecules with N atoms

◮ High-dimensional state space: 3N degrees of freedom ◮ Low-dimensional effective dynamics

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Molecular biology and drug design

Principle: structure ↔ function Protein aggregation

◮ Example: Blood clot formation ◮ Cause: Change in spatial structure

Features of macromolecules with N atoms

◮ High-dimensional state space: 3N degrees of freedom ◮ Low-dimensional effective dynamics ◮ Metastabilities: Almost-stable spatial structures

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Metastabilities: Almost-stable structures

n-pentane (C5H12): 3N = 51 degrees of freedom

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Metastabilities: Almost-stable structures

n-pentane (C5H12): 3N = 51 degrees of freedom

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Rare events: Structural changes

Effective dynamics in 2D landscape Metastable set 1 Structure 1

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Rare events: Structural changes

Effective dynamics in 2D landscape Metastable set 1 Structure 1

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Rare events: Structural changes

Effective dynamics in 2D landscape Metastable set 1 Structure 1

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Rare events: Structural changes

Transition to Metastable set 2 Change to Structure 2

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Some goals in studying molecules

Study changes between almost-stable structures

◮ Freidlin-Wentzell theory: Exit from attractor

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Some goals in studying molecules

Study changes between almost-stable structures

◮ Freidlin-Wentzell theory: Exit from attractor

Compute statistics

◮ Escape probabilities ◮ Escape times

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Why study molecules?

◮ Structure ↔ function: applications in medicine ◮ Rare events: changes in geometric structure ◮ High-dimensional data, low-dimensional phenomena

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Molecular dynamics (MD)

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Model: Position Langevin Dynamics

Molecule in equilibrium with heat bath: dXt = −∇V(Xt)dt +

  • 2kBTdBt

◮ Xt ∈ R3N - position coordinates for N atoms ◮ V - energy function ◮ T - temperature ◮ kB - Boltzmann’s constant ◮ Bt - Brownian motion w.r.t. P

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Model: Position Langevin Dynamics

Molecule in equilibrium with heat bath: dXt = −∇V(Xt)dt +

  • 2kBTdBt

◮ Xt ∈ R3N - position coordinates for N atoms ◮ V - energy function ◮ T - temperature ◮ kB - Boltzmann’s constant ◮ Bt - Brownian motion w.r.t. P

Numerical integration → time series data

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Rare events

Goal: Study A → B event Constraints:

  • Large barriers ∆FAB
  • Small thermal ‘kicks’ kBT

Example: amylases ∆FAB ≈ 208 kJ/mol kBT ≈ 2.4 kJ/mol Long waiting times

Graph of V

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An Application of Girsanov’s theorem

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Model: Position Langevin Dynamics

Molecule in equilibrium with heat bath: dXt = −∇V(Xt)dt + √ 2εdBt

◮ Bt - random fluctuations due to heat bath interactions

⇒ P - equilibrium measure

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Statistics of path functionals

Goal: Compute Px-expectation of W[X] = τ(S) f(Xt)dt + g(Xτ)

◮ f - running cost ◮ g - terminal cost ◮ τ(S) - first hitting time to some set S ◮ Px - path measure conditioned on {X0 = x}

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Model: Position Langevin Dynamics

Molecule not in equilibrium with heat bath: dXt = c(Xt) − ∇V(Xt)dt + √ 2εdBt

◮ c(Xt) - feedback control force ◮ Q - nonequilibrium measure due to control c

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Model: Position Langevin Dynamics

Molecule not in equilibrium with heat bath: dXt = c(Xt) − ∇V(Xt)dt + √ 2εdBt

◮ c(Xt) - feedback control force ◮ Q - nonequilibrium measure due to control c

Change of drift −∇V → (c − ∇V)

  • Change of measure P → Q
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Computing statistics by reweighting

Girsanov change of measure dPt dQt (X) = exp 1 √ 2ε t c(Xt)dBt − 1 4ε t |c(Xt)|2dt

  • Reweighting of nonequilibrium measurements:

Ex

P [W] = Ex Q

  • W dP

dQ

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Computing statistics by stochastic control

Duality relationships between free energy, relative entropy (Dai Pra, Meneghini and Runggaldier, 1996) − log EP [exp(−W/ε)] = inf

Q∈AC(P) {εR(QP) + EQ [W]} ◮ AC(P) - set of all Q abs. cont. w.r.t. P ◮ R(QP) = EQ[log dQ dP ] relative entropy of Q w.r.t. P ◮ W bounded

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Computing statistics by stochastic control

Since dPt dQt (X) = exp 1 √ 2ε t c(Xt)dBt − 1 4ε t |c(Xt)|2dt

  • we have

εR(QtPt) = EQ

  • log dQt

dPt

  • = EQ

1 4 t |c(Xs)|2ds

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Computing statistics by stochastic control

Since dPt dQt (X) = exp 1 √ 2ε t c(Xt)dBt − 1 4ε t |c(Xt)|2dt

  • we have

εR(QtPt) = EQ

  • log dQt

dPt

  • = EQ

1 4 t |c(Xs)|2ds

  • − log EP [exp(−W/ε)] =

inf

Q∈AC(P)

  • EQ
  • W + 1

4 t |c(Xs)|2ds

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Computing statistics by stochastic control

Objective: Minimise Ax[c] := Ex

Q

  • W + 1

4 t |c(Xs)|2ds

  • subject to dynamics

dXt = c(Xt) − ∇V(Xt)dt + √ 2εdBt Optimal control c∗(x) = −2∇F(x) (Hartmann and Sch¨ utte, J. Stat. Mech. Th. and Exp. 2012 )

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Computing statistics by stochastic control

Objective: Minimise Ax[c] := Ex

Q

  • W + 1

4 t |c(Xs)|2ds

  • subject to dynamics

dXt = c(Xt) − ∇V(Xt)dt + √ 2εdBt Optimal control: c∗(x) = −2∇x(−ε log Ex

P[exp(−W/ε)])

(Hartmann and Sch¨ utte, J. Stat. Mech. Th. and Exp. 2012 )

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Similarities with thermodynamics

Cumulant expansion of free energy difference ∆FAB ∆FAB := − log EP[exp(−W/ε)] Jensen’s inequality implies (second law) ∆FAB ≤ EQ[W] ‘No free lunch’ principle blah

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Similarities with thermodynamics

Cumulant expansion of free energy difference ∆FAB ∆FAB := − log EP[exp(−W/ε)] Jensen’s inequality implies (second law) ∆FAB ≤ EQ[W] Relative entropy representation: − log EP [exp(−W/ε)] ≤ EQ

  • W + 1

4 t |c(Xs)|2ds

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Illustrative results

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Statistics of path functionals

Feynman-Kac formula: for h(x) := Ex

P [exp(−W/ε)]

with W = τ

0 f(Xt)dt + g(Xτ), τ= first hitting time of A ∪ B

εLh(x) = f(x)h(x) x ∈ (A ∪ B)c h(x) = exp(−g(x)) x ∈ ∂(A ∪ B) for L = −∇V · ∇ + ε∆

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Example: commitment probability to B

qB(x) := Ex

P

  • 1B(Xτ(A∪B))
  • = P(Xτ(A∪B) ∈ B|X0 = x)

⇒ W[X] = −ε log 1B(Xτ) Goal: compute rare event probability (qB(x) ≪ 1) text

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Example: commitment probability to B

qB(x) := Ex

P

  • 1B(Xτ(A∪B))
  • V(x)
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Example: commitment probability to B

qB(x) := Ex

P

  • 1B(Xτ(A∪B))
  • = P(Xτ(A∪B) ∈ B|X0 = x)

⇒ W[X] = −ε log 1B(Xτ) Goal: compute rare event probability (qB(x) ≪ 1)

◮ Problem: Rare event slow convergence of simple MC ◮ Approach: Tilting landscape ↔ biasing force

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Computing statistics by stochastic control

Objective: Minimise Ax[c] := Ex

Q

  • W + 1

4 t |c(Xs)|2ds

  • subject to dynamics

dXt = c(Xt) − ∇V(Xt)dt + √ 2εdBt Optimal control: c∗(x) = −2∇x(−ε log Ex

P[exp(−W/ε)])

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Computing statistics by stochastic control

Objective: Minimise Ax[c] := Ex

Q

  • W + 1

4 t |c(Xs)|2ds

  • subject to dynamics

dXt = c(Xt) − ∇V(Xt)dt + √ 2εdBt Optimal control: c∗(x) = −2∇x(−ε log qB(x))

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

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Iterative minimisation

EQ

  • W + 1

2

  • |c(Xt)|2dt
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Closing remarks

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Summary

◮ Problem: Rare events in molecular dynamics ◮ Solution: relative entropy representation, stochastic control

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So what?

◮ Applications in biology, medicine ◮ Relations between physics and stochastics

◮ Equilibrium vs. nonequilibrium statistical mechanics ◮ Optimal control and geodesics on Riemannian manifolds

(Crooks, PRL 108, 190602 (2012); Zulkowski et. al., PRE 86 041148 (2012))

◮ Stochastic differential geometry for non-equilibrium

thermodynamics (Muratore-Ginnaneschi, J. Phys. A 46 (2013))

Statistical physics ↔ stochastics

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Acknowledgments

Carsten Hartmann, Christof Sch¨ utte (FU Berlin) Martin Vingron (MPI Molekulare Genetik, Berlin)

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Acknowledgments

Carsten Hartmann, Christof Sch¨ utte (FU Berlin) Martin Vingron (MPI Molekulare Genetik, Berlin) Giuseppe, Alberto, Giovanni and Sara

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Acknowledgments

Carsten Hartmann, Christof Sch¨ utte (FU Berlin) Martin Vingron (MPI Molekulare Genetik, Berlin) Giuseppe, Alberto, Giovanni and Sara Thank you!

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Questions