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

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

Why study molecules?

Molecular biology and drug design Principle: structure ↔ function

Molecular biology and drug design Principle: structure ↔ function Protein aggregation ◮ Example: Blood clot formation

Molecular biology and drug design Principle: structure ↔ function Protein aggregation ◮ Example: Blood clot formation ◮ Cause: Change in spatial structure

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: 3 N degrees of freedom

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

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

Metastabilities: Almost-stable structures n-pentane (C 5 H 12 ): 3 N = 51 degrees of freedom

Metastabilities: Almost-stable structures n-pentane (C 5 H 12 ): 3 N = 51 degrees of freedom

Rare events: Structural changes Structure 1 Effective dynamics in 2D landscape Metastable set 1

Rare events: Structural changes Structure 1 Effective dynamics in 2D landscape Metastable set 1

Rare events: Structural changes Structure 1 Effective dynamics in 2D landscape Metastable set 1

Rare events: Structural changes Change to Structure 2 Transition to Metastable set 2

Some goals in studying molecules Study changes between almost-stable structures ◮ Freidlin-Wentzell theory: Exit from attractor

Some goals in studying molecules Study changes between almost-stable structures ◮ Freidlin-Wentzell theory: Exit from attractor Compute statistics ◮ Escape probabilities ◮ Escape times

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

Molecular dynamics (MD)

Model: Position Langevin Dynamics Molecule in equilibrium with heat bath: � dX t = −∇ V ( X t ) dt + 2 k B TdB t ◮ X t ∈ R 3 N - position coordinates for N atoms ◮ V - energy function ◮ T - temperature ◮ k B - Boltzmann’s constant ◮ B t - Brownian motion w.r.t. P

Model: Position Langevin Dynamics Molecule in equilibrium with heat bath: � dX t = −∇ V ( X t ) dt + 2 k B TdB t ◮ X t ∈ R 3 N - position coordinates for N atoms ◮ V - energy function ◮ T - temperature ◮ k B - Boltzmann’s constant ◮ B t - Brownian motion w.r.t. P Numerical integration → time series data

Rare events Goal : Study A → B event Constraints: - Large barriers ∆ F AB - Small thermal ‘kicks’ k B T Example: amylases ∆ F AB ≈ 208 kJ/mol k B T ≈ 2 . 4 kJ/mol Long waiting times Graph of V

An Application of Girsanov’s theorem

Model: Position Langevin Dynamics Molecule in equilibrium with heat bath: √ dX t = −∇ V ( X t ) dt + 2 ε dB t ◮ B t - random fluctuations due to heat bath interactions ⇒ P - equilibrium measure

Statistics of path functionals Goal: Compute P x -expectation of � τ ( S ) W [ X ] = f ( X t ) dt + g ( X τ ) 0 ◮ f - running cost ◮ g - terminal cost ◮ τ ( S ) - first hitting time to some set S ◮ P x - path measure conditioned on { X 0 = x }

Model: Position Langevin Dynamics Molecule not in equilibrium with heat bath: √ dX t = c ( X t ) − ∇ V ( X t ) dt + 2 ε dB t ◮ c ( X t ) - feedback control force ◮ Q - nonequilibrium measure due to control c

Model: Position Langevin Dynamics Molecule not in equilibrium with heat bath: √ dX t = c ( X t ) − ∇ V ( X t ) dt + 2 ε dB t ◮ c ( X t ) - feedback control force ◮ Q - nonequilibrium measure due to control c Change of drift −∇ V → ( c − ∇ V ) � Change of measure P → Q

Computing statistics by reweighting Girsanov change of measure � 1 � t � t � dP t c ( X t ) dB t − 1 | c ( X t ) | 2 dt ( X ) = exp √ dQ t 4 ε 2 ε 0 0 Reweighting of nonequilibrium measurements: � W dP � E x P [ W ] = E x Q dQ

Computing statistics by stochastic control Duality relationships between free energy, relative entropy (Dai Pra, Meneghini and Runggaldier, 1996) − log E P [ exp ( − W /ε )] = Q ∈ AC ( P ) { ε R ( Q � P ) + E Q [ W ] } inf ◮ AC ( P ) - set of all Q abs. cont. w.r.t. P ◮ R ( Q � P ) = E Q [ log dQ dP ] relative entropy of Q w.r.t. P ◮ W bounded

Computing statistics by stochastic control Since � 1 � t � t dP t c ( X t ) dB t − 1 � | c ( X t ) | 2 dt √ ( X ) = exp dQ t 4 ε 2 ε 0 0 we have � t � � � 1 � log dQ t | c ( X s ) | 2 ds ε R ( Q t � P t ) = E Q = E Q dP t 4 0

Computing statistics by stochastic control Since � 1 � t � t dP t c ( X t ) dB t − 1 � | c ( X t ) | 2 dt √ ( X ) = exp dQ t 4 ε 2 ε 0 0 we have � t � � � 1 � log dQ t | c ( X s ) | 2 ds ε R ( Q t � P t ) = E Q = E Q dP t 4 0 � t � � W + 1 �� | c ( X s ) | 2 ds − log E P [ exp ( − W /ε )] = inf E Q 4 Q ∈ AC ( P ) 0

Computing statistics by stochastic control Objective: Minimise � t � W + 1 � A x [ c ] := E x | c ( X s ) | 2 ds Q 4 0 subject to dynamics √ dX t = c ( X t ) − ∇ V ( X t ) dt + 2 ε dB t Optimal control c ∗ ( x ) = − 2 ∇ F ( x ) (Hartmann and Sch¨ utte, J. Stat. Mech. Th. and Exp. 2012 )

Computing statistics by stochastic control Objective: Minimise � t � W + 1 � A x [ c ] := E x | c ( X s ) | 2 ds Q 4 0 subject to dynamics √ dX t = c ( X t ) − ∇ V ( X t ) dt + 2 ε dB t Optimal control: c ∗ ( x ) = − 2 ∇ x ( − ε log E x P [ exp ( − W /ε )]) (Hartmann and Sch¨ utte, J. Stat. Mech. Th. and Exp. 2012 )

Similarities with thermodynamics Cumulant expansion of free energy difference ∆ F AB ∆ F AB := − log E P [ exp ( − W /ε )] Jensen’s inequality implies (second law) ∆ F AB ≤ E Q [ W ] ‘No free lunch’ principle blah

Similarities with thermodynamics Cumulant expansion of free energy difference ∆ F AB ∆ F AB := − log E P [ exp ( − W /ε )] Jensen’s inequality implies (second law) ∆ F AB ≤ E Q [ W ] Relative entropy representation: � t � � W + 1 | c ( X s ) | 2 ds − log E P [ exp ( − W /ε )] ≤ E Q 4 0

Illustrative results

Statistics of path functionals Feynman-Kac formula: for h ( x ) := E x P [ exp ( − W /ε )] � τ 0 f ( X t ) dt + g ( X τ ) , τ = first hitting time of A ∪ B with W = x ∈ ( A ∪ B ) c ε Lh ( x ) = f ( x ) h ( x ) h ( x ) = exp ( − g ( x )) x ∈ ∂ ( A ∪ B ) for L = −∇ V · ∇ + ε ∆

Example: commitment probability to B q B ( x ) := E x � � = P ( X τ ( A ∪ B ) ∈ B | X 0 = x ) 1 B ( X τ ( A ∪ B ) ) P ⇒ W [ X ] = − ε log 1 B ( X τ ) Goal: compute rare event probability ( q B ( x ) ≪ 1) text

Example: commitment probability to B q B ( x ) := E x � � 1 B ( X τ ( A ∪ B ) ) P V ( x )

Example: commitment probability to B q B ( x ) := E x � � 1 B ( X τ ( A ∪ B ) ) = P ( X τ ( A ∪ B ) ∈ B | X 0 = x ) P ⇒ W [ X ] = − ε log 1 B ( X τ ) Goal: compute rare event probability ( q B ( x ) ≪ 1) ◮ Problem: Rare event � slow convergence of simple MC ◮ Approach: Tilting landscape ↔ biasing force

Computing statistics by stochastic control Objective: Minimise � t � � W + 1 A x [ c ] := E x | c ( X s ) | 2 ds Q 4 0 subject to dynamics √ dX t = c ( X t ) − ∇ V ( X t ) dt + 2 ε dB t Optimal control: c ∗ ( x ) = − 2 ∇ x ( − ε log E x P [ exp ( − W /ε )])

Computing statistics by stochastic control Objective: Minimise � t � W + 1 � A x [ c ] := E x | c ( X s ) | 2 ds Q 4 0 subject to dynamics √ dX t = c ( X t ) − ∇ V ( X t ) dt + 2 ε dB t Optimal control: c ∗ ( x ) = − 2 ∇ x ( − ε log q B ( x ))

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation

Iterative minimisation