Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics Peter R. Kramer Department of Mathematical Sciences Rensselaer Polytechnic Institute, Troy, NY and Institute for Mathematics and Its Applications, Minneapolis, MN Supported by NSF CAREER Grant DMS-0449717 and CMG OCE-0620956 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 1/29
Overview • Stochastic Drift-Diffusion Parameterization of Water Dynamics near Solute • with Adnan Khan and Shekhar Garde • Statistical mesoscale modeling for oceanic flows • with Banu Baydil and Shafer Smith (Courant, CAOS) λ ~ = 1 Poisson Vortex Streamlines, 10 9 8 7 6 y 2 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 y 1 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 2/29
Biological Disclaimer ( www.molecularium.com , S. Garde et al) Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 3/29
Stochastic Parameterization of Water Dynamics near Solute Simplified statistical description of water dynamics as possible basis for implicit solvent method to accelerate molecular dynamics simulations for proteins, etc. (with Adnan Khan (Lahore) and Shekhar Garde (Biochemical Engineering)) As a first step, we explore stochastic parameterization of water near C 60 buckyball molecule. • isotropic, chemically simple Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 4/29
Molecular Dynamics Snapshot of Buckyball Surrounded by Water Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 5/29
Statistical dynamics encoded in biophysical literature in terms of a diffusion coefficient: (Makarov et al, 1998; Lounnas et al, 1994,...) � � � | X ( t + 2 τ ) − X ( t ) | 2 � D B ( r ) ≡ � X ( t ) = r � 6 τ � � � � | X ( t + τ ) − X ( t ) | 2 � − � X ( t ) = r � 6 τ � But this seems to mix together inhomogeneities in mean and random motion. Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 6/29
Drift-Diffusion Framework We explore capacity of models of the form d X = U ( X ( t )) d t + Σ ( X ( t )) d W ( t ) , for water molecule center-of-mass position X ( t ) . • drift vector coefficient U ( r ) • diffusion tensor coefficient D ( r ) = 1 2 Σ ( r ) Σ † ( r ) For isometric solute (buckyball): • U ( r ) = U � ( | r | )ˆ r , • D ( r ) = D � ( | r | )ˆ r ⊗ ˆ r + D ⊥ ( | r | )( I − ˆ r ⊗ ˆ r ) , for position r = | r | ˆ r relative to center of symmetry. Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 7/29
Physically Inspired (DD-I) Model In analogy to Brownian dynamics simulations, take U ( r ) = − γ − 1 ∇ φ ( r ) , D ( r ) = D 0 I . Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 8/29
Physically Inspired (DD-I) Model In analogy to Brownian dynamics simulations, take U ( r ) = − γ − 1 ∇ φ ( r ) , D ( r ) = D 0 I . • Potential of mean force obtained from measuring concentration c ( r ) and Boltzmann distribution c ( r ) ∝ exp( − φ ( r ) /k B T ) . • Diffusivity unchanged from bulk value. • Friction coefficient from Einstein relation γ = k B T/D 0 . Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 8/29
Physically Inspired (DD-I) Model In analogy to Brownian dynamics simulations, take U ( r ) = − γ − 1 ∇ φ ( r ) , D ( r ) = D 0 I . Potential of mean force 14 12 10 8 6 φ (kJ/ mol) 4 2 0 −2 −4 −6 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 r (nm) Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 8/29
Physically Inspired (DD-I) Model In analogy to Brownian dynamics simulations, take U ( r ) = − γ − 1 ∇ φ ( r ) , D ( r ) = D 0 I . Mean Force (filtered) 700 600 500 F MD (kJ/ (mol nm)) 400 300 200 100 0 −100 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 r (nm) Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 8/29
Systematic, Data-Driven Parameterization (DD-II) Model Parametrize drift and diffusion functions from mathematical definitions: U � ( | r | ) = � X ( t + τ ) − X ( t ) � � � · ˆ lim � X ( t ) = r , r � τ τ ↓ 0 D � ( | r | ) = � 2 � � � � � ( X ( t + τ ) − X ( t )) · ˆ r − U � ( r ) τ � lim � 2 τ � τ ↓ 0 � X ( t ) = r � , Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 9/29
Systematic, Data-Driven Parameterization (DD-II) Model Parametrize drift and diffusion functions from mathematical definitions: D ⊥ ( | r | ) = � 1 � r ) | 2 � 4 τ | ( X ( t + τ ) − X ( t )) · ( I − ˆ r ⊗ ˆ lim � τ ↓ 0 � X ( t ) = r � . Obtain statistical data from MD simulations. Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 9/29
Time Difference τ must be chosen carefully Taking τ = ∆ t (time step of MD simulation) may not be appropriate • Limit τ ↓ 0 implicitly refers to times large enough for drift-diffusion approximation to be valid. Must choose T v ≪ τ ≪ T x , where: • T v is time scale of momentum. • T x is time scale of position. See also Pavliotis and Stuart (2007) about need to undersample. How choose τ in practice? Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 10/29
Explore simple Ornstein-Uhlenbeck (OU) model d X = V d t, � m d V = − γ V d t − α X d t + 2 k B Tγ d W ( t ) Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 11/29
Explore simple Ornstein-Uhlenbeck (OU) model d X = V d t, � m d V = − γ V d t − α X d t + 2 k B Tγ d W ( t ) Forces: friction, potential, and thermal. Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 11/29
Explore simple Ornstein-Uhlenbeck (OU) model d X = V d t, � m d V = − γ V d t − α X d t + 2 k B Tγ d W ( t ) Nondimensionalize: d X = V d t, d V = − a V d t − a X d t + a d W ( t ) Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 11/29
Explore simple Ornstein-Uhlenbeck (OU) model d X = V d t, � m d V = − γ V d t − α X d t + 2 k B Tγ d W ( t ) Nondimensionalize: d X = V d t, d V = − a V d t − a X d t + a d W ( t ) where a = γ 2 / ( mα ) is ratio of position to momentum time scale. Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 11/29
Explore simple Ornstein-Uhlenbeck (OU) model d X = V d t, � m d V = − γ V d t − α X d t + 2 k B Tγ d W ( t ) Nondimensionalize: d X = V d t, d V = − a V d t − a X d t + a d W ( t ) Exact drift-diffusion coarse-graining when a ≫ 1 : d X = − X d t + d W ( t ) Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 11/29
Explore simple Ornstein-Uhlenbeck (OU) model d X = V d t, � m d V = − γ V d t − α X d t + 2 k B Tγ d W ( t ) Nondimensionalize: d X = V d t, d V = − a V d t − a X d t + a d W ( t ) What if we try to obtain this from analysis of trajectories with finite but large a ? Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 11/29
Drift and diffusion coefficients of exact OU solution sampled with finite time difference τ . Longitudinal Drift Coefficient for OU, a=132 −0.3 Blue : Exact Red : Asymptotic −0.4 −0.5 U || (x) ⋅ x/|x| 2 −0.6 −0.7 −0.8 −0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 τ Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 12/29
Drift and diffusion coefficients of exact OU solution sampled with finite time difference τ . Longitudinal Diffusivity for OU, a=132 0.45 Blue : Exact 0.4 Red : Asymptotic 0.35 0.3 D || 0.25 0.2 0.15 0.1 0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 τ Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 12/29
Inferences from OU model • Good choice of τ may be the one which maximizes drift magnitude and diffusivity. • Beginning estimate obtained from OU model with same a value. Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 13/29
OU Model Insights → MD Data Parameteriza- tion in DD-II Model To obtain time scales, approximate main well in potential of mean force by quadratic. Potential of mean force 8 Red : Buckyball Potential 6 Green : Quadratic Fit to a characteristic part of the potential 4 φ (kJ/mol) 2 0 −2 −4 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 r (nm) This gives a = 132 . Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 14/29
OU Model Insights → MD Data Parameteriza- tion in DD-II Model Examine drift and diffusivity computed from various choices of τ . Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 14/29
OU Model Insights → MD Data Parameteriza- tion in DD-II Model Examine drift and diffusivity computed from various choices of τ . Longitudinal Drift 0.1 0.05 U || (nm/ps) 0 −0.05 Blue : τ =20fs Green : τ =40 fs Red : τ =60 fs Magenta : τ =80 fs Cyan : τ =100 fs −0.1 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 r (nm) Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 14/29
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