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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)

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

• f water near C60 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,...) DB(r) ≡

• |X(t + 2τ) − X(t)|2

6τ

• X(t) = r
• −
• |X(t + τ) − X(t)|2

6τ

• X(t) = r
• 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 dX = U(X(t)) dt + Σ(X(t)) dW (t), for water molecule center-of-mass position X(t).

• drift vector coefficient U(r)
• diffusion tensor coefficient D(r) = 1

2Σ(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) = D0I.

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) = D0I.

• Potential of mean force obtained from measuring

concentration c(r) and Boltzmann distribution c(r) ∝ exp(−φ(r)/kBT).

• Diffusivity unchanged from bulk value.
• Friction coefficient from Einstein relation

γ = kBT/D0.

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) = D0I.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −6 −4 −2 2 4 6 8 10 12 14

Potential of mean force r (nm) φ (kJ/ mol)

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) = D0I.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −100 100 200 300 400 500 600 700

r (nm) FMD (kJ/ (mol nm)) Mean Force (filtered)

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|) = lim

τ↓0

X(t + τ) − X(t) τ · ˆ r

• X(t) = r
• ,

D(|r|) = lim

τ↓0

(X(t + τ) − X(t)) · ˆ r − U(r)τ

• 2

2τ

• 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|) = lim

τ↓0

1 4τ |(X(t + τ) − X(t)) · (I − ˆ r ⊗ ˆ r)|2

• 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 Tv ≪ τ ≪ Tx, where:

• Tv is time scale of momentum.
• Tx 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 dX = V dt, mdV = −γV dt − αX dt +

• 2kBTγ dW (t)

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 11/29

Explore simple Ornstein-Uhlenbeck (OU) model dX = V dt, mdV = −γV dt − αX dt +

• 2kBTγ dW (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 dX = V dt, mdV = −γV dt − αX dt +

• 2kBTγ dW (t)

Nondimensionalize: dX = V dt, dV = −aV dt − aX dt + a dW (t)

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 11/29

Explore simple Ornstein-Uhlenbeck (OU) model dX = V dt, mdV = −γV dt − αX dt +

• 2kBTγ dW (t)

Nondimensionalize: dX = V dt, dV = −aV dt − aX dt + a dW (t) where a = γ2/(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 dX = V dt, mdV = −γV dt − αX dt +

• 2kBTγ dW (t)

Nondimensionalize: dX = V dt, dV = −aV dt − aX dt + a dW (t) Exact drift-diffusion coarse-graining when a ≫ 1: dX = −X dt + dW (t)

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 11/29

Explore simple Ornstein-Uhlenbeck (OU) model dX = V dt, mdV = −γV dt − αX dt +

• 2kBTγ dW (t)

Nondimensionalize: dX = V dt, dV = −aV dt − aX dt + a dW (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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3

τ U|| (x)⋅ x/|x|2 Longitudinal Drift Coefficient for OU, a=132

Blue : Exact Red : Asymptotic

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.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

τ D|| Longitudinal Diffusivity for OU, a=132

Blue : Exact Red : Asymptotic

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.

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 −4 −2 2 4 6 8

Potential of mean force r (nm) φ (kJ/mol)

Red : Buckyball Potential Green : Quadratic Fit to a characteristic part of the potential

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.

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 −0.1 −0.05 0.05 0.1

r (nm) U|| (nm/ps) Longitudinal Drift

Blue : τ=20fs Green : τ=40 fs Red : τ=60 fs Magenta : τ=80 fs Cyan : τ=100 fs

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.

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 x 10

−3

r (nm) D|| (nm2/ps) Longitudinal Diffusivity

Blue :τ=60 fs Green :τ=80 fs Magenta :τ=120 fs Cyan :τ=160 fs Black :τ=200 fs Yellow : τ=240 fs Red :τ=280 fs

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. Both desiderata about drift and diffusivity behavior not simultaneously satisfiable.

• Correct bulk diffusivity behavior more important

We choose τ = 0.2 ps = 200 fs.

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 14/29

Parameterization Used in DD-II Model

0.6 0.8 1 1.2 1.4 1.6 −0.04 −0.02 0.02 0.04 0.06 0.08

r (nm) U|| (nm/ps) Longitudinal Drift

Blue : DD−II output Red : DD−II input from MD data

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 15/29

Parameterization Used in DD-II Model

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1 2 3 4 5 6 7 x 10

−3

r (nm) D|| (nm2/ps) Longitudinal Diffusivity

Blue : DD−II output Red : DD−II input from MD Data

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 15/29

Parameterization Used in DD-II Model

0.6 0.8 1 1.2 1.4 1.6 1.8 2 3 4 5 6 7 8 x 10

−3

r (nm) D⊥ (nm2/ps) Lateral Diffusivity

Blue : DD−II output Red : DD−II input from MD data

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 15/29

Compare Predictions of Biophysical Diffusivity Formula DB(r) ≡

• |X(t + 2τ) − X(t)|2

6τ

• X(t) = r
• −
• |X(t + τ) − X(t)|2

6τ

• X(t) = r
• Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 16/29

Compare Predictions of Biophysical Diffusivity Formula

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 0.5 1 1.5

r (nm) DB/D0 DB (r) for DD−I Model

DDD−I DMD

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 16/29

Compare Predictions of Biophysical Diffusivity Formula

0.6 0.8 1 1.2 1.4 1.6 1.8 0.2 0.4 0.6 0.8 1 1.2 1.4

r (nm) DB/D0 DB (r) for DD−II Model

DMD

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 16/29

Future Work

Next steps

• anisotropies
• chemical heterogeneity

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 17/29

Unresolved Mesoscale Turbulence in Ocean Circulation Models Computational ocean models for climate prediction have resolution ∼ 100 km:

• does not adequately resolve mesoscale turbulent

structures on length scales 100 km

• even smaller scales 100 m : Kolmogorov

turbulence We focus on representing turbulent transport by unresolved mesoscale turbulence. Wind-driven double-gyre 2000 km basin-scale computational simulation, 5 km resolution, by Shafer Smith.

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 18/29

Mathematical Framework for Transport ∂T(x, t) ∂t + u(x, t) · ∇T(x, t) = κ∆T(x, t), T(x, t = 0) = Tin(x)

• Passive scalar field T(x, t)
• Velocity field u = V + v: large-scale mean flow

+ small-scale fluctuations

• “Molecular” diffusion coefficient κ

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 19/29

Parameterization Problem

• Obtain an equation for coarse-grained T:

∂T ∂t + V · ∇T = κ∆T − ∇ · F T(x, t = 0) = Tin(x),

• Turbulent Flux F = v T
• Problem of Parametrization:

F = F(T, V ) where F only involves a few parameters.

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 20/29

Parameterization of Small Scales One parameterization used in AOS and engineering (Gent and McWilliams 1990, Griffies 1998, Middleton and Loder 1989): F = −K∗(x, t) · ∇T, with generally nonsymmetric K∗ = S∗ + A∗.

• Symmetric part S∗: variably

enhanced diffusion

• Antisymmetric part A∗:

effective drift U ∗ = −∇ · A∗ relative to mean flow

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 21/29

Parameterization Approaches

Within the class of “effective diffusivity” or “eddy diffusivity” schemes, the way in which K∗ is modeled differs. Some examples:

• Mixing-length theory: K∗ ∼ ℓv.
• K − ε theory: parameterize based on local energy and

energy dissipation rate

• Gent-McWilliams: related to slope of surfaces of constant

potential density These are empirical, with varying degrees of success. Ocean circulation models often employ even simpler practice of choosing K∗ as constant multiple of identity, tuned a posteriori.

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 22/29

Homogenization-Based Parameterization Under assumption of scale separation (not so crazy for ocean), homogenization theory provides rigorous support and formula for effective diffusivity (Avellaneda, Majda, McLaughlin, Papanicolaou, Varadhan, Fannjiang, Pavliotis & K): K∗(x, t) = κ (I − v ⊗ χ)) where ∂χ ∂τ + (V + v) · ∇yχ − κ∆yχ = −v, is cell problem, solved on sub-grid scale with frozen mean flow V . · denotes subgrid-scale average.

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 23/29

Computational Homogenization

• Heterogenous multiscale methods (HMM)

natural, but perhaps too complex a modification to existing codes.

• More negotiable: parameterization formula for

K∗ based on statistical subgrid-scale model with small number of nondimensional parameters

• Our approach: build up from simple models,

combine numerical solutions with new and existing asymptotic results (Avellaneda, Majda, Vergassola, Bonn, McLaughlin, Kesten, Papanicolaou, etc.)

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 24/29

Example: Dynamical Poisson Vortex Model (Cinlar, Caglar,...) Stream function: ψ(y, t) =

• n

ψvor(y − Y(n)

c (t − ξ(n)), t − ξ(n))

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 25/29

Example: Dynamical Poisson Vortex Model (Cinlar, Caglar,...) Stream function: ψ(y, t) =

• n

ψvor(y − Y(n)

c (t − ξ(n)), t − ξ(n))

composed of simple vortex blobs, e.g., v(y, t) = ∂ψvor/∂y2 −∂ψvor/∂y1

• ,

ψvor(y, t = 0) = 12¯ vvorℓvor

• 1

9

• y

ℓvor

• 3

− 1 6

• y

ℓvor ]

• 2

for |y| ≤ ℓvor

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 25/29

Example: Dynamical Poisson Vortex Model (Cinlar, Caglar,...) Stream function: ψ(y, t) =

• n

ψvor(y − Y(n)

c (t − ξ(n)), t − ξ(n))

with centers Y(n)

c (t) obeying certain simple

dynamical law

• Brownian motion
• intervortical advection

and ψvor(y, t) → 0 for |t| → ∞,

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 25/29

Example: Dynamical Poisson Vortex Model (Cinlar, Caglar,...) Stream function: ψ(y, t) =

• n

ψvor(y − Y(n)

c (t − ξ(n)), t − ξ(n))

vortices nucleated according to space-time Poisson process (η(n), ξ(n)) of prescribed intensity λ, with Y(n)

c (0) = η(n)

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 25/29

Sample Snapshot of Poisson Vortex Field

y1

y2

Poisson Vortex Streamlines, λ~ = 1

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 26/29

Uncertainty Modeling Issues

How choose the statistical model parameters for dynamical parameterization?

• for observed subgrid-scale data, can try

parametric statistical fitting

• but want statistical subgrid-scale model to

represent unobserved small scales, with only coarse-scale data available

• so how predict small-scale statistical parameters

from large-scale observations?

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 27/29

References regarding Statistical Modeling in Molecular Dynamics

• T. Schlick, Molecular Modeling and Simulation:

An Interdisciplinary Guide, 2002.

• V. A. Makarov et al, Biophys J., 75 (1998)

150–158.

• biophysical modeling framework for water

near surface

• G. Pavliotis and A. M. Stuart, J. Stat. Phys. 124

(2007) 741–781.

• mathematical framework for computing

coarse-grained coefficients statistically from microscale data

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 28/29

References regarding Statistical Modeling in Ocean Turbulence

• Mathematics of Subgrid-Scale Phenomena in

Atmospheric and Oceanic Flows, Institute for Pure and Applied Mathematics,

www.ipam.ucla.edu/programs/atm2002

• A. J. Majda and P. R. Kramer, Phys. Rep. 314

(1999) 237–574.

• review of some statistical flow models
• M. Çaglar, et al, J. Atmos. Ocean. Tech. 23

(2006) 1745–1758.

• parameterically fitting observational data to

statistical vortex model

Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics – p. 29/29