princeton university ima, minneapolis, january 2008 i.g. kevrekidis -c. w. gear, g. hummer, r....

Princeton University

IMA, Minneapolis, January 2008

I.G. Kevrekidis -C. W. Gear, G. Hummer, R. Coifman and several other good people-

Department of Chemical Engineering, PACM & MathematicsPrinceton University, Princeton, NJ 08544

Computational Experimentsin coarse graining atomistic simulations

(Equation-free -& Variable Free- FrameworkFor Complex/Multiscale Systems)

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SIAM– July, 2004

Clustering and stirring in a plankton model

Young, Roberts and Stuhne, Nature 2001

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Dynamics of System with convection

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Simulation Method

• Random (equal) birth and death, probability: = .

• Brownian motion.

• Advective stirring. are random phases)

• IC: 20000 particles randomly placed in 1*1 box

• Analytical Equation for G(r):

)]()('cos[2)(')(

)]()('cos[2)(')(

ttkxUtyty

ttkyUtxtx

kkk

kkk

Dxtxxx kkkk 2);(' 2

)(2)(1

)(2)(1

2 3 r CGrr

GrGr

DG rrrrt

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Stirring by a random field (color = y)

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Dynamics of System with convection

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Projective Integration: From t=2,3,4,5 to 10

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RESTRICTION - a many-one mapping from a high-dimensional description (such as a collection of particles in Monte Carlo simulations) to a low-dimensional description - such as a finite element approximation to a distribution of the particles.

LIFTING - a one-many mapping from low- to high-dimensional descriptions.

We do the step-by-step simulation in the high-dimensional description.

We do the macroscopic tasks in the low-dimensional description.

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So, the main points:

• You have a “microscopic code”• Somebody tells you what are good coarse variable(s)• Somebody tells you what KIND of equation this variable

satisfies (deterministic, stochastic…) but NOT what the equation looks like.

• Then you can use the IDEA that such an equation exists and closes to accelerate the simulation/ extraction of information.

• KEY POINT – make micro states consistent with macro states

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Application to Micelle Formation

• Hydrophobic tail (T)

• Hydrophilic head (H)

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• Surfactant = chain of H and T beads (H4T4)

• No explicit solvent

• Hydrophobic-hydrophilic interactions modeled as direct interaction between H and T beads

• Pairwise interactions with nearest sites:

Lattice Model (Larson et al., 1985)

ji

ijE

2 ,0 TTHTHH

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Snapshot of Micellar System

T = 7.0, µ = - 47.40

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Kinetic Approach to Rare Events(Hummer and Kevrekidis, JCP 118, 10762 (2003))

• Evolution of coarse variables is slow– micelle birth/death are rare events

• Reconstruction of free energy surface:– long equilibrium simulation– series of short nonequilibrium simulations

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Reconstruction of Free Energy Surface

)(1

Gdt

d

Tk

dt

d B2)var(

)()( tFG

Obtain G() and () from short-scale nonequilibrium simulations

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Reconstructed free energy curve

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Results: Micelle Destruction Rate

Kramers’ formula

TkG BeminGsaddleGsaddle

k /)()()(2

1

• Result of nonequilibrium simulation: k = 5.58 x 10-9

• Equilibrium result: k = 7.70 x 10-9

• CPU time required: less than 7% of equilibrium simulation

• Extension to multi-dimensional systems (Hummer and Kevrekidis, 2003)

– Chapman-Kolmogorov equation

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Reverse Projective Integration – a sequence of outer integration steps backward; based on forward steps + estimation

We are studying the accuracy and stability of these methods

12

3

Reverse Integration: a little forward, and then a lot backward !

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Reverse coarse integration from both sides

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Reconstructed free energy curve

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Details of Multidimensional Dynamics

Small clusters: 2d dynamics Larger clusters: 1d dynamics

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Multidimensional Dynamics (2nd variable = E)

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Alanine DipeptideIn 700 tip3p waters

w/ Gerhard Hummer, NIDDK / J.Chem.Phys. 03

The waters The dipeptide and the Ramachandran plot

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G(ψ)

<ψ>ψ0

ψ0ψ1

var<ψ>

-180 180

)ψD((t)(dt

d

ψ

)ψG(

Tk

)ψD(

dt

ψd

B

2)var

0

3

6

1. Start with constrained MD2. Let 50 configurations free3. Estimate d/dt of average4. Perform projective step

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Alanine Dipeptide Energy Landscape

G.Hummer, I.G.Kevrekidis.Coarse molecular dynamics of a peptide fragment:

Free energy, kinetics, and long-time dynamics computations

J.Chem. Phys. 118 (2003).

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Protocols for Coarse MD (CMD) using Reverse Ring Integration

Step RingBACKWARD

in Energy

MD Simulationsrun FORWARD

in Time

Initialize ring nodes

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Fokker-Planck Equation (2D)for distribution P(x1,x2)

2 2

1 1 2 11 1 2 12 1 21 1 1 1 2

2 2

2 1 2 21 1 2 22 1 22 2 1 2 2

, , ,

, , ,

Pv x x D x x D x x

t x x x x x

v x x D x x D x xx x x x x

1 2 1 2

1 2

,P x x S S

t x x

2D FPE:

Probability Currents

1 21 2

i i i iS v P D P D Px x

Drift Coefficient Diffusion Coefficients

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Drift and Diffusion Coefficients

MD Simulations run FORWARD in Time

Ring node ICs

multiple replicas per nodecompute drift (v) and

diffusion (D) coefficients

use (v, D) estimates to check for existence of potential ()

and compute potential values at each node

Dihedral angles

P

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2x

1x

Ring at+

Ring at

Ring Stepping in Generalized Potential

Goal: Compute potential associatedwith ring nodes (ring “height” above x1-x2 plane)

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Right-handed -helical minimum

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Potential Conditions I

Case I: Diffusion matrix is proportional to unit matrix: ij ijD D ( D = scalar)

1

lni iS P v D Px

Probability Current:

Probability Current vanishes

1 1

lniv D P Dx x

Conditions for existence of generalized potential : 1 2

2 1

v v

x x

1 2,b bx x

1

1

2

2

11 1 2 11

12 1 2 2

, d

, d

b

a

b

a

xa

x

xb

x

D v x x x

D v x x x

1 2,a ax x

2x

1x

1 2,b ax x

Path

Path

Drift Coefficient

Potential Conditions

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CMD Exploration of Alanine Dipeptideusing drift coefficients

Reverse ring integration stagnates at saddle points

on coarse free energy landscape

Ring nodes

Short bursts of MD simulation initialized at each node in the ring

Data analysis of forward in time MD provides gradient informationsmall step FORWARD in time

large step BACKWARD in energy

ONLY drifts estimated herering step size is scaled

by unknown diffusivity D

Dstepsize ~ nodal “heights” unknown

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Src homology 3 (SH3) domain

Small Modular Domain 55-75 amino acids long

Characteristic fold consisting of five or six β-strands arranged as two tightly packed anti-parallel β sheets

blue (N-terminus) to red (C-terminus)

Distal Loop

n-Src

DivergingTurn

Protein modeled using off-lattice C representation associated with simplified

minimally frustrated Hamiltonian

P. Das, S. Matysiak, and C. Clementi, Proc. Natl. Acad. Sci. USA 102, 10141 (2005).

N-terminus

C-terminusFraction native contacts formed

Fra

ctio

n n

on-n

ativ

e co

ntac

ts f

orm

ed

FOLDED

UNFOLDED

TRANSITIONSTATE

Protein folding intrinsically low-dimensional

“Collective” (coarse, slow) coordinatesfully describe long time system dynamics

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Reverse Ring Integration and MDCoarse MD (CMD)

Step RingBACKWARD

in time

MD Simulationsrun FORWARD

in Time

Initialize ring nodes

Protein conformations “live”in high-dimensional space

described by Cartesian coordinates

Free energy landscape exploration using coarse reverse integration “backward-in-time” initialized near base of wells

reconstructed folding free energy surface of SH3

fraction of native contacts formed

frac

tion

of n

on-n

ativ

e co

ntac

ts f

orm

ed

0.05 ps MDforward

0.2 psbackwardtime step

Transition state identificationusing CMD

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So, again, the main points

• Somebody needs to tell you what the coarse variables are

• And what TYPE of equation they satisfy

• And then you can use this information

to bias the simulations “intelligently”

accelerating the extraction of information

(also need hypothesis testing).

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and now for something completely different: Little stars ! (well…. think fishes)

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Fish Schooling Models

tstvtc iii , ,

Initial State

Compute Desired Direction

ij ij

iji

tctc

tctcttd

1j j

j

ij ij

iji

tv

tv

tctc

tctcttd

ii

iii

gttd

gttdttd

ˆ

ˆ'

ttdi

Update Direction for Informed Individuals ONLY ttdi '

Zone of Deflection Rij< Zone of Attraction Rij<

Normalize ttdi ˆ

INFORMEDUNINFORMED

Update Positions

tsttvtcttc iiii

Position, Direction, Speed

Couzin, Krause, Franks & Levin (2005)

Nature (433) 513

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STUCK

~ typically aroundrxn coordinate

value of about 0.5

INFORMED DIRN

STICK STATES

INFORMED individualclose to front of group

away from centroid

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SLIP

~ wider range ofrxn coordinate values

for slip 00.35

INFORMED DIRN

SLIP STATES

INFORMED individualclose to group centroid

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Effective Fokker-Planck Equation

2

2

,,

P r tv r D r P r t

t r r

DriftCoefficient

DiffusionCoefficient

t

rtrrD

t

rtrrv

0

20 ,

2

1 ;

,

Constln'

'

'

0

rDdrrD

rv

Tk

r R

B

FPE:

Potential:

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Coarse Free Energy Calculation

t

XtXXv

0,

t

XtXXD

0

2 ,

2

1

Estimate Drift and Diffusion coefficients numerically from simulation “bursts”

Kln'

'

'

0

XDdXXD

Xv

Tk

X R

B

X0

X0X0

Kopelevich, Panagiotopoulos & KevrekidisJ Chem Phys 122 (2005)

Hummer & KevrekidisJ Chem Phys 118 (2003)

Improved estimates using Maximum Likelihood Estimation (MLE)

Y. Aït-Sahalia.Maximum Likelihood Estimation of Discretely Sampled Diffusions:

A Close-Form Approximation Approach.Econometrica 70 (2002).

Princeton UniversitySTUCK

SLIPSLIP

Energy Landscape – Fish Swarming Problem

CENTROIDInformed DIRN

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Using the computer to select variable

Rationale:

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Straight Line Distance

between locations

NOT representative of actual transition

difficulty\distance

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Using the computer to select variable

Rationale:

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Straight Line Distance

Actual transition difficulty represented

by curved path

Curved Transition Distance

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Using the computer to select good variables

Rationale:

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Straight Line Distance IS representative of

actual transition difficulty\distance

in small LOCAL patches

Patch size related to problem “geography”

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Using the computer to select variable

Rationale:

Lake Carnegie, Princeton, NJLake Carnegie, Princeton, NJ

Euclidean Distance

Selected Datapoint

XY

Z3D Dataset with 2D manifold

Euclidean distance in input spacemay be weak indicator

of INTRINSIC similarity of datapoints

Geodesic distance is good for this dataset

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1

2

3

W12

W23

vertices

edges

weights

2

expi j

ijWt

x x

parameter t

Dataset as Weighted Graph

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Multiple random walks through simulation data

initialized at

Unequal separation (Euclidean distance)

between IC ( ) and limits of random walk ( , )

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Parameter Local neighborhood size

Compute NN “neighborhood” matrix K

2

, exp

i j

i jK

x x=

{ }ix

Compute diagonal normalization matrix D

, ,1

N

i i i jj

D K=

1M D KCompute Markovian matrix M

N datapoints

1 1 1M

2 2 2M

N N NM

Require: Eigenvalues λ and Eigenvectors Φ of M

1 2 3 N

A few Eigenvalues\Eigenvectors provide meaningful information on dataset geometry

Top

2nd

Nth

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Dataset in x, y, z Dataset Diffusion Map

N datapoints N datapoints

eigencomputation

, , , 1,ii i ix y z i N x 2 3, , 1,i i i i N

Diffusion Maps

R. Coifman, S. Lafon, A. Lee, M. Maggioni, B. Nadler, F. Warner, and S. Zucker,Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps.PNAS 102 (2005).

B. Nadler, S. Lafon, R. Coifman, and I. G. Kevrekidis,Diffusion maps, spectral clustering and reaction coordinates

of dynamical systems.Appl. Comput. Harmon. Anal. 21 (2006).

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Diffusion Map (2, 3)

2

3

2

ABSOLUTE Coordinates SIGNED Coordinates

Report absolute distanceof all uninformed individuals

to informed individual to DMAP routine

Report (signed) distanceof all uninformed individuals

to informed individual to DMAP routine

STICK

SLIPSTICK

SLIP

Reaction Coordinate

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MAN

MA

CH

INE

MAN

MA

CH

INE

ABSOLUTE Coordinates SIGNED Coordinates

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So, again, the same simple theme

• If there is some reason to believe that there exist slow, effective dynamics in some smart collective variables

• Then this can be used to accelerate some features of the computation

• Tools for data-based detection of coarse variables…

• MAIN DIFFICULTY: finding physical initial conditions consistent with desired coarse initial conditions