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 & Mathematics Princeton University, Princeton, NJ Computational Experiments in coarse graining atomistic simulations (Equation-free -& Variable Free- Framework For Complex/Multiscale Systems)

Princeton University SIAM– July, 2004 Clustering and stirring in a plankton model Young, Roberts and Stuhne, Nature 2001

Princeton University Dynamics of System with convection

Princeton University Simulation Method Random (equal) birth and death, probability: = . Brownian motion. Advective stirring.  are random phases) IC: particles randomly placed in 1*1 box Analytical Equation for G(r):

Princeton University Stirring by a random field (color = y)

Princeton University Dynamics of System with convection

Princeton University

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

Princeton University 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.

Princeton University 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

Princeton University Application to Micelle Formation Hydrophobic tail (T) Hydrophilic head (H)

Princeton University Surfactant = chain of H and T beads (H 4 T 4 ) 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)

Princeton University Snapshot of Micellar System T = 7.0, µ =

Princeton University Kinetic Approach to Rare Events (Hummer and Kevrekidis, JCP 118, (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

Princeton University Reconstruction of Free Energy Surface Obtain G(  ) and  (  ) from short-scale nonequilibrium simulations

Princeton University Reconstructed free energy curve

Princeton University Results: Micelle Destruction Rate Kramers’ formula Result of nonequilibrium simulation: k = 5.58 x Equilibrium result: k = 7.70 x CPU time required: less than 7% of equilibrium simulation Extension to multi-dimensional systems (Hummer and Kevrekidis, 2003) – Chapman-Kolmogorov equation

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

Princeton University Reverse coarse integration from both sides

Princeton University Reconstructed free energy curve

Princeton University Details of Multidimensional Dynamics Small clusters: 2d dynamicsLarger clusters: 1d dynamics

Princeton University Multidimensional Dynamics (2 nd variable = E)

Princeton University Alanine Dipeptide In 700 tip3p waters w/ Gerhard Hummer, NIDDK / J.Chem.Phys. 03 The waters The dipeptide and the Ramachandran plot

Princeton University 1.Start with constrained MD 2.Let 50 configurations free 3.Estimate d/dt of average 4.Perform projective step

Princeton University 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). 

Princeton University

Protocols for Coarse MD (CMD) using Reverse Ring Integration Step Ring BACKWARD in Energy MD Simulations run FORWARD in Time Initialize ring nodes

Princeton University Fokker-Planck Equation (2D) for distribution P(x 1,x 2 ) 2D FPE: Probability Currents Drift Coefficient Diffusion Coefficients

Princeton University Drift and Diffusion Coefficients MD Simulations run FORWARD in Time Ring node ICs   multiple replicas per node compute 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

Princeton University Ring at  +  Ring at  Ring Stepping in Generalized Potential Goal: Compute potential  associated with ring nodes (ring “height” above x 1 - x 2 plane)

Princeton University Right-handed  -helical minimum

Princeton University Potential Conditions I Case I : Diffusion matrix is proportional to unit matrix: ( D = scalar) Probability Current: Probability Current vanishes  Conditions for existence of generalized potential  : Path Drift Coefficient Potential Conditions

Princeton University CMD Exploration of Alanine Dipeptide using 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 information small step FORWARD in time large step BACKWARD in energy   ONLY drifts estimated here ring step size is scaled by unknown diffusivity D stepsize ~ nodal “heights” unknown

Princeton University Src homology 3 (SH3) domain Small Modular Domain 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 Diverging Turn 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, (2005). N-terminus C-terminus Fraction native contacts formed Fraction non-native contacts formed FOLDED UNFOLDED TRANSITION STATE Protein folding intrinsically low-dimensional “Collective” (coarse, slow) coordinates fully describe long time system dynamics

Princeton University Reverse Ring Integration and MD Coarse MD (CMD) Transition state identification using CMD

Princeton University 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).

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

Princeton University

Fish Schooling Models Initial State Compute Desired Direction Update Direction for Informed Individuals ONLY Zone of Deflection R ij <  Zone of Attraction R ij <  Normalize  INFORMED UNINFORMED Update Positions Position, Direction, Speed Couzin, Krause, Franks & Levin (2005) Nature (433) 513

Princeton University STUCK ~ typically around rxn coordinate value of about 0.5 INFORMED DIRN STICK STATES INFORMED individual close to front of group away from centroid

Princeton University SLIP ~ wider range of rxn coordinate values for slip 0  0.35 INFORMED DIRN SLIP STATES INFORMED individual close to group centroid

Princeton University Effective Fokker-Planck Equation Drift Coefficient Diffusion Coefficient FPE: Potential:

Princeton University Coarse Free Energy Calculation Estimate Drift and Diffusion coefficients numerically from simulation “bursts” X0X0 X0X0 X0X0 Kopelevich, Panagiotopoulos & Kevrekidis J Chem Phys 122 (2005) Hummer & Kevrekidis J 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 University STUCK SLIP Energy Landscape – Fish Swarming Problem CENTROID Informed DIRN

Princeton University Using the computer to select variable Rationale: Lake Carnegie, Princeton, NJ Straight Line Distance between locations NOT representative of actual transition difficulty\distance

Princeton University Using the computer to select variable Rationale: Lake Carnegie, Princeton, NJ Straight Line Distance Actual transition difficulty represented by curved path Curved Transition Distance

Princeton University Using the computer to select good variables Rationale: Lake Carnegie, Princeton, NJ Straight Line Distance IS representative of actual transition difficulty\distance in small LOCAL patches Patch size related to problem “geography”

Princeton University Using the computer to select variable Rationale: Euclidean Distance Selected Datapoint X Y Z 3D Dataset with 2D manifold Euclidean distance in input space may be weak indicator of INTRINSIC similarity of datapoints Geodesic distance is good for this dataset

Princeton University W 12 W 23 vertices edges weights parameter t Dataset as Weighted Graph

Princeton University Multiple random walks through simulation data initialized at Unequal separation (Euclidean distance) between IC ( ) and limits of random walk (, )

Princeton University Parameter Local neighborhood size Compute N  N “neighborhood” matrix K Compute diagonal normalization matrix D Compute Markovian matrix M N datapoints Require: Eigenvalues λ and Eigenvectors Φ of M A few Eigenvalues\Eigenvectors provide meaningful information on dataset geometry Top 2 nd N th

Princeton University Dataset in x, y, zDataset Diffusion Map N datapoints eigencomputation 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).

Princeton University Diffusion Map (  2,  3 ) 22 33 22 ABSOLUTE Coordinates SIGNED Coordinates Report absolute distance of all uninformed individuals to informed individual to DMAP routine Report (signed) distance of all uninformed individuals to informed individual to DMAP routine STICK SLIP STICK SLIP Reaction Coordinate

Princeton University MAN MACHINE MAN MACHINE ABSOLUTE Coordinates SIGNED Coordinates

Princeton University 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