Stochastic Roadmap Simulation: An Efficient Representation and Algorithm for Analyzing Molecular Motion

Mehmet Serkan Apaydin, Douglas L. Brutlag, Carlos Guestrin, David Hsu, Jean-Claude Latombe

Presented by: Alan Chen

Outline

Introduction Stochastic Roadmap Simulation (SRS) First-step Analysis and Roadmap Query SRS vs. Monte Carlo Transmission Coefficients Results Discussions

Introduction: Protein Modeling

Pathways Native Structure Monte Carlo & Molecular

Dynamics Local minima Single pathways

Stochastic Roadmap Simulation (SRS) Random Multiple pathways Probabilistic Conformational

Roadmap Markov Chain Theory

SRS: Conformation Space (C)

Configuration Space Set of all conformations: (q) Parameters of protein

folding interactions between atoms van der Wall forces electrostatic forces Energy function: (E(q)) Backbone torsional angles:

(

SRS: Roadmap Construction

Pathways in C roadmap (G) Pij = probability of going from

conformation i to conformation j Protein

dE: Energy difference T: Temperature kB: Boltzmann Constant

C

SRS: Study Molecular Motion

Monte Carlo Random path through C

global E minimum Underlying continuous

conformation space Local minima problem

SRS Sampled conformations Discretized Monte Carlo No local minima problem

First-Step Analysis

Macrostate (F) Nodes that share a

common property

Transitions (t) Steps from a node to a

macrostate

SRS vs. Monte Carlo

1

3

2

Associated limiting distribution

Stationary distributioni = jPji

i > 0

i = 1

SRS vs. Monte Carlo

Monte Carlo

SRS

SRS vs. Monte Carlo

S subset of C Relative volume (S) > 0 Absolute error > 0 Relative error > 0 Confidence level > 0 N uniformly sampled

nodes

High probability, can approximate

Given certain constants, number of node:

Transmission Coefficients

Kinetic distance between conformations Macrostates

F: folded state U: unfolded state q in U; = 0; q in F; = 1;

Results: Synthetic energy landscape

2-D Conformation Space Radially Symmetric Gaussians Paraboloid Centered at Origin Two global minima

SRS Evaluating energy of nodes

8 sec, 10,000 nodes Solving linear equations

750 sec, solve linear system

Monte Carlo Est. 800,000 sec, 10,000 nodes

Results: Repressor of Primer Energy function

Hydrophobic interactions Excluded volume

Folded macrostate + 3 angstroms

Unfolded macrostate +10 angstroms

Time Monte Carlo: 3 days trasmission

coefficient of 1 conformation SRS: 1 hour transmission

coefficients of all nodes 5000 nodes

Discussions

SRS vs Monte Carlo multiple paths vs. single path In the limit, SRS converges to Monte Carlo One hour vs. three days

Improvements Better roadmaps

Reduce the dimension of C Better sampling strategy

Faster linear system solver Uses

Order of protein folding Overcoming energy barriers (catalytic sites)