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) P ij = probability of going from conformation i to conformation j Protein  dE: Energy difference  T: Temperature  k B : 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1 33 22 Associated limiting distribution  Stationary distribution   i =   j P ji   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)