Probabilistic Roadmaps: A Tool for Computing

Ensemble Properties of Molecular Motions

Serkan Apaydin, Doug Brutlag1

Carlos Guestrin, David Hsu2

Jean-Claude Latombe, Chris Varma

Computer Science DepartmentStanford University

1 Department of Biochemistry, Stanford University2 Computer Science Department, University of North Carolina

Goal of our ResearchGoal of our Research

Develop efficient computational representations and algorithms to study molecular pathways for protein folding and ligand-protein binding

Protein folding RECOMB ’02Ligand-protein binding ECCB ‘02

AcknowledgementsAcknowledgements

People: Leo Guibas Michael Levitt, Structural BiologyItay Lotan Vijay Pande, ChemistryFabian Schwarzer Amit SinghRohit Singh

Funding: NSF-ITR ACI-0086013Stanford’s Bio-X and Graduate Fellowship

programs

Analogy with RoboticsAnalogy with Robotics

Configuration SpaceConfiguration Space

Approximate the free space by random sampling

Probabilistic Roadmaps

Probabilistic RoadmapProbabilistic Roadmap

free space

[Kavraki, Svetska, Latombe,Overmars, 95][Kavraki, Svetska, Latombe,Overmars, 95]

Probabilistic CompletenessProbabilistic Completeness

The probability that a roadmap fails to correctly capture the connectivity of the

free space goes to 0 exponentially in the number of milestones (~ running time).

Random sampling is convenient incremental

scheme for approximating the free space

Computed ExamplesComputed Examples

Biology Biology Robotics Robotics

Energy field, instead of joint controlContinuous energy field, instead of binary free and in-collision spacesMultiple pathways, instead of single collision-free pathPotentially many more degrees of freedomRelation to real world is more complex

Initial WorkInitial Work[Singh, Latombe, Brutlag, 99][Singh, Latombe, Brutlag, 99]

Study of ligand-protein bindingProbabilistic roadmaps with edges weighted by energetic plausibilitySearch of most plausible paths

Initial WorkInitial Work[Singh, Latombe, Brutlag, 99][Singh, Latombe, Brutlag, 99]

Study of ligand-protein bindingProbabilistic roadmaps with edges weighted by energetic plausibilitySearch of most plausible pathsStudy of energy profiles along such paths

CatalyticSite

energy

Initial WorkInitial Work[Singh, Latombe, Brutlag, 99][Singh, Latombe, Brutlag, 99]

Study of ligand-protein bindingProbabilistic roadmaps with edges weighted by energetic plausibilitySearch of most plausible pathsStudy of energy profiles along such pathsExtensions to protein folding[Song and Amato, 01] [Apaydin et al., 01]

New Idea: New Idea: Capture the stochastic nature of molecular Capture the stochastic nature of molecular motion by assigning probabilities to edgesmotion by assigning probabilities to edges

vi

vj

Pij

Why is this a good idea?Why is this a good idea?

1) We can approximate Monte Carlo simulation as closely as we wish

2) Unlike with MC simulation, we avoid the local-minima problem

3) We can consider all pathways in the roadmap at once to compute ensemble properties

Edge probabilitiesEdge probabilities

Follow Metropolis criteria:

otherwise. ,

1

;0 if ,)/exp(

i

iji

Bij

ij

N

EN

TkE

P

Self-transition probability:

ijijii PP 1

vj

vi

Pij

Pii

Stochastic simulation on roadmap and Monte Carlo simulation converge to same Boltzmann distribution

S

Stochastic Roadmap SimulationStochastic Roadmap Simulation

Pij

Problems with Problems with Monte Carlo SimulationMonte Carlo Simulation

Much time is wasted in local minima Each run generates a single pathway

SolutionSolution

Pij

Treat roadmap as a Markov chain and use the First-Step Analysis tool

Example #1: Example #1:

Probability of Folding pProbability of Folding pfoldfold

Unfolded set Folded set

pfold1- pfold

“We stress that we do not suggest using pfold as a transition coordinate for practical purposes as it is

very computationally intensive.” Du, Pande, Grosberg, Tanaka, and Shakhnovich “On the Transition

Coordinate for Protein Folding” Journal of Chemical Physics (1998).

HIV integrase[Du et al. ‘98]

Pii

F: Folded setU: Unfolded set

First-Step AnalysisFirst-Step Analysis

Pij

i

k

j

l

m

Pik Pil

Pim

Let fi = pfold(i)After one step: fi = Pii fi + Pij fj + Pik fk + Pil fl + Pim fm

=1 =1

One linear equation per node Solution gives pfold for all nodes

No explicit simulation run All pathways are taken into account Sparse linear system

In Contrast …In Contrast …

Computing pfold with MC simulation requires:

Performing many MC simulation runs

Counting the number of times F is attained firstfor every conformation of interest:

Computational TestsComputational Tests• 1ROP (repressor of

primer)• 2 helices• 6 DOF

• 1HDD (Engrailed homeodomain)

• 3 helices• 12 DOF

H-P energy model with steric clash exclusion [Sun et al., 95]

1ROP

Correlation with MC ApproachCorrelation with MC Approach

Correlation with MC ApproachCorrelation with MC Approach

1HDD

Computation Times (1ROP)Computation Times (1ROP)

Monte Carlo:

49 conformations Over 11 days ofcomputer time

Over 106 energy

computations

Roadmap:

5000 conformations1 - 1.5 hours ofcomputer time

~15,000 energycomputations

~4 orders of magnitude speedup!

Example #2: Example #2: Ligand-Protein InteractionLigand-Protein Interaction

Computation of escape time from funnels of attraction around potential binding sites(funnel = ball of 10A rmsd)

Computing Escape Time with Computing Escape Time with RoadmapRoadmap

Funnel of Attraction

ij

kl

m

Pii

Pim

PilPikPij

i = 1 + Pii i + Pij j+ Pik k + Pil l + Pim m

(escape time is measured as number of stepsof stochastic simulation)

= 0

Similar Computation Similar Computation Through Simulation Through Simulation [Sept,

Elcock and McCammon `99]

10K to 30K independent simulations

ApplicationsApplications

1) Distinguishing catalytic site: Given several potential binding sites, which one is the catalytic site?

Complexes StudiedComplexes Studied

ligand protein # random nodes

# DOFs

oxamate 1ldm 8000 7

Streptavidin 1stp 8000 11

Hydroxylamine 4ts1 8000 9

COT 1cjw 8000 21

THK 1aid 8000 14

IPM 1ao5 8000 10

PTI 3tpi 8000 13

Distinction Based on Distinction Based on EnergyEnergy

Protein Bound state

Best potential binding site

1stp -15.1 -14.6

4ts1 -19.4 -14.6

3tpi -25.2 -16.0

1ldm -11.8 -13.6

1cjw -11.7 -18.0

1aid -11.2 -22.2

1ao5 -7.5 -13.1 (kcal/mol)

Able to distinguish

catalytic site

Not able

Distinction Based on Escape Distinction Based on Escape TimeTime

Protein Bound state

Best potential binding site

1stp 3.4E+9 1.1E+7

4ts1 3.8E+10 1.8E+6

3tpi 1.3E+11 5.9E+5

1ldm 8.1E+5 3.4E+6

1cjw 5.4E+8 4.2E+6

1aid 9.7E+5 1.6E+8

1ao5 6.6E+7 5.7E+6(# steps)

Able to distinguishcatalytic

site

Not able

ApplicationsApplications

1) Distinguishing catalytic site2) Computational mutagenesis

C

C

OO

O

GLN-101

ARG-106

ASP-195HIS-193

ASP-166

ARG-169

NADH

+

+

+

Loop

Chemical environment of LDH-NADH-substrate complex (pyruvate) (catalyzes conversion of pyruvate to lactate in the presence of NADH

CH3

Some amino acids aredeleted entirely, replaced by other amino acids, or sidechains altered

Binding of Pyruvate to LDHBinding of Pyruvate to LDH

ASP-195HIS-193

ASP-166

ARG-169

+

+

+

THR-245

C

C

OO

O

CH3

NADH

GLN-101

ARG-106Loop

ResultsResults

C

C

OO

O

GLN-101

ARG-106

ASP-195HIS-193

ASP-166

ARG-169

NADH

+

+

+

Loop

CH3

THR-245

Mutant Escape Time

Change

Wildtype 3.216E6 N/A

ResultsResults

C

C

OO

O

GLN-101

ALA-106

ASP-195ALA-193

ASP-166

ARG-169

NADH

+

Loop

CH3

Mutant Escape Time

Change

Wildtype 3.216E6 N/A

His193 AlaArg106 Ala

4.126E2

ResultsResults

Mutant Escape Time Change

Wildtype 3.216E6 N/A

His193 AlaArg106 Ala

4.126E2

His193 Ala 3.381E3

Arg106 Ala 2.550E2

Asp195 Asn 5.221E7

Gln101 Arg 1.669E6 No change

Thr245 Gly 4.607E5

C

C

OO

O

GLN-101

ARG-106

ASP-195HIS-193

ASP-166

ARG-169

NADH

+

+

+

Loop

CH3

GLY-245

ConclusionConclusion

Probabilistic roadmaps are a promising computational tool for studying ensemble properties of molecular pathwaysCurrent and future work: Better kinetic/energetic models Experimentally verifiable tests Non-uniform sampling strategies Encoding MD simulation

Stochastic simulation on a roadmap and MC simulation converge to the same distribution (Boltzman):For any set S, >0, >0,>0, there exists N such that a roadmap with N milestones has error bounded by:

with probability at least 1- )1)(()(ˆ)1)(( SSS

vs

vg

S

Stochastic Roadmap SimulationStochastic Roadmap Simulation

Ligand-Protein ModelingLigand-Protein Modeling

• DOF = 10 – 3 coordinates to position root atom;– 2 angles to specify first bond;– Angles for all remaining non-terminal atoms;– Bond angles are assumed constant;

• Protein assumed rigid[Singh, Latombe and Brutlag `99]

x,y,z

Energy of InteractionEnergy of Interaction

Ev

Rij

Ec

Rij

Ev = 0.2[(R0/Rij)12 - 2(R0/Rij)

6 ]Ec = 332 QiQj/(Rij)

Energy = van der Waals interaction (Ev)

+ electrostatic interaction (Ec)

Solvent Effects

• Is only valid for an infinite medium of uniform dielectric;• Dielectric discontinuities result in induced surface

charges;

Solution: Poisson-Boltzman equation

Ec = 332 QiQj/(Rij)

Use Delphi [Rocchia et al `01] Finite Difference solution is based on discretizing

the workspace into a uniform grid.

[(r) . (r)] - (r)k(r)2sinh([(r)] + 4rf(r)/kT = 0