Application of Probabilistic Roadmaps to

the Study of Protein Motion

Proteins Proteins are the workhorses of all living organisms They perform many vital functions, e.g:

• Catalysis of reactions• Transport of molecules• Building blocks of muscles• Storage of energy• Transmission of signals• Defense against intruders

They are large molecules (few 100s to several 1000s of atoms)

They are made of building blocks (amino acids) drawn from a small “library” of 20 amino-acids

They have an unusual kinematic structure: long serial linkage (backbone) with short side-chains

Protein Sequence

O

N

NN

N

OO

O

Long sequence of amino-acids (dozens to thousands), also called residues

Dictionary of 20 amino-acids (several billion years old)

(residue i-1)

Central Dogma of Molecular Biology

Physiological conditions: aqueous solution, 37°C, pH 7,atmospheric pressure

Mad cow disease is caused by mis-folding

Drug molecules act bybinding to proteins

Molecular motion is an essential process of life

So, studying molecular motion is of critical importance in

molecular biology

Stanford BioX cluster

NMR spectrometer

However, few tools are available

Computer simulation:- Monte Carlo simulation- Molecular Dynamics

Motion occurs at very different frequencies

HIV-1 protease

Low-frequency motions (diffusive motions) are more directly related to protein functions

I ntermediate states

I ntermediate states

Unfolded (denatured) state

Folded (native) stateMany pathwaysMany pathways

Two Major Drawbacks of MD and MC Simulation

1) Each simulation run yields a single pathway, while molecules tend to move along many different pathways

Interest in ensemble properties

Two Major Drawbacks ofMD and MC Simulation

1) Each simulation run yields a single pathway, while molecules tend to move along many different pathways

2) Each simulation run tends to waste much time in local minima

Kinematic Models Atomistic model: The position of each

atom is defined by its coordinates in 3-D space

(x4,y4,z4)

(x2,y2,z2)(x3,y3,z3)

(x5,y5,z5)

(x6,y6,z6)

(x8,y8,z8)(x7,y7,z7)

(x1,y1,z1)

p atoms 3p parameters

Drawback: The bond structure is not taken into account

Kinematic Models Linkage model: The protein consists of

atoms connected by rotatable bonds

NN

NN

C’

C’

C’

C’

O

O O

O

C

C

C

C

C

C C

C

Resi Resi+1 Resi+2 Resi+3

Roadmap-Based Representation

Compact representation of many motion pathways Coarse resolution relative to MC and MD simulation ( only low-frequency motions are represented) Efficient algorithms for analyzing multiple pathways

Initial Work A.P. Singh, J.C. Latombe, and D.L. Brutlag.

A Motion Planning Approach to Flexible Ligand Binding. Proc. 7th ISMB, pp. 252-261, 1999

Study of ligand-protein binding The ligand is a small flexible molecule, but the protein is assumed rigid A fixed coordinate system P is

attached to the protein and a moving coordinate system L is defined using three bonded atoms in the ligand

A conformation of the ligand is defined by the position and orientation of L relative to P and the torsional angles of the ligand

Roadmap Construction (Node Generation)

The nodes of the roadmap are generated by sampling conformations of the ligand uniformly at random in the parameter space (around the protein)

The energy E at each sampled conformation is computed: E = Einteraction + Einternal

Einteraction = electrostatic + van der Waals potentialEinternal = non-bonded pairs of atoms electrostatic + van der Waals

Roadmap Construction (Node Generation)

The nodes of the roadmap are generated by sampling conformations of the ligand uniformly at random in the parameter space (around the protein)

The energy E at each sampled conformation is computed: E = Einteraction + Einternal

Einteraction = electrostatic + van der Waals potentialEinternal = non-bonded pairs of atoms electrostatic + van der Waals

A sampled conformation is retained as a node of the roadmap with probability:

0 if E > Emax

Emax-EEmax-Emin

1 if E < Emin

Denser distribution of nodes in low-energy regions of conformational space

P = if Emin E Emax

Roadmap Construction (Edge Generation)

q q’

Each node is connected to its closest neighbors by straight edges

Each edge is discretized so that between qi and qi+1 no atom moves by more than some ε (= 1Å)

If any E(qi) > Emax , then the edge is rejected

qi qi+

1

E

Emax

Heuristic measureof energetic difficultyor moving from q to q’

Roadmap Construction (Edge Generation)

q q’

Any two nodes closer apart than some threshold distance are connected by a straight edge

Each edge is discretized so that between qi and qi+1 no atom moves by more than some ε (= 1Å)

If all E(qi) Emax , then the edge is retained and is assigned two weights w(qq’) and w(q’q)

where:

(probability that the ligand moves from qi to qi+1 when it is constrained to move along the edge)

qi qi+

1

i i+1i

w(q q') = -ln(P[q q ])

ii+1

i ii+1 i-1

-(E -E )/ kT

i i+1 -(E -E )/ kT -(E -E )/ kT

eP[q q ] =

e e

For a given goal node qg (e.g., binding conformation), the Dijkstra’s single-source algorithm computes the lowest-weight paths from qg to each node (in either direction) in O(N logN) time, where N = number of nodes

Various quantities can then be easily computed in O(N) time, e.g., average weights of all paths entering qg and of all paths leaving qg (~ binding and dissociation rates Kon and Koff)

Querying the Roadmap

Protein: Lactate dehydrogenaseLigand: Oxamate (7 degrees of freedom)

Computation of Potential Binding Conformations

1) Sample many (several 1000’s) ligand’s conformations at random around protein

2) Repeat several times: Select lowest-energy

conformations that are close to protein surface

Resample around them

3) Retain k (~10) lowest-energy conformations whose centers of mass are at least 5Å apart

lactate dehydrogenase

active site

Experiments on 3 Complexes

1) PDB ID: 1ldmReceptor: Lactate Dehydrogenase (2386 atoms, 309 residues)Ligand: Oxamate (6 atoms, 7 dofs)

2) PDB ID: 4ts1Receptor: Mutant of tyrosyl-transfer-RNA synthetase (2423

atoms, 319 residues)Ligand: L- leucyl-hydroxylamine (13 atoms, 9 dofs)

3) PDB ID: 1stpReceptor: Streptavidin (901 atoms, 121 residues)Ligand: Biotin (16 atoms, 11 dofs)

Results for 1ldm

Some potential binding sites have slightly lower energy than the active site Energy is not a discriminating factor

Average path weights (energetic difficulty) to enter and leave binding site are significantly greater for the active site Indicates that the active site is surrounded by an energy barrier that “traps” the ligand

Energy

ConformationPotential binding

site

Potential binding

site

Active site

Known native state Degrees of freedom: φ-ψ angles Energy: van der Waals, hydrogen bonds,

hydrophobic effect New idea: Sampling strategy Application: Finding order of SSE

formation

Application of Roadmaps to Protein Folding

N.M. Amato, K.A. Dill, and G. Song. Using Motion Planning to Map Protein Folding Landscapes and Analyze Folding Kinetics of

Known Native Structures. J. Comp. Biology, 10(2):239-255, 2003

High dimensionality non-uniform sampling

Conformations are sampled using Gaussian distribution around native state

Conformations are sorted into bins by number of native contacts (pairs of C atoms that are closeapart in native structure)

Sampling ends when all bins have minimum number of conformations “good” coverage of conformational space

Sampling Strategy(Node Generation)

The lowest-weight path is extracted from each denatured conformation to the folded one

The order of formation of SSE’s is computed along each path

The formation order that appears the most often over all paths is considered the SSE formation order of the protein

Application: Order of Formation of Secondary

Structures

1) The contact matrix showing the time step when each native contact appears is built

Method

Protein CI2 (1 + 4 )

Protein CI2(1 + 4 )

60

5

The native contact between residues 5 and 60 appears at step 216

1) The contact matrix showing the time step when each native contact appears is built

2) The time step at which a structure appears is approximated as the average of the appearance time steps of its contacts

Method

Protein CI2(1 + 4 )

forms at time step 122 (II)3 and 4 come together at 187 (V)2 and 3 come together at 210 (IV)1 and 4 come together at 214 (I) and 4 come together at 214 (III)

1) The contact matrix showing the time step when each native contact appears is built

2) The time step at which a structure appears is approximated as the average of the appearance time steps of its contacts

Method

Comparison with Experimental Data

CI2

1+5

31+4

1+4 5126, 70k

5471, 104k7975, 104k8357, 119k

roadmap sizeSSE’s

Stochastic Roadmaps M.S. Apaydin, D.L. Brutlag, C. Guestrin, D. Hsu, J.C. Latombe and C.

Varma. Stochastic Roadmap Simulation: An Efficient Representation and Algorithm for Analyzing Molecular Motion. J. Comp. Biol., 10(3-4):257-

281, 2003

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

vi

vj

Pij

Edge probabilities

Follow Metropolis criteria:

ijij

iij

i

exp(-ΔE / kT), if ΔE >0;

NP =

1, otherwise.

N

Self-transition probability:

ii ijj i

P=1- Pvj

vi

Pij

Pii

[Roadmap nodes are sampled uniformly at random and energy profilealong edges is not considered]

V

Stochastic Roadmap Simulation

Pij

Stochastic roadmap simulation and Monte Carlo simulation converge to the Boltzmann distribution, i.e., the number of times SRS is at a node in V converges towardwhen the number of nodes grows (and they are uniformly distributed)

-E/ kT

Ve dV

Roadmap as Markov Chain

Transition probability Pij depends only on i and j

Pijij

Example #1: Probability of Folding pfold

Unfolded state Folded state

pfold1- pfold

Pii

F: Folded stateU: Unfolded state

First-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

Number of Self-Avoiding Walks

on a 2D Grid

1, 2, 12, 184, 8512, 1262816,575780564, 789360053252, 3266598486981642,(10x10) 41044208702632496804, (11x11) 1568758030464750013214100,(12x12) 182413291514248049241470885236 > 1028 http://mathworld.wolfram.com/Self-AvoidingWalk.html

In contrast …

Computing pfold with MC simulation requires:

For every conformation q of interest

Perform many MC simulation runs from q

Count number of times F is attained first

Computational 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 Approach

pfold for ß hairpin

Immunoglobin binding protein

(Protein G)

Last 16 amino acids

Cα based representation

Go model energy function

42 DOFs

[Zhou and Karplus, `99]

Computation Times (ß hairpin)

Monte Carlo (30 simulations):

1 conformation ~10 hours ofcomputer time

Over 107 energy

computations

Roadmap:

2000 conformations23 seconds ofcomputer time

~50,000 energycomputations

~6 orders of magnitude speedup!

Using Path Sampling to Construct Roadmaps

N. Singhal, C.D. Snow, and V.S. Pande. Using Path Sampling to Build Better Markovian State Models: Predicting the Folding Rate

and Mechanism of a Tryptophan Zipper Beta Hairpin, J. Chemical Physics, 121(1):415-425, 2004

New idea:Paths computed with Molecular Dynamics simulation techniques are used to create the nodes of the roadmap

More pertinent/better distributed nodes

Edges are labeled with the time needed to traverse them

t

U

F

Sampling Nodes from Computed Paths (Path

Shooting)

Sampling Nodes from Computed Paths (Path

Shooting)

U

Fi

jtij

pij

Example: Langevin dynamics equation of motion is where R is a Gaussian random forceext

dxF -mγ +R=0

dt

Node Merging

If two nodes are closer apart than some , they are merged into one and merging rules are applied to update edge probabilities and times

4

1

5

3

2P12, t12

P14, t14

1

5

3

2’P12’, t12’

P12’ = P12 + P14 t12’ = P12xt12 + P14xt14

Node Merging

If two nodes are closer apart than some , they are merged into one and merging rules are applied to update edge probabilities and times

4

1

5

3

2P12, t12

P14, t14

1

5

3

2’P12’, t12’

P12’ = P12 + P14 t12’ = P12xt12 + P14xt14

Approximately uniform distribution of nodes over the reachable subset of

conformational space

Approximately uniform distribution of nodes over the reachable subset of

conformational space

Application: Computation of MFPT

Mean First Passage Time: the average time when a protein first reaches its folded state

First-Step Analysis yields: MPFT(i) = j Pij x (tij + MPFT(j)) MPFT(i) = 0 if i F

Assuming first-order kinetics, the probability that a protein folds at time t is:

where r is the folding rate

MFPT = =1/r

-rtfP(t) = 1 - e

f0

P(t) tdt

Computational Test

12-residue tryptophan zipper beta hairpin (TZ2)

Folding@Home used to generate trajectories (fully atomistic simulation) ranging from 10 to 450 ns

1750 trajectories (14 reaching folded state) 22,400-node roadmap MFPT ~ 2-9 s, which is similar to

experimental measurements (from fluorescence and IR)

Conclusion

Probabilistic roadmaps are a recent, but promising tool for exploring conformational space and computing ensemble properties of molecular pathways

Current/future research:• Better sampling strategies able to handle more

complex molecular models (protein-protein binding)• More work to include time information in roadmaps • More thorough experimental validation to compare

computed and measured quantitative properties