Molecular Motion Pathways: Computation of Ensemble
Properties with Probabilistic Roadmaps
1) A.P. Singh, J.C. Latombe, and D.L. Brutlag. A Motion Planning Approach to Flexible Ligand Binding. Proc. 7th Int. Conf. on Intelligent Systems for Molecular Biology (ISMB), AAAI Press, Menlo Park, CA, pp. 252-261, 1999.
2) 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.
3) 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. Biology, 10(3-4):257-281, 2003.
4) 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.
5) J. Cortés, T. Siméon, M. Renaud-Siméon, and V. Tran. Geometric Algorithms for the Conformational Analysis of Long Protein Loops. J. Comp. Chemistry, 25:956-967, 2004.
Mad cow disease is caused by misfolding Drug molecules act by
binding 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
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
Example of Ensemble Property:
Probability of Folding pfold
Unfolded state Folded state
pfold1- pfold
Measure kinetic distance to folded state Du, Pande, Grosberg, Tanaka,
and Shakhnovich. On the Transition Coordinate for Protein Folding. Journal of Chemical Physics (1998).
Other Examples of Ensemble Properties
Folding:• Order of formation of SSE’s• Folding rate / Mean first passage time• Key intermediates
Binding:• Average time to escape from active site• Average energy barrier
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
Roadmap-Based Representation
Compact representation of many motion pathways Coarse resolution relative to MC and MD simulation Efficient algorithms for analyzing multiple pathways
Roadmaps for Robot Motion Planning
free space
[Kavraki, Svetska, Latombe,Overmars, 96]
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)
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)
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
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!
Example #2: Ligand-Protein Interaction
Computation of escape time from funnels of attraction around potential binding sites
Funnel of attraction = ball of 10Å rmsd around bound state[Camacho and Vajda, 01]
Computation Through Simulation
[Sept, Elcock and McCammon `99]
10K to 30K independent simulations
Computing Escape Time with Roadmap
i = 1 + Pii i + Pij j+ Pik k + Pil l + Pim m
(escape time is measured as number of stepsof stochastic simulation)
= 0
Funnel of Attraction
ij
kl
m
Pii
Pim
PilPikPij
Distinguishing Active Site
Given several potential binding sites,
which one is the active one?
Energy: electrostatic + van der Waals + solvation free energy terms
Complexes 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 Using Escape Time
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
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)
Conformational Analysis of Protein Loops
J. Cortés, T. Siméon, M. Renaud-Siméon, and V. Tran. Geometric Algorithms for the Conformational Analysis of Long Protein Loops.
J. Comp. Chemistry, 25:956-967, 2004
New idea:Explore the clash-free subset of the conformational space of a loop, by building a tree-shaped roadmap
Kinematic model: - angles on the backbone + i torsional angles in side-chains
Amylosucrase (AS)- Only enzyme in its family that acts on sucrose substrate-The 17-residue loop (named loop 7) between Gly433 and Gly449 is believed to play a pivotal role
Roadmap Construction
A tree-shaped roadmap is created from a start conformation qstart
At each step of the roadmap construction, a conformation qrand of the loop is picked at random, and a new roadmap node is created by iteratively pulling toward it the existing node that is closest to qrand
Roadmap Construction
C CfreeCclosed
qstart
qrand
Roadmap Construction
C CfreeCclosed
qstart
qrand
Roadmap Construction
C Cfree
Cclosed
qstart
qrand
Roadmap Construction
C Cfree
Cclosed
qstart
qrand
Stops when one can’t get closer to qrand or a clash is detected
Computational Results Surprisingly, loop 7 can’t move much Main bottleneck is residue Asp231
Positions of theC atom of middleresidue (Ser441)
Computational Results Surprisingly, loop 7 can’t move much Main bottleneck is residue Asp231
Computational Results If residue Asp231 is “removed”, then loop
7’s mobility increases dramatically. The C atom of Ser441 can be displaced by more than 9Å from its crystallographic position
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
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