Transcript Efficient Energy Computation for Monte Carlo Simulation of Proteins Itay Lotan Fabian Schwarzer Jean-Claude Latombe Stanford University.

Efficient Energy Computation
for Monte Carlo Simulation of
Proteins
Itay Lotan
Fabian Schwarzer
Jean-Claude Latombe
Stanford University
Monte Carlo Simulation (MCS)
Popular method for studying the
conformation space of proteins:
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Estimation of thermodynamic quantities
over the space
Search for low-energy conformations, in
particular the native (folded) state
Preview of What’s to Come
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Method for speeding up MCS of proteins
Exploits the fact that a protein
backbone is a kinematic chain
Avoids the combinatorial explosion of
atomic interactions
Gives as much as 12X speed-up for
proteins we tested
MCS: What It Is
Random walk through the
conformation space of a protein that
samples conformations on its path.
Converges to the underlying
distribution of conformations after
enough time.
MCS: How It Works
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Propose random change in conformation
Compute energy E of new conformation
Accept new conformation with probability:
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P(accept )  min 1, e
E / kbT
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Energy Function
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Bonded terms:
Bond length, Bond angle, etc..
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Non-bonded terms
Van der Waals, Electrostatic and heuristic
Non-bonded terms depend on distances
between pairs of atoms  O(n2),
expensive to compute
Pairwise Interactions
Use cutoff distance (6 - 12Å)
Only O(n) interactions (Halperin & Overmars ’98)
O(1) interactions per atom
Find interacting pairs without
enumerating all pairs!
Reusing Energy Terms
Only few DOFs are changed at each step
1)
2)
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Large sub-chains remain rigid between steps
Many energy terms unaffected by change
Our Goal
Improve computational efficiency of
MCS by reducing average time to
accept/reject a new conformation
Independent of:
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Energy function
Step generator
Acceptance criterion
Exploiting: protein backbone is kinematic chain
Outline
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Related work
The ChainTree
Energy maintenance
Tests
Conclusion
Outline
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Related work
The ChainTree
Energy maintenance
Tests
Conclusion
Grid Method
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Subdivide space into cubic cells
Compute cell that contains each atom center
Store results in hash table
dcutof
f
Grid Method – cont.
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Θ(n) time to recompute
O(1) time to find interactions for each
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Θ(n) to find all interactions in all
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atom
cases
No way of detecting unchanged
interactions
Asymptotically optimal in worst-case!
Outline
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Related work
The ChainTree
Energy maintenance
Tests
Conclusion
The ChainTree
TNO= TJK*TKL
BV(A,B)
TJK
BV(C,D)
TKL
Updating the ChainTree
Update path to root:
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Recompute transforms that shortcut change
Recompute BVs that contain change
Finding Interacting Pairs
Test the ChainTree against itself
Finding Interacting Pairs
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Do not search inside rigid sub-chains (unmarked nodes)
Do not test two nodes with no marked node in between
Finding Interacting Pairs
Outline
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Related work
The ChainTree
Energy maintenance
Tests
Conclusion
Summing the Interactions
At
1.
2.
3.
each step need to sum contribution of:
New interactions
Changed interactions
Unchanged interactions
(1) & (2) are found by ChainTree search
How to retrieve (3) efficiently?
The EnergyTree
A caching scheme for partial energy sums:
• Efficient
to update
• Efficient
to query
Using the EnergyTree
E(N,N)
E(J,L)
E(K,L)
E(L,L)
E(M,M)
Outline
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Related work
The ChainTree
Energy maintenance
Tests
Conclusion
Test Setup
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Energy function:
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Van der Waals
Electrostatic
Attraction between native contacts
Cutoff at 12Å
300,000 steps MCS
Early rejection for large vdW terms
Results: 1-DOF change
(68)
(144)
(374)
(755)
Results: 5-DOF change
(68)
(144)
(374)
(755)
Outline
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Related work
The ChainTree
Energy maintenance
Tests
Conclusion
Conclusion
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Novel method to reduce average time
per step in MCS of proteins
Exploits kinematic chain nature of
protein
Significant speed-up for small number
of simultaneous DOF changes
Better for larger proteins
MCS Software
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EEF1 force field (Lazaridis & Karplus ’99)
Backbone DOFs (Φ,Ψ) and fixed rotamers
for side-chains (Dunbrack & Cohen ’97)
Classical MCS with simple move-set
Download and customize
http://robotics.stanford.edu/~itayl/mcs