Efficient Energy Computation for Monte Carlo Simulation of Proteins

1
Efficient Energy Computation for Monte Carlo
Simulation of Proteins

• Itay Lotan
• Fabian Schwarzer
• Jean-Claude Latombe

Stanford University
2
Monte Carlo Simulation (MCS)
Popular method for studying the conformation
space of proteins

• Estimation of thermodynamic quantities over the
  space
• Search for low-energy conformations, in
  particular the native (folded) state

3
Preview of Whats to Come

• 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

4
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.

5
MCS How It Works

• Propose random change in conformation

• Compute energy E of new conformation
• Accept new conformation with probability

6
Energy Function

• Bonded terms
• Bond length, Bond angle, etc..
• Non-bonded terms
• Van der Waals, Electrostatic and heuristic

Non-bonded terms depend on distances between
pairs of atoms ? O(n2), expensive to compute
7
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!
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Reusing Energy Terms

• Only few DOFs are changed at each step

1)
2)

• Large sub-chains remain rigid between steps
• Many energy terms unaffected by change

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Our Goal

• Improve computational efficiency of MCS by
  reducing average time to accept/reject a new
  conformation
• Independent of
• Energy function
• Step generator
• Acceptance criterion

Exploiting protein backbone is kinematic chain
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Outline

• Related work
• The ChainTree
• Energy maintenance
• Tests
• Conclusion

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Outline

• Related work
• The ChainTree
• Energy maintenance
• Tests
• Conclusion

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Grid Method

• Subdivide space into cubic cells
• Compute cell that contains each atom center
• Store results in hash table

dcutoff
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Grid Method cont.

• T(n) time to recompute
• O(1) time to find interactions for each atom
• T(n) to find all interactions in all cases
• No way of detecting unchanged interactions

Asymptotically optimal in worst-case!
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Outline

• Related work
• The ChainTree
• Energy maintenance
• Tests
• Conclusion

15
The ChainTree
TNO TJKTKL
TJK
TKL
16
Updating the ChainTree

• Update path to root
• Recompute transforms that shortcut change
• Recompute BVs that contain change

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Finding Interacting Pairs
Test the ChainTree against itself
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Finding Interacting Pairs

• Do not search inside rigid sub-chains (unmarked
  nodes)
• Do not test two nodes with no marked node in
  between

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Finding Interacting Pairs
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Outline

• Related work
• The ChainTree
• Energy maintenance
• Tests
• Conclusion

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Summing the Interactions

• At 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?
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The EnergyTree
A caching scheme for partial energy sums

• Efficient to update
• Efficient to query

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Using the EnergyTree
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Outline

• Related work
• The ChainTree
• Energy maintenance
• Tests
• Conclusion

25
Test Setup

• Energy function
• Van der Waals
• Electrostatic
• Attraction between native contacts
• Cutoff at 12Å
• 300,000 steps MCS
• Early rejection for large vdW terms

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Results 1-DOF change
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Results 5-DOF change
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Outline

• Related work
• The ChainTree
• Energy maintenance
• Tests
• Conclusion

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Conclusion

• 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

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MCS Software

• EEF1 force field (Lazaridis Karplus 99)
• Backbone DOFs (F,?) and fixed rotamers for
  side-chains (Dunbrack Cohen 97)
• Classical MCS with simple move-set
• Download and customize

http//robotics.stanford.edu/itayl/mcs