Modeling molecular dynamics from simulations

1
Modeling molecular dynamics from simulations

• Nina Singhal Hinrichs
• Departments of Computer Science and Statistics
• University of Chicago
• January 28, 2009

2
Motivation

• Proteins are essential parts of living organisms
• enzymes, cell signaling, membrane transport . . .
  
• Composed of chain of amino acids
• Fold to unique 3-dimensional structure
• Misfolding can cause diseases
• Alzheimers, Mad cow, Huntingtons . . .
• How do proteins fold?

3
Molecular dynamics

• Represent atoms of molecule and solvent
• Model forces on atoms
• Integrate laws of motion
• Small integration time step compared to motion
  timescales

4
Folding_at_Home Distributed computing for
biomolecular simulation

• Perform multiple simulations in parallel
• Total simulation times hundreds of microseconds
  (hundreds of CPU-years)

• Very powerful computational resource
• 200 Teraflops sustained performance
• gt1,000,000 total CPUs 200,000 active

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Challenge How to analyze?

• Enormous datasets
• Describe dynamics in microscopic detail
• Questions we want to answer
• Rate of folding, mechanism of folding . . .
• How can we extract these properties from our data?

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Outline

• Markovian state model for molecular motion
• Model description, uses, examples
• New algorithms for building these models
• Defining states and transition probabilities
• New methods for dealing with finite sampling
• Model complexity, uncertainty analysis, targeted
  sampling

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Chemical intuition
Chemical reactions often exhibit stochastic
behavior
n-butane
Chandler, Journal of Chemical Physics (1977)
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Markovian state model
Define states in the conformation space
Define transition probabilities, or edges,
between states
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Uses of the model
Chodera et al., Multiscale Modeling and
Simulation (2006)

• Populations of states over time
• Eigenvalues and eigenvectors
  conformational changes
• Kinetic properties virtually any kinetic
  property
• Mechanistic properties most likely path,
  probability of transitions as graph algorithms

p
t
10
Example models
Kasson et al., PNAS (2006)
alanine peptide
lipid vesicle fusion
Chodera et al., Multiscale Modeling and
Simulation (2006)
alpha helix
villin headpiece
Sorin and Pande, Biophysical Journal (2005)
Jayachandran et al., Journal of Structural
Biology (2006)
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Computational and statistical challenges

• Building Markovian state model
• Defining states that are Markovian
• Calculating the transition probabilities
• Refining Markovian state model
• Finding the best model
• Determining model uncertainty
• Designing new simulations

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Automatic state decomposition

• Building Markovian State Model
• Defining states that are Markovian
• Calculating the transition probabilities

• Challenge Find appropriate states
• Individual conformations as states does not scale
• Group conformations into discrete states
• Structural clustering is insufficient
• Basic algorithm combine structural and kinetic
  similarity

J. D. Chodera, N. Singhal, V. S. Pande, K. A.
Dill, and W. C. Swope. Automatic discovery of
metastable states for the construction of Markov
models of macromolecular conformational dynamics.
Journal of Chemical Physics, 126, 155101 (2007).
(These authors contributed equally to this work)
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Comparison of structural and kinetic clustering
trpzip2
Cochran et al. PNAS 985578, 2001.
structural clustering
kinetic clustering
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State decomposition splitting
Cluster conformations by root mean square
distance (RMSD)
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State decomposition lumping
group states which inter-convert quickly
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State decomposition resplitting
Cluster conformations, restricted to each state
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Blocked alanine peptide
60
y
-60
Chodera et al., Multiscale Modeling and
Simulation (2006)
f
-60
60
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Automatic state decomposition of alanine peptide
Black state sits on top of multiple other states!
y
Benefit of automatic algorithm
f
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Stability of decomposition
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TrpZip peptide
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Transition probabilities

• Building Markovian State Model
• Defining states that are Markovian
• Calculating the transition probabilities

Discretize trajectories into series of states
1?2?2?3?4?3?5
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 trp zipper beta hairpin. Journal of Chemical
Physics, 121(1), 415-425 (2004).
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Model selection

• Refining Markovian State Model
• Finding the best model
• Determining model uncertainty
• Designing new simulations

• Challenge How many states should we have?
• More states are more Markovian
• More states have more parameters
• How do we evaluate this tradeoff?

N. S. Hinrichs and V. S. Pande. Bayesian metrics
for validating and improving Markovian state
models for molecular dynamics simulations. (In
preparation)
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Hidden Markov Model formulation

• Formulate the problem as a Hidden Markov Model
  structure scoring question
• Different discretizations of continuous space
• Benefits of Bayesian scores
• Naturally handles tradeoff between complexity of
  model and amount of data
• Avoids over-fitting of parameters

States
Observations
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Alanine peptide results
Score of Hidden Markov models for different lag
times Last model is worse at shorter times but
preferred at longer times No previous evaluation
methods could distinguish these models
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Uncertainty analysis

• Refining Markovian State Model
• Finding the best model
• Determining model uncertainty
• Designing new simulations

Goal Once we have the states, what is the
uncertainty in the model?
Uncertainty caused by finite sampling
1
1
5
5
3
3
2
2
4
4
Both are reasonable but give different transition
probabilities ? Different MFPT, Pfold,
eigenvalues, eigenvectors ...
N. Singhal and V. S. Pande. Error analysis and
efficient sampling in Markovian state models for
protien folding. Journal of Chemical Physics,
123, 204909-204921 (2005). N. S. Hinrichs and V.
S. Pande. Calculation of the distribution of
eigenvalues and eigenvectors in Markovian state
models for molecular dynamics. Journal of
Chemical Physics, 126, 244101 (2007).
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Transition probabilities
Recall that we calculate transition probabilities
by counting

• Instead of getting a single value, we can talk
  about the distribution of transition
  probabilities
• Bayes Rule

pij
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Sampling approach
Possible solution to get distribution of
eigenvalues
Problem sampling can be expensive solving per
sample can be expensive
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Closed-form solution
Idea trade exact distribution for efficient
approximation
Eigenvalue equation
efficient to calculate using adjoint systems
Taylor series expansion
Multivariate normal approximation of Dpi
? Closed-form normal distribution for l
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Uncertainty results
5000 trajectories from each state
2
1
6
4
3
5
Alanine System
Transition Counts
Running times (6 states) Sampling-based 40
seconds Closed-form lt 0.01 seconds
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Sampling strategies

• Refining Markovian State Model
• Finding the best model
• Determining model uncertainty
• Designing new simulations

• Problem Simulations are expensive. Even with
  Folding_at_Home, we run simulations for months
• How to intelligently allocate our resources?
• Common approaches
• equilibrium sampling sample each conformation
  from its equilibrium distribution
• even sampling sample equally from each state
• New sequential approaches

N. Singhal and V. S. Pande. Error analysis and
efficient sampling in Markovian state models for
protien folding. Journal of Chemical Physics,
123, 204909-204921 (2005). N. S. Hinrichs and V.
S. Pande. Calculation of the distribution of
eigenvalues and eigenvectors in Markovian state
models for molecular dynamics. Journal of
Chemical Physics, 126, 244101 (2007).
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Adaptive sampling
Goal Reduce uncertainty of eigenvalue Uncertaint
y analysis decomposes by transitions from each
state
Variance depends on both uncertainty of and
sensitivity to transition probabilities
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Adaptive sampling alanine
On 6-state alanine system, select trajectories
randomly for 3 sampling strategies
Transition Counts
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Adaptive sampling villin
2454 states

• Benefits
• Very quickly reduce the variance
• Reduce the total number of simulations
• Need less computational power
• Can study more complex systems

Villin Headpiece
Jayachandran, et al., Journal of Chemical
Physics (2006)
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Summary

• Markovian state models are convenient methods to
  describe molecular motion
• Automatic state decomposition
• Scalable to large size systems
• Model selection
• Evaluate tradeoff between model complexity and
  amount of data
• Uncertainty analysis
• Efficient and decomposable
• Adaptive sampling
• Reduce number of simulations

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Acknowledgements

• Vijay Pande Stanford University adviser
• Bill Swope, Jed Pitera IBM collaborators
• John Chodera state decomposition work