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
Alzheimer’s, Mad cow, Huntington’s . . .
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@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
>1,000,000 total CPUs; 200,000 active

5. 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?

6. 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

7. Chemical intuition
Chemical reactions often exhibit stochastic behavior
n-butane
Chandler, Journal of Chemical Physics (1977)

8. Markovian state model Define states in the conformation space 1 5 3 24 5
Define transition probabilities, or edges, between states 1 2 3 4 5

9. Uses of the model 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
Chodera et al., Multiscale
Modeling and Simulation (2006) p t

10. Chodera et al., Multiscale Modeling
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)

11. 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 1 2 3 4 5 l 1 2 3 4 5

12. 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, (2007). (*These authors contributed equally to this work)

13. Comparison of structural and kinetic clustering
trpzip2
Cochran et al. PNAS 98:5578, 2001.
structural clustering
kinetic clustering

14. State decomposition – splitting
Cluster conformations by root mean square distance (RMSD)

15. State decomposition – lumping
group states which inter-convert quickly

16. State decomposition – resplitting
Cluster conformations, restricted to each state

17. Blocked alanine peptide1 2 3 4 6 5 60 y
-60
Chodera et al., Multiscale Modeling
and Simulation (2006)
-60 f 60

18. Automatic state decomposition of alanine peptide
Black state sits on top of multiple other states!
These conformations had an unusual peptide bond y
Benefit of automatic algorithm f

19. Stability of decomposition

20. TrpZip peptide

21. Transition probabilities
Building Markovian State Model
Defining states that are Markovian
Calculating the transition probabilities
Discretize trajectories into series of states 1 2 3 4 5
1223435
normalize
Count number of transitions between all pairs of states
transition counts
transition probabilities
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), (2004).

22. Model selection Challenge: How many states should we have?
Refining Markovian State Model
Finding the best model
Determining model uncertainty
Designing new simulations
Model selection
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)

23. 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

24. 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

25. Refining Markovian State Model
Finding the best model
Determining model uncertainty
Designing new simulations
Uncertainty analysis
Goal: Once we have the states, what is the uncertainty in the model?
Uncertainty caused by finite sampling 1 1 5 5 2 3 2 3 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, (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, (2007).

26. Transition probabilities
Recall that we calculate transition probabilities by counting: i
700
300 k j
Instead of getting a single value, we can talk about the distribution of transition probabilities
Bayes’ Rule: i 70 30 k j
pij

27. [pij] [pij] Sampling approach l l
Possible solution to get distribution of eigenvalues:
solve for eigenvalue
[pij] l
solve for eigenvalue
[pij] l
Problem:
sampling can be expensive
solving per sample can be expensive

28. efficient to calculate using adjoint systems
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

29. Uncertainty results 5000 trajectories from each state 1 2 6 3 4 5
Alanine System
Transition Counts
Running times (87 states)
Sampling-based: 3600 seconds
Closed-form: < 0.07 seconds
Running times (6 states)
Sampling-based: 40 seconds
Closed-form: < 0.01 seconds

30. Refining Markovian State Model
Finding the best model
Determining model uncertainty
Designing new simulations
Sampling strategies
Problem: Simulations are expensive. Even with 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, (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, (2007).

31. Adaptive sampling Goal: Reduce uncertainty of eigenvalue
Uncertainty analysis decomposes by transitions from each state
Variance depends on both uncertainty of and sensitivity to transition probabilities

32. Adaptive sampling – alanine
On 6-state alanine system, select trajectories randomly for 3 sampling strategies
Transition Counts

33. 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)

34. 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

35. Acknowledgements Vijay Pande – Stanford University adviser
Bill Swope, Jed Pitera – IBM collaborators
John Chodera – state decomposition work