1. A Bayesian Approach to Calculating Free Energies in Chemical and Biological Systems Andrew Pohorille NASA-Ames Research Center

2. Outline Why do we care about free energies? Some relevant statistical mechanics And now something different - a Bayesian approach to calculating free energies What does it all mean (to a Bayesian)?

3. K  exp(-  A/kT)

4. Poly-LQ at the water- membrane interface

5. Helmholtz free energy We really care only about free energy differences partition function

7. excess free energy 1-dimensional integral

8. The algorithm Perform MD or MC simulations of system 0 Calculate U 0 and U 1 ; store  U = U 1 - U 0 every so often (independent samples) Construct P(  U) Calculate  A by numerically integrating

9. Where is the problem?

10. Stratification

11. Importance sampling

12. Model of the Probability Distribution Gram-Charlier (normalized) Hermite normalization!

13. posterior prior likelihood function uniform prior marginalize C N expand P(X | C N, N) around optimal P(X | C N 0,N)

14. Finding ML coefficients Find extremum of lnP(X | C N,N) use Lagrange multipliers Statistically independent sample

15. Easy to solve using gradient-based non-linear solvers

16. What does it mean? N+1 conditions for orthogonality of {  n } = -M

17. bad idea! expand P(X | C N, N) around the ML solution P(X | C N 0,N)

18. second-order approximation  c m = c m - c m 0

20. And what does this mean? recall that orthogonality of {  n }

21. and the final result is… uniform prior of a N-dimensional unit hypersphere

22. Numerical simulations Linear combination of 3 Gaussian functions Gaussian with  = 8; 20 x 100,000 20 x 100,000 20 x 1,000

23. The ML free energy as a function of N

24. The ML choice of N

25. Free energies calculated using different methods

26. There is also a non-Bayesian solution The results are similar but somewhat ambiguous

27. A Big Picture? A common view is that free energy can be properly calculated only if microstates from the low-energy tail of the pdf are adequately sampled. But this can’t be right - see harmonic systems Theory-based model for the pdf is lacking. Is information-theoretical model possible? (there is precedence)

28. Conclusions An expansion of P(x) for Gaussian-like functions was proposed. The ML degree and coefficients of the expansion were determined. The approach is quite successful for calculating free energies from statistical simulations. Can we extract information about low-U tail of the pdf from its peak?