A Bayesian Approach to Calculating Free Energies in

Chemical and Biological Systems

Andrew Pohorille

NASA-Ames Research Center

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)?

K exp(-A/kT)

Poly-LQ at the water-membrane interface

Helmholtz free energy

We really care only about free energy differences

partition function

excess free energy

1-dimensional integral

The algorithm

• Perform MD or MC simulations of system 0

• Calculate U0 and U1; store U = U1 - U0 every so often (independent samples)

• Construct P(U)

• Calculate A by numerically integrating

Where is the problem?

Stratification

Importance sampling

Model of the Probability Distribution

Gram-Charlier(normalized)

Hermite

normalization!

posterior prior

likelihoodfunction

uniform prior

marginalize CN

expand P(X | CN, N) around optimal P(X | CN0,N)

Finding ML coefficientsFind extremum of lnP(X | CN,N)use Lagrange multipliers

Statistically independentsample

Easy to solve using gradient-based non-linear solvers

What does it mean?

N+1 conditions for orthogonality of {n}

= -M

bad idea!

expand P(X | CN, N) around the ML solution P(X | CN0,N)

second-order approximation

cm = cm - cm0

And what does this mean?

recall that

orthogonality of {n}

and the final result is…

uniform prior of a N-dimensional unit hypersphere

Numerical simulations

Linear combination of 3 Gaussian functions

Gaussian with = 8; 20 x 100,000

20 x 100,000

20 x 1,000

The ML free energy as a function of N

The ML choice of N

Free energies calculated using different methods

There is also a non-Bayesian solution

The results are similar but somewhat ambiguous

A Big Picture?A common view is that free energy can be properly calculated only if microstates from the low-energy tail of the pdfare 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)

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?