E. C. Cyr, Numerical Methods for Computing the Free-Energy of
Coarse-Grained Molecular Systems, PhD Thesis, Department
of Computer Science, University of Illinois at Urbana-Champaign,
Coarse-grained models reduce the number of degrees of freedom in a molecular system.
Key to these models is accounting for the mean effect of the fine degrees of freedom decoupled
from the coarse system. The mean effect is expressed as a thermodynamic quantity known
as the free energy. In this thesis, we develop numerical methods for computing the free energy
of two distinct coarse-grained systems.
We first consider computing the potential of mean force (PMF) along a single degree of freedom.
Our approach is to develop algorithms based on the method of weighted
residuals (WRM) and maximum likelihood estimation (MLE). We show that traditional methods, like
and the direct histogram method, are specific instances of the WRM and MLE. The efficacy of the WRM
and MLE is demonstrated using two sample systems. Results indicate that methods based on WRM are
more robust with respect to the size of the solution space, while those based on MLE, are more accurate.
We also show how both the MLE and WRM can be used to perform adaptive sampling. This leads to
the development of the ABF-WRM, which combines the flexibility of the WRM framework with the
enhanced sampling of the adaptive biasing force (ABF) method.
The second coarse-grained system reduces a fully explicit
solvent-solute system to an explicit solute in an implicitly represented
solvent environment. The critical step in this reduction is to solve a
partial differential equation known as the Poisson-Boltzmann equation. We
develop algorithms that accurately compute the solvation free
energy by using goal-oriented mesh refinement. Results indicating
the benefits of goal-oriented refinement are presented.
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