Entropy-preserving transformation method

Numerous biological effects may strongly be influenced by solvent entropy, such as the hydrophobic effect or the formation of lipid bilayers. To explain such effects we need to compute thermodynamic quantities as the entropy and the free energy. Calculation of both quantities requires knowledge of the density of all possible spatial configurations of solvent molecules. The density, however, is not directly accessible by simple numerical methods. Thus, we need easily manageable analytical estimation methods for the solvent density. For proteins many established estimation methods are available, such as the Principle Component Analysis [1], which require a condensed distribution of configurations in the vicinity of a single stable configuration. However, the density estimation was originally tailored for proteins, thus, the solvent cannot easily be incorporated. Two major problems occur when treating solvents. First, the diffusive motion of the solvent leads to a large configurational space that has to be sampled. Second, the motion of the solvent molecules is governed by a very shallow energy landscape. Hence, the configurational density has a complex topology excluding it from a straightforward analytical estimation. Friedemann Reinhard et al. developed a transformation, exploiting the permutation symmetry of the solvent [2]. Their approach rendered established estimation methods applicable. Whereas this permutation algorithm provides a promising method to locally condense the configurational density, the topology stays complex. Thus, the transformed configurational density cannot be optimally fitted by a Gaussian distribution allowing a simple entropy estimation. Here, we present a new method to improve Reinhard's permutation reduction by deforming the density in an entropy-preserving fashion such that we can make use of established entropy estimations. In order to deform the density we solve the convection equation in an incompressible flow, which we construct by means of divergence-free wavelets. With this method we want to contribute to the understanding of biological processes such as protein folding. We have developed a method that lays the ground for solvent entropy calculations and likewise enables to estimate entropies from highly unharmonic systems.