Distribution functions in Ising patches: a test case for protein denaturation

Many proteins exhibit a sharp maximum in the heat capacity as a function of temperature as a result of the denaturation process. We have recently shown that the temperature dependence of the heat capacity can be converted into a finite set of moments of the enthalpy distribution for the protein. Using the maximum-entropy method one can then use these moments to construct approximations to the enthalpy distribution function, the more moments used the better the approximation. We find that for many proteins the enthalpy distribution is bimodal when one expands the heat capacity in the neighborhood of the maximum in this function reflecting the presence of two distinctly different populations of molecules. In the present paper we analyze a soluble model system to test the accuracy of the approximate distribution functions obtained from the maximum-entropy method using a finite set of moments. For this purpose we pick the two-dimensional Ising model near the critical point, treating finite patches of lattice sites. In the finite system there is no true phase transition, but there is a sharp change in density and enthalpy in the neighborhood of the critical point. The distribution functions for the Ising model are in fact very similar to those for proteins. For the Ising model, we can then compare the exact distribution functions with those calculated using a finite number of moments and the maximum-entropy method and we find that the latter give excellent agreement with the exact results thus lending credibility to similar results obtained for distribution functions describing protein denaturation.