Maximum-entropy calculation of free energy distributions for two forms of myoglobin

The temperature dependence of the heat capacity of myoglobin depends dramatically on pH. At low pH (near 4.5), there are two weak maxima in the heat capacity at low and intermediate temperatures, respectively, whereas at high pH (near 10.7), there is one strong maximum at high temperature. Using literature data for the low-pH form (Hallerbach and Hinz, 1999) and for the high-pH form (Makhatadze and Privalov, 1995), we applied a recently developed technique (Poland, 2001d) to calculate the free energy distributions for the two forms of the protein. In this method, the temperature dependence of the heat capacity is used to calculate moments of the protein enthalpy distribution function, which in turn, using the maximum-entropy method, are used to construct the actual distribution function. The enthalpy distribution function for a protein gives the fraction of protein molecules in solution having a given value of the enthalpy, which can be interpreted as the probability that a molecule picked at random has a given enthalpy value. Given the enthalpy distribution functions at several temperatures, one can then construct a master free energy function from which the probability distributions at all temperatures can be calculated. For the high-pH form of myoglobin, the enthalpy distribution function that is obtained exhibits bimodal behavior at the temperature corresponding to the maximum in the heat capacity (Poland, 2001a), reflecting the presence of two populations of molecules (native and unfolded). For this form of myoglobin, the temperature evolution of the relative probabilities of the two populations can be obtained in detail from the master free energy function. In contrast, the enthalpy distribution function for the low-pH form of myoglobin does not show any special structure at any temperature. In this form of myoglobin the enthalpy distribution function simply exhibits a single maximum at all temperatures, with the position of the maximum increasing to higher enthalpy values as the temperature is increased, indicating that in this case there is a continuous evolution of species rather than a shift between two distinct population of molecules.