Correlations in thermodynamics and evolution of proteins

An important prerequisite for the biological function of a protein is the thermodynamic stability of its three-dimensional structure, the so-called native state. By adjusting the amino acid sequence the stability can be optimized by two different strategies. While positive design increases the stability with respect to unfolding by decreasing the free energy of the native state, negative design increases the free energy of misfolded structures in order to optimize the stability against misfolding. One stability can be optimized only at the expense of the other, thus optimal stability demands a trade-off between the two strategies. In the first part of this work, negative design in naturally occurring proteins was investigated using a simple energy model based on contact interactions of amino acids. The calculation of the free energy of the misfolded ensemble is difficult due to the large number of misfolded structures. A widely used model to describe the free energy of the misfolded ensemble is the Random Energy Model (REM), which assumes contacts to be uncorrelated and to occur with equal frequency. This is, however, an inaccurate description, as the probability of contact decreases with increasing distance in the sequence and the formation of a contact in a misfolded structure is correlated with other contacts. The first part of the thesis investigates how contact frequency and contact correlation affect negative design. Here, the free energy of the misfolded ensemble is approximated by a cumulant expansion, where contact frequency and contact correlation are explicitly included. In addition, it is investigated how the description of optimal hydrophobicity profiles, which have maximal stability in the native state, can be enhanced by the inclusion of contact correlations. The detailed description of the misfolded ensemble can help to improve the design of sequences or allows a more accurate modeling of protein evolution. Since protein sequences change during evolution, correlated substitutions of amino acids at different sites in the protein --- in the literature often referred to as correlated mutations --- give insight into the native structure and function of a protein. However, there was no theoretical description to quantify the effects of the physical constraints of structure and folding stability on correlated mutations in protein sequences. In the second part, a model is studied which quantitatively predicts the correlated mutations from constraints on the folding stability. The model is based on maximizing the sequence entropy, which is approximated by a cluster expansion up to second order. The model is tested using data from computer simulations and a statistical analysis of proteins from the Protein Data Bank. In particular, the determination of the model parameters allows an interpretation of the correlations in terms of both design strategies that characterize sequence evolution. The model can help to distinguish native from non-native contacts based on correlated mutations, thus improving the prediction of contacts and hence the prediction of protein structures. In addition, the model could be helpful to distinguish between correlated mutations that result from the folding stability or other selective pressures.