Statistical mechanics analysis of sparse data.

Inferential structure determination uses Bayesian theory to combine experimental data with prior structural knowledge into a posterior probability distribution over protein conformational space. The posterior distribution encodes everything one can say objectively about the native structure in the light of the available data and additional prior assumptions and can be searched for structural representatives. Here an analogy is drawn between the posterior distribution and the canonical ensemble of statistical physics. A statistical mechanics analysis assesses the complexity of a structure calculation globally in terms of ensemble properties. Analogs of the free energy and density of states are introduced; partition functions evaluate the consistency of prior assumptions with data. Critical behavior is observed with dwindling restraint density, which impairs structure determination with too sparse data. However, prior distributions with improved realism ameliorate the situation by lowering the critical number of observations. An in-depth analysis of various experimentally accessible structural parameters and force field terms will facilitate a statistical approach to protein structure determination with sparse data that avoids bias as much as possible.