Reweighting ensemble probabilities with experimental histogram data constraints using a maximum entropy principle

Entropy maximization methods that update a probability distribution P ₀(x) to a new distribution P(x) with the use of externally known, averaged constraints find use in diverse areas. Jaynes developed a Maximum Entropy Procedure (MEP) that is an objective approach to incorporate external data to update P ₀(x) to P(x). In this work, we consider the MEP in the context of external data known from a probability distribution versus that from a mean and a few higher moments. An immediate problem is that the conventional iterative Lagrange multiplier method, which relies on inverting a certain covariance matrix, is not applicable here because the covariance matrix is not invertible. We introduce an indicator function method that does not suffer from this problem. It leads to an analytic solution to this version of a MEP. As an example, a previously generated ensemble of peptide conformations used to characterize an intrinsically disordered protein is analyzed. The external constraint is on the radius of gyration probability distribution, p(RG), of this peptide. Ensemble observables such as geometric, shape characteristics, the residue end-to-end distance distribution, the all atom-pair distribution function related to the scattering intensity, the polyproline II content, and NMR ³JHNHα three bond couplings are evaluated with the initial and updated ensembles. Some observables are found to be insensitive and others sensitive to the external information. An example of a 24-residue peptide, histatin 5, where an experimentally derived p(RG) is available, is also analyzed.