Molecular Transfer Model for pH effects on Intrinsically Disordered Proteins: Theory and Applications

Abstract

We present a theoretical method to study how changes in pH shape the heterogeneous conformational ensemble explored by intrinsically disordered proteins (IDPs). The theory is developed in the context of coarse-grained models, which enable a fast, accurate, and extensive exploration of conformational space at a given protonation state. In order to account for pH effects, we generalize the Molecular Transfer Model (MTM), in which conformations are re-weighted using the transfer free energy, which is the free energy necessary for bringing to equilibrium in a new environment a "frozen" conformation of the system. Using the semi-grand ensemble, we derive an exact expression of the transfer free energy, which amounts to the appropriate summation over all the protonation states. Because the exact result is computationally too demanding to be useful for large polyelectrolytes or IDPs, we introduce a mean-field (MF) approximation of the transfer free energy. Using a lattice model, we compare the exact and MF results for the transfer free energy and a variety of observables associated with the model IDP. We find that the precise location of the charged groups (the sequence), and not merely the net charge, determines the structural properties. We demonstrate that some of the limitations previously noted for MF theory in the context of globular proteins are mitigated when disordered polymers are studied. The excellent agreement between the exact and MF results poises us to use the method presented here as a computational tool to study the properties of IDPs and other biological systems as a function of pH.