Quantifying charge state heterogeneity for proteins with multiple ionizable residues

Ionizable residues can release and take up protons and this has an influence on protein structure and function. The extent of protonation is linked to the overall pH of the solution and the local environments of ionizable residues. Binding or unbinding of a single proton generates a distinct charge microstate defined by a specific pattern of charges. Accordingly, the overall partition function is a sum over all charge microstates and Boltzmann weights of all conformations associated with each of the charge microstates. This ensemble-of-ensembles description recast as a q-canonical ensemble allows us to analyze and interpret potentiometric titrations that provide information regarding net charge as a function of pH. In the q-canonical ensemble, charge microstates are grouped into mesostates where each mesostate is a collection of microstates of the same net charge. Here, we show that leveraging the structure of the q-canonical ensemble allows us to decouple contributions of net proton binding and release from proton arrangement and conformational considerations. Through application of the q-canonical formalism to analyze potentiometric measurements of net charge in proteins with repetitive patterns of Lys and Glu residues, we are able to determine the underlying mesostate pKa values and, more importantly, we estimate relative mesostate populations as a function of pH. This is a strength of using the q-canonical approach and cannot be obtained using purely site-specific analyses. Overall, our work shows how measurements of charge equilibria, decoupled from measurements of conformational equilibria, and analyzed using the framework of the q-canonical ensemble, provide protein-specific quantitative descriptions of pH-dependent populations of mesostates. This method is of direct relevance for measuring and understanding how different charge states contribute to conformational, binding, and phase equilibria of proteins. STATEMENT OF SIGNIFICANCEThe net charge of a protein in solution is governed by the overall pH as well as context and conformational contexts. Measurements of net charge are accessible via techniques such as potentiometry that quantify the buffering capacity of a protein solution. Here, we use the formal structure of the q-canonical ensemble to identify charge states that are compatible with a measured net charge profile as a function of pH. Our approach highlights how measurements of charge, decoupled from measurements of conformation, can be used to identify the ensembles of charge states that contribute to the overall population for given solution conditions. The methods introduced will be useful for measuring charge states and interpreting these measurements in different contexts.