Protonation of interacting residues in a protein by a Monte Carlo method: application to lysozyme and the photosynthetic reaction center of Rhodobacter sphaeroides.

Abstract

We used Monte Carlo methods to treat the statistical problem of electrostatic interactions among many titrating amino acids and applied these methods to lysozyme and the photosynthetic reaction center of Rhodobacter sphaeroides, including all titrating sites. We computed the average protonation of residues as a function of pH from an equilibrium distribution of states generated by random sampling. Electrostatic energies were calculated from a finite difference solution to the linearized Poisson-Boltzmann equation using the coordinates from solved protein structures. For most calculations we used the Metropolis algorithm to sample protonation states; for strongly coupled sites, we substantially reduced sampling errors by using a modifIed algorithm that allows multiple site transitions. The Monte Carlo method agreed with calculations for a small test system, lysozyme, for which the complete partition function was calculated. We also calculated the pH dependence of the free energy change associated with electron transfer from the primary to the secondary quinone in the photosynthetic reaction center. The shape of the resulting curve agreed fairly well with experiment, but the proton uptake from which the free energy was calculated agreed only to within a factor of two with the observed values. We believe that this discrepancy resulted from errors in the individual electrostatic energy calculations rather than from errors in the Monte Carlo sampling.Electrostatic interactions in proteins are important for protein structure and function. The largest contribution to the electrostatic potential within a protein arises from protonatable amino acids that can carry a net charge. The problem of determining the average charges on protonatable residues can be separated into two parts. (i) The energies of protonation of the individual amino acids and the interaction energies between pairs of charged residues must be calculated. Much progress along these lines has been made (1-4). (ii) The average protonation of each residue must be determined from the electrostatic energies. Since the protonation of a site depends on the protonation of all other sites (in a typical protein there may be hundreds of titrating sites), an exact statistical calculation becomes too time consuming for more than -25 titrating residues. In this paper we present a Monte Carlo method to solve the statistical problem of finding the protonation of many interacting protonatable residues.Previous methods used to solve this problem can handle only a small number of sites or are inaccurate. Exact values of average protonations calculated from a partition function work well when the number of sites is below -25, and the reduced-site approximation can treat twice as many sites for some systems (5). The Tanford-Roxby approximation (6) ignores fluctuations in the protonation of residues and has been shown to be inaccurate for strongly interacting titrating residues (5).We used a Monte Carlo technique for determining the protonation of many interacting sit...

Computational Methods for Biomolecular Electrostatics

“…However, a more complete treatment of ionizable residues in proteins considers the coupling between all the ionizable sites. There are a number of techniques for treating such coupling (Tanford and Roxby, 1972;Beroza et al, 1991;Antosiewicz et al, 1996a;Antosiewicz et al, 1996b;Bashford, 2004) and thereby determining the complete titration state of the protein. Unfortunately, such methods are complex and are beyond the scope of the current discussion.…”

Structure of the non-redox-active tungsten/[4Fe:4S] enzyme acetylene hydratase

“…For the model compounds, the electrostatic potential was calculated by focusing with two grids of 121 3 and with 121 3 grid points and grid spacings of 1.0 Å and 0.25 Å, respectively, again with the larger grid being centered on the protein and the smaller one being centered on the titratable group. We used a Monte Carlo approach (40) to calculate the protonation probabilities of all titratable sites of AH by using our program CMCT (41).…”

Electrostatic coupling to pH‐titrating sites as a source of cooperativity in protein‐ligand binding

“…Several approximation schemes have been worked out that allow one to calculate the total protonation, or the protonation O,, of particular sites as a function of pH in such situations (Bashford & Karplus, 1991;Beroza et al, 1991;Gilson, 1993;Yang et al, 1993). It is then possible to use an alternative formalism for the pH dependence of GD(c,pH) based on integration over a binding isotherm:…”

“…This is closely related to the problem of calculating microscopic protonation state equilibria in proteins that have a large number of protonation sites (Rashford & Karplus, 1991: Beroza et al. 1991: Gilson.…”