Calculating the binding free energies of charged species based on explicit-solvent simulations employing lattice-sum methods: An accurate correction scheme for electrostatic finite-size effects

Abstract

The calculation of a protein-ligand binding free energy based on molecular dynamics (MD) simulations generally relies on a thermodynamic cycle in which the ligand is alchemically inserted into the system, both in the solvated protein and free in solution. The corresponding ligand-insertion free energies are typically calculated in nanoscale computational boxes simulated under periodic boundary conditions and considering electrostatic interactions defined by a periodic lattice-sum. This is distinct from the ideal bulk situation of a system of macroscopic size simulated under non-periodic boundary conditions with Coulombic electrostatic interactions. This discrepancy results in finite-size effects, which affect primarily the charging component of the insertion free energy, are dependent on the box size, and can be large when the ligand bears a net charge, especially if the protein is charged as well. This article investigates finite-size effects on calculated charging free energies using as a test case the binding of the ligand 2-amino-5-methylthiazole (net charge +1 e) to a mutant form of yeast cytochrome c peroxidase in water. Considering different charge isoforms of the protein (net charges -5, 0, +3, or +9 e), either in the absence or the presence of neutralizing counter-ions, and sizes of the cubic computational box (edges ranging from 7.42 to 11.02 nm), the potentially large magnitude of finite-size effects on the raw charging free energies (up to 17.1 kJ mol(-1)) is demonstrated. Two correction schemes are then proposed to eliminate these effects, a numerical and an analytical one. Both schemes are based on a continuum-electrostatics analysis and require performing Poisson-Boltzmann (PB) calculations on the protein-ligand system. While the numerical scheme requires PB calculations under both non-periodic and periodic boundary conditions, the latter at the box size considered in the MD simulations, the analytical scheme only requires three non-periodic PB calculations for a given system, its dependence on the box size being analytical. The latter scheme also provides insight into the physical origin of the finite-size effects. These two schemes also encompass a correction for discrete solvent effects that persists even in the limit of infinite box sizes. Application of either scheme essentially eliminates the size dependence of the corrected charging free energies (maximal deviation of 1.5 kJ mol(-1)). Because it is simple to apply, the analytical correction scheme offers a general solution to the problem of finite-size effects in free-energy calculations involving charged solutes, as encountered in calculations concerning, e.g., protein-ligand binding, biomolecular association, residue mutation, pKa and redox potential estimation, substrate transformation, solvation, and solvent-solvent partitioning.

All-Atom Continuous Constant pH Molecular Dynamics With Particle Mesh Ewald and Titratable Water

“…45–47 The finite-size error has several physical origins, including the periodicity induced net charge interaction and undersolvation, discrete solvent effects, as well as residual integratd potential effects. 47 Rocklin et al recently showed that, for a system neutralized with counter-ions, the finite-size error in the charging free energy of a ligand (with +1 or −1 charge) is dominated by the discrete solvent effects due to an offset potential that compensates for the potential generated by discrete solvent. 47 The corresponding offset energy in charging a titratable group is given as 47 …”

“…47 Rocklin et al recently showed that, for a system neutralized with counter-ions, the finite-size error in the charging free energy of a ligand (with +1 or −1 charge) is dominated by the discrete solvent effects due to an offset potential that compensates for the potential generated by discrete solvent. 47 The corresponding offset energy in charging a titratable group is given as 47 …”

A GPU-Accelerated Parameter Interpolation Thermodynamic Integration Free Energy Method

“…Furthermore, some readers may question the appropriateness of using the Ewald background plasma correction for charged systems. 1,118–120 Previous works have suggested that one should perform many, short simulations to obtain an average Δ G from independent estimates rather than from running a few, long simulations. 121–124 The error estimation described in the Computational Details section accounts for correlation within a simulation, but it does not attempt to estimate the error incurred from finite sampling.…”

Proton-Coupled Conformational Allostery Modulates the Inhibitor Selectivity for β-Secretase

“…Δ G finite–size represents the correction to the electrostatic energy calculated with particle mesh Ewald under periodic boundary conditions for charged systems. 22 No error bars are given for $\mathrm{\Delta }{G}_{\text{res}}^{\mathrm{normalL}}$ and Δ G finite–size , as they were calculated analytically. The error bar of Δ G (pH 4.6) was estimated by combining the errors in Δ G (pH ref ) and the pH-dependent corrections.…”