Charged binding calculations

Authored by Gabriel Rocklin, last updated June 15 2014

Overview

Many thermodynamically interesting processes involve ions changing chemical environment, such as the transfer of an ion from the gas phase to the solvated phase, the protonation of a protein side chain, or the binding of a charged ligand. Free energy simulations of these processes can be done in two ways: (1) the same simulated system can be used for both chemical environments, such as when a charged ligand is "pulled" from a protein binding site into the bulk solvent, or (2) different simulated systems can be used for different chemical environments, such as when a charged ligand is alchemically decoupled from its own "free in solution" box and alchemically inserted/coupled into the protein binding site in a different simulated system. In both cases, the computed free energies can depend on details of the calculation setup, such as the boundary conditions applied at the edge of the explicit solvent system (periodic, implicit solvent, or vacuum), the size of the periodic unit cell, and even completely arbitrary details of the solvent molecular topology.

These dependences on setup parameters are referred to as "finite size effects" because they are caused by the finite and extremely small size of the simulated system/periodic unit cell relative to the macroscopic size of the experimental system. Finite size effects are basically computational artifacts because they contribute to the simulated free energy change but not to a bulk experimental measurements. It is thus desirable to remove these free energy contributions before comparing simulated free energy results with experimental results.

The conventional way to do this is to introduce correction terms to convert the raw simulation results (which are specific to the particular choices of boundary condition, system size, molecular topology, etc) into corrected results based on ideal physical conditions, meaning that the solute is infinitely diluted in a nonperiodic bulk of solvent. Once simulation results have been corrected to match these ideal conditions, different simulation methods can be compared to each other without confounding artifacts, and simulation results can be compared with experimental results. (Of course, bulk experiments are not exactly performed under true ideal conditions either, but the non-idealities in experiments have little to do with finite size artifacts in simulations (periodic or otherwise) so it is still best to correct simulation results for these artifacts prior to comparisons with experimental results.)

This page provides short guidance on how to remove finite size artifacts for several common simulation approaches. For full details see the associated references. Note that the only types of artifacts considered here are those relating to the net charge of the investigated species (for instance, the charged ligand). For now, this page only covers simulations performed under periodic boundary conditions.

Periodic boundary conditions alter the electric field in the unit cell

The key difference between periodic boundary conditions and nonperiodic boundary conditions is that under periodic boundary conditions

[math]\displaystyle{ &phi = 0 }[/math]

Scenario 1: Protein-ligand binding in the same box

A common approach for calculating protein-ligand binding is to compute the potential of mean force (PMF) of "pulling" the ligand out of the protein binding side using artificial restraining potentials. The ligand is considered "unbound" once it is "suitably far" from the protein as to be considered non-interacting.