Concepts and protocols for electrostatic free energies

Abstract

Electrostatic free energies play an essential role in numerous biomolecular processes occurring in solution. Difficulties arise when the long-range Coulomb interaction is computed for idealised infinite simulation models with periodic boundary conditions. To maintain a neutral simulation box and a finite per-box energy, a neutralising charge density or ‘gellium’ is commonly used, leading to a mean box potential that is constrained to be rigorously equal to zero at all times. Thus, in considering quantities such as ion solvation free energy, the potential drop to move from solvent into the usual, gas phase reference state is missing. In fact, for an infinite molecular system, the electrostatic potential itself is not uniquely defined, but takes the form of an infinite series that is only conditionally convergent. This leads to several possible computational conventions that give different values for the potential and field, all mathematically valid. For experimentally measurable quantities, however, unique results are obtained when sufficiently large simulation boxes are utilised. These concepts are detailed, as well as a fundamental, linear response theoretical framework that provides qualitative understanding of the physical processes involved, especially dielectric relaxation of the environment in response to a new solute charge. Illustrative applications to ligand binding and biomolecular electron transfer are described.

Keywords:

• Free energy simulation
• Coulomb potential
• molecular dynamics
• Ewald summation
• dielectric relaxation

Acknowledgements

We thank Nicolas Panel for doing the PDZ calculations.

Notes

No potential conflict of interest was reported by the authors.

1 There was a sign error in Equation (16) of Lin et al. [56].

Additional information

Funding

This work was supported by [grant number MCB-1517221] from the National Science Foundation (B.R.).