Calculation of energies and forces

Sharp, K. A.

doi:10.1107/97809553602060000712

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecues
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 22.3, p. 555 | 1 | 2 |

Section 22.3.2.4. Calculation of energies and forces

K. A. Sharp^a ^*

^aE. R. Johnson Research Foundation, Department of Biochemistry and Biophysics, University of Pennsylvania, Philadelphia, PA 19104-6059, USA
Correspondence e-mail: sharp@crystal.med.upenn.edu

22.3.2.4. Calculation of energies and forces

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Once the electrostatic potential distribution has been obtained, calculation of experimental properties usually requires evaluation of the electrostatic energy or force. For a linear system (where the dielectric and ionic responses are linear) the electrostatic free energy is given by $[\Delta G^{\rm el} = 1/2 {\textstyle\sum\limits_{i}} \varphi_{i} q_{i}, \eqno(22.3.2.6)]$ where $[\varphi_{\rm i}]$ is the potential at an atom with charge $[{q}_{i}]$ . The most common source of nonlinearity is the Boltzmann term in the PB equation (22.3.2.4 ) for highly charged molecules such as nucleic acids. The total electrostatic energy in this case is (Reiner & Radke, 1990; Sharp & Honig, 1990; Zhou, 1994) $[{\Delta G^{\rm el} = {\textstyle\int\limits_{V}} \{\rho^{e}\varphi - (\varepsilon E^{2}/8\pi) - kT {\textstyle\sum\limits_{i}} c_{i}^{0} [\exp (- z_{i}e \varphi/kT) - 1] \}\hbox{ d}{\bf r}}, \eqno(22.3.2.7)]$ where the integration is now over all space.

The general expression for the electrostatic force on a charge q is given by the gradient of the total free energy with respect to that charge's position, $[{\bf f}_{q} = - \nabla{_{{\bf r}{q}}} (G^{\rm el}). \eqno(22.3.2.8)]$ If the movement of that charge does not affect the potential distribution due to the other charges and dipoles, then equation (22.3.2.8) can be evaluated using the `test charge' approach, in which case the force depends only on the gradient of the potential or the field at the charge: $[{\bf f} = q{\bf E}. \eqno(22.3.2.9)]$ However, in a system like a macromolecule in water, which has a non-homogeneous dielectric, forces arise between a charge and any dielectric boundary due to image charge (reaction potential) effects. A similar effect to the `dielectric pressure' force arises from solvent-ion pressure at the solute–solvent boundary. This results in a force acting to increase the solvent exposure of charged and polar atoms. An expression for the force that includes these effects has been derived within the PB model (Gilson et al., 1993): $[{{\bf f} = \rho^{e} {\bf E} - (1/2) E^{2} \nabla \varepsilon - kT {\textstyle\sum\limits_{i}} c_{i}^{0} [\exp (-z_{i}e\varphi/kT) - 1] \nabla A,} \eqno(22.3.2.10)]$ where A is a function describing the accessibility to solvent ions, which is 0 inside the protein, and 1 in the solvent, and whose gradient is nonzero only at the solute–solvent surface. Similarly, in a two-dielectric model (solvent plus molecule) the gradient of ɛ is nonzero only at the molecular surface. The first term accounts for the force acting on a charge due to a field, as in equation (22.3.2.9), while the second and third terms account for the dielectric surface pressure and ionic atmosphere pressure terms respectively. Equation (22.3.2.10) has been used to combine the PB equation and molecular mechanics (Gilson et al., 1995).

References

Gilson, M., Davis, M., Luty, B. & McCammon, J. (1993). Computation of electrostatic forces on solvated molecules using the Poisson–Boltzmann equation. J. Phys. Chem. 97, 3591–3600.Google Scholar

Gilson, M., McCammon, J. & Madura, J. (1995). Molecular dynamics simulation with continuum solvent. J. Comput. Chem. 16, 1081–1095.Google Scholar

Reiner, E. S. & Radke, C. J. (1990). Variational approach to the electrostatic free energy in charged colloidal suspensions. J. Chem. Soc. Faraday Trans. 86, 3901.Google Scholar

Sharp, K. & Honig, B. (1990). Calculating total electrostatic energies with the non-linear Poisson–Boltzmann equation. J. Phys. Chem. 94, 7684–7692.Google Scholar

Zhou, H. X. (1994). Macromolecular electrostatic energy within the nonlinear Poisson–Boltzmann equation. J. Phys. Chem. 100, 3152–3162.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 22.3, p. 555