The response of the system to electrostatic fields

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, pp. 553-554 | 1 | 2 |

Section 22.3.2.1. The response of the system to electrostatic fields

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

22.3.2.1. The response of the system to electrostatic fields

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The response to the electrostatic field arising from the molecular charge distribution arises from three physical processes: electronic polarization, reorientation of permanent dipolar groups and redistribution of mobile ions in the solvent. Movement of ionized side chains, if significant, is sometimes viewed as part of the dielectric response of the protein, and sometimes explicitly as a conformational change of the molecule.

Electronic polarizability can be represented either by point inducible dipoles (Warshel & Åqvist, 1991) or by a dielectric constant . The latter approach relates the electrostatic polarization , P(r) (the mean dipole moment induced in some small volume V) to the Maxwell (total) field, E(r), and the local dielectric constant representing the response of that volume, $[\varepsilon({\bf r})]$ , according to $[{\bf P(r)} = [\varepsilon ({\bf r}) - 1] {\bf E(r)} / 4\pi. \eqno(22.3.2.1)]$ The contribution of electronic polarizability to the dielectric constant of most organic material and water is fairly similar. It can be evaluated by high-frequency dielectric measurements or the refractive index, and it is in the range 2–2.5.

The reorientation of groups such as the peptide bond or surrounding water molecules which have large permanent dipoles is an important part of the overall response. This response too may be treated using a dielectric constant, i.e. using equation (22.3.2.1) with a larger value of the dielectric constant that incorporates the additional polarization from dipole reorientation. An alternative approach to equation (22.3.2.1) for treating the dipole reorientation contribution of water surrounding the macromolecules is the Langevin dipole model (Lee et al., 1993; Warshel & Åqvist, 1991; Warshel & Russell, 1984). Four factors determine the degree of response from permanent dipoles: (i) the dipole-moment magnitude; (ii) the density of such groups in the protein or solvent; (iii) the freedom of such groups to reorient; and (iv) the degree of cooperativity between dipole motions. Thus, water has a high dielectric constant (ɛ = 78.6 at 25 °C). For electrostatic models based on dielectric theory, the experimental solvent dielectric constant, reflecting the contribution of electronic polarizability and dipole reorientation, is usually used. From consideration of the four factors that determine the dielectric response, macromolecules would be expected to have a much lower dielectric constant than the solvent. Indeed, theoretical studies of the dielectric behaviour of amorphous protein solids (Gilson & Honig, 1986; Nakamura et al., 1988) and the interior of proteins in solution (Simonson & Brooks, 1996; Simonson & Perahia, 1995; Smith et al., 1993), and experimental measurements (Takashima & Schwan, 1965) provide an estimate of ɛ = 2.5–4 for the contribution of dipolar groups to the protein dielectric.

The Langevin model can account for the saturation of the response at high fields that occurs if the dipoles become highly aligned with the field. The dielectric model can also be extended to incorporate saturation effects (Warwicker, 1994), although there is a compensating effect of electrostriction, which increases the local dipole density (Jayaram, Fine et al., 1989). While the importance of saturation effects would vary from case to case, linear solvent dielectric models have proven sufficiently accurate for most protein applications to date.

Charge groups on molecules will attract solvent counter-ions and repel co-ions. The most common way of treating this charge rearrangement is via the Boltzmann model , where the net charge density of mobile ions is given by $[\rho^{m} ({\bf r}) = {\textstyle\sum\limits_{i}}z_{i}ec_{i}^{o}\exp[-z_{i} e \varphi ({\bf r})/kT], \eqno(22.3.2.2)]$ where $[c_{i}^{o}]$ is the bulk concentration of an ion of type i, valence $[z_{i}]$ , and $[\varphi ({\bf r})]$ is the average potential (an approximation to the potential of mean force) at position r. The Boltzmann approach neglects the effect of ion size and correlations between ion positions. Other models for the mobile-ion behaviour that account for these effects are integral equation models and MC models (Bacquet & Rossky, 1984; Murthy et al., 1985; Olmsted et al., 1989, 1991; Record et al., 1990). These studies show that ion size and correlation effects do not compromise the Boltzmann model significantly for monovalent (1–1) salts at mid-range concentrations 0.001–0.5 M, and consequently it is widely used for modelling salt effects in proteins and nucleic acids.

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International Tables for Crystallography (2006). Vol. F. ch. 22.3, pp. 553-554