International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 22.4, pp. 730-735   | 1 | 2 |
https://doi.org/10.1107/97809553602060000888

Chapter 22.4. Electrostatic interactions in proteins

K. A. Sharpa*

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

Electrostatic interactions play a key role in determining the structure, stability, binding affinity and chemical properties, and hence the biological reactivity, of proteins and nucleic acids. The goal of electrostatic calculations is to take the structural information provided by crystallography or NMR and obtain a realistic description of the electrostatic potential distribution ϕ(r), energies and forces. Three problems must be solved to obtain the electrostatic potential distribution: modelling the macromolecular charge distribution; modelling the response of the macromolecule, water and solvent ions; and rapidly and accurately solving the electrostatic equations that determine the potential. The response arises from electronic polarization, reorientation of permanent dipolar groups and redistribution of mobile ions in the solvent. The most common theoretical approaches to modelling this response are briefly described, along with the methods used to obtain potential distributions, energies and forces. A summary of the theory behind application to three broad areas is presented: electrostatic potential distributions, charge (proton and electron) transfer equilibria, and electrostatic contributions to binding energies.

22.4.1. Introduction

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Electrostatic interactions play a key role in determining the structure, stability, binding affinity, chemical properties, and hence the biological reactivity, of proteins and nucleic acids. Interactions where electrostatics play an important role include:

  • (1) Ligand/substrate association. Long-range electrostatic forces can considerably enhance association rates by facilitating translational and rotational diffusion or by reduction in the dimensionality of the diffusion space.

  • (2) Binding affinity. Tight specific binding is often a prerequisite for biological activity, and electrostatics make important contributions to desolvation and formation of chemically complementary interactions during binding.

  • (3) Modification of chemical and physical properties of functional groups such as cofactors (haems, metal ions etc.), alteration of the ionization energy pKa of side chains and shifting of redox midpoints.

  • (4) The creation of potentials or fields in the active sites to stabilize functionally important charged or dipolar intermediates in processes such as catalysis.

In this chapter I will discuss, within the framework of classical electrostatics, how such effects can be modelled starting from the structural information provided by X-ray crystallography. Nevertheless, many of the concepts of classical electrostatics can be used in combination with molecular dynamics (MD), quantum mechanics (QM) and other computational methods to study a wider range of macromolecular properties, for example specific protein motions, the breaking or forming of bonds, determination of intrinsic pKa's, determination of electronic energy levels etc.

The central aim in studying the electrostatic properties of macromolecules is to take the structural information provided by crystallography (typically the atomic coordinates, although B-factor information may also be of use) and obtain a realistic description of the electrostatic potential distribution [\varphi({\bf r})]. The electrostatic potential distribution can then be used in a variety of ways: (i) graphical analysis may reveal deeper aspects of the structure and help identify functionally important regions or active sites; (ii) the potentials may be used to calculate energies and forces, which can then be used to calculate equilibrium or kinetic properties; and (iii) the electrostatic potentials may be used in conjunction with other computational methods such as QM and MD.

Three problems must be solved to obtain the electrostatic potential distribution. The first is to model the macromolecular charge distribution, usually by specifying the location and charge of all its atoms. Although the coordinates of the molecule are determined by crystallographic methods, the charge distribution is not. A number of atomic charge distributions have been developed for proteins and nucleic acids using quantum mechanical methods and/or parameterization to different experimental data. The second problem is that the positions of the water molecules and solvent ions are generally not known. (Water molecules and ions seen in even the best crystal structures usually constitute a small fraction of the total important in solvating the molecule. Moreover, the orientation of the crystallographic water molecules, crucial in determining the electrostatic potential, is rarely known.) Within the framework of classical electrostatics, inclusion of the effect of the solvating water molecules and ions is handled not by treating them explicitly, but implicitly in terms of an `electrostatic response' to the field created by the molecular charge distribution. The third problem is that incorporation of the available structural information at atomic resolution results in a complicated spatial distribution of charge, dielectric response etc. Numerical methods for rapidly and accurately solving the electrostatic equations that determine the potential are therefore essential.

22.4.2. Theory

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22.4.2.1. 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[link]) 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(22.4.2.1)]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 (22.4.2.1[link]) with a larger value of the dielectric constant that incorporates the additional polarization from dipole reorientation. An alternative approach to equation (22.4.2.1[link]) for treating the dipole reorientation contribution of water surrounding the macromolecules is the Langevin dipole model (Lee et al., 1993[link]; Warshel & Åqvist, 1991[link]; Warshel & Russell, 1984[link]). 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 ([epsilon] = 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[link]; Nakamura et al., 1988[link]) and the interior of proteins in solution (Simonson & Brooks, 1996[link]; Simonson & Perahia, 1995[link]; Smith et al., 1993[link]), and experimental measurements (Takashima & Schwan, 1965[link]) provide an estimate of [epsilon] = 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[link]), although there is a compensating effect of electrostriction, which increases the local dipole density (Jayaram, Fine et al., 1989[link]). 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(22.4.2.2)]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[link]; Murthy et al., 1985[link]; Olmsted et al., 1989[link], 1991[link]; Record et al., 1990[link]). 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.

22.4.2.2. Dependence of the potential on the charge distribution

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The potential at a point in space, r, arising from some charge density distribution [\rho ({\bf s})] and some dipole density distribution P(s) (which includes polarization) is given by [\varphi ({\bf r}) = {\textstyle\int} \rho ({\bf s})/(|{\bf s - r}|) + {\bf P}({\bf s})({\bf s - r})/(|{\bf s - r}|^{3}) \hbox{ d}{\bf s}. \eqno(22.4.2.3)]The total charge distribution is the sum of the explicit charge distribution on the molecule and that from the mobile solvent ion distribution, [\rho = \rho^{e} + \rho^{m}]. Substituting for the dielectric polarization using equation (22.4.2.1[link]) and for the mobile ion charge distribution using equation (22.4.2.2[link]), the potential may be expressed in terms of a partial differential equation, the Poisson–Boltzmann (PB) equation: [{\nabla \varepsilon ({\bf r}) \nabla \varphi ({\bf r}) + 4\pi {\textstyle\sum\limits_{i}} z_{i} ec_{i}^{o} \exp [-z_{i} e \varphi ({\bf r}) / kT]+ 4 \pi \rho^{e} ({\bf r}) = 0,\hfill} \eqno(22.4.2.4)]which relates the potential, molecular charge and dielectric distributions, [\varphi ({\bf r})], [\rho^{e}({\bf r})] and [\varepsilon ({\bf r})], respectively. Contributions to the polarizability from electrons, a molecule's permanent dipoles and solvent dipoles are incorporated into this model by using an appropriate value for the dielectric for each region of protein and solvent. Values for protein atomic charges, radii and dielectric constants suitable for use with the Poisson–Boltzmann equation are available in the literature (Jean-Charles et al., 1990[link]; Mohan et al., 1992[link]; Simonson & Brünger, 1994[link]; Sitkoff et al., 1994[link]). For protein applications, the Boltzmann term in equation (22.4.2.4[link]) is usually linearized to become [-8\pi \varphi ({\bf r})I/kT] where I is the ionic strength, whereas for nucleic acids and molecules of similarly high charge density the full nonlinear equation is used.

22.4.2.3. The concepts of screening, reaction potentials, solvation, dielectric, polarity and polarizability

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Application of a classical electrostatic view to macromolecular electrostatics involves a number of useful concepts that describe the physical behaviour. It should first be recognized that the potential at a particular charged atom i includes three physically distinct contributions. The first is the direct or Coulombic potential of j at i. The second is the potential at i generated by the polarization (of a molecule, water and ion atmosphere) induced by j. This is often referred to as the screening potential, since it opposes the direct Coulombic potential. The third arises from the polarization induced by i itself. This is often referred to as the reaction or self-potential, or if solvent is involved, as the solvation potential.

When using models that apply the concept of a dielectric constant (a measure of polarizability) to a macromolecule, it is important to distinguish between polarity and polarizability. Briefly, polarity may be thought of as describing the density of charged and dipolar groups in a particular region. Polarizability, by contrast, refers to the potential for reorganizing charges, orienting dipoles and inducing dipoles. Thus polarizability depends both on the polarity and the freedom of dipoles to reorganize in response to an applied electric field. When a protein is folding or undergoing a large conformational rearrangement, the peptide groups may be quite free to reorient. In the folded protein, these may become spatially organized so as to stabilize another charge or dipole, creating a region with high polarity, but with low polarizability, since there is much less ability to reorient the dipolar groups in response to a new charge or dipole without significant disruption of the structure. Thus, while there is still some discussion about the value and applicability of a protein dielectric constant, it is generally agreed that the interior of a macromolecule is a less polarizable environment compared to solvent. This difference in polarizability has a significant effect on the potential distribution.

Formally charged groups on proteins, particularly the longer side chains on the surface of proteins, Arg, Lys, and to a lesser extent Glu and Asp, have the ability to alter their conformation in response to electrostatic fields. In addition, information about fluctuations about their mean position may need to be included in calculating average properties. Three approaches to modelling protein formal charge movements can be taken. The first is to treat the motions within the dielectric response. In this approach, the protein may be viewed as having a dielectric higher than 2.5–4 in the regions of these charged groups, particularly at the surface, where the concentration and mobility of these groups may give an effective dielectric of 20 or more (Antosiewicz et al., 1994[link]; Simonson & Perahia, 1995[link]; Smith et al., 1993[link]). A second approach is to model the effect of charge motions on the electrostatic quantity of interest explicitly, e.g. with MD simulations (Langsetmo et al., 1991[link]; Wendoloski & Matthew, 1989[link]). This involves generating an ensemble of structures with different atomic charge distributions. The third approach is based on the fact that one is often interested in a specific biological process [A \rightarrow B] in which one can evaluate the structure of the protein in states A and B (experimentally or by modelling), and any change in average charge positions is incorporated at the level of different average explicit charge distribution inputs for the calculation, modelling only the electronic, dipolar and salt contributions as the response.

The term `effective' dielectric constant is sometimes used in the literature to describe the strength of interaction between two charges, [q_{1}] and [q_{2}]. This is defined as the ratio of the observed or calculated interaction strength, U, to that expected between the same two charges in a vacuum: [\varepsilon^{\rm eff} = [(q_{1} q_{2})/r_{12}]/U, \eqno(22.4.2.5)]where [r_{12}] is the distance between the charges. If the system were completely homogeneous in terms of its electrostatic response and involved no charge rearrangement then [\varepsilon^{\rm eff}] would describe the dielectric constant of the medium containing the charges. This is generally never the case: the strength of interaction in a protein system is determined by the net contribution from protein, solvent and ions, so [\varepsilon^{\rm eff}] does not give information about the dielectric property of any particular region of space. In fact, in the same system different charge–charge interactions will generally yield different values of [\varepsilon^{\rm eff}]. Thus [\varepsilon^{\rm eff}] is really no more than its definition – a measure of the strength of interaction – and it cannot be used directly to answer questions about the protein dielectric constant, for example. Rather, it is one of the quantities that one aims to extract from theoretical models to compare with an experiment.

22.4.2.4. 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(22.4.2.6)]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 (22.4.2.4[link] ) for highly charged molecules such as nucleic acids. The total electrostatic energy in this case is (Reiner & Radke, 1990[link]; Sharp & Honig, 1990[link]; Zhou, 1994[link]) [{\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(22.4.2.7)]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(22.4.2.8)]If the movement of that charge does not affect the potential distribution due to the other charges and dipoles, then equation (22.4.2.8[link]) 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(22.4.2.9)]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[link]): [{{\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(22.4.2.10)]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 [epsilon] 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 (22.4.2.9[link]), while the second and third terms account for the dielectric surface pressure and ionic atmosphere pressure terms respectively. Equation (22.4.2.10[link]) has been used to combine the PB equation and molecular mechanics (Gilson et al., 1995[link]).

22.4.2.5. Numerical methods

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A variety of numerical methods exist for calculating electrostatic potentials of macromolecules. These include numerical solution of self-consistent field electrostatic equations, which has been used in conjunction with the protein dipole–Langevin dipole method (Lee et al., 1993[link]). Numerical solution of the Poisson–Boltzmann equation requires the solution of a three-dimensional partial differential equation, which can be nonlinear. Many numerical techniques, some developed in engineering fields to solve differential equations, have been applied to the PB equation. These include finite-difference methods (Bruccoleri et al., 1996[link]; Gilson et al., 1988[link]; Nicholls & Honig, 1991[link]; Warwicker & Watson, 1982[link]), finite-element methods (Rashin, 1990[link]; Yoon & Lenhoff, 1992[link]; Zauhar & Morgan, 1985[link]), multigridding (Holst & Saied, 1993[link]; Oberoi & Allewell, 1993[link]), conjugate-gradient methods (Davis & McCammon, 1989[link]) and fast multipole methods (Bharadwaj et al., 1994[link]; Davis, 1994[link]). Methods for treating the nonlinear PB equation include under-relaxation (Jayaram, Sharp & Honig, 1989[link]) and powerful inexact Newton methods (Holst et al., 1994[link]). The nonlinear PB equation can also be solved via a self-consistent field approach, in which one calculates the potential using equation (22.4.2.5[link]), then the mobile charge density is calculated using equation (22.4.2.3[link]), and the procedure is repeated until convergence is reached (Pack & Klein, 1984[link]; Pack et al., 1986[link]). The method allows one to include more elaborate models for the ion distribution, for example incorporating the finite size of the ions (Pack et al., 1993[link]). Approximate methods based on spherical approximations (Born-type models) have also been used (Schaeffer & Frommel, 1990[link]; Still et al., 1990[link]). Considerable numerical progress has been made in finite methods, and accurate rapid algorithms are available. The reader is referred to the original references for numerical details.

22.4.3. Applications

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An exhaustive list of applications of classical electrostatic modelling to macromolecules is beyond the scope of this chapter. Three general areas of application are discussed.

22.4.3.1. Electrostatic potential distributions

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Graphical analysis of electrostatic potential distributions often reveals features about the structure that complement analysis of the atomic coordinates. For example, Fig. 22.4.3.1(a)[link] shows the distribution of charged residues in the binding site of the proteolytic enzyme thrombin. Fig. 22.4.3.1(b)[link] shows the resulting electrostatic potential distribution on the protein surface. The basic (positive) region in the fibrinogen binding site, which could be inferred from close inspection of the distribution of charged residues in Fig. 22.4.3.1(a)[link], is clearly more apparent in the potential distribution. Fig. 22.4.3.1(c)[link] shows the effect of increasing ionic strength on the potential distribution, shrinking the regions of strong potential. Fig. 22.4.3.1(d)[link] is calculated assuming the same dielectric for the solvent and protein. The more uniform potential distribution compared to Fig. 22.4.3.1(b)[link] shows the focusing effect that the low dielectric interior has on the field emanating from charges in active sites and other cleft regions.

[Figure 22.4.3.1]

Figure 22.4.3.1 | top | pdf |

(a) The proteolytic enzyme thrombin (yellow backbone worm) complexed with an inhibitor, hirudin (blue backbone worm). The negatively charged (red) and positively charged (blue) side chains of thrombin are shown in bond representation. (b) Solvent-accessible surface of thrombin coded by electrostatic potential (blue: positive, red: negative). Hirudin is shown as a blue backbone worm. Potential is calculated at zero ionic strength. (c) Solvent-accessible surface of thrombin coded by electrostatic potential (blue: positive, red: negative). Hirudin is shown as a blue backbone worm. Potential is calculated at physiological ionic strength (0.145 M). (d) Solvent-accessible surface of thrombin coded by electrostatic potential (blue: positive, red: negative). Hirudin is shown as a blue backbone worm. Potential is calculated using the same polarizability for protein and solvent.

22.4.3.2. Charge-transfer equilibria

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Charge-transfer processes are important in protein catalysis, binding, conformational changes and many other functions. The primary examples are acid–base equilibria, electron transfer and ion binding, in which the transferred species is a proton, an electron or a salt ion, respectively. The theory of the dependence of these three equilibria within the classical electrostatic framework can be treated in an identical manner, and will be illustrated with acid–base equilibria. A titratable group will have an intrinsic ionization equilibrium, expressed in terms of a known intrinsic [\hbox{p}K^{0}_{a}], where [\hbox{p}K^{0}_{a} = -\log_{10}(K^{0}_{a})], [K^{0}_{a}] is the dissociation constant for the reaction [\hbox{H}^{+}A = \hbox{H}^{+}+A] and A can be an acid or a base. The [\hbox{p}K^{0}_{a}] is determined by all the quantum-chemical, electrostatic and environmental effects operating on that group in some reference state. For example, a reference state for the aspartic acid side-chain ionization might be the isolated amino acid in water, for which [\hbox{p}K^{0}_{a} = 3.85]. In the environment of the protein, the [\hbox{p}K_{a}] will be altered by three electrostatic effects. The first occurs because the group is positioned in a protein environment with a different polarizability, the second is due to interaction with permanent dipoles in the protein, the third is due to charged, perhaps titratable, groups. The effective [\hbox{p}K_{a}] is given by [{\hbox{p}K_{a} = \hbox{p}K^{0}_{a} + (\Delta \Delta G^{\rm rf} + \Delta \Delta G^{\rm perm} + \Delta \Delta G^{{\rm tit}})/2.303kT,} \eqno(22.4.3.1)]where the factor of 1/2.303kT converts units of energy to units of [\hbox{p}K_{a}]. The first contribution, [\Delta \Delta G^{\rm rf}], arises because the completely solvated group induces a strong favourable reaction field (see Section 22.4.2.3[link]) in the high dielectric water, which stabilizes the charged form of the group. (The neutral form is also stabilized by the solvent reaction field induced by any dipolar groups, but to a lesser extent.) Desolvating the group to any degree by moving it into a less polarizable environment will preferentially destabilize the charged form of that group, shifting the [\hbox{p}K_{a}] by an amount [\Delta \Delta G^{\rm rf} = (1/2) {\textstyle\sum\limits_{i}} \left(q_{i}^{d}\Delta \varphi_{i}^{{\rm rf}, \, d} - q\>_{i}^{p}\Delta \varphi_{i}^{{\rm rf}, \, p}\right), \eqno(22.4.3.2)]where [q\>_{i}^{p}] and [q_{i}^{d}] are the charge distributions on the group, [\Delta \varphi_{i}^{{\rm rf}, \, p}] and [\Delta \varphi_{i}^{{\rm rf}, \, d}] are the changes in the group's reaction potential upon moving it from its reference state into the protein, in the protonated (superscript p) and deprotonated (superscript d) forms, respectively, and the sum is over the group's charges. The contribution of the permanent dipoles is given by [\Delta \Delta G^{\rm tit} = {\textstyle\sum\limits_{i}} \left(q_{i}^{d} - q\>_{i}^{p}\right)\varphi_{i}^{{\rm perm}}, \eqno(22.4.3.3)]where [\varphi_{i}^{\rm perm}] is the interaction potential at the ith charge due to all the permanent dipoles in the protein, including the effect of screening. It is observed that intrinsic [\hbox{p}K_{a}]'s of groups in proteins are rarely shifted by more than [1\ \hbox{p}K_{a}] unit, indicating that the effects of desolvation are often compensated to a large degree by the [\Delta \Delta G^{\rm perm}] term (Antosiewicz et al., 1994[link]). The final term accounts for the contribution of all the other charged groups: [\Delta \Delta G^{\rm tit} = {\textstyle\sum\limits_{i}} \left(q_{i}^{d} \langle \varphi_{i} \rangle_{{\rm pH}, \, c, \, \Delta V}^{d} - q\>_{i}^{p} \langle\varphi_{i} \rangle_{{\rm pH}, \, c, \, \Delta V}^{p}\right),\eqno(22.4.3.4)]where [\langle\varphi_{i} \rangle] is the mean potential at group charge i from all the other titratable groups. The charge states of the other groups in the protein depend in turn on their intrinsic `[\hbox{p}K_{a}]'s', on the external pH if they are acid–base groups, the external redox potential, [\Delta V], if they are redox groups and the concentration of ions, c, if they are ion-binding sites, as indicated by the subscript to [\langle\varphi_{i} \rangle]. Moreover, the charge state of the group itself will affect the equilibrium at the other sites. Because of this linkage, exact determination of the complete charged state of a protein is a complex procedure. If there are N such groups, the rigorous approach is to compute the titration-state partition function by evaluating the relative electrostatic free energies of all [2^{N}] ionization states for a given set of pH, c, [\Delta V]. From this one may calculate the mean ionization state of any group as a function of pH, [\Delta V] etc. For large N this becomes impractical, but various approximate schemes work well, including a Monte Carlo procedure (Beroza et al., 1991[link]; Yang et al., 1993[link]) or partial evaluation of the titration partition function by clustering the groups into strongly interacting sub-domains (Bashford & Karplus, 1990[link]; Gilson, 1993[link]; Yang et al., 1993[link]).

Calculation of ion-binding and electron-transfer equilibria in proteins proceeds exactly as for calculation of acid–base equilibria, the results usually being expressed in terms of an association constant, [K_{a}], or a redox midpoint potential [E_{m}] (defined as the external reducing potential at which the group is half oxidized and half reduced, usually at pH 7), respectively.

22.4.3.3. Electrostatic contributions to binding energy

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The electrostatic contribution to the binding energy of two molecules is obtained by taking the difference in total electrostatic energies in the bound (AB) and unbound [A + B] states. For the linear case, [{\Delta \Delta G_{\rm bind}^{\rm elec} = (1/2) {\textstyle\sum\limits_{i}^{N_{A}}} q_{i}^{A} (\varphi_{i}^{AB} - \varphi_{i}^{A}) + (1/2) {\textstyle\sum\limits_{j}^{N_{B}}} q_{i}^{B} (\varphi_{j}^{AB} - \varphi_{j}^{B}),} \eqno(22.4.3.5)]where the first and second sums are over all charges in molecule A and B, respectively, and [\varphi^{x}] is the total potential produced by x = A, B, or AB. From equation (22.4.3.5[link]), it should be noted that the electrostatic free energy change of each molecule has contributions from intermolecular charge–charge interactions, and from changes in the solvent reaction potential of the molecule itself when solvent is displaced by the other molecule. Equation (22.4.3.5[link]) allows for the possibility that the conformation may change upon binding, since different charge distributions may be used for the complexed and uncomplexed forms of A, and similarly for B. However, other energetic terms, including those involved in any conformational change, have to be added to equation (22.4.3.5[link]) to obtain net binding free energy changes. Nevertheless, changes in binding free energy due to charge modifications or changes in external factors such as pH and salt concentration may be estimated using equation (22.4.3.5[link]) alone. For the latter, salt effects are usually only significant in highly charged molecules, for which the nonlinear form for the total electrostatic energy, equation (22.4.2.4[link]), must be used. The salt dependence of binding of drugs and proteins to DNA has been studied using this approach (Misra, Hecht et al., 1994[link]; Misra, Sharp et al., 1994[link]; Sharp et al., 1995[link]), including the pH dependence of drug binding (Misra & Honig, 1995[link]). Other applications include the binding of sulfate to the sulfate binding protein (Åqvist et al., 1991[link]) and antibody and antigen interactions (Lee et al., 1992[link]; Slagle et al., 1994[link]).

References

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