Charge-transfer equilibria

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. 555-556 | 1 | 2 |

Section 22.3.3.2. Charge-transfer equilibria

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

22.3.3.2. 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 $[{\hbox{p}K_{a} = \hbox{p}K^{0}_{a} + (\Delta \Delta G^{\rm rf} + \Delta \Delta G^{\rm perm} + \Delta \Delta G^{{\rm tit}})/2.303kT,} \eqno(22.3.3.1)]$ 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 22.3.2.3) 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 $[\Delta \Delta G^{\rm rf} = (1/2) {\textstyle\sum\limits_{i}} \left(q_{i}^{d}\Delta \varphi_{i}^{{\rm rf}, \, d} - q\>_{i}^{p}\Delta \varphi_{i}^{{\rm rf}, \, p}\right), \eqno(22.3.3.2)]$ 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 $[\Delta \Delta G^{\rm tit} = {\textstyle\sum\limits_{i}} \left(q_{i}^{d} - q\>_{i}^{p}\right)\varphi_{i}^{{\rm perm}}, \eqno(22.3.3.3)]$ 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). The final term accounts for the contribution of all the other charged groups: $[\Delta \Delta G^{\rm tit} = {\textstyle\sum\limits_{i}} \left(q_{i}^{d} \langle \varphi_{i} \rangle_{{\rm pH}, \, c, \, \Delta V}^{d} - q\>_{i}^{p} \langle\varphi_{i} \rangle_{{\rm pH}, \, c, \, \Delta V}^{p}\right),\eqno(22.3.3.4)]$ 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; Yang et al., 1993) or partial evaluation of the titration partition function by clustering the groups into strongly interacting sub-domains (Bashford & Karplus, 1990; Gilson, 1993; Yang et al., 1993).

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.

References

Antosiewicz, J., McCammon, J. A. & Gilson, M. K. (1994). Prediction of pH-dependent properties of proteins. J. Mol. Biol. 238, 415–436.Google Scholar

Bashford, D. & Karplus, M. (1990). pK_a's of ionizable groups in proteins: atomic detail from a continuum electrostatic model. Biochemistry, 29, 10219–10225.Google Scholar

Beroza, P., Fredkin, D., Okamura, M. & Feher, G. (1991). Protonation of interacting residues in a protein by a Monte-Carlo method. Proc. Natl Acad. Sci. USA, 88, 5804–5808.Google Scholar

Gilson, M. (1993). Multiple-site titration and molecular modeling: 2. Rapid methods for computing energies and forces for ionizable groups in proteins. Proteins Struct. Funct. Genet. 15, 266–282.Google Scholar

Yang, A., Gunner, M., Sampogna, R., Sharp, K. & Honig, B. (1993). On the calculation of pK_a in proteins. Proteins Struct. Funct. Genet. 15, 252–265.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 22.3, pp. 555-556