Tanford-Kirkwood model

The charge of a number of proteins has been measured by titration. The early experimental work focused on the determination of charge as a function of pH later work focused on comparing the experimental and theoretical results the latter obtained from the extensions of the Tanford-Kirkwood models on the electrostatic behavior of proteins. Ed-sall and Wyman [104] discuss the early work on the electrostatics of polar molecules and ions in solution, considering fundamental coulombic interactions and accounting for the dielectric properties of the media. Tanford [383,384], and Tanford and Kirkwood [387] describe the development of the Tanford-Kirkwood theories of protein electrostatics. For more recent work on protein electrostatics see Lenhoff and coworkers [64,146,334]. 

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

The first computations of ionization constants of residues in proteins for structures derived from molecular dynamics trajectories were described by Wendoloski and Matthew for tuna cytochrome c. In that study, conformers were generated using molecular dynamics simulations with a range of solvents, simulating macroscopic dielectric formalisms, and one solvent model that explicitly included solvent water molecules. The authors calculated individual pR values, overall titration curves, and electrostatic potential surfaces for average structures and structures along each simulation trajectory. However, the computational scheme for predicting electrostatic interactions in proteins used by Wendoloski and Matthew was not based on a FDPB model but on the modified Tanford-Kirkwood approach, which is not discussed in this chapter. 

J. J. Flavranek and P. B. Harbury, Proc. Natl. Acad. Sci. (USA), 96,11145 (1999). Tanford-Kirkwood Electrostatics for Protein Modeling. 

Note that the separability of the two contributions of the polar effect is attained with difficulty in fact, attempts to separate the polar effect contributions have been unsuccessful, so they are usually considered together. However, from a theoretical point of view, field effects have been studied using the Kirkwood-Westheimer [Kirkwood and Westheimer, 1938 Westheimer and Kirkwood, 1938] and Tanford models [Tanford, 1957]. 

Tanford, C. (1957). The Location of Electrostatic Charges in Kirkwood s Model of Organic Ions. J.Am.ChemSoc., 79,5348-5352. 

If all of the atoms and charges in the system of interest are explicitly represented and atomic polarization is included, the use of a dielectric constant other than unity would be inappropriate. A variety of models has been used, however, to approximate the dielectric behavior of a macromolecular system where the solvent was not explicitly included. Dielectric constants for the protein interior between 2 and 10 have been employed, as has a distance-dependent dielectric response equal to the distance in angstroms.78 Also, simple forms of the Kirkwood-Westheimer-Tanford model79 have been used to approximate the effect of the aqueous solvent. An approach that may improve our understanding in this area employs linear response theory to evaluate the spatially dependent dielectric response.80 In any such model it is necessary to consider the frequency dependence of the dielectric constant relative to the time scale of the dynamic process under consideration. 

In the more realistic discrete-charge electrostatic theory (Tanford and Kirkwood ), the amino acid groups are point charges positioned at fixed sites on the surface of the protein or are buried at a short distance within the interior of the molecule which is assumed to be a continuous medium of low dielectric constant. The theory was successfully tested on a variety of model compounds. However, this calculation was also limited to the mutual effect of two groups only, such as the iron atom and the amino acid in a hemoglobin molecule. 

As noted by Warshel and coworkers,continuum models for the calculation of electrostatic effects in proteins and, in particular pK s of ionizable groups, have undergone significant modifications since the classical work of Tanford and Kirkwood and at present should rather be named discretized continuum (DC) models. Another possibility is to name them mesoscopic models because many elements of the microscopic structure of proteins enter the continuum models. First, the atomic structure is used to define the dielectric boundary between the protein and the solvent, with fixed charges at positions of the protein atoms. Second, since hydrogen atoms are not resolved in X-ray structures, they are added later by the modeler and their positions minimized by... 

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