The recent application of numerical Poisson−Boltzmann methods to the determination of the electrostatic potential and counterion distributions around polyelectrolytes such as DNA has prompted the determination of accurate solvent dielectric constants. The previous assumption for proteins of using a constant value of about 80 for the solvating environment appears inadequate when dealing with the much higher potential gradients and local ion concentrations of charged polyelectrolytes in solution. Approximations using lower dielectric values near 30 at the surface and increasing away to a bulk value of 78.5 have been incorporated into modern finite-difference Poisson−Boltzmann techniques and have led to more reasonable estimates of counterion distributions. However, choosing this dielectric constant “field” has been more a matter of guessing than scientific analysis. To put these techniques on a firmer foundation, we present a simple calculational procedure for determining dielectric constants near the surface of macromolecules and lipid membranes. To demonstrate the soundness of the calculation, we compare calculated dielectric constants near the surface of DNA with data from recent experiments measuring this quantity in both the major and minor grooves of B-form DNA. This procedure can be useful for proteins as well and may be particularly applicable in solvent pockets where strong potential gradients lead to the electrostatic guiding of ions or ligands toward specific binding sites.

The standard gas phase normal mode analysis is generalized to condensed phases within the framework of the classical Langevin equation. The solution of this equation for a system of 3N atoms moving on a harmonic potential surface and subject to viscous damping described by a friction matrix (which is nondiagonal in the presence of hydrodynamic interactions) is reduced to the diagonalization of a 6N X 6N real nonsymmetric matrix obtained from the massweighted force constant and friction matrices. Alternative formulations of the problem requiring either the diagonalization of a complex symmetric matrix or the solution to a generalized eigenvalue problem involving real symmetric matrices are also considered. The resulting eigenvalues, in general complex, are real when the motion is overdamped and occur in complex conjugate pairs in the underdamped case. It is shown that when the eigenvectors are normalized in a particular way, all quantities of interest (e.g., correlation functions) have simple spectral representations and can be expressed as summations over the eigenvalues and eigenvectors ("Langevin modes") of the system. To treat situations in which the exact matrix solution is computationally prohibitive, a perturbation approach is developed which utilizes the gas phase normal mode results for the system. In zeroth-order each normal coordinate is a Langevin oscillator with a friction constant equal to a diagonal element of the normal modetransformed friction matrix. This description is analytically tractable, can be improved perturbatively, and is exact in the limit that all atoms experience the same friction. The formalism developed in this paper provides a viable way of studying the influence of solvent on the dynamics of collective motions in macromolecules. 7334

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Reaction-diffusion equations, in which the reaction is described by a sink term consisting of a sum of delta functions, are studied. It is shown that the Laplace transform of the reactive Green's function can be analytically expressed in terms of the Green's function for diffusion in the absence of reaction. Moreover, a simple relation between the Green's functions satisfying the radiation boundary condition and the reflecting boundary condition is obtained. Several applications are presented and the formalism is used to establish the relationship between the time-dependent geminate recombination yield and the bimolecular reaction rate for diffusion-influenced reactions. Finally, an analogous development for lattice random walks is presented.