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Review
. 2007 Mar 28;126(12):124114.
doi: 10.1063/1.2714528.

Polarizable Atomic Multipole Solutes in a Poisson-Boltzmann Continuum

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Free PMC article
Review

Polarizable Atomic Multipole Solutes in a Poisson-Boltzmann Continuum

Michael J Schnieders et al. J Chem Phys. .
Free PMC article

Abstract

Modeling the change in the electrostatics of organic molecules upon moving from vacuum into solvent, due to polarization, has long been an interesting problem. In vacuum, experimental values for the dipole moments and polarizabilities of small, rigid molecules are known to high accuracy; however, it has generally been difficult to determine these quantities for a polar molecule in water. A theoretical approach introduced by Onsager [J. Am. Chem. Soc. 58, 1486 (1936)] used vacuum properties of small molecules, including polarizability, dipole moment, and size, to predict experimentally known permittivities of neat liquids via the Poisson equation. Since this important advance in understanding the condensed phase, a large number of computational methods have been developed to study solutes embedded in a continuum via numerical solutions to the Poisson-Boltzmann equation. Only recently have the classical force fields used for studying biomolecules begun to include explicit polarization in their functional forms. Here the authors describe the theory underlying a newly developed polarizable multipole Poisson-Boltzmann (PMPB) continuum electrostatics model, which builds on the atomic multipole optimized energetics for biomolecular applications (AMOEBA) force field. As an application of the PMPB methodology, results are presented for several small folded proteins studied by molecular dynamics in explicit water as well as embedded in the PMPB continuum. The dipole moment of each protein increased on average by a factor of 1.27 in explicit AMOEBA water and 1.26 in continuum solvent. The essentially identical electrostatic response in both models suggests that PMPB electrostatics offers an efficient alternative to sampling explicit solvent molecules for a variety of interesting applications, including binding energies, conformational analysis, and pK(a) prediction. Introduction of 150 mM salt lowered the electrostatic solvation energy between 2 and 13 kcalmole, depending on the formal charge of the protein, but had only a small influence on dipole moments.

Figures

Figure 1
Figure 1
Normalized 5th order B-spline on the interval [0, 5].
Figure 2
Figure 2
The sum of two 4th order B-splines (dashed) equal the first derivative of a normalized 5th order B-spline (solid).
Figure 3
Figure 3
The sum of three 3rd order B-splines (dashed) equal the second derivative of a normalized 5th order B-spline (solid).
Figure 4
Figure 4
Comparison of cubic, quintic and heptic characteristic functions for an atom with radius 3 Å using a total window width of 0.6 Å.
Figure 5
Figure 5
Analytic and finite-difference gradients for a neutral cavity fixed at the origin and a sphere with unit positive charge vs. separation. Both spheres have a radius of 3.0 Å and the solvent dielectric is 78.3. The gradient of the neutral cavity is due entirely to the dielectric boundary force and cancels exactly the force on the charged sphere.
Figure 6
Figure 6
Analytic and finite-difference gradients for a neutral cavity fixed at the origin and a sphere with dipole moment components of (2.54, 2.54, 2.54) debye vs. separation. Both spheres have a radius of 3.0 Å and movement of the dipole is along the x-axis. The gradient of the neutral cavity is due entirely to the dielectric boundary force and cancels exactly the sum of the forces on the dipole and a third site (that has no charge density or dielectric properties) that defines the local coordinate system of the dipole.
Figure 7
Figure 7
Analytic and finite-difference gradients for a neutral cavity fixed at the origin and a sphere with quadrupole moment components of (5.38, 2.69, 2.69, 2.69, −2.69, 2.69, 2.69, 2.69, −2.69) Buckinghams vs. separation. Both spheres have a radius of 3.0 Å and movement of the quadrupole is along the x-axis. The gradient of the neutral cavity cancels exactly the sum of the forces on the quadrupole and a third site (that has no charge density or dielectric properties) that defines the local coordinate system of the quadrupole.
Figure 8
Figure 8
Analytic and finite-difference gradients for a neutral, polarizable cavity fixed at the origin and a sphere with unit positive charge vs. separation using the direct polarization model. Both spheres have a radius of 3.0 Å. The gradient can be seen to approach zero at a number of points, notably when the spheres are separated by approximately 1.5 Å leading to a maximum in the reaction field produced by the charge at the polarizable site, and again when the spheres are superimposed and the reaction field is zero at the polarizable site.
Figure 9
Figure 9
Analytic and finite-difference gradients for a neutral, polarizable cavity fixed at the origin and a polarizable sphere with unit positive charge vs. separation using the mutual polarization model. Both spheres have a radius of 3.0 Å and a polarizability of 1.0 Å −3. Note that the mutual polarization gradients are smaller than those of Figure 8 for the otherwise equivalent direct polarization model.
Figure 10
Figure 10
The dielectric of the solvent and test spheres are both set to 1 in this case, while a salt concentration of 150 mM is used to isolate the ionic boundary gradients. Analytic and finite-difference gradients for a neutral, polarizable cavity fixed at the origin (3.0 Å radius) and a polarizable sphere with a unit positive charge (1.0 Å radius) vs. separation using the mutual polarization model. Both spheres have a polarizability of 1.0 Å −3, and the ionic radius is set to zero Å.

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