Continuum and Dipole-Lattice Models of Solvation

Protein Dielectrophoresis: A Tale of Two Clausius-Mossottis-Or Something Else?

Standard DEP theory, based on the Clausius-Mossotti (CM) factor derived from solving the boundary-value problem of macroscopic electrostatics, fails to describe the dielectrophoresis (DEP) data obtained for 22 different globular proteins over the past three decades. The calculated DEP force appears far too small to overcome the dispersive forces associated with Brownian motion. An empirical theory, employing the equivalent of a molecular version of the macroscopic CM-factor, predicts a protein's DEP response from the magnitude of the dielectric β-dispersion produced by its relaxing permanent dipole moment. A new theory, supported by molecular dynamics simulations, replaces the macroscopic boundary-value problem with calculation of the cross-correlation between the protein and water dipoles of its hydration shell. The empirical and formal theory predicts a positive DEP response for protein molecules up to MHz frequencies, a result consistently reported by electrode-based (eDEP) experiments. However, insulator-based (iDEP) experiments have reported negative DEP responses. This could result from crystallization or aggregation of the proteins (for which standard DEP theory predicts negative DEP) or the dominating influences of electrothermal and other electrokinetic (some non-linear) forces now being considered in iDEP theory.

Molecular modeling of nucleic Acid structure: electrostatics and solvation

This unit presents an overview of computer simulation techniques as applied to nucleic acid systems, ranging from simple in vacuo molecular modeling techniques to more complete all-atom molecular dynamics treatments that include an explicit representation of the environment. The third in a series of four units, this unit focuses on critical issues in solvation and the treatment of electrostatics. UNITS 7.5 & 7.8 introduced the modeling of nucleic acid structure at the molecular level. This included a discussion of how to generate an initial model, how to evaluate the utility or reliability of a given model, and ultimately how to manipulate this model to better understand its structure, dynamics, and interactions. Subject to an appropriate representation of the energy, such as a specifically parameterized empirical force field, the techniques of minimization and Monte Carlo simulation, as well as molecular dynamics (MD) methods, were introduced as a way of sampling conformational space for a better understanding of the relevance of a given model. This discussion highlighted the major limitations with modeling in general. When sampling conformational space effectively, difficult issues are encountered, such as multiple minima or conformational sampling problems, and accurately representing the underlying energy of interaction. In order to provide a realistic model of the underlying energetics for nucleic acids in their native environments, it is crucial to include some representation of solvation (by water) and also to properly treat the electrostatic interactions. These subjects are discussed in detail in this unit.

The electrostatic response of water to neutral polar solutes: implications for continuum solvent modeling

Continuum solvation models are widely used to estimate the hydration free energies of small molecules and proteins, in applications ranging from drug design to protein engineering, and most such models are based on the approximation of a linear dielectric response by the solvent. We used explicit-water molecular dynamics simulations with the TIP3P water model to probe this linear response approximation in the case of neutral polar molecules, using miniature cucurbituril and cyclodextrin receptors and protein side-chain analogs as model systems. We observe supralinear electrostatic solvent responses, and this nonlinearity is found to result primarily from waters' being drawn closer and closer to the solutes with increased solute-solvent electrostatic interactions; i.e., from solute electrostriction. Dielectric saturation and changes in the water-water hydrogen bonding network, on the other hand, play little role. Thus, accounting for solute electrostriction may be a productive approach to improving the accuracy of continuum solvation models.

Biomolecular electrostatics and solvation: a computational perspective

An understanding of molecular interactions is essential for insight into biological systems at the molecular scale. Among the various components of molecular interactions, electrostatics are of special importance because of their long-range nature and their influence on polar or charged molecules, including water, aqueous ions, proteins, nucleic acids, carbohydrates, and membrane lipids. In particular, robust models of electrostatic interactions are essential for understanding the solvation properties of biomolecules and the effects of solvation upon biomolecular folding, binding, enzyme catalysis, and dynamics. Electrostatics, therefore, are of central importance to understanding biomolecular structure and modeling interactions within and among biological molecules. This review discusses the solvation of biomolecules with a computational biophysics view toward describing the phenomenon. While our main focus lies on the computational aspect of the models, we provide an overview of the basic elements of biomolecular solvation (e.g. solvent structure, polarization, ion binding, and non-polar behavior) in order to provide a background to understand the different types of solvation models.

A simple polarizable continuum solvation model for electrolyte solutions

We propose a Debye-Hückel-like screening model (DESMO) that generalizes the familiar conductor-like screening model (COSMO) to solvents with non-zero ionic strength and furthermore provides a numerical generalization of the Debye-Hückel model that is applicable to non-spherical solute cavities. The numerical implementation of DESMO is based upon the switching/Gaussian (SWIG) method for smooth cavity discretization, which we have recently introduced in the context of polarizable continuum models (PCMs). This approach guarantees that the potential energy is a smooth function of the solute geometry and analytic gradients for DESMO are reported here. The SWIG formalism also facilitates analytic implementation of two other PCMs that are based on a screened Coulomb potential: the "integral equation formalism" (IEF-PCM) and the "surface and simulation of volume polarization for electrostatics" [SS(V)PE] method. Fully analytic implementations of these screened PCMs are reported here for the first time. Numerical results, for model systems where an exact solution of the linearized Poisson-Boltzmann equation is available, demonstrate that these screened PCMs are highly accurate. In realistic test cases, they are as accurate as the best available three-dimensional finite-difference methods. In polar solvents, DESMO is nearly as accurate as more sophisticated screened PCMs, but is significantly simpler and more efficient.

Coarse-grained (multiscale) simulations in studies of biophysical and chemical systems

Recent years have witnessed an explosion in computational power, leading to attempts to model ever more complex systems. Nevertheless, there remain cases for which the use of brute-force computer simulations is clearly not the solution. In such cases, great benefit can be obtained from the use of physically sound simplifications. The introduction of such coarse graining can be traced back to the early usage of a simplified model in studies of proteins. Since then, the field has progressed tremendously. In this review, we cover both key developments in the field and potential future directions. Additionally, particular emphasis is given to two general approaches, namely the renormalization and reference potential approaches, which allow one to move back and forth between the coarse-grained (CG) and full models, as these approaches provide the foundation for CG modeling of complex systems.

Multiscale modeling of biological functions

Recent years have witnessed a tremendous explosion in computational power, which in turn has resulted in great progress in the complexity of the biological and chemical problems that can be addressed by means of all-atom simulations. Despite this, however, our computational time is not infinite, and in fact many of the key problems of the field were resolved long before the existence of the current levels of computational power. This review will start by presenting a brief historical overview of the use of multiscale simulations in biology, and then present some key developments in the field, highlighting several cases where the use of a physically sound simplification is clearly superior to a brute-force approach. Finally, some potential future directions will be discussed.

Effect of the ordered interfacial water layer in protein complex formation: A nonlocal electrostatic approach

Using a nonlocal electrostatic approach that incorporates the short-range structure of the contacting media, we evaluated the electrostatic contribution to the energy of the complex formation of two model proteins. In this study, we have demonstrated that the existence of an ordered interfacial water layer at the protein-solvent interface reduces the charging energy of the proteins in the aqueous solvent, and consequently increases the electrostatic contribution to the protein binding (change in free energy upon the complex formation of two proteins). This is in contrast with the finding of the continuum electrostatic model, which suggests that electrostatic interactions are not strong enough to compensate for the unfavorable desolvation effects.

Semi-continuum electrostatic calculations of redox potentials in photosystem I

The midpoint redox potentials (E(m)) of all cofactors in photosystem I from Synechococcus elongatus as well as of the iron-sulfur (Fe(4)S(4)) clusters in two soluble ferredoxins from Azotobacter vinelandii and Clostridium acidiurici were calculated within the framework of a semi-continuum dielectric approach. The widely used treatment of proteins as uniform media with single dielectric permittivity is oversimplified, particularly, because permanent charges are considered both as a source for intraprotein electric field and as a part of dielectric polarizability. Our approach overcomes this inconsistency by using two dielectric constants: optical epsilon(o)=2.5 for permanent charges pre-existing in crystal structure, and static epsilon(s) for newly formed charges. We also take into account a substantial dielectric heterogeneity of photosystem I revealed by photoelectric measurements and a liquid junction potential correction for E(m) values of relevant redox cofactors measured in aprotic solvents. We show that calculations based on a single permittivity have the discrepancy with experimental data larger than 0.7 V, whereas E(m) values calculated within our approach fall in the range of experimental estimates. The electrostatic analysis combined with quantum chemistry calculations shows that (i) the energy decrease upon chlorophyll dimerization is essential for the downhill mode of primary charge separation between the special pair P(700) and the primary acceptor A(0); (ii) the primary donor is apparently P(700) but not a pair of accessory chlorophylls; (iii) the electron transfer from the A branch quinone Q(A) to the iron-sulfur cluster F(X) is most probably downhill, whereas that from the B branch quinone Q(B) to F(X) is essentially downhill.

Modeling electrostatic effects in proteins

Electrostatic energies provide what is perhaps the most effective tool for structure-function correlation of biological molecules. This review considers the current state of simulations of electrostatic energies in macromolecules as well as the early developments of this field. We focus on the relationship between microscopic and macroscopic models, considering the convergence problems of the microscopic models and the fact that the dielectric 'constants' in semimacroscopic models depend on the definition and the specific treatment. The advances and the challenges in the field are illustrated considering a wide range of functional properties including pK(a)'s, redox potentials, ion and proton channels, enzyme catalysis, ligand binding and protein stability. We conclude by pointing out that, despite the current problems and the significant misunderstandings in the field, there is an overall progress that should lead eventually to quantitative descriptions of electrostatic effects in proteins and thus to quantitative descriptions of the function of proteins.

New Theoretical and Experimental Infrared Results on Formaldehyde in Solution

An extension of our combined procedure to determine a complete quartic force field and to resolve a vibrational problem thanks to a variational treatment is proposed for quantitative calculations of vibrational spectra in solution. Energies and gradients are obtained through a polarizable continuum model (PCM), the so-called self-consistent isodensity (SCI)-PCM. We present in this paper new experimental results dealing with formaldehyde in solution in cyclohexane, chloroform, THF, acetonitrile, DMSO and water; the obtained vibrational spectra are then compared with CCSD(T)/cc-pVQZ calculations. In addition, density functional theory (DFT) calculations have been carried out with the aim of both anticipating and positioning these approaches for larger sized molecules.

Influence of the solvent structure on the electrostatic interactions in proteins

The proper estimation of the influence of the many-body dynamic solvent microstructure on a pairwise electrostatic interaction (PEI) at the protein-solvent interface is very important for solving many biophysical problems. In this work, the PEI energy was calculated for a system that models the interface between a protein and an aqueous solvent. The concept of nonlocal electrostatics for interfacial electrochemical systems was used to evaluate the contribution of a solvent orientational polarization, correlated by the network of hydrogen bonds, into the PEI energy in proteins. The analytical expression for this energy was obtained in the form of Coulomb's law with an effective distance-dependent dielectric function. The asymptotic and numerical analysis carried out for this function revealed several features of dielectric heterogeneity at the protein-solvent interface. For charges located in close proximity to this interface, the values of the dielectric function for the short-distance electrostatic interactions were found to be remarkably smaller than those determined by the classical model, in which the solvent was considered as the uniform dielectric medium of high dielectric constant. Our results have shown that taking into consideration the dynamic solvent microstructure remarkably increases the value of the PEI energy at the protein-solvent interface.

Density Functional Theory of Solvation and Its Relation to Implicit Solvent Models

We describe a density functional theory approach to solvation in molecular solvents. The solvation free energy of a complex solute can be obtained by direct minimization of a density functional, instead of the thermodynamic integration scheme necessary when using atomistic simulations. In the homogeneous reference fluid approximation, the expression of the free-energy functional relies on the knowledge of the direct correlation function of the pure solvent. After discussing general molecular solvents, we present a generic density functional describing a dipolar solvent and we show how it can be reduced to the conventional implicit solvent models when the solvent microscopic structure is neglected. With respect to those models, the functional includes additional effects such as the microscopic structure of the solvent, the dipolar saturation effect, and the nonlocal character of the dielectric constant. We also show how this functional can be minimized numerically on a three-dimensional grid around a solute of complex shape to provide, in a single shot, both the average solvent structure and the absolute solvation free energy.

A semi-implicit solvent model for the simulation of peptides and proteins

We present a new model of biomolecules hydration based on macroscopic electrostatic theory, that can both describe the microscopic details of solvent-solute interactions and allow for an efficient evaluation of the electrostatic hydration free energy. This semi-implicit model considers the solvent as an ensemble of polarizable pseudoparticles whose induced dipole describe both the electronic and orientational solvent polarization. In the presented version of the model, there is no mutual dipolar interaction between the particles, and they only interact through short-ranged Lennard-Jones interactions. The model has been integrated into a molecular dynamics code, and offers the possibility to simulate efficiently the conformational evolution of biomolecules. It is able to provide estimations of the electrostatic solvation free energy within short time windows during the simulation. It has been applied to the study of two small peptides, the octaalanine and the N-terminal helix of ribonuclease A, and two proteins, the bovine pancreatic trypsin inhibitor and the B1 immunoglobin-binding domain of streptococcal protein G. Molecular dynamics simulations of these biomolecules, using a slightly modified Amber force field, provide stable and meaningful trajectories in overall agreement with experiments and all-atom simulations. Correlations with respect to Poisson-Boltzmann electrostatic solvation free energies are also presented to discuss the parameterization of the model and its consequences.

Fast prediction of hydration free energies from molecular interaction fields

A novel empirical model is presented that allows the fast computation of hydration free energies with high accuracy. The linear model is based upon the separation of the free energy of hydration into a cavity and an interaction term. The cavity contribution is modeled as a linear combination of molecular volume and surface terms. The interaction part is derived from the statistical three-dimensional (3D) free energy density and is modeled approximately as a molecular interaction field using the program GRID. A compression scheme is employed to represent this 3D information on the molecular surface by means of a linear combination of surface functions. A set of 81 small organic molecules with known experimental hydration free energies is used to determine the coefficients of the linear model by least squares regression. The fit is statistically significant yielding a correlation coefficient of 0.99, a root mean square error of 0.27 kcal/mol for the 81 molecules belonging to the training set, and 0.63 kcal/mol for an independent test set of 10 molecules.

Molecular modeling of nucleic acid structure: electrostatics and solvation

This unit presents an overview of computer simulation techniques as applied to nucleic acid systems, ranging from simple in vacuo molecular modeling techniques to more complete all-atom molecular dynamics treatments that include an explicit representation of the environment. The third in a series of four units, this unit focuses on critical issues in solvation and the treatment of electrostatics.

A critical investigation of the Tanford-Kirkwood scheme by means of Monte Carlo simulations

Monte Carlo simulations are used to assess the adequacy of the Tanford-Kirkwood prescription for electrostatic interactions in macromolecules. Within a continuum dielectric framework, the approach accurately describes salt screening of electrostatic interactions for moderately charged systems consistent with common proteins at physiological conditions. The limitations of the Debye-Hückel theory, which forms the statistical mechanical basis for the Tanford-Kirkwood result, become apparent for highly charged systems. It is shown, both by an analysis of the Debye-Hückel theory and by numerical simulations, that the difference in dielectric permittivity between macromolecule and surrounding solvent does not play a significant role for salt effects if the macromolecule is highly charged. By comparison to experimental data, the continuum dielectric model (combined with either an approximate effective Hamiltonian as in the Tanford-Kirkwood treatment or with exact Monte Carlo simulations) satisfactorily predicts the effects of charge mutation on metal ion binding constants, but only if the macromolecule and solvent are assigned the same or similar permittivities.

Monte Carlo simulation towards ripple phase modelling

We present a novel approach to analysis of the gel-fluid transition of lipid membrane. The method is based on the Pink's model but in contrast to its standard version the dipole character of the lipid molecules polar part is considered. Moreover, less constrained movement of entire molecules is allowed. Such an approach includes into the model conditions imposed by the adjacent medium such as ionic strength, pH, and other factors affecting biological membranes via the polar part. The results obtained contribute to the explanation of the ripple phase phenomenon.

Generalized born models of macromolecular solvation effects

It would often be useful in computer simulations to use a simple description of solvation effects, instead of explicitly representing the individual solvent molecules. Continuum dielectric models often work well in describing the thermodynamic aspects of aqueous solvation, and approximations to such models that avoid the need to solve the Poisson equation are attractive because of their computational efficiency. Here we give an overview of one such approximation, the generalized Born model, which is simple and fast enough to be used for molecular dynamics simulations of proteins and nucleic acids. We discuss its strengths and weaknesses, both for its fidelity to the underlying continuum model and for its ability to replace explicit consideration of solvent molecules in macromolecular simulations. We focus particularly on versions of the generalized Born model that have a pair-wise analytical form, and therefore fit most naturally into conventional molecular mechanics calculations.

Electrostatic effects in macromolecules: fundamental concepts and practical modeling

The past few years have seen an exponential growth in the calculations of electrostatic energies of macromolecules and an increased recognition of the crucial role of electrostatic effects. This review considers the current state of the field. Focus is placed on calculations of pKas, redox potentials and binding energies in macromolecules and clarification of the fact that the value of the dielectric 'constant' of a protein depends on its definition and that small dielectric constants should not be used in describing charge-charge interactions by current continuum models.