Jan Hermans

Interaction Models for Water in Relation to Protein Hydration

For molecular dynamics simulations of hydrated proteins a simple yet reliable model for the intermolecular potential for water is required. Such a model must be an effective pair potential valid for liquid densities that takes average many-body interactions into account. We have developed a three-point charge model (on hydrogen and oxygen positions) with a Lennard-Jones 6–12 potential on the oxygen positions only. Parameters for the model were determined from 12 molecular dynamics runs covering the two-dimensional parameter space of charge and oxygen repulsion. Both potential energy and pressure were required to coincide with experimental values. The model has very satisfactory properties, is easily incorporated into protein-water potentials, and requires only 0.25 sec computertime per dynamics step (for 216 molecules) on a CRAY-1 computer.

Hydrophilicity of cavities in proteins

Water molecules inside cavities in proteins constitute integral parts of the structure. We have sought a quantitative measure of the hydrophilicity of the cavities by calculating energies and free energies of introducing a water molecule into these cavities. A threshold value of the water‐protein interaction energy at −12 kcal/mol was found to be able to distinguish hydrated from empty cavities. It follows that buried waters have entropy comparable to that of liquid water or ice. A simple consistent picture of the energetics of the buried waters provided by this study enabled us to address the reliability of buried waters assigned in experiments.

Comparison of a QM/MM force field and molecular mechanics force fields in simulations of alanine and glycine "dipeptides" (Ace-Ala-Nme and Ace-Gly-Nme) in water in relation to the problem of modeling the unfolded peptide backbone in solution.

We compare the conformational distributions of Ace‐Ala‐Nme and Ace‐Gly‐Nme sampled in long simulations with several molecular mechanics (MM) force fields and with a fast combined quantum mechanics/molecular mechanics (QM/MM) force field, in which the solutes intramolecular energy and forces are calculated with the self‐consistent charge density functional tight binding method (SCCDFTB), and the solvent is represented by either one of the well‐known SPC and TIP3P models. All MM force fields give two main states for Ace‐Ala‐Nme, β and α separated by free energy barriers, but the ratio in which these are sampled varies by a factor of 30, from a high in favor of β of 6 to a low of 1/5. The frequency of transitions between states is particularly low with the amber and charmm force fields, for which the distributions are noticeably narrower, and the energy barriers between states higher. The lower of the two barriers lies between α and β at values of ψ near 0 for all MM simulations except for charmm22. The results of the QM/MM simulations vary less with the choice of MM force field; the ratio β/α varies between 1.5 and 2.2, the easy pass lies at ψ near 0, and transitions between states are more frequent than for amber and charmm, but less frequent than for cedar. For Ace‐Gly‐Nme, all force fields locate a diffuse stable region around ϕ = π and ψ = π, whereas the amber force field gives two additional densely sampled states near ϕ = ±100° and ψ = 0, which are also found with the QM/MM force field. For both solutes, the distribution from the QM/MM simulation shows greater similarity with the distribution in high‐resolution protein structures than is the case for any of the MM simulations. Proteins 2003;50:451–463.

Discrimination between native and intentionally misfolded conformations of proteins: ES/IS, a new method for calculating conformational free energy that uses both dynamics simulations with an explicit solvent and an implicit solvent continuum model†

A new method for calculating the total conformational free energy of proteins in water solvent is presented. The method consists of a relatively brief simulation by molecular dynamics with explicit solvent (ES) molecules to produce a set of microstates of the macroscopic conformation. Conformational energy and entropy are obtained from the simulation, the latter in the quasi‐harmonic approximation by analysis of the covariance matrix. The implicit solvent (IS) dielectric continuum model is used to calculate the average solvation free energy as the sum of the free energies of creating the solute‐size hydrophobic cavity, of the van der Waals solute‐solvent interactions, and of the polarization of water solvent by the solutes charges. The reliability of the solvation free energy depends on a number of factors: the details of arrangement of the proteins charges, especially those near the surface; the definition of the molecular surface; and the method chosen for solving the Poisson equation. Molecular dynamics simulation in explicit solvent relaxes the proteins conformation and allows polar surface groups to assume conformations compatible with interaction with solvent, while averaging of internal energy and solvation free energy tend to enhance the precision. Two recently developed methods—SIMS, for calculation of a smooth invariant molecular surface, and FAMBE, for solution of the Poisson equation via a fast adaptive multigrid boundary element—have been employed. The SIMS and FAMBE programs scale linearly with the number of atoms. SIMS is superior to Connollys MS (molecular surface) program: it is faster, more accurate, and more stable, and it smooths singularities of the molecular surface. Solvation free energies calculated with these two programs do not depend on molecular position or orientation and are stable along a molecular dynamics trajectory. We have applied this method to calculate the conformational free energy of native and intentionally misfolded globular conformations of proteins (the EMBL set of deliberately misfolded proteins) and have obtained good discrimination in favor of the native conformations in all instances. Proteins 32:399–413, 1998.

Quantum mechanics simulation of protein dynamics on long timescale.

Protein structure and dynamics are the keys to a wide range of problems in biology. In principle, both can be fully understood by using quantum mechanics as the ultimate tool to unveil the molecular interactions involved. Indeed, quantum mechanics of atoms and molecules have come to play a central role in chemistry and physics. In practice, however, direct application of quantum mechanics to protein systems has been prohibited by the large molecular size of proteins. As a consequence, there is no general quantum mechanical treatment that not only exceeds the accuracy of state‐of‐the‐art empirical models for proteins but also maintains the efficiency needed for extensive sampling in the conformational space, a requirement mandated by the complexity of protein systems. Here we show that, given recent developments in methods, a general quantum mechanical‐based treatment can be constructed. We report a molecular dynamics simulation of a protein, crambin, in solution for 350 ps in which we combine a semiempirical quantum‐mechanical description of the entire protein with a description of the surrounding solvent, and solvent‐protein interactions based on a molecular mechanics force field. Comparison with a recent very high‐resolution crystal structure of crambin (Jelsch et al., Proc Natl Acad Sci USA 2000;102:2246–2251 ) shows that geometrical detail is better reproduced in this simulation than when several alternate molecular mechanics force fields are used to describe the entire system of protein and solvent, even though the structure is no less flexible. Individual atomic charges deviate in both directions from “canonical” values, and some charge transfer is found between the N and C‐termini. The capability of simulating protein dynamics on and beyond the few hundred ps timescale with a demonstrably accurate quantum mechanical model will bring new opportunities to extend our understanding of a range of basic processes in biology such as molecular recognition and enzyme catalysis. Proteins 2001;44:484–489.

Excess free energy of liquids from molecular dynamics simulations. Application to water models

Thermodynamic integration and perturbation methods have been applied to calculate the excess free energy, /Delta//Alpha//sub e/, of several models of liquid water. The results for the SPC and TIP models agree well with the experimental excess free energy of liquid water computed from the vapor pressure. The precision of the calculation, in which the nonbonded interaction of all molecules is coupled to a forcing parameter, is demonstrated by the agreement of the computed temperature dependence of /Delta//Alpha//sub e/ with that expected. Some results are presented that show the apparent equivalence of the stepwise perturbation and continuous integration algorithms for this calculation. Finally, it is suggested that the excess free energy of liquids should be among the most important observables used in the parameterization of interatomic potential functions for application in free energy simulations of biological molecules.

SIMS: computation of a smooth invariant molecular surface

SIMS, a new method of calculating a smooth invariant molecular dot surface, is presented. The SIMS method generates the smooth molecular surface by rolling two probe spheres. A solvent probe sphere is rolled over the molecule and produces a Richards-Connolly molecular surface (MS), which envelops the solvent-excluded volume of the molecule. In deep crevices, Connollys method of calculating the MS has two deficiencies. First, it produces self-intersecting parts of the molecular surface, which must be removed to obtain the correct MS. Second, the correct MS is not smooth, i.e., the direction of the normal vector of the MS is not continuous, and some points of the MS are singular. We present an exact method for removing self-intersecting parts and smoothing the singular regions of the MS. The singular MS is smoothed by rolling a smoothing probe sphere over the inward side of the singular MS. The MS in the vicinity of singularities is replaced with the reentrant surface of the smoothing probe sphere. The smoothing method does not disturb the topology of a singular MS, and the smooth MS is a better approximation of the dielectric border between high dielectric solvent and the low dielectric molecular interior. The SIMS method generates a smooth molecular dot surface, which has a quasi-uniform dot distribution in two orthogonal directions on the molecular surface, which is invariant with molecular rotation and stable under changes in the molecular conformation, and which can be used in a variety of implicit methods of modeling solvent effects. The SIMS program is faster than the Connolly MS program, and in a matter of seconds generates a smooth dot MS of a 200-residue protein. The program is available from the authors on request (see http:@femto.med.unc.edu/SIMS).

Excluded‐volume theory of polymer–protein interactions based on polymer chain statistics

A theory is presented of mutual steric exclusion of protein molecules and randomly coiled polymer molecules, in which the former are presented as impenetrable rigid particles and the latter as flexible segmented chains obeying Gaussian statistics. The limiting case of small polymer chains and large protein molecules is treated first, by approximating the protein surface as planar; this gives an excluded volume equal to the sum of protein volume V and a term in V h 0/R, i.e., protein volume times the ratio of rms end‐to‐end distance of the polymer and radius of the protein. Several cases of steric exclusion are treated next, using a quite general approach, by which excluded volume can be expressed as V/〈m〉, the ratio of protein volume and the average number of elements that are simultaneously excluded by, or that simultaneously exclude, one protein molecule. Expressions for 〈m〉 are successively obtained for thin rods arbitrarily finely divided into segments, for uniformly randomly distributed segments, for segments linked into very long polymers, and for segments linked into finite polymers. In all these cases the protein is represented as a sphere; in addition an expression for the excluded volume of very long polymers and cylindrical protein molecules is obtained. Excluded volume of spherical proteins and very long polymers is found to be simply proportional to the radius of the protein and the square of the rms end‐to‐end distance of the polymer, while the larger excluded volume of proteins and finite chains is conveniently expressed as the product of the result for very long chains and a series expansion in powers of R/h 0. Comparison with results of Monte Carlo simulation shows all predictions of theory to be excellent, with two exceptions. As a result of the use of Gaussian statistics for polymers of any chain length, the theory does not adequately describe the case of stiff polymer chains and small protein molecules. As a result of a second, at present unavoidable, approximation in the calculation of 〈m〉 for polymers of finite chain length, theory systematically predicts excluded volumes that are too low by a factor that varies from 1 to 2. As a result, this theory makes useful predictions for short polymers, as long as h 0/R does not exceed 1 and for long polymers, as long as R/h 0 does not exceed 0.5. Comparison with experimental data is found to be good; the prediction of proportionality of excluded volumes of globular proteins and high molecular weight polymers with protein radius agrees with recent observations with poly(ethylene oxide) and predicted excluded volumes of serum albumin and this polymer at three different polymer molecular weights agree within 10% with recently observed values.

Effects of calcium ion and covalent crosslinking on formation and elasticity of fibrin gels

Abstract The elastic properties of fibrin gels prepared under a variety of conditions were studied using a Couette elastometer. Fibrin gels formed either by addition of thrombin to fibrinogen or by dilution of previously prepared fibrin monomer were seen to be dependent on calcium ion for the expression of maximum elasticity. By dialyzing a fibrinogen solution it was shown that two calcium ions per mole are tightly bound. Factor XIIIa was shown to have an effect on the elastic modulus by means of its crosslinking the α-chains to form α-polymer. γ-γ crosslinking caused no additional increase over calcium in elasticity of fibrin gels. The relative effects of calcium concentration on the rates of crosslinking of α-chains and γ-chains to form α-polymers and γ-γ dimers were determined by means of SDS gel electrophoresis in the presence of dithiothreitol. At calcium concentrations below 2×10 −4 M α- and γ-chain crosslinking proceed at the same rate which depends strongly on the calcium concentration, while at higher concentrations the rate of α-crosslinking remains constant and the γ-chain crosslinking rate is dependent on calcium concentration. Factor XIIIa was shown also to lower the gel point of dilute solutions of fibrinogen clotted with thrombin in the presence of calcium. This demonstrates that Factor XIIIa promotes gelation in very dilute fibrin solution, probably by irreversibly crosslinking intermediate polymers of fibrin.

Effect of Inertia on the Brownian Motion of Rigid Particles in a Viscous Fluid

From Boussinesqs work it is known that the frictional resistance of a particle in a viscous, inert fluid, depends on its history. This plays an important part in Brownian motion. The general theory of fluctuations in fluid dynamics leads to explicit expressions for the autocorrelation function G(t) for the random force acting on a Brownian particle and the autocorrelation function φ(t) for its velocity. It is demonstrated that G(t) contains an inertial term which depends on the velocity distribution in the liquid surrounding the particle. The results obtained lead to the correct value of the diffusivity.