H.J.C. Berendsen

ALGORITHMS FOR MACROMOLECULAR DYNAMICS AND CONSTRAINT DYNAMICS

The application of the computer simulation method of molecular dynamics to macromolecules is investigated. The protein trypsin inhibitor (BPTI), consisting of 454 united atoms, is used as an example. Different algorithms for integrating the equations of motion are compared, both theoretically and in practice. It is examined to what extent the chain structure of a macromolecule allows a reduction of the computational effort by the introduction of constraints in the dynamics of the chain. A calculational scheme is proposed, by which constraints can be incorporated in predictor-corrector algorithms. The optimum choice of an algorithm depends on the desired accuracy of the solution and on the character of the forces acting on the molecule, viz. whether these are noisy or not. For nonconstraint dynamics a Gear predictor-corrector algorithm yields the best results, whereas for constraint dynamics the Gear and Verlet algorithms produce comparable results. The application of bond-length constraints reduces the re...

ALGORITHMS FOR BROWNIAN DYNAMICS

A third order algorithm for brownian dynamics (BD) simulations is proposed, which is identical to the powerful molecular dynamics (MD) algorithm due to Verlet in the limit of infinitely small friction coefficient γ. In contrast to most BD algorithms used up till now, the integration time step Δt is not limited by the condition Δt ≪ γ-1. It is shown how constraints, such as bondlength or bond angle constraints, can be incorporated in the computational scheme. For the molecule considered here the proposed BD algorithm is about ten times more efficient than the second order ones that are generally used. The introduction of bondlength constraints saves about a factor of three in computing time, as in the MD case. The application of bond angle constraints is not recommended.

Free energy of ionic hydration: Analysis of a thermodynamic integration technique to evaluate free energy differences by molecular dynamics simulations

The thermodynamic integration technique to evaluate free energy differences by molecular dynamics simulations is analyzed. The hydration of the ions Na+ , K+ , Ca++ , F−, Cl−, and Br− is used as the process to illustrate the potential utility of the method. A neon–water system is used as a reference system. The parameters that influence the performance and accuracy of the thermodynamic integration, in which the potential interaction parameters are gradually and continuously changed, are studied. These parameters include the total simulation time, the magnitude of the time step for the numerical integration of the equations of motion, the system size, and the cutoff radii for the intermolecular interactions. Fast convergence is found for the Gibbs free energy difference between Ne and Na+ with respect to total simulation time. The time step and system size are relatively unimportant. The use of cutoff radii, for the ion–water but especially unfortunately also the water–water intermolecular interactions, seriously influences the results obtained. A simple correction for the use of cutoff radii cannot be made. Results are compared to experimental values.

PREDICTION OF PROTEIN CONFORMATIONAL FREEDOM FROM DISTANCE CONSTRAINTS

A method is presented that generates random protein structures that fulfil a set of upper and lower interatomic distance limits. These limits depend on distances measured in experimental structures and the strength of the interatomic interaction. Structural differences between generated structures are similar to those obtained from experiment and from MD simulation. Although detailed aspects of dynamical mechanisms are not covered and the extent of variations are only estimated in a relative sense, applications to an IgG‐binding domain, an SH3 binding domain, HPr, calmodulin, and lysozyme are presented which illustrate the use of the method as a fast and simple way to predict structural variability in proteins. The method may be used to support the design of mutants, when structural fluctuations for a large number of mutants are to be screened. The results suggest that motional freedom in proteins is ruled largely by a set of simple geometric constraints. Proteins 29:240–251, 1997.

MOLECULAR-DYNAMICS SIMULATION OF A SMECTIC LIQUID-CRYSTAL WITH ATOMIC DETAIL

A molecular dynamics simulation of a sodium–decanoate/decanol/water system is reported. The system is treated in full atomic detail, with the exception of CH2 and CH3 groups that are considered to be ‘‘united atoms,’’ and is a refinement of a previous model membrane [Mol. Phys. 11, 1 (1983)]. The long‐range Coulomb interactions were included specifically. The order parameters of the chain units of the lipids and diffusion constants of components in the system calculated from the simulation agree well with those reported in experiments on this model membrane. The overall structure of the membrane shows considerable disorder, with a broad lipid–water interface, extending over approximately 1 nm. The distribution of the components is such that an almost complete charge cancellation occurs throughout the system, which is in contradiction with the generally assumed electrical double layer structure for membranes. A counterion condensation of 70% is observed. Both the translational and the rotational motions of...

An efficient method for sampling the essential subspace of proteins

A method is presented for a more efficient sampling of the configurational space of proteins as compared to conventional sampling techniques such as molecular dynamics. The method is based on the large conformational changes in proteins revealed by the essential dynamics analysis. A form of constrained dynamics is performed, forcing the system to move along some of the essential coordinates. This results in a broader sampling of the essential subspace than in a comparable conventional molecular dynamics simulation without constraints. The new sampling method (essential dynamics sampling) was applied to the histidine-containing phosphocarrier protein HPr. The results indicate that the essential dynamics sampling method produces physically allowed structures, as estimated by the evaluation of many geometrical properties. In addition, a study of the motions in the essential subspace reveals a diffusion-like behavior.

Free energy of hydrophobic hydration : A molecular dynamics study of noble gases in water

The potential utility and limitations of two methods to determine free energy differences from molecular dynamics simulations (MD) are studied. The computation of the free energy of hydration of the inert gases serves as a simple but illustrative example. Good results are obtained for the inert gases from a perturbation treatment, using a reference ensemble obtained from a MD simulation of a cavity in water, if these atoms are comparable in size to the cavity and the calculated free energy differences are small. This limits the applicability of the perturbation treatment of a small number of cases. Larger free energy differences can be obtained with reasonable accuracy from MD simulations with continuously changing interaction parameters. This integration method is more generally applicable, but makes an additional simulation necessary.

STOCHASTIC DYNAMICS FOR MOLECULES WITH CONSTRAINTS BROWNIAN DYNAMICS OF NORMAL-ALKANES

It is shown how constraints, such as bondlength or bond angle constraints, can be incorporated in the algorithms currently being used in brownian dynamics (BD) simulations of molecular liquids or solutions. The validity of the BD model, in which the stochastic and frictional force fields possess neither time correlations nor space correlations between different atoms of the molecule, is investigated by comparing the BD results with those of molecular dynamics (MD) simulations for liquid n-butane and n-decane. The BD model appears to yield a good approximation to the dynamics of a chain molecule in the liquid state. For butane, solvent packing effects play an important role in the condensed phase. For decane, the equilibrium conformation and dynamics are determined mainly by intramolecular interactions. When analysing the conformational transitions occurring in these molecules, one observes a correlation between consecutive transitions both of one dihedral angle and of two adjacent dihedral angles, due to ...

Molecular Dynamics Simulation of the Transport of Small Molecules Across a Polymer Membrane

The transport of small molecules through a polymer membrane is modeled using the computer simulation technique of molecular dynamics (MD). The transport coefficient is derived from a combination of the excess free energy and the diffusion constant. Both properties are derived from MD simulations, applied to helium and methane in polydimethylsiloxane (PDMS). The diffusional process appears to have the character of a jump diffusion for methane and less so for helium. Jumps are allowed by fluctuations of the size and shape of holes. Experimental diffusion constants are well reproduced. The excess free energies, determined by a particle insertion method, are lower by 5–7 kJ/mol than experimental values. It is shown that, as a result of a higher solubility, methane has a higher permeability constant than helium, despite its lower diffusion constant.

ESTIMATION OF STATISTICAL ERRORS IN MOLECULAR SIMULATION CALCULATIONS

The naive estimation of errors in averages obtained from molecular simulation calculations may lead to serious underestimates due to the strong correlations that usually exist within the series. A commonly employed procedure to overcome this difficulty is to estimate the error from the deviation in the averages over subseries that are considered uncorrelated. However, without knowledge of the correlation the choice of the number of subseries cannot be soundly based. A method to estimate the statistical error using the correlation explicitly without serious additional computational effort and without the need for a tedious examination of the correlation behaviour itself is presented. Results of a test of this method for a molecular dynamics study on the polarization energy of xenon in liquid water are given and a comparison is made with the procedure based on subaverages and with a recently suggested method by Smith and Wells [1].