Constraints In Molecular Dynamics Simulations Research Articles

A method of incorporating rate constants as kinetic constraints in molecular dynamics simulations

From the point of view of statistical mechanics, a full characterization of a molecular system requires an accurate determination of its possible states, their populations, and the respective interconversion rates. Toward this goal, well-established methods increase the accuracy of molecular dynamics simulations by incorporating experimental information about states using structural restraints and about populations using thermodynamics restraints. However, it is still unclear how to include experimental knowledge about interconversion rates. Here, we introduce a method of imposing known rate constants as constraints in molecular dynamics simulations, which is based on a combination of the maximum-entropy and maximum-caliber principles. Starting from an existing ensemble of trajectories, obtained from either molecular dynamics or enhanced trajectory sampling, this method provides a minimally perturbed path distribution consistent with the kinetic constraints, as well as modified free energy and committor landscapes. We illustrate the application of the method to a series of model systems, including all-atom molecular simulations of protein folding. Our results show that by combining experimental rate constants and molecular dynamics simulations, this approach enables the determination of transition states, reaction mechanisms, and free energies. We anticipate that this method will extend the applicability of molecular simulations to kinetic studies in structural biology and that it will assist the development of force fields to reproduce kinetic and thermodynamic observables. Furthermore, this approach is generally applicable to a wide range of systems in biology, physics, chemistry, and material science.

A method to apply bond-angle constraints in molecular dynamics simulations.

An algorithm to apply bond-angle constraints in molecular dynamics simulations of macromolecules or molecular liquids is presented. It uses Cartesian coordinates and determines the Lagrange multipliers required for maintaining the constraints iteratively. It constitutes an alternative to the use of only distance constraints (DCs) between particles to maintain a particular geometry. DCs are unsuitable to maintain particular, for example, linear or flat, geometries of molecules. The proposed algorithm can easily handle bond-length, bond-angle, and dihedral-angle constraints simultaneously, as when calculating a potential of mean force along a dihedral-angle degree of freedom.

General framework for constraints in molecular dynamics simulations

ABSTRACTThe article presents a theoretical framework for molecular dynamics simulations of complex systems subject to any combination of holonomic and non-holonomic constraints. Using the concept of constrained inverse matrices both the particle accelerations and the associated constraint forces can be determined from given external forces and kinematical conditions. The formalism enables in particular the construction of explicit kinematical conditions which lead to the well-known Nosé–Hoover type equations of motion for the simulation of non-standard molecular dynamics ensembles. Illustrations are given for a few examples and an outline is presented for a numerical implementation of the method.

Molecular dynamics simulations on PGLa using NMR orientational constraints

NMR data obtained by solid state NMR from anisotropic samples are used as orientational constraints in molecular dynamics simulations for determining the structure and dynamics of the PGLa peptide within a membrane environment. For the simulation the recently developed molecular dynamics with orientational constraints technique (MDOC) is used. This method introduces orientation dependent pseudo-forces into the COSMOS-NMR force field. Acting during a molecular dynamics simulation these forces drive molecular rotations, re-orientations and folding in such a way that the motional time-averages of the tensorial NMR properties are consistent with the experimentally measured NMR parameters. This MDOC strategy does not depend on the initial choice of atomic coordinates, and is in principle suitable for any flexible and mobile kind of molecule; and it is of course possible to account for flexible parts of peptides or their side-chains. MDOC has been applied to the antimicrobial peptide PGLa and a related dimer model. With these simulations it was possible to reproduce most NMR parameters within the experimental error bounds. The alignment, conformation and order parameters of the membrane-bound molecule and its dimer were directly derived with MDOC from the NMR data. Furthermore, this new approach yielded for the first time the distribution of segmental orientations with respect to the membrane and the order parameter tensors of the dimer systems. It was demonstrated the deuterium splittings measured at the peptide to lipid ratio of 1/50 are consistent with a membrane spanning orientation of the peptide.

Correcting for the free energy costs of bond or angle constraints in molecular dynamics simulations

BackgroundFree energy simulations are an important tool in the arsenal of computational biophysics, allowing the calculation of thermodynamic properties of binding or enzymatic reactions. This paper introduces methods to increase the accuracy and precision of free energy calculations by calculating the free energy costs of constraints during post-processing. The primary purpose of employing constraints for these free energy methods is to increase the phase space overlap between ensembles, which is required for accuracy and convergence. MethodsThe free energy costs of applying or removing constraints are calculated as additional explicit steps in the free energy cycle. The new techniques focus on hard degrees of freedom and use both gradients and Hessian estimation. Enthalpy, vibrational entropy, and Jacobian free energy terms are considered. ResultsWe demonstrate the utility of this method with simple classical systems involving harmonic and anharmonic oscillators, four-atomic benchmark systems, an alchemical mutation of ethane to methanol, and free energy simulations between alanine and serine. The errors for the analytical test cases are all below 0.0007kcal/mol, and the accuracy of the free energy results of ethane to methanol is improved from 0.15 to 0.04kcal/mol. For the alanine to serine case, the phase space overlaps of the unconstrained simulations range between 0.15 and 0.9%. The introduction of constraints increases the overlap up to 2.05%. On average, the overlap increases by 94% relative to the unconstrained value and precision is doubled. ConclusionsThe approach reduces errors arising from constraints by about an order of magnitude. Free energy simulations benefit from the use of constraints through enhanced convergence and higher precision. General significanceThe primary utility of this approach is to calculate free energies for systems with disparate energy surfaces and bonded terms, especially in multi-scale molecular mechanics/quantum mechanics simulations. This article is part of a Special Issue entitled Recent developments of molecular dynamics.

Applying efficient implicit nongeometric constraints in alchemical free energy simulations.

Several strategies have been developed for satisfying bond lengths, angle, and other geometric constraints in molecular dynamics simulations. Advanced variations of alchemical free energy perturbation simulations, however, also require nongeometric constraints. In our recently developed multisite λ-dynamics simulation method, the conventional λ parameters that are associated with the progress variables in alchemical transformations are treated as dynamic variables and are constrained such that: 0 ≤ λ(i) ≤ 1 and Σ(i = 1)(N) λ(i) = 1. Here, we present four functional forms of λ that implicitly satisfy these nongeometric constraints, whose values and forces are facile to compute and that yield stable simulations using a 2 fs integration timestep. Using model systems, we present the sampling characteristics of these functional forms and demonstrate the enhanced sampling profiles and improved convergence rates that are achieved by the functional form: λ(i) = e(c sinθ(i))/Σ(j = 1)(N) e(c sinθ(j)) that oscillates between λ(i) = 0 and λ(i) = 1 and has relatively steep transitions between these endpoints.

ϑ-SHAKE: An extension to SHAKE for the explicit treatment of angular constraints

This paper presents ϑ-SHAKE, an extension to SHAKE, an algorithm for the resolution of holonomic constraints in molecular dynamics simulations, which allows for the explicit treatment of angular constraints. We show that this treatment is more efficient than the use of fictitious bonds, significantly reducing the overlap between the individual constraints and thus accelerating convergence. The new algorithm is compared with SHAKE, M-SHAKE, the matrix-based approach described by Ciccotti and Ryckaert and P-SHAKE for rigid water and octane.

Mapping Peptide Hormone−Receptor Interactions Using a Disulfide-Trapping Approach

Efforts to elucidate the nature of the bimolecular interaction of parathyroid hormone (PTH) with its cognate receptor, the PTH receptor type 1 (PTHR1), have relied heavily on benzoylphenylalanine- (Bpa-) based photoaffinity cross-linking. However, given the flexibility, size, and shape of Bpa, the resolution at the PTH-PTHR1 interface appears to be reaching the limit of this technique. Here we employ a disulfide-trapping approach developed by others primarily for use in screening compound libraries to identify novel ligands. In this method, cysteine substitutions are introduced into a specific site within the ligand and a region in the receptor predicted to interact with each other. Upon ligand binding, if these cysteines are in close proximity, they form a disulfide bond. Since the geometry governing disulfide bond formation is more constrained than Bpa cross-linking, this novel approach can be employed to generate a more refined molecular model of the PTH-PTHR1 complex. Using a PTH analogue containing a cysteine at position 1, we probed 24 sites and identified 4 in PTHR1 to which cross-linking occurred. Importantly, previous photoaffinity cross-linking studies using a PTH analogue with Bpa at position 1 only identified a single interaction site. The new sites identified by the disulfide-trapping procedure were used as constraints in molecular dynamics simulations to generate an updated model of the PTH-PTHR1 complex. Mapping by disulfide trapping extends and complements photoaffinity cross-linking. It is applicable to other peptide-receptor interfaces and should yield insights about yet unknown sites of ligand-receptor interactions, allowing for generation of more refined models.

P-SHAKE: A quadratically convergent SHAKE in [formula omitted

An algorithm for solving arbitrary linear constraints in molecular dynamics simulations of rigid and semi-rigid molecules is presented. The algorithm – P-SHAKE – is a modified version of the SHAKE [J.-P. Ryckaert, G. Ciccotti, H.J.C. Berendsen, Numerical integration of the cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes, J. Comput. Phys. 23 (1977) 327–341.] algorithm with a preconditioner applied which effectively de-couples the constraint equations. It achieves quadratic convergence, as does M-SHAKE [V. Kräutler, W.F. van Gunsteren, P.H. Hünenberger, A fast SHAKE algorithm to solve distance constraint equations for small molecules in molecular dynamics simulations. J. Comput. Chem. 22 (5) (2001) 501–508.], yet at a cost of only O ( n 2 ) operations per iteration, as opposed to O ( n 3 ) per iteration for M-SHAKE. The algorithm is applied to simulations of rigid water, DMSO, chlorophorm and non-rigid ethane and cyclohexane and is shown to be faster than M-SHAKE by up to a factor of three for relatively small error tolerances.

An approximate but fast method to impose flexible distance constraints in molecular dynamics simulations

A fast but approximative method to apply flexible constraints to bond lengths in molecular dynamics simulations is presented and the effects of the approximation are investigated. The method is not energy conserving, but coupling to a temperature bath results in stable simulations. The high frequencies from bond-length vibrations are successfully removed from the system while maintaining the flexibility of the bonds. As a test liquid neopentane is simulated at different pressures. Energetic and dynamic properties are not affected by the new flexible constraint simulation method.