Henk Bekker

LINCS : A linear constraint solver for molecular simulations

In this article, we present a new LINear Constraint Solver (LINCS) for molecular simulations with bond constraints. The algorithm is inherently stable, as the constraints themselves are reset instead of derivatives of the constraints, thereby eliminating drift. Although the derivation of the algorithm is presented in terms of matrices, no matrix matrix multiplications are needed and only the nonzero matrix elements have to be stored, making the method useful for very large molecules. At the same accuracy, the LINCS algorithm is three to four times faster than the SHAKE algorithm. Parallelization of the algorithm is straightforward. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1463–1472, 1997

Depth-Dependent Halos: Illustrative Rendering of Dense Line Data

We present a technique for the illustrative rendering of 3D line data at interactive frame rates. We create depth-dependent halos around lines to emphasize tight line bundles while less structured lines are de-emphasized. Moreover, the depth-dependent halos combined with depth cueing via line width attenuation increase depth perception, extending techniques from sparse line rendering to the illustrative visualization of dense line data. We demonstrate how the technique can be used, in particular, for illustrating DTI fiber tracts but also show examples from gas and fluid flow simulations and mathematics as well as describe how the technique extends to point data. We report on an informal evaluation of the illustrative DTI fiber tract visualizations with domain experts in neurosurgery and tractography who commented positively about the results and suggested a number of directions for future work.

Illustrative molecular visualization with continuous abstraction

Molecular systems may be visualized with various degrees of structural abstraction, support of spatial perception, and ‘illustrativeness.’ In this work we propose and realize methods to create seamless transformations that allow us to affect and change each of these three parameters individually. The resulting transitions give viewers a dedicated control of abstraction in illustrative molecular visualization and, consequently, allow them to seamlessly explore the resulting abstraction space for obtaining a fundamental understanding of molecular systems. We show example visualizations created with our approach and report informal feedback on our technique from domain experts.

FORCE AND VIRIAL OF TORSIONAL-ANGLE-DEPENDENT POTENTIALS

Simple expressions for the forces due to dihedral‐angle interactions are derived using first principles of mechanics. The expressions require significantly fewer numerical operations than those generally used in the literature and provide insight into the physics of dihedral‐angle interactions. It is also shown that the scalar virial due to angle‐dependent interactions is zero.

An Efficient Algorithm to Calculate the Minkowski Sum of Convex 3D Polyhedra

A new method is presented to calculate the Minkowski sum of two convex polyhedra A and B in 3D. The method works as follows. The slope diagrams of A and B are considered as graphs. These graphs are given edge attributes. From these attributed graphs the attributed graph of the Minkowski sum is constructed. This graph is then transformed into the Minkowski sum of A and B. The running time of the algorithm is linear in the number of edges of the Minkowski sum.

Using bipartite and multidimensional matching to select the roots of a system of polynomial equations

Assume that the system of two polynomial equations f(x,y) = 0 and g(x,y) = 0 has a finite number of solutions. Then the solution consists of pairs of an x-value and an y-value. In some cases conventional methods to calculate these solutions give incorrect results and are complicated to implement due to possible degeneracies and multiple roots in intermediate results. We propose and test a two-step method to avoid these complications. First all x-roots and all y-roots are calculated independently. Taking the multiplicity of the roots into account, the number of x-roots equals the number of y-roots. In the second step the x-roots and y-roots are matched by constructing a weighted bipartite graph, where the x-roots and the y-roots are the nodes of the graph, and the errors are the weights. Of this graph the minimum weight perfect matching is computed. By using a multidimensional matching method this principle may be generalized to more than two equations.

A method to obtain a near-minimal-volume molecular simulation of a macromolecule, using periodic boundary conditions and rotational constraints

If the rotational motion of a single macromolecule is constrained during a molecular dynamics simulation with periodic boundary conditions it is possible to perform such simulations in a computational box with a minimal amount of solvent. In this article we describe a method to construct such a box, and test the approach on a number of macromolecules, randomly chosen from the protein databank. The essence of the method is that the molecule is first dilated with a layer of at least half the cut‐off radius. For the enlarged molecule a near‐densest lattice packing is calculated. From this packing the simulation box is derived. On average, the volume of the resulting box proves to be about 50% of the volume of standard boxes. In test simulations this yields on average a factor of about two in simulation speed.

Selecting the roots of a small system of polynomial equations by tolerance based matching

The roots of a system of two bivariate polynomial equations are calculated using a two-step method. First all x-roots and y-roots are determined independently. Then tolerance based weighted matching is used to form (x,y)-pairs that together form a minimum-error solution to the system.

Similarity measure computation of convex polyhedra revisited

We study the computation of rotation-invariant similarity measures of convex polyhedra, based on Minkowskis theory of mixed volumes. To compute the similarity measure, a (mixed) volume functional has to be minimized over a number of critical orientations of these polyhedra. These critical orientations are those relative configurations where faces and edges of the two polyhedra are as much as possible parallel. Two types of critical orientations exist for two polyhedra A and B. Type-1 critical orientations are those relative orientations where a face of B is parallel to a face of A, and an edge of B is parallel to a face of A, or vice versa. Type-2 critical orientations correspond to the case that three edges of A are parallel to three faces of B, or vice versa. It has been conjectured that to perform minimization of the volume functional, it is sufficient to consider Type-1 critical orientations only. Here we present experimental proof showing this conjecture to be false.

SQUEEZE-E: The Optimal Solution for Molecular Simulations with Periodic Boundary Conditions

In molecular simulations of macromolecules, it is desirable to limit the amount of solvent in the system to avoid spending computational resources on uninteresting solvent-solvent interactions. As a consequence, periodic boundary conditions are commonly used, with a simulation box chosen as small as possible, for a given minimal distance between images. Here, we describe how such a simulation cell can be set up for ensembles, taking into account a priori available or estimable information regarding conformational flexibility. Doing so ensures that any conformation present in the input ensemble will satisfy the distance criterion during the simulation. This helps avoid periodicity artifacts due to conformational changes. The method introduces three new approaches in computational geometry: (1) The first is the derivation of an optimal packing of ensembles, for which the mathematical framework is described. (2) A new method for approximating the α-hull and the contact body for single bodies and ensembles is presented, which is orders of magnitude faster than existing routines, allowing the calculation of packings of large ensembles and/or large bodies. 3. A routine is described for searching a combination of three vectors on a discretized contact body forming a reduced base for a lattice with minimal cell volume. The new algorithms reduce the time required to calculate packings of single bodies from minutes or hours to seconds. The use and efficacy of the method is demonstrated for ensembles obtained from NMR, MD simulations, and elastic network modeling. An implementation of the method has been made available online at http://haddock.chem.uu.nl/services/SQUEEZE/ and has been made available as an option for running simulations through the weNMR GRID MD server at http://haddock.science.uu.nl/enmr/services/GROMACS/main.php .