International Journal of Computational Geometry & Applications

Explicit solvent molecular dynamics simulations of a macromolecule are slow as the number of solvent atoms considered typically increases by order of magnitude. Implicit methods introduce surface-dependent corrections to the force field, gaining speed at the expense of accuracy. Properties such as molecular interface surfaces, volumes and cavities are captured by Laguerre tessellations of macromolecules. However, Laguerre cells of exterior atoms tend to be overly large or unbounded. Our method, the inclusion-exclusion based Laguerre-Intersection method, caps cells in a physically accurate manner by considering the intersection of the space-filling diagram with the Laguerre tessellation. We optimize an adjustable parameter, the weight, to ensure the areas and volumes of capped cells exposed to solvent are as close as possible, on average, to those computed from equilibrated explicit solvent simulations. The contact planes are radical planes, meaning that as the solvent weight is varied, interior cells remain constant. We test the consistency of our model using a high-quality trajectory of HIV-protease, a dimer with flexible loops and open-close transitions. We also compare our results with interval-arithmetic Gauss-Bonnet based method. Optimal solvent parameters quickly converge, which we use to illustrate the increased fidelity of the Laguerre-Intersection method over two recently proposed methods as compared to the explicit model.

Let $R$ and $B$ be two disjoint sets of red points and blue points, respectively, in the plane in general position. Assign a weight $\alpha$ to each red point and a weight $\beta$ to each blue point, where $\alpha$ and $\beta$ are positive integers. Define the weight of a region in the plane as the sum of the weights of red and blue points in it. We give necessary and sufficient conditions for the existence of a line that bisects the weight of the plane whenever the total weight $\alpha \left|R\right|+\beta \left|B\right|$ is $2\omega$, for some integer $\omega \ge 1$. Moreover, we look closely into the special case where $\alpha =2$ and $\beta =1$ since this case is important to generate a weight-equitable subdivision of the plane. Among other results, we show that for any configuration of $R\cup B$ with total weight $2\left|R\right|+\left|B\right|=n\omega$, for some integer $n\ge 2$ and odd integer $\omega \ge 1$, the plane can be subdivided into $n$ convex regions of weight $\omega$ if and only if $\left|B\right|\ge n$. Using the proofs of the main result, we also give a polynomial time algorithm in finding a weight-equitable subdivision in the plane.

This paper studies the influence of the definition of neighborhoods and methods used for creating point connectivity on topological analysis of scalar functions. It is assumed that a scalar function is known only at a finite set of points with associated function values. In order to utilize topological approaches to analyze the scalar-valued point set, it is necessary to choose point neighborhoods and, usually, point connectivity to meaningfully determine critical-point behavior for the point set. Two distances are used to measure the difference in topology when different point neighborhoods and means to define connectivity are used: (i) the bottleneck distance for persistence diagrams and (ii) the distance between merge trees. Usually, these distances define how different scalar functions are with respect to their topology. These measures, when properly adapted to point sets coupled with a definition of neighborhood and connectivity, make it possible to understand how topological characteristics depend on connectivity. Noise is another aspect considered. Five types of neighborhoods and connectivity are discussed: (i) the Delaunay triangulation; (ii) the relative neighborhood graph; (iii) the Gabriel graph; (iv) the $k$-nearest-neighbor (KNN) neighborhood; and (v) the Vietoris–Rips complex. It is discussed in detail how topological characterizations depend on the chosen connectivity.