Molecular Skin Surface-Based Transformation Visualization between Biological Macromolecules

Molecular skin surface (MSS), proposed by Edelsbrunner, is a C2 continuous smooth surface modeling approach of biological macromolecules. Compared to the traditional methods of molecular surface representations (e.g., the solvent exclusive surface), MSS has distinctive advantages including having no self-intersection and being decomposable and transformable. For further promoting MSS to the field of bioinformatics, transformation between different MSS representations mimicking the macromolecular dynamics is demanded. The transformation process helps biologists understand the macromolecular dynamics processes visually in the atomic level, which is important in studying the protein structures and binding sites for optimizing drug design. However, modeling the transformation between different MSSs suffers from high computational cost while the traditional approaches reconstruct every intermediate MSS from respective intermediate union of balls. In this study, we propose a novel computational framework named general MSS transformation framework (GMSSTF) between two MSSs without the assistance of union of balls. To evaluate the effectiveness of GMSSTF, we applied it on a popular public database PDB (Protein Data Bank) and compared the existing MSS algorithms with and without GMSSTF. The simulation results show that the proposed GMSSTF effectively improves the computational efficiency and is potentially useful for macromolecular dynamic simulations.

Estimating Multidimensional Persistent Homology Through a Finite Sampling

An exact computation of the persistent Betti numbers of a submanifold [Formula: see text] of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of [Formula: see text] is available. We show that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of [Formula: see text] from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain. Using these inequalities we improve a previous lower bound for the natural pseudodistance to assess dissimilarity between the shapes of two objects from a sampling of them. Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it.

APPROXIMATING GREEN'S FUNCTIONS ON ℙ1 POSITIVE CHARACTERISTIC

Fix a finite K-symmetric set [Formula: see text] and a K-symmetric probability vector [Formula: see text]. Let 𝔇v be a finite union of balls [Formula: see text] for some ah ∈ Kv and some [Formula: see text], where the balls 𝔅(ah, rh) are disjoint from 𝔛. Put 𝔈v := 𝔇v ∩ ℙ1(Kv). Then there exists a positive integer Nv such that for each sufficiently large integer N divisible by Nv, there are a number Rv, with [Formula: see text], and an [Formula: see text]-function fv(z) ∈ Kv(z) of degree N whose zeros form a "well-distributed" sequence in 𝔈v such that [Formula: see text] is a disjoint union of balls centered at the zeros of fv(z) and for all z ∉ 𝔇v, [Formula: see text]