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Proteins are essential macromolecules defined by their amino acid sequences, which determine their three-dimensional structures and, consequently, their functions in all living organisms. Therefore, generative protein modeling necessitates…
Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our…
In this work we intend to report on some results obtained by an analytical modelling of biomacromolecular structures as driven by the study of Steiner points and Steiner trees with an Euclidean definition of distance.
In this paper, we discuss some problems of elementary plane differential geometry and kinematics. Although the results are not new, the consistent use of complex-valued functions (plane curves) of a real variable (parameter) allows to…
Given native 2D contact map, protein 3D structure could be reconstructed with accuracy of 2A or better, and such reconstruction is a feasible computational approach for protein folding problem. The prediction accuracy from traditional…
We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The…
Inferring the structural properties of a protein from its amino acid sequence is a challenging yet important problem in biology. Structures are not known for the vast majority of protein sequences, but structure is critical for…
Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical…
Subdivision methods such as quadtrees, octrees, and higher-dimensional orthrees are standard practice in different domains of computer science. We can use these methods to represent given geometries, such as curves, meshes, or surfaces.…
The remarkable structural diversity of modern proteins reflects millions of years of evolution, during which sequence space has expanded while many structural features remain conserved. This conservation is evident not only among homologous…
Proteins are miniature machines whose function depends on their three-dimensional (3D) structure. Determining this structure computationally remains an unsolved grand challenge. A major bottleneck involves selecting the most accurate…
Intrinsically disordered proteins and regions are increasingly appreciated for their abundance in the proteome and the many functional roles they play in the cell. In this short review, we describe a variety of approaches used to obtain…
The purpose of the present paper is to present the main applications of a new method for the determination of the fractal structure of plane curves. It is focused on the inverse problem, that is, given a curve in the plane, find its fractal…
Proteins are linear chain molecules that play a central role in life and health. Protein native state folds are modular assemblies of space-filling building blocks of {\alpha}-helices, \{beta}-sheets and tight turns. Here we deduce the…
Massive molecular simulations of drug-target proteins have been used as a tool to understand disease mechanism and develop therapeutics. This work focuses on learning a generative neural network on a structural ensemble of a drug-target…
Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Previous efforts for exact algorithms have been unable to avoid structural problems that appear for…
We show that cluster expansions (CE), previously used to model solid-state materials with binary or ternary configurational disorder, can be extended to the protein design problem. We present a generalized CE framework, in which properties…
In this paper we study the algebraic geometry of any two-point code on the Hermitian curve and reveal the purely geometric nature of their dual minimum distance. We describe the minimum-weight codewords of many of their dual codes through…
We propose a protein model based on a hierarchy of constraints that force the protein to follow certain pathways when changing conformation. The model exhibits a first order phase transition, cooperativity and is exactly solvable. It also…
Thinning is the removal of contour pixels/points of connected components in an image to produce their skeleton with retained connectivity and structural properties. The output requirements of a thinning procedure often vary with…