Estratégia Algorítmica para a Reconstrução e Validação da Estrutura Molecular de Variantes do SARS-CoV-2

Aspectos matemático-computacionais e físico-químicos envolvidos na reconstrução da estrutura molecular tridimensional de proteínas do SARS-CoV-2 são abordados neste artigo, envolvendo a variante P.1 detectada em pacientes infectados em solo brasileiro, principalmente as do estado do Amazonas. Foi realizado um estudo sobre o impacto teórico da P.1 por intermédio da reconstrução estrutural de proteínas onde a mutagênese foi realizada computacionalmente e com o auxílio da implementação de um algoritmo de enumeração implícita de factibilidade, Branch-and-Prune, cujas soluções foram validadas através do uso do gráfico de Ramachandran. Desta forma, mesmo com a ausência de estruturas cristalográficas caracterizando estas mutações, pôde-se computacionalmente modelar uma estrutura tridimensional onde ao fim realizou-se o alinhamento estrutural com a cristalografia do complexo ACE2-RBD

Analysis of reachable workspace for spatial 3-cable under-constrained suspended cable driven parallel robots

Workspace is an important reference for design of cable-driven parallel robots (CDPRs). Most current researches focus on calculating the workspace of redundant CDPRs. However, few literatures study the workspace of under-constrained CDPRs. In this paper, the static equilibrium reachable workspace (SERW) of spatial 3-cable under-constrained CDPRs is solved numerically since expressions describing workspace boundaries cannot be obtained in closed form. The analysis steps to solve the SERW are as follows. First, expressions which describe the SERW and its boundaries are proposed. Next, these expressions are instantiated through the novel anchor points model composed of linear equations, quadratic equations and limits of tension in cables. Then, based on the reformulated linearization technique (RLT), the constraint system is transformed into a system containing only linear equality constraints and linear inequality constraints. Finally, the framework of branch-and-prune (BP) algorithm is adopted to solve this system. The effect of the algorithm is verified by 2 examples. One is a special 3-cable CDPR in which the anchor points layouts both on the moving platform (MP) and on the base are equilateral triangles, followed by the method to extract the SERW boundary where cables do not interfere with each other. The other is a general case with randomly selected geometry arrangement. The presented method in this paper is universal for spatial 3-cable CDPRs with arbitrary geometry parameters.

NMR Protein Structure Calculation and Sphere Intersections

Nuclear Magnetic Resonance (NMR) experiments can be used to calculate 3D protein structures and geometric properties of protein molecules allow us to solve the problem iteratively using a combinatorial method, called Branch-and-Prune (BP). The main step of BP algorithm is to intersect three spheres centered at the positions for atoms i − 3, i − 2, i − 1, with radii given by the atomic distances di−3,i, di−2,i, di−1,i, respectively, to obtain the position for atom i. Because of uncertainty in NMR data, some of the distances di−3,i should be represented as interval distances [{\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}}], where {\underline{d}_{i - 3,i}} \le {d_{i - 3,i}} \le {\bar d_{i - 3,i}}. In the literature, an extension of the BP algorithm was proposed to deal with interval distances, where the idea is to sample values from [{\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}}]. We present a new method, based on conformal geometric algebra, to reduce the size of [{\underline{d}_{i - 3,i}},{\bar d_{i - 3,i}}], before the sampling process. We also compare it with another approach proposed in the literature.

Systematic exploration of protein conformational space using a Distance Geometry approach

The optimisation approaches classically used during the determination of protein structure encounter various diffculties, specially when the size of the conformational space is large. Indeed, in such case, algorithmic convergence criteria are more difficult to set up. Moreover, the size of the search space makes it difficult to achieve a complete exploration. The interval Branch-and-Prune (iBP) approach, based on the reformulating of the Distance Geometry Problem (DGP) provides a theoretical frame for the generation of protein conformations, by systematically sampling the conformational space. When an appropriate subset of inter-atomic distances is known exactly, this worst-case exponential-time algorithm is provably complete and fixed-parameter tractable. These guarantees, however, immediately disappear as distance measurement errors are introduced. Here we propose an improvement of this approach: the threading-augmented interval Branch-and-Prune (TAiBP), where the combinatorial explosion of the original iBP approach arising from its exponential complexity is alleviated by partitioning the input instances into consecutive peptide fragments and by using Self-Organizing Maps (SOMs) to obtain clusters of similar solutions. A validation of the TAiBP approach is presented here on a set of proteins of various sizes and structures. The calculation inputs are: a uniform covalent geometry extracted from force field covalent terms, the backbone dihedral angles with error intervals, and a few long-range distances. For most of the proteins smaller than 50 residues and interval widths of 20°, the TAiBP approach yielded solutions with RMSD values smaller than 3 Å with respect to the initial protein conformation. The efficiency of TAiBP approach for proteins larger than 50 residues will require the use of non-uniform covalent geometry, and may have benefits from the recent development of residue-specific force-fields.