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Linear Algebra in Biomolecular Modeling
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
A fundamental problem in protein modeling is to find the three-dimensional structure of a protein and its relationship with the protein’s biological function. One of the experimental techniques for structure determination is to use the nuclear magnetic resonance (NMR) to obtain some information on the distances for certain pairs of atoms in the protein and then find the coordinates of the atoms based on the obtained distance information. Mathematically, the second part of the work requires the solution of a so-called distance geometry problem, i.e., determine the coordinates for a set of points in a given topological space, given the distances for a subset of all pairs of points. We consider such a problem with the distances for all pairs of points assumed to be given.
A review on bio-functional models of catechol oxidase probed by less explored first row transition metals
Published in Journal of Coordination Chemistry, 2022
Rashmi Rekha Tripathy, Shuvendu Singha, Sohini Sarkar
The catalytic efficiency and turnover numbers obtained for different complexes depend on structural parameters like M…M distance, geometry around the metal, steric congestion provided by the ligands, etc. Ideal Zn…Zn separation should vary in the range 3.0–3.2 Å which is in correlation with Cu…Cu separation (2.9–3.2 Å) [13]. Complexes with intermetallic separation more than 3.4 Å can show catalytic activity. The central metal ion can adopt various kinds of geometries like square pyramidal [145], trigonal bipyramidal [151], tetrahedral [142], or elongated octahedral [146]. Square pyramidal geometry with one vacant coordination site may provide an appropriate environment for substrate binding and thus leads to high values of rate constants (3.52 × 103–1.0 × 104 h−1) [145]. Tetrahedral complexes can also act as efficient catalysts as shown in the work of Hens and Mandal et al. [152,153]. A trinuclear Schiff base complex, reported by Ghosh et al., has a high turnover number (30.40 × 103 h−1) [154]. In this case both the terminal metal ions were present in square pyramidal environment, while the central one was octahedral. However, coordination environment around the centers should not have any influence on the catecholase activity since upon treatment with DTBC each underwent decomposition to a four-coordinate mononuclear fragment for binding the catechol moiety.
A dynamic geometry system approach to analyse distance geometry problems based on partial Latin squares
Published in International Journal of Mathematical Education in Science and Technology, 2020
In order to endow partial Latin squares with the mentioned dynamism, the Distance Geometry Problem (DGP) [19, 20] arises as an interesting approach. This problem consists of determining the existence of a set of points within a given k-dimensional space so that the distance matrix among all these points holds certain constraints. Due to the NP-hard computational complexity of the DGP, Saxe [21] suggested to put emphasis on highly constrained distance matrices. In this regard, the requirement of non-repetition of symbols within any row or column of a partial Latin square establishes a series of geometric constraints that make any of these combinatorial structures to constitute an interesting distance matrix to be considered [22, 23].
Discrete-time interval optimal control problem
Published in International Journal of Control, 2019
J. R. Campos, E. Assunção, G. N. Silva, W. A. Lodwick, M. C. M. Teixeira
The interval arithmetic we will use to solve the DTIOCP is called single-level constraint interval arithmetic (SLCIA) and was proposed recently by Chalco-Cano, Lodwick, and Bede (2014). The SLCIA always consider the same level for all intervals involved in the operations as we shall see. Costa, Bouwmeester, Lodwick, and Lavor (2017) used the SLCIA to calculate the molecular distance geometry. Furthermore, it can also be used to study the practical problems presented by Assunção, Teixeira, Faria, Da Silva, and Cardim (2007) and Buzachero, Assunção, Teixeira, and Silva (2015) and, we will show that in our case, SLCIA reduces the complexity of the DTIOCP rendering it computationally tractable.