Application of geometric algebra for the description of polymer conformations

Abstract

In this paper a Clifford algebra-based method is applied to calculate polymer chain conformations. The approach enables the calculation of the position of an atom in space with the knowledge of the bond length (l), valence angle (theta), and rotation angle (phi) of each of the preceding bonds in the chain. Hence, the set of geometrical parameters {l(i),theta(i),phi(i)} yields all the position coordinates p(i) of the main chain atoms. Moreover, the method allows the calculation of side chain conformations and the computation of rotations of chain segments. With these features it is, in principle, possible to generate conformations of any type of chemical structure. This method is proposed as an alternative for the classical approach by matrix algebra. It is more straightforward and its final symbolic representation considerably simpler than that of matrix algebra. Approaches for realistic modeling by means of incorporation of energetic considerations can be combined with it. This article, however, is entirely focused at showing the suitable mathematical framework on which further developments and applications can be built.

Polymer conformations in internal (polyspherical) coordinates

“…The computational cost of the present approach is summarized in Table 1, together with the estimates of other methods found in the literature. These methods require a similar amount of operations, or far more, depending on which source one uses 4, 7, 16, 17. In the table, we present the cheapest versions.…”

“…Because it is easy to constrain, or not to constrain, bond lengths bond angles, and torsion angles in the present approach, it is more versatile than most other approaches 4–7. For example, the rotation matrix method parametrizes conformations only in the torsion space, and offers no way to take into account the changes in bond lengths and bond angles.…”

High‐performance transformation of protein structure representation from internal to Cartesian coordinates

“…Abagyan and Mazur have conducted the most extensive development of IC, [12][13][14][15][16][17][18] beginning with a flexible and general approach to IC dynamics [12,13] and continuing through recent work on a molecular force field optimized for use in IC. [18] Recently, Chys et al have advanced the use of spinors and geometric algebra as a formalism for converting between Cartesian and IC [19][20][21] (an analysis of different approaches may be found here [22] ). Conducting molecular simulations (whether MD or MC) purely in IC raises a variety of algorithmic and theoretical challenges, thermostatting algorithms, [23] time integrators, [24,23] the equipartition principle, [25] the imposition of constraints, and the computation of entropies.…”

“…Conducting molecular simulations (whether MD or MC) purely in IC raises a variety of algorithmic and theoretical challenges, thermostatting algorithms, [23] time integrators, [24,23] the equipartition principle, [25] the imposition of constraints, and the computation of entropies. [8] Even computing transformations between coordinate representations presents some challenges, motivating the development of multiple approaches [3,12,19,[26][27][28][29][30] that are often tailored for specific applications, such as normal mode analysis. [3] The present work is not intended to advance capabilities for MD simulations that use IC; the primary aim of such methods is, of course, to avoid the coordinate transformations of interest here.…”

Spinor product computations for protein conformations

“…Efficient and fast algorithms for generating and updating conformations in Cartesian from IC have been studied fairly intensive in the past. [9–15] Most coordinate methods use matrices, quaternions, or hybrid combinations of them. [10–13] Recently, two other approaches have as well been proposed.…”