Structural regularities of helicoidally-like biopolymers in the framework of algebraic topology: I. Special class of sta
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Structural Regularities of HelicoidallyLike Biopolymers in the Framework of Algebraic Topology: I. Special Class of Stable Linear Structures Defined by the Sequence of Algebraic Polytopes M. I. Samoylovicha and A. L. Talisb a
OAO Central Research Technological Institute Technomash, ul. Ivana Franko 4, Moscow, 121108 Russia email: [email protected] b Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28, Moscow, 119991 Russia Received October 30, 2012
Abstract—It is shown that the subsystems of the second coordination sphere of 8dimensional lattice E8 define the closed sequence of 4dimensional polyhedra (polytopes). A chain of constructions of algebraic topology, which makes it possible to select a class of discrete helicoidallylike structures that are topologically stable in 3dimensional Euclidean space, is mapped to the structural level. A structure belonging to this class is limited by a minimal surface, the singular points of which are related by transformations determined by polytope symmetry. It will be shown in the second part of the paper that the developed apparatus of “struc tural application of algebraic geometry” makes it possible to determine a priori structures setting the symme try parameters of biopolymers, in particular, α helices and some forms of DNA structures. DOI: 10.1134/S1063774513040160
INTRODUCTION Living nature is sheltered from the crystalline type of ordering in a 3dimensional Euclidean space of Е3 by passing from the invariance of a relatively infinite translational lattice to a local periodicity determined by invariance with respect to constructions of alge braic topology. The need for these constructions is determined by the fact that Е3 forms a Lie algebra with respect to vector multiplication. These constructions include actually used locally periodic groups, all sub groups of which are finite [1–3]. The need for these groups is caused by the requirement for a partition into polyhedra of the first and second coordination spheres, which is in agreement with the physically implemented interaction between neighboring atoms. The 3dimensional structure under consideration can be limited by a surface, the automorphisms of which allow one to select certain classes of 3dimen sional varieties. A surface with zero mean curvature is referred to as minimal (locally minimal). Below we will restrict ourselves to the consideration of topologi cally stable systems having a minimal surface, as deter mined by the tubular neighborhood theorem [1]. Sin gular points of this surface can be related by the sym metry transformations of the “algebraic” polytope [4– 9], which is determined by the corresponding sub system of the 8dimensional lattice of octaves Е8, which closes the series of possible numbers: real num bers–complex numbers–quaternions–octaves. Cell complexes in Е3, which are constructed using mixed
Abelian groups, can also be limited by a minimal sur face. The finiteorder elements of these groups are
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