Abstract
In this paper we provide a review of selected mathematical ideas that can help us better understand the boundary between living and non-living systems. We focus on group theory and abstract algebra applied to molecular systems biology. Throughout this paper we briefly describe possible open problems. In connection with the genetic code we propose that it may be possible to use perturbation theory to explore the adjacent possibilities in the 64-dimensional space-time manifold of the evolving genome.With regards to algebraic graph theory, there are several minor open problems we discuss. In relation to network dynamics and groupoid formalism we suggest that the network graph might not be the main focus for understanding the phenotype but rather the phase space of the network dynamics. We show a simple case of a C6 network and its phase space network. We envision that the molecular network of a cell is actually a complex network of hypercycles and feedback circuits that could be better represented in a higher-dimensional space. We conjecture that targeting nodes in the molecular network that have key roles in the phase space, as revealed by analysis of the automorphism decomposition, might be a better way to drug discovery and treatment of cancer.
Highlights
In 1944 Erwin Schrödinger published a series of lectures in What is Life? [1]
The motivation of this paper is to examine an alternative set of mathematical abstractions applied to biology, and in particular systems biology
The Genetic Code we review some work describing the genetic code in groupoid and group theory terms
Summary
In 1944 Erwin Schrödinger published a series of lectures in What is Life? [1]. This small book was a major inspiration for a generation of physicists to enter microbiology and biochemistry, with the goal of attempting to define life by means of physics and chemistry. The curves on this manifold should map, in a complex way, to the symmetry breaking described below, or bifurcation, and give a second route to the differential geometry of Findley et al [18] Another approach to understanding the evolution of the genetic code is based on analogies with particle physics and symmetry breaking from higher-dimensional space. Hornos and Hornos [22] and Forger et al [23] use group theory to describe the evolution of the genetic code from a higher-dimensional space They propose a dynamical system algebra or Lie algebra [24]–the Lie algebra is a structure carried by the tangent space at the identity element of a Lie group.
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