Conception of Biologic System: Basis Functional Elements and Metric Properties

TL;DR

This paper proposes a mathematical framework for biological systems using Lie algebra and group theory, highlighting the role of feedback and reciprocal links as fundamental regulatory elements with a Minkowski-like metric.

Contribution

It introduces a novel algebraic and geometric model of biological regulation based on Lie algebra sl(2,R) and Minkowski space, unifying feedback and reciprocal interactions.

Findings

Biological regulatory elements form a Lie algebra structure.

The space of biological variables has a Minkowski-like metric.

Regulatory elements create autonomous subsystems within biological systems.

Abstract

A notion of biologic system or just a system implies a functional wholeness of comprising system components. Positive and negative feedback are the examples of how the idea to unite anatomical elements in the whole functional structure was successfully used in practice to explain regulatory mechanisms in biology and medicine. There are numerous examples of functional and metabolic pathways which are not regulated by feedback loops and have a structure of reciprocal relationships. Expressed in the matrix form positive feedback, negative feedback, and reciprocal links represent three basis elements of a Lie algebra sl(2,R)of a special linear group SL(2,R). It is proposed that the mathematical group structure can be realized through the three regulatory elements playing a role of a functional basis of biologic systems. The structure of the basis elements endows the space of biological variables with indefinite metric. Metric structure resembles Minkowski's space-time (+, -, -) making the carrier spaces of biologic variables and the space of transformations inhomogeneous. It endows biologic systems with a rich functional structure, giving the regulatory elements special differentiating features to form steady autonomous subsystems reducible to one-dimensional components.