Hierarchical evolutive systems, fuzzy categories and the living single cell

TL;DR

This paper extends the theory of hierarchical evolutive systems by incorporating fuzzy categories to model a living single cell across molecular, cellular, and structural levels, using dynamical systems to analyze its behavior at the edge of chaos.

Contribution

It introduces a fuzzy categorical framework for hierarchical evolutive systems and applies it to model the dynamics of a living cell at the edge of chaos.

Findings

Cell modeled as a hierarchical fuzzy category with three levels.

Cell dynamics driven by a generalized Ricker map at the edge of chaos.

Toy model emphasizes cellular fission with fewer parameters.

Abstract

In this article, the theory of hierarchical evolutive systems of Ehresmann and Vandremeersch [Bull. Math. Bio. 49, 13-50 (1987)] is improved by considering the categories of the theory as fuzzy sets whose elements are the composite objects formed by the arrows and corresponding vertices of their embedded graphs. This way each category can be represented as a point in the states space [0.1]**N. The introduction of a diffeomorphism, that acts in this context as a functor between categories, allows to define a measure preserving dynamical system. In particular, we apply this formalism to describe a living single cell. We propose for its state at a given time a hirerchical category with three levels (molecular, coarse-grained and cellular levels) related by adequate colimits. Each level involves the main functional and structural modules in which the cell can be partitioned. The time evolution of the cell is drived by a transformation which is a N-dimensional generalization of the Ricker map whose parameters we propose to be determined by requiring that, as hallmark of its behavior, the living cell to evolve at the edge of chaos. From the dynamical point of view this property manifests in the fact that the largest Lyapunov exponent is equal to zero. Since in a rather complete model of the living cell the huge number of involved parameters can make of the calculations a hard task, we also propose a toy model, with fewer parameters to be determined, which emphasizes the cellular fission.