A tropical geometry for bounded biochemical state spaces

TL;DR

This paper introduces a tropical algebra framework to accurately model bounded, absorbing, and irreversible biochemical state spaces, addressing limitations of traditional linear methods and revealing underlying biological structures.

Contribution

It formalizes the algebraic structure of bounded biochemical spaces using tropical algebra, providing a principled foundation for their analysis.

Findings

Tropical algebra naturally models non-invertible, absorbing biochemical states.

Linear algebra is incompatible with bounded, irreversible biochemical data.

The framework clarifies biological structure obscured by traditional methods.

Abstract

Many biochemical measurements define state spaces that are bounded, absorbing, and physically irreversible, yet are routinely analysed using linear and Euclidean frameworks that assume global invertibility, symmetry, and translation invariance. This mismatch can irretrievably obscure biological structure, independent of data quality, scale, or preprocessing. This work formalises the structure of bounded biochemical state spaces using cysteine redox regulation as a representative example and identify the minimal algebraic properties required for categorically correct representation. Hard boundaries, absorbing states, and irreversible ensemble dynamics render linear algebra incompatible with these objects. This work demonstrates that tropical algebra provides a natural realisation of the required properties by replacing additive linear structure with order-based, piecewise-linear operations that encode dominance, saturation, and path dependence without contradiction. By making non-invertibility and absorption explicit rather than implicit, this framework resolves a fundamental algebraic mismatch and establishes a principled foundation for the representation and analysis of bounded biochemical data.