A tropical geometry for bounded biochemical state spaces

TY - JOUR

T1 - A tropical geometry for bounded biochemical state spaces

AU - Cobley, James N.

N1 - © 2026 The Author.

PY - 2026/7/9

Y1 - 2026/7/9

N2 - Many biochemical measurements define state spaces that are bounded, absorbing, and generically noninvertible, yet are routinely analysed using categorically mismatched linear and Euclidean frameworks. This categorical mismatch can irretrievably obscure biological structure. This work formalises the minimal algebraic properties imposed by bounded biochemical state spaces, demonstrating that a bounded tropical semiring provides a natural category match by replacing additive linear structure with order-based, piecewise-linear operations that encode dominance, saturation, and path dependence without contradiction. To instantiate this framework, measurement-native tropical geometric analysis was applied to cysteine redox proteomic data (Oxi-Shapes). Oxi-Shapes represents cysteine oxidation occupancies as a bounded scalar field over a discrete lattice of invariant residue identities, enabling analysis of both ensemble-level organisation and site-wise admissible transitions. As demonstrated on empirical cysteine redox proteomic data, categorically correct algebraic structures exposed geometric features that remained inaccessible under unbounded linear representations.

AB - Many biochemical measurements define state spaces that are bounded, absorbing, and generically noninvertible, yet are routinely analysed using categorically mismatched linear and Euclidean frameworks. This categorical mismatch can irretrievably obscure biological structure. This work formalises the minimal algebraic properties imposed by bounded biochemical state spaces, demonstrating that a bounded tropical semiring provides a natural category match by replacing additive linear structure with order-based, piecewise-linear operations that encode dominance, saturation, and path dependence without contradiction. To instantiate this framework, measurement-native tropical geometric analysis was applied to cysteine redox proteomic data (Oxi-Shapes). Oxi-Shapes represents cysteine oxidation occupancies as a bounded scalar field over a discrete lattice of invariant residue identities, enabling analysis of both ensemble-level organisation and site-wise admissible transitions. As demonstrated on empirical cysteine redox proteomic data, categorically correct algebraic structures exposed geometric features that remained inaccessible under unbounded linear representations.

KW - tropical

KW - entropy

KW - data

KW - proteomics

KW - biochemical

KW - cysteine

KW - geometry

KW - oxi-shapes

KW - order

KW - redox

UR - https://www.scopus.com/pages/publications/105044322301

U2 - 10.1016/j.patter.2026.101613

DO - 10.1016/j.patter.2026.101613

M3 - Article

SN - 2666-3899

JO - Patterns

JF - Patterns

M1 - 101613

ER -