Towards mathematical spaces for biological processes

TL;DR

This paper introduces a novel mathematical space tailored for biological processes, capturing context dependence, partial observability, and irreversibility, to enable structured, quantitative analysis similar to physical theories.

Contribution

It develops a unified mathematical framework using locally convex spaces indexed by context, incorporating biological constraints and partial observability, with explicit constructions and a practical cancer example.

Findings

Mapped single-cell data into the new framework

Calibrated thresholds from literature for predictions

Formulated testable predictions on cancer cell states

Abstract

Physics relies on mathematical spaces carefully matched to the phenomena under study. Phase space in classical mechanics, Hilbert space in quantum theory, configuration spaces in field theory all provide representations in which physical laws, stability and invariants become expressible and testable. In contrast, biology lacks an agreed-upon notion of space capturing context dependence, partial observability, degeneracy and irreversible dynamics. To address this gap, we introduce a unified mathematical space tailored to biological processes where states are represented in locally convex spaces indexed by context, where context includes both environment and history. Within our setting, proximity is defined through families of seminorms rather than a single global metric, allowing biological relevance to vary across conditions. Admissible sets encode biological constraints, observation maps formalize partial observability and many-to-one relations between state and dynamics capture irreversibility without requiring convergence to fixed points. Stabilization is characterized by neighborhood inclusion and degeneracy arises naturally through quotient structures induced by observation. We develop explicit constructions, operators and bounds within this space, yielding quantitative predictions dictated by its structure. A worked example based on EGFR-mutant non-small-cell lung cancer shows how single-cell data can be mapped into our framework, how numerical thresholds can be calibrated from the literature and how testable predictions can be formulated concerning rare tolerant states, context-dependent proximity and early stabilization. Overall, by providing biology with a space playing a role analogous to those used in physics, we aim to support structurally grounded and quantitative analyses of biological systems across contexts.