Redox biology underpins signalling, metabolism, immunity, and adaptation, yet lacks a unifying theoretical framework capable of formalising structure, function, and dynamics. Current interpretations rely on descriptive catalogues of molecules and reactions, obscuring how redox behaviour emerges from constrained biochemical organisation. Here, we present a mathematical theory of redox biology that resolves this gap by treating redox systems as finite, compositional, dynamical, and spatially embedded objects. We define a structured redox state space in which admissible molecular transformations form a neutral algebra of possibilities. Biological function emerges when this structure is embedded within a wider molecular network and interpreted through weighted flux distributions. Time-dependent reweighting of these transformations generates redox dynamics, while spatial embedding enforces locality and causality, yielding a distributed redox field. Within this framework, context dependence, nonlinearity, hysteresis, and memory arise naturally from bounded state spaces and irreversible transformations, without requiring ad hoc assumptions. This theory provides a working, predictive interpretative basis for redox biology: it constrains admissible states and trajectories, clarifies the meaning of redox measurements, and links chemical transformation to biological behaviour. Redox biology emerges as a geometric, dynamical process governed by lawful organisation.
Redox biology is now widely recognised as essential for life, playing a central role in signalling, metabolism, immunity, adaptation, and the biological basis of health and disease across the lifespan [1][2][3][4][5][6][7][8][9][10]. Experimentally, the field has advanced to the point where detailed measurements of small molecules, protein residues, and pathway-level responses can be made across conditions, time, and space [11][12][13][14][15][16][17][18]. These measurements, however, probe not isolated molecules but local chemical environments-structured contexts in which redox reactions occur, interact, and propagate. Despite this experimental maturity, redox biology lacks a theoretical framework capable of formalising the structure, function, and dynamics of these environments [19]. As a result, data interpretation relies on descriptive inventories and narrative integration rather than on the principled unification of experiment with theory [20,21].
Objects of study in redox biology are typically organised through descriptive molecular catalogues, such as the grouping of small molecules into lists including “reactive species”, “reactive oxygen species”, or “reactive sulfur species”. While useful heuristics [22][23][24], these catalogues abstract molecules from the environments in which they act and do not expose the underlying structure of redox systems [25]. Similarly, grouping molecules by functional action-such as “oxidant”, “antioxidant”, or “reductant”-assigns context-dependent roles as if they were invariant properties. Even though the same molecular entity can act in multiple, and opposing, ways depending on its local environment and reaction partners [26]. These representations succeed at description but fail to define a coherent state space for the environment itself, obscuring how objects relate, compose, and transform.
Redox function is an emergent, context-dependent property of this environment. Functional properties are determined relationally by how local molecular transformations interact within a dynamic network, rather than being intrinsic and invariant labels (e.g., “good”). In this framing, redox biology is naturally understood as a spatially embedded, dynamic interaction network, in which biological function emerges from the configuration and evolution of redox transformations across space and time [27,28]. Function is therefore relative and not fixed, free to change as the structure and dynamics of the environment evolve. Despite this implicit network-dynamical and geometric nature, redox biology currently lacks a mathematical formalism capable of representing the environment as such an object. Without such a framework, structure, state, and dynamics remain implicit, and functional interpretation relies on narrative reasoning rather than derivation from a defined state space.
In this work, we formalise redox biology as a geometrically embedded dynamical network of local transformations. We first define a minimal compositional structure for small-molecule redox chemistry (the Core Redox Module, CRM) as a symmetric monoidal category of objects and morphisms. Here, “symmetric monoidal” simply means that redox reactions can be composed in parallel and that the order of independent reactions does not matter. We then embed the CRM into a wider cellular interaction graph that includes a Modifying Redox Module (MRM) and the redoxome, and define redox state as a functorial readout of weighted flux on this graph. Next, we extend this mapping to dynamics by evolving edge weights in time under a constrained flow law. Finally, we embed the resulting algebra into physical space as a field of local categories coupled by transport, yielding a redox field theory that unifies structure, function, and dynamics under locality and causality (see Box 1 for a plain-language theoretical guide).
The mathematics in this paper is not about solving equations for single reactions. Instead, it provides a way to describe what redox chemistry can do, how it combines, and how biological meaning emerges from its use.
The first step is structural. The theory treats key redox-active molecules as belonging to a finite set of physically allowed molecular states, and it treats every chemically possible transformation between those states-such as oxidation, reduction, proton transfer, or excitation-as an allowed process linking one state to another. This turns redox chemistry into a structured network of states and transformations, rather than an unorganised list of reactions.
Crucially, this network is compositional. Individual transformations can be chained into pathways, or they can occur independently and at the same time. The mathematics used to formalise this comes from category theory, which is a precise language for describing how processes connect, combine, and remain independent when they should. This does not assume linearity or equilibrium; it simply encodes which transformations are physically admissible
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