Applications of Wedderburn's Theorem in Modelling Non-Commutative Biological and Evolutionary Systems

Abstract

A wide range of biological and evolutionary processes is determined not merely by the occurrence of specific events, but by the exact order in which those events unfold. Gene regulation, developmental pathways, metabolic cascades and genetic evolution often display non-commutative behaviour, in which reversing the sequence of events results in functionally distinct outcomes. Conventional modelling approaches often fail to account for such directionality and sequence dependence, thereby limiting their capacity to capture the complexity of regulatory logic. We represent ordered sequences of biological operations--such as transcription factor binding or mutational trajectories--as elements of a non-commutative algebra designed to encode the functional logic of systems where the order of events is critically determinant. Subsequently, we apply Wedderburns Theorem to decompose the algebra into a direct sum of matrix modules, each capturing an irreducible and functionally distinct component of the underlying sequence-dependent system. We provide examples from gene expression regulation and evolutionary dynamics, focusing on scenarios where trait development is determined by the specific order of underlying molecular or mutational events. Our results demonstrate that the algebraic framework effectively maps intricate biological processes onto smaller, linear components, facilitating clearer interpretation and analysis. Simplifying non-commutative biological systems into interpretable submodules, Wedderburn-style decomposition may clarify gene regulatory logic, capture behavioural outcomes, reduce computational burden and uncover pathway redundancies and structural symmetries. Overall, by unifying diverse biological processes within a coherent algebraic structure, our method may improve the tractability of complex, order-dependent systems.