A reactant-dependent algebraic framework to define, construct, characterize and assess the redundant forms of a biochemical network

Abstract

The intracellular milieu, replete with concomitantly occurring biochemical networks, constitutes a complex physicochemical environment wherein macromolecules interact and whose activities can be modified by small molecules. These metabolism-contributing, signal-transducing and response-regulating networks, and the redundant forms that result, thereof, are responsible for complex biochemical-, physiological- and clinical-outcomes. It is unclear whether a unique mathematical structure, which is concurrently resolvable into redundant, unequal and functionally relevant substructures, can be ascribed to a biochemical network. Here, we develop a reactant-dependent algebraic framework to define, construct, characterize and assess the redundant forms of a biochemical network. We deploy a constraint-based approach to populate and select plausible reaction vectors in accordance with known paradigms of biochemically relevant outcomes. There is a strict lower bound for the number of participating reactants with integral changes, across all molecules of a reactant, which are Boolean (zero, signed non-zero) and combinatorially distributed across linearly independent plausible reaction vectors. The reactants and the selected plausible reaction vectors form a set of unique stoichiometry number matrices, each of which represents a distinct biochemical network which when multiplied by a set of positive and non-unitary rational numbers will generate a library of equivalent rational number matrices. The matrices that comprise this library are unequal, disjoint, form a semigroup with respect to addition, share the same null space and represent the redundant forms of a biochemical network. The presented framework is theoretically sound, mathematically rigorous, readily implementable, easily parameterized and can be used to assess the biomedical relevance for the redundant forms of a biochemical network. We conclude our thesis with plausible explanations into the genesis of formaldehyde resistance and thence nosocomial infections by Klebsiella spp.