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Several land biogeochemical models used for studying carbon–climate\udfeedbacks have begun explicitly representing microbial dynamics. However, to\udour knowledge, there has been no theoretical work on how to achieve a\udconsistent scaling of the complex biogeochemical reactions from microbial\udindividuals to populations, communities, and interactions with plants and\udmineral soils. We focus here on developing a mathematical formulation of the\udsubstrate–consumer relationships for consumer-mediated redox reactions of the\udform A + BE→  products, where products could be, e.g., microbial biomass or bioproducts. Under the quasi-steady-state\udapproximation, these substrate–consumer relationships can be formulated as\udthe computationally difficult full equilibrium chemistry problem or\udapproximated analytically with the dual Monod (DM) or synthesizing unit (SU)\udkinetics. We find that DM kinetics is scaling inconsistently for reaction\udnetworks because (1) substrate limitations are not considered,\ud(2) contradictory assumptions are made regarding the substrate processing\udrate when transitioning from single- to multi-substrate redox reactions, and\ud(3) the product generation rate cannot be scaled from one to multiple\udsubstrates. In contrast, SU kinetics consistently scales the product\udgeneration rate from one to multiple substrates but predicts unrealistic\udresults as consumer abundances reach large values with respect to their\udsubstrates. We attribute this deficit to SU's failure to incorporate\udsubstrate limitation in its derivation. To address these issues, we propose\udSUPECA (SU plus the equilibrium chemistry approximation – ECA) kinetics, which\udconsistently imposes substrate and consumer mass balance constraints. We show\udthat SUPECA kinetics satisfies the partition principle, i.e., scaling\udinvariance across a network of an arbitrary number of reactions (e.g., as in\udNewton's law of motion and Dalton's law of partial pressures). We tested\udSUPECA kinetics with the equilibrium chemistry solution for some simple\udproblems and found SUPECA outperformed SU kinetics. As an example\udapplication, we show that a steady-state SUPECA-based approach predicted an\udaerobic soil respiration moisture response function that agreed well with\udlaboratory observations. We conclude that, as an extension to SU and ECA\udkinetics, SUPECA provides a robust mathematical representation of complex\udsoil substrate–consumer interactions and can be applied to improve Earth system model (ESM) land\udmodels.</p

SUPECA kinetics for scaling redox reactions in networks of mixed substrates and consumers and an example application to aerobic soil respiration