Theoretical insights from upscaling Michaelis–Menten microbial dynamics in biogeochemical models: a dimensionless approach

Leading an effective response to the accelerating crisis of anthropogenic climate change will require improved understanding of global carbon cycling. A critical source of uncertainty in Earth system models (ESMs) is the role of microbes in mediating both the formation and decomposition of soil organic matter, and hence in determining patterns of CO2 efflux. Traditionally, ESMs model carbon turnover as a first-order process impacted primarily by abiotic factors, whereas contemporary biogeochemical models often explicitly represent the microbial biomass and enzyme pools as the active agents of decomposition. However, the combination of non-linear microbial kinetics and ecological heterogeneity across space and time guarantees that upscaled dynamics will violate mean-field assumptions via Jensen's inequality. Violations of mean-field assumptions mean that parameter estimates from models fit to upscaled data (e.g., eddy covariance towers) are likely systematically biased. Likewise, predictions of CO2 efflux from models conditioned on mean-field values will also be biased. Here we present a generic mathematical analysis of upscaling Michaelis–Menten kinetics under heterogeneity and provide solutions in dimensionless form. We illustrate how our dimensionless form facilitates qualitative insight into the significance of this scale transition and argue that it will facilitate cross-site intercomparisons of flux data. We also identify the critical terms that need to be constrained in order to unbias parameter estimates.