Metabolic scaling in small life forms

Metabolic scaling is one of the most important patterns in biology. Theory explaining the 3/4-power size-scaling of biological metabolic rate does not predict the non-linear scaling observed for smaller life forms. Here we present a new model for cells $<10^{-8}$ m$^{3}$ that maximizes power from the reaction-displacement dynamics of enzyme-catalyzed reactions. Maximum metabolic rate is achieved through an allocation of cell volume to optimize a ratio of reaction velocity to molecular movement. Small cells $< 10^{-17}$ m$^{3}$ generate power under diffusion by diluting enzyme concentration as cell volume increases. Larger cells require bulk flow of cytoplasm generated by molecular motors. These outcomes predict curves with literature-reported parameters that match the observed scaling of metabolic rates for unicells, and predicts the volume at which Prokaryotes transition to Eukaryotes. We thus reveal multiple size-dependent physical constraints for microbes in a model that extends prior work to provide a parsimonious hypothesis for how metabolism scales across small life.