  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.
