Optimal Allocation Ratios : A Square Root Relationship between the Ratios of Symbiotic Costs and Benefits

Abstract

All organisms struggle to make sense of environmental stimuli in order to maximize their fitness. For animals, the responses of single cells and superorganisms to stimuli are generally proportional to stimulus ratios, a phenomenon described by Weber’s law. However, Weber’s law has not yet been used to predict how plants respond to stimuli generated from their symbiotic partners. Here we develop a model for quantitatively predicting the ratios of carbon (C) allocation to symbionts that provide nutrients to their plant host. Consistent with Weber’s law, our model demonstrates that the optimal ratio of resources allocated to a less beneficial relative to a more beneficial symbiont scale to the ratio of the growth benefits of the two strains. As C allocation to symbionts increases, the ratio of C allocation to two strains approaches the square root of the ratio of symbiotic growth benefits (e.g., a worse symbiont providing one-fourth the benefits gets 1/4=1/2 the C of a better symbiont). We document a compelling correspondence between our square root model prediction and a meta-analysis of experimental literature on C allocation. This type of preferential allocation can promote coexistence between more beneficial and less beneficial symbionts, offering a potential mechanism behind the high diversity of microbial symbionts observed in nature.