Optimal allocation ratios: A square root relationship between the ratios of symbiotic costs and benefits

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Brian Steidinger, University of Konstanz

Background/Question/Methods
All organisms struggle to make sense of environmental stimuli in order to maximize their fitness. For animals, single cells and superorganisms responses 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 carbon (C) allocation ratios into symbionts that provide nutrients to their plant host.
Results/Conclusions
Consistent with Weber’s Law, our model demonstrates the optimal ratio of resources allocated into a less- relative to the more-beneficial symbiont scale to the ratio of the growth benefits of the two strains. As C allocation into symbionts increases, the ratio of C allocation into two strains approaches the square root of the ratio of symbiotic growth benefits (e.g., a worse symbiont providing ¼ the benefits gets √¼ =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- and less-beneficial symbionts, offering a potential mechanism behind the high diversity of microbial symbionts observed in nature.