Skip to main content
Dryad logo

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


Steidinger, Brian (2021), Data from: Optimal allocation ratios: A square root relationship between the ratios of symbiotic costs and benefits, Dryad, Dataset,


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. 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 sqrt(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.


We conducted two related meta-analysis in order to (i) validate a key simplifying assumption in the model and (ii) test whether empirical data fits our model predictions. First, we evaluated the model assumption that the growth benefits with multiple symbiont strains are equal to the linear average of growth with each of the constituent single-symbionts.

To test whether growth benefits with multiple symbiont strains are equal to the linear average of growth with each of the constituent single-symbionts, we evaluated papers from the MycoDataBase, which was published to evaluate the context-dependency of plant response to inoculation with mycorrhizal fungi (Hoeksema et al. 2010). We subset a supplemental file of all studies included in MycoDataBase so that it included only those studies that fit the following two criteria: (1) they quantified the growth benefit to the plant of associating with multiple symbionts in pure culture; (2) they quantified the growth benefit to the plant when inoculated with multiple-strains. Because only a small number of studies in MycoDataBase matched our criterion for meta-analysis, we expanded our search using Google Scholar and the terms “plant growth” and “multiple-inoculum.” We recorded for each study evaluated why it was or was not included in the meta-analysis – a frequent reason for non-inclusion was that multiple-strain inoculation treatments included some strains that were not provided to the plant as a single-inoculum treatment.

For the first metaanalysis, we compared the actual growth (g) with multiple symbionts strains to the expectation using simple linear averaging (g_bar) . The goodness-of-fit was evaluated by manually computing the coefficient of determination (c.o.d) using the following equation: c.o.d= 1-SSres/SStot, where SStot is the total sum of squares and SSres is the residual sum of squares using g_bar as the model prediction. To assess the % error of model, we calculated the mean of the residual growth over the predicted growth using the following equation: %error=100*(g-g_bar)/ g . Although the meta-analysis results were plotted a on a log scale, both c.o.d and %error were calculated on untransformed data.

For the second meta-analysis, we evaluated studies that measured resource allocation into different strains that colonized a single root system. For these studies we quantified both the value of (the ratio of growth benefits) and of α (the ratio of allocation) as follows: (1) = μBA, or the ratio of plant biomass when inoculated with the lesser vs greater growth promoting symbiont strain in isolation and (2) α = CB/CA, where CB and CA are the carbon allocation into the lesser vs greater growth promoting symbiont strain.

Because MycoDataBase does not include resource allocation to symbionts, we searched for studies that fit these criteria using Google Scholar and the following terms: “carbon allocation,” “partner choice,” “split root,” and “symbiosis.” Studies included in our meta-analysis had to fit the following criteria:  (1) they quantified the growth benefit to the plant of associating of each of the multiple symbionts in pure culture (same as criterion one from the first analysis); and (2) they presented data on resource allocation to symbionts in a way that made it possible to compute the ratio of allocation into the different strains.

We made an exception of these two inclusion criteria, including in our meta-analysis one study from the legume / N-fixing rhizobium mutualism that quantified the benefit ratios, B,  in terms of the potential % N-fixing capacity (Kiers et al. 2006). The reduction in potential N-fixation was accomplished by substituting a percentage of N2 gas with ArO2. In this case, rather than measure C allocation directly, the authors measured the fitness of rhizobium as the number of bacteria that emerged from leguminous nodules. Although this study uses a different methodology as the others in our meta-analysis, it was able to stratify its treatments along a gradient of symbiotic benefit ratio in a manner ideal for comparison with our model. Additionally, we omitted one outlier where the plant allocated more C to the less beneficial symbiont (which occurred under a high phosphorus fertilizer treatment, where overall C allocation was low and small differences in allocation between strains lead to highly skewed allocation ratios  (Ji and Bever 2016)

For both meta-analyses, we either transcribed data from tables directly into a .csv file or used WebPlotDigitizer to estimate mean or median responses from published figures (point, bar, or box-and-whisker plots). For some studies, plant biomass was measured in roots, stems, and leaves separately, whereas in others only stem or overall biomass were recorded. We used either the whole biomass or only stem-biomass, depending on data-availability.


Hoeksema, J. D., V. B. Chaudhary, C. A. Gehring, N. C. Johnson, J. Karst, R. T. Koide, A. Pringle, et al. 2010. A meta-analysis of context-dependency in plant response to inoculation with mycorrhizal fungi. Ecology Letters 13:394–407.

Ji, B., and J. D. Bever. 2016. Plant preferential allocation and fungal reward decline with soil phosphorus: implications for mycorrhizal mutualism. Ecosphere 7:e01256.

Kiers, E. T., * Robert A. Rousseau, and R. F. Denison. 2006. Measured sanctions: legume hosts detect quantitative variation in rhizobium cooperation and punish accordingly. Evolutionary Ecology Research 8:1077–1086.


Usage Notes

The data is provided in .csv files and the code required to complete the analysis is in the annotated .Rmd files (Rstudio notebook). The .Rmd files require the user to specify a home directory (location where the unzipped files are stored).