DOI: 10.5061/dryad.vt4b8gts7
title: Data from: Optimal allocation ratios: A square root relationship between the ratios of symbiotic costs and benefits

description: 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.
description: 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 (<i>g</i>) with multiple symbionts
 strains to the expectation using simple linear averaging (<i>g_bar</i><i>)</i> . The goodness-of-fit was evaluated by manually computing the coefficient of determination (c.o.d) using the following equation: c.o.d= 1-<i>SS<sub>res</sub></i>/<i>SS<sub>tot</sub></i>, where <i>SS<sub>tot</sub></i> is the total sum of squares and <i>SS<sub>res</sub></i> is the residual sum of squares using <i>g_bar</i> 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*(<i>g</i>-<i>g</i>_bar)/<i> </i><i>g</i> . 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 <i>B </i>(the ratio of growth benefits) and of <i>α</i> (the ratio of allocation) as follows: (1) <i>B </i>= <i>μ<sub>B</sub>/μ<sub>A</sub></i>, or the ratio of plant biomass when inoculated with the lesser vs greater growth promoting symbiont strain in isolation and (2) <i>α</i> = <i>C<sub>B</sub>/C<sub>A</sub></i>, where <i>C<sub>B</sub></i> and <i>C<sub>A</sub></i> 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, <i>B</i><i>,</i>  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 N<sub>2</sub> gas with ArO<sub>2</sub>. 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. <strong>Refs</strong> 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.  
description: 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).