Philippa Kaur

To derive the relationship among macro-level ecological variables, which would constitute an ecological analog of the thermodynamic equation of state, we introduce a fourth state variable, B, the total biomass in the community. The ecological analog of the thermodynamic equation of state, an expression for biomass, B, in terms of S, N, and E, arises if we combine METE with a scaling result from the metabolic theory of ecology (MTE)18,21. In particular, we assume the MTE scaling relationship between the metabolic rate, (varepsilon ,) of an individual organism and its mass, m: (varepsilon sim {m}^{3/4}). Without loss of generality22, units are normalized such that the smallest mass and the smallest metabolic rate within a censused plot are each assigned a value of 1. With this units convention, the proportionality constant in this scaling relationship can be assigned a value of 1. From the definition of the structure-function, it follows23 that averaging the biomass of individuals times the abundance of species, nε4/3, over the distribution R and multiplying by the number of species gives the total ecosystem biomass as a function of S, N, and E. Explicitly:$$B=Smathop{sum}limits_{n}nint dvarepsilon ,{varepsilon }^{4/3}R(n,varepsilon {{{{{rm{|}}}}}}S,N,E)$$
(1)
Both the sum and integral in the above equation can be calculated numerically, and Python code to do so for a given set of state variables S, N, and E, is available at github.com/micbru/equation of_ state/.We can also approximate the solution to Eq. 1 analytically (Supplementary Note 2) to reveal the predicted functional relationship among the four state variables. If E > > N > > S > > 1:$$B=cfrac{{E}^{4/3}}{{S}^{1/3}{{{{{rm{ln}}}}}}(1/beta )}$$
(2)
where (capprox (7/2)Gamma (7/3)) ≈ 4.17 and (beta) = ({lambda }_{1}+{lambda }_{2}) is estimated13,22 from the relationship (beta {{{{{rm{ln}}}}}}(1/beta )approx S/N). Equation 2 approximates the numerical result to within 10% for 5 of the 42 datasets analyzed here, corresponding to N/S greater than ~100 and E/N greater than ~25. Multiplying the right-hand side of Eq. 2 by (1-1.16{beta }^{1/3}) approximates the numerical result to within 10% for 33 of the 42 datasets analyzed here, corresponding to N/S greater than ~3 and E/N greater than ~5. The inequality requirements are not necessary for the numerical solution of Eq. 1, which is what is used below to test the prediction.Empirical values of E and B can be estimated from the same data. In particular, if measured metabolic rates of the individuals are denoted by ({varepsilon }_{i},) where i runs from 1 to N, then E is given by the sum over the ({varepsilon }_{i}) and B is given by the sum over the ({{varepsilon }_{i}}^{4/3}.) Similarly, if the mass, mi, of each individual is measured, then B is the sum over the mi and E is the sum over the mi3/4. In practice, for animal data, metabolic rate is often estimated by measuring mass and then using metabolic scaling, while for tree data, metabolic rate is estimated from measurements of individual tree basal areas, which are estimators5 of the ({varepsilon }_{i}).With E and B estimated from the same measurements, the question naturally arises as to whether a simple mathematical relationship holds between them, such as E = B3/4. If all the measured m’s, are identical, then all the calculated individual (varepsilon {{hbox{‘}}}s) are identical, and with our units convention we would have E = B. More generally, with variation in masses and metabolic rates, the only purely mathematical relationship we can write is inequality between E and B3/4: (E=sum {varepsilon }_{i}ge (sum {{{varepsilon }_{i}}^{4/3}})^{3/4}={B}^{3/4}). Our derived equation of state (Eq. 2) can be interpreted as expressing the theoretical prediction for the quantitative degree of inequality between E and B3/4 as a function of S and N.A test of Eq. 1 that compares observed and predicted values of biomass with data from 42 censused plots across a variety of habitats, spatial scales, and taxa is shown in Fig. 1. The 42 plots are listed and described in Table S2 and Supplementary Note 3. The communities censused include arthropods and plants, the habitats include both temperate and tropical, and the census plots range in area from 0.0064 to 50 ha. As seen in the figure, 99.4% of the variance in the observed values of B is explained by the predicted values of B.Fig. 1: A test of the ecological equation of state.Observed biomass is determined by either summing empirical masses of individuals or summing empirical metabolic rates raised to the ¾ power of each individual. Predicted biomass is determined from Eq. 1 using observed values of S, N, and E. The quantity ln(predicted biomass) explains 99.4% of the variance in observed biomass. Units of mass and metabolism are chosen such that the masses of the smallest individuals in each dataset are set to 1 and those individuals are also assigned a metabolic rate of 1. The shape of the marker indicates the type of data, and the lighter color corresponds to higher species richness. Data for all analyses come from tropical trees39,40,41,42,43,44,45, temperate trees30,31,32,33,46,47,48, temperate forest communities27,49, subalpine meadow flora28, and tropical island arthropods50.Full size imageFigure 2 addresses the possible concern that the success of Eq. 1 shown in Fig. 1 might simply reflect an approximate constancy, across all the datasets, of the ratio of E to B3/4. If that ratio were constant, then S and N would play no effective role in the equation of state. Equation 1 predicts that variation in the ratio depends on S and N in the approximate combination S1/4ln3/4(1/(beta (N/S))). In Fig. 2, the observed and predicted values of E/B3/4 calculated from Eq. 1, are compared, showing a nearly fourfold variation in that ratio across the datasets. The equation of state predicts 60% of the variance in the ratio.Fig. 2: The explanatory power of diversity and abundance.The observed ratio E/B3/4 is plotted against the ratio predicted by Eq. 1. Of the fourfold variability across ecosystems in that ratio, 60% is explained by the variability in the predicted combination of diversity and abundance. The shape of the marker indicates the type of data, and the lighter color corresponds to higher species richness. Data for all analyses come from tropical trees39,40,41,42,43,44,45, temperate trees30,31,32,33,46,47,48, temperate forest communities27,49, subalpine meadow flora28, and tropical island arthropods50.Full size imageFigure 3 shows the dependence on S and N of the predicted ratio E/B3/4 over empirically observed values of S, N, and E. We examined the case in which S is varied for two different fixed values of each of N and E (Fig. 3a) and N is varied for two different fixed values of S and E (Fig. 3b). The value of E does not have a large impact on the predicted ratio, particularly when E > > N. On the other hand, the predicted ratio depends more strongly on N and S.Fig. 3: The theoretical prediction for the ratio E/B3/4 as a function of S and N.The biomass B is predicted by holding E fixed along with one other state variable. In a N is fixed and S is varied, and in b S is fixed and N is varied. The fixed values are chosen to be roughly consistent within a range of the data considered. The color of the lines represents the corresponding fixed value of N or S, while the solid and dashed lines represent different fixed values of E.Full size imageThe total productivity of an ecological community is a focus of interest in ecology1, as a possible predictor of species diversity24 and more generally as a measure of ecosystem functioning25. By combining the METE and MTE frameworks, we can now generate explicit predictions for certain debated ecological relationships, including one between productivity and diversity. Interpreting total metabolic rate E in our theory as gross productivity, then in the limit 1 More

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The generalist accumulates extracellular nitriteWe first tested whether the generalist accumulates substantial extracellular nitrite under our experimental conditions, and thus creates a niche for the specialist. To accomplish this, we grew the generalist alone in bioreactors with anoxic ACS medium amended with 12 mM nitrate as the growth-limiting substrate and measured the extracellular concentrations of nitrate and nitrite over time. We performed these experiments at pH 6.5 (strong nitrite toxicity) and 7.5 (weak nitrite toxicity).We observed a substantial accumulation of extracellular nitrite regardless of the pH (Fig. 3A, B). When grown at pH 6.5 (strong nitrite toxicity), extracellular nitrite accumulated to a concentration comparable to the initial nitrate concentration (measured maximum extracellular nitrite concentration, 11.8 mM; measured initial nitrate concentration, 12.0 mM) and was subsequently consumed to below the detection limit (Fig. 3A). When grown at pH 7.5 (weak nitrite toxicity), extracellular nitrite again accumulated to a concentration comparable to the initial nitrate concentration (measured maximum extracellular nitrite concentration, 11.7 mM; measured initial nitrate concentration, 12.9 mM) and was subsequently consumed to below the detection limit (Fig. 3B). During growth at pH 6.5, substantial nitrite consumption did not begin until a prolonged period of time after nitrate consumption was complete, resulting in a relatively long duration of nitrite availability (Fig. 3A). During growth at pH 7.5, in contrast, substantial nitrite consumption began immediately after nitrate consumption was complete, resulting in a relatively short duration of nitrite availability (Fig. 3B). The longer duration of nitrite availability at pH 6.5 indicates that the duration of the niche created by the generalist for the specialist depends on pH.Fig. 3: Growth and nitrogen oxide dynamics of the generalist in batch culture.We grew the generalist alone in a bioreactor at A pH 7.5 (weak nitrite toxicity) or B pH 6.5 (strong nitrite toxicity) under anoxic conditions with nitrate as the growth-limiting substrate. Blue squares are measured extracellular nitrate concentrations, yellow triangles are measured extracellular nitrite concentrations, and black circles are measured cell densities. We measured extracellular nitrate and nitrite concentrations with IC and cell densities with FC. C Measured durations of nitrite availability for the generalist growing in batch culture. We grew the generalist alone in 96-well microtiter plates under anoxic conditions with nitrate as the growth-limiting substrate. Open symbols are durations of nitrite availability at pH 6.5 and closed symbols are durations of nitrite availability at pH 7.5. Each symbol is an independent biological replicate.Full size imageTo routinely quantify the duration of nitrite availability, we grew the generalist alone with varying amounts of nitrate as the growth-limiting substrate. We then quantified the length of time from when the growth rate with nitrate was maximum to when the growth rate with nitrite was maximum. This cell density-based proxy measure is valid because the growth of the generalist is directly linked to the consumption of nitrate and nitrite (Fig. 3A, B). The cell density of the generalist was initially linearly correlated with nitrate consumption at both pH 6.5 (strong nitrite toxicity) (two-sided Pearson correlation test; r = −0.96, p = 1.5 × 10–8, n = 15) (Fig. 3A) and 7.5 (weak nitrite toxicity) (two-sided Pearson correlation test; r = −1.00, p = 2.2 × 10–16, n = 30) (Fig. 3B). After nitrate was depleted, the cell density of the generalist became linearly correlated with nitrite consumption at both pH 6.5 (strong nitrite toxicity) (two-sided Pearson correlation test; r = −0.97, p = 3 × 10–4, n = 7) (Fig. 3A) and 7.5 (weak nitrite toxicity) (two-sided Pearson correlation test; r = −0.97, p = 6.8 × 10–10, n = 16) (Fig. 3B). We further validated our cell density-based approach by testing for concordance with our IC-based direct measures of the duration of nitrite availability. We observed a significant positive and linear relationship between the cell density- and IC-based measures (two-sided Pearson correlation test; r = 0.999, p = 0.023, n = 3) (linear regression model; slope = 1.19, intercept = −2.31, r2 = 0.99) (Supplementary Fig. S2), which further validates our cell density-based approach to routinely estimate the duration of nitrite availability.Using our cell density-based approach, we found that the duration of nitrite availability was significantly longer at pH 6.5 (strong nitrite toxicity) than at 7.5 (weak nitrite toxicity) regardless of the initial nitrate concentration (two-sample two-sided t-tests; Holm-adjusted p  0.92, Holm-adjusted p  0.6), and thus followed model predictions (Fig. 4A). However, when the specialist was initially rare (measured initial log rS/Gs of –3.19, –2.65, and –0.88), the relative abundances of the specialist continuously decreased between the third and twelfth transfers (Mann–Kendall trend tests; tau = –0.61 to –0.89, p  0 were dominated by phenotype C (dominant ancestral phenotype with a long time delay between nitrate and nitrite consumption), while generalist isolates from co-cultures with initial rS/Gs  More