Fig 1.
An ecological landscape framework for community optimization.
We consider the problem of assembling an ecological community from a pool of S candidate species that is optimal at performing a desired function. The candidate species in the pool are indexed from 1 to S. A test community is seeded with a subset of the candidate species. The composition of the community is encoded in using 0 for absence and 1 for presence for each of the species. The test communities are allowed to grow until they reach a steady state. The species abundances in the steady-state community is
, and the community function is
. The growth of the community to steady-state is simulated by consumer-resource models, illustrated on the right. The growth rate depends on its ability to consume resources and death occurs by dilution or mortality. In models with leakage (gray), metabolic byproducts can be leaked and used by other species. Equations shown correspond to the scenario without metabolic leakage; see methods for further explanation and model equations with leakage.
Table 1.
Microbial communities often converge to a unique stable steady state, but more complex dynamics are possible.
By a single stable steady-state, we mean that the dynamics converge to a steady-state that is stable to invasion by species that had gone extinct en route. In other words, if a species present initially goes extinct and is absent from the final steady-state, then it can not invade the final community successfully. Thus species presence-absences in the initial community fully determine the outcome. Note that this table only approximately condenses a vast and complex literature; for a more thorough understanding we encourage the interested reader to consult the cited references. We have provided additional details regarding the cited studies in S1 Text Sec.1. Macroscopic organisms are discussed in Ref. [64].
Fig 2.
Niche overlap determines the linearity of ecological landscapes.
The top panels in (A) and (B) show the consumption matrices for species in each species pool. In (A), the niche overlap is low because the species are specialists and consume only a few resources. In (B), the niche overlap is high because the species are generalists and consume many resources. The bottom panels show how well a linear model (Eq 1) fits the steady-state abundances across all possible species combinations. Color bar quantifies the local density of points, as measured by gaussian kernel density estimation. The goodness of the model fit, quantified by , is higher when the niche overlap is lower. This conclusion is robust to varying the size of the community and the degree of niche overlap. (C) shows how
varies with the number of species in the pool, S, and the average number of resources that they consume, Mconsumed. The degree of niche overlap is determined by Mconsumed/Mtotal. Letters indicate the parameters used in panels A and B. Note that
was computed by averaging R2 of the linear model fitted separately for each species, and the number of resources supplied was fixed to Mtotal = S. For each set of parameters, results were averaged over 10 independently generated species pools. Simulations were of consumer-resource models without leakage. Other simulation parameters are provided in Methods, and the robustness of the conclusions to certain modeling assumptions is further discussed in S1 Text and S2 Fig.
Fig 3.
Search efficacy decreases with niche overlap.
Search efficacy was quantified as the ratio of the best community function found by the search to the highest value
across the entire landscape. The plots report the average and the standard error of the mean obtained from ten independently generated species pools. Different panels correspond to the four different community functions. The complexity of the landscape was controlled by varying the average number of resources that each species consumed Mconsumed while keeping the total number of resources Mtotal fixed at Mtotal = S = 16. Data was from consumer-resource model simulations with 16 species shown in Fig 2. See Methods for a complete description of these simulations.
Fig 4.
Example of the workflow for calculating the ruggedness of ecological landscapes.
The roughness-slope ratio r/s quantifies the deviation from linearity of the landscape. It is calculated by fitting the measured community function landscape with a linear model and then estimating r and s from the fit. The roughness, r, measures the average error of a linear fit to the community function landscape and the slope, s, measures the magnitude of the change in
. The calculation of other ruggedness metrics follows a similar pattern, but with different mathematical quantities being computed.
Fig 5.
Ruggedness of community function landscapes.
All metrics of ruggedness increased with niche overlap. Different panels correspond to different metrics of ruggedness. The plots report the average and the standard error of the mean obtained from ten independently generated species pools. The complexity of the landscape was controlled by varying the average number of resources that each species consumed, Mconsumed, while keeping the total number of resources, Mtotal, fixed at Mtotal = S = 16. See Methods for the complete description of these simulations. The community function here was community diversity; see S3 Fig for other community functions.
Fig 6.
Contribution of higher-order interactions.
Including higher order terms in the fitting model reduced the variance in the data not explained by the fit. First order fitting models are linear, second—quadratic, etc. The decrease in unexplained variance was approximately exponential. We used S = 16, Mconsumed = 2 for specialists and Mconsumed = 14 for generalists; the community function was Shannon diversity. All other simulation parameters are the same as in Fig 5.
Fig 7.
Landscape ruggedness inhibits search success.
Search efficacy is anti-correlated with the landscape ruggedness. Panels (A), (B), (C), (D) show examples for different community functions and ruggedness metrics. Each point corresponds to a separate S = 16 species pool studied in Fig 5. The systematic effects of the ruggedness metric and the type of community function is tabulated in (E), which reports the Spearman’s correlation coefficient ρ between search efficacy and ruggedness. The correlations were highly significant (p ≪ 10−4) in all cases. Darkness of table background color indicates magnitude of correlation.
Fig 8.
Estimating ruggedness from limited data.
(A) True and estimated ruggedness are tightly correlated when only 1% of the data is available for the estimation. The data are from Fig 5, and each point corresponds to a landscape with a different degree of niche overlap. (B) The accuracy of the estimation improves with the amount of available data for all ruggedness measures. (C) The estimated ruggedness remain highly informative of search efficacy. The utility of the ruggedness estimate, |ρ|, is measured as the magnitude of the Spearman’s correlation coefficient between search efficacy and estimated ruggedness. Note that |ρ| remained close to one even for very low fractions of the data for which R2 in panel B started to decrease rapidly suggesting that deviations between predicted and actual roughness (see A and B) do not impede the prediction of search success. For this figure, the community function is the Shannon diversity. Similar results were obtained for other functions tested.
Fig 9.
Ruggedness is predictive of search efficacy in models with cross-feeding.
Cross-feeding altered the composition of microbial communities (A), but had no systematic effect on the success of the heuristic search (B) or ruggedness (C). Importantly, the search efficacy and ruggedness remained strongly anti-correlated in communities with cross-feeding (D). Simulated species pools had S = 12 species that leaked a fraction l of the resources they consumed as consumable byproducts. Community function used in panels B, C, D is the abundance of a focal species. A strong negative anti-correlation was found for other choices of community function as well (See S6 Fig). Only one out of 12 modeled resources was supplied externally; the rest were present only due to metabolic leakage. Each species was able to consume 6 resources. Ruggedness was quantified by r/s. See Methods for further simulation details.
Fig 10.
Ruggedness is informative for a variety of search protocols.
(A) shows the search efficacy of three different community optimization protocols. Despite differences in the performance of different protocols, ruggedness remained informative of search success across protocols (panels B, C, D). Community function was pair-productivity. All searches started from the best of q = 13 communities. Dilution rate in the dilution-based protocols was adjusted for the highest performance. Simulated communities were the same as in Fig 9.
Fig 11.
Ruggedness of experimental landscapes.
(A) shows the ruggedness and search efficacy in the seven environments studied in Ref. [32] labeled by the carbon sources used. There is an apparent anticorrelation between the efficacy and ruggedness, but it does not reach statistical significance (p = 0.29). Note that the range of ruggedness and search efficacy is quite narrow presumably because of the overlap in the carbon sources in different environments and the choice of species that grow well on these sugars. Nevertheless, the search efficacy was highest on the least rugged landscape (xylose). Search efficacy and ruggedness were anti-correlated with ruggedness on the three pure carbon sources as well. (B) The decrease in the variance unexplained with increasing model complexity shows behavior similar to that of the simulated landscapes in Fig 6. Dashed lines in panel B show a linear fit on the semi-logarithmic plot.(C) The estimated ruggedness of experimental landscapes of five different community functions studied in Ref. [65]. The ruggedness was estimated of the 25 species landscape was estimated from experimental measurements of community function of 577 different species combinations.