• Teemu Kuosmanen, 
• Juhani Rantanen, 
• Dovydas Kičiatovas, 
• Sanna Pausio, 
• Ville-Petri Friman, 
• Teppo Hiltunen, 
• Ville Mustonen

x

Abstract

Citation: Kuosmanen T, Rantanen J, Kičiatovas D, Pausio S, Friman V-P, Hiltunen T, et al. (2026) Normative assembly rule reveals fairness in microbial communities. PLoS Biol 24(6): e3003872. https://doi.org/10.1371/journal.pbio.3003872

Academic Editor: Britt Koskella, University of California Berkeley, UNITED STATES OF AMERICA

Received: January 28, 2026; Accepted: June 8, 2026; Published: June 24, 2026

Copyright: © 2026 Kuosmanen et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All data and analysis notebooks are available at: https://osf.io/a7yzp/overview.

Funding: This work was funded in part by the Research Council of Finland (Multidisciplinary Center of Excellence in Antimicrobial Resistance Research, grants 346126 and 364232 to TH; grants 346128 and 364234 to VM). https://www.aka.fi/en/ The funders did not play any role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Abbreviations: AMP, ampicillin; CON, control condition; MOX, moxifloxacin; PPY, proteose-peptone-yeast; STR, streptomycin

Introduction

Organisms interact with each other and their surrounding environment to grow, adapt, and assemble to biological populations, ecological communities, and economic coalitions which in turn self-organize to functioning ecosystems and technological societies with advanced economies [1]. Such complex systems display various emergent phenomena ranging from cyclicity and chaos [2,3] to robustness and resilience [4,5], and evolution and diversification [6,7]. By exploiting available free energy reservoirs, all living organisms produce surplus output that is distributed in the community context by some distinct mechanisms. In nature, the ecological game of growth is largely based on competitive success in utilizing limited shared resources whereas cooperative interactions among social groups, such as seen in human societies, might lead to a different payoff distribution.

By studying competition in its purest form, ecological and evolutionary theory have since their adoption broadly influenced economic thought by shedding light to properties of the ‘state of nature’. While the distribution of income, wealth, and consumption in human societies has been a prominent theme in economic literature [8,9], species abundance distributions [10,11] have similarly been some of the most intensively studied macroecological patterns ever since Preston’s observation that different physical, economical, and ecological ‘wealth distributions’ all seem to take similar form [12,13]. But whereas the distributive properties of political and economic systems have also given rise to normative theories addressing questions of distributive justice in terms of fair resource allocation, corresponding ecological theories have remained mainly descriptive, characterizing biodiversity with statistical scaling laws [14,15]. However, redistribution of resources has also been observed in non-human species, where uncertainty about kinship can reduce inequality among offspring, a finding consistent with Rawlsian notion of fairness [16].

The field of game theory can be roughly divided to non-cooperative game theory, which focuses on modeling individual strategic behavior in a competitive setting, and cooperative game theory, which in contrast focuses solely on the outputs of different coalitions of players and how the total output is distributed among them (either by bargaining or fair redistribution). While game theory has been extensively applied, and indeed also developed, within the broader fields of evolution and ecology, existing eco-evolutionary theory is predominantly making use of the concepts of non-cooperative game theory by attempting to model microscopic interactions and strategy dynamics either at the level of individuals, species, or genes as players [17]. However, measuring these underlying mechanistic interactions can be generally very difficult in case of complex biological systems even for simple minimal models, such as the generalized Lotka-Volterra model commonly employed for studying ecological communities [18,19]. Here, we demonstrate, perhaps counterintuitively, that concepts of cooperative game theory can be meaningfully applied also in an inherently non-cooperative biological setting, where competition and exploitation are the norm [20,21]. As we will show, this opens new, top-down methods to model and predict assembly of complex ecological communities.

We introduce the ecological Game of Growth where we first measure the biomass growth—the universal currency of living systems—in different species assemblages and then analyze the game’s distributional properties using the concept of Shapley value (see Fig 1). The Shapley value [22] is an established solution concept in cooperative game theory, which provides an objective way to distribute the total payoff between the players fairly in the sense that, among all possible payoff distributions, the Shapley distribution is the only one satisfying the mathematical properties of efficiency, symmetry, and linearity [22] as well as admitting a unique potential function [23]. The Shapley value has been extensively applied and extended in the economics game theory literature [24] as well as more recently in machine learning as a principled way to quantify the feature importances of different predictors [25]. In the context of biology, analysis inspired by the Shapley value has found applications, for example, in the fair proportion index used in constructing phylogenetic trees, to assess the importance of genes within a metabolic pathway, or to determine the individual contribution of each taxon within a multi-taxa setting to microbiome functional profiles [26–32].

In this paper, we investigate how biomass as a proxy for cumulative consumption of shared resources is distributed in microbial communities and further pose the question of how fair the resulting ecological outcome is from a normative point of view. By systematically measuring the growth of 14 distinct 4-species communities across all possible subcommunities, we can compute the Shapley value of each species and assess the fairness of the community assembly by comparing these to the actual end-population shares of the species. We also applied varying antibiotic stress to each community to see whether the observed species abundance distribution and fairness change in more challenging growth conditions, where cooperation can readily emerge via production of public goods, detoxification, or nutrient availability promotion, and thus potentially play a pivotal role in the community outcomes [33]. By using the established concept of Shapley value, we first empirically quantify the fairness of altogether 56 community outcomes and then develop a normative assembly rule that can also predict unfairly assembled communities if the realized relative abundances in lower-level subcommunities are known.

Results

The Shapley value of player i in a game with n players forming a grand coalition is given by [22]

(1)

where is a function that gives the output of every 2ⁿ possible (sub)coalitions that can be formed from the members of the grand coalition. In the context of ecology, we can interpret as some community function of community which we assume can be directly measured. Here, we focus on the community yield, denoted by , which is the total biomass at the end of the experiment. Now, to compute the Shapley values, we need not make any assumptions about the ecological dynamics that give rise to the species abundances, but instead simply experimentally measure the observable community yield in all subcommunities (see Fig 1).

In practice, the Shapley value Eq. (1) can be computed more efficiently using the following recursive formula

(2)

with initial condition for empty coalitions (for a proof, see Ref. [34]). This more interpretable formula shows how the Shapley value is built bottom-up from the subcoalition Shapley values based on the species’ marginal value on the community function.

We formed 14 distinct communities by randomly selecting four species to each community from a pool of 16 microbial species that are known to be able to coexist as a community in laboratory microcosm experiments (see Materials and methods for more details). We then measured the growth curves and determined the yields from the combinatorially complete set of experiments in four different growth conditions, which allowed us to compute the Shapley values of the community members across the different communities and environments. These results are summarized in Fig 2.

First, we investigated the yield distribution within each community without focusing on the species identities: by ranking the Shapley values from largest to smallest independently for each community and environment, we gain a first look on how the fair yield shares based on the Shapley value (Fig 2A) compare to the actual yield distribution (Fig 2B). Note that by the property of efficiency, the Shapley value distributes the total yield fully among the community members, so we are comparing two different ways to distribute the same surplus output. By averaging these proportions across all communities (Fig 2C), we noticed that the most prevalent species in each community typically attains higher abundance compared to ranking based on Shapley value, the second-ranked species obtains roughly the same share under both distribution rules, and the third and fourth-ranked species are left with lower shares than what their Shapley values would warrant. Therefore, the observed yield shares are more concentrated to the most abundant species than the corresponding yield distribution obtained from the Shapley values which would distribute the biomass more equally. However, in the more challenging growth condition of moxifloxacin stress, we noticed that both the Shapley values and the observed yield shares become increasingly concentrated on a single species. In this case, two of the communities produce no observable biomass, and some species even have negative values, meaning that their presence lowered the community output.

We then turned to analyze the roles of the individual species by collecting their Shapley values from all the communities they were present. Note that by the property of linearity, averaging the Shapley values of any given coalition across different environments constitutes a Shapley value of the entire game across environments. However, in our experiment, some species can belong to multiple different coalitions, and averaging over them is not anymore formally a Shapley value. Nevertheless, by averaging over all the individual Shapley values per species, the obtained mean Shapley value gives a measure of the species overall deserved share of the games’ output (performance). We show that the mean Shapley value of each species is correlated quite strongly with the true abundances (r = 0.754), but note that some species receive on average less than what their objective contribution to the community output is, while others gain a larger share than what could be considered fair (see Figs 2D and S1 where the same is shown separately for each environment).

Next, we investigated how closely the observed frequency distribution follows the game-theoretically fair end-distribution , which can be obtained from the optical density data by normalizing with the sum of the Shapley values, which equals the yield of the grand coalition (see Materials and methods for more details). For example, in the case of only two species, the fair end-distribution is given by

(3.1)(3.2)

where and are the monoculture yields of species 1 and 2, respectively, and is the biculture yield. This uniquely defines the fair distribution in case of two species for any ecological dynamics, assigning a higher fair proportion for the species with more efficient resource conversion efficiency while the biculture value scales the monoculture difference more strictly in case of antagonism (lower biculture yield) and more loosely in case of synergy (higher biculture yield) allowing for more equal proportions. The fair distribution therefore has the interesting feature that as the total output increases, the fair proportion of the less abundant species increases not only in absolute, but also in relative terms.

In reality, the true end-frequencies after the community growth can deviate from the fair distribution because, in a competitive setting, resource efficiency (monoculture yield) is not the only important factor: for example, shorter lag time, higher resource uptake rate, consumption of secondary metabolites, exploitation of other public goods without reciprocal contribution to their production (cheating), and direct negative interactions, such as bacterial warfare [20], can all significantly change the competitive dynamics such that the monoculture yield alone is not predictive enough for success in a community setting. However, species that reach high abundances mainly by exploiting others without contributing themselves cannot grow successfully in harsher environments without the help of the contributing species. Thus, by observing growth systematically across all possible coalitions, we can effectively dissect the causal impact and role of each species on the community output, which defines the fair proportions of the species. By sequencing the community after growth, we can estimate the true end-frequencies, which allows us to compare how ecological reality relates to what would be fair community outcomes.

To quantify the fairness of each community across different environments, we computed the cosine similarity of the fair distribution and the true frequency vector (Fig 3A). We found communities that systematically followed the fair distribution across all environments; communities that were fair in some conditions, but not others; and communities that deviated significantly from the fair distribution across all conditions. We then quantified both the actual and implied community diversity of the true and fair distributions, respectively, using the Gini–Simpson index, which can be interpreted as the probability of observing two different species when randomly sampling a pair of individuals from the population, i.e., the probability of interspecific encounter (note that the Gini–Simpson index is inversely related to the Gini index with higher values meaning higher diversity). First, we found that the implied community diversity of the fair distribution could be either high or low, with the latter being the default mode in the presence of moxifloxacin (Fig 3D). However, in other conditions, the fair distribution displayed non-trivial coexistence as showcased in Fig 3C, which makes the observed fairness of these communities intriguing. We then compared the properties of the fair distribution and the true ecological distribution by contrasting their community diversity against each other (Fig 3E) and found that the Gini–Simpson index of actual ecological communities was systematically lower or at most as high as that of the fair distribution. This suggests that the ecological reality seems to be more unequal than the fair distribution.

A property of the fair distribution is that any discrepancies present with respect to true ecological outcomes already in the biculture competitions get propagated forward via the recursive logic of the Shapley value as shown by Eq. (2). However, if we have access to such additional biologically relevant information besides the yields, can we use this information to predict the full community assembly?

To test this, we also sequenced all the subcommunities and modified the recursive formula Eq. (2) to take the estimated true abundances at some subcommunity size as the initial condition for the recursion, and then computed the higher-order Shapley values as before. This leads to the following recursive algorithm

(4)

which can be used to predict the species abundances in the full community as . Note that this fair assembly rule contains no free parameters but instead assumes that after the initial unfairness present in the lower subcommunities has been accounted for, the community assembly thereafter follows the rules of the fair assembly.

By incorporating the true biculture and triplet frequencies, we found that the fair assembly rule Eq. (4) was able to predict the observed full community outcomes up to very high accuracy (see Fig 4). We quantified the prediction accuracy using the Euclidean distance between the predicted and observed community composition and found that the mean squared error of the fair distribution with only yields was 0.367 0.034 (SE, N = 54). By using the fair assembly rule with the correct pairwise frequencies, this improved to 0.233 0.025 (SE, N = 54), and to 0.132 0.015 (SE, N = 54) with triplet frequencies. This suggests that the observed assembly process is consistent with the recursive logic of the fair assembly rule: the improved predictive performance is not driven by simply feeding in more data, but by adjusting the initial condition of the recursion based on observed, as opposed to fair, subcommunity outcomes.

Discussion

Although much empirical research has been devoted to characterizing biodiversity in terms of species abundance distributions, there is surprisingly little knowledge about the normative status of the “state of nature.” Here, we asked how fair ecological outcomes are in microbial communities by comparing the observed species abundance distribution to a fair distribution where each species claims only their fair share of the total surplus given by their Shapley value. We found examples of communities where the true ecological distribution followed the fair distribution very closely both in the control conditions and under antibiotic stress, while in other cases some significant species reached unfairly high abundances at the expense of “exploited species” which had a higher objective contribution to the community function than their observed share. One potential mechanism leading to unfairness in our system could be the varying lag time before the onset of growth: we found that species with shorter lag time were generally able to claim their fair share according to their Shapley value while species with a longer lag grew either better or worse after the initial growth of other species (see S2 Fig in the Supplementary material). This observation may be explained by the varying degree of niche overlap, where the growth of the species with longer lag can be hindered in case of high overlap, but improved in case of low overlap, for example, due to cross-feeding.

While very unequal payoff distributions can also be fair, as seen in the stronger selective environment (Fig 3D), we found that the community diversity of the true ecological distribution was systematically lower or at most the same as the corresponding diversity of the fair distribution as measured by the Gini–Simpson index. This tentatively suggests that at least in our microbial model system, the ecological reality seems to be even harsher than that of a purely meritocratic system. However, an important open question remains, whether this applies also in real ecosystems where species have co-evolved together as opposed to the random assemblages used here. Therefore, it may well be plausible that in the absence of “unfair” asymmetries, such as historical priority effects, nature is indeed organized increasingly ‘fairly’ in the Shapley sense. To what extent species sorting and evolutionary mechanisms invoking community state shifts [35] can act on fairness remains an intriguing topic for future studies, especially if fairness indeed affects community function. For example, it has been shown that community evenness can facilitate higher robustness in metrics such as invasion resistance and functional redundancy [36,37].

We believe that the introduced framework based on ideas of cooperative game theory and the Shapley value can become broadly useful in analyzing ecological data and developing ecological theory based on the several key strengths highlighted by our work. First, it provides a fully general way to make a non-trivial, game-theoretical baseline prediction for community outcomes and as such potentially serve as a useful null model that is driven by experimental data rather than any assumptions about the underlying ecological dynamics. Indeed, the fair assembly rule Eq. (4) based on the recursive definition of the Shapley value Eq. (2) could accurately predict complex community outcomes with zero free parameters. To emphasize, we did not fit any mathematical or statistical model to data but rather derived the predictions from the basic principles of fair distribution, further showing that the recursive logic of the Shapley value provided a very parsimonious description for the observed data and a new lens for understanding community composition. This success is particularly interesting as it remains unclear when microbial assembly follows general rules in contrast to cases where the community member species succeed in “surprising” the experimenters by exhibiting the so-called emergent properties [38,39]. Thus, our assembly rule based on fairness opens a new way to identify at what level of hierarchy the possible emergent properties arise. The methodology can also be directly applied to other community functions or properties of interest besides the yield, which was our focus here. By construction, the Shapley value compares community outcomes with and without the presence of each species, thus allowing to make causal inferences of the species contributions as well as their interactions in a model-agnostic way.

There are, of course, also some limitations to the approach taken here, the most notable being the fact that to compute the Shapley value in practice, one must experimentally measure the yield (or other value of interest) from all the 2ⁿ possible subcommunities, which grows quickly with the number of present species. However, the technology to make combinatorially complete experiments even for moderately large communities already exists [40] and is likely to become easier in the future. Similarly, computer scientists actively develop approximative analysis ideas trying to overcome enumeration challenges related to Shapley values [30] which may be transferable to experimental design also in biology. Additionally, if one has access to the realized species distribution at some level, the introduced fair assembly rule can then be used to make predictions for the full community of any size without having to measure all the smaller subcommunities. This makes leave-one-out experiments (see, e.g., Ref. [41]) particularly interesting for testing how these ideas might translate to larger communities.

In conclusion, we have demonstrated how a purely normative concept can be successfully applied to studying a complex system, where a central problem in the typical bottom-up modeling approaches has been the number of unknown parameters that are very difficult to infer even for a minimal model. The suggested top-down approach of using the Shapley value and fairness to understand and predict community assembly is analogous to methods based on optimization principles [42] or principle of maximum entropy [43], which avoid this problem by deriving the model parameters and predictions from first principles. We anticipate that community assembly theory could be advanced with new types of normative models and identify the evolution of fairness as a particularly interesting area of future research both theoretically related to the evolutionary stability and reachability of fair community compositions as well as empirically in experimental evolution setups.

Materials and methods

Supporting information

Acknowledgments

We would like to acknowledge members of the Bioinformatics and Evolution group for helpful discussions and comments as well as Shane Hogle and Johannes Cairns for their help related to bioinformatics. The authors wish to acknowledge CSC—IT Center for Science, Finland, for computational resources.

References

1. 1. Smil V. Growth: from microorganisms to megacities. MIT Press; 2019.
2. 2. Hastings 2 A, Hom CL, Ellner S, Turchin P, Godfray HCJ. Chaos in ecology: is mother nature a strange attractor?. Annu Rev Ecol Syst. 1993;24:1–33.
   • View Article
   • Google Scholar
3. 3. Grinin L, Korotayev A, Tausch A. Economic cycles, crises, and the global periphery. 1st ed. Cham, Switzerland: Springer International Publishing; 2016.
4. 4. Thorogood R, Mustonen V, Aleixo A, Aphalo PJ, Asiegbu FO, Cabeza M, et al. Understanding and applying biological resilience, from genes to ecosystems. NPJ Biodivers. 2023;2(1):16. pmid:39242840
   • View Article
   • PubMed/NCBI
   • Google Scholar
5. 5. Arani BMS, Carpenter SR, Lahti L, van Nes EH, Scheffer M. Exit time as a measure of ecological resilience. Science. 2021;372(6547):eaay4895. pmid:34112667
   • View Article
   • PubMed/NCBI
   • Google Scholar
6. 6. Anderson PW, Arrow K, Pines D, editors. The economy as an evolving complex system. Addison-Wesley; 1988.
7. 7. Geritz SAH, Kisdi E, Mesze´NA G, Metz JAJ. Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol Ecol. 1998;12(1):35–57.
   • View Article
   • Google Scholar
8. 8. Kuznets S. Economic growth and income inequality. Am Econ Rev. 1955;45:1–28.
   • View Article
   • Google Scholar
9. 9. Piketty T, Saez E. Inequality in the long run. Science. 2014;344(6186):838–43. pmid:24855258
   • View Article
   • PubMed/NCBI
   • Google Scholar
10. 10. MacArthur R. On the relative abundance of species. Am Nat. 1960;94(874):25–36.
    • View Article
    • Google Scholar
11. 11. Locey KJ, White EP. How species richness and total abundance constrain the distribution of abundance. Ecol Lett. 2013;16(9):1177–85. pmid:23848604
    • View Article
    • PubMed/NCBI
    • Google Scholar
12. 12. Nekola JC, Brown JH. The wealth of species: ecological communities, complex systems and the legacy of Frank Preston. Ecol Lett. 2007;10(3):188–96. pmid:17305802
    • View Article
    • PubMed/NCBI
    • Google Scholar
13. 13. Preston FW. The commonness, and rarity, of species. Ecology. 1948;29:254–83.
    • View Article
    • Google Scholar
14. 14. Grilli J. Macroecological laws describe variation and diversity in microbial communities. Nat Commun. 2020;11(1):4743. pmid:32958773
    • View Article
    • PubMed/NCBI
    • Google Scholar
15. 15. Gao Y, Abdullah A, Wu M. The powerbend distribution provides a unified model for the species abundance distribution across animals, plants and microbes. Nat Commun. 2025;16(1):4035. pmid:40301372
    • View Article
    • PubMed/NCBI
    • Google Scholar
16. 16. Marshall HH, Johnstone RA, Thompson FJ, Nichols HJ, Wells D, Hoffman JI, et al. A veil of ignorance can promote fairness in a mammal society. Nat Commun. 2021;12(1):3717. pmid:34162841
    • View Article
    • PubMed/NCBI
    • Google Scholar
17. 17. Broom M, Rychtář J. Game-theoretical models in biology. Boca Raton: Chapman and Hall/CRC; 2022.
18. 18. Meacock OJ, Mitri S. Environment-organism feedbacks drive changes in ecological interactions. Ecol Lett. 2025;28(1):e70027. pmid:39737705
    • View Article
    • PubMed/NCBI
    • Google Scholar
19. 19. Cui W, Marsland R, Mehta P. Les Houches lectures on community ecology: From niche theory to statistical mechanics. arXiv Preprint. 2024.
    • View Article
    • Google Scholar
20. 20. Granato ET, Meiller-Legrand TA, Foster KR. The evolution and ecology of bacterial warfare. Curr Biol. 2019;29(11):R521–37. pmid:31163166
    • View Article
    • PubMed/NCBI
    • Google Scholar
21. 21. Palmer JD, Foster KR. Bacterial species rarely work together. Science. 2022;376(6593):581–2. pmid:35511986
    • View Article
    • PubMed/NCBI
    • Google Scholar
22. 22. Shapley LS. A value for n-person games. In: Kuhn HW, Tucker AW, editors. Contributions to the theory of games. Princeton: Princeton University Press; 1953. p. 307–17.
23. 23. Hart S, Mas-Colell A. Potential, value, and consistency. Econometrica. 1989;57:589.
    • View Article
    • Google Scholar
24. 24. Roth AE. The shapley value: essays in honor of Lloyd S. Shapley. Cambridge, England: Cambridge University Press; 1988.
25. 25. Scott M, Su-In L. A unified approach to interpreting model predictions. Adv Neural Inf Process Syst. 2017;30:4765–74.
    • View Article
    • Google Scholar
26. 26. Bohl K, Hummert S, Werner S, Basanta D, Deutsch A, Schuster S, et al. Evolutionary game theory: molecules as players. Mol Biosyst. 2014;10(12):3066–74. pmid:25248454
    • View Article
    • PubMed/NCBI
    • Google Scholar
27. 27. Deutscher D, Meilijson I, Schuster S, Ruppin E. Can single knockouts accurately single out gene functions?. BMC Syst Biol. 2008;2:50. pmid:18564419
    • View Article
    • PubMed/NCBI
    • Google Scholar
28. 28. Keinan A, Sandbank B, Hilgetag CC, Meilijson I, Ruppin E. Fair attribution of functional contribution in artificial and biological networks. Neural Comput. 2004;16(9):1887–915. pmid:15265327
    • View Article
    • PubMed/NCBI
    • Google Scholar
29. 29. Manor O, Borenstein E. Systematic characterization and analysis of the taxonomic drivers of functional shifts in the human microbiome. Cell Host Microbe. 2017;21(2):254–67. pmid:28111203
    • View Article
    • PubMed/NCBI
    • Google Scholar
30. 30. Pham G, Milazzo P. A comprehensive review of the use of Shapley value to assess node importance in the analysis of biological networks. Comput Methods Programs Biomed Update. 2025;7:100185.
    • View Article
    • Google Scholar
31. 31. Waldock C, Wegscheider B, Josi D, Calegari BB, Brodersen J, Jardim de Queiroz L, et al. Deconstructing the geography of human impacts on species’ natural distribution. Nat Commun. 2024;15(1):8852. pmid:39402017
    • View Article
    • PubMed/NCBI
    • Google Scholar
32. 32. Hartmann K. The equivalence of two phylogenetic biodiversity measures: the Shapley value and Fair Proportion index. J Math Biol. 2013;67(5):1163–70. pmid:22986892
    • View Article
    • PubMed/NCBI
    • Google Scholar
33. 33. Piccardi P, Vessman B, Mitri S. Toxicity drives facilitation between 4 bacterial species. Proc Natl Acad Sci U S A. 2019;116(32):15979–84. pmid:31270235
    • View Article
    • PubMed/NCBI
    • Google Scholar
34. 34. Dragan I, Some recursive definitions of the Shapley value and other linear values of cooperative TU games. Working Paper 328 of the University of Texas at Arlington; 1997.
35. 35. Kivikoski M, Cairns J, Hogle SL, Pausio S, Becks L, Mustonen V, et al. Evolution induced state shifts in a long-term microbial community experiment. Proc Natl Acad Sci U S A. 2026;123(22):e2533269123. pmid:42201971
    • View Article
    • PubMed/NCBI
    • Google Scholar
36. 36. Wittebolle L, Marzorati M, Clement L, Balloi A, Daffonchio D, Heylen K, et al. Initial community evenness favours functionality under selective stress. Nature. 2009;458(7238):623–6. pmid:19270679
    • View Article
    • PubMed/NCBI
    • Google Scholar
37. 37. De Roy K, Marzorati M, Negroni A, Thas O, Balloi A, Fava F, et al. Environmental conditions and community evenness determine the outcome of biological invasion. Nat Commun. 2013;4:1383. pmid:23340423
    • View Article
    • PubMed/NCBI
    • Google Scholar
38. 38. Chang C-Y, Bajić D, Vila JCC, Estrela S, Sanchez A. Emergent coexistence in multispecies microbial communities. Science. 2023;381(6655):343–8. pmid:37471535
    • View Article
    • PubMed/NCBI
    • Google Scholar
39. 39. Friedman J, Higgins LM, Gore J. Community structure follows simple assembly rules in microbial microcosms. Nat Ecol Evol. 2017;1(5):109. pmid:28812687
    • View Article
    • PubMed/NCBI
    • Google Scholar
40. 40. Diaz-Colunga P, Catalan M, San Roman A, Arrabal A, Sanchez . Full factorial construction of synthetic microbial communities. eLife. 2024;13:RP101906.
    • View Article
    • Google Scholar
41. 41. Ansari AF, Reddy YBS, Raut J, Dixit NM. Predicting structures of microbial communities. Nat Comput Sci 2021;1:619–28.
    • View Article
    • Google Scholar
42. 42. Sokolowski TR, Gregor T, Bialek W, Tkačik G. Deriving a genetic regulatory network from an optimization principle. Proc Natl Acad Sci U S A. 2025;122(1):e2402925121. pmid:39752518
    • View Article
    • PubMed/NCBI
    • Google Scholar
43. 43. Harte J, Newman EA. Maximum information entropy: a foundation for ecological theory. Trends Ecol Evol. 2014;29(7):384–9. pmid:24863182
    • View Article
    • PubMed/NCBI
    • Google Scholar
44. 44. Cairns J, Jokela R, Hultman J, Tamminen M, Virta M, Hiltunen T. Construction and characterization of synthetic bacterial community for experimental ecology and evolution. Front Genet. 2018;9:312. pmid:30154827
    • View Article
    • PubMed/NCBI
    • Google Scholar
45. 45. Glenn TC, Pierson TW, Bayona-Vásquez NJ, Kieran TJ, Hoffberg SL, Thomas Iv JC, et al. Adapterama II: universal amplicon sequencing on Illumina platforms (TaggiMatrix). PeerJ. 2019;7:e7786. pmid:31616589
    • View Article
    • PubMed/NCBI
    • Google Scholar
46. 46. Martin M. Cutadapt removes adapter sequences from high-throughput sequencing reads. EMBnet J. 2011;17(10).
    • View Article
    • Google Scholar
47. 47. Gaspar JM. NGmerge: merging paired-end reads via novel empirically-derived models of sequencing errors. BMC Bioinformatics. 2018;19(1):536. pmid:30572828
    • View Article
    • PubMed/NCBI
    • Google Scholar
48. 48. Rognes T, Flouri T, Nichols B, Quince C, Mahé F. VSEARCH: a versatile open source tool for metagenomics. PeerJ. 2016;4:e2584. pmid:27781170
    • View Article
    • PubMed/NCBI
    • Google Scholar
49. 49. Zhang P, Spaepen S, Bai Y, Hacquard S, Garrido-Oter R. Rbec: a tool for analysis of amplicon sequencing data from synthetic microbial communities. ISME Commun. 2021;1(1):73. pmid:37938657
    • View Article
    • PubMed/NCBI
    • Google Scholar