Defining coarse-grainability in a model of structured microbial ecosystems

Despite their complexity, microbial ecosystems appear to be at least partially "coarse-grainable" in that some properties of interest can be adequately described by effective models of dimension much smaller than the number of interacting lineages (frequently in the dozens or hundreds). This is especially puzzling since recent studies demonstrate that a surprising amount of functionally relevant diversity is present at all levels of resolution, down to strains differing by 100 nucleotides or fewer. Rigorously defining coarse-grainability and understanding the conditions for its emergence is of critical importance for understanding microbial ecosystems. To begin addressing these questions, we propose a minimal model for investigating hierarchically structured ecosystems within the framework of resource competition. We use our model to operationally define coarse-graining quality based on reproducibility of the outcomes of a specified experiment and show that a coarse-graining can be operationally valid despite grouping together functionally diverse strains. We further show that, at least within our model, a high diversity of strains (while nominally more complex) may in fact facilitate coarse-grainability. However, this only applies if the pool of interacting strains is sampled from the appropriate "native" environment, as we discuss.