Spatial ecology of territorial populations.

Many ecosystems, from vegetation to biofilms, are composed of territorial populations that compete for both nutrients and physical space. What are the implications of such spatial organization for biodiversity? To address this question, we developed and analyzed a model of territorial resource competition. In the model, all species obey trade-offs inspired by biophysical constraints on metabolism; the species occupy nonoverlapping territories, while nutrients diffuse in space. We find that the nutrient diffusion time is an important control parameter for both biodiversity and the timescale of population dynamics. Interestingly, fast nutrient diffusion allows the populations of some species to fluctuate to zero, leading to extinctions. Moreover, territorial competition spontaneously gives rise to both multistability and the Allee effect (in which a minimum population is required for survival), so that small perturbations can have major ecological effects. While the assumption of trade-offs allows for the coexistence of more species than the number of nutrients-thus violating the principle of competitive exclusion-overall biodiversity is curbed by the domination of "oligotroph" species. Importantly, in contrast to well-mixed models, spatial structure renders diversity robust to inequalities in metabolic trade-offs. Our results suggest that territorial ecosystems can display high biodiversity and rich dynamics simply due to competition for resources in a spatial community.

Metabolic Trade-Offs Promote Diversity in a Model Ecosystem.

In nature, a large number of species can coexist on a small number of shared resources; however, resource-competition models predict that the number of species in steady coexistence cannot exceed the number of resources. Motivated by recent studies of phytoplankton, we introduce trade-offs into a resource-competition model and find that an unlimited number of species can coexist. Our model spontaneously reproduces several notable features of natural ecosystems, including keystone species and population dynamics and abundances characteristic of neutral theory, despite an underlying non-neutral competition for resources.

Symmetry, gauge freedoms, and the interpretability of sequence-function relationships.

Quantitative models that describe how biological sequences encode functional activities are ubiquitous in modern biology. One important aspect of these models is that they commonly exhibit gauge freedoms, i.e., directions in parameter space that do not affect model predictions. In physics, gauge freedoms arise when physical theories are formulated in ways that respect fundamental symmetries. However, the connections that gauge freedoms in models of sequence-function relationships have to the symmetries of sequence space have yet to be systematically studied. Here we study the gauge freedoms of models that respect a specific symmetry of sequence space: the group of position-specific character permutations. We find that gauge freedoms arise when model parameters transform under redundant irreducible matrix representations of this group. Based on this finding, we describe an "embedding distillation" procedure that enables analytic calculation of the number of independent gauge freedoms, as well as efficient computation of a sparse basis for the space of gauge freedoms. We also study how parameter transformation behavior affects parameter interpretability. We find that in many (and possibly all) nontrivial models, the ability to interpret individual model parameters as quantifying intrinsic allelic effects requires that gauge freedoms be present. This finding establishes an incompatibility between two distinct notions of parameter interpretability. Our work thus advances the understanding of symmetries, gauge freedoms, and parameter interpretability in sequence-function relationships.

Gauge fixing for sequence-function relationships.

Quantitative models of sequence-function relationships are ubiquitous in computational biology, e.g., for modeling the DNA binding of transcription factors or the fitness landscapes of proteins. Interpreting these models, however, is complicated by the fact that the values of model parameters can often be changed without affecting model predictions. Before the values of model parameters can be meaningfully interpreted, one must remove these degrees of freedom (called "gauge freedoms" in physics) by imposing additional constraints (a process called "fixing the gauge"). However, strategies for fixing the gauge of sequence-function relationships have received little attention. Here we derive an analytically tractable family of gauges for a large class of sequence-function relationships. These gauges are derived in the context of models with all-order interactions, but an important subset of these gauges can be applied to diverse types of models, including additive models, pairwise-interaction models, and models with higher-order interactions. Many commonly used gauges are special cases of gauges within this family. We demonstrate the utility of this family of gauges by showing how different choices of gauge can be used both to explore complex activity landscapes and to reveal simplified models that are approximately correct within localized regions of sequence space. The results provide practical gauge-fixing strategies and demonstrate the utility of gauge-fixing for model exploration and interpretation.

MAVE-NN: learning genotype-phenotype maps from multiplex assays of variant effect.

Multiplex assays of variant effect (MAVEs) are a family of methods that includes deep mutational scanning experiments on proteins and massively parallel reporter assays on gene regulatory sequences. Despite their increasing popularity, a general strategy for inferring quantitative models of genotype-phenotype maps from MAVE data is lacking. Here we introduce MAVE-NN, a neural-network-based Python package that implements a broadly applicable information-theoretic framework for learning genotype-phenotype maps-including biophysically interpretable models-from MAVE datasets. We demonstrate MAVE-NN in multiple biological contexts, and highlight the ability of our approach to deconvolve mutational effects from otherwise confounding experimental nonlinearities and noise.

Selection for Protein Stability Enriches for Epistatic Interactions.

A now classical argument for the marginal thermodynamic stability of proteins explains the distribution of observed protein stabilities as a consequence of an entropic pull in protein sequence space. In particular, most sequences that are sufficiently stable to fold will have stabilities near the folding threshold. Here, we extend this argument to consider its predictions for epistatic interactions for the effects of mutations on the free energy of folding. Although there is abundant evidence to indicate that the effects of mutations on the free energy of folding are nearly additive and conserved over evolutionary time, we show that these observations are compatible with the hypothesis that a non-additive contribution to the folding free energy is essential for observed proteins to maintain their native structure. In particular, through both simulations and analytical results, we show that even very small departures from additivity are sufficient to drive this effect.

Microbial consortia at steady supply.

Metagenomics has revealed hundreds of species in almost all microbiota. In a few well-studied cases, microbial communities have been observed to coordinate their metabolic fluxes. In principle, microbes can divide tasks to reap the benefits of specialization, as in human economies. However, the benefits and stability of an economy of microbial specialists are far from obvious. Here, we physically model the population dynamics of microbes that compete for steadily supplied resources. Importantly, we explicitly model the metabolic fluxes yielding cellular biomass production under the constraint of a limited enzyme budget. We find that population dynamics generally leads to the coexistence of different metabolic types. We establish that these microbial consortia act as cartels, whereby population dynamics pins down resource concentrations at values for which no other strategy can invade. Finally, we propose that at steady supply, cartels of competing strategies automatically yield maximum biomass, thereby achieving a collective optimum.