Jacopo Grilli

Information-based fitness and the emergence of criticality in living systems

Significance Recently, evidence has been mounting that biological systems might operate at the borderline between order and disorder, i.e., near a critical point. A general mathematical framework for understanding this common pattern, explaining the possible origin and role of criticality in living adaptive and evolutionary systems, is still missing. We rationalize this apparently ubiquitous criticality in terms of adaptive and evolutionary functional advantages. We provide an analytical framework, which demonstrates that the optimal response to broadly different changing environments occurs in systems organizing spontaneously—through adaptation or evolution—to the vicinity of a critical point. Furthermore, criticality turns out to be the evolutionary stable outcome of a community of individuals aimed at communicating with each other to create a collective entity. Empirical evidence suggesting that living systems might operate in the vicinity of critical points, at the borderline between order and disorder, has proliferated in recent years, with examples ranging from spontaneous brain activity to flock dynamics. However, a well-founded theory for understanding how and why interacting living systems could dynamically tune themselves to be poised in the vicinity of a critical point is lacking. Here we use tools from statistical mechanics and information theory to show that complex adaptive or evolutionary systems can be much more efficient in coping with diverse heterogeneous environmental conditions when operating at criticality. Analytical as well as computational evolutionary and adaptive models vividly illustrate that a community of such systems dynamically self-tunes close to a critical state as the complexity of the environment increases while they remain noncritical for simple and predictable environments. A more robust convergence to criticality emerges in coevolutionary and coadaptive setups in which individuals aim to represent other agents in the community with fidelity, thereby creating a collective critical ensemble and providing the best possible tradeoff between accuracy and flexibility. Our approach provides a parsimonious and general mechanism for the emergence of critical-like behavior in living systems needing to cope with complex environments or trying to efficiently coordinate themselves as an ensemble.

Modularity and stability in ecological communities

Networks composed of distinct, densely connected subsystems are called modular. In ecology, it has been posited that a modular organization of species interactions would benefit the dynamical stability of communities, even though evidence supporting this hypothesis is mixed. Here we study the effect of modularity on the local stability of ecological dynamical systems, by presenting new results in random matrix theory, which are obtained using a quaternionic parameterization of the cavity method. Results show that modularity can have moderate stabilizing effects for particular parameter choices, while anti-modularity can greatly destabilize ecological networks.

Statistical mechanics of ecological systems: Neutral theory and beyond

It is of societal importance to advance the understanding of emerging patterns of biodiversity from biological and ecological systems. The neutral theory offers a statistical-mechanical framework that relates key biological properties at the individual scale with macroecological properties at the community scale. This article surveys the quantitative aspects of neutral theory and its extensions for physicists who are interested in what important problems remain unresolved for studying ecological systems.

Higher-order interactions stabilize dynamics in competitive network models

Ecologists have long sought a way to explain how the remarkable biodiversity observed in nature is maintained. On the one hand, simple models of interacting competitors cannot produce the stable persistence of very large ecological communities. On the other hand, neutral models, in which species do not interact and diversity is maintained by immigration and speciation, yield unrealistically small fluctuations in population abundance, and a strong positive correlation between a species’ abundance and its age, contrary to empirical evidence. Models allowing for the robust persistence of large communities of interacting competitors are lacking. Here we show that very diverse communities could persist thanks to the stabilizing role of higher-order interactions, in which the presence of a species influences the interaction between other species. Although higher-order interactions have been studied for decades, their role in shaping ecological communities is still unclear. The inclusion of higher-order interactions in competitive network models stabilizes dynamics, making species coexistence robust to the perturbation of both population abundance and parameter values. We show that higher-order interactions have strong effects in models of closed ecological communities, as well as of open communities in which new species are constantly introduced. In our framework, higher-order interactions are completely defined by pairwise interactions, facilitating empirical parameterization and validation of our models.

Feasibility and coexistence of large ecological communities

The role of species interactions in controlling the interplay between the stability of ecosystems and their biodiversity is still not well understood. The ability of ecological communities to recover after small perturbations of the species abundances (local asymptotic stability) has been well studied, whereas the likelihood of a community to persist when the conditions change (structural stability) has received much less attention. Our goal is to understand the effects of diversity, interaction strengths and ecological network structure on the volume of parameter space leading to feasible equilibria. We develop a geometrical framework to study the range of conditions necessary for feasible coexistence. We show that feasibility is determined by few quantities describing the interactions, yielding a nontrivial complexity–feasibility relationship. Analysing more than 100 empirical networks, we show that the range of coexistence conditions in mutualistic systems can be analytically predicted. Finally, we characterize the geometric shape of the feasibility domain, thereby identifying the direction of perturbations that are more likely to cause extinctions.

Effect of localization on the stability of mutualistic ecological networks

The relationships between the core–periphery architecture of the species interaction network and the mechanisms ensuring the stability in mutualistic ecological communities are still unclear. In particular, most studies have focused their attention on asymptotic resilience or persistence, neglecting how perturbations propagate through the system. Here we develop a theoretical framework to evaluate the relationship between the architecture of the interaction networks and the impact of perturbations by studying localization, a measure describing the ability of the perturbation to propagate through the network. We show that mutualistic ecological communities are localized, and localization reduces perturbation propagation and attenuates its impact on species abundance. Localization depends on the topology of the interaction networks, and it positively correlates with the variance of the weighted degree distribution, a signature of the network topological heterogeneity. Our results provide a different perspective on the interplay between the architecture of interaction networks in mutualistic communities and their stability.

Spatial aggregation and the species–area relationship across scales

There has been a considerable effort to understand and quantify the spatial distribution of species across different ecosystems. Relative species abundance (RSA), beta diversity and species-area relationship (SAR) are among the most used macroecological measures to characterize plants communities in forests. In this paper we introduce a simple phenomenological model based on Poisson cluster processes which allows us to exactly link RSA and beta diversity to SAR. The framework is spatially explicit and accounts for the spatial aggregation of conspecific individuals. Under the simplifying assumption of neutral theory, we derive an analytical expression for the SAR which reproduces tri-phasic behavior as sample area increases from local to continental scales, explaining how the tri-phasic behavior can be understood in terms of simple geometric arguments. We also find an expression for the endemic area relationship (EAR) and for the scaling of the RSA.

Large-scale dynamics of horizontal transfers

The widespread exchange of genes between bacteria must have consequences on the global architecture of their genomes, which are being found in the abundant genomic data available today. Most of the expansion of bacterial protein families can be attributed to transfer events, which are positively biased for smaller evolutionary distances between genomes, and more frequent for classes that are larger, when summed over all known bacteria. Moreover, “innovation” events where horizontal transfers carry exogenous evolutionary families appear to be less frequent for larger genomes. This dynamic expansion of evolutionary families is interconnected with the acquisition of new biological functions and thus with the size and distribution of the genes’ functional categories found on a genome. This commentary presents our recent contributions to this line of work and possible future directions.

Metapopulation Persistence in Random Fragmented Landscapes

Habitat destruction and land use change are making the world in which natural populations live increasingly fragmented, often leading to local extinctions. Although local populations might undergo extinction, a metapopulation may still be viable as long as patches of suitable habitat are connected by dispersal, so that empty patches can be recolonized. Thus far, metapopulations models have either taken a mean-field approach, or have modeled empirically-based, realistic landscapes. Here we show that an intermediate level of complexity between these two extremes is to consider random landscapes, in which the patches of suitable habitat are randomly arranged in an area (or volume). Using methods borrowed from the mathematics of Random Geometric Graphs and Euclidean Random Matrices, we derive a simple, analytic criterion for the persistence of the metapopulation in random fragmented landscapes. Our results show how the density of patches, the variability in their value, the shape of the dispersal kernel, and the dimensionality of the landscape all contribute to determining the fate of the metapopulation. Using this framework, we derive sufficient conditions for the population to be spatially localized, such that spatially confined clusters of patches act as a source of dispersal for the whole landscape. Finally, we show that a regular arrangement of the patches is always detrimental for persistence, compared to the random arrangement of the patches. Given the strong parallel between metapopulation models and contact processes, our results are also applicable to models of disease spread on spatial networks.

Cross-species gene-family fluctuations reveal the dynamics of horizontal transfers

Prokaryotes vary their protein repertoire mainly through horizontal transfer and gene loss. To elucidate the links between these processes and the cross-species gene-family statistics, we perform a large-scale data analysis of the cross-species variability of gene-family abundance (the number of members of the family found on a given genome). We find that abundance fluctuations are related to the rate of horizontal transfers. This is rationalized by a minimal theoretical model, which predicts this link. The families that are not captured by the model show abundance profiles that are markedly peaked around a mean value, possibly because of specific abundance selection. Based on these results, we define an abundance variability index that captures a familys evolutionary behavior (and thus some of its relevant functional properties) purely based on its cross-species abundance fluctuations. Analysis and model, combined, show a quantitative link between cross-species family abundance statistics and horizontal transfer dynamics, which can be used to analyze genome ‘flux’. Groups of families with different values of the abundance variability index correspond to genome sub-parts having different plasticity in terms of the level of horizontal exchange allowed by natural selection.