Stochastic Turing patterns of trichomes in Arabidopsis leaves.

Biological patterns that emerge during the morphogenesis of multicellular organisms can display high precision at large scales, while at cellular scales, cells exhibit large fluctuations stemming from cell-cell differences in molecular copy numbers also called demographic noise. We study the conflicting interplay between high precision and demographic noise in trichome patterns on the epidermis of wild-type Arabidopsis thaliana leaves, as a two-dimensional model system. We carry out a statistical characterization of these patterns and show that their power spectra display fat tails-a signature compatible with noise-driven stochastic Turing patterns-which are absent in power spectra of patterns driven by deterministic instabilities. We then present a theoretical model that includes demographic noise stemming from birth-death processes of genetic regulators which we study analytically and by stochastic simulations. The model captures the observed experimental features of trichome patterns.

Controlling the Mean Time to Extinction in Populations of Bacteria.

Populations of ecological systems generally have demographic fluctuations due to birth and death processes. At the same time, they are exposed to changing environments. We studied populations composed of two phenotypes of bacteria and analyzed the impact that both types of fluctuations have on the mean time to extinction of the entire population if extinction is the final fate. Our results are based on Gillespie simulations and on the WKB approach applied to classical stochastic systems, here in certain limiting cases. As a function of the frequency of environmental changes, we observe a non-monotonic dependence of the mean time to extinction. Its dependencies on other system parameters are also explored. This allows the control of the mean time to extinction to be as large or as small as possible, depending on whether extinction should be avoided or is desired from the perspective of bacteria or the perspective of hosts to which the bacteria are deleterious.

Effect of compositional fluctuation on the survival of bet-hedging species.

Understanding the coexistence of diverse species in a changing environment is an important problem in community ecology. Bet-hedging is a strategy that helps species survive in such changing environments. However, studies of bet-hedging have often focused on the expected long-term growth rate of the species by itself, neglecting competition with other coexisting species. Here we study the extinction risk of a bet-hedging species in competition with others. We show that there are three contributions to the extinction risk. The first is the usual demographic fluctuation due to stochastic reproduction and selection processes in finite populations. The second, due to the fluctuation of population growth rate caused by environmental changes, may actually reduce the extinction risk for small populations. Besides those two, we reveal a third contribution, which is unique to bet-hedging species that diversify into multiple phenotypes: The phenotype composition of the population will fluctuate over time, resulting in increased extinction risk. We compare such compositional fluctuation to the demographic and environmental contributions, showing how they have different effects on the extinction risk depending on the population size, generation overlap, and environmental correlation.

Allee Effects Plus Noise Induce Population Dynamics Resembling Binary Markov Highs and Lows.

We show that the combination of Allee effects and noise can produce a stochastic process with alternating sudden decline to a low population phase, followed, after a random time, by abrupt increase in population density. We introduce a new, flexible, deterministic model of attenuated Allee effects, which interpolates between the logistic and a usual Allee model. Into this model, we incorporate environmental and demographic noise. The solution of the resulting Kolmogorov forward equation shows a dichotomous distribution of residence times with heavy occupation of high, near saturation, and low population states. Investigation of simulated sample paths reveals that indeed attenuated Allee effects and noise, acting together, produce alternating, sustained, low and high population levels. We find that the transition times between the two types of states are approximately exponentially distributed, with different parameters, rendering the embedded hi-low process approximately Markov.

Exclusion of the fittest predicts microbial community diversity in fluctuating environments.

Microorganisms live in environments that inevitably fluctuate between mild and harsh conditions. As harsh conditions may cause extinctions, the rate at which fluctuations occur can shape microbial communities and their diversity, but we still lack an intuition on how. Here, we build a mathematical model describing two microbial species living in an environment where substrate supplies randomly switch between abundant and scarce. We then vary the rate of switching as well as different properties of the interacting species, and measure the probability of the weaker species driving the stronger one extinct. We find that this probability increases with the strength of demographic noise under harsh conditions and peaks at either low, high, or intermediate switching rates depending on both species' ability to withstand the harsh environment. This complex relationship shows why finding patterns between environmental fluctuations and diversity has historically been difficult. In parameter ranges where the fittest species was most likely to be excluded, however, the beta diversity in larger communities also peaked. In sum, how environmental fluctuations affect interactions between a few species pairs predicts their effect on the beta diversity of the whole community.

Robust, coherent, and synchronized circadian clock-controlled oscillations along Anabaena filaments.

Circadian clocks display remarkable reliability despite significant stochasticity in biomolecular reactions. We study the dynamics of a circadian clock-controlled gene at the individual cell level in Anabaena sp. PCC 7120, a multicellular filamentous cyanobacterium. We found significant synchronization and spatial coherence along filaments, clock coupling due to cell-cell communication, and gating of the cell cycle. Furthermore, we observed low-amplitude circadian oscillatory transcription of kai genes encoding the post-transcriptional core oscillatory circuit and high-amplitude oscillations of rpaA coding for the master regulator transducing the core clock output. Transcriptional oscillations of rpaA suggest an additional level of regulation. A stochastic one-dimensional toy model of coupled clock cores and their phosphorylation states shows that demographic noise can seed stochastic oscillations outside the region where deterministic limit cycles with circadian periods occur. The model reproduces the observed spatio-temporal coherence along filaments and provides a robust description of coupled circadian clocks in a multicellular organism.

Cooperation in Microbial Populations: Theory and Experimental Model Systems.

Cooperative behavior, the costly provision of benefits to others, is common across all domains of life. This review article discusses cooperative behavior in the microbial world, mediated by the exchange of extracellular products called public goods. We focus on model species for which the production of a public good and the related growth disadvantage for the producing cells are well described. To unveil the biological and ecological factors promoting the emergence and stability of cooperative traits we take an interdisciplinary perspective and review insights gained from both mathematical models and well-controlled experimental model systems. Ecologically, we include crucial aspects of the microbial life cycle into our analysis and particularly consider population structures where ensembles of local communities (subpopulations) continuously emerge, grow, and disappear again. Biologically, we explicitly consider the synthesis and regulation of public good production. The discussion of the theoretical approaches includes general evolutionary concepts, population dynamics, and evolutionary game theory. As a specific but generic biological example, we consider populations of Pseudomonas putida and its regulation and use of pyoverdines, iron scavenging molecules, as public goods. The review closes with an overview on cooperation in spatially extended systems and also provides a critical assessment of the insights gained from the experimental and theoretical studies discussed. Current challenges and important new research opportunities are discussed, including the biochemical regulation of public goods, more realistic ecological scenarios resembling native environments, cell-to-cell signaling, and multispecies communities.

The Effects of Stochasticity on Pattern Formation in a Space- and Time-Discrete Predator-Prey System with Strong Allee Effect in the Prey.

The effects of demographic and environmental noise on the vital dynamics and spatial pattern formation are studied for a predator-prey system with strong Allee effect in the prey species. Time and space are taken discrete. It is shown that noise can promote extinction depending on the growth and interaction parameters as well as the noise type and amplitude. The extinction risk increases with the noise amplitude; however, the environmental and demographic noise can have different effects on the risk of extinction. In space, the spatial structures obtained are blurred versions of the deterministic ones in most scenarios. In particular, the complex spatial structures that appear in the parameter domains where the deterministic local dynamics leads to extinction are robust to the density-dependent stochastic fluctuations but are disrupted with environmental noise.

Noise⁻Seeded Developmental Pattern Formation in Filamentous Cyanobacteria.

Under nitrogen-poor conditions, multicellular cyanobacteria such as Anabaena sp. PCC 7120 undergo a process of differentiation, forming nearly regular, developmental patterns of individual nitrogen-fixing cells, called heterocysts, interspersed between intervals of vegetative cells that carry out photosynthesis. Developmental pattern formation is mediated by morphogen species that can act as activators and inhibitors, some of which can diffuse along filaments. We survey the limitations of the classical, deterministic Turing mechanism that has been often invoked to explain pattern formation in these systems, and then, focusing on a simpler system governed by birth-death processes, we illustrate pedagogically a recently proposed paradigm that provides a much more robust description of pattern formation: stochastic Turing patterns. We emphasize the essential role that cell-to-cell differences in molecular numbers-caused by inevitable fluctuations in gene expression-play, the so called demographic noise, in seeding the formation of stochastic Turing patterns over a much larger region of parameter space, compared to their deterministic counterparts.

Demographic noise and cost of greenbeard can facilitate greenbeard cooperation.

Cooperation among organisms, where cooperators suffer a personal cost to benefit others, is ubiquitous in nature. Greenbeard is a key mechanism for the evolution of cooperation, where a single gene or a set of linked genes codes for both cooperation and a phenotypic tag (metaphorically called "green beard"). Greenbeard cooperation is typically thought to decline over time since defectors can also evolve the tag. However, models of tag-based cooperation typically ignore two key realistic features: populations are finite, and that phenotypic tags can be costly. We develop an analytical model for coevolutionary dynamics of two evolvable traits in finite populations with mutations: costly cooperation and a costly tag. We show that an interplay of demographic noise and cost of the tag can induce coevolutionary cycling, where the evolving population does not reach a steady state but spontaneously switches between cooperative tag-carrying and noncooperative tagless states. Such dynamics allows the tag to repeatedly reappear even after it is invaded by defectors. Thus, we highlight the surprising possibility that the cost of the tag, together with demographic noise, can facilitate the evolution of greenbeard cooperation. We discuss implications of these findings in the context of the evolution of quorum sensing and multicellularity.

From noise to knowledge: how randomness generates novel phenomena and reveals information.

Noise, as the term itself suggests, is most often seen a nuisance to ecological insight, a inconvenient reality that must be acknowledged, a haystack that must be stripped away to reveal the processes of interest underneath. Yet despite this well-earned reputation, noise is often interesting in its own right: noise can induce novel phenomena that could not be understood from some underlying deterministic model alone. Nor is all noise the same, and close examination of differences in frequency, colour or magnitude can reveal insights that would otherwise be inaccessible. Yet with each aspect of stochasticity leading to some new or unexpected behaviour, the time is right to move beyond the familiar refrain of "everything is important" (Bjørnstad & Grenfell ). Stochastic phenomena can suggest new ways of inferring process from pattern, and thus spark more dialog between theory and empirical perspectives that best advances the field as a whole. I highlight a few compelling examples, while observing that the study of stochastic phenomena are only beginning to make this translation into empirical inference. There are rich opportunities at this interface in the years ahead.

Species coexistence in a neutral dynamics with environmental noise.

Environmental fluctuations have important consequences in the organization of ecological communities, and understanding how such a variability influences the biodiversity of an ecosystem is a major question in ecology. In this paper, we analyze the case of two species competing for the resources within the framework of the neutral theory in the presence of environmental noise, devoting special attention on how such a variability modulates species fitness. The environment is dichotomous and stochastically alternates between periods favoring one of the species while disfavoring the other one, preserving neutrality on the long term. We study two different scenarios: in the first one species fitness varies linearly with the environment, and in the second one the effective fitness is re-scaled by the total fitness of the individuals competing for the same resource. We find that, in the former case environmental fluctuations always reduce the time of species coexistence, whereas such a time can be enhanced or reduced in the latter case, depending on the correlation time of the environment. This phenomenon can be understood as a direct consequence of Chesson's storage effect.

Spatial synchrony in population dynamics: The effects of demographic stochasticity and density regulation with a spatial scale.

We generalize a previous simple result by Lande et al. (1999) on how spatial autocorrelated noise, dispersal rate and distance as well as strength of density regulation determine the spatial scale of synchrony in population density. It is shown how demographic noise can be incorporated, what effect it has on variance and spatial scale of synchrony, and how it interacts with the point process for locations of individuals under random sampling. Although the effect of demographic noise is a rather complex interaction with environmental noise, migration and density regulation, its effect on population fluctuations and scale of synchrony can be presented in a transparent way. This is achieved by defining a characteristic area dependent on demographic and environmental variances as well as population density, and subsequently using this area to define a spatial demographic coefficient. The demographic noise acts through this coefficient on the spatial synchrony, which may increase or decrease with increasing demographic noise depending on other parameters. A second generalization yields the modeling of density regulation taking into account that regulation at a given location does not only depend on the density at that site but also on densities in the whole territory or home range of individuals. It is shown that such density regulation with a spatial scale reduces the scale of synchrony in population fluctuations relative to the simpler model with density regulation at each location determined only by the local point density, and may even generate negative spatial autocorrelations.