We investigate a type of bistability occurring in population systems where noise not only causes transitions between stable states, but also constructs the states themselves. We focus on the experimentally well-studied system of ants choosing between two food sources to illustrate the essential points, but the ideas are more general. The mean time for switching between the two bistable states of the system is calculated. This suggests a procedure for estimating, in a real system, the critical population size above which bistability ceases to occur.

Mutualisms between species play an important role in ecosystem function and stability. However, in some environments, the competitive aspects of an interaction may dominate the mutualistic aspects. Although these transitions could have far-reaching implications, it has been difficult to study the causes and consequences of this mutualistic–competitive transition in experimentally tractable systems. Here, we study a microbial cross-feeding mutualism in which each yeast strain supplies an essential amino acid for its partner strain. We find that, depending upon the amount of freely available amino acid in the environment, this pair of strains can exhibit an obligatory mutualism, facultative mutualism, competition, parasitism, competitive exclusion, or failed mutualism leading to extinction of the population. A simple model capturing the essential features of this interaction explains how resource availability modulates the interaction and predicts that changes in the dynamics of the mutualism in deteriorating environments can provide advance warning that collapse of the mutualism is imminent. We confirm this prediction experimentally by showing that, in the high nutrient competitive regime, the strains rapidly reach a common carrying capacity before slowly reaching the equilibrium ratio between the strains. However, in the low nutrient regime, before collapse of the obligate mutualism, we find that the ratio rapidly reaches its equilibrium and it is the total abundance that is slow to reach equilibrium. Our results provide a general framework for how mutualisms may transition between qualitatively different regimes of interaction in response to changes in nutrient availability in the environment.

A stochastic version of the Brusselator model is proposed and studied via the system size expansion. The mean-field equations are derived and shown to yield to organized Turing patterns within a specific parameters region. When determining the Turing condition for instability, we pay particular attention to the role of cross-diffusive terms, often neglected in the heuristic derivation of reaction-diffusion schemes. Stochastic fluctuations are shown to give rise to spatially ordered solutions, sharing the same quantitative characteristic of the mean-field based Turing scenario, in term of excited wavelengths. Interestingly, the region of parameter yielding to the stochastic self-organization is wider than that determined via the conventional Turing approach, suggesting that the condition for spatial order to appear can be less stringent than customarily believed.

We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics which generalise the more familiar macroscopic equations. We apply this formalism to the analysis of two specific noiseinduced phenomena observed in biologically-inspired models. In the first example, we show how the stochastic amplification of a Turing instability gives rise to spatial and temporal patterns which may be understood within the linear noise approximation. The second example concerns the spontaneous emergence of cell polarity, where we make analytic progress by exploiting a separation of time-scales.

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The observed single-handedness of biological amino acids and sugars has long been attributed to autocatalysis. However, the stability of homochiral states in deterministic autocatalytic systems relies on cross inhibition of the two chiral states, an unlikely scenario for early life self-replicators. Here, we present a theory for a stochastic individual-level model of autocatalysis due to early life selfreplicators. Without chiral inhibition, the racemic state is the global attractor of the deterministic dynamics, but intrinsic multiplicative noise stabilizes the homochiral states, in both well-mixed and spatially-extended systems. We conclude that autocatalysis is a viable mechanism for homochirality, without imposing additional nonlinearities such as chiral inhibition.PACS numbers: 87.23. Kg, 87.18.Tt, One of the very few universal features of biology is homochirality: every naturally occurring amino acid is left-handed ( L-chiral) while every sugar is right-handed (d-chiral) [1, 2]. Although such unexpected broken symmetries are well-known in physics, for example in the weak interaction, complete biological homochirality still defies explanation. In 1953, Charles Frank suggested that homochirality could be a consequence of chemical autocatalysis [3], frequently presumed to be the mechanism associated with the emergence of early life selfreplicators. Frank introduced a model in which the d and L enantiomers of a chiral molecule are autocatalytically produced from an achiral molecule A in reactions A + d → 2d and A + L → 2 L, and are consumed in a chiral inhibition reaction, d + L → 2A [4]. The state of this system can be described by the chiral order parameter ω defined as ω ≡ (d − l)/(d + l), where d and l are the concentrations of d and L. The order parameter ω is zero at the racemic state, and ±1 at the homochiral states. Frank's model has three deterministic fixed points of the dynamics; the racemic state is an unstable fixed point, and the two homochiral states are stable fixed points. Starting from almost everywhere in the d-L plane, the system converges to one of the homochiral fixed points (Fig. 1a).In the context of biological homochirality, extensions of Frank's idea have essentially taken two directions. On the one hand, the discovery of a synthetic chemical system of amino alcohols that amplifies an initial excess of one of the chiral states [5] has motivated several autocatalysisbased models (see [6] and references therein). On the other hand, ribozyme-driven catalyst experiments [7], have inspired theories based on polymerization and chiral inhibition that minimize [8-10] or do not include at all [11,12] autocatalysis. In contrast, a recent experimental realization of RNA replication using a novel ribozyme shows such efficient autocatalytic behavior that chiral inhibition does not arise [13]. Further extensions accounting for both intrinsic noise [6,14] and diffusion [15][16][17][18] build further upon Frank's work.Regardless of the specific model details, all these models share the three-fixed-po...

The amplitude of fluctuation-induced patterns might be expected to be proportional to the strength of the driving noise, suggesting that such patterns would be difficult to observe in nature. Here, we show that a large class of spatially-extended dynamical systems driven by intrinsic noise can exhibit giant amplification, yielding patterns whose amplitude is comparable to that of deterministic Turing instabilities. The giant amplification results from the interplay between noise and non-orthogonal eigenvectors of the linear stability matrix, yielding transients that grow with time, and which, when driven by the ever-present intrinsic noise, lead to persistent large amplitude patterns. This mechanism provides a robust basis for fluctuation-induced biological pattern formation based on the Turing mechanism, without requiring fine tuning of diffusion constants.

The origin of homochirality, the observed single-handedness of biological amino acids and sugars, has long been attributed to autocatalysis, a frequently assumed precursor for early life self-replication. However, the stability of homochiral states in deterministic autocatalytic systems relies on cross-inhibition of the two chiral states, an unlikely scenario for early life self-replicators. Here we present a theory for a stochastic individual-level model of autocatalytic prebiotic self-replicators that are maintained out of thermal equilibrium. Without chiral inhibition, the racemic state is the global attractor of the deterministic dynamics, but intrinsic multiplicative noise stabilizes the homochiral states. Moreover, we show that this noise-induced bistability is robust with respect to diffusion of molecules of opposite chirality, and systems of diffusively coupled autocatalytic chemical reactions synchronize their final homochiral states when the self-replication is the dominant production mechanism for the chiral molecules. We conclude that nonequilibrium autocatalysis is a viable mechanism for homochirality, without imposing additional nonlinearities such as chiral inhibition. DOI: 10.1103/PhysRevE.95.032407 Homochirality, the single-handedness of all biological amino acids and sugars, is one of two major universal features of life on Earth. The other is the canonical genetic code. Their universality transcends all categories of life, up to and including the three domains, and thus requires an explanation that transcends the idiosyncrasies of individual organisms and particular environments. The only universal process common to all life is, of course, evolution, and so it is natural to seek an explanation for biological homochirality in these terms, just as has been done to account for the universality and error-minimization aspects of the genetic code [1]. This paper is just such an attempt, using the simplest and most general commonly accepted attributes of living systems.The origin of biological homochirality has been one of the most debated topics since its discovery by Louis Pasteur in 1848 [2]. There are those who argue that homochirality must have preceded the first chemical systems undergoing Darwinian evolution, and there are those who believe homochirality is a consequence of life, but not a prerequisite [3]. There are even those who argue that homochirality is a consequence of underlying asymmetries from the laws of physics, invoking complicated astrophysical scenarios for the origin of chiral organic molecules [4] or even the violation of parity from the weak interactions [5,6]! In fact, explanations that are based on physical asymmetries can only predict an enantiomeric excess of one handedness over another, and not the 100% effect observed in nature [7]. * Corresponding author: fjafarpo@purdue.eduThe most influential class of theories for biological homochirality rest on an idea of F.C. Frank's, in which there is a kinetic instability of a racemic (50% right handed and 50% left handed) mi...