The emergence of waves in random discrete systems

Essential criteria for the emergence of wave-like manifestations occurring in an entirely discrete system are identified using a simple model for the movement of particles through a network. The dynamics are entirely stochastic and memoryless involving a birth-death-migration process. The requirements are that the network should have at least three nodes, that migration should have a directional bias, and that the particle dynamics have a non-local dependence. Well defined bifurcations mark transitions between amorphous, wave-like and collapsed states with an intermittent regime between the latter two

A Simple Stochastic Model with Environmental Transmission Explains Multi-Year Periodicity in Outbreaks of Avian Flu

Avian influenza virus reveals persistent and recurrent outbreaks in North American wild waterfowl, and exhibits major outbreaks at 2–8 years intervals in duck populations. The standard susceptible-infected- recovered (SIR) framework, which includes seasonal migration and reproduction, but lacks environmental transmission, is unable to reproduce the multi-periodic patterns of avian influenza epidemics. In this paper, we argue that a fully stochastic theory based on environmental transmission provides a simple, plausible explanation for the phenomenon of multi-year periodic outbreaks of avian flu. Our theory predicts complex fluctuations with a dominant period of 2 to 8 years which essentially depends on the intensity of environmental transmission. A wavelet analysis of the observed data supports this prediction. Furthermore, using master equations and van Kampen system-size expansion techniques, we provide an analytical expression for the spectrum of stochastic fluctuations, revealing how the outbreak period varies with the environmental transmission

Finite-size and correlation-induced effects in Mean-field Dynamics

The brain's activity is characterized by the interaction of a very large number of neurons that are strongly affected by noise. However, signals often arise at macroscopic scales integrating the effect of many neurons into a reliable pattern of activity. In order to study such large neuronal assemblies, one is often led to derive mean-field limits summarizing the effect of the interaction of a large number of neurons into an effective signal. Classical mean-field approaches consider the evolution of a deterministic variable, the mean activity, thus neglecting the stochastic nature of neural behavior. In this article, we build upon two recent approaches that include correlations and higher order moments in mean-field equations, and study how these stochastic effects influence the solutions of the mean-field equations, both in the limit of an infinite number of neurons and for large yet finite networks. We introduce a new model, the infinite model, which arises from both equations by a rescaling of the variables and, which is invertible for finite-size networks, and hence, provides equivalent equations to those previously derived models. The study of this model allows us to understand qualitative behavior of such large-scale networks. We show that, though the solutions of the deterministic mean-field equation constitute uncorrelated solutions of the new mean-field equations, the stability properties of limit cycles are modified by the presence of correlations, and additional non-trivial behaviors including periodic orbits appear when there were none in the mean field. The origin of all these behaviors is then explored in finite-size networks where interesting mesoscopic scale effects appear. This study leads us to show that the infinite-size system appears as a singular limit of the network equations, and for any finite network, the system will differ from the infinite system

Nonlinear Sigma Model for Disordered Media: Replica Trick for Non-Perturbative Results and Interactions

In these lectures, given at the NATO ASI at Windsor (2001), applications of the replicas nonlinear sigma model to disordered systems are reviewed. A particular attention is given to two sets of issues. First, obtaining non-perturbative results in the replica limit is discussed, using as examples (i) an oscillatory behaviour of the two-level correlation function and (ii) long-tail asymptotes of different mesoscopic distributions. Second, a new variant of the sigma model for interacting electrons in disordered normal and superconducting systems is presented, with demonstrating how to reduce it, under certain controlled approximations, to known ``phase-only'' actions, including that of the ``dirty bosons'' model.Comment: 25 pages, Proceedings of the NATO ASI "Field Theory of Strongly Correlated Fermions and Bosons in Low - Dimensional Disordered Systems", Windsor, August, 2001; to be published by Kluwe

A new hammer to crack an old nut : interspecific competitive resource capture by plants is regulated by nutrient supply, not climate

Peer reviewedPublisher PD

Investigating the robustness of the classical enzyme kinetic equations in small intracellular compartments

<p>Abstract</p> <p>Background</p> <p>Classical descriptions of enzyme kinetics ignore the physical nature of the intracellular environment. Main implicit assumptions behind such approaches are that reactions occur in compartment volumes which are large enough so that molecular discreteness can be ignored and that molecular transport occurs via diffusion. Though these conditions are frequently met in laboratory conditions, they are not characteristic of the intracellular environment, which is compartmentalized at the micron and submicron scales and in which active means of transport play a significant role.</p> <p>Results</p> <p>Starting from a master equation description of enzyme reaction kinetics and assuming metabolic steady-state conditions, we derive novel mesoscopic rate equations which take into account (i) the intrinsic molecular noise due to the low copy number of molecules in intracellular compartments (ii) the physical nature of the substrate transport process, i.e. diffusion or vesicle-mediated transport. These equations replace the conventional macroscopic and deterministic equations in the context of intracellular kinetics. The latter are recovered in the limit of infinite compartment volumes. We find that deviations from the predictions of classical kinetics are pronounced (hundreds of percent in the estimate for the reaction velocity) for enzyme reactions occurring in compartments which are smaller than approximately 200 nm, for the case of substrate transport to the compartment being mediated principally by vesicle or granule transport and in the presence of competitive enzyme inhibitors.</p> <p>Conclusion</p> <p>The derived mesoscopic rate equations describe subcellular enzyme reaction kinetics, taking into account, for the first time, the simultaneous influence of both intrinsic noise and the mode of transport. They clearly show the range of applicability of the conventional deterministic equation models, namely intracellular conditions compatible with diffusive transport and simple enzyme mechanisms in several hundred nanometre-sized compartments. An active transport mechanism coupled with large intrinsic noise in enzyme concentrations is shown to lead to huge deviations from the predictions of deterministic models. This has implications for the common approach of modeling large intracellular reaction networks using ordinary differential equations and also for the calculation of the effective dosage of competitive inhibitor drugs.</p

Effects of bursty protein production on the noisy oscillatory properties of downstream pathways

Experiments show that proteins are translated in sharp bursts; similar bursty phenomena have been observed for protein import into compartments. Here we investigate the effect of burstiness in protein expression and import on the stochastic properties of downstream pathways. We consider two identical pathways with equal mean input rates, except in one pathway proteins are input one at a time and in the other proteins are input in bursts. Deterministically the dynamics of these two pathways are indistinguishable. However the stochastic behavior falls in three categories: (i) both pathways display or do not display noise-induced oscillations; (ii) the non-bursty input pathway displays noise-induced oscillations whereas the bursty one does not; (iii) the reverse of (ii). We derive necessary conditions for these three cases to classify systems involving autocatalysis, trimerization and genetic feedback loops. Our results suggest that single cell rhythms can be controlled by regulation of burstiness in protein production

Asymmetry in Species Regional Dispersal Ability and the Neutral Theory

The neutral assumption that individuals of either the same or different species share exactly the same birth, death, migration, and speciation probabilities is fundamental yet controversial to the neutral theory. Several theoretical studies have demonstrated that a slight difference in species per capita birth or death rates can have a profound consequence on species coexistence and community structure. Whether asymmetry in migration, a vital demographic parameter in the neutral model, plays an important role in community assembly still remains unknown. In this paper, we relaxed the ecological equivalence assumption of the neutral model by introducing differences into species regional dispersal ability. We investigated the effect of asymmetric dispersal on the neutral local community structure. We found that per capita asymmetric dispersal among species could reduce species richness of the local community and result in deviations of species abundance distributions from those predicted by the neutral model. But the effect was moderate compared with that of asymmetries in birth or death rates, unless very large asymmetries in dispersal were assumed. A large difference in species dispersal ability, if there is, can overwhelm the role of random drift and make local community dynamics deterministic. In this case, species with higher regional dispersal abilities tended to dominate in the local community. However, the species abundance distribution of the local community under asymmetric dispersal could be well fitted by the neutral model, but the neutral model generally underestimated the fundamental biodiversity number but overestimated the migration rate in such communities

The frequency-dependent Wright-Fisher model: diffusive and non-diffusive approximations

We study a class of processes that are akin to the Wright-Fisher model, with transition probabilities weighted in terms of the frequency-dependent fitness of the population types. By considering an approximate weak formulation of the discrete problem, we are able to derive a corresponding continuous weak formulation for the probability density. Therefore, we obtain a family of partial differential equations (PDE) for the evolution of the probability density, and which will be an approximation of the discrete process in the joint large population, small time-steps and weak selection limit. If the fitness functions are sufficiently regular, we can recast the weak formulation in a more standard formulation, without any boundary conditions, but supplemented by a number of conservation laws. The equations in this family can be purely diffusive, purely hyperbolic or of convection-diffusion type, with frequency dependent convection. The particular outcome will depend on the assumed scalings. The diffusive equations are of the degenerate type; using a duality approach, we also obtain a frequency dependent version of the Kimura equation without any further assumptions. We also show that the convective approximation is related to the replicator dynamics and provide some estimate of how accurate is the convective approximation, with respect to the convective-diffusion approximation. In particular, we show that the mode, but not the expected value, of the probability distribution is modelled by the replicator dynamics. Some numerical simulations that illustrate the results are also presented