Probing Noise in Gene Expression and Protein Production

We derive exact solutions of simplified models for the temporal evolution of the protein concentration within a cell population arbitrarily far from the stationary state. We show that monitoring the dynamics can assist in modeling and understanding the nature of the noise and its role in gene expression and protein production. We introduce a new measure, the cell turnover distribution, which can be used to probe the phase of transcription of DNA into messenger RNA.Comment: 10 pages, 3 figures, supplementary information on reques

Upscaling Species Richness and Abundances in Tropical Forests

The quantification of tropical tree biodiversity worldwide remains an open and challenging problem. More than two-fifths of the number of worldwide trees can be found either in tropical or in subtropical forests, but only ≈0.000067% of species identities are known. We introduce an analytical framework that provides robust and accurate estimates of species richness and abundances in biodiversity-rich ecosystems, as confirmed by tests performed on both in silico–generated and real forests. Our analysis shows that the approach outperforms other methods. In particular, we find that upscaling methods based on the log-series species distribution systematically overestimate the number of species and abundances of the rare species. We finally apply our new framework on 15 empirical tropical forest plots and quantify the minimum percentage cover that should be sampled to achieve a given average confidence interval in the upscaled estimate of biodiversity. Our theoretical framework confirms that the forests studied are comprised of a large number of rare or hyper-rare species. This is a signature of critical-like behavior of species-rich ecosystems and can provide a buffer against extinction

Growth-rate distributions of gut microbiota time series: neutral models and temporal dependence

Logarithmic growth-rates are fundamental observables for describing ecological systems and the characterization of their distributions with analytical techniques can greatly improve their comprehension. Here a neutral model based on a stochastic differential equation with demographic noise, which presents a closed form for these distributions, is used to describe the population dynamics of microbiota. Results show that this model can successfully reproduce the log-growth rate distribution of the considered abundance time-series. More significantly, it predicts its temporal dependence, by reproducing its kurtosis evolution when the time lag $\tau$ is increased. Furthermore, its typical shape for large $\tau$ is assessed, verifying that the distribution variance does not diverge with $\tau$. The simulated processes generated by the calibrated stochastic equation and the analysis of each time-series, taken one by one, provided additional support for our approach. Alternatively, we tried to describe our dataset by using a logistic model with an environmental stochastic term. Analytical and numerical results show that this model is not suited for describing the leptokurtic log-growth rates distribution found in our data. These results effectively support a neutral model with demographic stochasticity for describing the growth-rate dynamics and the stationary abundance distribution of the considered microbiota. This suggests that there are no significant parametric demographic differences among the species, which can be statistically characterized by the same vital rates.Comment: 14 pages, 6 figure

Large system population dynamics with non-Gaussian interactions

We investigate the Generalized Lotka-Volterra (GLV) equations, a central model in theoretical ecology, where species interactions are assumed to be fixed over time and heterogeneous (quenched noise). Recent studies have suggested that the stability properties and abundance distributions of large disordered GLV systems depend, in the simplest scenario, solely on the mean and variance of the distribution of species interactions. However, empirical communities deviate from this level of universality. In this article, we present a generalized version of the dynamical mean field theory for non-Gaussian interactions that can be applied to various models, including the GLV equations. Our results show that the generalized mean field equations have solutions which depend on all cumulants of the distribution of species interactions, leading to a breakdown of universality. We leverage on this informative breakdown to extract microscopic interaction details from the macroscopic distribution of densities which are in agreement with empirical data. Specifically, in the case of sparse interactions, which we analytically investigate, we establish a simple relationship between the distribution of interactions and the distribution of species population densities

Asymmetric Stochastic Resetting: Modeling Catastrophic Events

In the classical stochastic resetting problem, a particle, moving according to some stochastic dynamics, undergoes random interruptions that bring it to a selected domain, and then, the process recommences. Hitherto, the resetting mechanism has been introduced as a symmetric reset about the preferred location. However, in nature, there are several instances where a system can only reset from certain directions, e.g., catastrophic events. Motivated by this, we consider a continuous stochastic process on the positive real line. The process is interrupted at random times occurring at a constant rate, and then, the former relocates to a value only if the current one exceeds a threshold; otherwise, it follows the trajectory defined by the underlying process without resetting. We present a general framework to obtain the exact non-equilibrium steady state of the system and the mean first passage time for the system to reach the origin. Employing this framework, we obtain the explicit solutions for two different model systems. Some of the classical results found in symmetric resetting such as the existence of an optimal resetting, are strongly modified. Finally, numerical simulations have been performed to verify the analytical findings, showing an excellent agreement.Comment: 10 pages including: main text with 6 figures and appendice

Spatial Patterns Emerging from a Stochastic Process Near Criticality

There is mounting empirical evidence that many communities of living organisms display key features which closely resemble those of physical systems at criticality. We here introduce a minimal model framework for the dynamics of a community of individuals which undergoes local birth-death, immigration, and local jumps on a regular lattice. We study its properties when the system is close to its critical point. Even if this model violates detailed balance, within a physically relevant regime dominated by fluctuations, it is possible to calculate analytically the probability density function of the number of individuals living in a given volume, which captures the close-to-critical behavior of the community across spatial scales. We find that the resulting distribution satisfies an equation where spatial effects are encoded in appropriate functions of space, which we calculate explicitly. The validity of the analytical formulae is confirmed by simulations in the expected regimes. We finally discuss how this model in the critical-like regime is in agreement with several biodiversity patterns observed in tropical rain forests

Linear Response Theory and Fluctuation Dissipation Theorem for Systems with Absorbing States

The Fluctuation Dissipation Theorem (FDT) is one of the fundamental results of statistical mechanics and is a powerful tool that connects the macroscopic properties to microscopic dynamics. The FDT and the linear response theory are mainly restricted to systems in the vicinity of stationary states. However, frequently, physical systems do not conserve the total probability. In systems with absorbing states, the net flux out of the system is positive, and the total probability decays with time. In this case the stationary distribution is trivially zero throughout and the tools provided by standard linear response theory fail. Here we present a new FDT for decaying systems which connects the response of observables conditioned on survival to conditional correlations without perturbations. The results have been verified through simulations and numerics in various important examplesComment: 38 pages, 13 figure

Ginzburg-Landau amplitude equation for nonlinear nonlocal models

Regular spatial structures emerge in a wide range of different dynamics characterized by local and/or nonlocal coupling terms. In several research fields this has spurred the study of many models, which can explain pattern formation. The modulations of patterns, occurring on long spatial and temporal scales, can not be captured by linear approximation analysis. Here, we show that, starting from a general model with long range couplings displaying patterns, the spatio-temporal evolution of large scale modulations at the onset of instability is ruled by the well-known Ginzburg-Landau equation, independently of the details of the dynamics. Hence, we demonstrate the validity of such equation in the description of the behavior of a wide class of systems. We introduce a novel mathematical framework that is also able to retrieve the analytical expressions of the coefficients appearing in the Ginzburg-Landau equation as functions of the model parameters. Such framework can include higher order nonlocal interactions and has much larger applicability than the model considered here, possibly including pattern formation in models with very different physical features.Comment: 14 pages including appendice