Optimal first-passage time in gene regulatory networks

The inherent probabilistic nature of the biochemical reactions, and low copy number of species can lead to stochasticity in gene expression across identical cells. As a result, after induction of gene expression, the time at which a specific protein count is reached is stochastic as well. Therefore events taking place at a critical protein level will see stochasticity in their timing. First-passage time (FPT), the time at which a stochastic process hits a critical threshold, provides a framework to model such events. Here, we investigate stochasticity in FPT. Particularly, we consider events for which controlling stochasticity is advantageous. As a possible regulatory mechanism, we also investigate effect of auto-regulation, where the transcription rate of gene depends on protein count, on stochasticity of FPT. Specifically, we investigate for an optimal auto-regulation which minimizes stochasticity in FPT, given fixed mean FPT and threshold. For this purpose, we model the gene expression at a single cell level. We find analytic formulas for statistical moments of the FPT in terms of model parameters. Moreover, we examine the gene expression model with auto-regulation. Interestingly, our results show that the stochasticity in FPT, for a fixed mean, is minimized when the transcription rate is independent of protein count. Further, we discuss the results in context of lysis time of an \textit{E. coli} cell infected by a $\lambda$ phage virus. An optimal lysis time provides evolutionary advantage to the $\lambda$ phage, suggesting a possible regulation to minimize its stochasticity. Our results indicate that there is no auto-regulation of the protein responsible for lysis. Moreover, congruent to experimental evidences, our analysis predicts that the expression of the lysis protein should have a small burst size.Comment: 8 pages, 3 figures, Submitted to Conference on Decision and Control 201

Halo stochasticity in global clustering analysis

In the present work we study the statistics of haloes, which in the halo model determines the distribution of galaxies. Haloes are known to be biased tracer of dark matter, and at large scales it is usually assumed there is no intrinsic stochasticity between the two fields. Following the work of Seljak & Warren (2004), we explore how correct this assumption is and, moving a step further, we try to qualify the nature of stochasticity. We use Principal Component Analysis applied to the outputs of a cosmological N-body simulation to: (1) explore the behaviour of stochasticity in the correlation between haloes of different masses; (2) explore the behaviour of stochasticity in the correlation between haloes and dark matter. We show results obtained using a catalogue with 2.1 million haloes, from a PMFAST simulation with box size of 1000h^{-1}Mpc. In the relation between different populations of haloes we find that stochasticity is not-negligible even at large scales. In agreement with the conclusions of Tegmark & Bromley (1999) who studied the correlations of different galaxy populations, we found that the shot-noise subtracted stochasticity is qualitatively different from `enhanced' shot noise and, specifically, it is dominated by a single stochastic eigenvalue. We call this the `minimally stochastic' scenario, as opposed to shot noise which is `maximally stochastic'. In the correlation between haloes and dark matter, we find that stochasticity is minimized, as expected, near the dark matter peak (k ~ 0.02 h Mpc^{-1} for a LambdaCDM cosmology) and, even at large scales, it is of the order of 15 per cent above the shot noise. Moreover, we find that the reconstruction of the dark matter distribution is improved when we use eigenvectors as tracers of the bias. [Abridged]Comment: 9 pages, 12 figures. Submitted to MNRA

On stochasticity in nearly-elastic systems

Nearly-elastic model systems with one or two degrees of freedom are considered: the system is undergoing a small loss of energy in each collision with the "wall". We show that instabilities in this purely deterministic system lead to stochasticity of its long-time behavior. Various ways to give a rigorous meaning to the last statement are considered. All of them, if applicable, lead to the same stochasticity which is described explicitly. So that the stochasticity of the long-time behavior is an intrinsic property of the deterministic systems.Comment: 35 pages, 12 figures, already online at Stochastics and Dynamic

Wave Propagation in Stochastic Spacetimes: Localization, Amplification and Particle Creation

Here we study novel effects associated with electromagnetic wave propagation in a Robertson-Walker universe and the Schwarzschild spacetime with a small amount of metric stochasticity. We find that localization of electromagnetic waves occurs in a Robertson-Walker universe with time-independent metric stochasticity, while time-dependent metric stochasticity induces exponential instability in the particle production rate. For the Schwarzschild metric, time-independent randomness can decrease the total luminosity of Hawking radiation due to multiple scattering of waves outside the black hole and gives rise to event horizon fluctuations and thus fluctuations in the Hawking temperature.Comment: 26 pages, 1 Postscript figure, submitted to Phys. Rev. D on July 29, 199

Local stochastic non-Gaussianity and N-body simulations

Large-scale clustering of highly biased tracers of large-scale structure has emerged as one of the best observational probes of primordial non-Gaussianity of the local type (i.e. f_{NL}^{local}). This type of non-Gaussianity can be generated in multifield models of inflation such as the curvaton model. Recently, Tseliakhovich, Hirata, and Slosar showed that the clustering statistics depend qualitatively on the ratio of inflaton to curvaton power \xi after reheating, a free parameter of the model. If \xi is significantly different from zero, so that the inflaton makes a non-negligible contribution to the primordial adiabatic curvature, then the peak-background split ansatz predicts that the halo bias will be stochastic on large scales. In this paper, we test this prediction in N-body simulations. We find that large-scale stochasticity is generated, in qualitative agreement with the prediction, but that the level of stochasticity is overpredicted by ~30%. Other predictions, such as \xi independence of the halo bias, are confirmed by the simulations. Surprisingly, even in the Gaussian case we do not find that halo model predictions for stochasticity agree consistently with simulations, suggesting that semi-analytic modeling of stochasticity is generally more difficult than modeling halo bias.Comment: v3: minor changes matching published versio

Effect of Noise on the Standard Mapping

The effect of a small amount of noise on the standard mapping is considered. Whenever the standard mapping possesses accelerator modes (where the action increases approximately linearly with time), the diffusion coefficient contains a term proportional to the reciprocal of the variance of the noise term. At large values of the stochasticity parameter, the accelerator modes exhibit a universal behavior. As a result the dependence of the diffusion coefficient on the stochasticity parameter also shows some universal behavior.Comment: Plain TeX, 18 pages, 4 figure