The small number of reactant molecules involved in gene regulation can lead to significant fluctuations in intracellular mRNA and protein concentrations, and there have been numerous recent studies devoted to the consequences of such noise at the regulatory level. Theoretical and computational work on stochastic gene expression has tended to focus on instantaneous transcriptional and translational events, whereas the role of realistic delay times in these stochastic processes has received little attention. Here, we explore the combined effects of time delay and intrinsic noise on gene regulation. Beginning with a set of biochemical reactions, some of which are delayed, we deduce a truncated master equation for the reactive system and derive an analytical expression for the correlation function and power spectrum. We develop a generalized Gillespie algorithm that accounts for the non-Markovian properties of random biochemical events with delay and compare our analytical findings with simulations. We show how time delay in gene expression can cause a system to be oscillatory even when its deterministic counterpart exhibits no oscillations. We demonstrate how such delay-induced instabilities can compromise the ability of a negative feedback loop to reduce the deleterious effects of noise. Given the prevalence of negative feedback in gene regulation, our findings may lead to new insights related to expression variability at the whole-genome scale.master equation ͉ stochastic delay equations ͉ noise ͉ time delay ͉ systems biology T here is considerable experimental evidence that noise can play a major role in gene regulation (1-10). These fluctuations can arise from either intrinsic sources, which are related to the small numbers of reactant biomolecules, or extrinsic sources, which are attributable to a noisy cellular environment. Although the importance of fluctuations in gene regulation was stressed Ͼ30 years ago (11), recent experimental advances have renewed interest in the stochastic modeling of the biochemical reactions that underlie gene regulatory networks (12-16). The most typical approaches are the utilization of the Gillespie algorithms (17)(18)(19)(20), the direct analysis of the master equation, or the development of simplified descriptions based on the Fokker-Planck or Langevin equations (see ref. 21 for a review). A common thread in many of these approaches has been to consider intrinsic noise as the dominant source of variability in gene expression.One major difficulty often encountered in the analysis of gene regulatory networks is the vast separation of time scales between what are typically the fast reactions (dimerization, protein-DNA binding͞unbinding) and the slow reactions (transcription, translation, degradation). There have been many studies devoted to the development of reduced descriptions of these systems using the idea of quasiequilibrium for the fast processes compared with the slow dynamics (cf. ref. 22 and references therein). These approaches have thus far implicitly assumed that all...
We report new chemoconvective pattern formation phenomena observed in a two-layer system of miscible fluids filling a vertical Hele-Shaw cell. We show both experimentally and theoretically that the concentration-dependent diffusion coupled with the frontal acid-base neutralization can give rise to formation of the local unstable zone low in density resulting in a perfectly regular cell-type convective pattern. The described effect gives an example of yet another powerful mechanism which allows the reaction-diffusion processes to govern the flow of reacting fluids under gravity condition.
This article is devoted to the investigation of Marangoni-driven pattern formation at the interface between two immiscible fluids filling a Hele-Shaw cell, each of them containing a reactant of an exothermic neutralization reaction. In such a system, convective patterns arise when one reactant diffuses through the interface to react with the other chemical species in one of the fluids. A chemo-hydrodynamical pattern appears due to Marangoni instabilities taking place because of heat and solutal driven changes of the surface tension. The mathematical model we develop consists in a set of reaction-diffusion-advection equations ruling the evolution of concentrations and temperature coupled to Navier-Stokes equation, written in a Hele-Shaw approximation. In our analysis, the time-dependent convectionless reaction-diffusion base state is first obtained and studied in detail. Next, we perform a linear stability analysis of this base state with regard to thermal and solutal Marangoni effects to determine the parameter values beyond which convection occurs. Finally, we perform numerical simulations of the fully nonlinear system and study the influence of the different parameters on pattern formation.
a b s t r a c tBuoyancy-driven convection can be induced by concentration and temperature gradients near the interface between two immiscible fluids filling a vertical Hele-Shaw cell, each of them containing a reactant of an exothermic A þ B-C reaction taking place in the bulk of the lower layer. A chemohydrodynamical pattern appears then due to the combined action of Rayleigh-Taylor, diffusive layer convection and Rayleigh-Bé nard instabilities occurring either below or above the interface. The mathematical model we develop to describe such dynamics consists in a set of reaction-diffusionadvection equations ruling the evolution of concentrations and temperature coupled to Navier-Stokes equation, written in a Hele-Shaw approximation. We perform numerical simulations of the non-linear system and study the influence on pattern formation of changes in the Damköhler number of the problem and in the ratio of initial reactant concentrations.
We report shock-wave-like structures that are strikingly different from previously observed fingering instabilities, which occur in a two-layer system of miscible fluids reacting by a second-order reaction A+B→S in a vertical Hele-Shaw cell. While the traditional analysis expects the occurrence of a diffusion-controlled convection, we show both experimentally and theoretically that the exothermic neutralization reaction can also trigger a wave with a perfectly planar front and nearly discontinuous change in density across the front. This wave propagates fast compared with the characteristic diffusion times and separates the motionless fluid and the area with anomalously intense convective mixing. We explain its mechanism and introduce a new dimensionless parameter, which allows to predict the appearance of such a pattern in other systems. Moreover, we show that our governing equations, taken in the inviscid limit, are formally analogous to well-known shallow-water equations and adiabatic gas flow equations. Based on this analogy, we define the critical velocity for the onset of the shock wave which is found to be in the perfect agreement with the experiments.
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