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.

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.

We report on 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 frontal acid-base neutralization can give rise to the formation of a 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 conditions.

The buoyancy-driven instabilities triggered by neutralization reaction were studied experimentally in a miscible two-layer system placed in a vertically oriented Hele-Shaw cell. The initial density stratification was always set to exclude the development of the Rayleigh–Taylor instability. The problem was examined for a few reactant pairs formed by a strong acid and a strong base. To classify the numerous experimental observations we introduced a dimensionless parameter, namely, a reaction-induced buoyancy number , the unstable density stratification above the reaction front leads to the development of vigorous convection in the upper layer, forcing the reaction front to move downwards fast, so that it takes just a few minutes for reagents to burn out. We show that a new parameter can be used to plot the stability maps, which allows us to predict reaction–diffusion–convection processes in similar systems prior to experiment.