How can we describe density evolution under delayed dynamics?

Although the theory of density evolution in maps and ordinary differential equations is well developed, the situation is far from satisfactory in continuous time systems with delay. This paper reviews some of the work that has been done numerically, the interesting dynamics that have emerged, and the largely unsuccessful attempts that have been made to analytically treat the evolution of densities in differential delay equations. We also present a new approach to the problem and illustrate it with a simple example.

Response of an oscillatory differential delay equation to a single stimulus.

Here we analytically examine the response of a limit cycle solution to a simple differential delay equation to a single pulse perturbation of the piecewise linear nonlinearity. We construct the unperturbed limit cycle analytically, and are able to completely characterize the perturbed response to a pulse of positive amplitude and duration with onset at different points in the limit cycle. We determine the perturbed minima and maxima and period of the limit cycle and show how the pulse modifies these from the unperturbed case.

The limiting dynamics of a bistable molecular switch with and without noise.

We consider the dynamics of a population of organisms containing two mutually inhibitory gene regulatory networks, that can result in a bistable switch-like behaviour. We completely characterize their local and global dynamics in the absence of any noise, and then go on to consider the effects of either noise coming from bursting (transcription or translation), or Gaussian noise in molecular degradation rates when there is a dominant slow variable in the system. We show analytically how the steady state distribution in the population can range from a single unimodal distribution through a bimodal distribution and give the explicit analytic form for the invariant stationary density which is globally asymptotically stable. Rather remarkably, the behaviour of the stationary density with respect to the parameters characterizing the molecular behaviour of the bistable switch is qualitatively identical in the presence of noise coming from bursting as well as in the presence of Gaussian noise in the degradation rate. This implies that one cannot distinguish between either the dominant source or nature of noise based on the stationary molecular distribution in a population of cells. We finally show that the switch model with bursting but two dominant slow genes has an asymptotically stable stationary density.

The utility of simple mathematical models in understanding gene regulatory dynamics.

In this review, we survey work that has been carried out in the attempts of biomathematicians to understand the dynamic behaviour of simple bacterial operons starting with the initial work of the 1960's. We concentrate on the simplest of situations, discussing both repressible and inducible systems and then turning to concrete examples related to the biology of the lactose and tryptophan operons. We conclude with a brief discussion of the role of both extrinsic noise and so-called intrinsic noise in the form of translational and/or transcriptional bursting.

Molecular distributions in gene regulatory dynamics.

Extending the work of Friedman et al. (2006), we study the stationary density of the distribution of molecular constituents in the presence of noise arising from either bursting transcription or translation, or noise in degradation rates. We examine both the global stability of the stationary density as well as its bifurcation structure. We have compared our results with an analysis of the same model systems (either inducible or repressible operons) in the absence of any stochastic effects, and shown the correspondence between behaviour in the deterministic system and the stochastic analogs. We have identified key dimensionless parameters that control the appearance of one or two stable steady states in the deterministic case, or unimodal and bimodal densities in the stochastic systems, and detailed the analytic requirements for the occurrence of different behaviours. This approach provides, in some situations, an alternative to computationally intensive stochastic simulations. Our results indicate that, within the context of the simple models we have examined, bursting and degradation noise cannot be distinguished analytically when present alone.