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A classical result about Markov jump processes states that a certain class of dynamical systems given by ordinary differential equations are obtained as the limit of a sequence of scaled Markov jump processes. This approach fails if the scaling cannot be carried out equally across all entities. In the present paper we present a convergence theorem for such an unequal scaling. In contrast to an equal scaling the limit process is not purely deterministic but still possesses randomness. We show that these processes constitute a rich subclass of piecewise-deterministic processes. Such processes apply in molecular biology where entities often occur in different scales of numbers.

We develop a simple model for the random distribution of a gene product. It is assumed that the only source of variance is due to switching transcription on and off by a random process. Under the condition that the transition rates between on and off are constant we find that the amount of mRNA follows a scaled Beta distribution. Additionally, a simple positive feedback loop is considered. The simplicity of the model allows for an explicit solution also in this setting. These findings in turn allow, e.g., for easy parameter scans. We find that bistable behavior translates into bimodal distributions. These theoretical findings are in line with experimental results.

The authors propose piecewise deterministic Markov processes as an alternative approach to model gene regulatory networks. A hybrid simulation algorithm is presented and discussed, and several standard regulatory modules are analysed by numerical means. It is shown that despite of the partial simplification of the mesoscopic nature of regulatory networks such processes are suitable to reveal the intrinsic noise effects because of the low copy numbers of genes.

In the present work we propose an alternative approach to model autocatalytic networks, called piecewise-deterministic Markov processes. These were originally introduced by Davis in 1984. Such a model allows for random transitions between the active and inactive state of a gene, whereas subsequent transcription and translation processes are modeled in a deterministic manner. We consider three types of autoregulated networks, each based on a positive feedback loop. It is shown that if the densities of the stationary distributions exist, they are the solutions of a system of equations for a one-dimensional correlated random walk. These stationary distributions are determined analytically. Further, the distributions are analyzed for different simulation periods and different initial concentration values by numerical means. We show that, depending on the network structure, beside a binary response also a graded response is observable.

We analyze the reduction of intrinsic noise caused by transition of a promoter between its active and inactive state in a negatively regulated genetic network, i.e., transcription of the gene is inhibited by its own gene product. To measure the noise attenuation, we compare its behavior to an inducible gene for which activation and deactivation of the gene take place at constant rates. As a model, we choose a hybrid approach in which some of the reaction channels are modeled as discrete events, and other reactions are modeled as continuous processes. Such a model is appropriate for investigations of noise caused by low reactant numbers. By focusing on intrinsic noise originating from the switching behavior of the regulatory system of a particular gene, we model only the transition between two different promoter states as a discrete event. We show that the stationary distributions of the unregulated and the autoregulated system are given as a solution of two coupled ordinary differential equations. Also, beside the distribution densities, the first two central moments are derived in closed analytical forms. We give conditions on the parameters when one or the other system shows lower fluctuations.

Loss of Hes7 function leads to irregular somite formation demonstrating that Hes7 is a crucial component of the segmentation clock during somitogenesis. Experiments revealed that not only the repressor functionality but also the half-life of the protein is crucial for oscillatory expression of Hes7 and regular somite formation. Numerical integration of a delay equation system supported this finding. However, in a recent paper it was shown that the number of binding sites is also decisive for damped or undamped oscillations. It was shown that for more than one binding site the Hill coefficient increases. This leads to a completely different behavior. The oscillations are undamped and thus the mathematical model can no longer explain the results observed in the experiments. In this paper we propose a more sophisticated model for the Hes7 oscillator. Since Hes7 is degraded by the ubiquitin-proteasome pathway we include Michaelis-Menten kinetics for the ubiquitination of Hes7. We identify the Michaelis-Menten constant as an additional model parameter for oscillatory behavior. By increasing the Michaelis-Menten constant we found damped oscillations even if the Hill coefficient is increased.

The Notch signaling pathway is an evolutionarily conserved transduction pathway involved in embryonic patterning and regulation of cell fates during development. Recent studies have demonstrated that this pathway is integral to a complex system of interactions, which are also involved in distinct human diseases. Delta1 is one of the known ligands of the Notch receptors. Mice homozygous for a loss-of-function allele of the Delta1 gene Dll1lacZ/lacZ die during embryonic development. Here, we present the results of two phenotype-driven modifier screens. Heterozygous Dll1lacZ knockout animals were crossed with ENU-mutagenized mice and screened for dysmorphological, clinical chemical, and immunological variants that are dependent on the Delta1 loss-of-function allele. First, we show that mutagenized heterozygous Dll1lacZ offspring have reduced body weight and altered specific clinical chemical parameters, including changes in metabolites and electrolytes relevant for kidney function. In our mutagenesis screen we have successfully generated 35 new mutant lines. Of major interest are 7 mutant lines that exhibit a Dll1lacZ/+-dependent phenotype. These mutant mouse lines provide excellent in vivo tools for studying the role of Notch signaling in kidney and liver function, cholesterol and iron metabolism, cell-fate decisions, and during maturation of T cells in the immune system.

To model dynamic expression patterns in somitogenesis we developed a Java-application for simulating gene regulatory networks in many cells in parallel and visualising the results using the Java3D API, thus simulating the collective behaviour of many thousand cells. According to the ‘clock-and-wave-front’ model mesodermal segmentation of vertebrate embryos is regulated by a ‘segmentation clock’, which oscillates with a period of about 2 h in mice, and a ‘wave front’ moving back with the growing caudal end of the presomitic mesoderm. The clock is realised through cycling expression of genes such as Hes1 and Hes7, whose gene products repress the transcription of their encoding genes in a negative feedback loop. By coupling the decay of the Hes1 mRNA to a gradient with the same features and mechanism of formation as the mesodermal Fgf8 gradient we can simulate typical features of the dynamic expression pattern of Hes1 in the presomitic mesoderm. Furthermore, our program is able to synchronise Hes1 oscillations in thousands of cells through simulated Delta–Notch signalling interactions.

Background It is commonly accepted that embryonic segmentation of vertebrates is regulated by a segmentation clock, which is induced by the cycling genes Hes1 and Hes7. Their products form dimers that bind to the regulatory regions and thereby repress the transcription of their own encoding genes. An increase of the half-life of Hes7 protein causes irregular somite formation. This was shown in recent experiments by Hirata et al. In the same work, numerical simulations from a delay differential equations model, originally invented by Lewis, gave additional support. For a longer half-life of the Hes7 protein, these simulations exhibited strongly damped oscillations with, after few periods, severely attenuated the amplitudes. In these simulations, the Hill coefficient, a crucial model parameter, was set to 2 indicating that Hes7 has only one binding site in its promoter. On the other hand, Bessho et al. established three regulatory elements in the promoter region. Results We show that – with the same half life – the delay system is highly sensitive to changes in the Hill coefficient. A small increase changes the qualitative behaviour of the solutions drastically. There is sustained oscillation and hence the model can no longer explain the disruption of the segmentation clock. On the other hand, the Hill coefficient is correlated with the number of active binding sites, and with the way in which dimers bind to them. In this paper, we adopt response functions in order to estimate Hill coefficients for a variable number of active binding sites. It turns out that three active transcription factor binding sites increase the Hill coefficient by at least 20% as compared to one single active site. Conclusion Our findings lead to the following crucial dichotomy: either Hirata's model is correct for the Hes7 oscillator, in which case at most two binding sites are active in its promoter region; or at least three binding sites are active, in which case Hirata's delay system does not explain the experimental results. Recent experiments by Chen et al. seem to support the former hypothesis, but the discussion is still open.