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The segmentation of the vertebrate body is laid down during early embryogenesis. The formation of signaling gradients, the periodic expression of genes of the Notch-, Fgf- and Wnt-pathways and their interplay in the unsegmented presomitic mesoderm (PSM) precedes the rhythmic budding of nascent somites at its anterior end, which later develops into epithelialized structures, the somites. Although many in silico models describing partial aspects of somitogenesis already exist, simulations of a complete causal chain from gene expression in the growth zone via the interaction of multiple cells to segmentation are rare. Here, we present an enhanced gene regulatory network (GRN) for mice in a simulation program that models the growing PSM by many virtual cells and integrates WNT3A and FGF8 gradient formation, periodic gene expression and Delta/Notch signaling. Assuming Hes7 as core of the somitogenesis clock and LFNG as modulator, we postulate a negative feedback of HES7 on Dll1 leading to an oscillating Dll1 expression as seen in vivo. Furthermore, we are able to simulate the experimentally observed wave of activated NOTCH (NICD) as a result of the interactions in the GRN. We esteem our model as robust for a wide range of parameter values with the Hes7 mRNA and protein decays exerting a strong influence on the core oscillator. Moreover, our model predicts interference between Hes1 and HES7 oscillators when their intrinsic frequencies differ. In conclusion, we have built a comprehensive model of somitogenesis with HES7 as core oscillator that is able to reproduce many experimentally observed data in mice.

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.

Somitogenesis describes the segmentation of vertebrate embryonic bodies, which is thought to be induced by ultradian clocks (i.e., clocks with relatively short cycles compared to circadian clocks). One candidate for such a clock is the bHLH factor Hes1, forming dimers which repress the transcription of its own encoding gene. Most models for such small autoregulative networks are based on delay equations where a Hill function represents the regulation of transcription. The aim of the present paper is to estimate the Hill coefficient in the switch of an Hes1 oscillator and to suggest a more detailed model of the autoregulative network. The promoter of Hes1 consists of three to four binding sites for Hes1 dimers. Using the sparse data from literature, we find, in contrast to other statements in literature, that there is not much evidence for synergistic binding in the regulatory region of Hes1, and that the Hill coefficient is about three. As a model for the negative feedback loop, we use a Goodwin system and find sustained oscillations for systems with a large enough number of linear differential equations. By a suitable variation of the number of equations, we provide a rational lower bound for the Hill coefficient for such a system. Our results suggest that there exist additional nonlinear processes outside of the regulatory region of Hes1.

While it is known that a large fraction of vertebrate genes are under the control of a gene regulatory network (GRN) forming a clock with circadian periodicity, shorter period oscillatory genes like the Hairy-enhancer-of split (Hes) genes are discussed mostly in connection with the embryonic process of somitogenesis. They form the core of the somitogenesis-clock, which orchestrates the periodic separation of somites from the presomitic mesoderm (PSM). The formation of sharp boundaries between the blocks of many cells works only when the oscillators in the cells forming the boundary are synchronized. It has been shown experimentally that Delta-Notch (D/N) signaling is responsible for this synchronization. This process has to happen rather fast as a cell experiences at most five oscillations from its ‘birth’ to its incorporation into a somite. Computer simulations describing synchronized oscillators with classical modes of D/N-interaction have difficulties to achieve synchronization in an appropriate time. One approach to solving this problem of modeling fast synchronization in the PSM was the consideration of cell movements. Here we show that fast synchronization of Hes-type oscillators can be achieved without cell movements by including D/N cis-inhibition, wherein the mutual interaction of DELTA and NOTCH in the same cell leads to a titration of ligand against receptor so that only one sort of molecule prevails. Consequently, the symmetry between sender and receiver is partially broken and one cell becomes preferentially sender or receiver at a given moment, which leads to faster entrainment of oscillators. Although not yet confirmed by experiment, the proposed mechanism of enhanced synchronization of mesenchymal cells in the PSM would be a new distinct developmental mechanism employing D/N cis-inhibition. Consequently, the way in which Delta-Notch signaling was modeled so far should be carefully reconsidered., Author Summary During vertebrate embryonic development, the segmented structure of the axial skeleton is laid down by the process of somitogenesis. Periodically, blocks of cells separate at the anterior end of a mesenchymal tissue (PSM) on either side of the neural tube and develop later into spinal vertebrae. Cellular oscillators operating in each cell of the PSM control this process. Their synchronization is essential, and is effected by direct cell-to-cell signaling of the Delta/Notch (D/N) pathway. To better understand the regulation of the genes involved, we employ computer modeling. In this case, the fast synchronization of the oscillators represents a special challenging and worked so far only by the integration of cell movements. Now, we have succeeded in accelerating the synchronization for the first time without cell movements by the interposition of the novel mechanism of intracellular reciprocal inhibition termed D/N cis-inhibition into our computer simulations.

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.

The segmentation of the vertebrate body is laid down during early embryogenesis. The formation of signaling gradients, the periodic expression of genes of the Notch-, Fgf- and Wnt-pathways and their interplay in the unsegmented presomitic mesoderm (PSM) precedes the rhythmic budding of nascent somites at its anterior end, which later develops into epithelialized structures, the somites. Although many in silico models describing partial aspects of somitogenesis already exist, simulations of a complete causal chain from gene expression in the growth zone via the interaction of multiple cells to segmentation are rare. Here, we present an enhanced gene regulatory network (GRN) for mice in a simulation program that models the growing PSM by many virtual cells and integrates WNT3A and FGF8 gradient formation, periodic gene expression and Delta/Notch signaling. Assuming Hes7 as core of the somitogenesis clock and LFNG as modulator, we postulate a negative feedback of HES7 on Dll1 leading to an oscillating Dll1 expression as seen in vivo. Furthermore, we are able to simulate the experimentally observed wave of activated NOTCH (NICD) as a result of the interactions in the GRN. We esteem our model as robust for a wide range of parameter values with the Hes7 mRNA and protein decays exerting a strong influence on the core oscillator. Moreover, our model predicts interference between Hes1 and HES7 oscillators when their intrinsic frequencies differ. In conclusion, we have built a comprehensive model of somitogenesis with HES7 as core oscillator that is able to reproduce many experimentally observed data in mice., Author Summary Somitogenesis is a process in embryonic development establishing the segmentation of the vertebrate body by the periodic separation of small balls of epithelialized cells called somites from a growing mesenchymal tissue, the presomitic mesoderm (PSM). The basic mechanisms are often discussed in terms of the clock-and-wave-front model, which was proposed already in 1976. Candidate genes for this model were found only in the last fifteen years with the cyclically expressed Hairy/Hes genes functioning as the clock and posteriorly expressed Fgf, Tbx6, and Wnt genes establishing the gradient(s). In addition, the Delta/Notch signal transduction pathway seems to be important for boundary formation between forming somites and the remaining PSM by inducing Mesp2 expression just behind a future somitic boundary. Although many in silico models describing partial aspects of somitogenesis already exist, there are still conflicts regarding the mechanisms of the somitogenesis clock. Furthermore, a simulation that fully integrates clock and gradient was only recently published for chicken. Here, we propose a cell- and gene-based computer model for mammalian somitogenesis, simulating a gene regulatory network combining clock (Hes1/7) and gradient (Tbx6, Fgf8, Wnt3a) with Delta/Notch signaling resulting in dynamic gene expression patterns as observed in vivo finally leading to boundary formation.