A spatio-temporal model of Notch signalling in the zebrafish segmentation clock: conditions for synchronised oscillatory dynamics

Abstract

In the vertebrate embryo, tissue blocks called somites are laid down in head-to-tail succession, a process known as somitogenesis. Research into somitogenesis has been both experimental and mathematical. For zebrafish, there is experimental evidence for oscillatory gene expression in cells in the presomitic mesoderm (PSM) as well as evidence that Notch signalling synchronises the oscillations in neighbouring PSM cells. A biological mechanism has previously been proposed to explain these phenomena. Here we have converted this mechanism into a mathematical model of partial differential equations in which the nuclear and cytoplasmic diffusion of protein and mRNA molecules is explicitly considered. By performing simulations, we have found ranges of values for the model parameters (such as diffusion and degradation rates) that yield oscillatory dynamics within PSM cells and that enable Notch signalling to synchronise the oscillations in two touching cells. Our model contains a Hill coefficient that measures the co-operativity between two proteins (Her1, Her7) and three genes (her1, her7, deltaC) which they inhibit. This coefficient appears to be bounded below by the requirement for oscillations in individual cells and bounded above by the requirement for synchronisation. Consistent with experimental data and a previous spatially non-explicit mathematical model, we have found that signalling can increase the average level of Her1 protein. Biological pattern formation would be impossible without a certain robustness to variety in cell shape and size; our results possess such robustness. Our spatially-explicit modelling approach, together with new imaging technologies that can measure intracellular protein diffusion rates, is likely to yield significant new insight into somitogenesis and other biological processes.

Figure 1. Top view micrograph of formed and forming somites in a zebrafish embryo.

The embryo is at 10-somite stage, stained by in situ hybridization for deltaC mRNA. The deltaC gene exhibits oscillatory expression in the presomitic mesoderm (PSM). The oscillations are quickest in cells at the tail end (posterior) PSM. Cells in the posterior PSM enter the anterior PSM and then the somitic mesoderm (the region of formed somites) as the tail end of the embryo grows away from them. As the tail end grows it releases a Wnt and FGF signal. This gives rise to a morphogen gradient, which causes the oscillations to slow down in cells the further they are from the tail end. Cells cease oscillating altogether and form somites as they exit the anterior PSM. The spatio-temporal expression pattern of the her1 and her7 genes is very similar to the expression pattern for deltaC. Reproduced from figure 1a in under the Creative Commons Attribution License.

Figure 2. Cell circuitry schematic for the zebrafish segmentation clock core oscillator mechanism.

In the nucleus, the her1, her7, and deltaC genes are transcribed to produce her1, her7, and deltaC mRNA respectively. These mRNAs diffuse into the cytoplasm where they are translated to produce Her1, Her7, and DeltaC proteins. The Her1 and Her7 proteins then diffuse into the nucleus and combinatorially inhibit the transcription of the her1, her7, and deltaC mRNAs. The DeltaC protein diffuses to the cell membrane and does not enter the nucleus. Zebrafish PSM cells vary in shape. For simplicity we chose a square cell for this schematic.

Figure 3. Core oscillator model simulation results showing sustained oscillations.

Here we plot Her1 protein (red) and her1 mRNA (blue) concentrations over time. The left plot shows concentrations in the cytoplasm and the right plot shows concentrations in the nucleus. The concentrations are scaled by reference values. Thus, multiplying the Her1 protein concentration by and the her1 mRNA concentration by gives the true concentrations. All parameter values are stated in the Supporting Information. The other species in the core oscillator model show qualitatively similar behaviour.

Figure 4. Core oscillator model simulation results showing spatial distributions of her1 mRNA.

The plots show her1 mRNA concentration at six minute intervals over the fourth period of oscillations by which time transient behaviour has died down. The concentration is scaled by a reference value (multiplying the concentration by gives the true concentration). Parameter values are stated in the Supporting Information.

Figure 5. Core oscillator model simulation results showing spatial distributions of Her1 protein.

The plots show Her1 protein concentration at six minute intervals over the fourth period of oscillations by which time transient behaviour has died down. The concentration is scaled by a reference value (multiplying the concentration by gives the true concentration). Parameter values are stated in the Supporting Information.

Figure 6. Cell circuitry schematic for Notch signalling between two neighbouring zebrafish PSM cells.

Inside each cell, the core oscillator mechanism described in figure 2 holds with the exception that Notch signalling positively regulates the expression of the her1 and her7 genes. The signalling mechanism is as follows. In the nucleus, notch mRNA is transcribed, which then diffuses into the cytoplasm and produces Notch proteins by translation. These Notch molecules diffuse to the cell membrane where they function as receptors. DeltaC proteins from one cell bind to Notch receptors in the neighbouring cell. Each Notch-DeltaC complex is cleaved in two separate locations, causing an intracellular fragment of Notch (called Notch intracellular domain or NICD) to break off and translocate to the nucleus where it upregulates her1 and her7 gene expression. Zebrafish PSM cells vary in shape. For simplicity we chose square cells for this schematic.

Figure 7. Plots of Her1 protein concentration produced from simulations of the Notch signalling model.

Each plot shows the total concentration of Her1 protein in cell 1 (blue) and cell 2 (red) over time. The concentration is scaled by a reference value (multiplying the concentration by gives the true concentration). In each plot, transcription begins in cell 1 at time 0 and in cell 2 when half the core oscillator period (15 minutes) has elapsed. All parameter values are stated in the Supporting Information. Top: no signalling. The cells oscillate perfectly out of synchrony. Middle: Notch signalling begins at time 150 minutes. Oscillations in the two cells synchronise by 550 minutes and the average concentration is increased by approximately 20% compared to the case (top plot) in which no signalling occurs. Bottom: Notch signalling begins at time 150 minutes but here we reduce by a factor of 10 (relative to the middle plot) the threshold concentration of NICD above which its impact on transcription is stronger. The oscillations synchronise by 400 minutes, the average concentration is further increased, and the amplitude of the oscillations is notably reduced.

Figure 8. Notch signalling model simulation results showing spatial distributions of her1 mRNA.

The plots show her1 mRNA concentration at 100 minute intervals. The concentration is scaled by a reference value (multiplying the concentration by gives the true concentration). The parameters used to create the middle plot in figure 7 are used here (see Supporting Information). Signalling between the cells begins at 150 minutes and has clearly synchronised their behaviour by 600 minutes.

Figure 9. Notch signalling model simulation results showing spatial distributions of Her1 protein.

The plots show Her1 protein concentration at 100 minute intervals. The concentration is scaled by a reference value (multiplying the concentration by gives the true concentration). The parameters used to create the middle plot in figure 7 are used here (see Supporting Information). Signalling between the cells begins at 150 minutes and has clearly synchronised their behaviour by 600 minutes.

Figure 10. Plots of DeltaC protein concentration produced from simulations of the Notch signalling model.

Each plot shows the total concentration of DeltaC protein in cell 1 (blue) and cell 2 (red) over time. The concentration is scaled by a reference value (multiplying the concentration by gives the true concentration). In each plot, transcription begins in cell 1 at time 0 and in cell 2 when half the core oscillator period (15 minutes) has elapsed. All parameter values are stated in the Supporting Information. Top: no signalling. The cells oscillate perfectly out of synchrony. Middle: Notch signalling begins at time 150 minutes. Oscillations in the two cells synchronise by 550 minutes and, compared to the case (top plot) in which no signalling occurs, the average concentration is decreased by a factor of 3 and the amplitude of the oscillations is notably reduced. Bottom: Notch signalling begins at time 150 minutes but here we reduce by a factor of 10 (relative to the middle plot) the threshold concentration of NICD above which its impact on transcription is stronger. The oscillations synchronise by 400 minutes, the average concentration is further decreased, and the amplitude of the oscillations is further reduced.

Figure 11. Notch signalling model simulation results showing spatial distributions of deltaC mRNA.

The plots show deltaC mRNA concentration at 100 minute intervals. The concentration is scaled by a reference value (multiplying the concentration by gives the true concentration). The parameters used to create the middle plot in figures 7 and 10 are used here (see Supporting Information). Signalling between the cells begins at 150 minutes and has synchronised their behaviour by 600 minutes.

Figure 12. Notch signalling model simulation results showing spatial distributions of DeltaC protein.

The plots show DeltaC protein concentration at 100 minute intervals. The concentration is scaled by a reference value (multiplying the concentration by gives the true concentration). The protein is absent from the nucleus by our modelling assumptions. The parameters used to create the middle plot in figures 7 and 10 are used here (see Supporting Information). Signalling between the cells begins at 150 minutes and has synchronised their behaviour by 600 minutes.