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

Terry AJ, Sturrock M, Dale JK, Maroto M, Chaplain MA - PLoS ONE (2011)

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Division of Mathematics, University of Dundee, Dundee, United Kingdom. aterry@maths.dundee.ac.uk

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.

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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.
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pone-0016980-g003: 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.

Mentions: Simulations for our non-dimensionalised core oscillator model, performed on a hexagonal cell with a circular nucleus, are demonstrated in figures 3, 4, and 5, and also in two Supporting Information files (see “Animation S1” and “Animation S2”). Figure 3 demonstrates stable oscillations over time in the concentrations of Her1 protein and her1 mRNA both in the cytoplasm and the nucleus. Notice that the concentration of Her1 protein is higher in the cytoplasm than the nucleus. We will be interested to see if this balance holds when we consider the active transport of Her1 (and Her7) into the nucleus in future work.


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

Terry AJ, Sturrock M, Dale JK, Maroto M, Chaplain MA - PLoS ONE (2011)

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.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3046134&req=5

pone-0016980-g003: 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.
Mentions: Simulations for our non-dimensionalised core oscillator model, performed on a hexagonal cell with a circular nucleus, are demonstrated in figures 3, 4, and 5, and also in two Supporting Information files (see “Animation S1” and “Animation S2”). Figure 3 demonstrates stable oscillations over time in the concentrations of Her1 protein and her1 mRNA both in the cytoplasm and the nucleus. Notice that the concentration of Her1 protein is higher in the cytoplasm than the nucleus. We will be interested to see if this balance holds when we consider the active transport of Her1 (and Her7) into the nucleus in future work.

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Division of Mathematics, University of Dundee, Dundee, United Kingdom. aterry@maths.dundee.ac.uk

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.

Show MeSH