Limits...
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.

Show MeSH
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.
© Copyright Policy
Related In: Results  -  Collection


getmorefigures.php?uid=PMC3046134&req=5

pone-0016980-g007: 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.

Mentions: Simulations for our signalling model, performed on hexagonal cells with circular nuclei, are demonstrated in figures 7, 8, 9, 10, 11, and 12, and also in two Supporting Information files (see “Animation S3” and “Animation S4”). We denote as cells 1 and 2 the cells that are on the left and right respectively in figures 8, 9, 11, and 12.


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)

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.
© Copyright Policy
Related In: Results  -  Collection

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

pone-0016980-g007: 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.
Mentions: Simulations for our signalling model, performed on hexagonal cells with circular nuclei, are demonstrated in figures 7, 8, 9, 10, 11, and 12, and also in two Supporting Information files (see “Animation S3” and “Animation S4”). We denote as cells 1 and 2 the cells that are on the left and right respectively in figures 8, 9, 11, and 12.

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