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
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.
© Copyright Policy
Related In: Results  -  Collection


getmorefigures.php?uid=PMC3046134&req=5

pone-0016980-g002: 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.

Mentions: Let us begin by considering a mathematical model for the core oscillator, that is, for the oscillatory dynamics within an individual PSM cell. A cell circuitry schematic is given in figure 2. This schematic is consistent with the mechanism of Lewis, as decribed in the previous section (but see also figure 1B in [1] and figure 1B in [5]). There are, additionally, some spatial assumptions. Proteins are translated from mRNA by ribosomes in the cytoplasm. This process is likely to occur at least some minimal distance from the nucleus, so we assume that Her1, Her7, and DeltaC are translated some minimal distance from the nucleus. Since Her1 and Her7 proteins function as transcription factors, we assume that they can diffuse in both the cytoplasm and the nucleus. By contrast, DeltaC is synthesised for export to the cell membrane, so it is unlikely to diffuse into the nucleus. Hence we assume that DeltaC is absent from the nucleus. Finally, unlike the DDE model of Lewis [1], [6], we will not include time delays for transport, transcription, and translation in our model. The duration of these processes will be accounted for in our diffusion, transcription, and translation rates.


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)

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.
© Copyright Policy
Related In: Results  -  Collection

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

pone-0016980-g002: 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.
Mentions: Let us begin by considering a mathematical model for the core oscillator, that is, for the oscillatory dynamics within an individual PSM cell. A cell circuitry schematic is given in figure 2. This schematic is consistent with the mechanism of Lewis, as decribed in the previous section (but see also figure 1B in [1] and figure 1B in [5]). There are, additionally, some spatial assumptions. Proteins are translated from mRNA by ribosomes in the cytoplasm. This process is likely to occur at least some minimal distance from the nucleus, so we assume that Her1, Her7, and DeltaC are translated some minimal distance from the nucleus. Since Her1 and Her7 proteins function as transcription factors, we assume that they can diffuse in both the cytoplasm and the nucleus. By contrast, DeltaC is synthesised for export to the cell membrane, so it is unlikely to diffuse into the nucleus. Hence we assume that DeltaC is absent from the nucleus. Finally, unlike the DDE model of Lewis [1], [6], we will not include time delays for transport, transcription, and translation in our model. The duration of these processes will be accounted for in our diffusion, transcription, and translation rates.

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