Stochastic Regulation of her1/7 Gene Expression Is the Source of Noise in the Zebrafish Somite Clock Counteracted by Notch Signalling

Abstract

The somite segmentation clock is a robust oscillator used to generate regularly-sized segments during early vertebrate embryogenesis. It has been proposed that the clocks of neighbouring cells are synchronised via inter-cellular Notch signalling, in order to overcome the effects of noisy gene expression. When Notch-dependent communication between cells fails, the clocks of individual cells operate erratically and lose synchrony over a period of about 5 to 8 segmentation clock cycles (2-3 hours in the zebrafish). Here, we quantitatively investigate the effects of stochasticity on cell synchrony, using mathematical modelling, to investigate the likely source of such noise. We find that variations in the transcription, translation and degradation rate of key Notch signalling regulators do not explain the in vivo kinetics of desynchronisation. Rather, the analysis predicts that clock desynchronisation, in the absence of Notch signalling, is due to the stochastic dissociation of Her1/7 repressor proteins from the oscillating her1/7 autorepressed target genes. Using in situ hybridisation to visualise sites of active her1 transcription, we measure an average delay of approximately three minutes between the times of activation of the two her1 alleles in a cell. Our model shows that such a delay is sufficient to explain the in vivo rate of clock desynchronisation in Notch pathway mutant embryos and also that Notch-mediated synchronisation is sufficient to overcome this stochastic variation. This suggests that the stochastic nature of repressor/DNA dissociation is the major source of noise in the segmentation clock.

Fig 1. Modelling the effects of inter-cellular variability in transcription, translation and degradation rates and delays.

1A: Her1/7 feedback loop in which Her1/7 protein inhibits expression of her1/7 genes. There is a delay in transcription of ${\tau }_{m_{h1/7}}$ and a delay in translation of ${\tau }_{p_{h1/7}}.$ Stimulation is given by → and inhibition by ⊣. 1B-1H: The oscillating Her1 population levels for five independent cells in the system (blue, red, green, magenta and cyan) and overlaid mean population level of all 100 cells in the system (black) versus time for variability in reaction rate, delay and number of Hes6 molecules. The mean population levels reflect how synchronous the oscillations of neighbouring cells are in addition to the amplitude of oscillation of the individual cells. Inter-cellular variability is distributed as a Gaussian distribution with mean, μ, and standard deviation, σ, i.e. as N(μ, σ ²). The mean value is given by the fitted values for each parameter (see S2 Text) with the maximum variation possible before qualitative changes in oscillatory behaviour occur. Quantification methods for desynchronisation can be found in Methods and S3 Fig. 1B: Variability in transcription rate, α _h, distributed as α _h ∼ N(33,9²). The system is very robust to changes in transcription rate. Cells are still oscillating in synchrony after 30 oscillations. 1C: Variability in translation rate, β _h, distributed as β _h ∼ N(9.2, 1.5²). The system undergoes severe damping when translation rate falls below 4.6 min^-1. Cells are still oscillating in synchrony after 30 oscillations. 1D: Variability in degradation rate, λ _h, distributed as λ _h ∼ N(0.23, 0.025²). The system undergoes severe damping when degradation rate falls below 0.15 min^-1. Cell clocks desynchronise in 20–21 oscillations. 1E: Variability in transcription delay, ${\tau }_{m_{h1/7}},$ for both her1 and her7, distributed as ${\tau }_{m_{h1/7}}\sim N\left(7,{0.5}^2\right)$. Damping of oscillation on individual cells occurs for values below 5 min. Cell clocks desynchronise in 11–12 oscillations. 1F: Variability in translation delay, ${\tau }_{p_{h1}}$ and ${\tau }_{p_{h7}}$ for her1 and her7 respectively, distributed as ${\tau }_{p_{h1}}\sim N\left(1.1,{0.3}^2\right)$ and ${\tau }_{p_{h7}}\sim N\left(0.7,{0.2}^2\right)$. Cell clocks desynchronise in 20–21 oscillations. 1G: Variability in transcription delay, ${\tau }_{m_{h1/7}},$ for both her1 and her7, distributed as Generalised Pareto distributions with parameters, location of 5.84, scale of 1.10 and shape of 0.05, resulting in expected values 7 and variances 1.5. In this instance a small number of cells have much increased transcription delay, resulting in the increased variance for the sample. Cell clocks desynchronise in 14–15 oscillations. 1H: Variability in cellular numbers of Hes6 molecules, p _h6, distributed as p _h6 ∼ N(100, 25²). Cells are still oscillating in synchrony after 33 oscillations. Inter-cellular variability in the reaction rate and delay constants and number of Hes6 molecules is not of the right magnitude to explain the desynchronisation of oscillation in Notch mutants.

Fig 2. Modelling the effects of stochastic dissociation.

2A: Reaction kinetics of Her1/7 proteins binding to the inhibitory sites on her1/7 DNA to switch off expression of her1/7. 2B-F: The oscillating Her1 population levels for five independent cells in the system (blue, red, green, magenta and cyan) and overlaid mean population level of all 100 cells in the system (black) versus time for differing $k_{of{f}_{Her1/7}}$ values relating to the her1/7 inhibitory reaction described in 2A. The mean population levels of all 100 cells primarily reflect how synchronous the oscillations of neighbouring cells are. Quantification methods for desynchronisation can be found in Methods and S3 Fig. 2B: $k_{of{f}_{Her1/7}}=1$ min^-1. Cell clocks are still in synchrony after 14 oscillations. 2C: $k_{of{f}_{Her1/7}}=1/2$ min^-1. Cell clocks desynchronise in 11–12 oscillations. 2D: $k_{of{f}_{Her1/7}}=1/3$ min^-1. Cell clocks desynchronise in 6–7 oscillations. 2E: $k_{of{f}_{Her1/7}}=1/4$ min^-1. Cell clocks desynchronise in 5–6 oscillations. 2F: $k_{of{f}_{Her1/7}}=1/6$ min^-1. Cell clocks desynchronise in 2–3 oscillations. 2G: Mean population levels of 100 cells for seven different simulations, for $k_{of{f}_{Her1/7}}=1/3$ min^-1. There is variability in the embryos and cell clocks desynchronise in 6–8 oscillations. The figure demonstrates that noisy gene regulation can drive desynchronisation in Notch mutants.

Fig 3. Plots of multiple cells oscillatory clocks versus time.

Each lattice is a single cell. Each five minute interval is marked by two columns of 32 cells. The mean phases of oscillation over all cells are marked by the magenta lines. The data is that of Fig 2. A: $k_{of{f}_{Her1/7}}=1$ min^-1. The cells remain in synchrony and no salt and pepper pattern is generated. B: $k_{of{f}_{Her1/7}}=1/3$ min^-1. The cells gradually drift out of synchrony to a salt and pepper pattern over 6–7 oscillation cycles. C: $k_{of{f}_{Her1/7}}=1/6$ min^-1. Cell oscillations quickly desynchronise and tend to a salt and pepper pattern in 2–3 oscillation cycles. The figures demonstrate that, gradually, the number of cells oscillating in phase, within the magenta periods, decreases as noisy gene regulation causes the cell clocks to gradually drift out of synchrony from one another. The result is a salt and pepper pattern. As this salt and pepper pattern is reached, somites will not form correctly. (Compare to the wildtype case of Fig 5E.)

Fig 4. Quantification of delay in expression between her1 gene copies in a PSM cell.

4A: Schematic of quantification. Oscillatory gene expression in the posterior of the PSM is shown in cyan. The PSM is divided into intervals in order to quantify this gene expression, spatially. 4B: A single slice of a zebrafish embryo with nuclei in magenta and her1 mRNA in cyan. 4C: The zoomed in square of 4B to closer demonstrate the cytoplasmic her1 mRNA molecules and her1 transcripts in the course of synthesis. In both figures, the cyan stripes of her1 mRNA are apparent. Intervals are overlaid, based on the gradient of the her1 mRNA waves, and the frequency of nuclei with one her1 transcript in the course of synthesis and those with two is quantified per interval. 4Ci: A cell containing cytoplasmic her1 mRNA molecules. 4Cii: A cell with one her1 gene copy expressed. 4Ciii: A cell with two her1 gene copies expressed. 4D: Plot of frequency of nuclei with one dot (blue) and frequency with two dots (red) per interval. The two dot signal is delayed behind the one dot signal. 4E: The signals smoothed and the interval scale transformed to distance over the anterior axis (scaled from zero to one). The large black circles demonstrate the inflection points used to calculate the delay. This specific delay is selected as the increase in frequency of active her1 genes is a reflection of the repressing Her1/7 protein dissociating from the gene. The smaller symbol, between the two circles, gives the x coordinate used for further calculation in each case. 4F: Further example of smoothed dot count signals for an additional embryo. 4G: Box and whisker plot of delay between one and two dot signals, as a proportion of time to make one somite (left axis) and in minutes assuming the time to make one somite is thirty minutes (right axis). The minimum and maximum are given by the whiskers, the lower quartile and upper quartile by the box. The solid red line gives the median and the dot-dash red line gives the mean. The predicted delay in expression between the two her1 gene copies in a cell is three minutes. The mathematical model predicts cells will desynchronise in 6–8 oscillations when inserting the dissociation rate, $k_{of{f}_{Her1/7}}=1/3$ min^-1, corresponding to this delay.

Fig 5. Modelling the effects of Notch signalling on the synchrony of neighbouring cell oscillations.

5A: Notch signalling network between neighbouring cells. The light blue section corresponds to the intra-cellular her1/7 feedback loop of Fig 1A. The green section corresponds to Notch inter-cellular signalling. Her1/7 protein inhibits expression of delta in addition to her1/7. Transcription and translation of delta occurs with respective delays ${\tau }_{m_{\delta }}$ and ${\tau }_{p_{\delta }}.$ Delta activates Notch in the neighbouring cell and NICD is produced with delay, ${\tau }_{p_N}.$ When Notch binds to the her1/7 genes this leaves them in an active state, influencing the her1/7 intra-cellular feedback loop. 5B: Competitive binding reaction kinetics of Her1/7 and Notch proteins to the sites on her1/7 DNA. When the her1/7 gene is free or bound to NICD then the gene is active. When Her1/7 binds to the her1/7 gene then expression of the gene is inhibited. It is these reaction kinetics that we model with a modified Gillespie Algorithm. 5C: Plot of oscillating Her1 levels versus time when Notch signalling has been incorporated into the model. The coloured lines correspond to five randomly selected individual cells; the black line corresponds to the mean of all 64 cells. The cells oscillate in synchrony. 5D: The normalised mean population levels of her1 mRNA, her7 mRNA, Her1 protein, Her7 protein, delta mRNA, Delta protein and Notch protein demonstrating the phase of each population. Notch is almost totally out of phase with Her1/7 so will be high when Her1/7 is low and vice versa. 5E:Plots of multiple cells oscillatory clocks versus time (compare to Fig 3). Each lattice is a single cell. Each five minute interval is marked by two columns of 32 cells. The mean phases of oscillation over all cells are marked by the magenta lines. The cells remain in synchrony throughout and generate well defined somites. The modelling demonstrates that Notch signalling is able to override the effects of stochastic gene regulation and keep neighbouring cell clocks oscillating in synchrony.

Fig 6. Modelling Notch signalling’s ability to combat increasing levels of stochasticity.

In each case we plot the mean Her1 protein levels for 64 cells to illustrate how synchronous the clock oscillations are. Notch signalling is active in each case. 6A: Dissociation parameter, $k_{of{f}_{Her1/7}}=1/6$ min^-1. Notch signalling can cope with double the magnitude of stochasticity. 6B: Dissociation parameter, $k_{of{f}_{Her1/7}}=1/15$ min^-1. Notch signalling begins to struggle to keep neighbouring cells oscillating in synchrony. 6C: Dissociation parameter, $k_{of{f}_{Her1/7}}=1/30$ min^-1. Notch signalling is not strong enough to counteract the levels of stochasticity in her1/7 gene regulation. Neighbouring cells quickly desynchronise oscillation. 6D: Dissociation parameter, $k_{of{f}_{Her1/7}}=1/30$ min^-1 while delay (transcription and translation delays and delay in activation of Notch) has been increased threefold. Neighbouring cells once again oscillate in synchrony, but with longer period. As the levels of stochasticity in her1/7 gene regulation increase, Notch signalling is no longer able to override stochastic effects and keep neighbouring cells oscillating in synchrony. To rescue synchrony, further changes to the system are required, for example, by increasing the magnitude of delay.