Figures and data
Computational model of the clock and the PSM.
A Stills of a simulation of the model of Uriu et al. (2021). Kinematic phase (θi) waves emerge in the posterior (right) and travel towards the tissue anterior (left, x = xa), where phase is arrested. The model is parameterised to data from zebrafish, and accordingly the clock oscillates every 30 minutes. B Insets illustrating the key processes driving cell movements in the PSM within the model. Top: Cells advect towards the anterior of the tissue, simulating elongation of the PSM. Middle: New cells are added to replenish the loss of cellular material as cells advect towards the anterior. Bottom: Cells undergo motility-driven rearrangements. C Functions in the model describing (top) the intrinsic oscillation frequency, (middle) the advection velocity, (bottom) and the motility, of each cell depending on its normalised position along the anterior-posterior axis, (x − xa/L). Plots were generated using the parameters given by Uriu et al. (2021).
The effect of cell ingression position on clock frequency and synchrony.
A Diagram highlighting the position of cell addition across the three ingression scenarios tested here. Magenta shading shows where cells are added onto the tissue surface. The pale magenta and blue planes respectively correspond to the anterior limit of the PSM, x = xa, and the anterior limit of cell addition, x = xd. In the ‘Random’ condition cells are added at random positions within the PSM posterior to the plane x = xd, and accordingly no magenta surface is shown. Similarly in the dorso-posterior case (‘DP’) cells are added at random positions in the two lateral cylinders to maintain density, and no magenta surface is shown there. In the dorso-posterior + lateral-ventral (‘DP+LV’) case cells are only added on the tissue surface at the positions shown by magneta shading. B Oscillation synchrony (r) at the PSM anterior after 1000 mins, for the three tested scenarios of cell ingression. N=100 simulations. C Mean frequency of oscillations for cells at the PSM anterior after 1000 mins of simulation, for the three tested scenarios of cell ingression. N=100 simulations. D Kymographs of synchrony along the x-axis on the left-hand side of the PSM for single simulations from each of the three scenarios of cell ingression tested. Black arrowheads highlight strongly asynchronous populations of cells being transported to the tissue anterior by advection.
Effect of cell motility profile on clock frequency and synchrony.
A Overview of how intrinsic cell motion is encoded in the model. Each cell is given a random direction vector v0(xi)ni(t) (black arrows) whose magnitude v0(xi) increases towards the PSM posterior. B Magnitude of intrinsic cell motion v0(x) for the parameters used by Uriu et al. (2021) (blue), and how the shape of the function can change when increasing (yellow) or decreasing (green) the inflexion point and curve steepness parameters, Xv and h, respectively. C Clock synchrony r (top) and difference from expected mean frequency Δdθ/dt (bottom) for different v0(x) specified by combinations of Xv and h. The corresponding pixels display the synchrony or frequency at the PSM anterior after 1000 mins of simulation using the specified motility profile, averaged across N = 100 simulations. Parameter ranges used are Xv ∈ {0, 0.1, 0.2, …, 1} and h ∈ {0, 0.5, 1, …, 5}.
Effect of tissue density and length.
A Stills from exemplar simulations illustrating the impact of changes in tissue density ρ0. B Anterior synchrony after 1000 mins for changing tissue density ρ0 and varying position of cell ingression. N=100 simulations. C Anterior mean frequency after 1000 mins for changing tissue density ρ0 and varying position of cell ingression. N=100 simulations. D Stills from exemplar simulations illustrating changes in tissue length. E Anterior synchrony after 1000 mins for changing tissue length L and varying position of cell ingression. N=100 simulations. F Anterior mean frequency after 1000 mins for changing tissue length L and varying position of cell ingression. N=100 simulations.
Effect of co-varying coupling strength κ, tissue length L, and position of cell ingression on the median anterior synchrony of N=100 simulations. A black + corresponds to the parameter pair used elsewhere in this paper, unless otherwise stated (L = 385 µm, κ = 0.07 min−1).
Top Anterior synchrony after 1000 mins for changing tissue length L and varying position of cell ingression, where xd is fixed at xd = L − R − r − 100. N=100 simulations. Bottom Anterior mean frequency after 1000 mins for changing tissue length L and varying position of cell ingression where x[d] is fixed at xd = L − R − r − 100. N=100 simulations.
Effect of compaction-extension on clock synchrony and frequency.
A Snapshots of an exemplar simulation showing how the PSM shrinks in length and diameter as time progresses. B Functions for PSM length L, radius r, density ρ, and cell diameter dc, derived from (Thomson et al. (2021)) (yellow), and the constant functions (blue) with which the effect of these functions is compared. C Anterior synchrony and D mean anterior frequency over time, for N=100 simulations. The solid line indicates the median and the inter-quartile range is given by a shaded band either side of this line. Blue shows simulations where the tissue does not undergo compaction-extension after tshrink, and yellow shows the results for simulations where after tshrink the tissue undergoes compaction-extension according to the functions shown in B. Results are plotted until the time at which at least one of the replicate simulations encounters a gap in the tissue at the tissue anterior (see methods).
Fitted function interpolating the data for zebrafish somitogenesis from Schröter et al. (2008) (black), against the raw data (magenta).
Comparing the choice of a ‘stepwise’ function for decreasing cell diameter dc against a function where dc decreases continuously until the simulation’s end. A The stepwise (blue) and continuous (yellow) functions compared here. The intercept and gradient for these functions are derived from data from Thomson et al. (2021). B Anterior synchrony (r) over time for stepwise shrinking cells (blue) and continuously shrinking cells (yellow). Dark line shows the median of N=100 simulations, and the shaded area either side of this line shows the inter-quartile range (IQR). Results are plotted until the time at which at least one of the replicate simulations encounters a gap in the tissue at the tissue anterior (see methods). C Mean anterior frequency (dθ/dt) over time for stepwise shrinking cells (blue) and continuously shrinking cells (yellow). Dark line shows the median of N=100 simulations, and the shaded area either side of this line shows the IQR. Results are plotted until the time at which at least one of the replicate simulations encounters a gap in the tissue at the tissue anterior.
Effect of starting tissue density d0 on anterior synchrony and frequency over time, for a compacting tissue. A Functions showing tissue density ρ(t) for different intercept values d0. B Anterior synchrony (r) over time. Dark line shows the median of N=100 simulations, and the shaded area either side of this line shows the inter-quartile range (IQR). Results are plotted until the time at which at least one of the replicate simulations encounters a gap in the tissue at the tissue anterior (see methods). C Mean anterior frequency (dθ/dt) over time. Dark line shows the median of N=100 simulations, and the shaded area either side of this line shows the IQR. Results are plotted until the time at which at least one of the replicate simulations encounters a gap in the tissue at the tissue anterior.
Effect of cell division on clock frequency and synchrony.
A Diagram showing how during cell division, clock phase θ arrests, causing a cell to fall out of phase with its neighbour. On the right hand side a still from a simulation for TM = 15 mins is shown, illustrating how this creates asynchrony of oscillations. The effect of cell division on anterior synchrony and frequency is shown in figures B and C, respectively, for TM = 15 mins after 1000 mins. N = 100. D The effect on anterior synchrony when division is restricted to only occur posterior to x = xdiv, for TM = 15 mins. N = 100. E The effect on anterior synchrony after 1000 mins when TM is varied. In each case, the total length of the cell cycle is maintained at a constant length, i.e. TM + TG = 187.5 mins. N = 100. To rule out trivial changes in synchrony and frequency, in all analysis here we restrict measurement to non-dividing cells, i.e. cells such that τ ∈ [0, T G).
The effect of cell division on frequency. A Effect on median frequency when varying xdiv after 1000 mins for N=100 simulations. B Effect on median frequency when varying TM after 1000 mins for N=100 simulations.
A Dependence of anterior synchrony on xdiv and TM. The median anterior synchrony after 1000 mins for N=100 simulations is plotted. B Dependence of anterior synchrony on TM and intrinsic clock frequency ω0. The median anterior synchrony after 1000 mins for N=100 simulations is plotted.
Anterior synchrony after 1000 mins, for varying maximum magnitude of intrinsic cell motion vs and clock phase coupling strength k, for three different scenarios of cell ingression. Each pixel corresponds to the median value of anterior synchrony across N = 100 simulations. A black + marks the experimental values for zebrafish, vs = 1 µm · min−1, k = 0.07 min−1, derived in Uriu et al. (2017) and Riedel-Kruse et al. (2007) respectively, that are used elsewhere in this paper. All other parameters are held constant at their normal values (see table 1).
Top: schematic of the PSM in the xy plane. The PSM is comprised of two cylinders, centred at y = r and y = 2R + r, respectively, with radius r. The ‘Tailbud’ is represented as a half-torus domain centred at x = (Xc, Yc, Zc)T, with internal radius r and radius from torus centre R. Cross-sections of the PSM and Tailbud are highlighted in magenta, and illustrated on the bottom, showing how a point within the tissue xi is assigned the polar coordinates ri and qi. Adapted from Uriu et al. (2021).
Parameter values used in our work, unless otherwise stated. Values derived from Kanki and Ho (1997); Horikawa et al. (2006); Riedel-Kruse et al. (2007); Uriu et al. (2017, 2021), and Thomson et al. (2021).
Top Trace of synchrony r for a dc -wide domain of cells at the left-hand anterior edge of the PSM over 1000 mins. Data drawn from a single simulation with random cell addition, using parameters as per Uriu et al. (2021). Bottom Trace of mean frequency dθ/dt for a dc -wide domain of cells at the left-hand anterior edge of the PSM over 1000 mins. Data drawn from the same simulation as above. Plotted with a black dashed line is the average intrinsic frequency ω(x) across the domain, calculated using the formula shown.
