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How to use
Pick a preset, or move the sliders: the model is live. Cells flow from the tail bud (right in the 2D page) toward the anterior; the colour is the Her1 protein level (dark = low, bright yellow = high). The dashed yellow line is the determination front. Behind it the pattern is frozen, and each rising edge becomes a somite boundary (white lines, numbered S1, S2...). The kymograph below the tissue shows position against time: slanted stripes are travelling waves. Things to try: (1) raise the elongation speed v and read the somite length S = v T; (2) switch the coupling to 0 and add noise: the Delta-Notch mutant, ragged boundaries; (3) set gamma = 0 and watch the whole PSM flash together; (4) shorten lambda (FGF inhibitor) and the PSM shrinks; (5) press Run sweep S(v) for the measured against the predicted somite length. Drag to orbit, wheel to zoom. Space pauses, R resets.
The glass-bottom dish with its agarose bed and the yolk (with baked oil-globule texture), the embryo head with eyes and ear vesicles, the chevron somite block with muscle-fibre texture, the notochord and neural-tube segments, the tail bud, the PSM cell, the fluorescence microscope (column, arm, turret, objective), the magnifier and the bench props were modelled in Blender 5.2 with baked PBR textures and exported as one Draco GLB. Three.js r160 instances the segment meshes along the coiled body axis and drives the colours from the same clock model as the 2D page. The magnified window shows the same lattice of cells, the FGF mRNA particles and the front.
The science
Our spine, ribs and trunk muscles are built from repeated blocks of tissue called somites. They bud off in pairs from the front of the unsegmented presomitic mesoderm (PSM) at a steady rhythm: about one every 30 minutes in a zebrafish embryo at 28 C, about 2 hours in a mouse and roughly 5 hours in human cells grown in a dish (typical literature values, not measurements from this page). Two ideas explain the rhythm and the spacing.
1. The clock. Cells of the PSM switch the genes her1 and her7 (Hes7 in mouse) on and off. The Her protein represses its own gene, but only after two delays: the time needed to make and export the mRNA (Tm) and to translate it (Tp). Delayed negative feedback is a textbook oscillator (Lewis 2003, Monk 2003). In the model each lattice site carries
dm/dt = f(p(t − Tm)) − c m, f(p) = k / (1 + (p/p0)2)
dp/dt = a m(t − Tp) − b p
Without delay the system just settles on the fixed point p* that solves p* (1 + (p*/p0)2) = a k/(b c). With delay it oscillates; for strong repression the period lies between two and four times the total delay Tm + Tp, longer when the degradation rates are slow (the page checks this). The parameter values are illustrative: the equation has the form used for zebrafish by Lewis (2003) but the rates were retuned so that the free-running period is about 31 min.
2. The wavefront. Cells are born in the tail bud and the axis grows backward at a speed v. A signal (FGF8/Wnt) is made in the tail bud and decays as the cells age, F(x) = exp(−x/λ) where x is the distance from the tail bud (Dubrulle and Pourquie 2004: the mRNA is inherited and decays). Where F falls below a threshold Fc the cell stops cycling and is committed (the determination front, Cooke and Zeeman 1976):
xc = −λ ln Fc, S = v T, delay factor g(x) = 1 + γ (1 − F(x))
In the model the delays grow anteriorly by the factor g (γ = 0 switches this off), which slows the clock of older cells. A slower clock behind a faster one is exactly what makes waves run from the tail bud toward the front and stall there: a cell that has an older age falls behind in phase. In the steady state the rhythm seen at any fixed position is that of the tail bud, so the somite length is the distance the axis grows in one tail-bud period, S = v T0. The page measures S from the coupled tissue and compares it with v T0 computed from an independent single-oscillator integration.
3. Coupling. Neighbouring cells talk through the Delta-Notch pathway, which keeps them in step; without it the cells drift out of phase and boundaries become ragged (zebrafish deltaC, deltaD, notch1a mutants, Jiang 2000, Horikawa 2006). Here the promoter of each cell sees peff = (1 − κ) pself + κ ⟨pneighbours⟩ (four neighbours: two along the axis, two around the tube), a simple diffusive coupling. The boundary roughness readout is the spread of the boundary time between the eight rows, converted to micrometres with v.
Numerics. The PSM is a chain of 110 sites 5 µm apart (about half a cell diameter; each site stands for a small group of cells) times 8 rows, plus a reservoir ring that stands for the tail bud and feeds a new ring every 5 µm / v. Time step 0.1 min, explicit Euler, delays rounded to the grid; the delayed values are read from a ring buffer. The checks compare this scheme with a Heun integrator (second order, dt down to 0.0125 min, interpolated delays).
Numerical checks run in the page. (1) The zero-delay run reaches the closed-form fixed point p*; (2) the lattice period agrees with the independent Heun reference; (3) period / (Tm + Tp) lies in [2, 4]; (4) the same at the front, where the delays are longer; (5) the measured somite length equals v T0; (6) time-step convergence of the period.
Simplifications. One-dimensional chain of coupled oscillators with eight rows instead of a three-dimensional tissue; no cell division, cell mixing or tissue shortening; the front is a sharp threshold, while real determination uses a bistable Mesp2/Ripply/Tbx6 switch and an FGF-Wnt-retinoic-acid network; the noise is a multiplicative factor on transcription, not a full stochastic gene-expression model; the FGF decay and the delay slow-down are modelled as simple exponentials; rates are illustrative. The preset names (FGF inhibitor, Delta-Notch mutant) describe the qualitative direction of the real experiment only. In the 3D scene the embryo is drawn at 0.0055 scene units per micrometre (a drawing scale, not an anatomical one).
Sources to spot-check (from memory, flagged): Cooke and Zeeman 1976 (J Theor Biol); Palmeirim et al. 1997 (Cell); Lewis 2003 (Curr Biol); Dubrulle and Pourquie 2004 (Nature); Horikawa et al. 2006 (Cell); Oates, Morelli and Ares 2012 (Development).