Fig 1.
The anterior part of the sagittal suture in an adult skull shows stochastic small amplitude interdigitation.
The suture in the anterior skull near the bregma (red box) has a rough surface with less prominent curvature and a more stochastic appearance than that of the lambda suture (black box). Scale bars (bottom right of boxes) = 0.5 cm.
Fig 2.
Schematic diagram of the present study.
First, we reduced our previous model [19] into graph form h(x, t). Next, we added a noise term to the model based on our experimental observation of noise (Fig 6). Then we mathematically derived the scaling law from this equation and confirmed it using numerical simulations. Finally, we showed experimentally that both noise and scaling do exist in skull sutures (Fig 7).
Fig 3.
Original model and its reduction.
(a) The original model ([19], Eqs (1) and (2)). The model considered a band-like solution with width 2y0. Growth speed of the interface (V) is determined by the effect of substrate molecule (v(= K * u), inlet) and surface tension bκ. (b) The present model (Eq (3)). This model focuses on the dynamics of the centerline h(x, t) of the band-like solution, considering only the onset of pattern formation without overhangs.
Fig 4.
Log–log plot of the (k, −1/λ(k)) of full model, which should reflect the power spectrum .
The distribution shows linearity in the parameter range without spontaneous pattern formation, indicating λ ∝ k−γ.
Fig 5.
Numerical simulation of the reduced model showed the scaling predicted by the mathematical analysis.
(a) Dispersion relation of the full model (blue) and its approximation by λ(k) ∝ kγ (red). The parameter set was a = 1, b = 0.1, c = 0.48, and r = 1. Domain size = 20π and Δx = 0.2π. We defined H(x, y, t) as a white noise and obtained at each time step. We used cutoff frequency ωc = 0.8 to obtain this differentiation. (b) Result of the numerical simulation. The upper panel shows the initial shape (t = 0); the lower panel shows the curved shape obtained after sufficient time had passed (t = 2000). (c) Log–log plots of the average of
obtained by numerical simulations. A region of linear scaling was observed in the high wavenumber region. (d–f) Time course of the log–log plots of the power spectrum obtained by numerical simulations. (d) t = 0, (e) t = 1000, and (f) t = 10000. (g) The relationship between surface roughness w and system size L. (h) The relationsip between surface roughness w and time t.
Fig 6.
Direct observation of the intrinsic noise of the osteogenic signal in a newborn mouse skull.
(a) Experimental system setup. A newborn skull of an ERK-FRET transgenic mice was dissected and then set on the glass bottom dish, which was suitable for organ culture. (b) Brightfield Image of the observation area. We chose posterior fontanelle region for observation. Observation areas in (c-f) is shown by black box. Scale bar = 100 μm. (c) Spatial distribution of the FRET signal u (CFP/YFP ratio) at specific timepoint. (d) Fourier transformation of u(x, t). There was no specific characteristic in the distribution, indicating the noise was white noise. (e) Time course of FRET signal intensity at a specific line. The signal showed fluctuation. (f) Fourier transformation of (e) at single point. No specific trend was observed, indicating that the spatial noise could be regarded as white noise.
Fig 7.
Scaling in actual skull sutures.
(a) A traced suture line converted into coordinates and plotted on the x–h plane. (b) Relationship between k and on a log–log plot. (c) The relationship between surface roughness w and system size L. (d) Histogram of γ obtained by separate data.