(a) The model scheme. We included two factors, osteogenic differentiation state of a tissue (u) and substrate concentration (v). Mesenchyme (u = −1) and bone (u = 1) are two stable states and do not interchange easily. Mesenchymal cells produce substrate molecules which promote the differentiation of undifferentiated mesenchyme. (b) Governing equations of the system and explanations of each term.

The model reproduces suture width maintenance. (a) Scheme of the human skull at birth. The suture maintenance simulation region is depicted as a red line. (b) One-dimensional simulation showed the maintenance of thin sutural tissue. The red line represents u and blue line represents v. When the sutural tissue width was changed, substrate production was changed accordingly, which resulted in the maintenance of constant tissue thickness (see text). Simulation parameters: a₀ = 0.05, a₁ = 0.5, a₂ = 0.1, a₃ = 0.1, d = 4.0. The unit of x-axis is 50 μm. c, coronal suture; s, sagittal suture; l, lambda suture; f, frontal suture.

Formation of a fractal structure in the model. (a) Lambdoid suture of a human skull showed quite a complex fractal pattern. (b) Simulation result of original model, which ended up ‘‘labyrinthine“ and looked different from the in vivo pattern. (c) Modified form of the model. We introduced a time-dependent parameter in the diffusion coefficient to implement fibrosis of the sutural tissue (Cohen & MacLean 2000). (d) Simulation result of the modified model. The pattern became one continuous line, which looked similar to the in vivo pattern. (f) Intuitive explanation of the pattern difference. In the modified model, the generated pattern becomes gradually smaller with time, and detailed structure is added onto the original pattern. This procedure is analogous to the formation of a Koch curve, which results in the formation of a fractal structure. Simulation parameters: a₀ = 0.05, a₁ = 0.5, a₂ = 0.1, a₃ = 0.1, d = 4.0.

(a) Difference of sutural interdigitation between the superficial and deep surfaces of the skull. Mouse skull data were obtained using microCT. In both humans and mice, the suture line is thinner and interdigitation is less prominent on the deeper side of the skull. (b) Effect of substrate basal activity a₀. When a₀ was increased, undifferentiated sutural tissue became thinner and interdigitation less prominent. (c) Working hypothesis of the difference. If there is a source of substrate on the deeper side of the skull, the basal effect of the substrate is stronger on the deeper side, resulting in the observed morphological difference. (d–f) Immunohistochemical localization of substrate molecules in 3-week-old sagittal sutural tissue. Distribution of substrate molecules showed that they were mainly produced in the dura mater, confirming the prediction. In all the specimens tested, we observed stronger staining in the dura mater (arrowheads) than in the pericranium (arrows). Simulation parameters: a₀ = 0.05−−0.1, a₁ = 0.5, a₂ = 0.1, a₃ = 0.1, d = 4.0.

Development of skull suture interdigitation. (a) In the newborn human skull, the sagittal suture is straight. (b) In the adult human skull, the sagittal suture shows interdigitation. (c) Time course of fractal dimension in the human skull suture. The correlation is moderate according to Cohen's scale (Cohen 1988). (d) Time course of average amplitude change in the human skull suture. The correlation is moderate according to Cohen's scale (Cohen 1988). (e) MicroCT observation of a living mouse skull at 3 weeks. The sagittal suture remains straight. (f) MicroCT observation of the same mouse at 7 weeks. Interdigitation is present. c, coronal suture; s, sagittal suture; l, lambda suture; f, frontal suture; pf, posterior frontal suture.

The model reproduces suture interdigitation. (a) Scheme of the human skull at birth. The interdigitation simulation region is depicted by a dashed box. (b) Two-dimensional simulation of the model with a straight line initial condition faithfully mimicked the pattern formation of a suture line. The white area represents a higher value. Sometimes ‘‘sprouting” patterns were observed in simulation results (red circle). (c) Actual pattern formation dynamics in the mouse skull from 4 to 7 weeks. The formation of interdigitation resembles results of simulation. (d) Actual pattern in a human skull specimen. The sprouting pattern was also found in actual specimens (red circle). Simulation parameters: a₀ = 0.05, a₁ = 0.5, a₂ = 0.1, a₃ = 0.1, d = 4.0. The width and height of the simulation area roughly corresponds to 1 cm. c, coronal suture; s, sagittal suture; l, lambda suture; f, frontal suture.

Retraction of osteogenic front occurs during suture development. (a) Cultured skull specimen after 4 days. (b) Cultured skull specimen after 7 days. Compared with (a), some parts of the osteogenic front have retracted (arrowheads). (c) TRAP staining of a mouse skull (3 weeks old). Staining was detected in suture lines which undergo active interdigitation. (d) Osteoclast activity was also detected in the suture by MMP9 immunohistochemistry (arrow). (e–j) The time course of the development of sutural tissue in a single specimen. (e–g) represents a surface-rendering view and (h–j) shows the observation of a specific section. The suture line was relatively straight at 3 weeks (e,h) but formed interdigitation at 6 weeks (f,i). Superimposition of (e–f) and (h–i) revealed that the suture line retracted in the interdigitating region (arrows in g, j). In (j), plate (h) in red and plate (i) in blue are superimposed. Therefore, red regions are sites of retraction (arrows).

(a) Time course of posterior frontal suture closure. The surface view and frontal section of a single mouse skull specimen are presented. The posterior frontal (PF) suture was closed while the sagittal suture remained patent at 7 weeks. The posterior frontal suture did not undergo interdigitation. Frontal bones were in general thicker than parietal bones, and edges of the frontal bones became thicker with age. (b) Working hypothesis. The amount of substrate from the dura mater is greater in the posterior frontal suture than in the sagittal suture. (c) Numerical simulation of the model with very high substrate basal activity (a₀). If we increased a₀ too much, the sutural tissue disappeared, mimicking craniosynostosis. (d) Production of substrate molecules in the dura mater. mRNA levels of G3PDH (positive control), BMP4, FGF2 and TGFb1 were compared in the dura mater underlying the sagittal and posterior frontal sutures. Stronger expression of substrate molecules was observed in the posterior frontal region, indicating that the amount of substrate from the dura mater determines the sagittal-posterior frontal difference. Simulation parameters: a₁ = 0.5, a₂ = 0.1, a₃ = 0.1, d = 4.0. c, coronal suture; s, sagittal suture; l, lambda suture; pf, posterior frontal suture.