Reaction-diffusion systems have been widely used to model pattern formation in biological systems. However, the emergence of Turing patterns in three-dimensional (3D) domains remains relatively unexplored. A few studies on this topic have shown that extending pattern formation from 2D to 3D is not straightforward. Linear stability analysis, which is commonly used to associate admissible wave modes with predicted patterns in 1D and 2D, has yet to be applied in 3D. We have used this approach, together with finite element modeling of a Turing system with Schnakenberg kinetics, to investigate the effects of initial conditions and growing domains on the competition between admissible modes in 3D Turing pattern emergence. Our results reveal that non-random initial conditions on the activator play a stronger role than those on the substrate. We also observe a path dependency of the evolving pattern within a growing domain. Our findings shed new light on the mechanisms ensuring reliable pattern formation in 3D domains and have important implications for the development of more robust models of morphogen patterning in developmental processes.

Adolescent obesity has risen dramatically in the last few decades. While adult obesity may be osteoprotective, the effects of obesity during adolescence, which is a period of massive bone accrual, are not clear. We used a murine model of induced adolescent obesity to examine the structural, mechanical, and compositional differences between obese and healthy weight bone in 16‐week‐old female C57Bl6 mice. We also examined the effects of a return to normal weight after skeletal maturity (24 weeks old). We found obese adolescent bone exhibited decreased trabecular bone volume, increased cortical diameter, increased ultimate stress, and increased brittleness (decreased plastic energy to fracture), similar to an aging phenotype. The trabecular bone deficits remained after return to normal weight after skeletal maturity. However, after returning to normal diet, there was no difference in ultimate stress nor plastic energy to fracture between groups as the normal diet group increased ultimate stress and brittleness. Interestingly, compositional changes appeared in the former high‐fat diet mice after skeletal maturity with a lower mineral to matrix ratio compared to normal diet mice. In addition there was a trend toward increased fluorescent advanced glycation endproducts in the former high‐fat diet mice compared to normal diet mice but this did not reach significance (p < 0.05) due to the large variability. The skeletal consequences of adolescent obesity may have lasting implications for the adult skeleton even after return to normal weight. Given the rates of adolescent obesity, skeletal health should be a concern.