A Universal Model to Align Heterochrony of Chondrogenesis Across Species

Although embryonic chondrogenesis follows conserved developmental stages across mammalian species, the rate of this process varies significantly. We hypothesized that these varying developmental rates could be normalized and modeled by aligning key developmental milestones of articular cartilage: mesenchymal condensation, chondrogenic differentiation, joint interzone formation, cavitation, and primary ossification center formation. A mathematical model was developed in MATLAB by fitting the developmental curves of these key stages in mice, rats, pigs, horses, and humans. Among various mathematical models, a sigmoid curve model demonstrated the best fit, with the highest R² (0.98 ± 0.02) and lowest residual norm (0.36 ± 0.21) after normalizing for gestation period and femoral condylar width to account for species-specific differences. The model's high predictive accuracy (R² = 0.98) was further validated using data from rabbits, canines, and sheep. By differentiating the developmental curve equation, we determined the developmental rate for each species. Our results indicate that embryonic chondrogenesis occurs within a relatively narrow window of gestation (11-32%), with rodents exhibiting faster rates than larger animals. This study presents the first mathematical model to quantitatively align embryonic chondrogenesis across mammals. This model has the potential to fill gaps in developmental timelines through rational calculation and by improving the precision of developmental timing, factors that have often been overlooked in the field. As a pioneering effort, this work opens new avenues for cross-species alignment and enhances the accuracy of data related to embryonic chondrogenesis.Impact Statement As the first mathematical model to align chondrogenesis across different mammals, this work has the potential to enhance the consistency and accuracy of data from various species by correcting inaccuracies and supplementing missing information.