Comparing six nonlinear equations describing the 2-D profiles of apical meristems

Shoot and root apical meristems (SAMs and RAMs, respectively) drive primary vascular plant growth, yet their 2-D profile geometries remain poorly quantified. Shoot and root apical meristems differ in evolutionary origin, cellular organization, and developmental context, prompting investigation into whether their shapes differ within and across angiosperms, gymnosperms, and representative seedless vascular plants.

SAM and RAM profiles from 11 representative taxa were extracted from histological images. Six nonlinear equations (the catenary, parabolic, hybrid catenary-parabolic, performance, superparabolic, and superellipse equations) were fitted to normalized profile coordinates. Model performance was evaluated using the Akaike information criterion (AIC).

The superparabolic equation provided the best fit for eight of 16 of the SAM and RAM profiles, as evidenced by the lowest AIC values, whereas the hybrid catenary-parabolic equation performed best for five profiles. These two equations outperformed the other four, though no single model was universally superior across all profiles. Although meristem shapes differed, SAM and RAM geometries showed no consistent differences across the three plant groupings.

Both the superparabolic and hybrid catenary-parabolic equations provide robust descriptions of SAM and RAM profiles, perhaps reflecting a convergence in apical meristem geometry across otherwise divergent vascular plant lineages. This quantitative approach offers a potential tool for comparing meristem geometry and shape that can be extended to the study of nonvascular plants to increase our understanding of plant form, evolution, and meristem functionality.