10:45 AM - 12:15 PM
[BPT04-P01] An inverse analysis on the estimation of growth trajectory of shell shape in ammonoids
Keywords:ammonoids, ontogenetic trajectory, inverse analysis, Bayesian inference
Ammonoid shell shape has been one of the most popular targets for evolutionary morphology among fossil organisms. In particular, ontogenetic variation in conch geometry has been focused with attention to its relation to phylogeny. Since a change in shape with growth is a consequence of different rates and/or timings of ontogeny among traits, a path through the trait-time space, ontogenetic trajectory, provides essential information to describe ontogenetic variation. However, ontogenetic trajectory has rarely been analyzed in ammonoids because age information is not available for the animals. In the present study, the ontogenetic trajectories of shell coiling in ammonoids were estimated by using Bayesian inference.
The shell shape of normally coiled ammonoids can be represented by a planispiral tube and the geometry of shell coiling is defined by the relationship between growth of spiral radius (whorl expansion) and increase in spiral length (whorl elongation). The increase of each dimension follows a logistic growth model which is defined by its initial value (x0), intrinsic growth rate (γ), and timing of growth deceleration (τ). This assumption allows us to define a relative time interval over the duration of growth of the spiral radius without information of an absolute time. The growth curve for whorl elongation was defined as a logistic function of the relative time with respect to whorl expansion with a set of given parameter values. The parameters of the logistic model for whorl elongation were optimized in such a way as to maximize the probability density of the posterior distribution when the likelihood function for the regression residuals and the prior distributions of the parameters follow normal distributions. The inverse analyses on the parameter estimation were carried out for 26 Jurassic and Cretaceous ammonitid species.
A grid search over the three-dimensional parameter space revealed that the posterior distribution has a unimodal peak on the long ridge in a γ-τ subspace. The theoretical morphologic models based on the estimated parameters well reproduced ontogenetic changes in shell shape of actual ammonites. The results of parameter estimation indicate a phylogenetic signal in ontogenetic trajectory of the manner of shell coiling. The species belonging to the Phylloceratoidea or the Tetragonitoidea tend to show earlier start of growth deceleration of whorl elongation relative to whorl expansion. However, the timing of growth deceleration is not considerably different between whorl expansion and whorl elongation in most of the desmoceratoid and perisphinctoid species examined. The results infer that local change in timing of growth deceleration caused allometric repatterning observed at the global scale across ammonitid taxa. In contrast, the estimated initial values of spiral length were highly variable within each superfamily. This result suggests that the variability of coiling geometry in the initial growth stage allowed ammonitids to evolve a wide variety of adult shell forms within each clade.
The shell shape of normally coiled ammonoids can be represented by a planispiral tube and the geometry of shell coiling is defined by the relationship between growth of spiral radius (whorl expansion) and increase in spiral length (whorl elongation). The increase of each dimension follows a logistic growth model which is defined by its initial value (x0), intrinsic growth rate (γ), and timing of growth deceleration (τ). This assumption allows us to define a relative time interval over the duration of growth of the spiral radius without information of an absolute time. The growth curve for whorl elongation was defined as a logistic function of the relative time with respect to whorl expansion with a set of given parameter values. The parameters of the logistic model for whorl elongation were optimized in such a way as to maximize the probability density of the posterior distribution when the likelihood function for the regression residuals and the prior distributions of the parameters follow normal distributions. The inverse analyses on the parameter estimation were carried out for 26 Jurassic and Cretaceous ammonitid species.
A grid search over the three-dimensional parameter space revealed that the posterior distribution has a unimodal peak on the long ridge in a γ-τ subspace. The theoretical morphologic models based on the estimated parameters well reproduced ontogenetic changes in shell shape of actual ammonites. The results of parameter estimation indicate a phylogenetic signal in ontogenetic trajectory of the manner of shell coiling. The species belonging to the Phylloceratoidea or the Tetragonitoidea tend to show earlier start of growth deceleration of whorl elongation relative to whorl expansion. However, the timing of growth deceleration is not considerably different between whorl expansion and whorl elongation in most of the desmoceratoid and perisphinctoid species examined. The results infer that local change in timing of growth deceleration caused allometric repatterning observed at the global scale across ammonitid taxa. In contrast, the estimated initial values of spiral length were highly variable within each superfamily. This result suggests that the variability of coiling geometry in the initial growth stage allowed ammonitids to evolve a wide variety of adult shell forms within each clade.