12:00 PM - 12:15 PM
[BPT02-06] A unification of absolute growth, allometric, and theoretical morphologic models: a case of molluscan shells
Keywords:theoretical morphology, absolute growth, allometry
Study on growth and form of past creatures is a fundamental part of paleobiology. There are three types of approaches: absolute growth analysis, allometry, and theoretical morphology. An absolute growth curve describing the pattern of growth with time provides information about the life history strategy of the animal. However, such information is often unavailable for fossil specimens, and paleontologists commonly focus on the relationship of relative growth rate between measurable morphological traits. Allometric equations are commonly used to describe the relative growth. However, allometric models do not fully represent the geometric characteristics of biological forms and abstraction of geometric information prevents us to perform an inversion analysis of the properties of absolute growth curves from morphological data. A theoretical morphologic model which mimics growth and form of an organism represents the relationship between elementary growth processes and geometric properties. However, theoretical morphology has developed independently from absolute and relative growth analyses in terms of methodology. In the present study, I introduce a new theoretical morphologic model of molluscan shell growth which combines absolute growth and allometric models. It allows us to estimate the properties of absolute growth curves based on an inversion analysis from morphological measurements. A pilot inversion analysis was carried out using artificial data sets to assess reliability of the estimation.
The theoretical morphologic model introduced herein is based on a logistic growth model represented by a sigmoid function of time. It is approximately exponential in the early growth stage, but shows a convex upward curve in the later growth stage, and finally approaches the upper bound of the curve. The logistic function can be defined by its initial value, the intrinsic growth rate, and the inflection point of the logistic curve. These parameters describe the rate and timing of growth processes that underlie shell geometries. The theoretical morphologic model is represented by growth of a helicospiral tube which consists of increases in length of the helicospiral, radius of the spiral, height of the helix, and radii of the elliptic cross-section of the tube. Anisomorphic shell growth can be represented by the theoretical morphologic model if growth of each dimension is approximated by a logistic growth function. A total of 14 dimensionless parameters uniquely define the ontogenetic trajectory of a computer-generated shell form. When intrinsic growth rate and the inflection point of the logistic curve are equal among dimensions, shell shape remains constant throughout ontogeny, even though growth rate changes during growth. If the timing of the inflection point of the logistic curve is nearly equal between a pair of dimensions and the intrinsic growth rate is not greatly different between them, the relative growth between the two dimensions can be approximated by an allometric model. Various patterns of ontogenetic variation in shell shape observed in actual mollusks were reproduced by the theoretical model.
The theoretical morphologic model was designed as a forward model for inversion of parameters of the logistic growth function that are ecologically and developmentally important. A computer-generated planispiral shell form with known parameter values which mimics anisomorphic shell growth of an ammonoid was used to generate data for a pilot inversion analysis. A grid search over the 11-dimensional parameter space revealed that the topography of the parameter hyper-space is not intractably complicated, although the global minimum corresponding to the true values of parameters is in a narrow spike.
The theoretical morphologic model introduced herein is based on a logistic growth model represented by a sigmoid function of time. It is approximately exponential in the early growth stage, but shows a convex upward curve in the later growth stage, and finally approaches the upper bound of the curve. The logistic function can be defined by its initial value, the intrinsic growth rate, and the inflection point of the logistic curve. These parameters describe the rate and timing of growth processes that underlie shell geometries. The theoretical morphologic model is represented by growth of a helicospiral tube which consists of increases in length of the helicospiral, radius of the spiral, height of the helix, and radii of the elliptic cross-section of the tube. Anisomorphic shell growth can be represented by the theoretical morphologic model if growth of each dimension is approximated by a logistic growth function. A total of 14 dimensionless parameters uniquely define the ontogenetic trajectory of a computer-generated shell form. When intrinsic growth rate and the inflection point of the logistic curve are equal among dimensions, shell shape remains constant throughout ontogeny, even though growth rate changes during growth. If the timing of the inflection point of the logistic curve is nearly equal between a pair of dimensions and the intrinsic growth rate is not greatly different between them, the relative growth between the two dimensions can be approximated by an allometric model. Various patterns of ontogenetic variation in shell shape observed in actual mollusks were reproduced by the theoretical model.
The theoretical morphologic model was designed as a forward model for inversion of parameters of the logistic growth function that are ecologically and developmentally important. A computer-generated planispiral shell form with known parameter values which mimics anisomorphic shell growth of an ammonoid was used to generate data for a pilot inversion analysis. A grid search over the 11-dimensional parameter space revealed that the topography of the parameter hyper-space is not intractably complicated, although the global minimum corresponding to the true values of parameters is in a narrow spike.