Allometries and the morphogenesis of the molluscan shell: a quantitative and theoretical model

Abstract

This article explores the close relationships between growth rate and allometries of molluscan shells. After reviewing the previous theoretical approaches devoted to the understanding of shell form and its morphogenesis, we present a free-form vector model which can simulate apertural shape changes and nonlinear allometries. Shell morphology is generated by iteratively adding a growth increment onto the last computed aperture. The first growth increment defines so-called growth vectors which are assumed to be constant in direction (relative to the last computed aperture position) during a simulation of a shell (ontogeny). These growth vectors are uniformly scaled at each time step according to various growth rate curves that are used to simulate the mantle growth over time. From the model, we derive morphometric variables that illustrate the ontogenetic trajectories in time-size-shape space. We investigate the effects of changing the growth curves types, growth rate parameters and growth vector maps on the direction, speed and patterns of ontogenetic allometries. Because this model focuses the issue on time, it highlights a plausible effect of growth rate on shell shape and illustrates some fundamental geometrical properties of the logarithmic spiral, in particular the close relationship between the size and the geometry of growth increments. This model could be used to develop a mathematically data-driven approach where experimentally obtained growth curves could be used as inputs in the model. More generally, our study recalls the role of growth rates in the generation of allometries.

A method for quantifying, visualising, and analysing gastropod shell form

“…For the empirical approach, the quantification methods of shell form have evolved from traditional linear measurement to landmark-based geometric morphometrics and outline analyses (for an overview see Van Bocxlaer & Schultheiß, 2010). At the same time, for the theoretical approach, the simulations of shell form have evolved from simple geometry models that aimed to reproduce the form, to more comprehensive models that simulate shell ontogenetic processes (for an overview see Urdy et al, 2010). Hence, each of the two approaches has been moving forward but away from each other, where synthesis between the two schools of shell morphologists has become more challenging.…”

“…Picado, 2009, Stępień, 2009Meinhardt, 2009;Urdy et al, 2010;Harary & Tal, 2011;Moulton, Goriely & Chirat, 2012;Faghih Shojaei et al, 2012;Chacon, 2012). Here, we will not further discuss the details of the at least 29 published shell models, but refer to the comprehensive overviews and descriptions of these models in Dera et al (2009) and Urdy et al (2010).…”

What can we learn from confusing Olivella columellaris and O. semistriata (Olivellidae, Gastropoda), two key species in panamic sandy beach ecosystems?

“…In contrast, O. semistriata grows isometrically up to 23 mm (Figure 6a), which significantly exceeds the size given in recent identification keys (15 mm; Burch & Burch 1963, Keen 1971. Differences in gastropod shell shape and allometric growth can, in many cases, be explained as a function or even a direct consequence of differences and changes in shell growth rate (Kemp & Bertness 1984, Urdy et al 2010a, 2010b. Our analysis of shell morphometrics ( Figure 6) allows conclusions regarding geometric aspects of growth, but lack the temporal component required to evaluate growth rates.…”

Surface growth kinematics via local curve evolution

“…With regards to seashell models, the inclusion of time as an explicit parameter is a fairly recent development (Rice 1998), and is of great importance in understanding the growth process (Urdy et al 2010). The growth governed by Equations (27) is not time invariant and so the coiling rate -the time it takes to coil around one time, or to complete one whorl in the terminology of seashells -is not fixed; rather the time to coil around increases exponentially while the aperture size increases linearly.…”

“…These models have the advantage that they do not rely on an artificial fixed frame, however this description does not incorporate the actual growth process associated with material deposition rates. More recent models use growth vectors defined on an aperture (Hammer and Bucher 2005;Urdy et al 2010). Such models can create fairly general shell evolutions and explicitly include the aspect of time in shell growth; however, they are still largely based on global definitions and typically cannot provide a convenient mathematical description in terms of a minimal number of parameters.…”