Theoretical morphological models of coiling#

The ktch.coiling module provides morphological analyses using theoretical morphological models of shell coiling. For hands-on use, see the tutorials linked at the end.

Theoretical morphological models#

A theoretical morphological model is a model that mimics morphologies and morphogenesis: it maps (a few interpretable) parameters to a form (parameters to form). The parameters span a theoretical morphospace, so the model can describe not only forms that exist but also those that are geometrically possible yet unobserved.

In practice the main interest is usually generating forms (generative map). ktch currently provides the generative map as inverse_transform, and parameter estimation from measurement data is planned.

Coiling models#

The ktch.coiling module describes an unbranched accretionary tube: a generating curve (the aperture shape) swept along a generating spiral (also called the growth trajectory), the space curve traced by a reference point as the organism grows.

The module covers forms from a straight tube to a tightly coiled tube, including shells, horns, tusks, and beaks.

Raup’s model#

Raup’s model (Raup & Michelson 1965) represents a shell as a generating curve growing along a logarithmic (equiangular) spiral.

The tube radius and the locus of the generating curve’s reference point are

\[ r_R(\theta) = r_0\, W_R^{\theta / 2\pi}, \]
\[\begin{split} p_R(\theta) = r_0\, W_R^{\theta / 2\pi}\, R_z(\theta) \begin{pmatrix} \dfrac{2 D_R}{1 - D_R} + 1 \\ 0 \\ 2 T_R \left( \dfrac{D_R}{1 - D_R} + 1 \right) \end{pmatrix}, \end{split}\]

where \(\theta\) is the coiling angle and \(R_z(\theta)\) is the rotation about the coiling axis (Noshita 2014). The form is determined by the whorl expansion rate \(W_R\) (w_r, \(W_R > 1\)), the translation rate \(T_R\) (t_r), and the relative distance of the generating curve from the axis \(D_R\) (d_r, \(-1 < D_R < 1\)); \(r_0\) is the initial radius (r0). ktch implements this model as the raup function and the RaupModel class.

Raup’s model, and the morphospace studies it enabled, are the origin of theoretical morphology (Raup & Michelson 1965, Raup 1966).

Growing tube model#

The growing tube model takes a local, differential-geometric view: the shell is a tube of radius \(r(s)\) swept along a trajectory, and the form follows from how the trajectory and the radius change at each growth stage \(s\). A Frenet frame \((\xi_1, \xi_2, \xi_3)\) and the radius evolve as

\[\begin{split} \frac{d}{ds} \begin{pmatrix} \xi_1 \\ \xi_2 \\ \xi_3 \end{pmatrix} = \begin{pmatrix} 0 & C_G & 0 \\ -C_G & 0 & T_G \\ 0 & -T_G & 0 \end{pmatrix} \begin{pmatrix} \xi_1 \\ \xi_2 \\ \xi_3 \end{pmatrix}, \qquad \frac{dr}{ds} = E_G\, r, \end{split}\]

and the trajectory integrates the tangent, \(p_G(s) = p_0 + \int_0^s r(s')\, \xi_1(s')\, ds'\), with the arc length \(l\) satisfying \(dl/ds = r\). Here \(E_G\) is the expansion rate (e_g), and \(C_G\) (c_g) and \(T_G\) (t_g) are the standardized curvature and torsion, that is, curvature and torsion expressed relative to the tube radius. ktch implements this as the growing_tube function and the GrowingTubeModel class.

The model was introduced by Okamoto (1988) to analyse heteromorph ammonoids (\(E_G\) is the logarithm of Okamoto’s original expansion rate \(E\); Noshita 2014). Because the parameters are specified locally, each may vary along growth, passed as a function of \(s\) rather than a constant. This yields heteromorph (irregularly coiled) shells, such as the meandering ammonite Nipponites, which no constant parameter set can describe.

Relationships between models#

The coiling models are closely related. A future revision of this page will describe their correspondence, including the conversion between their parameters.

References#

  • Okamoto, T., 1988. Analysis of heteromorph ammonoids by differential geometry. Palaeontology 31, 35–52.

  • Noshita, K., 2014. Quantification and geometric analysis of coiling patterns in gastropod shells based on 3D and 2D image data. Journal of Theoretical Biology 363, 93–104.

  • Raup, D.M., 1966. Geometric analysis of shell coiling: general problems. Journal of Paleontology 40, 1178–1190.

  • Raup, D.M., Michelson, A., 1965. Theoretical morphology of the coiled shell. Science 147, 1294–1295.

See also#