Shell morphology and its conceptual implications have attracted the attention of scientists in a wide range of disciplines1,2,3,4. Computational investigations have aimed at providing realistic descriptions of shapes and patterns of coiled shells5,6,7,8,9, while empirical investigations have concentrated on the analysis of the adaptive nature of various morphologies10,11,12,13,14. Although the degree to which evolution is predictable is under debate, the phenomenon of convergent evolution demonstrates that evolutionary pathways are more or less constrained above the level of species15,16,17. In effect, there is evidence that shell shapes are adaptive, for they have evolved independently more than once18. Convergent evolution of form is addressed from a modeling perspective of theoretical morphology12, in which the concept of theoretical morphospace is introduced10. Each point in a morphospace represents a hypothetical form and the evolution of an actual form is visualized as a predictable process in the morphospace12,13.

The seminal work of Raup showed that natural shells are not randomly distributed in the morphospace of theoretically possible forms, but rather they are confined to restricted regions10. The biased distribution is explained in terms of functional and developmental factors in the manner that theoretically possible but naturally not occurring forms would be biologically impossible or functionally inefficient13. By investigating various functional factors, Raup concluded that the observed distribution of normal ammonoids, an extinct group of marine mollusks (cephalopods), is not explained by a single factor. Indeed, the ideal form to optimize the utilization of shell material resources did not correspond to any natural species19.

The concern of the present study is that this accepted opinion is based on a plausible and convenient assumption of isometric growth that shell thickness increases in proportion to shell size. Although this assumption is often made for spiral shells in general20,21, as remarked by Raup19, biometric data of Trueman indicate rather that shell thickness of ammonoids does not increase as rapidly as shell size22. If this observation is accepted, it is not intuitively clear whether and how allometric variation of thickness influences the economy of curved surface construction. Fortunately, this problem is reduced to a well-defined mathematical problem. Here we revisit how shell shape affects shell use efficiency, particularly concerning the effect of allometric scaling of shell thickness. The neglected factor of thickness variation is shown to have a significant impact on the evolutionary perspective of shell form.

Shell form is represented by the growth trajectory of the mouth aperture23. We investigate hypothetical shells generated from a circular aperture with radius 1 and thickness h0 (Fig. 1a). Each form is specified by the center coordinate (x0, y0) of this initial aperture, the whorl expansion rate (W), and the scaling exponent of thickness (ε). The parameter W is used by Raup, while the first two parameters x0 and y0 correspond to Raup’s T and D by T = x0/y0 and D = (y0 − 1)/(y0 + 1) (ref. 10). Various shapes are represented by means of these three parameters (x0, y0, W) (Fig. 1b). The last parameter (ε) for thickness variation is a new feature of this study. When the scaling is isometric (ε = 1), thickness and size of the aperture grow at the same rate. Allometric scaling (ε < 1) means that shell thickness does not increase as rapidly as the apertural size. For a given volume of shell material (Vs), different shells with different sets of “genetic” instructions (x0, y0, W and ε) end up with different interior volumes (V). The present problem is to find optimal shape to maximize the inner volume (V) for a fixed volume of shell material (Vs). A scaling argument indicates that these volumes V and Vs are proportional to the 3rd and (2 + ε)-th power of linear size L, respectively. Roughly speaking, the latter is understood as surface area (S ∝ L2) times thickness (h ∝ Lε), i.e., Vs ∝ L2+ε. We are interested in shape dependence, i.e., the problem independent of the size L. Maximizing inner volume (V) for a given value of shell volume (Vs) is equivalent to maximizing , where a factor F is introduced by noting that Vs is proportional to h0, the initial thickness (Fig. 1a). This factor is determined by the shell form (x0, y0, W, and ε) (Supplementary Information). Accordingly, it is interpreted as a measure of the efficient use of shell materials. Below we show how this factor (F) varies depending on the morphological parameters (x0, y0, W, and ε).

Figure 1: Theoretical representation of coiled shells.
figure 1

(a) A coiled shell is described by geometrical parameters x0, y0, W, h0 and ε. The first two parameters (x0, y0) are the x and y-coordinates of the center of an initial aperture of radius 1 and thickness h0. The expansion rate of successive whorls is W, whereas thickness varies in proportion to W raised to the power of ε. As shown in this figure, successive whorls overlap when the expansion rate W (>1) is small. (b) Coiled shell forms in the three-dimensional parameter space (morphospace) of x0 (>0), y0 (>0) and W (>1). This is a schematic representation of varying morphology. The central form is continuously deformed into each of three forms at the end of axes as one of the three parameters x0, and W is increased while the others are fixed.

Figure 2 shows contour plots of F in the x0-y0 plane for various values of ε and W, where the expansion rate W is expressed in terms of the natural logarithm logW. Three bottom panels for isometric growth (ε = 1) indicate that the peak of F does not lie in the shown range of W. Indeed, F is maximized for logW = 0 (W = 1). Shell shapes in this limit are unrealistically tightly coiled. Most importantly, a sublinear variation of thickness (ε < 1) brings an optimal shape in a realistic region of the parameter space (morphospace). For ε = 0.5 (the second row of Fig. 2), a peak of F lies at (x0, y0, logW) = (0, 1, 2.83).

Figure 2: Shape dependence of shell material efficiency.
figure 2

The efficiency (F) is shown as contour plots in the x0-y0 plane for ε = 0, 0.5, 0.8, 1 and logW = 6.28, 2.93, 0.55. For some representative points (indicated with arrows), the aperture (circle) and shell shape (image) are shown along with the coiling axis and the expansion center (a dot with a dashed line).

Optimal values of the morphological parameters (x0, y0, W) vary depending on the thickness exponent ε in an interesting manner. As shown in Fig. 3a, optimal shape makes transition around from a plane-spiral form (x0 = 0) to a conical-spiral form (x0 > 0). For ε < 0.6, F is maximized at x0 = 0 and y0 = 1 (Fig. 3b). Then the expansion center O lies at the edge of the aperture (D = 0). This is a basic characteristic of bivalve shell form (ε = 0 of Fig. 3a)1. As ε increases, a steep rise in y0 sets in while keeping x0 = 0 for a while. A leftward rise in Fig. 3b is continued to a dashed line in Fig. 3c. Note that W on the horizontal axes of Fig. 3b and c decreases as ε increases. Figure 3d is a contour plot of F in the W- y0 plane for x0 = 0, where an adaptive ridge of plane spiral (planispiral) forms is seen. As a matter of fact, for ε > 0.8, the planispiral form is barely stable if it were not for the constraint of bilateral symmetry (x0 = 0). The true optimum has a conispiral form (ε = 0.9 in Fig. 3a). Two ridges radiate from the summit of F, one circling to the planispiral x0 = 0 and the other flowing along the x0-axis (see panels for logW = 0.55 in Fig. 2). As ε increases further, the optimum shifts in the latter direction. Figure 3e shows the W-dependence of the semi-angle β at the cone apex of optimal shape. In the limit of isometric growth (ε → 1), optimal shape is infinitely highly spired, i.e., W → 1, x0 → ∞, β → 0, and F → 0.85. This is consistent with the above observation that the ideal isometry gives an unreasonable result. In effect, this limit is a mathematical singularity (Supplementary Information). Even a slight deviation from isometry (ε = 1) has a significant effect on optimal shape. Figure 3f shows the locus of optimal points in the morphospace.

Figure 3: Optimal shape depends on the thickness exponent ε.
figure 3

(a) Optimal shapes resemble three basic shapes of natural shells, namely the bivalve shell, the flat-coiled shell, and the conical-coiled shell. (Left) ε = 0: (x0, y0, logW) = (0, 1, 4.45) (F = 0.175). (Left center) ε = 0.8: (F = 0.580). (Right center) ε = 0.8: (x0, y0, logW) = (1.61, 1.12, 0.93) (F = 0.580). (Right) ε = 0.9: (x0, y0, logW) = (5.67, 1.27, 0.29) (F = 0.683). (b) Optimal values of y0 and W (x0 = 0). (c) A dashed curve is the locus of y0 and W that maximize F under the constraint of x0 = 0. Ammonoid suborders are indicated according to Raup’s biometric data19. (d) A contour plot of F in the W-y0 plane (ε = 0.88). (e) The semi-angle β at the apex of optimal shapes is plotted against W (ε ≥ 0.8). Biometric data are due to Thompson1. (f) Optimal values of (x0, y0, logW) are plotted in a three-dimensional morphospace. According to Raup10, the regions in which representatives of four major taxonomic groups are concentrated are indicated by letters.

Even though there is no adjustable parameter, the results conform with general tendencies observed in the frequency distribution of actual species. In Fig. 3c and e, biological data are indicated for comparison1,19. Raup remarked that the bulk of species in four taxonomic groups, brachiopods, bivalves, gastropods, and cephalopods, are confined to non-overlapping regions, which if taken together comprise a relatively small part of the parameter space10. These regions are indicated in Fig. 3f. The result in Fig. 3c is consistent with the prior result W → 1 and D → 1 for isometric ammonoids (ε = 1)19. For conical spiral shells, the result in Fig. 3e accords with the tendency that high-spired gastropod shells are constrained to have a low expansion ratio (negative correlation between T and W)24. In the continuous spectrum of our solutions, the basic forms of snails and bivalves lie at the opposite ends (Fig. 3). In prior studies, it was necessary to fix W at a realistic value20,25. Assuming from the empirical facts, McGhee showed that fossil biconvex brachiopods tend to optimize the surface-to-volume ratio12,13. In contrast to these studies, optimal values of W and D are uniquely determined without ad hoc assumptions (Fig. 3f).

In the present model, the shell volume increases as Vs ∝ L2 + ε in terms of a linear size L, which may be of use to evaluate the thickness exponent ε empirically. The exponent appears to have not been investigated except for ammonoids19,22. As remarked by Raup19, their thickness exponent ε is certainly less than 1 and most likely ε ≅ 0.8 (Supplementary Fig. S2). Indeed, Trueman22 observed that the volume of ammonoid shells increases in proportion to the diameter raised to the power of 2.7–2.8. In comparing with actual shells, however, the thickness variation h ∝ Lε (Fig. 1) does not necessarily correspond to the size dependence of actual thickness if the animal overlays or redistributes shell material as it grows. In fact, thickness of a clam shell is made approximately constant by accretion of an inner shell layer, whereas total thickness increases as size increases26. This observation is consistent with , while the constant thickness h0 in this case is actually not constant for shells of different sizes. Biology is often regarded as a science of exceptions, defying a unified theory. As a pure theory of a general nature, the present study is primitive in many other regards. Additional parameters are necessary to describe more detailed aspects of real forms8. At a low phylogenetic level, ontogenetic variation in the parameters may not be negligible19,27. The efficiency of shell material use is only one factor in the evolution of coiled organisms. In fact, diverse forms in nature signify the concomitant presence of competitive driving forces and various constraints. To name a few, a small value of W is unfit for bivalves to hold the two valves together12,13. Hydrodynamic efficiency should contribute to ammonoid forms (Fig. 3c and d)11,12,13. Postural stability may compete with the shell use efficiency depending on the mode of life (Fig. 3e)14. The actual optimality of shell forms can thus depend on various life history traits and environmental factors, such as predatory animals, food habits and habitat conditions.

It has long been considered that (i) isometric growth is a sound assumption in coiled shells and (ii) shell shape has to do with optimal utilization of shell material19,20,25. We showed that these two conceptions are compatible only if non-isometric thickness variation is taken into account. The more the ideal isometry is approached, the more curved the optimal shell surface.

Additional Information

How to cite this article: Okabe, T. and Yoshimura, J. Optimal designs of mollusk shells from bivalves to snails. Sci. Rep. 7, 42445; doi: 10.1038/srep42445 (2017).

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