DISCUSSION

These results do not provide support for the tie-point model of septal formation. Suture lobes are not more constrained in their position than suture saddles, nor are saddle tops more rounded than lobe tips, as predicted by the tie-point model (Seilacher 1973, 1975, 1988; Westermann 1975). Rather, sutures show constraint along the entire septal margin, implying that whatever controls septal shape affects the entire margin equally.

In addition, the widely assumed symmetry of suture patterns across the ammonoid's midline simply does not exist. Sutural asymmetry, while not extreme, is the norm and must be taken into account by those proposing alternative models of septal formation, such as the viscous fingering (García-Ruiz et al. 1990; García-Ruiz and Checa 1993; Checa and García-Ruiz 1996; Checa 2003) and diffusion-reaction (Hammer 1999) models. The lack of symmetry may be especially distressing for Paleozoic ammonoid workers, who, by tradition, invert drawings of left sutures so that only "rights" are figured for publication (Mapes, personal commun., 2003). Indeed, we found it difficult to locate published drawings of left sutures for Paleozoic goniatites. Hopefully, the results of the present work will encourage all ammonoid workers to specify whether they are figuring a right or a left suture pattern, and ideally figure both sides of an individual suture. We also urge paleontologists to include multiple sutures for each taxon in figures, and especially an ontogenetic series for each species, with sutures from multiple individuals at about the same ontogenetic stage included. Including more information about suture variation in the paleontological literature would greatly benefit analyses like those presented here and help us develop and test models for septal formation.

The sutural asymmetries may reflect fundamental asymmetries of the shell and/or soft parts. Modern cephalopods have asymmetrical reproductive and digestive systems (largely located in the area just in front of the septum in Nautilus) and even sometimes grow one eye much larger than the other (Nesis 1987; Ward 1987), so it is certainly biologically plausible for asymmetries to exist in ancient cephalopods. Indeed, numerous other authors have noted the existence of asymmetry in the suture line of ammonoids (e.g., Swinnerton and Truemann 1917; Spath 1919b; Hölder 1956; Hengsbach 1979, 1986a, 1986b; Guex and Rakus 1991; Guex 1992; Longridge et al. 2009). In most of these cases, the sutural asymmetry was produced (1) by a shift in the position of the siphuncle away from the midline, (2) as a result of a heteromorphic, helical shell shape, in which one side of the septum was enlarged relative to the other, or (3) in response to shell damage or other individual pathology. None of these explanations applies in the cases presented here. Rather, the present results suggest a more fundamental anatomical cause for the sutural asymmetry.

Notably, the asymmetry in suture pattern documented here is directed, meaning one side is consistently different from the other. Directed asymmetry (DA) is contrasted with fluctuating asymmetry (FA), in which the deviations from symmetry are randomly distributed on either side of the midline, with a mean deviation of zero when specimens are pooled (Dongen 2006). Fluctuating asymmetry has long been used as a window into developmental instability in both extant and fossil organisms, including cephalopods (e.g., Smith 1998; Gowland et al. 2003; Dongen 2006; Vogt et al. 2008). Directed asymmetry, on the other hand, does not indicate developmental instability (Dongen 2006). However, directed asymmetry may stem from developmental regulatory mechanisms operating very early in ontogeny. Recent work has identified a potential molecular pathway, involving the transforming growth factor-beta molecule Nodal, for the establishment of left-right asymmetry in the development of snails, including the position of the initial cells of the shell gland (Grande and Patel 2009).

While the tie-point model of septal formation is not supported by the analyses presented above, the results can be used to place some constraints on alternative models for septal growth and function. One option being explored centers on work by applied mathematicians and physicists studying complexly folded thin sheets; these workers have shown how the edges of flexible membranes deform as a function of the growth rate of the membrane (Sharon et al. 2002, 2004; Marder 2003; Marder et al. 2003; Nath et al. 2003; Coen et al. 2004; Efrati et al. 2009). Applying these concepts to ammonoid septa, and relating them to Hammer's (1999) reaction-diffusion model, this model would enable septa to fold in a consistent way along their entire margin without requiring specific genetically controlled tie-points. It also blends the emphasis of the tie-point model on biological processes with the importance of considering the material properties of the septal membrane stressed by the viscous-fingering model (García-Ruiz et al. 1990; García-Ruiz and Checa 1993; Checa and García-Ruiz 1996; Checa 2003). Work is underway to develop specific predictions of this edge buckling model that can be tested using GIS-based analyses of real ammonoid sutures. Of particular interest is combining GIS and geometric morphometric techniques, such as sliding semilandmarks, to study septal surfaces in three dimensions (Bookstein 1997; Adams et al. 2004).