ADAPTATION FROM RESTRICTED GEOMETRIES: THE SHELL INCLINATION OF TERRESTRIAL GASTROPODS

Authors


Abstract

The adaptations that occur for support and protection can be studied with regard to the optimal structure that balances these objectives with any imposed constraints. The shell inclination of terrestrial gastropods is an appropriate model to address this problem. In this study, we examined how gastropods improve shell angles to well-balanced ones from geometrically constrained shapes. Our geometric analysis and physical analysis showed that constantly coiled shells are constrained from adopting a well-balanced angle; the shell angle of such basic shells tends to increase as the spire index (shell height/width) increases, although the optimum angle for stability is 90° for flat shells and 0° for tall shells. Furthermore, we estimated the influences of the geometric rule and the functional demands on actual shells by measuring the shell angles of both resting and active snails. We found that terrestrial gastropods have shell angles that are suited for balance. The growth lines of the shells indicated that this adaptation depends on the deflection of the last whorl: the apertures of flat shells are deflected downward, whereas those of tall shells are deflected upward. Our observations of active snails demonstrated that the animals hold their shells at better balanced angles than inactive snails.

What baggage shape and what method of carrying it are optimal for transporting goods? The issue of ergonomics has become more important since animals became terrestrial and, thus, subjected more directly to the effects of gravity. Members of the class Gastropoda are ideal model organisms to use to study such a problem. The Gastropoda appeared during the Cambrian period and are highly diverse in both shell morphology and habitat. This class contains fresh-water, marine, and terrestrial species, and comparisons between the aquatic and terrestrial species are very useful for understanding the adaptations to each environment. Additionally, the parts of the animal's body that carry (soft tissue) and are carried (shell) can be considered separately from each other, and the morphology of Gastropod shells can be treated quantitatively because several models have been developed to mimic them (e.g., Raup 1966; McGhee 1980; Illert 1982, 1989; Okamoto 1988; Ackerly 1989; Stone 1995).

The development of a well-balanced body shape is one effective method for reducing the cost of locomotion under the force of gravity, and there have been various studies on shell balance and shell shapes of terrestrial gastropods. The spire index, defined as a shell's height divided by its diameter, shows a bimodal distribution in terrestrial gastropods: there are many high- and low-spired gastropods, whereas middle-spired gastropods are rare (Cain 1977, 1978a,b; Cameron 1981; Cowie 1995). In contrast, aquatic gastropods show a continuous distribution of spire indices (Cain 1977). Field and laboratory studies have revealed that gastropods with tall shells are generally active on high-angled or vertical surfaces and that gastropods with flat shells are active on low-angled or horizontal surfaces (Cain and Cowie 1978; Cameron 1978; Cook and Jaffer 1984). Therefore, this bimodality was proposed to be related to the mechanics of shell balance during locomotion on different substrates (Cain and Cowie 1978), a hypothesis that was supported by theoretical analyses and empirical data (Okajima and Chiba 2009). It was also shown that both tall and flat shells are good for stability, whereas middle-spired shells are not well balanced. Noshita et al. (2012) demonstrated that land snails are more highly constrained than marine species with regard to achieving a balance between postural stability and the available space for their soft body.

However, the evolution of form is dependent upon both adaptations and constraints (Raup 1966; Seilacher 1970; Gould 1989; Ubukata 1994). Most of the basic forms of coiling shells exhibit logarithmic spiral growth, but shell shapes are basically limited by the manner of their production because the possible shapes of constantly growing shells are limited. The shell outline (cylindricality) of land snails displays features that are both geometrically constrained and suited for stability; whereas the shells of freshwater snails simply correspond to geometric constraints (Okajima and Chiba 2011). This observation implies that terrestrial gastropod shells have been modified to well-balanced shapes from restricted shapes since the transition to land. However, no studies have undertaken to quantify how the shapes of organisms adapt from constrained shapes.

To study the adjustment of shell balance from constrained shapes, we focus on another essential character of shells, the shell angle (Fig. 1), which has a decisive influence on the shell balance on land and the resistance to flow in water. Shell balance may be more important during locomotion than during inactivity. Cain and Cowie (1978) suggested that holding the shell in a balanced position is more difficult for snails while moving. In fact, terrestrial gastropods have the stronger preference for the angle of activity sites than that of resting sites (Cameron 1978).

Figure 1.

Measurements of gastropod shells. (A) h is the shell height, and d is the maximum diameter of the shells. (B) The shell angle of the resting snails (θrest) is defined as the angle between the coiling axis and shell aperture. (C) The shell angle during activity (θact) is the angle of the coiling axis with respect to the surface on which the snail moves. (D) The deflection during the last phase of shell growth is denoted by Δθrest. Δθact is the amount of change in the shell angle caused by the snail's body tissue.

Vermeij (1971) has measured shell morphology in many different habitats and reported that many high-spired shells have apertures with low angles with the shell axis and various forms of generating curves. Linsley (1977) studied marine gastropods and proposed “the law of tangential apertures,” which states that the aperture plane is tangential to the body whorl of the univalve such that the aperture and ventral-most part of the body whorl lie together in one plane. Shells with such apertures can clamp tightly to the substrate and reduce the risk of predation; this law was supported by detailed measurements of gastropods in diverse habitats (Noshita et al. 2012). The aperture of many land snails becomes deflected downward during the last phase of shell growth (Mazek-Fialla 1934; Mcnair et al. 1981; Goodfriend 1986). However, there are no data on the amount and the direction of the deflection. A comprehensive understanding of the ecological relevance of the deflection is still lacking.

This study built the following working hypothesis: the shell angle of terrestrial gastropods is adjusted to the well-balanced angle from the geometrically constrained angle. If this hypothesis is true, the shell angle of empirical land snails is close to the optimum angle for stability than the constrained angle, and the adjustment of shell angle is found in the developmental process of shells or in the way of holding shells or both. Then, we estimate the shell inclination constrained by the shell geometry and the shell inclination suited to the stability, by developing geometrical and physical models. Additionally, to test whether the adjustment exists, the development of shells and snails’ behaviors was measured.

Materials and Methods

We defined two angles of shell inclination: the shell angle of resting snails (θrest) and that of active snails (θact). The former is the angle of the columella (the center axis of the shell) relative to the shell aperture (Fig. 1B), and the latter is the angle of the columella relative to the surface on which the snails are moving (Fig. 1C). By considering the θrest of a constantly coiling shell as the basic angle of resting snails (θrest:b), the empirical shell angle of resting snails (θrest:e) is given by

image(1)

where, Δθrest is the amount of deflection during the last phase of shell growth (Fig. 1D). Then, the empirical angle of acting snails (θact:e) is

image(2)

where Δθact is the amount of leaning by the soft body. If the structure of the soft tissue limits the degree that the shell can lean (or tilt), the shell inclination depends on the shell shape (the angle between the center axis of the shell and the shell aperture) and the tilt of the soft tissue.

We estimated θrest:b using a geometric model and calculated the optimum θact for stability using a physical model; we then investigated θrest:e and Δθrest by measuring the morphology of actual shells and estimated θact:e and Δθact by observing the behavior of snails. Our results are expressed as the means ± the SD of the mean. Data analysis included the two-sided Student's t-test and two-sided Welch's t-test, depending on the distribution pattern of the data, and nonparametric correlation analysis (Spearman). Differences with P < 0.05 were considered significant.

BASIC ANGLE OF RESTING SNAILS

Gastropod shells are produced by the accretion of calcium carbonate to the apertures, and most shell forms are accurately approximated by a tube expanding and coiling around an imaginary coiling axis. During the growth process, the shell angle (θrest) and spire index (h/d) are decided simultaneously; thus, the shell angle should exhibit a relationship with the spire index. To assess the basic relationship, this study theoretically drew shell shapes of constantly growing shells. We implemented Okamoto's growing tube model (Okamoto 1988) in Mathematica 6, according to Okajima and Chiba (2011).

The shell tube grows as follows “1, During growth of the tube from s to s+ɛ, the center of generating curve Qs moves ɛrs in length (θ radians) around the point Os, in the tangent plane; 2, in the normal plane, the maximum growth point MGPs revolves ϕradians around Qs; 3, the radius of generating curve increases from rs to rs+ɛ” (Okamoto 1988) (Fig. 2A). This model consists of three parameters: E, the radius-enlarging ratio; C, the standardized curvature; and T, the standardized torsion. In this study, the value of E was fixed at 1.025, and C and T remained constant during all of the growth stages (0 < C≤ 1; 0 ≤T≤ 1). Under these conditions, Okamoto's model draws the logarithmic spiral-grown shells, similar to the Raup model (Raup 1966). T defines the amount of bending of the shell tube along the coiling axis, and C defines the tightness of shell coiling around the axis. We assumed that the plane of the shell aperture is perpendicular to the direction of shell growth. When the whorl of the generated shell was separated from the preceding whorl, the values of these parameters were neglected in further analyses because this type of “nonoverlapping” shell is unlikely to exist in nature (Raup 1966).

Figure 2.

The basic relationship between the shell angle and spire index (h/d), as estimated with Okamoto's model (Okamoto 1988). (A) Three differential parameters, E, C, and T, of the growing tube model (Okamoto 1988). (B) Representative examples of theoretical models of coiled shell forms with the E= 1.025, C= 0.4, 0.6, 0.8, 1.0, and T= 0, 0.05, 0.1, 0.2. The gray cells correspond to “nonoverlapping shells.” (C) The basic angles (θrest:b) of the drawn shells plotted against the spire indices (E= 1.025, 0 < C≤ 1, and 0 ≤T≤ 1), exhibiting a significant positive correlation (r= 0.94, P < 0.01).

OPTIMUM SHELL ANGLE FOR STABILITY

The shell balance was estimated by calculating the moment of force at the fulcrum point on the horizontal and vertical surfaces. Moment is obtained by multiplying the magnitude of force by the moment arm. A gastropod shell with a large moment is poorly balanced, because the larger moment works to detach the snail from the surface on which it is crawling. For simplicity, shell shapes were approximated as a cone, a frustum of a cone, and a column, ignoring the portion of the shell that lies under the periphery of the body whorl (the region marked by oblique lines in Fig. 3A). Globular shells and conical shells can be regarded as columns and cones, respectively, and shells with intermediate shapes between column and cone can be regarded as frustums of a cone. To focus on the relationship between shell shape and shell balance, the shell volume and shell thickness were held constant and the thickness of the foot was ignored.

Figure 3.

(A) Approximations of shell shapes for the calculation of shell moment. The shape of shell portions under the maximum diameters (marked by oblique lines) is neglected. Shell shapes are approximated as a column, frustum of a cone, and cone according to shell outline (cylindricality). (B) Schematic diagrams how snails hold their shells. G, the center of gravity; ZG, the distance between G and the center of shell bottom; F, the fulcrum point. (C) Contour plots of the calculated shell moment as a function of spire index and shell angle. (D) Cases in which the shell moment reaches a local minimum with column-shaped (solid lines) and cone-shaped (dashed lines) shells: (1) θ≈ 90° on the horizontal surface (horizontal line), (2) negative SI and θ≈ 90° on the vertical surface (rising curve), and (3) positive SI and θ≈ 0° on the vertical surface (falling curve). The moment caused by a small deviation (1°) of the angle from the above cases is shown by the gray curves.

To assess the shell moment, we estimated the moment arm, that is, the perpendicular distance between the fulcrum and the center of gravity. We assumed that the columellar muscle takes the shortest route from the foot to the center of the shell bottom, and the fulcrum (F) corresponds to the point where the columellar muscle reaches the surface (Fig. 3B). We designated the center of gravity as G and the distance between G and the center of the shell bottom as ZG (e.g., ZG=h/2 for the column and h/4 for the cone). The shell moments on the horizontal surface (Mh) and vertical surface (Mv) are given by

image(3)
image(4)

respectively, where V is the shell volume, ρ is the density of the shell, and g is the acceleration of gravity.

EMPIRICAL ANGLE OF RESTING SNAILS

To test the influence of the geometric rule and shell stability on the shell angle, we used the photographs of snail specimens owned by the Kyoto University Museum (99 species) and the Japanese National Museum of Nature and Science (68 species). All specimens were adult Japanese Stylommatophora. Stylommatophora is a taxon that includes the majority of land snails. The specimens used for measurements were all of the specimens of Stylommatophora collected from Japan and stored in these museums. The spire indices (h/d) and angles of shell aperture (θrest:e) were measured using these photographs.

We also calculated the aperture deflections from the photographs. Deflection is defined as the amount of increase in the shell angle in the last whorl (Δθrest):θrest:e minus the angle of the previous aperture (Fig. 1D). The angle of the previous aperture can be measured based on the growth line, which is a record of the aperture at each growth step. We exclude photographs of shells in which the growth lines are hard to see. The number of species whose shell aperture deflections could be measured is 125.

EMPIRICAL ANGLE OF ACTIVE SNAILS

To study the behaviors of terrestrial gastropods, we measured the shell angles of moving snails (θact:e) (Fig. 1C) on horizontal and vertical surfaces. In these experiments, the snails were placed, one by one, into a 10 cm square clear plastic box. We captured images of moving land snails from three dimensions and calculated θact:e every 10 sec for 10 repetitions. The shell inclination of each individual was expressed as the mean ± SE of these 10 datapoints. The snails used in this experiment were 33 individuals of 15 species (Table 2). Shells of these species extend a wide range of spire index. The specimens measured were all of the live snails collected in our field surveys in Japan. The shell height, shell width, and angle of the columella against the shell aperture (θrest:e) were also measured. Additionally, we measured the amount of change in the shell angle between inactive and active animals: (Δθactact:e–θrest:e) (Fig. 1D).

Table 2.  Shell characteristics and angles of empirical gastropods locomoting on horizontal and vertical surfaces.
SpeciesShell height [cm]Shell width [cm]Shell angle during inactivity (θrest:e) [degree]Shell angle during activity (θact:e) [degree]
Horizontal surface (mean ± SD)Vertical surface (mean ± SD)
 
Euhadra peliomphala 12.103.7045.2 77.8 ± 7.27 68.1 ± 9.97
Euhadra peliomphala 2 2.00 3.80 46.4  70.3 ± 4.78  73.8 ± 10.1
Euhadra peliomphala 32.103.8542.8 67.0 ± 9.82 72.5 ± 5.92
Euhadra quaesita 1 2.35 3.60 41.9  63.6 ± 5.29  71.9 ± 1.20
Euhadra quaesita 22.453.8542.7 75.4 ± 2.68 70.4 ± 6.73
Euhadra subnimbosa 1 1.95 3.15 58.7  64.5 ± 4.37  68.5 ± 9.97
Euhadra subnimbosa 21.903.0054.6 79.2 ± 4.54 78.8 ± 4.65
Euhadra grata 1 2.00 3.10 40.2  67.3 ± 3.61  66.9 ± 7.20
Euhadra grata 21.903.0541.3 69.7 ± 3.95 69.6 ± 1.86
Euhadra grata 3 1.85 3.00 45.8  77.0 ± 1.06  67.8 ± 6.94
Cantareus aspersus 12.502.7547.0 37.9 ± 6.80 29.7 ± 4.97
Cantareus aspersus 2 2.30 2.85 42.0  54.1 ± 2.13  51.1 ± 6.20
Cantareus aspersus 32.502.5041.1 36.6 ± 1.25 38.4 ± 14.2
Ainohelix editha 1 1.20 2.15 51.2  77.3 ± 7.38  70.5 ± 1.96
Ainohelix editha 21.302.2050.4 66.2 ± 11.0 62.3 ± 1.98
Ainohelix editha 3 1.20 2.00 50.5  71.2 ± 3.34  59.1 ± 4.01
Acusta despecta sieboldiana 12.102.1735.9 57.0 ± 3.45 42.2 ± 12.6
Acusta despecta sieboldiana 2 1.93 1.92 28.1  49.3 ± 7.77  51.4 ± 3.63
Bradybaena similaris 11.151.6029.9 65.1 ± 5.49 47.3 ± 8.38
Bradybaena similaris 2 0.90 1.20 32.7  64.1 ± 13.7  61.2 ± 15.8
Aegista pannosa 10.811.6354.1 65.8 ± 6.52 77.4 ± 2.20
Aegista pannosa 2 0.90 1.85 59.1  79.8 ± 2.63  68.5 ± 4.30
Ezohelix gainesi 12.793.4225.9 49.0 ± 2.99 41.5 ± 3.97
Ezohelix gainesi 2 2.40 3.05 26.1  54.5 ± 7.17  57.4 ± 14.4
Pinguiphaedusa platydera 2.84 .568.87 2.30 ± 1.11−4.19 ± 4.58
Stereophaedusa japonica 2.70 0.65 5.14  7.37 ± 1.13 −5.59 ± 0.98
Succinea lauta 11.801.0017.6 13.1 ± 3.11 10.2 ± 3.05
Succinea lauta 2 1.85 1.10 17.6  7.98 ± 2.60  13.2 ± 4.84
Succinea lauta 31.901.1719.3 15.1 ± 2.87 13.7 ± 2.91
Mirus japonicus ugoensis 1 2.40 0.88 18.0  5.83 ± 3.70  4.20 ± 2.34
Mirus japonicus ugoensis 22.430.8421.7−7.15 ± 0.52−8.40 ± 1.86
Mirus japonicus hokkaidonis 1 2.64 0.74 10.6 −0.14 ± 4.50 −2.91 ± 3.67
Mirus japonicus hokkaidonis 22.520.7210.8−8.72 ± 1.33−3.01 ± 0.74

Results

BASIC ANGLES OF RESTING SNAILS

Representative examples of theoretically drawn shells are shown in Figure 2B. Table 1 shows the spire index and shell angles (degree) of models of constantly grown shells. The figure and table show that the effect of T is greater on both the spire index and the basic angle than the effect of C: a larger T produces a taller shell and a higher angled aperture (Fig. 2C). We found a significant positive correlation between the basic shell angle and the logarithmic spire index (Spearman's correlation coefficient [two-tailed]: r= 0.94, P < 0.001, n= 40). Essentially, a high-spired shell is produced by sufficient whorls of a tube bent downward. Note that if E is set at a different value (E > 1.0), the basic angle of the theoretically drawn shells shows the same relationship with the spire index.

Table 1.  Shell angles of theoretically drawn shells.
Standardized curvature CE= 1.0250Standardized torsion T
0.0250.050.10.150.20.250.30.350.4
 
  1. Spire indices are shown in parentheses. Shells with low C and high T are removed because they are “nonoverlapping” shells.

1.00.23 (0.53)1.8 (0.53)2.5 (0.74)6.0 (1.2)9.9 (1.5)12 (1.9)12 (2.3)18 (2.7)19 (3.1)20 (3.5)
0.9 0.33 (0.50) 3.5 (0.52) 4.7 (0.73) 6.5 (1.1) 12 (1.5) 15 (2.0) 16 (2.4) 18 (2.9)   
0.80.49 (0.48)2.8 (0.51)4.7 (0.73)8.4 (1.2)10 (1.6)13 (2.1)    
0.7 0.96 (0.45) 4.3 (0.51) 5.9 (0.75) 7.0 (1.3) 12 (1.8)      
0.60.42 (0.41)2.4 (0.50)3.3 (0.93)10 (1.3)      
0.5 0.51 (0.36) 1.5 (0.53) 4.2 (0.79)        
0.40.67 (0.33)2.9 (0.52)8.4 (0.80)       
0.3 0.73 (0.26)          

We used Okamoto's model to describe shell morphology. If we use other modeling methods (e.g., Raup 1966; McGhee 1980; Illert 1982, 1989; Okamoto 1988; Ackerly 1989; Stone 1995) on the assumption that the shell aperture, which expands with shell growth, is perpendicular to the direction of growth, and the shell is formed based on a constant rule during all of the growth stages, these methods will produce the same tendency as Okamoto's model.

OPTIMUM SHELL ANGLE FOR STABILITY

Figure 3C shows that the shell moment was minimized (1) when θ≈ 90° on the horizontal surface, whereas on the vertical surface, it was minimized (2) when SI (h/d) was minimized and θ≈ 90° and (3) when SI was maximized and θ≈ 0°. The local optimum for shell balance on each of the horizontal and vertical surfaces was obtained by overlapping the curves that show the moment of the column- and cone-shaped shells (Fig. 3D). Note that the shell approximated as a frustum of a cone has an intermediate moment in between the column- and cone-shaped shells (Okajima and Chiba 2011). The optimum angle for each-spired shells, therefore, can be obtained by comparison between two extremes: column- and cone-shaped shells (Fig. 3D). On the horizontal surface, when the angle between the coiling axis and the surface was changed slightly, the rate of increase of the moment was greater in shells with a higher spire index than those with a lower spire index.

Therefore, for gastropods to have the best-balanced spire index and cylindricality, the optimum θact of the low-spired gastropods with SI ≤ 1.4 is 90° on both horizontal and vertical surfaces. For tall shells where SI > 1.4, only 0° is defined as the optimum angle because high-spired gastropods are generally active on high-angled or vertical surfaces (Cain and Cowie 1978; Cameron 1978; Cook and Jaffer 1984) and because a shell held at an angle is very problematic for these animals (Fig. 3D).

EMPIRICAL ANGLES OF RESTING SNAILS

Figure 4A shows the empirical angles of resting snails (θrest:e) plotted against spire index. The mean angle of flat shells with an SI ≤ 1.4 was 46 ± 9.2° and that of tall shells with SI > 1.4 was 11 ± 6.9°. The angle of flat shells is significantly larger than that of tall shells (Welch's t-test, P < 0.001).

Figure 4.

The empirical shell angle and aperture deflection of resting snails. (A) The empirical shell angle of resting snails (θrest:e), as measured using shell specimens of Japanese Stylommatophora (the Kyoto University Museum and Japan National Museum of Nature and Science). (B) Deflections (Δθrest) during the last whorl are plotted against the spire index. (C) Shell angles during inactivity without the deflection: θrest:e–Δθrest. (D) The deflection shows a significant positive correlation with the shell angle after the deflection.

Figure 4B shows the aperture's deflection (Δθrest) during the last phase of shell growth. The mean Δθrest of the shells with an SI ≤ 1.4 is 14 ± 9.3° and that of the shells with an SI > 1.4 is –13 ± 7.9°. A significant difference was found by the Student's t-test analysis between these mean values (P < 0.001). The shell angles of resting snails without the deflection, θrest:e–Δθrest, of each species are shown in Figure 4C. For shells with an SI ≤ 1.4, the mean inclination before adjustment is 33 ± 8.1°, and in shells with an SI > 1.4, the mean inclination before adjustment is 24 ± 6.7°. Even before the deflection, a flat shell has a larger shell angle than a tall shell (Student's t-test, P < 0.001). In flat shells, however, the mean angle of deflected shells (θrest:e) is significantly larger than that of shells before deflection (θrest:e–Δθrest), whereas in tall shells, θrest:e is smaller than θrest:e–Δθrest (Student's t-test, P < 0.001). Figure 4D shows the relationship between the shell angles after the deflection (θrest:e) and the deflection (Δθrest). Statistically significant (Spearman, P < 0.001) positive correlations were observed between θrest:e and Δθrest in both flat shells (r= 0.54) and tall shells (r= 0.52). Therefore, these deflections increase the angles of flat shells and decrease those of tall shells.

EMPIRICAL ANGLES OF ACTIVE SNAILS

Table 2 shows the shell characteristics and angles of gastropods crawling on the horizontal and vertical surfaces. However, no significant difference was found between the shell angle of snails moving on the horizontal surface and on the vertical surface. The following analyses were then conducted without considering the surface inclination. The empirical angles of active snails are shown in Figure 5A. The mean θact:e of shells with an SI ≤ 1.4 is 63 ± 13° and that of shells with an SI > 1.4 is 2.9 ± 8.2°. Flat shells have significantly larger θact:e values than tall shells (Welch's t-test, P < 0.01).

Figure 5.

Shell inclinations of active gastropods. (A) The shell angles (θact:e) of snails moving under laboratory conditions. The filled circles show the shell angle of snails on a 0° surface, and the crosses show the shell angle of snails on a 90° surface. (B) The amount of change in the shell angle between inactivity and activity: (Δθactact:e−θrest:e).

We then studied how a snail's body changes the shell angle during activity (Δθact) (Fig. 5B). The mean Δθact of low-spired gastropods with an SI ≤ 1.4 is 20 ± 12° and that of high-spired gastropods is –11 ± 8.2°. There is a significant difference between them (Student's t-test, P < 0.01). Thus, when terrestrial gastropods are moving, their soft bodies hold flat shells in an erect position, whereas tall shells are laid flat.

Discussion

The important findings of this study are that (1) constantly coiled shells are geometrically constrained from having a well-balanced angle and (2) the empirical shell angle of land snails is improved with respect to stability by the deflection in the last whorl and the snails manner of carrying the shell.

Our geometric analysis suggests that taller shells tend to have larger angles when the shells are constantly growing during all of the animal's growth stages. This trend is primarily due to the relationship between shell shapes (angle and spire index) and the parameter T: the shell aperture is gradually bent down along the shell axis as a function of T as the shell grows. Thus, when the direction of shell growth is perpendicular to the shell aperture at each growth step, the larger T produces taller shells with larger shell angle. This result is in conflict with the optimum shell angle for stability: 90° for flat shells and 0° for tall shells (Fig. 3D). Furthermore, this result indicates that it is difficult for constantly growing shells to maintain well-balanced shell angles.

Our analyses suggest that the actual shell angle of terrestrial gastropods decreases with an increase in the spire index, a trend that agrees with the relationship that well-balanced shells should have. Additionally, the stability of the actual shell is improved by the deflection of the shell aperture during the last phase of growth. Regarding flat shells, the deflection has been discussed and has been interpreted as an adaptation to reduce water loss in resting snails (Mazek-Fialla 1934). McNair et al. (1981) proposed that the deflection reduces the probability of the dislodgement of the shell of resting individuals. The downward deflection of flat shells, and the upward deflection of tall shells suggest that shell deflections are also adaptations to improve shell balance. If there is some upper limit to the amount of leaning that can be accomplished by the soft tissue, this improvement in the shell angle of resting animals also benefits more active animals.

The shell angle without deflection is not positively correlated with the spire index, even though the geometric model predicted that it would be. For simplicity, we assumed that the shell aperture is perpendicular to the direction of the shell growth in our geometric model; although this assumption is approximately true, there may be an improvement in shell balance with respect to the aperture's angle with the direction of growth. Most flat shells appear to have growth lines that are oriented more downward than 90° with respect to the growth direction, thus increasing the shell angle of flat shells, which may be effective for improving shell balance.

While terrestrial gastropods are moving, shell stability may strongly influence the cost of locomotion. Cameron (1978) has shown that active snails have a stronger preference for well-balanced sites than resting snails. During activity, terrestrial gastropods hold their shells at a better angle for stability: low-spired shells are held up, and high-spired shells are laid downward by the soft tissue. However, the amount of change in the shell angle is less than 40° in almost all of the species evaluated in this research, suggesting that the largest possible degree of change in the shell angle is limited, which may be why low-spired gastropods do not hold their shells at the optimally balanced angle of 90°.

This study defines the optimum angle for the balance of tall shells as 0°. Strictly speaking, this is the optimum when the animals are moving on a vertical surface (Okajima and Chiba 2009, 2011). Without considering the vibration of the shell, 90° is the optimum angle for movement on a horizontal surface; however, carrying a shell without shaking it is impossible. To carry a tall shell in an upright stance is very unstable because any deviation from 90° throws the shell out of balance, which is consistent with the fact that the only snails that raise tall shells straight upward are juveniles or small snails. Because most of the high-spired gastropods are generally active on high-angled and vertical surfaces, we simplified our definition to specify that the optimum angle of high-spired gastropods is 0°. However, there are species of large gastropods with very elongated shells that drag their shells on horizontal surfaces. In our experiment, the shells of Pinguiphaedusa platydera and Stereophaedusa japonica touched the horizontal surface. For these species, the cost of dragging their shells, including the friction associated with dragging, may be smaller than the cost of holding the shell upright.

To evaluate the adaptation of snails to gravity, a comparison of terrestrial and aquatic species (on which gravity imposes little effect) is valuable. The results of Noshita et al. (2012) demonstrate that, in contrast with terrestrial species, there is no general correlation between the shell angle of aquatic species and the spire index, indicating that the shell features of land snails are caused by adaptations regarding postural stability. Such a trend in the shell angles of aquatic gastropods is also inconsistent with the prediction of the geometric rule, unlike the shell outline: fresh-water gastropods exhibit a negative correlation between the outline and the spire index, corresponding to the result of the geometric model (Okajima and Chiba 2011). The shell angle of terrestrial gastropods may be adapted from restricted shapes for the purpose of stability under the effect of gravity, whereas the shell angle of aquatic gastropods may have adapted for other reasons, for example, to reduce the risk of dislodgement, for protection against predation, and for resistance to water flow (Vermeij 1971; Linsley 1977).


Associate Editor: P. D. Polly

ACKNOWLEDGMENTS

We are grateful to Y. J. Wakano, T. Ubukata, and K. Yamasaki for fruitful discussions. We also thank M. Kowalewski, J. W. Huntley, and an anonymous reviewer for critically reviewing the manuscript. This study was supported by the Meiji University Global COE Program “Formation and Development of Mathematical Sciences Based on Modeling and Analysis.”

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