Effects of shell morphology on mechanics of zebra and quagga mussel locomotion

Zebra mussels (Dreissena polymorpha Pallas 1771) and quagga mussels (Dreissena bugensis Andrusov 1897) are two highly invasive species that have become established in the Great Lakes of North America within the past 25 years (Gelembiuk et al., 2006; Karatayev et al., 1998; May et al., 2006), causing much ecological destruction and economic loss (Pimentel et al., 2005). Although zebra mussels have inflicted negative ecological impacts as the earlier colonizer, quagga mussels might eventually become more widespread and destructive than zebra mussels. Quagga mussels initially became prevalent in deep (>50 m) soft sedimentary habitats (e.g. sand, silt), where zebra mussels tend to be rare (Dermott and Munawar, 1993; Mills et al., 1993). However, quagga mussels have more recently been displacing zebra mussels in shallow (<15 m) hard substrate habitats (e.g. rocks and mollusc shells), where zebra mussels previously had been more dominant (Jarvis et al., 2000; Mills et al., 1996; Mills et al., 1999; Stoeckmann, 2003). Differences in shell morphology between zebra and quagga mussels have been proposed to contribute to differences in their ability to colonize diverse substrate types among habitats (e.g. shallow hard versus deep soft sedimentary substrates), potentially affecting competition between the two species (Claxton et al., 1998; Peyer et al., 2010). However, functional consequences of the morphological differences had not been thoroughly examined.

Zebra mussels possess only a shallow-water morphotype, whereas quagga mussels have distinct shallow- and deep-water morphotypes (Fig. 1) (Claxton et al., 1998; Dermott and Munawar, 1993; Peyer et al., 2010). Differences in shell morphology between shallow- and deep-water quagga mussels were found to result from phenotypic plasticity, rather than genetic differentiation, in a common-garden experiment in which the mussels were reared under various environmental conditions (Peyer et al., 2010). Although zebra and shallow quagga mussels resemble each other in overall shell morphology (Fig. 1A,versus 1B) [e.g. in terms of shell height to width ratio (Claxton et al., 1998)], they differ in ventral surface morphology, which is distinctly flattened for zebra mussels and slightly rounded for shallow quagga mussels (Fig. 1A,versus 1B, anterior–posterior views) (Claxton et al., 1998; Dermott and Munawar, 1993). In contrast to zebra and shallow quagga mussels (Fig. 1A,B), deep quagga mussels (Fig. 1C) have shells that are more flattened and compressed together (Fig. 1C, dorsal–ventral view), more ovular in shape (Claxton et al., 1998; Dermott and Munawar, 1993; Peyer et al., 2010) and less dense (Claxton et al., 1998; Roe and MacIsaac, 1997). The ventral surface morphology of deep quagga mussels (Fig. 1C, anterior–posterior view) is much more pointed than that of zebra and shallow quagga mussels (Fig. 1A,B, anterior–posterior views). Such differences in shell morphology among the three morphotypes (Fig. 1) have been suggested to affect their interaction with different substrate types, influencing the distribution of each species within the Great Lakes (Claxton et al., 1998; Dermott and Munawar, 1993; Mackie, 1991; Roe and MacIsaac, 1997) (see Discussion).

A shell morphology that facilitates locomotion across particular substrate types might affect fitness and survival of zebra and quagga mussels on those substrates. Zebra mussels are known to move in response to various environmental factors (e.g. light, nitrogenous waste, oxygen and substrate type) that influence habitat selection and survival (see Burks et al., 2002; Kilgour and Mackie, 1993; Kobak, 2001; Kobak and Nowacki, 2007; Marsden and Lansky, 2000; Toomey et al., 2002).

The objective of this study was to directly examine the effects of shell morphology on locomotion of different mussel morphotypes (i.e. zebra, shallow quagga and deep quagga mussels) on hard versus soft sedimentary substrates. We determined the effects of shell shape on locomotory function by calculating translational and rotational kinetic energy (K_trans and K_rot, respectively) for zebra and quagga mussels moving across hard and soft substrate types. K_trans and K_rot directly account for effects of shell size and shape on locomotory function, as K_trans is a function of mussel mass and velocity whereas K_rot is a function of mussel mass moment of inertia and angular velocity. For the mass moment of inertia of the mussel, we used as a surrogate the polar moment of inertia (see Peyer et al., 2010), which was our descriptor of shell morphology. Thus, K_rot includes specific information on shell morphology, allowing us to observe direct links between shell shape and locomotion.

Although moments of inertia have been used in studies of animal locomotion, they have often been applied to specific anatomical features (e.g. vertebrate limbs) rather than to whole-animal morphology and locomotion (e.g. Walter and Carrier, 2002; Carrier et al., 2001). In contrast to previous studies that examined locomotion of rigid-bodied (i.e. shelled) animals (e.g. Alexander, 1993; Amyot and Downing, 1998; Uryu et al., 1996; Waller et al., 1999), we quantified whole-shell morphology using the polar moment of inertia, and then directly linked this geometrical measure to the mechanics of mussel locomotion. This approach enabled us to directly observe the relationship between zebra and quagga mussel shell shape and locomotion.

To quantify shell morphology, we collected zebra and quagga mussels 5–30 mm in size from Lake Ontario, North America. We collected the zebra and shallow quagga mussels in June 2006 from rocky substrate (1–2 m depth) in Oswego Harbor, NY, USA (∼43°47′N, 76°49′W). We randomly sampled and removed mussels from rocks by cutting their byssal threads with a knife. We collected deep quagga mussels in April 2005 at Thirty Mile Point (43°24′N, 78°33′W), offshore from Niagara Falls, NY, USA, with a motor driven ponar grab (65 m depth).

We also collected zebra and quagga mussels in July 2008 from Lake Michigan, North America, for the analysis of movement across hard and soft sedimentary substrates. We collected zebra and shallow quagga mussels from several submerged poles made of polyvinyl chloride (1–3 m depth) at the University of Wisconsin Great Lakes WATER Institute in Milwaukee, WI, USA (∼43°04′N, 87°95′W), stationed on the shore of Lake Michigan. We randomly sampled and removed these mussels from the poles by cutting their byssal threads with a knife. We collected deep quagga mussels offshore from Milwaukee, WI, USA (43°01′N, 87°21′W) using a trawl (60 m depth).

After collection, we wrapped mussels in damp paper towels and transported them on ice within sealed plastic bags. In the laboratory, we fed mussels a commercial shellfish diet (Isochrysis sp., Pavlova sp., Tetraselmis sp. and Thalassiosira weissflogii) from Reeds Mariculture Inc. (Campbell, CA, USA), an algal mixture that has been used to support zebra and quagga mussels previously (Vanderploeg et al., 1996; Peyer et al., 2009; Peyer et al., 2010). We housed mussels in aquaria at an experimental temperature range of ∼18–20°C. Throughout this study we used water collected from Lake Michigan, at Racine Harbor, WI, USA, where both zebra and quagga mussels occur.

We calculated polar moments of inertia for the dorsal–ventral and anterior–posterior shell views in the same manner as described above for the lateral shell view. For the dorsal–ventral shell view, we calculated J_yy about the y-axis as the sum of I_xx and I_zz (Fig. 2B), and for the anterior–posterior shell view, we calculated J_xx about the x-axis as the sum of I_yy and I_zz (Fig. 2C). We captured images of the three shell views for mussels of various size (5–30 mm length), 60 per morphotype. We weighed all mussels because mass is used to calculate the mass moment of inertia (Eqn 2).

We examined zebra, shallow quagga and deep quagga mussel movement across either hard glass or soft sedimentary substrates. For this analysis, we used mussels collected from Lake Michigan. We observed mussels in two treatments, either in 20 liters aquaria with a 3 cm soft sediment layer or in aquaria without sediment, exposing a glass bottom. The aquaria were filled with ∼10 liters of Lake Michigan water and gently aerated. The sedimentary substrate was collected from Lake Ontario with a motor driven ponar grab (65 m depth) in April 2005 at Thirty Mile Point, where we collected the deep quagga mussels. For the hard substrate treatment, we used the glass bottom of the aquarium because its surface was level, uniform and had low friction relative to the sedimentary substrate. Using glass enabled us to quantify movement on a flat hard surface alone without the extraneous effects of jagged or unlevel surfaces (e.g. rocks and mollusc shells) that would make the interaction with the hard surface more difficult to observe clearly. For both treatments, we placed six mussels of the same morphotype into each aquarium (five to 14 replicates per morphotype per substrate type), with the lateral shell view (Fig. 2A) towards (i.e. z-axis normal to) the substrate. We distributed the mussels, with roughly equal spacing, across each substrate. We used mussels of medium size (10–20 mm length, 0.3–0.8 g). Zebra and quagga mussels move across a substrate by pushing themselves with their foot, which extends out of their shells from their ventral surface. We captured images of mussels moving across each substrate with a Dragonfly IEEE-1394 digital camera and IMAQ programming software in LabVIEW at a rate of 1 frame min^–1. We recorded images over a 20 h period under dim light produced with a red 25 W bulb. The red bulb provided sufficient light for imaging the mussels, while providing limited visible spectrum, which mussels were found to prefer over white light conditions (Kobak, 2001; Kobak and Nowacki, 2007; Toomey et al., 2002). After each imaging session, we collected data on shell morphometrics and mussel mass as described above. We placed six mussels in each experimental replicate (i.e. six mussels per morphotype per substrate type). However, not all mussels moved during the observation period, resulting in a different number of observations per treatment. For those mussels moving on soft sedimentary substrate, we collected data on zebra mussels (N=24 total) from five replicate aquaria, shallow quagga mussels (N=24 total) from five replicate aquaria and deep quagga mussels (N=24 total) from nine replicate aquaria. For those mussels moving on hard substrate, we collected data on zebra mussels from nine replicate aquaria (N=32 total), shallow quagga mussels from seven replicate aquaria (N=30 total) and deep quagga mussels from 14 replicate aquaria (N=26 total).

In order to calculate the rotational kinetic energy, we determined the orientation of each mussel for each frame of an entire walk. While in motion, a mussel was able to position itself in different orientations (local reference frame) with respect to the substrate (global reference frame). We defined these orientations as the lateral, dorsal–ventral and anterior–posterior shell views (Fig. 1). It was important to determine the orientation of each mussel for calculating rotational kinetic energy (Eqn 9), because the variable, J, differed for each shell view. Thus, in Eqn 9, J was a principal moments tensor that consisted of one of J_zz, J_yy or J_xx and was represented by a 3×3 matrix (Fig. 5; Eqn 10), indicating changes in orientation. For example, if at time T_i a mussel was positioned on its lateral surface, at T_i+1 it could have rotated about the z-axis (yawed) and remained positioned on its lateral surface (Fig. 5, A to A), rotated about the x-axis (rolled) onto its ventral surface (Fig. 5, A to B) or rotated about the y-axis (pitched) onto its posterior surface (i.e. hinge; Fig. 5, A to C).

We also tested whether the polar moment of inertia depended on mussel morphotype and mussel mass using an ANOVA, with Tukey's post hoc pairwise comparisons, in R. We tested for differences in polar moment of inertia among the morphotypes used in the locomotion analysis (from Lake Michigan) to determine whether shell morphology was correlated with locomotion. As the mussel populations used to collect morphometric data above were from Lake Ontario, this additional morphometric analysis was necessary for Lake Michigan samples. We also tested for differences in mass among morphotypes using an ANOVA, with Tukey's post hoc pairwise comparisons, to confirm that all mussels were of similar size.

Shell morphology, quantified by the polar moment of inertia, differed significantly among the three morphotypes (i.e. zebra, shallow quagga and deep quagga) for all three shell views (Fig. 6, Table 1). Distribution of shell area and resistance to rotation about the z-axis (Fig. 2A, lateral shell view J_zz) was greatest for deep quagga mussels relative to the other morphotypes [assuming ρ(r)=m/V in Eqn 1] for a given mass (Fig. 6A; Table 1, comparisons 1, 2; Eqn 5). Therefore, greater work was required by deep quagga mussels than by zebra or shallow quagga mussels to turn a given amount (of degrees or radians) about the z-axis for a given mass (Fig. 2A, J_zz only). However, distribution of shell area and presumed resistance to rotation about the y-axis (Fig. 2B, dorsal–ventral shell view J_yy) was greatest for zebra mussels (Fig. 6B; Table 1, comparisons 4, 5; Eqn 6), indicating greater work required by zebra mussels to turn a given amount about the y-axis (Fig. 2B, J_yy only). Finally, resistance to rotation about the x-axis (Fig. 2C, anterior–posterior shell view J_xx) was greatest for shallow quagga mussels relative to the other morphotypes (Fig. 6C; Table 1, comparisons 7, 8; Eqn 7), indicating greater work for shallow quagga mussels to turn about the x-axis (Fig. 2C, J_xx only).

In addition to the morphometric analysis above, using mussels collected from Lake Ontario, we performed an additional comparison of morphotypes that were independently collected (from Lake Michigan) for the locomotion analysis. This second data set of mussels possessed a narrower size range. Results from the Lake Michigan samples were concordant with those above, except that deep quagga mussels showed significantly greater J_yy (dorsal–ventral shell view) for a given mass than the other morphotypes (Table 2, comparisons 4, 5). The deep quagga mussels from Lake Michigan were visibly longer than those from Lake Ontario, which would account for their greater J_yy. The different morphotypes from Lake Michigan did not differ significantly in mass (ANOVA: F=0.88, d.f.=2, P=0.42), indicating that differences in mass were unlikely to contribute to differences in locomotion between the morphotypes.

The ratio of rotational to translational kinetic energy (K_rot:K_trans) depended significantly on mussel morphotype (ANOVA: F=8.57, d.f.=2, P=0.0003) and substrate type (ANOVA: F=6.29, d.f.=1, P=0.013) but not on replicate (ANOVA: F=3.34, d.f.=1, P=0.070) (Table 3, Fig. 7). On glass substrate, K_rot:K_trans was significantly greater for deep quagga than for zebra and shallow quagga mussels (Table 3A, comparisons 1, 2; Fig. 7). In addition, deep quagga mussels had significantly greater K_rot:K_trans on glass relative to soft sedimentary substrate (Table 3B, comparison 1; Fig. 7), whereas shallow quagga and zebra mussels did not differ significantly in K_rot:K_trans between the two substrates (Table 3B, comparisons 2, 3; Fig. 7). Overall, these results indicated that deep quagga mussels expended more work in rotational relative to translational movement than either zebra or shallow quagga mussels on glass but not on soft sedimentary substrate. The ω²:v²_c ratio, a component within K_rot:K_trans (Eqns 8, 9), did not differ significantly among morphotypes on each substrate (ANOVA: F=1.63, d.f.=2, P=0.20), between substrate types for each morphotype (ANOVA: F=0.43, d.f.=1, P=0.51) or among replicates (ANOVA: F=0.35, d.f.=2, P=0.56; Fig. 8). Therefore, the significant differences in K_rot:K_trans among morphotypes were caused primarily by differences in mussel shell morphology (i.e. magnitude of the polar moment of inertia) rather than by ω²:v²_c, the degree of turning relative to translation of a mussel during movement.

The ratio of net to total distance traveled (i.e. effective distance) differed significantly among morphotypes (ANOVA: F=12.7, d.f.=2, P<0.00001; Fig. 9). Zebra mussels exhibited significantly greater effective distance traveled than deep quagga mussels on glass (Tukey: P=0.0044) and on soft sedimentary substrates (Tukey: P=0.0091). Shallow quagga mussels did not differ significantly from zebra or deep quagga mussels in effective distance traveled on either substrate (Tukey: P>0.24). Therefore, only zebra mussels were more directed in their movement than deep quagga mussels on the two substrates. Effective distance traveled did not differ significantly for a given morphotype on the two substrates (ANOVA: F=0.015, d.f.=1, P=0.90) or among replicates (ANOVA: F=0.93, d.f.=1, P=0.34).

During movement on glass, the morphotypes differed significantly in their orientation, with either their lateral (Fig. 2A) or ventral shell surface (Fig. 2B) towards (i.e. parallel to) the substrate. Most notably, zebra mussels were oriented with their ventral surface (from which their foot protrudes) towards the glass substrate significantly more often than deep quagga mussels (Table 4A, comparisons 1–3; Fig. 10, Fig. 11A). However, deep quagga mussels were primarily oriented with their lateral shell surface towards the glass substrate, with the foot extended out laterally, significantly more often than the other morphotypes (Fig. 10, Fig. 11A). Therefore, the polar moment of inertia of the dorsal–ventral shell view (Fig. 2B, i.e. J_yy) contributed primarily to K_rot:K_trans of zebra mussels, whereas the polar moment of inertia of the lateral shell view (Fig. 2A, i.e. J_zz) mostly contributed to K_rot:K_trans of deep quagga mussels.

During movement on soft sediment, mussels were oriented in one of three positions, with the lateral (Fig. 2A), ventral (Fig. 2B) or posterior shell surface (i.e. hinge, Fig. 2C) towards the substrate. However, all morphotypes were oriented mostly with their ventral surface towards the substrate (Fig. 10, Fig. 11B). Zebra mussels were positioned on their ventral surface significantly more often than deep quagga mussels but did not differ significantly from shallow quagga mussels in this regard (Table 4B, comparisons 1–3; Fig. 10). Mean percentages of steps for which mussels were oriented with their ventral surface towards the substrate were greater on soft sedimentary substrate than on glass substrate for all three morphotypes (Fig. 10). Most notably, deep quagga mussels were oriented on their ventral surface for ∼70% of the steps while moving on sedimentary substrate (Fig. 10) but for only ∼20% of the steps while moving on glass substrate (Fig. 10). Therefore, unlike on glass substrate, on sedimentary substrate the polar moment of inertia of the dorsal–ventral shell view (i.e. J_yy) contributed primarily to K_rot:K_trans of deep quagga mussels, similar to that of zebra and shallow quagga mussels.

Shell morphology serves as a functional trait that affects mussel locomotion on distinct substrate types. We found that shell morphology differed significantly among the three morphotypes (Figs 1, 6), and that shell morphology affected movement of the mussels differentially across hard relative to soft sedimentary substrates (Figs 7, 9). Specifically, the polar moment of inertia, our descriptor of shell morphology, directly contributed to differences in work expenditure during movement on glass substrate (i.e. K_rot:K_trans) between morphotypes (i.e. deep quagga versus zebra and shallow quagga mussels). This effect of shell morphology on locomotion provides the first biomechanically quantifiable insights into the ability of each morphotype to interact with and colonize different substrates within the Great Lakes.

We examined whether zebra, shallow quagga and deep quagga mussels differ in their movement (K_rot:K_trans; see Eqns 8, 9) on hard versus soft sedimentary substrate. High K_rot:K_trans (greater than one) indicates greater work expenditure in the form of rotation (i.e. K_rot; turning) relative to translation (i.e. K_trans; directed movement). On hard (glass) substrate, deep quagga mussels expended a significantly greater proportion of work than zebra and shallow quagga mussels in the form of rotation, with approximately fourfold greater K_rot:K_trans than zebra and shallow quagga mussels (Fig. 7). This difference in K_rot:K_trans of deep quagga relative to zebra or shallow quagga mussels was primarily attributed to the difference in the magnitude of their polar moment of inertia rather than the degree of turning relative to translating, as we found no statistically significant difference in the ω²:v²_c ratio among morphotypes (Fig. 8; see Eqns 8, 9).

The similarity in energy expenditure among the three morphotypes on soft sedimentary substrate was largely due to a decrease in K_rot:K_trans of deep quagga mussels to approximately one-third of that on glass substrate (Fig. 7). Such a reduction in K_rot:K_trans of deep quagga mussels between the differing substrate types was also attributed to shell morphology and mussel orientation. Whereas on glass substrate deep quagga mussels were positioned on and rotated about their lateral shell surface (z-axis) (i.e. relatively high value for J_zz; Fig. 10, Fig. 11A), on soft sedimentary substrate deep quagga mussels were able to position themselves on and rotate about their ventral surface (y-axis) (i.e. relatively low value for J_yy; Fig. 10, Fig. 11B), similar to zebra and shallow quagga mussels. Therefore, on soft sedimentary substrate, all three morphotypes were able to minimize their polar moment of inertia with respect to the y-axis (analogous to a figure skater spinning with arms held close to the body), consequently reducing K_rot. In contrast to glass substrate, soft sediment appeared to provide lateral support for deep quagga mussels as they generated a path for themselves, enabling them to remain positioned on their ventral surface more often (Fig. 1C, anterior–posterior view). Consequently, deep quagga mussels were able to attain equal performance, with respect to the amount of work needed for rotation and translation, relative to zebra and shallow quagga mussels on soft sedimentary substrate.

We also examined whether zebra, shallow quagga and deep quagga mussels differed in their effective distance traveled on hard versus soft sedimentary substrate. Zebra mussels showed the greatest and deep quagga mussels showed the lowest effective distance traveled on both hard and soft sedimentary substrates (Fig. 9). Thus, of all morphotypes, zebra mussels achieved the most directed movement in translation, although shallow quagga mussels were not significantly different from zebra mussels in terms of this aspect of locomotion.

Directed movement is likely to be a fitness-related trait that affects fitness and survival. As mentioned previously (see Introduction), zebra mussels have been found to move in response to environmental factors, including light, nitrogenous waste, oxygen and substrate type (see Burks et al., 2002; Kilgour and Mackie, 1993; Kobak, 2001; Kobak and Nowacki, 2007; Marsden and Lansky, 2000; Toomey et al., 2002). A previous study found that zebra mussels moved 22 cm within a 2 h period (Toomey et al., 2002). In our study we observed zebra, shallow quagga and deep quagga mussels moving mean distances of 20 cm (s.e.m.=0.051 cm), 29 cm (s.e.m.=0.084 cm) and 21 cm (s.e.m.=0.065 cm), respectively, within a 20 h time frame across the different substrates. Zebra mussels have also been found to seek dimmer over brighter lit habitats (Kobak, 2001; Kobak and Nowacki, 2007; Toomey et al., 2002) and show preference for specific substrate types (Kilgour and Mackie, 1993; Marsden and Lansky, 2000) and substrate color (Kobak, 2001). Other bivalves have been shown to aggregate in the presence of predators [e.g. Mytilus edulis (Côte and Jelnikar, 1999); see also Tridacna squamosa (Huang et al., 2007)]. Some (e.g. unionids) have been thought to move in response to water flow or food conditions (Schwalb and Pusch, 2007) and during spawning events (Watters et al., 2001). Thus, the ability of zebra and shallow quagga mussels to achieve greater directed movement than deep quagga mussels, in addition to the lesser energy that they expend in transit on hard substrates (i.e. lesser K_rot:K_trans), is likely to have functional benefits, possibly enabling more effective movement in response to such environmental conditions.

Achieving directed movement is likely to depend on ventral surface morphology, which is highly flattened for zebra mussels (Fig. 2A, anterior–posterior view) and appears to provide a stable platform during locomotion on flatter hard substrates. Deep quagga mussels that lack such a flattened ventral surface (Fig. 1C, anterior–posterior view) might be at a disadvantage with respect to seeking the most favorable environmental conditions in a directed manner on flat hard surfaces.

Our results suggest that shell morphology has functional consequences for mussel movement on hard versus soft sedimentary substrates. These consequences might contribute to differences in colonization and range expansion of zebra versus quagga mussels. For example, developmental plasticity has been found to contribute significantly to the divergence in shell morphology between shallow and deep quagga mussels (Peyer et al., 2010). Such a plastic response in morphology to environmental factors might allow quagga mussels greater ability than zebra mussels to utilize diverse substrates and colonize both shallow- and deep-water habitats of the Great Lakes. Although the deep quagga mussel morphotype appeared to be maladaptive with respect to movement on hard substrate (i.e. high K_rot:K_trans), the shallow quagga mussel achieved values for K_rot:K_trans that were more similar to that of the zebra mussel, resulting in similar energy expenditure between these two morphotypes (Fig. 7). In addition, values for K_rot:K_trans and effective distance traveled were not significantly different between zebra and shallow quagga mussels on both substrates (Figs 7, 9). Thus, shallow quagga mussels were similar to zebra mussels in our calculated metrics of locomotion on both hard and soft sedimentary substrates. Although deep quagga mussels had the least effective distance traveled on both substrates, the energy that they expended in terms of K_rot:K_trans was comparable to that of zebra and shallow quagga mussels on soft sedimentary substrate. It is unclear why effective distance traveled was lowest for deep quagga mussels on soft sedimentary as well as on glass substrate. It is possible that effective distance traveled is affected by ventral surface morphology (Fig. 1, anterior–posterior view) on soft sedimentary as well as on glass substrate. Frictional forces, which our study did not take into account, might also play an important role in such movement.

Within their invaded ranges, zebra and quagga mussels come into direct contact and have direct competitive interactions in shallow-water habitats with hard substrates. Yet the colonization of deep soft sedimentary substrate by quagga mussels is also of concern, because the substrate of the Great Lakes is predominantly comprised of soft sediment [e.g. ∼80% of the area in Lake Erie (Berkman et al., 1998)]. Although the deep quagga mussel morphotype appears to exact a cost to locomotion on flatter hard substrates, the shells might be adaptive for living on the soft sedimentary substrate typical of deeper waters, although studies on such effects have yet to be performed. If the deep quagga mussel shells are adaptive for living on soft sedimentary substrate, quagga mussels might have the advantage of having the ability to adopt both shallow- and deep-water morphotypes (Fig. 1B,C) [i.e. via developmental plasticity (Peyer et al., 2010)], providing greater versatility for utilizing both hard and soft sedimentary substrates. This versatility might confer quagga mussels with a greater ability to colonize a wider range of habitats than zebra mussels, and might lead to an overall fitness advantage in the long run.

Shell morphology might have additional functional consequences that affect the distribution of the two species. For example, the flattened ventral surface morphology of zebra mussels might aid in resisting dislodgment in flow, by enabling secure attachment with byssal threads to hard substrates in shallow-water habitats (Claxton et al., 1998; Dermott and Munawar, 1993; Mackie, 1991; Peyer et al., 2009). In addition, the high-density shells of zebra and shallow quagga mussels relative to deep quagga mussels (see Claxton et al., 1998; Roe and MacIsaac, 1997) have been considered less prone to damage in shallow-water habitats with considerable wave impact (Claxton et al., 1998). In deep-water habitats, deep quagga mussels with shells that are flattened and compressed together might be able to burrow and attach their byssal threads more deeply into the sediment, possibly anchoring themselves more securely or avoiding predation.

Numerous studies have examined various morphological features and their effects on locomotory ability of diverse animal taxa. However, a relatively small subset of these locomotion studies included an examination of different morphological attributes of shelled animals (e.g. Alexander, 1993; Amyot and Downing, 1998; Uryu et al., 1996; Waller et al., 1999). In some studies, moments of force, which depend on shell morphology, were found to affect locomotion of various gastropods (e.g. see Huryn and Denny, 1997; Okajima and Chiba, 2009). Moments of inertia have also been proposed to play an important role in locomotion of the painted turtle, Chrysemys picta, although these values were not specifically calculated (Rivera et al., 2006).

Most notably, we are aware of no study that has directly quantified whole-shell morphology using a geometric measure (e.g. moment of inertia) that has direct input into physical models of motion (e.g. K_rot:K_trans). In this study, we applied a unique approach by directly linking moments of inertia to the energetics of locomotion of different mussel morphotypes. Our approach indicated novel constraints of shell morphology on locomotory ability. In particular, such constraints were revealed by the greater work expenditure on hard substrate of deep quagga mussels relative to shallow quagga and zebra mussels. These constraints might have crucially important implications for range expansions of zebra and quagga mussels onto diverse substrate types in novel environments.

 
• A

  mussel shell area

 
• C

  shell centroid

 
• dA

  differential element of shell area

 
• D_eff

  effective distance traveled

 
• dV

  differential element of shell volume

 
• I_aa

  mass moment of inertia about the a-axis

 
• I_xx

  moment of inertia about the principal x-axis

 
• I_yy

  moment of inertia about the principal y-axis

 
• I_zz

  moment of inertia about the principal z-axis

 
• J

  polar moment of inertia tensor for K_rot (i.e. Trace[J]={J_xx, J_yy, J_zz})

 
• J_xx

  polar moment of inertia about the principal x-axis

 
• J_yy

  polar moment of inertia about the principal y-axis

 
• J_zz

  polar moment of inertia about the principal z-axis

 
• K_rot

  rotational kinetic energy

 
• K_trans

  translational kinetic energy

 
• m

  mussel mass

 
• r

  position vector from a point on a particular axis to dA or dV

 
• V

  shell volume

 
• v_c

  velocity of shell centroid

 
• x

  distance from y-axis to dA

 
• x_m

  mussel mass (statistical notation)

 
• x_r

  experimental replicate (statistical notation)

 
• x_s

  substrate type (statistical notation)

 
• x_t

  mussel type (statistical notation)

 
• y

  distance from x-axis to dA

 
• β

  maximum likelihood parameter estimate

 
• Δx

  displacement vector of shell centroid

 
• Δθ

  angle through which a mussel rotates

 
• ρ(r)

  density of an object at position r relative to a-axis

 
• ω_c

  angular velocity about instantaneous center of rotation