SUMMARY

Although zebra mussels (Dreissena polymorpha) initially colonized shallow habitats within the North American Great Lakes, quagga mussels (Dreissena bugensis) are becoming dominant in both shallow- and deep-water habitats. Shell morphology differs among zebra, shallow quagga and deep quagga mussels but functional consequences of such differences are unknown. We examined effects of shell morphology on locomotion for the three morphotypes on hard (typical of shallow habitats) and soft (characteristic of deep habitats) sedimentary substrates. We quantified morphology using the polar moment of inertia, a parameter used in calculating kinetic energy that describes shell area distribution and resistance to rotation. We quantified mussel locomotion by determining the ratio of rotational (Krot) to translational kinetic energy (Ktrans). On hard substrate, Krot:Ktrans of deep quagga mussels was fourfold greater than for the other morphotypes, indicating greater energy expenditure in rotation relative to translation. On soft substrate, Krot:Ktrans of deep quagga mussels was approximately one-third of that on hard substrate, indicating lower energy expenditure in rotation on soft substrate. Overall, our study demonstrates that shell morphology correlates with differences in locomotion (i.e. Krot:Ktrans) among morphotypes. Although deep quagga mussels were similar to zebra and shallow quagga mussels in terms of energy expenditure on sedimentary substrate, their morphology was energetically maladaptive for linear movement on hard substrate. As quagga mussels can possess two distinct morphotypes (i.e. shallow and deep morphs), they might more effectively utilize a broader range of substrates than zebra mussels, potentially enhancing their ability to colonize a wider range of habitats.

INTRODUCTION

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).

Fig. 1.

Lateral, dorsal–ventral and anterior–posterior shell views of (A) zebra, (B) shallow quagga and (C) deep quagga mussels. Shells were 2–3 cm in length.

Fig. 1.

Lateral, dorsal–ventral and anterior–posterior shell views of (A) zebra, (B) shallow quagga and (C) deep quagga mussels. Shells were 2–3 cm in length.

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 (Ktrans and Krot, respectively) for zebra and quagga mussels moving across hard and soft substrate types. Ktrans and Krot directly account for effects of shell size and shape on locomotory function, as Ktrans is a function of mussel mass and velocity whereas Krot 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, Krot 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.

MATERIALS AND METHODS

Population sampling

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.

Collection of morphometric data

We digitally imaged zebra, shallow quagga and deep quagga mussels to quantify differences in shell morphology among the three morphotypes from Lake Ontario. We captured images of lateral, dorsal–ventral and anterior–posterior shell views of each mussel (Fig. 1) with a Dragonfly IEEE-1394 digital camera (Point Grey Research, Vancouver, BC, Canada). To quantify shell morphology, we used IMAQ programming software in LabVIEW (National Instruments, Austin, TX, USA) to obtain the polar moment of inertia as a surrogate for the mass moment of inertia of a mussel. The mass moment of inertia is a physical measurement that describes the distribution of the mass of an object and its resistance to rotation about an arbitrary axis (Hibbeler, 1989). The mass moment of inertia, Iaa, about an arbitrary axis, a, is defined as:
formula
(1)
where ua is the unit vector defining the a-axis, r is the vector from the a-axis to a differential element of the object volume, dV, and ρ(r) is the density of the object at position r relative to the a-axis. Greater distribution of mass for an object (greater Iaa) results in greater resistance to rotation, analogous to a spinning figure skater that has greater resistance to rotation and spins more slowly with outstretched arms than with arms held close to the body (i.e. lesser distribution of mass and lesser Iaa). In our study, we simplified the three-dimensional morphology reconstruction by using two-dimensional images of mussel shells, and by assuming the density distribution, ρ(r), and thickness of mussels to be constants. With these assumptions, we simplified Iaa in Eqn 1 to the mussel mass multiplied by the polar moment of inertia (Fig. 2). We established a local orthogonal coordinate system on the mussel with the origin at its centroid and with the unit vector defined along the principal axis as calculated by the moment of inertia. Thus, for the lateral shell view (Fig. 2A), the polar moment of inertia about the z-axis, Jzz, (coming out of the page) at the shell centroid would then be defined by:
formula
(2)
where rz is the distance from the centroid to an element of the shell area, dA, and m is the mass of the mussel (see also Peyer et al., 2010). Jzz describes the distribution of the shell area and, assuming that ρ(r)=m/V (V is shell volume) (Eqn 1), its resistance to rotation about the z-axis. Thus, a mussel with larger shell area has greater resistance to rotation than a mussel of smaller shell area for the same total mass.
Fig. 2.

Three shell views that were morphologically analyzed for zebra, shallow quagga and deep quagga mussels. (A) Lateral view (mussel's hinge at left, siphon at right): moments of inertia, Ixx and Iyy, were about the x- and y-axes and polar moment of inertia (Jzz) was about the z-axis (out of page). (B) Dorsal–ventral view (hinge at left, siphon at right, foot parallel to the page): moments of inertia, Ixx and Izz, were about the x- and z-axes and polar moment of inertia (Jyy) was about the y-axis (out of page). (C) Anterior–posterior view (hinge and siphon were aligned with the x-axis, perpendicular to the page): moments of inertia, Iyy and Izz, were about the y- and z-axes and polar moment of inertia (Jxx) was about the x-axis (out of page).

Fig. 2.

Three shell views that were morphologically analyzed for zebra, shallow quagga and deep quagga mussels. (A) Lateral view (mussel's hinge at left, siphon at right): moments of inertia, Ixx and Iyy, were about the x- and y-axes and polar moment of inertia (Jzz) was about the z-axis (out of page). (B) Dorsal–ventral view (hinge at left, siphon at right, foot parallel to the page): moments of inertia, Ixx and Izz, were about the x- and z-axes and polar moment of inertia (Jyy) was about the y-axis (out of page). (C) Anterior–posterior view (hinge and siphon were aligned with the x-axis, perpendicular to the page): moments of inertia, Iyy and Izz, were about the y- and z-axes and polar moment of inertia (Jxx) was about the x-axis (out of page).

Using LabVIEW, we calculated Jzz as the sum of the moments of inertia, Ixx and Iyy, about the centroid (see Beer and Johnston, 1981). From a two-dimensional perspective, Ixx and Iyy are the resistances of the shell area to rotation about the principal x- and y-axes, respectively, which intersect at the centroid (Fig. 2A) (Beer and Johnston, 1981). Ixx and Iyy are defined by the equations:
formula
(3)
and
formula
(4)
respectively, where y and x are the distances from the x- and y-axes to dA.

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 Jyy about the y-axis as the sum of Ixx and Izz (Fig. 2B), and for the anterior–posterior shell view, we calculated Jxx about the x-axis as the sum of Iyy and Izz (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).

Analysis of morphometric data

We performed an analysis of covariance (ANCOVA), using the statistical package R (R Development Core Team, 2008), to test whether the polar moment of inertia depended on mussel morphotype (i.e. zebra, shallow quagga and deep quagga) and mussel mass. Our models for the polar moments of inertia for each shell view, Jzz, Jyy and Jxx, as dependent variables were:
formula
(5)
formula
(6)
and
formula
(7)
where independent variables were mussel morphotype (xt) and mussel mass (xm). Maximum likelihood parameter estimates were represented by β0, β1 and β2. All variables were treated as fixed effects. As mass is a function of length cubed and moment of inertia is a function of length quartic, we raised mussel mass to the 4/3 power in order to non-dimensionalize its relationship with polar moments of inertia. Following the ANCOVA, we performed pairwise comparisons of Jzz, Jyy and Jxx between morphotypes.

Collection of mussel locomotion data

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).

Fig. 3.

Characterization of mussel movement between two imaging frames, with lateral shell surface parallel to the substrate. Coordinates (x, y) of the shell hinge and siphon were obtained, frame by frame, for each step taken by a mussel during its entire walk. These coordinates were used to quantify the displacement of the mussel hinge (ΔxH) and siphon (ΔxS) between frames.

Fig. 3.

Characterization of mussel movement between two imaging frames, with lateral shell surface parallel to the substrate. Coordinates (x, y) of the shell hinge and siphon were obtained, frame by frame, for each step taken by a mussel during its entire walk. These coordinates were used to quantify the displacement of the mussel hinge (ΔxH) and siphon (ΔxS) between frames.

We used time-lapse photographic images to quantify movement of mussels in translation and rotation. For each series of images of individual mussels moving across a given substrate, we used IMAQ programming software in LabVIEW to obtain x- and y-coordinates of the hinge and siphon of each mussel frame by frame (Fig. 3). With these coordinates we quantified translational and rotational movement of each mussel during an entire ‘walk’ within each 20 h imaging period. Translational movement is the displacement vector (Δx) of the centroid of the mussel shell from frame to frame, whereas rotational movement is the angle vector (Δθ) of the hinge-siphon axis of the mussel shell from frame to frame (Fig. 4). Using Δx, we calculated translational kinetic energy of mussels, Ktrans, as:
formula
(8)
where m is the mussel mass and vc is the velocity vector of the centroid of the mussel shell (i.e.Δx per frame, expressed in cm min–1 and representing one step per walk) (Hibbeler, 1989). We summed Ktrans over the entire walk. Using Δθ, we calculated rotational kinetic energy, Krot, as:
formula
(9)
where
formula
(10)
and
formula
(11)
The angular velocity vector (ω) was about the centroid of the mussel (i.e. Δθ in rad min–1) (Hibbeler, 1989). We summed Krot over a mussel's entire walk. For calculations of Ktrans and Krot we used data only for the time period during which a mussel was unobstructed by neighboring mussels or by walls of the aquarium.
Fig. 4.

Variables used to calculate translational [Ktrans=1/2(m·v2c)] and rotational kinetic energy [Krot=1/2(J·ω2c)]. Images of mussel movement were captured once per minute. Thus, mussel velocity (vc) in the calculation of Ktrans was defined as displacement, Δxc, in cm min–1 of the centroid and mussel angular velocity (ωc) in the calculation of Krot was defined as Δθ in rad min–1.

Fig. 4.

Variables used to calculate translational [Ktrans=1/2(m·v2c)] and rotational kinetic energy [Krot=1/2(J·ω2c)]. Images of mussel movement were captured once per minute. Thus, mussel velocity (vc) in the calculation of Ktrans was defined as displacement, Δxc, in cm min–1 of the centroid and mussel angular velocity (ωc) in the calculation of Krot was defined as Δθ in rad min–1.

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 Jzz, Jyy or Jxx and was represented by a 3×3 matrix (Fig. 5; Eqn 10), indicating changes in orientation. For example, if at time Ti a mussel was positioned on its lateral surface, at Ti+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).

Fig. 5.

Matrix of possible changes in mussel orientation during movement across a substrate. A mussel could rotate from being positioned on its (A) lateral, (B) ventral or (C) posterior surface at time Ti to being positioned on its (A) lateral, (B) ventral or (C) posterior surface at time Ti+1. In the process of moving from one position to another, a mussel rotated about different axes. The polar moments of inertia that were used as a mussel rotated from an orientation at Ti to another orientation at Ti+1 are indicated in each cell of the matrix. For example, movement of a mussel from the lateral to the lateral surface (A to A) used Jzz of the lateral shell view, from lateral to ventral surface (A to B) used Jxx of the anterior–posterior shell view and from lateral to posterior surface (A to C) used Jyy of the dorsal–ventral shell view.

Fig. 5.

Matrix of possible changes in mussel orientation during movement across a substrate. A mussel could rotate from being positioned on its (A) lateral, (B) ventral or (C) posterior surface at time Ti to being positioned on its (A) lateral, (B) ventral or (C) posterior surface at time Ti+1. In the process of moving from one position to another, a mussel rotated about different axes. The polar moments of inertia that were used as a mussel rotated from an orientation at Ti to another orientation at Ti+1 are indicated in each cell of the matrix. For example, movement of a mussel from the lateral to the lateral surface (A to A) used Jzz of the lateral shell view, from lateral to ventral surface (A to B) used Jxx of the anterior–posterior shell view and from lateral to posterior surface (A to C) used Jyy of the dorsal–ventral shell view.

Analysis of mussel locomotion

We characterized mussel locomotion using the ratio of rotational to translational kinetic energy (Krot:Ktrans), mean ω2 to mean v2c (ω2:v2c) and net to total distance traveled (i.e. effective distance). We tested for differences in these values among the three morphotypes using ANOVA in the software package R. Mussels with high Krot:Ktrans values expended more energy in rotational relative to translational movement. By using Krot:Ktrans in our analyses rather than mean values for Krot and Ktrans, we normalized energy expenditure across all mussels. This normalization was necessary in order to determine the effect of shell morphology on locomotion, because individual mussels moved different distances, in turn affecting mean values for Krot and Ktrans. We also tested for differences in the ω2:v2c ratio to determine the extent to which this value might have affected Krot:Ktrans. For example, a high value for Krot:Ktrans could result from a high degree of turning relative to translating (i.e. high ω2:v2c ratio) or from a large polar moment of inertia (the measure we used to characterize morphology) for a given mussel mass (see Eqns 8, 9). Therefore a significant effect of the ω2:v2c ratio could mask the effect of morphology on Krot:Ktrans. Finally, we tested for differences in the effective distance traveled to determine whether the three morphotypes differed in their tendency toward directed movement on the differing substrates. Effective distance was defined as the net distance traveled over the 20 h period divided by the total distance traveled. Therefore, effective distance values close to one would indicate strong directional preference. Our models for Krot:Ktrans, ω2:v2c and effective distance traveled (Deff) as dependent variables were:
formula
(12)
formula
(13)
and
formula
(14)
where independent variables were represented by mussel morphotype (xt), substrate type (xs) and replicate (xr). Maximum likelihood parameter estimates were represented by β0, β1, β2 and β3. All variables were treated as fixed effects.

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.

We tested for differences in orientation of the three morphotypes during movement on hard glass and soft sedimentary substrates using logistic regression in R with Tukey's post hoc pairwise comparisons. The model for percentage of steps of each walk, p, for which a mussel was orientated with its lateral, dorsal–ventral or anterior–posterior shell view parallel to the substrate was:
formula
(15)
with mussel morphotype (xt) and replicate (xr) as independent variables. Maximum likelihood parameter estimates were represented by β0, β1 and β2. Our model used a logit link function, because our data consisted of proportions, and quasibinomial errors to account for overdispersion. Our independent variables, xt and xr, were treated as fixed effects.
Fig. 6.

Differences in polar moment of inertia among morphotypes [zebra (dashed line), shallow quagga (solid line), deep quagga mussels (dot-dashed line)] across mass, shown for the (A) lateral (Jzz), (B) dorsal–ventral (Jyy) and (C) anterior–posterior (Jxx) shell views (see Fig. 2). N=60 for each morphotype. Mussel mass was raised to the 4/3 power to non-dimensionalize its relationship with Jzz, Jyy and Jxx.

Fig. 6.

Differences in polar moment of inertia among morphotypes [zebra (dashed line), shallow quagga (solid line), deep quagga mussels (dot-dashed line)] across mass, shown for the (A) lateral (Jzz), (B) dorsal–ventral (Jyy) and (C) anterior–posterior (Jxx) shell views (see Fig. 2). N=60 for each morphotype. Mussel mass was raised to the 4/3 power to non-dimensionalize its relationship with Jzz, Jyy and Jxx.

Fig. 7.

Differences in mechanics of motion among morphotypes on different substrate types (i.e. hard glass versus soft sediment) (***, P<0.001). Mechanics of mussel motion was measured as mean ratio of rotational to translational kinetic energy (Krot:Ktrans). On glass substrate, N=32 for zebra, N=30 for shallow quagga and N=26 for deep quagga mussels. On soft sedimentary substrate, N=24 per morphotype. Error bars are ±s.e.m.

Fig. 7.

Differences in mechanics of motion among morphotypes on different substrate types (i.e. hard glass versus soft sediment) (***, P<0.001). Mechanics of mussel motion was measured as mean ratio of rotational to translational kinetic energy (Krot:Ktrans). On glass substrate, N=32 for zebra, N=30 for shallow quagga and N=26 for deep quagga mussels. On soft sedimentary substrate, N=24 per morphotype. Error bars are ±s.e.m.

RESULTS

Moment of inertia of zebra and quagga mussel morphotypes

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 Jzz) 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, Jzz only). However, distribution of shell area and presumed resistance to rotation about the y-axis (Fig. 2B, dorsal–ventral shell view Jyy) 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, Jyy only). Finally, resistance to rotation about the x-axis (Fig. 2C, anterior–posterior shell view Jxx) 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, Jxx only).

Table 1.

Pairwise comparisons of polar moment of inertia between mussel morphotypes (zebra, shallow quagga and deep quagga), for lateral, dorsal–ventral and anterior–posterior shell views (Eqns 5–7; d.f.=175)

Pairwise comparisons of polar moment of inertia between mussel morphotypes (zebra, shallow quagga and deep quagga), for lateral, dorsal–ventral and anterior–posterior shell views (Eqns 5–7; d.f.=175)
Pairwise comparisons of polar moment of inertia between mussel morphotypes (zebra, shallow quagga and deep quagga), for lateral, dorsal–ventral and anterior–posterior shell views (Eqns 5–7; d.f.=175)

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 Jyy (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 Jyy. 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.

Table 2.

Pairwise comparisons of polar moment of inertia for a given mussel mass between mussel morphotypes for lateral, dorsal–ventral and anterior–posterior shell views

Pairwise comparisons of polar moment of inertia for a given mussel mass between mussel morphotypes for lateral, dorsal–ventral and anterior–posterior shell views
Pairwise comparisons of polar moment of inertia for a given mussel mass between mussel morphotypes for lateral, dorsal–ventral and anterior–posterior shell views

Locomotion of the three morphotypes on differing substrates

The ratio of rotational to translational kinetic energy (Krot:Ktrans) 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, Krot:Ktrans 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 Krot:Ktrans on glass relative to soft sedimentary substrate (Table 3B, comparison 1; Fig. 7), whereas shallow quagga and zebra mussels did not differ significantly in Krot:Ktrans 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:v2c ratio, a component within Krot:Ktrans (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 Krot:Ktrans among morphotypes were caused primarily by differences in mussel shell morphology (i.e. magnitude of the polar moment of inertia) rather than by ω2:v2c, the degree of turning relative to translation of a mussel during movement.

Table 3.

Pairwise comparisons of rotational to translational kinetic energy (Krot:Ktrans) (A) between mussel morphotypes on hard (glass) and sedimentary substrates and (B) between substrate types for each morphotype

Pairwise comparisons of rotational to translational kinetic energy (Krot:Ktrans) (A) between mussel morphotypes on hard (glass) and sedimentary substrates and (B) between substrate types for each morphotype
Pairwise comparisons of rotational to translational kinetic energy (Krot:Ktrans) (A) between mussel morphotypes on hard (glass) and sedimentary substrates and (B) between substrate types for each morphotype

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).

Fig. 8.

Differences in the amount of turning relative to translation (ω2:v2c ratio) among morphotypes on different substrate types (glass, sediment). On glass substrate, N=32 for zebra, N=30 for shallow quagga and N=26 for deep quagga mussels. On soft sedimentary substrate, N=24 per morphotype. Error bars are ±s.e.m.

Fig. 8.

Differences in the amount of turning relative to translation (ω2:v2c ratio) among morphotypes on different substrate types (glass, sediment). On glass substrate, N=32 for zebra, N=30 for shallow quagga and N=26 for deep quagga mussels. On soft sedimentary substrate, N=24 per morphotype. Error bars are ±s.e.m.

Fig. 9.

Differences in effective distance traveled (i.e. ratio of net to total distance traveled) among morphotypes on different substrate types (glass, sediment) (**, P<0.01). On glass substrate, N=32 for zebra, N=30 for shallow quagga and N=26 for deep quagga mussels. On soft sedimentary substrate, N=24 per morphotype. Error bars are ±s.e.m.

Fig. 9.

Differences in effective distance traveled (i.e. ratio of net to total distance traveled) among morphotypes on different substrate types (glass, sediment) (**, P<0.01). On glass substrate, N=32 for zebra, N=30 for shallow quagga and N=26 for deep quagga mussels. On soft sedimentary substrate, N=24 per morphotype. Error bars are ±s.e.m.

Mussel orientation during locomotion

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. Jyy) contributed primarily to Krot:Ktrans of zebra mussels, whereas the polar moment of inertia of the lateral shell view (Fig. 2A, i.e. Jzz) mostly contributed to Krot:Ktrans of deep quagga mussels.

Table 4.

Pairwise comparisons of the percentage of steps of each walk where the ventral shell surface was oriented toward the substrate between mussel morphotypes (A) on glass (Eqn 15; d.f.=87) and (B) on soft sedimentary substrate (Eqn 15; d.f.=70)

Pairwise comparisons of the percentage of steps of each walk where the ventral shell surface was oriented toward the substrate between mussel morphotypes (A) on glass (Eqn 15; d.f.=87) and (B) on soft sedimentary substrate (Eqn 15; d.f.=70)
Pairwise comparisons of the percentage of steps of each walk where the ventral shell surface was oriented toward the substrate between mussel morphotypes (A) on glass (Eqn 15; d.f.=87) and (B) on soft sedimentary substrate (Eqn 15; d.f.=70)
Fig. 10.

Differences in shell orientation during movement among morphotypes (***, P<0.0001). Graph shows percentage of steps of each ‘walk’ for which a mussel was positioned on its lateral, ventral and posterior (see Fig. 1) surfaces. On hard (glass) substrate, mussels were oriented in two possible positions, either on lateral (L) or ventral (D-V) surfaces. On soft sedimentary substrate, mussels were oriented in three possible positions, either on lateral, ventral or posterior (A-P) surfaces.

Fig. 10.

Differences in shell orientation during movement among morphotypes (***, P<0.0001). Graph shows percentage of steps of each ‘walk’ for which a mussel was positioned on its lateral, ventral and posterior (see Fig. 1) surfaces. On hard (glass) substrate, mussels were oriented in two possible positions, either on lateral (L) or ventral (D-V) surfaces. On soft sedimentary substrate, mussels were oriented in three possible positions, either on lateral, ventral or posterior (A-P) surfaces.

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. Jyy) contributed primarily to Krot:Ktrans of deep quagga mussels, similar to that of zebra and shallow quagga mussels.

DISCUSSION

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. Krot:Ktrans) 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.

Rotational versus translational kinetic energy of mussels

We examined whether zebra, shallow quagga and deep quagga mussels differ in their movement (Krot:Ktrans; see Eqns 8, 9) on hard versus soft sedimentary substrate. High Krot:Ktrans (greater than one) indicates greater work expenditure in the form of rotation (i.e. Krot; turning) relative to translation (i.e. Ktrans; 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 Krot:Ktrans than zebra and shallow quagga mussels (Fig. 7). This difference in Krot:Ktrans 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:v2c ratio among morphotypes (Fig. 8; see Eqns 8, 9).

The extent to which the polar moment of inertia (Jzz, Jyy or Jxx) contributed to Krot:Ktrans (see Eqn 9) depended on shell morphology (Fig. 6) and the orientation of the mussels during movement (Figs 5, 10, 11). On glass substrate, deep quagga mussels were primarily positioned on and rotated about the axis defining their lateral shell surface (z-axis; Jzz) during movement (Fig. 10, Fig. 11A). Therefore, Krot for deep quagga mussels was dominated by the equation:
formula
(16)
This lateral shell view exhibited the greatest polar moment of inertia (Jzz) for all morphotypes of a given mass (Fig. 6A). However, during movement on glass substrate, zebra and shallow quagga mussels were positioned on their ventral surface and rotated mostly about the axis defining their dorsal–ventral shell surface (y-axis; Jyy) more often than deep quagga mussels (Fig. 10, Fig. 11A). Therefore, relative to deep quagga mussels, Krot for zebra and shallow quagga mussels was more dominated by the equation:
formula
(17)
Values of Jyy for the dorsal–ventral shell view for zebra and shallow quagga mussels were less than values of Jzz for the lateral shell view for deep quagga mussels (Fig. 6A,versusFig. 6B). Such greater polar moment of inertia (Jzz) of deep quagga mussels relative to the lesser Jyy of zebra and shallow quagga mussels (Eqn 16 versus Eqn 17) resulted in greater Krot:Ktrans for deep quagga mussels. Thus, to achieve the same amount of rotation as zebra and shallow quagga mussels on hard substrate, deep quagga mussels were required to expend a greater amount of work. In essence, the movement of deep quagga mussels was analogous to a figure skater spinning with outstretched arms, increasing the resistance to rotation, whereas the movement of zebra and shallow quagga mussels was analogous to a figure skater spinning with arms held close to the body. Even if all three morphotypes were positioned on their lateral shell surface, deep quagga mussels would still have the greatest Krot:Ktrans (assuming similar ω2:v2c among morphotypes) because they had the greatest Jzz (Fig. 6A). If a mussel's capacity to turn has functional and fitness consequences, deep quagga mussels might be at an energetic disadvantage, in terms of locomotion on hard substrate, as a consequence of shell morphology and mussel orientation relative to the substrate.
Fig. 11.

Most common shell orientation and mode of locomotion of shallow (left) and deep quagga (right) mussels on (A) hard (glass) and (B) soft sedimentary substrates. Shell orientation and mode of locomotion of zebra mussels (not shown) on the two substrates were similar to that of shallow quagga mussels. All mussels were primarily positioned on their ventral shell surface on the two substrates, except for deep quagga mussels on glass substrate, which were positioned on their lateral shell surface.

Fig. 11.

Most common shell orientation and mode of locomotion of shallow (left) and deep quagga (right) mussels on (A) hard (glass) and (B) soft sedimentary substrates. Shell orientation and mode of locomotion of zebra mussels (not shown) on the two substrates were similar to that of shallow quagga mussels. All mussels were primarily positioned on their ventral shell surface on the two substrates, except for deep quagga mussels on glass substrate, which were positioned on their lateral shell surface.

The similarity in energy expenditure among the three morphotypes on soft sedimentary substrate was largely due to a decrease in Krot:Ktrans of deep quagga mussels to approximately one-third of that on glass substrate (Fig. 7). Such a reduction in Krot:Ktrans 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 Jzz; 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 Jyy; 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 Krot. 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.

Functional consequences of shell morphology for locomotion and other activities

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 Krot:Ktrans), 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 Krot:Ktrans), the shallow quagga mussel achieved values for Krot:Ktrans 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 Krot:Ktrans 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 Krot:Ktrans 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. Krot:Ktrans). 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.

LIST OF ABBREVIATIONS

     
  • A

    mussel shell area

  •  
  • C

    shell centroid

  •  
  • dA

    differential element of shell area

  •  
  • Deff

    effective distance traveled

  •  
  • dV

    differential element of shell volume

  •  
  • Iaa

    mass moment of inertia about the a-axis

  •  
  • Ixx

    moment of inertia about the principal x-axis

  •  
  • Iyy

    moment of inertia about the principal y-axis

  •  
  • Izz

    moment of inertia about the principal z-axis

  •  
  • J

    polar moment of inertia tensor for Krot (i.e. Trace[J]={Jxx, Jyy, Jzz})

  •  
  • Jxx

    polar moment of inertia about the principal x-axis

  •  
  • Jyy

    polar moment of inertia about the principal y-axis

  •  
  • Jzz

    polar moment of inertia about the principal z-axis

  •  
  • Krot

    rotational kinetic energy

  •  
  • Ktrans

    translational kinetic energy

  •  
  • m

    mussel mass

  •  
  • r

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

  •  
  • V

    shell volume

  •  
  • vc

    velocity of shell centroid

  •  
  • x

    distance from y-axis to dA

  •  
  • xm

    mussel mass (statistical notation)

  •  
  • xr

    experimental replicate (statistical notation)

  •  
  • xs

    substrate type (statistical notation)

  •  
  • xt

    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

  •  
  • ωc

    angular velocity about instantaneous center of rotation

FOOTNOTES

This work was supported by a grant from the Wisconsin Sea Grant Institute (R/LR-91, C.E.L.); an Anna Grant Birge Memorial Scholarship from the Center for Limnology and Marine Sciences at the University of Wisconsin (S.M.P.); and a Dr and Mrs. Carl A. Bunde Award from the Department of Zoology at the University of Wisconsin (S.M.P.). Dawn Dittman, Robert O'Gorman, Theodore Strang, Terry Lewchanin, Maureen Walsh and Edward Perry of the U.S. Geological Survey, and John Janssen of the University of Wisconsin WATER Institute assisted with mussel collections. Mark Eriksson provided programming assistance for processing data files. Amir Assadi, Janette Boughman, Warren Porter, Gregory Gelembiuk, Cécile Ané, Marijan Posavi, Davorka Gulisija and two anonymous reviewers offered helpful comments.

REFERENCES

Alexander
R. R.
(
1993
).
Correlation of shape and habit with sediment grain-size for selected species of the bivalve Anadara
.
Lethaia
26
,
153
-
162
.
Amyot
J.-P.
,
Downing
J. A.
(
1998
).
Locomotion in Elliptio complanata (Mollusca: Unionidae): a reproductive function?
Freshw. Biol.
39
,
351
-
358
.
Beer
F. P.
,
Johnston
E. R.
Jr
(
1981
).
Mechanics of Materials
,
616
pp.
New York
:
McGraw-Hill, Inc.
Berkman
P. A.
,
Haltuch
M. A.
,
Tichich
E.
,
Garton
D. W.
,
Kennedy
G. W.
,
Gannon
J. E.
,
Mackey
S. D.
,
Fuller
J. A.
,
Liebenthal
D. L.
(
1998
).
Zebra mussels invade Lake Erie muds
.
Nature
393
,
27
-
28
.
Burks
R. L.
,
Tuchman
N. C.
,
Call
C. A.
,
Marsden
J. E.
(
2002
).
Colonial aggregates: effects of spatial position on zebra mussel responses to vertical gradients in interstitial water quality
.
J. N. Am. Benthol. Soc.
21
,
64
-
75
.
Carrier
D. R.
,
Walter
R. M.
,
Lee
D. V.
(
2001
).
Influence of rotational inertia on turning performance of theropod dinosaurs: clues from humans with increased rotational inertia
.
J. Exp. Biol.
204
,
3917
-
3926
.
Claxton
W. T.
,
Wilson
A. B.
,
Mackie
G. L.
,
Boulding
E. G.
(
1998
).
A genetic and morphological comparison of shallow- and deep-water populations of the introduced dreissenid bivalve Dreissena bugensis
.
Can. J. Zool.
76
,
1269
-
1276
.
Côté
I. M.
,
Jelnikar
E.
(
1999
).
Predator-induced clumping behaviour in mussels (Mytilus edulis Linnaeus)
.
J. Exp. Mar. Biol. Ecol.
235
,
201
-
211
.
Dermott
R.
,
Munawar
M.
(
1993
).
Invasion of Lake Erie offshore sediments by Dreissena, and its ecological implications
.
Can. J. Fish. Aquat. Sci.
50
,
2298
-
2304
.
Gelembiuk
G. W.
,
May
G. E.
,
Lee
C. E.
(
2006
).
Phylogeography and systematics of zebra mussels and related species
.
Mol. Ecol.
15
,
1033
-
1050
.
Hibbeler
R. C.
(
1989
).
Engineering Mechanics: Dynamics
,
552
pp.
New York
:
Macmillan Publishing Company
.
Huang
D.
,
Todd
P. A.
,
Guest
J. R.
(
2007
).
Movement and aggregation in the fluted giant clam (Tridacna squamosa L.)
.
J. Exp. Mar. Biol. Ecol.
342
,
269
-
281
.
Huryn
A. D.
,
Denny
M. W.
(
1997
).
A biomechanical process explaining upstream movements by the freshwater snail Elimia
.
Funct. Ecol.
11
,
472
-
483
.
Jarvis
P.
,
Dow
J.
,
Dermott
R.
,
Bonnell
R.
(
2000
).
Zebra (Dreissena polymorpha) and quagga mussel (Dreissena bugensis) distribution and density in Lake Erie, 1992-1998
.
Can. Tech. Rep. Fish. Aquat. Sci.
2304
,
1
-
46
.
Karatayev
A. Y.
,
Burlakova
L. E.
,
Padilla
D. K.
(
1998
).
Physical factors that limit the.distribution and abundance of Dreissena polymorpha (Pall.)
.
J. Shellfish Res.
17
,
1219
-
1235
.
Kilgour
B. W.
,
Mackie
G. L.
(
1993
).
Colonization of different construction materials by the zebra mussel (Dreissena polymorpha)
. In
Zebra Mussels: Biology, Impacts, and Control
(ed.
Nalepa
T. F.
,
Schloesser
D. W.
), pp.
167
-
173
.
Boca Raton
:
CRC Press, Inc.
Kobak
J.
(
2001
).
Light, gravity and conspecifics as cues to site selection and attachment behaviour of juvenile and adult Dreissena polymorpha Pallas, 1771
.
J. Molluscan Stud.
67
,
183
-
189
.
Kobak
J.
,
Nowacki
P.
(
2007
).
Light-related behavior of the zebra mussel (Dreissena polymorpha, Bivalvia)
.
Fund. Appl. Limnol.
169
,
341
-
352
.
Mackie
G. L.
(
1991
).
Biology of the exotic zebra mussel, Dreissena polymorpha, in relation to native bivalves and its potential impact in Lake St. Clair
.
Hydrobiologia
219
,
251
-
268
.
Marsden
J. E.
,
Lansky
D. M.
(
2000
).
Substrate selection by settling zebra mussels, Dreissena polymorpha, relative to material, texture, orientation, and sunlight
.
Can. J. Zool.
78
,
787
-
793
.
May
G. E.
,
Gelembiuk
G. W.
,
Panov
V. E.
,
Orlova
M. I.
,
Lee
C. E.
(
2006
).
Molecular ecology of zebra mussel invasions
.
Mol. Ecol.
15
,
1021
-
1031
.
Mills
E. L.
,
Dermott
R. M.
,
Roseman
E. F.
,
Dustin
D.
,
Mellina
E.
,
Conn
D. B.
,
Spidle
A. P.
(
1993
).
Colonization, ecology, and population structure of the “quagga” mussel, (Bivalvia:Dreissenidae) in the lower Great Lakes
.
Can. J. Fish. Aquat. Sci.
50
,
2305
-
2314
.
Mills
E. L.
,
Rosenberg
G.
,
Spidle
A. P.
,
Ludyanskiy
M.
,
Pligin
Y.
,
May
B.
(
1996
).
A review of the biology and ecology of the quagga mussel (Dreissena bugensis), a second species of freshwater dreissenid introduced to North America
.
Am. Zool.
36
,
271
-
286
.
Mills
E. L.
,
Chrisman
J. R.
,
Baldwin
B.
,
Owens
R. W.
,
O’Gorman
R.
,
Howell
T.
,
Roseman
E. F.
,
Raths
M. K.
(
1999
).
Changes in the dreissenid community in the lower Great Lakes with emphasis on southern Lake Ontario
.
J. Great Lakes Res.
25
,
187
-
197
.
Okajima
R.
,
Chiba
S.
(
2009
).
Cause of bimodal distribution in the shape of a terrestrial gastropod
.
Evolution
63
,
2877
-
2887
.
Peyer
S. M.
,
McCarthy
A. J.
,
Lee
C. E.
(
2009
).
Zebra mussels anchor byssal threads faster and tighter than quagga mussels in flow
.
J. Exp. Biol.
212
,
2027
-
2036
.
Peyer
S. M.
,
Hermanson
J. C.
,
Lee
C. E.
(
2010
).
Developmental plasticity of shell morphology in quagga mussels from shallow and deep-water habitats in the Great Lakes
.
J. Exp. Biol.
213
,
2602
-
2609
.
Pimentel
D.
,
Zuniga
R.
,
Morrison
D.
(
2005
).
Update on the environmental and economic costs associated with alien-invasive species in the United States
.
Ecol. Econ.
52
,
273
-
288
.
R Development Core Team
. (
2008
).
R: A Language and Environment for Statistical Computing
.
Vienna
:
R Foundation for Statistical Computing
.
Rivera
G.
,
Rivera
A. R. V.
,
Dougherty
E. E.
,
Blob
R. W.
(
2006
).
Aquatic turning performance of painted turtles (Chrysemys picta) and functional consequences of a rigid body design
.
J. Exp. Biol.
209
,
4203
-
4213
.
Roe
S. L.
,
MacIsaac
H. J.
(
1997
).
Deepwater population structure and reproductive state of quagga mussels (Dreissena bugensis) in Lake Erie
.
Can. J. Fish. Aquat. Sci.
54
,
2428
-
2433
.
Schwalb
A. N.
,
Pusch
M. T.
(
2007
).
Horizontal and vertical movements of unionid mussels in a lowland river
.
J. N. Am. Benthol. Soc.
26
,
261
-
272
.
Stoeckmann
A.
(
2003
).
Physiological energetics of Lake Erie dreissenid mussels: a basis for the displacement of Dreissena polymorpha by Dreissena bugensis
.
Can. J. Fish. Aquat. Sci.
60
,
126
-
134
.
Toomey
M. B.
,
McCabe
D.
,
Marsden
J. E.
(
2002
).
Factors affecting the movement of adult zebra mussels (Dreissena polymorpha)
.
J. N. Am. Benthol. Soc.
21
,
468
-
475
.
Uryu
Y.
,
Iwasaki
K.
,
Hinoue
M.
(
1996
).
Laboratory experiments on behaviour and movement of a freshwater mussel, Limnoperna fortunei (Dunker)
.
J. Molluscan Stud.
62
,
327
-
341
.
Vanderploeg
H. A.
,
Liebig
J. R.
,
Gluck
A. A.
(
1996
).
Evaluation of different phytoplankton for supporting development of zebra mussel larvae (Dreissena polymorpha): the importance of size and polyunsaturated fatty acid content
.
J. Great Lakes Res.
22
,
36
-
45
.
Waller
D. L.
,
Gutreuter
S.
,
Rach
J. J.
(
1999
).
Behavioral responses to disturbance in freshwater mussels with implications for conservation and management
.
J. N. Am. Benthol. Soc.
18
,
381
-
390
.
Walter
R. M.
,
Carrier
D. R.
(
2002
).
Scaling of rotational inertia in marine rodents and two species of lizard
.
J. Exp. Biol.
205
,
2135
-
2141
.
Watters
G. T.
,
O’Dee
S. H.
,
Chordas
S.
(
2001
).
Patterns of vertical migration in freshwater mussels (Bivalvia: Unionoida)
.
J. Freshw. Ecol.
16
,
541
-
549
.