Kinematics and hydrodynamics analysis of swimming anurans reveals striking inter-specific differences in the mechanism for producing thrust

For decades, frogs have served as model systems to understand the link between morphology and performance. For example, the morphological and ecological variety of frogs has generated much interest in how musculoskeletal anatomy relates to diversity in jumping performance among species (Rand, 1952; Emerson, 1978; Zug, 1978). More recently, single species studies of jumping mechanics have relied on the rich body of anuran muscle physiology work (Hill, 1970) to establish an integrated understanding of the limits of jumping performance in the context of muscle mechanics (Lutz and Rome, 1994; Peplowski and Marsh, 1997; Roberts and Marsh, 2003). Such studies have led to exploration of the unique demands of terrestrial versus aquatic locomotion met through modulating motor control (Peters et al., 1996; Gillis and Biewener, 2000) and the propulsive impulse from the hind limbs (Nauwelaerts and Aerts, 2003). Furthermore, species that are both excellent swimmers and jumpers potentially have musculoskeletal compromises that benefit swimming at the expense of jumping performance [and vice versa (Nauwelaerts et al., 2007)]. However, despite much comparative work on jumping frogs, there are no studies comparing the swimming mechanics of ‘generalized’ semi-aquatic species with either ‘specialized’ jumpers or swimmers. The current study combines previously developed analytical tools (Gal and Blake, 1988; Nauwelaerts and Aerts, 2003; Richards, 2008) with novel approaches to investigate the hind limb kinematics and hydrodynamics of two aquatic species (Xenopus laevis and Hymenochirus boettgeri) compared with the semi-aquatic Rana pipiens and the terrestrial Bufo americanus.

In addition to exploring the relationship between morphology and function, frogs are ideal models for addressing the link between kinematics and performance. Similar to other tetrapods, anuran hind limbs have joints that enable a large range of various leg kinematics patterns (Kargo and Rome, 2002). In principle, any individual frog may vary its limb kinematics within this potential limb ‘workspace’ [e.g. to achieve different swimming speeds (Richards, 2008)]. Likewise, an entire species may use a particular set of kinematics (within the workspace) according to the species' required range of swimming tasks. For example, an ambush predator that feeds in the water may use a different set of kinematics from a semi-aquatic frog that only enters the water to dive to the bottom upon escaping predation. Previous workers have established that use of synchronous leg motions produces higher thrust and faster swimming speed than kicking the legs asynchronously (Johansson and Lauder, 2004). However, variation in time-varying kinematics patterns (within an individual frog) has only been measured in detail for two species, Rana esculenta (Nauwelaerts and Aerts, 2003) and X. laevis (Richards, 2008). Moreover, hind limb kinematics during synchronous swimming have been suggested to be species specific (Johansson and Lauder, 2004), motivating the current study, which investigates the kinematic diversity of synchronous swimming kicks among anurans.

Given the diversity of available kinematics patterns within individuals and among species, attempting to relate the many joint kinematics parameters (e.g. joint extension amplitudes, phases and velocities) is potentially difficult. In the current study, I use a recently developed method for decomposing foot motion into aft-directed translational velocity and cranio-caudal foot rotational velocity (Richards, 2008) to relate time-varying joint kinematics to foot kinematics and hydrodynamic performance. Prior work on ranid frogs suggests that the feet generate propulsion mainly by translation. Although neither measurements of translational versus rotational velocity nor hydrodynamic modeling have been done, workers have observed that the feet are held at 90 deg. to flow during power strokes (Peters et al., 1996; Johansson and Lauder, 2004; Nauwelaerts et al., 2005). Nauwelaerts et al. speculated that a ranid foot may be biased toward translational motion because of a higher moment of inertia (Nauwelaerts et al., 2007). Consequently, strong jumpers may be expected to have a high foot moment of inertia (relative to fully aquatic species) in order to maximize the period of ground contact during jumps. By contrast, X. laevis were found to produce thrust mostly by rapid foot rotation, with foot translation contributing much less significantly to total thrust impulse (Richards, 2008). Similarly, the aquatic H. boettgeri rotate their feet continuously throughout the stroke, unlike R. pipiens, which maintain the feet nearly perpendicular to flow [e.g. compare fig. 5B from Gal and Blake (Gal and Blake, 1988) to fig. 4 from Peters et al. (Peters et al., 1996)]. I therefore suggest that rotational velocity is important for both X. laevis and H. boettgeri.

Based on prior work, the current study uses kinematics and hydrodynamic modeling to test the hypothesis that purely aquatic species, Xenopus laevis and Hymenochirus boettgeri (belonging to Pipidae), generate thrust by foot rotation, and semi-aquatic (or terrestrial) species, Rana pipiens (Ranidae) and Bufo americanus (Bufonidae), generate propulsion mainly by foot translation. The current study is the first detailed comparative analysis of the hydrodynamic mechanism of swimming among frog species and within individuals of a given species. Findings will give insight into how individual frogs operate within the anuran ‘kinematics workspace’. Moreover, the analysis will reveal that kinematic variation is species specific. Additionally, methods introduced in the current study will provide an analytical framework for integrating joint kinematics and hydrodynamics to help understand the function of multi-joint limbs during swimming.

Xenopus laevis (Daudin 1802) were obtained from Xenopus Express, Inc. (Plant City, FL, USA). Rana pipiens (Schreber 1782) and Hymenochirus boettgeri (Tornier 1896) were purchased from Carolina Biological (Burlington, NC, USA). Bufo americanus (Holbrook 1836) toads were wild-caught in Bedford, MA, USA. All frogs were housed in aquaria at the Concord Field Station and maintained at 20-22°C under a 12 h:12 h light:dark cycle.

Small (~0.5 cm diameter) plastic markers were placed on the skin above the snout, hip, knee, ankle and tarsometatarsal (tmt) joints (Fig. 1A) using a cyanoacrylate adhesive. Frogs were filmed from above swimming in a still water tank (27×52 cm Plexiglas™ for H. boettgeri and 80 cm×96 cm Plexiglas™ for all other species) at 250 frames s⁻¹ at a 1/1000 s shutter speed with a high-speed Photron Fastcam camera (Photron Ltd, San Diego, CA, USA). Each swimming tank was large enough for several swimming strokes (roughly 6, 17, 7, 14 swimming strokes for X. laevis, H. boettgeri, R. pipiens and B. americanus, respectively). Due to behavioral differences between species, R. pipiens and B. americanus were filmed swimming at the surface, whereas X. laevis and H. boettgeri swam submerged. For X. laevis and H. boettgeri, a water depth of 5-6 cm was used to ensure horizontal swimming. For R. pipiens and B. americanus, deeper water (~20 cm) was used to avoid hydrodynamic interactions between the feet and the bottom. X—Y marker coordinates were digitized in Matlab (The MathWorks, Natick, MA, USA) using a customized routine (DLTdataviewer 2.0 written by Tyson Hedrick). To obtain foot angle measurements, the position of the tip of the second toe was digitized in addition to the joint markers. Only synchronous strokes with a straight swimming trajectory were analyzed. A range of slow to fast swimming sequences was elicited by tapping slightly behind the frog leg. Several swimming sequences were recorded to capture the entire range of swimming velocities. Only the power stroke (i.e. defined as the period of positive center of mass acceleration) was analyzed for each swimming stroke (Fig. 1B; see Discussion).

Time-varying hydrodynamic thrust produced by translational and rotational velocity and acceleration of frog feet was estimated by a blade element model following earlier studies (Gal and Blake, 1988; Richards, 2008). For each species, frog body shape was approximated by an ellipsoid with axes lengths corresponding to lengths of the snout—vent, medio-lateral and dorso-ventral axes measured from individual frogs used in the video recordings. For each individual frog, digital images of feet (held flat on a sheet of Plexiglas™ and photographed from below) were traced in Scion image (Scion Corporation, Frederick, MD, USA) to digitally measure foot area. Foot shape was modeled as a trapezoidal flat plate with the same area as the actual frog foot. Using a previously described and verified forward dynamics approach (Richards, 2008), these morphological measurements and foot kinematics were used to simulate time-varying hydrodynamic thrust and forward swimming velocity using a numerical differential equation solver in Mathematica 6.0 (Wolfram Research, Inc., Champaign, IL, USA). Simulated swimming COM velocity profiles (based on actual foot kinematics) were then compared with recorded COM velocity profiles to verify that the blade element model is applicable across species.

Data for statistical analysis were gathered from measurements made on the power stroke portion of individual swimming strokes within each frog species. Multiple comparisons of means (pooled data from individual frogs) were done by ANOVA using the Tukey—Kramer HSD post-hoc test. Alpha values from multiple paired t-tests were adjusted by the sequential Bonferroni method (Quinn and Keough, 2002). Two-way ANOVAs were performed to account for variation between individual frogs when comparing means among species. For the two-way ANOVAs, species was used as a fixed factor, and individual frog was used as a random factor (nested within species). For the hydrodynamic modeling, data was split into slow swimming trials (<30% maximum swimming velocity) and fast swimming trials (>60% maximum swimming velocity). All statistical analyses were done in SPSS 16.0 (SPSS Inc., Chicago, IL, USA).

Each species swam by extending the hip, knee, ankle and tarsometatarsal (tmt) joints (Table 2; see Movie 1 in supplementary material). In each species, onset times of joint extension were staggered, with the knee beginning to extend first, followed by the hip then the ankle (Fig. 2). Extension onset times for the tmt and foot varied among species. Peak swimming stroke velocity in X. laevis ranged from 3.3±1.1 to 20.9±2.5 body lengths s⁻¹ (BL s⁻¹; mean ± s.d., N=4 frogs). Similarly, H. boettgeri and R. pipiens peak velocities ranged from 6.8±2.1 to 28.6±3.7 BL s⁻¹ (N=4 frogs) and from 4.9±0.5 to 20.9±4.1 BL s⁻¹ (N=4 frogs), respectively. The velocity range of 3.1±0.3 to 7.0±2.0 BL s⁻¹ for B. americanus was significantly narrower (N=4 frogs, ANOVA, Tukey—Kramer, P≤0.01) than the other species. The velocity ranges of the three other species did not differ significantly (P>0.2). Joint kinematics were unique for each species. Mean minimum and maximum joint angles differed significantly for most comparisons between species (Table 2). Most notably, the mean minimum foot angle was highly variable among species, ranging from 54.2±8.1 to 79.3±21.8 to 83.4±10.4 to 103.9±15.6, in X. laevis, H. boettgeri, R. pipiens and B. americanus, respectively (mean ± s.e.m.; Table 2). Therefore, the range of foot rotation was highest in X. laevis and lowest in B. americanus. Despite species differences in joint angular excursions, mean joint extension patterns followed qualitatively similar sinusoidal increases in joint angle throughout the power stroke (Fig. 3). This general pattern was observed in all joints except the tmt joint in R. pipiens and B. americanus, which each showed minimal (<30 deg.) net joint extension (Fig. 3Diii,iv; Table 2). Most joint angles peaked near 100% of the power stroke duration; however, mean peak hip, knee and ankle angles in B. americanus were phase-shifted to 65%, 52% and 67% stroke duration, respectively (Fig. 3Aiv,Biv,Civ). For each multiple comparison in the current study, a two-way ANOVA was used to verify that significant differences among species were not confounded by differences between individual frogs (P<0.05).

As a consequence of variation in hind limb kinematics, EFV profiles were distinct among species. In each species, EFV increased during most of the power stroke (the period of positive COM, acceleration; Fig. 1B) and decreased late in the stroke (Fig. 4). Because EFV was still positive at the end of the power stroke, the COM began decelerating before full hind limb extension in all species (Fig. 4A). In X. laevis, peak EFV phase averaged 70±8% (Fig. 4Ai) with respect to the onset of the power stroke, which was significantly earlier compared with 88±3, 87±4 and 81±6% in H. boettgeri, R. pipiens and B. americanus, respectively (mean ± s.e.m.; ANOVA, Tukey—Kramer, P≤0.001 for all comparisons). The translational component of EFV in X. laevis peaked at 62±8% (Fig. 4Ai), which was significantly earlier than 83±8% in H. boettgeri, 78±2% in R. pipiens and 80±8% in B. americanus. Similarly, rotational EFV in X. laevis peaked significantly earlier in the power stroke, at 71±8% compared with H. boettgeri, R. pipiens and B. americanus means of 88±2%, 92±5% and 82±5%, respectively. Magnitudes of relative translational EFV (100% peak translational EFV/peak total EFV) as well as relative rotational EFV were highly variable among species (Table 3). In H. boettgeri, there was no statistical difference between relative translational and rotational EFV (55±3% versus 53±6%; P=0.98). Relative rotational EFV was 74±2% in X. laevis and 62±2% in B. americanus, which was significantly higher than mean relative translational EFV of 32±5% and 40±3%, respectively (P<0.0001). However, in R. pipiens relative translational EFV was 64±5%, which was significantly higher than mean rotational EFV of 51±7% (P<0.0001;Table 3). In H. boettgeri and B. americanus, rotational EFV was negative early in the power stroke. This was likely due to slight asynchrony in the limbs for some swimming strokes of H. boettgeri. For B. americanus, however, a small burst of COM acceleration occurred late in the limb recovery phase, which is likely due to positive thrust caused by added mass effects (Nauwelaerts et al., 2005). Consequently, the COM began to accelerate slightly before (~80 ms) the onset of joint extension in most swimming strokes. Additionally, ‘net translational EFV’ (translational EFV — COM velocity) was calculated to obtain foot velocity in the global frame of reference. Strikingly, net EFV was almost entirely negative in X. laevis and H. boettgeri (peaking at ~0 m s⁻¹) compared with peak net EFV of 0.23±0.13 and 0.11±0.1 in R. pipiens and B. americanus, respectively (Fig. 4B).

Throughout all recorded power strokes, total angular excursions at each leg joint contributed to the total translational displacement of the foot (Fig. 5A). In each species, the hip contributed most to total translational foot displacement. In X. laevis, the relative hip contribution to total translational displacement (100%×d_t,hip/d_t) was 61±6% compared with 8±6% and 31±5% for knee and ankle contributions, respectively. In H. boettgeri, R. pipiens and B. americanus, ankle extension contributed least to total translational displacement. Within each of the four species, all comparisons of group mean joint contributions to foot translation were significant (mean ± s.e.m.; ANOVA, Tukey—Kramer, P<0.003). However, the hip versus knee contributions to translational displacement were not significantly different in B. americanus (P=0.105).

Total rotational foot displacement was also a function of excursions at each limb joint (Fig. 5B). Each of the four species showed a similar pattern where the knee and tmt contributed least to the total change in foot angle. Most strikingly, the relative knee contribution to foot rotational displacement (100%×d_r,knee/d_r) was −75±14%, −256±67%, −170±38% and −138±18%, which decreased total foot angle change in X. laevis, H. boettgeri, R. pipiens and B. americanus, respectively. Since the knee angle is oriented opposite the other angles (i.e. the knee points laterally; Fig. 1A), knee extension reduces foot angle, causing negative knee contributions to peak angular displacement in all species. Notably, the tmt also contributed 61±17% and 75±22% to foot rotation in X. laevis and H. boettgeri, whereas in R. pipiens and B. americanus the tmt contributed negligibly (−32±24% and 8±30%, respectively) to total foot rotation. The relative contributions of the hip and ankle to foot rotation also varied among species. The hip contribution to foot rotation ranged from 52±21% to 64±11% to 90±23% to 160±37% in B. americanus, X. laevis, R. pipiens and H. boettgeri, respectively. Similarly, the ankle contribution to foot rotation ranged from 94±3% to 100±15% to 149±24% to 165±37% in B. americanus, X. laevis, H. boettgeri and R. pipiens, respectively. Within each species, differences among all joints were significant (ANOVA, Tukey—Kramer, P<0.012) except for hip versus tmt in X. laevis and hip versus ankle in H. boettgeri (Fig. 5B).

For all four species, joint extension at the hip, knee and ankle each contributed to foot translational velocity (Fig. 6). In R. pipiens, the relative hip contribution to translational velocity (100%×peak v_t,hip/peak v_t) was 43±5% (Fig. 6Aiii), which was significantly lower than 62±7%, 66±2% and 60±5% in X. laevis, H. boettgeri and B. americanus, respectively (mean ± s.e.m.; ANOVA, Tukey—Kramer, P<0.001 for all comparisons). The relative knee contribution to translational velocity varied considerably among species, with the highest mean of 56±19% in B. americanus (Fig. 6Biv) and lowest mean of 21±11% in X. laevis (Fig. 6Bi). Differences in the relative knee contribution to translational velocity were significant in both X. laevis and B. americanus compared with H. boettgeri and R. pipiens (P≤0.01). In all four species, ankle extension contributed less to translational velocity than hip extension. Mean contributions of ankle extension to translational velocity of 39±8%, 35±7% and 34±4% in X. laevis, R. pipiens and B. americanus (Fig. 5C), respectively, were statistically similar (P≥0.1), but significantly different from 26±6% in H. boettgeri.

Rotational foot velocity was also broken down into contributions from hip, knee, ankle and tmt joint extension (Fig. 7). The relative hip contribution to peak rotational velocity (100%×peak v_r,hip/peak v_r) was 49±11% in R. pipiens (Fig. 7Aiii), which was statistically similar to 50±10% in B. americanus (Fig. 7Aiv; P=0.62). Mean contribution of hip extension was 70±5% in H. boettgeri and 34±3% in X. laevis, each differing significantly from R. pipiens and B. americanus (P≤0.001). In X.laevis, the mean knee contribution to peak rotational velocity, −49±11% (Fig. 7Bi), was significantly higher (P≤0.001) than −125±16%, −106±14% and −120±10% in H. boettgeri, R. pipiens and B. americanus, respectively, which did not differ significantly (P>0.7). Mean contributions of ankle extension to peak rotational velocity were 70±4% in X. laevis, 111±29% in H. boettgeri, 156±10% in R. pipiens and 118±10% in B. americanus. Differences among all mean ankle contributions were statistically significant (P≤0.001) except for the difference between H. boettgeri and B. americanus (P=0.89). For each species, the ankle showed the highest relative contribution to peak rotational velocity compared with hip, knee and tmt joints. The relative tmt contribution to peak rotational velocity averaged −15±24% in R. pipiens (Fig. 7Diii), which was statistically similar to −1±47% in B. americanus (Fig. 7Div; P=0.97). Mean peak tmt extension contributed 57±20% to peak rotational velocity in X. laevis and 51±30% in H. boettgeri, each significantly higher than R. pipiens and B. americanus (P≤0.001).

Net hydrodynamic thrust impulse (i.e. the time integral of estimated hydrodynamic thrust) in H. boettgeri, R. pipiens and B. americanus was produced both by translational and rotational foot motion (Fig. 8A; Table 3). By contrast, net thrust impulse was only derived from rotational foot motion in X. laevis. For data pooled across all swimming speeds, the mean relative translational impulse (100% × translational impulse/maximum impulse) was 15±10%, 25±9% and 20±16% in H. boettgeri, R. pipiens and B. americanus, respectively, but only −2±9% in X. laevis. The mean relative rotational impulse was 36±13%, 25±10%, 13±6% and 23±3% in X. laevis, H. boettgeri, R. pipiens and B. americanus. Paired t-tests within each species revealed that relative translational and rotational impulses were significantly different in X. laevis, H. boettgeri and R. pipiens (P<0.0001) but statistically similar in B. americanus (P=0.64). Inter-specific differences in translational versus rotational thrust impulses were nearly independent of swimming speed. For slow swimming trials (<30% peak swimming velocity), mean relative translational versus rotational impulses were −1±3% versus 18±7%, 0.4±6% versus 17±15% and 10±5% versus 5±3% for X. laevis, H. boettgeri and R. pipiens, respectively (Fig. 8B). Similarly, relative translational versus rotational impulses for fast swimming trials (>60% peak swimming velocity) were −5±18% versus 61±30%, 24±14% versus 33±12% and 33±13% versus 19±9% for X. laevis, H. boettgeri and R. pipiens (Fig. 8C). Only pooled data were analyzed for B. americanus because most of the swimming trials were within 60% of maximum swimming speed.

Peak velocity for swimming strokes positively correlated with thrust impulse in all four species observed (Table 4). X. laevis, H. boettgeri and R. pipiens showed the strongest correlation between peak velocity and thrust impulse, compared with the relatively weak relationships observed in B. americanus frogs (Table 4).

The present study aimed to quantify hind limb kinematics patterns and estimate the hydrodynamic forces produced by the feet in four anuran species. Based on prior studies (Peters et al., 1996; Johansson and Lauder, 2004; Nauwelaerts et al., 2005), I expected that semi-aquatic/terrestrial species (represented by B. americanus and R. pipiens) would primarily use proximal leg joints to produce translational foot motion to power swimming. By contrast, purely aquatic frogs (represented by X. laevis and H. boettgeri) were expected to swim by rotating their feet via distal joint extension (Richards, 2008). Findings from the current study do not fully support either hypothesis. As expected, thrust produced by foot rotation is significantly higher than translational-based thrust in the two aquatic species X. laevis and H. boettgeri. However, leg and foot kinematics are not consistent within aquatic or semi-aquatic/terrestrial groups, despite similar maximum swimming performance among them. Instead, the four species examined represent a continuum between propulsion driven exclusively by rotational motion (X. laevis), mostly rotational motion (H. boettgeri), rotational and translational motion (B. americanus), and mostly translational motion (R. pipiens).

In all four species, frogs increased swimming speed from one stroke to the next by increasing the thrust impulse produced at the feet (Table 4). However, the differential use of translational versus rotational components of thrust varies among these species (Fig. 8). Moreover, inter-specific differences in thrust mechanism do not differ significantly between slow and fast swimming speeds (Fig. 8B,C). The mechanism by which B. americanus modulates swimming speed remains unclear, given that swimming speed is only weakly correlated with thrust impulse in most of the frogs observed (Table 4). Perhaps the weak correlation is explained by a very narrow range of swimming speeds in B. americanus compared with the other three species. Additional peculiarities observed in B. americanus swimming are discussed in further detail below.

Body kinematics patterns are superficially similar among all four species. Frogs swim by a ‘kick-and-glide’ mechanism, causing rapid acceleration of the COM followed by a glide phase in which the COM velocity gradually decreases due to the drag on the body and the limbs. After the glide, the leg joints flex to retract the limb in preparation for the subsequent stroke (Fig. 1B). During the recovery phase, the COM velocity decreases sharply. The glide and recovery strokes are highly variable within individuals (Nauwelaerts et al., 2001). Moreover, the length of the glide phase, relative to the propulsive phase, varies among the species observed (discussed in further detail below). Therefore, only the power stroke portion is analyzed in the current study to avoid the confounding variation in glide and recovery strokes among species.

All species observed employ a similar power stroke pattern driven by temporally staggered extension of the leg joints (Fig. 2), resulting in cranio-caudal foot translation and rotation. Despite temporally staggered onsets of joint extension in all species, the hip, knee and ankle generate translational velocity of the feet (Fig. 6A—C). Similarly, the hip and ankle contribute rotational velocity to the feet in all species (Fig. 7A,C). Importantly, because the knee angle faces medially (unlike all other joints, which face laterally), knee extension velocity is negative with respect to the other joints. For example, on the left leg, hip, ankle, and tmt extend in the anticlockwise direction whereas knee extension is clockwise (Fig. 1A). Therefore, as a consequence of the conserved limb morphology among anurans, knee extension contributes negatively to foot rotational velocity in all four species (Fig. 7B).

Due to coordinated joint extension, all species achieve net positive EFV during power strokes (Fig. 4A). Since EFV is both a function of foot velocity and the foot's projected area in flow, it is a useful measure of how translational and rotational foot velocity interact to generate drag-based thrust (drag-based thrust ∞ EFV²). In all species tested, EFV peaks prior to maximum COM velocity. Consistent with observations in R. pipiens (Johansson and Lauder, 2004) and H. boettgeri (Gal and Blake, 1988), the leg joints continue to extend after the end of the power stroke (i.e. after the period of positive COM acceleration). The duration of the limb extension phase (the period between the beginning of the power stroke and the point where EFV reaches zero; Fig. 1B) is roughly twice the duration of the power stroke, which does not significantly differ among species (P>0.05). Consequently, in frogs, only approximately the first half of limb extension duration provides enough thrust at the feet to overcome hydrodynamic drag and added mass on the body. The remaining portion of limb extension serves to straighten the leg to reduce drag on the animal's profile (Peters et al., 1996), as well as to assist shedding of the attached vortices on the foot (Johansson and Lauder, 2004) to minimize retarding added-mass-based force (see Daniel, 1984).

All four frog species shared superficial similarities in body and limb morphology (Table 1), joint angle patterns (Fig. 3) and foot velocity patterns (Fig. 4). Despite these similarities, hydrodynamic modeling suggests dramatic differences in the propulsion mechanics of these species. A recently developed forward-inverse dynamics approach (Richards, 2008) allows the estimation of total thrust as the sum of thrust produced by translational and rotational foot motion. In X. laevis and H. boettgeri, most of the thrust required for swimming comes from rotational foot motion whereas R. pipiens generate thrust mainly by foot translation (Fig. 8). Thrust estimates from B. americanus, however, were highly variable and, consequently, the hydrodynamic mechanism in toads is unclear from the current study and will be discussed below.

Although EFV traces are useful for relating joint extension to resultant foot motion in a local coordinate system, predicting species-specific thrust patterns also requires consideration of foot motion in the global frame of reference. After Gal and Blake (Gal and Blake, 1988), translational EFV was converted to ‘net translational EFV’ (translational EFV — COM velocity) to approximate foot velocity relative to the background flow surrounding the foot as the body moves forward. For example, if translational EFV=COM velocity, the feet are pushing backward against the water at the same rate that the COM progresses forward. Therefore, no translational thrust is produced because the tmt joint (i.e. the base of the foot) is stationary in the global reference frame. Importantly, when net translational EFV=0, rotational EFV generates enough thrust to overcome drag and added mass on the body, enabling the COM to move forward. Foot translational velocity in X. laevis and H. boettgeri seldom exceeds the forward COM velocity. Consequently, net translational EFV is negative throughout most swimming strokes (Fig. 4B). In the latter half of the power stroke, R. pipiens and B. americanus show the opposite pattern of rapid translational velocity and relatively slower forward COM velocity due to the backward ‘slip’ of the foot.

Combining the above hydrodynamic modeling with joint kinematics measurements enables a detailed analysis of how individual joints contribute to propulsion. In all four species examined, hip extension contributes largely to both foot translation and rotation (Fig. 5). In R. pipiens, the hip generates propulsive motion primarily by translating the foot. However, because net translational EFV is low in X. laevis and H. boettgeri, the hip contributes directly to propulsion only by helping to rotate the foot. Although the knee diminishes rotational velocity (therefore likely reducing rotational thrust) in all four species, the translational component of knee extension contributes to thrust most strongly in R. pipiens, but also in H. boettgeri and B. americanus. The roles of the ankle and tmt joints are highly variable among species. X. laevis and R. pipiens use ankle extension both to translate and rotate the feet. However, the ankle of X. laevis likely contributes to propulsion by assisting foot rotation whereas R. pipiens likely uses the ankle to produce translational thrust (helping to maintain the foot at ~90 deg. to flow). By contrast, the ankle does not contribute to translational velocity in H. boettgeri and B. americanus but likely contributes substantially to thrust by rotating the foot.

The varied use of individual leg joints among species suggests that different species employ unique joint coordination strategies to power swimming. Variation in the underlying joint kinematics can be understood in the context of hydrodynamic differences among species. For example, R. pipiens frogs effectively generate translational thrust by maintaining ~90 deg. foot angle in flow for the first approximately half of the power stroke (Peters et al., 1996; Nauwelaerts et al., 2005; present study; Fig. 3Eiii), maximizing the foot area projected into flow. Pure translation (i.e. no net change in foot angle) requires three aspects of coordination. Firstly, hip and knee extension must be synchronized to cause the foot to move mainly in the cranio-caudal axis (see supplementary material, Movie 2versus Movie 3). If the knee were to extend further (or earlier) than the hip, the feet would move laterally instead of caudally (Movie 4 in supplementary material). Additionally, if the hip were to extend further (or earlier) than the knee, the foot would rotate instead of translate. R. pipiens, unlike the three other species observed, extend the hip and knee synchronously (Fig. 2), enabling the transfer of joint rotation to foot translation. Secondly, extension velocities of the knee and ankle should be similar. This allows foot rotation caused by the ankle to cancel foot counter-rotation caused by knee extension, enabling cranio-caudal foot translation at a fixed foot angle (Movie 5 in supplementary material). Lastly, hip extension (important for foot translation) also causes foot rotation. Hip-generated foot rotation is likely balanced by negative rotation (i.e. flexion) at the tmt joint in the first approximately half of the stroke (Fig. 3Diii), possibly explaining why R. pipiens is the only species observed to generate negative net rotational motion at the tmt joint (Fig. 5B; Fig. 7Diii). Speculation on the functional significance of the ‘translational’ swimming stroke in ranid frogs is discussed below.

Instead of minimizing foot rotation (as seen in R. pipiens), X. laevis appear to coordinate limb joints to enhance rotational foot velocity. As noted above, knee extension (synchronized with hip extension) produces positive foot translational velocity at the expense of foot rotational velocity. This tradeoff suggests that increased extension of distal joints (ankle and tmt) may compensate for counter-rotation at the knee to produce the necessary foot rotation. In H. boettgeri, R. pipiens and B. americanus, the knee is the second most important joint contributing to translational foot motion. Interestingly, in X. laevis, the mean knee angle range is ~50 deg. compared with ~100 deg. in all other species tested (Fig. 3; Table 2). Because X. laevis rely heavily on rotational foot velocity for thrust (Fig. 8), knee extension may be kept minimal to reduce foot counter-rotation. X. laevis, however, require a small amount of translational thrust to accelerate the COM early in the stroke (Richards, 2008) and therefore must produce the necessary foot translation by substantial ankle rotation (to compensate for reduced knee extension). Similarly, B. americanus kinematics also reveal strategies to enhance foot rotation. In B. marinus, the knee is the first joint to extend, pushing the feet laterally early in the stroke (Gillis and Biewener, 2000). Later in the stroke, the ankle and hip extend to rotate the foot. If the hip and ankle were in phase with the knee, extension at the knee would cancel foot rotation from hip and ankle extension (see supplementary material, Movie 5versus Movie 6). Importantly, however, rapid swimming likely requires all joints to extend in phase to maximize both translational and rotational based thrust (Richards, 2008). Consequently, the out-of-phase coordination of B. americanus joints may partially explain their relatively slow maximum swimming speed (~7 BL s⁻¹ compared with >20 BL s⁻¹ in the three other species tested).

The blade element model used for the current study has been verified previously (Richards, 2008); however, limitations arise using this approach. All hydrodynamic calculations in the current study assume that the feet are the principal propulsive surfaces on the hind limbs. This assumption is valid for X. laevis, H. boettgeri and R. pipiens, which each have large ratios of foot area:frontal body area (0.56±0.09, 0.42±0.02 and 0.15±0.07, respectively; mean ± s.d.) compared with B. americanus (0.08±0.05). This suggests that B. americanus feet are less effective at producing thrust for any given stroke pattern. Toads may rely on thrust produced by their leg segments in addition to the foot, possibly explaining why thrust impulse from the feet is a poor predictor of swimming performance. Future modifications to the blade element model could include hydrodynamic reaction forces from the upper and lower leg to account for all possible propulsive surfaces. An additional limitation is that the feet are approximated as flat plates with constant area. However, the feet change their shape and area throughout the course of a swimming stroke (C.R., personal observations), implying that animals may benefit by controlling their propulsor flexibility (Lauder et al., 2006). This problem could be addressed by a 3-D analysis of foot kinematics coupled with more advanced modeling tools (e.g. computational fluid dynamics) as well as robotic models (Lauder et al., 2007) to determine how dynamic propulsor shape influences hydrodynamic performance in frogs.

In addition to the limitations of the blade element model, there are three important species differences not accounted for by the analytical approaches used in the current study. Firstly, R. pipiens and B. americanus most often swim at the surface, whereas X. laevis and H. boettgeri swim fully submerged. Frogs swimming at the surface potentially incur additional drag on the body due to bow wave formation. The importance of drag due to wave formation can be assessed from the Froude number [Fr=swimming velocity/(gravitational acceleration × outstretched length of the frog)^0.5 (Hoerner, 1965; Johansson and Lauder, 2004). Such surface drag effects are substantial at approximately 0.4<Fr<0.8 (Hoerner, 1965; Johansson and Lauder, 2004). B. americanus frogs swim slowly (maximum Fr~0.4), suggesting that surface wave drag is negligible. During rapid swimming (Fr>~0.8), R. pipiens advance forward faster than the time required for a bow wave to develop (Johansson and Lauder, 2004). At intermediate speeds (Fr~0.5), formation of bow waves likely causes the current model to underestimate body drag. However, increasing the body coefficient of drag by up to 4-fold in the hydrodynamic model did not significantly affect calculations of translational versus rotational components thrust. Therefore, behavioral differences of surface versus submerged swimming are not likely to account for the species differences in hydrodynamic mechanism reported in the current study. Secondly, X. laevis and H. boettgeri move their legs in the horizontal plane during normal, level swimming (C.R., personal observations). However, ranid frogs extend their legs slightly downwards in addition to the backward propulsive motion (Johansson and Lauder, 2004; Nauwelaerts et al., 2005), resulting in a downward-directed vortex ring and an upward force on the frog body (Johansson and Lauder, 2004). Since all of the calculations in the current study are from joint angles projected onto the horizontal plane, the analytical approach used here does not account for the downward motion of the legs. However, the slight (~14 deg.) angle of the frog body relative to horizontal [see fig. 2 in Johansson and Lauder (Johansson and Lauder, 2004)] would cause only a slight underestimate (~4%) of translational foot displacements, which is unlikely to bias the calculations used in the current study. Thirdly, the four species observed show drastic differences in the length of the glide phase relative to the propulsive phase. X. laevis and H. boettgeri individuals often (~30-50% of trials) truncate the glide phase whereas R. pipiens usually (~80% of trials) extend the glide phase until swimming velocity has reached zero. B. americanus, however, moves the legs continuously without gliding. The functional significance of such glide length differences is unclear. Perhaps glide length is correlated with the propulsive strategy used. For example, the extended glide phase (as seen in R. pipiens) may be a gliding strategy that is used for translational-driven strokes, whereas the shortened glide phase may be associated with rotational-driven strokes. Future modeling that accounts for the hydrodynamics of propulsive, glide and recovery stroke phases could be used to determine how propulsion and glide strategies can be coupled to enhance swimming performance (e.g. mean speed and efficiency over several swimming strokes).

Variation in kinematics and estimated hydrodynamics among species is potentially explained by three effects. Firstly, shared similarities between aquatic frogs (e.g. foot rotation-powered swimming in X. laevis and H. boettgeri) may reflect specializations for swimming. Despite similarities in morphology (Table 1) and performance among the ranid and pipid species observed, differences in hind limb kinematics and hydrodynamics suggest underlying differences in joint mechanics (e.g. muscle work, joint work, mechanical efficiency). Recent hydrodynamic modeling of swimming frogs proposed that mechanical efficiency (work done on the COM/net work done at the joints) is highly dependent on foot kinematics. At any given swimming speed, the most efficient swimming strokes are predicted to require a high ratio of foot rotational velocity to translational velocity [i.e. peak efficiency is predicted at a rotational to translational velocity ratio of ~1.7:1 (Richards, 2008)]. Perhaps obligatorily aquatic frogs (represented by X. laevis and H. boettgeri) rely heavily on rotational foot motion to increase swimming efficiency. Future studies could apply inverse dynamics to model mechanical power and work produced at the leg joints to estimate if mechanical efficiency is higher in aquatic versus semi-terrestrial and terrestrial frogs.

Secondly, characteristics of X. laevis and H. boettgeri may represent an ancestral (generalized) propulsive strategy in anurans that has become modified in more derived species that excel in jumping. This hypothesis is supported by the fact that, as pipid frogs, X. laevis and H. boettgeri are basal to the neobatrachia, which comprise most other frog species (Tree of Life; http://tolweb.org/Anura/16963). The more highly derived Ranidae and Bufonidae (belonging to neobatrachia) are much stronger jumpers than X. laevis or H. boettgeri (C.R., personal observations). Although jumping evolved very early in anurans (Shubin and Jenkins, 1995; Jenkins and Shubin, 1998), jumping ability is highly variable among species (Rand, 1952; Jug, 1978). Among the species in the current study, R. pipiens is the strongest jumper, reaching maximum jump distances (relative to body mass) ~1.7 fold higher than in B. americanus (Emerson, 1977). R. pipiens also shows the most extreme dependence on foot translational velocity during swimming. Recent work on R. esculenta predicted that a high moment of inertia of the feet (relative to other leg segments) would benefit swimming by helping to maintain the foot at 90 deg. to flow by minimizing foot rotation (Nauwelaerts et al., 2007). Counter to their predictions, these workers found a positive correlation between performance and foot moment of inertia in jumping, but not in swimming. During jumping, increased foot moment of inertia may help to stabilize the feet as the proximal joints extend, maximizing the period of ground contact. If this is true, the translational-driven swimming seen in ranids (and possibly other powerful jumpers) may simply be a consequence of each foot's inertial resistance to rotation. A future analysis could include Arcis crepitans, a semi-aquatic hylid frog that is capable of jumps ~2-3× further than R. pipiens (Rand, 1952; Zug, 1978). Analysis of leg segment masses and moments of inertia among aquatic and semi-aquatic species might also help explain variation in swimming mechanics among frogs.

Thirdly, variation among species may correlate with phylogenetic differences rather than functional differences between purely aquatic and semi-terrestrial species. My findings demonstrate that foot-rotation-powered swimming only appears in the two aquatic species observed here. However, these findings cannot resolve whether ‘foot rotational’ swimming is a specialization for the aquatic habitat or, rather, is a shared trait among aquatic pipid species. Future studies on additional pipid, as well as neobatrachian taxa, could apply similar kinematics methods as employed here to achieve a wider sample of hind limb function. Independent contrasts analysis could then be used to establish if functional differences among species correlate with habitat (aquatic, semi-aquatic, terrestrial, arboreal) or with phylogenetic distance (Felsenstein, 1985).

Data from the current study demonstrate striking functional differences between anuran species during swimming. Within the study of frog swimming, these findings shed new light both on earlier important work (e.g. Gal and Blake, 1988; Peters et al., 1996), as well as more recent investigations. For example, Gal and Blake proposed a novel propulsion mechanism in H. boettgeri whereby the right and left feet produce a central backward jet as they approach the midline (Gal and Blake, 1988). Recent work in ranid frogs using digital particle image velocimetry (DPIV) (Johansson and Lauder, 2004; Nauwelaerts et al., 2005) shows no evidence for a central jet. However, findings from the current study show highly divergent kinematics and hydrodynamics between H. boettgeri and R. pipiens. A new study using DPIV in Hymenochirus would be required to reevaluate Gal and Blake's original claim.

The current study offers simple analytical tools for integrating joint kinematics and hydrodynamics in a multi-joint limb. Given the apparent complexity of limb motion by multiple joints, one is tempted to only present descriptions of the joint extension patterns in the absence of any functional interpretation. I propose that resolving limb kinematics into translational versus rotational EFV is useful for presenting limb kinematics in a functional context. Moreover, the relative contributions of individual joints to propulsion are reasonably well predicted by analyzing joint contributions to translational and rotational velocity. In the current study, positive net translational EFV (e.g. R. pipiens) results from the feet translating backward faster than the animal advances forward. This suggests that the observed hip, knee and ankle extension, with minimized tmt extension, produces thrust from translational foot velocity. Conversely, negative net translational EFV (e.g. X. laevis and H. boettgeri) results from the body moving forward faster than the feet translate backward. This suggests an alternative mechanism for producing thrust; the observed rapid extension of the hip, ankle and tmt, with minimized knee extension, produces thrust from rotational velocity. As confirmed by hydrodynamic modeling, the kinematic observations suggest that pipids swim mainly by rotation-driven thrust whereas ranids swim mainly by translation-driven thrust.

Despite assumptions of prior literature, kinematics of swimming anurans are highly species specific, even among aquatic and semi-aquatic/terrestrial groups. Although time-varying patterns of joint kinematics appear superficially similar, differences in the magnitude and timing of joint velocity accumulate from proximal to distal. This results in dramatic differences in effective foot velocity when comparing patterns among species. Consequently, the hydrodynamic function of anuran feet varies from producing mainly translational thrust (unique to ranid frogs) to producing mainly rotational thrust (unique to pipid frogs). The current study, therefore, motivates future studies in which a broader sample of anuran diversity can be compared by the simple kinematic analyses presented here. Such a study would further illuminate how frog morphological diversity (both at the level of the musculoskeletal system and the whole body) affects the kinematics and hydrodynamics used to power swimming.

I thank Pedro Ramirez for animal care and I greatly thank my PhD advisor, Andrew Biewener, for invaluable advice and mentorship in addition to the lab space and equipment necessary for this work. I thank Tim Higham for statistics advice and Brian Joo for assistance with data collection. I also thank members of my thesis committee: George Lauder, Farish Jenkins, Jr, Jack Dennerlein and Anna Ahn for helpful comments on this work. I am thankful for helpful comments on the preparation of this manuscript provided by Andrew Biewener, Em Standen, Carolyn Eng and anonymous reviewers. This work was funded by the National Science Foundation's Integrative Graduate Education and Research Traineeship (IGERT) program, the Chapman Fellowship and the Department of Organismic and Evolutionary Biology at Harvard.

 
• BL

  body lengths

 
• COM

  center of mass

 
• d_r

  angular displacement

 
• d_t

  translational displacement

 
• EFV

  effective foot velocity

 
• L_f

  foot length

 
• L_fem

  femur length

 
• L_tars

  proximal tarsal length

 
• L_tib

  tibia—fibula length

 
• tmt

  tarsometatarsal

 
• v_l

  lateral translational velocity

 
• v_r

  rotational velocity

 
• v_t

  translational velocity

 
• θ_ankle

  ankle angle

 
• θ_f

  foot angle

 
• θ_hip

  hip angle

 
• θ_knee

  knee angle

 
• θ_tmt

  tarsometatarsal angle