Size-Dependence of Water-Running Ability in Basilisk Lizards (Basiliscus Basiliscus) (2024)

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1

Department of Organismic and Evolutionary Biology

2

Division of Applied Sciences

,

Harvard University

,

Cambridge, MA 02138, USA

*

Present address: McKinsey and Company, Inc., Two Crossroads Drive, Bedminster, NJ 07921, USA (e-mail: james–glasheen@mcKinsey.com).

Accepted:

14 Aug 1996

Fig. 1.

Gross morphological measurements made on live Basiliscus basiliscus lizards. The relationship between snout–vent length (SVL in m) and body mass (M_b in g) of live lizards was then used to estimate the live body masses of museum specimens (SVL=0.0317M_b^0.332, r=0.972, P_slope<0.0001, P_y-int<0.0001; 41 measurements).

Fig. 1.

Gross morphological measurements made on live Basiliscus basiliscus lizards. The relationship between snout–vent length (SVL in m) and body mass (M_b in g) of live lizards was then used to estimate the live body masses of museum specimens (SVL=0.0317M_b^0.332, r=0.972, P_slope<0.0001, P_y-int<0.0001; 41 measurements).

Fig. 2.

(A) The missile apparatus. A foot model was attached directly to a transducer, which was connected to a weight (W) and a long rod. In impact experiments, the transducer was an accelerometer, while in cavity-drag experiments a force transducer was used. The missile was connected via fishing line to a pulley mounted on a rotary optical encoder. (B) A diagrammatic force record. When a foot model drops into water, a transient, high-amplitude force impulse is generated as water in the vicinity of the foot model is accelerated from rest to the foot model’s speed. Following impact, an air cavity, which extends up from the foot model and is open to atmospheric air, is expanded by the downward movement of the foot model. Drag forces rise during this period due to increasing pressure differences between atmospheric pressure in the air cavity and the hydrostatic pressure encountered by the foot model. Finally, the air cavity seals below the water surface, resulting in high-amplitude force oscillations. Cavity-drag analysis begins only after impact forces have been dissipated and ends immediately before the cavity seals. T_seal is the period between impact and cavity closure.

Fig. 2.

(A) The missile apparatus. A foot model was attached directly to a transducer, which was connected to a weight (W) and a long rod. In impact experiments, the transducer was an accelerometer, while in cavity-drag experiments a force transducer was used. The missile was connected via fishing line to a pulley mounted on a rotary optical encoder. (B) A diagrammatic force record. When a foot model drops into water, a transient, high-amplitude force impulse is generated as water in the vicinity of the foot model is accelerated from rest to the foot model’s speed. Following impact, an air cavity, which extends up from the foot model and is open to atmospheric air, is expanded by the downward movement of the foot model. Drag forces rise during this period due to increasing pressure differences between atmospheric pressure in the air cavity and the hydrostatic pressure encountered by the foot model. Finally, the air cavity seals below the water surface, resulting in high-amplitude force oscillations. Cavity-drag analysis begins only after impact forces have been dissipated and ends immediately before the cavity seals. T_seal is the period between impact and cavity closure.

In experiments designed to measure the drag force as the foot model continued downwards after the impact, the accelerometer was replaced by a force transducer (model ELF-TC500-20, Entran Devices, Fairfield, NJ, USA). Following the force peak associated with impact, the drag force rose as the foot model descended at a nearly constant velocity (the change in model-foot velocity was less than 5 % of the root-mean squared velocity of the foot model; Fig. 2B). A high-speed video camera (NAC, HSV-200, Tokyo, Japan; 500 fields s⁻¹) recorded the depth of the foot model during the continuing downward stroke after impact. During this time (the cavity-drag phase), an air-filled cavity extended from the foot model to the surface. The drag was measured up to the moment at which the cavity collapsed and was sealed off from the atmospheric air. A water-entry drag coefficient (C_D^*) was defined as:

where D(t) is the time-varying drag, S is the foot area, u is the velocity, g is the gravitational acceleration, and h(t) is the time-varying depth of the foot model below the water surface. For a flat, circular disk, C_D^* has been shown to be constant, regardless of size, speed or cavity depth within the parameter ranges appropriate for basilisk water-running (Glasheen and McMahon, 1996a). In the present experiments, C_D^* had a (slightly) different constant value for each foot. T_seal was obtained by measuring the period between impact and cavity closure (Fig. 2B). Further details regarding measurement of the impact and drag forces as well as the timing of cavity collapse are explained in Glasheen and McMahon (1996a). A method for estimating the relative force contribution of surface tension is presented in Appendix A.

We developed a hydrodynamic model which uses results from our previous study of the dynamics of flat, circular disks entering the water (Glasheen and McMahon, 1996a) to predict several observable measures of how lizards of different body sizes run on water. The model consists of three phases: (1) slap; (2) stroke; and (3) protraction. The maximum slap impulse that a lizard could produce was estimated using the equation:

The allometry of the effective disk radius, r_eff, was determined from the hydrodynamic experiments using the 10 model feet. Analysis of the video records showed that the peak downward velocity u_peak was extremely variable, even for a single lizard, and was not a regular function of body size. To fix the parameters of our model, a value of 3.75 m s⁻¹ was taken as a reasonable upper bound for u_peak in lizards of all body sizes. Once r_eff had been determined for a lizard of a given body size, the maximum possible slap impulse was calculated using equation 3, taking the upper-bound value of 3.75 m s⁻¹ for u_peak. In addition to calculating the maximum possible slap impulse, we also calculated the slap impulse for a given step by using the actual u_peak measured for that step.

A model for the maximum stroke impulse was also developed. The model assumes a vertical downward stroke at a constant velocity (u_rms) to a depth equal to that of the lizard’s leg (L_L), while keeping the foot orthogonal to the direction of travel throughout the entire stroke. The stroke impulse may be defined as the product of the average drag force and the time over which the drag force is applied:

The total drag force is due to two sources of a pressure differential across the foot. First, there is a constant drag due to the inertia of the fluid (0.5C_D^*Sαu_rms²). Second, as depth increases, the pressure difference between the hydrostatic pressure at a particular level in the water and the atmospheric pressure in the air cavity increases. Thus, the average drag due to hydrostatic pressure is 0.5C_D^*SαgL_L. The time over which the stroke force is applied is L_L/u_rms. To obtain a general model for how the stroke impulse varies with size, allometric relationships were substituted for C_D^*, S and L_L. Kinematic analysis of the video images provided an estimate of the greatest value for u_rms in all runs of all animals (2.5 m s⁻¹ ). Estimates of maximum stroke impulse contributions obtained using equation 4 with parameters determined as explained above agree well with detailed measurements in which the variation of foot angle and foot speed are taken into account (see Appendix B).

Finally, in addition to estimating the upper bounds for the maximum slap and stroke impulses, we developed a model which predicts the upward impulse needed to maintain the center of mass at a constant height. We assume that downward drag forces produced during foot protraction are negligible, because our video records show that the lizard protracts its foot within the air cavity. Protraction drag is further minimized by the feathering of the lateral toe fringes and the plantar flexion of the foot such that the foot’s long axis is parallel to the direction of travel. Thus, the minimum needed impulse is simply the product of the body weight and the period between steps (T_step):

Analysis of the video records showed that stride frequencies between 5 and 10 Hz were used by all animals, with no clear dependence on body mass. Therefore, for the model calculations, we assumed an upper bound of 10 Hz on f_str, independent of M_b. Since the video records show that there is rarely a time within a stride when either two feet or no feet are in the water, we assumed that the minimum T_step is 0.05 s (T_step=0.5f_str⁻¹). In addition to calculating the minimum needed impulse, we also calculated the actual needed impulse for a given step by using the measured step period for that sequence.

Fig. 3.

Morphological allometry. All measurements are from museum specimens. (A) The distance from the hip to the heel, leg length (L_L), and (B) from the heel to the tip of the fourth toe, foot length (L_F), increase isometrically over nearly two orders of magnitude in body mass (L_L=0.0165M_b^0.338, r=0.986, P_slope<0.0001, P_y-int<0.0001; L_F=0.0164M_b^0.302, r=0.990, P_slope<0.0001, P_y-int<0.0001). The filled symbols in B represent specimens that were selected for detailed foot measurements.

Fig. 3.

Morphological allometry. All measurements are from museum specimens. (A) The distance from the hip to the heel, leg length (L_L), and (B) from the heel to the tip of the fourth toe, foot length (L_F), increase isometrically over nearly two orders of magnitude in body mass (L_L=0.0165M_b^0.338, r=0.986, P_slope<0.0001, P_y-int<0.0001; L_F=0.0164M_b^0.302, r=0.990, P_slope<0.0001, P_y-int<0.0001). The filled symbols in B represent specimens that were selected for detailed foot measurements.

Fig. 4.

Tracings of three of the ten foot models. Detailed foot measurements were used to construct physical models of the lizards’ feet. Impact and drag forces as well as the timing of air-cavity collapse were measured in hydrodynamic experiments using these foot models. Although the shape of each foot model is quite different from that of a disk, calculations based on measurements from disks were able to predict many of the results from hydrodynamic experiments using the foot models (Glasheen and McMahon, 1996a).

Fig. 4.

Tracings of three of the ten foot models. Detailed foot measurements were used to construct physical models of the lizards’ feet. Impact and drag forces as well as the timing of air-cavity collapse were measured in hydrodynamic experiments using these foot models. Although the shape of each foot model is quite different from that of a disk, calculations based on measurements from disks were able to predict many of the results from hydrodynamic experiments using the foot models (Glasheen and McMahon, 1996a).

Fig. 5.

Impact allometry. Measurements obtained from physical models of the lizards’ feet. The effective radius (r_eff) of a lizard’s foot increases less rapidly than L_L or L_F as body mass increases (r_eff=4.37×10⁻³M_b^0.252; r=0.995, P_slope<0.0001, P_y-int<0.0001). r_eff is defined as the radius of a disk that would generate the same relationship between slap impulse and impact speed as that of the foot model.

Fig. 5.

Impact allometry. Measurements obtained from physical models of the lizards’ feet. The effective radius (r_eff) of a lizard’s foot increases less rapidly than L_L or L_F as body mass increases (r_eff=4.37×10⁻³M_b^0.252; r=0.995, P_slope<0.0001, P_y-int<0.0001). r_eff is defined as the radius of a disk that would generate the same relationship between slap impulse and impact speed as that of the foot model.

Fig. 6.

Cavity-drag allometry. For a given speed and depth, the product of the water-entry drag coefficient (C_D^*) and the foot area (S) determines the drag forces during the cavity-drag period for the foot model experiments that produced these results. C_D^* values were determined from equation 2 using measurements of D(t) from the force transducer (Fig. 2B) and measurements of u from the polynomial fits to data from video images showing the vertical position of a foot model versus time; foot area S was determined by weighing the aluminum-sheet foot models. The points (means ± S.D., 20 measurements for each foot model) are compared with a calculated line which is based on the drag coefficient of a disk and the effective foot area calculated from the impact experiments [broken bold line; C_D^*(disk)S_eff=0.703πr_eff²; where r_eff=4.37×10⁻³M_b^0.252]. The solid line is a power fit to the mean C_D^*S data.

Fig. 6.

Cavity-drag allometry. For a given speed and depth, the product of the water-entry drag coefficient (C_D^*) and the foot area (S) determines the drag forces during the cavity-drag period for the foot model experiments that produced these results. C_D^* values were determined from equation 2 using measurements of D(t) from the force transducer (Fig. 2B) and measurements of u from the polynomial fits to data from video images showing the vertical position of a foot model versus time; foot area S was determined by weighing the aluminum-sheet foot models. The points (means ± S.D., 20 measurements for each foot model) are compared with a calculated line which is based on the drag coefficient of a disk and the effective foot area calculated from the impact experiments [broken bold line; C_D^*(disk)S_eff=0.703πr_eff²; where r_eff=4.37×10⁻³M_b^0.252]. The solid line is a power fit to the mean C_D^*S data.

Fig. 7.

Cavity-seal allometry. Larger foot models create cavities which remain open for longer than do the cavities created by the smaller foot models. The period between impact and cavity closure (T_seal) is shown (mean ± S.D.; N=20). The results are compared with a calculated curve (broken line) based on the r_eff derived from the impact experiments and the results from cavity closure experiments with disks [broken bold line; T_seal=2.285(r_eff/g)^0.5; where g is gravitational acceleration and r_eff=4.37×10⁻³M_b^0.252]. The solid light line is a power fit to the mean T_seal data.

Fig. 7.

Cavity-seal allometry. Larger foot models create cavities which remain open for longer than do the cavities created by the smaller foot models. The period between impact and cavity closure (T_seal) is shown (mean ± S.D.; N=20). The results are compared with a calculated curve (broken line) based on the r_eff derived from the impact experiments and the results from cavity closure experiments with disks [broken bold line; T_seal=2.285(r_eff/g)^0.5; where g is gravitational acceleration and r_eff=4.37×10⁻³M_b^0.252]. The solid light line is a power fit to the mean T_seal data.

Fig. 8.

Kinematic allometry. Stride frequency, f_str, is measured as the inverse of the period between water strikes of the same foot of lizards running in a large tank. (A) Larger lizards tended to use a f_str that is above a theoretical minimum (dashed curve). The minimum f_str is defined as the lowest stride frequency which would still allow a lizard to pull its foot out of the water before the air cavity collapses. The assumed maximum stride frequency used in the hydrodynamic model is also shown (dotted line). (B) The peak downward velocity of the midpoint of the foot (u_peak) is extremely variable. The assumed maximum u_peak used in the hydrodynamic model is also shown (dotted line).

Fig. 8.

Kinematic allometry. Stride frequency, f_str, is measured as the inverse of the period between water strikes of the same foot of lizards running in a large tank. (A) Larger lizards tended to use a f_str that is above a theoretical minimum (dashed curve). The minimum f_str is defined as the lowest stride frequency which would still allow a lizard to pull its foot out of the water before the air cavity collapses. The assumed maximum stride frequency used in the hydrodynamic model is also shown (dotted line). (B) The peak downward velocity of the midpoint of the foot (u_peak) is extremely variable. The assumed maximum u_peak used in the hydrodynamic model is also shown (dotted line).

Fig. 9.

The relative contribution of the slap impulse. (A) The percentage of the needed impulse that is generated by the slap impulse varies with body mass. Small lizards display a large range in the degree to which they can use the slap impulse. In contrast, large lizards are constrained to use a relatively small range of slap impulses. The hatched area shows the range of possible slap-impulse contributions for steady locomotion. The vertical distance between the maximum (upper dashed curve) and minimum (lower dashed curve) slap-impulse contribution curves for a lizard of a given body mass (M_b) illustrates the maximum relative impulse surplus (maximum impulse surplus/minimum needed impulse) that a lizard of that size could generate. For instance, the vertical distance between the rightmost vertical arrows indicates that a 200 g lizard has the capacity to generate an 11 % impulse surplus (i.e. 111 % of what is needed to remain at the water surface). In contrast, the distance between the leftmost vertical arrows represents the prediction that a 2 g lizard has the capacity to generate a 125 % force surplus (225 % of the needed impulse). The upper dashed curve represents a prediction from a hydrodynamic model for the maximum slap-impulse contribution (maximum slap impulse/minimum needed impulse). The lower dashed curve was generated using a hydrodynamic model of the minimum slap-impulse contribution required for the lizard to avoid sinking into the water from one step to the next. The minimum slap-impulse contribution model is based on a prediction of the maximum possible stroke-impulse contribution (minimum slap impulse/needed impulse = 1 − maximum stroke impulse/needed impulse). The open circles are calculations of the slap-impulse contributions from individual steps. Three sequences in which lizards generated near-maximal slap-impulse contributions were identified (filled triangles). From these sequences, tracings were drawn of (B) 2.25 g, (C) 25.5 g and (D) 103.7 g lizards at the moment at which their feet reached their maximum depth below the water surface. These tracings demonstrate that, in spite of the fact that all three lizards are generating near-maximal slap-impulse contributions for their respective sizes, larger lizards must stroke more deeply into the water in order to generate the necessary stroke impulse to support their body weight. Tracings are not drawn to the same scale. Filled circles represent data from Glasheen and McMahon (1996b).

Fig. 9.

The relative contribution of the slap impulse. (A) The percentage of the needed impulse that is generated by the slap impulse varies with body mass. Small lizards display a large range in the degree to which they can use the slap impulse. In contrast, large lizards are constrained to use a relatively small range of slap impulses. The hatched area shows the range of possible slap-impulse contributions for steady locomotion. The vertical distance between the maximum (upper dashed curve) and minimum (lower dashed curve) slap-impulse contribution curves for a lizard of a given body mass (M_b) illustrates the maximum relative impulse surplus (maximum impulse surplus/minimum needed impulse) that a lizard of that size could generate. For instance, the vertical distance between the rightmost vertical arrows indicates that a 200 g lizard has the capacity to generate an 11 % impulse surplus (i.e. 111 % of what is needed to remain at the water surface). In contrast, the distance between the leftmost vertical arrows represents the prediction that a 2 g lizard has the capacity to generate a 125 % force surplus (225 % of the needed impulse). The upper dashed curve represents a prediction from a hydrodynamic model for the maximum slap-impulse contribution (maximum slap impulse/minimum needed impulse). The lower dashed curve was generated using a hydrodynamic model of the minimum slap-impulse contribution required for the lizard to avoid sinking into the water from one step to the next. The minimum slap-impulse contribution model is based on a prediction of the maximum possible stroke-impulse contribution (minimum slap impulse/needed impulse = 1 − maximum stroke impulse/needed impulse). The open circles are calculations of the slap-impulse contributions from individual steps. Three sequences in which lizards generated near-maximal slap-impulse contributions were identified (filled triangles). From these sequences, tracings were drawn of (B) 2.25 g, (C) 25.5 g and (D) 103.7 g lizards at the moment at which their feet reached their maximum depth below the water surface. These tracings demonstrate that, in spite of the fact that all three lizards are generating near-maximal slap-impulse contributions for their respective sizes, larger lizards must stroke more deeply into the water in order to generate the necessary stroke impulse to support their body weight. Tracings are not drawn to the same scale. Filled circles represent data from Glasheen and McMahon (1996b).

On a qualitative level, we evaluated the effect of surface tension by adding a small amount of detergent to the water tank in which the lizards ran. Detergent reduces the surface tension at the air–water interface to about half of its usual value (approximately 0.03 Nm⁻¹ as opposed to 0.072 Nm⁻¹) and therefore could reduce the lizards’ ability to support their body weight on the water surface. However, we could not detect any change in performance due to a reduced surface tension. An analysis of the physics of water entry reveals why surface tension plays such a small role in the lizards’ locomotion. To determine the maximal contribution of surface tension to the overall upward force, we calculated the ratio of surface tension force to inertial force for the conditions in which surface tension would play the largest role: the smallest lizard’s foot (r_eff≈0.005 m) moving at the slowest speed (u_rms≈0.5m s⁻¹). The upward force due to surface tension (D_γ) can be approximated as:

The variable γ is the value for surface tension of pure water in air at 25 °C (0.072 N m⁻¹). Excluding unsteady and gravitational effects, the upward force due to the fluid’s inertia is:

where C_D^* is the water-entry drag coefficient (C_D^*≈0.7) and α is the density of water. Thus, the ratio between D_γ and D_steady is:

Thus, if unsteady forces and hydrostatic pressure played no role in the lizards’ locomotion, and the lizards were to support themselves using only dynamic pressure and surface tension, surface tension would still account for only 5 % of the overall support force.

To determine whether our simple model for the stroke impulse (equation 4) is appropriate, we calculated the model’s prediction of the maximum contribution of the stroke impulse to the overall impulse needed. We compared the model’s prediction for a 90 g lizard with detailed measurements from water-running observations on moderately sized lizards (approximately 90 g; Glasheen and McMahon, 1996b). We analyzed only the sequences in which the lizards stroked downwards into the water to their full leg length, to ensure that the detailed measurements were approximating maximum stroke efforts (12 sequences). The contribution from the stroke impulse to supporting body weight was calculated as follows:

The time-varying velocity of the mid-foot u(t) was obtained by fitting a seventh-order polynomial to a plot of foot displacement versus time h(t) and evaluating the derivative. Similarly, the angle of the normal of the foot relative to the vertical ϕ(t) was determined from a seventh-order polynomial fit of foot angle versus time. To calculate the stroke impulse, stroke force was integrated from the time when the lizard’s foot first slapped the water surface until the time at which the mid-foot reached its greatest depth. We found that the results from our simple model (equation 4) agree rather well with the detailed calculations (equation B1); only a slight trend to overpredict the maximum stroke impulse contribution is evident (equation 4 / equation B1 = model stroke impulse contribution / detailed calculations of stroke impulse contribution = 1.06±0.10).