Variability of Structural and Biomechanical Parameters of Pelophylax Esculentus (Amphibia, Anura) Limb Bones

Abstract Variability of Structural and Biomechanical Prameters of Pelophylax esculentus (Amphibia, Anura) Limb Bones. Broshko Ye. O. — Structural and biomechanical parameters of Edible Frog, Pelophylax esculentus (Linnaeus, 1758), limb bones, namely, mass, linear dimensions, parameters of the shaft’s cross-sectional shape (cross-sectional area, moments of inertia, radiuses of inertia) were investigated. Some coefficients were also estimated: diameters ratio (df/ds), cross-sectional index (ik), principal moments of inertia ratio (Imax/Imin.). Coefficients of variation of linear dimensions (11.9-20.0 %) anrelative bone mass (22-35 %) were established. Moments of inertia of various bones are more variable (CV = 41.67-56.35 %) in relation to radii of inertia (CV = 9.68-14.67 %). Shaft’s cross-sectional shape is invariable in all cases. However, there is high individual variability of structural and biomechanical parameters of P. esculentus limb bones. Variability of parameters was limited by the certain range. We suggest the presence of stable norm in bone structure. Stylopodium bones have the primary biomechanical function among the elements of limb skeleton, because their parameters most clearly responsive to changes in body mass.

Several hypotheses on the origin of terrestrial tailless amphibians jumping locomotion are discussed in the literature (Gans, Parsons, 1966;Sukhanov, 1968): 1) jumping as an adaptation to life on land (Inger, 1962), 2) the occurrence of aquatic primary synchronous propulsion of hindlimbs when swimming, which later could also be used for terrestrial locomotion (Griffi ths, 1963), and 3) as an optional method of danger avoiding (Schmalhausen 1964;Gans, Parsons, 1966). Th e primary tailless amphibians in adulthood were hardly fully terrestrial animals. Th ey were of semi-aquatic lifestyle, so both forms of locomotion could evolve in parallel. In addition, hindlimbs of tailless amphibians do not have signifi cant diff erences in the overall scheme of kinematics when moving in the water and on land (Nauwelaerts, Aerts, 2003;Nauwelaerts et al., 2005) except their orientation in the horizontal plane when swimming (Sukhanov, 1968). It should be added that the synchrony and symmetry of the limb movements are the most eff ective way to swim with a shortened trunk and lenghthened limb (Griffi ths, 1963;Sukhanov, 1968).
A ground leap of a frog includes the following phases (Nauwelaerts, Aerts, 2006): 1) propulsion (sudden straightening of hindlimbs), 2) fl ight (with bringing of hindlimbs under the body) 3) landing (amortization on forelimbs), 4) recovery (adoption limbs starting position). Propulsion phase, in turn, comprises two parts (Sukhanov, 1968;Nauwelaerts et al., 2005): 1) extension of the knee joint and 2) extension of ankle and metatarsal joints. At the same time during swimming all the joints act more or less synchronously. Propulsion impact of hindlimbs during a jump is considerably stronger as compared with swimming (Nauwelaerts, Aerts, 2003;Nauwelaerts et al., 2005). Th us, the push force, developed in the fi rst phase of repulsion is 6.5 times higher than that in its second phase (Sukhanov, 1968).
Forelimbs function is in body amortisation in the fi nal phase of the jump. As the surface resistance force when landing on average three times higher than at a push, forelimb bones must also have a suffi cient safety margin (Nauwelaerts, Aerts, 2006).
While jumping and swimming at fi rst glance are quite similar in their kinematics, they are used in environments with diff erent density, and hence with diff erent mechanical impact on various parts of limbs. Th erefore the distribution of mechanical loads on the elements of the limb should be distinguished. Th us when jumping the proximal limb elements are most involved, and when swimming -distal ones (Nauwelaerts et al., 2007). Because of this, various morphological and functional properties of skeletal limbs refl ect adaptation to a particular method of locomotion. Some of them can display adaptation to both types of locomotion, anotheronly to one of them (Nauwelaerts et al., 2007). We assume that the properties of limb bones signifi cant for both types of locomotion should have a relatively high variability in comparison with those that are relevant for only a single type.
As far as swimming and jumping are realized in diff erent environments, "a confl ict of optimization" of locomotion apparatus may occur (Nauwelaerts et al., 2007). Th eoretically extension of variability range of limb bones parameters can lead to the animal transition to only swimming or jumping, and accordingly, the existence of adults in defi nite environment that occur in many species of anurans.
Th e aim of this work is the investigation of individual variability of structural and biomechanical parameters of long bones of edible frog Pelophylax esculentus (Linnaeus, 1758) concerning the changes in body weight, as well as the interrelationship between these characteristics (variability relative to each other). Emphasis is biased towards consideration of adaptations to terrestrial locomotion as in contact with a substrate harder than water limbs skeleton susceptible to mechanical stress more intense than in the aquatic environment.

Material and methods
Skeletal elements of P. esculentus of one population (n = 18), namely: bones of stylopodium -humerus and the femur, zeugopodium -forearm bone (os antebrachii), and tibia (os cruris) were studied. Th e following parameters: bone mass -absolute (m) and the relative body mass (m r ); linear dimensions -length (l), frontal (df) and sagittal (ds) diameters; geometry parameters of mid-diaphyseal cross-section -compact area (Sk), principal (I max , I min ) and polar (J) moments of inertia, radii of inertia (i max , i min ). Additionally such coeffi cients: diameter ratio (df/ds), the index of compacts (ik), the ratio of the principal moments of inertia (Imax/Imin) were calculated. To determine the indices of parameters of the diaphyseal bone cross-section geometry special technique was used.
Preview of the section was applied to the grid, where coordinates of individual points on the outer and inner contours of the section were determined. Th e obtained coordinates were processed with a special computer program (Melnik, Klykov, 1991) for calculation of indicated geometry section parameters (Melnik, Klykov, 1991).
For each of the parameters its average value (M), the rms-parameter deviation (SD), coeffi cients of variation (CV) and the standard deviation of coeffi cient of variation (SCV) by standard methods were calculated.
To determine the allometric dependences of each bone parameters on body weight, as well as the diameters of the bones to their length, the equation of nonlinear regression y = ax b , where x -independent variable, y -separate feature index, a -initial growth constant, b -allometric constant was used (Schmidt-Nielsen, 1987;Melnik, Klykov, 1991).
According to generally accepted theoretical calculations, in isometric of parameters to their body mass their allometric constant will be: for bone mass -1, linear parameters -0.33, cross-sectional area -0.67, the moments of inertia -1.33. If the corresponding allometric constants larger than these values, it is evidence of an indication as to the positive allometry of body weight, if less -about the negative allometry. Calculations of indicators were done by nonlinear regression with the help of the program SigmaPlot 11.0.
We also calculated the coeffi cients of multivariate allometry (A) for the set of equivalent one-dimensional features of all bones (separately for bone mass with body weight, relative bone mass, linear dimensions, crosssectional area, moments of inertia). In this case the estimation of mutual changes in equal parameters is performed. Th e multivariate allometry coeffi cient is the ratio of the fi rst principal component factor load on the sign to the arithmetic mean of all loads. Calculating of main component is based on variance-covariance matrix of the log-transformed values (Jolicoeur, 1963(Jolicoeur, , 1984. Here positive allometry characteristics are shown at A > 1, and negative -for A < 1. Calculating of the multivariate allometry coeffi cient was performed using the program PAST (Hammer et al., 2001).

Results and discussion
The cross-sectional shape of the bones. Cross-sectional shape of the limb bones illustrates their adaptation to loads of diff erent character. Th us, hindlimb bones (femur and shin bone) have a round cross-sectional shape (fi g. 1, C, D), indicating the adaptation of the latter to bending and torsion, and is associated with segmental position of limbs in amphibians. Although the basic movements of the hindlimb are exercised along the sagittal plane, the rotation around the front axis (especially of the femur) is also presented. Рис. 1. Форма поперечного сечения диафиза длинных костей конечностей P. esculentus: А -плечевой; B -кости предплечья; C -бедренной; D -кости голени (d -дорсальный маркер; m -медиальный маркер; I max -ось максимальных моментов инерции; I min -ось минимальных моментов инерции). Cross-sectional shape of the humerus is extended in the frontal plane (fi g. 1, A), indicating more intensive bending loads in the frontal plane due to the damping function of the bone (acting as a lever) in the fi nal phase of the jump.
Th e forearm bone also has features of adaptation to loads in this phase, resulting in a fusion of two primary elements (ulna and radius) with forming a coherent rigid structure (fi g. 1, B).
O v e r a l l p a r a m e t e r s v a r i a b i l i t y o f l i m b b o n e s. Overall in P. esculentus, a rather large variability of the studied parameters (table 1; fi g. 2) is observed. Unfortunately, published data on the variability of structural and biomechanical parameters of bone in tailless amphibians are virtually unknown.
It should be also noted that, according to our data, the parameters of geometry of the diaphysis cross section (especially the moments of inertia) are generally very variable val-  Th e high individual variability of the species and morphological characteristics of the postcranial skeleton of Anura is described in papers of E. E. Kovalenko (2011). Th is author, on the basis of a range of possible options for the structure of the body, marks out its norm -the most common variants. In our case, the individual variability of structural and biomechanical parameters of P. esculentus limb bones is quite high (fi g. 2). In this case the diaphysis cross-sectional shape in all individuals remains constant in accordance with its adaptations to a certain kind of mechanical load.
Based on the foregoing, and works of E. E. Kovalenko in particular, we can speak of a sustainable limb bones building standards in P. esculentus, and the high variability of com- Рис. 2. Коэффициенты вариации (CV) морфометрических и структурно-биомеханических параметров длинных костей конечностей P. esculentus (1 -плечевая; 2 -кость предплечья; 3 -бедренная; 4кость голени). pact area and moments of inertia -is among other things a manifestation of individual morphological features, which, however, do not go beyond the norm, as are compensated by stability form of the diaphysis cross section.
To assess the constant cross-sectional shape compact index and indexes diameter ratio and principal moments of inertia ratio are indicative, showing in this case a relatively small variation, which confi rms the existence of some rules in the structure of bones. Relatively broad scope of "normal structure" is in the forearm bone due to the high variability of its frontal diameter. Accordingly, the higher variability is shown for the ratio of its diameters.
A l l o m e t r i c p a r a m e t e r s d e p e n d e n c e s. Mass and linear dimensions of limb bones have a high correlation with body weight (r > 0.7;

T a b l e 3. Allometric dependencies of bone diameters on bone length Т а б л и ц а 3. Аллометрические зависимости диаметров костей от их длины
parameters of cross section geometry of hindlimb bones. For forelimb bones these options have an average level of correlation (r = 0.5-0.7). In assessing the null hypothesis for the allometric constants is revealed that their values do not exceed the limits of confi dence intervals variation.
Th is can indirectly testify of signs isometric to body weight. But the experience of many researchers shows that, in determining the dependency of the relative parameters growth should be guided by the averages values of allometric constants. Allometric constants of bone parameters relative to body mass indicate the varied nature of the dependency (table 2). Th us, the lengths and diameters of the most bones vary isometrically with the exception of the forearm, where a negative allometry in all respects is observed.
Th e latter indicates that changes in body weight are not determining factor for forearm, since by its sophisticated design, it is quite stable. In other respects these tendencies can be traced according to body weight: positive allometry of bone mass, negative allometry radii of inertia and positive allometry for stylopodium bones compact area.   In the study of allometric dependencies of diaphysis diameters to the length of the humerus bone negative diameters allometry is clearly expressed, for the other bones this dependence is close to isometry (table 3).

T a b l e 4. Coeffi cients of multivariate allometry of bone mass and body mass Т а б л и ц а 4. Коэффициенты многомерной аллометрии массы костей и тела
We conclude that stylopodium play a leading role in the bio-mechanics of the frogs' limbs as the femur and the humerus most likely respond to changes in body weight.
Th is manifests itself in a positive allometry to body weight of these bones lengths and geometry parameters of the humerus section.
Overall, these results are consistent with the available literature data describing the relationship of the studied parameters on the interspecifi c level in mammals. Th us, the linear dimensions of isometry of bone corresponds to the model of geometric similarity (McMahon, 1975). Deviations from geometry parameters isometry of section are described by the elastic similarity model justifying their dependence on the elastic properties of bone (Alexander, 1977). Th is indicates the possibility of using allometry methods for studying the intraspecifi c variation.
Multivariate parameters allometry. Multivariate mass indices allometry (body and separate bones; table 4) shows a signifi cant negative allometry of body mass relative to bone mass (A = 0.715), a small positive allometry for the forelimb bone (1.102-1.119), and isometry for hindlimb bones (0.994-1.071).
Th ese data are consistent with the data of simple allometry, where the bone mass has a positive allometry relative to body weight. Th is is probably indicative of a more intensive increase in mechanical load on the skeleton limbs relative to body weight gain.
For the relative weights of the limbs skeleton elements a tendency toward positive allometry for forelimb bones (1.11-1.217), the negative allometry for hindlimb bones (0.769-0.904; table 5) were recorded. Th us for the tibia bone negative allometry is quite high (0.769).
According to multivariate allometry moments of inertia (table 8) isometry in all bones (0.968-1.096) is observed, only in the forearm it is negative (0.794-0.903).
In general, mutual allometry of homologous parameters of diff erent bones coincides with the results of simple allometry signs to body mass. So, for stylopodium higher relative growth of linear parameters and compact area are also marked, for the forearm -the lower growth of the moments of inertia is noted.

Conclusion
Analyzing the morphometric and structural -biomechanical parameters of skeleton limbs elements of P. esculentus and their variability, we obtain the following generalizations.
Retaining the relative stability of certain parameters on the background of elevated variability of others the Anura limbs skeleton achieved the desired balance of its structure: it does not lose its adaptations to aquatic locomotion and gained stability to excess mechanical load accompanying terrestrial locomotion .
A certain range of variability inherent in diff erent structural and biomechanical parameters of the limbs skeletal elements of P. esculentus.
Meanwhile variability is present in certain limits, without going out the boundary of the extreme values, which suggests to talk about a certain norm of structural organization of the bones associated with both habitat conditions and characteristics of locomotion.
Existing range of variability of separate parameters are considered as saving the potential for the realization of other ways of adaptive evolution in the case of changes in living conditions.