to the scaling of an animal and can be overcome by an individual's body design. the metabolic scope for an animal is the ratio of resting and maximum rate of metabolism for that particular species as determined by oxygen consumption. oxygen consumption vo2 and maximum oxygen consumption vo2 max. oxygen consumption in species that differ in body size and organ system dimensions show a similarity in their charted vo2 distributions indicating that, despite the complexity of their systems, there is a power law dependence of similarity; therefore, universal patterns are observed in diverse animal taxonomy. across a broad range of species, allometric relations are not necessarily linear on a log-log scale. for example, the maximal running speeds of mammals show a complicated relationship with body mass, and the fastest sprinters are of intermediate body size. allometric muscle characteristics the muscle characteristics of animals are similar in a wide range of animal sizes, though muscle sizes and shapes can and often do vary depending on environmental constraints placed on them. the muscle tissue itself maintains its contractile characteristics and does not vary depending on the size of the animal. physiological scaling in muscles affects the number of muscle fibers and their intrinsic speed to determine the maximum power and efficiency of movement in a given animal. the speed of muscle recruitment varies roughly in inverse proportion to the cube root of the animal's weight (compare the intrinsic frequency of the sparrow's flight muscle to that of a stork). f r e q u e n c y = 1 m a s s 1 / 3 = ^ for inter-species allometric relations related to such ecological variables as maximal reproduction rate, attempts have been made to explain scaling within the context of dynamic energy budget theory and the metabolic theory of ecology. however, such ideas have been less successful. methods of study allometry has been used to study patterns in locomotive principles across a broad range of species. such research has been done in pursuit of a better understanding of animal locomotion, including the factors that different gaits seek to optimize. allometric trends observed in extant animals have even been combined with evolutionary algorithms to form realistic hypotheses concerning the locomotive patterns of extinct species. these studies have been made possible by the remarkable similarities among disparate species' locomotive kinematics and dynamics, "despite differences in morphology and size". allometric study of locomotion involves the analysis of the relative sizes, masses, and limb structures of similarly shaped animals and how these features affect their movements at different speeds. patterns are identified based on dimensionless froude numbers, which incorporate measures of animals' leg lengths, speed or stride frequency, and weight. alexander incorporates froude-number analysis into his "dynamic similarity hypothesis" of gait patterns. dynamically similar gaits are those between which there are constant coefficients that can relate linear dimensions, time intervals, and forces. in other words, given a mathematical description of gait a and these three coefficients, one could produce gait b, and vice versa. the hypothesis itself is as follows: "animals of different sizes tend to move in dynamically similar fashion whenever the ratio of their speed allows it." while the dynamic similarity hypothesis may not be a truly unifying principle of animal gait patterns, it is a remarkably accurate heuristic. it has also been shown that living organisms of all shapes and sizes utilize spring mechanisms in their locomotive systems, probably in order to minimize the energy cost of locomotion. the allometric study of these systems has fostered a better understanding of why spring mechanisms are so common, how limb compliance varies with body size and speed, and how these mechanisms affect general limb kinematics and dynamics. principles of legged locomotion identified through allometry alexander found that animals of different sizes and masses traveling with the same froude number consistently exhibit similar gait patterns. duty factors—percentages of a stride during which a foot maintains contact with the ground—remain relatively constant for different animals moving with the same froude number. the dynamic similarity hypothesis states that "animals of different sizes tend to move in dynamically similar fashion whenever the ratio of their speed allows it". body mass has even more of an effect than speed on limb dynamics. leg stiffness, k leg = peak force peak displacement = , is proportional to m 0.67 , where m is body mass. peak force experienced throughout a stride is proportional to m 0.97 . the amount by which a leg shortens during a stride (i.e. its peak displacement) is proportional to m 0.30 . the angle swept by a leg during a stride is proportional to m − 0.034 . the mass-specific work rate of a limb is proportional to m 0.11 . drug dose scaling the physiological effect of drugs and other substances in many cases scales allometrically. for example, plasma concentration of carotenoids scales to the three-quarter power of mass in nine predatory and scavenger raptor species. west, brown, and enquist in 1997 derived a hydrodynamic theory to explain the universal fact that metabolic rate scales as the 3⁄4 power with body weight. they also showed why lifespan scales as the +1⁄4 power and heart rate as the -1⁄4 power. blood flow (+3⁄4) and resistance (-3⁄4) scale in the same way, leading to blood pressure being constant across species. hu and hayton in 2001 discussed whether the basal metabolic rate scale is a 2⁄3 or 3⁄4 power of body mass. the exponent of 3⁄4 might be used for substances that are eliminated mainly by metabolism, or by metabolism and excretion combined, while 2⁄3 might apply for drugs that are eliminated mainly by renal excretion. an online allometric scaler of drug doses based on the above work is available. the us food and drug administration (fda) published guidance in 2005 giving a flow chart that presents the decisions and calculations used to generate the maximum recommended starting dose in drug clinical trials from animal data. allometric scaling in fluid locomotion the mass and density of an organism have a large effect on the organism's locomotion through a fluid. for example, a tiny organism uses flagella and can effectively move through a fluid it is suspended in, while on the other end of the scale, a blue whale is much more massive and dense relative to the viscosity of the fluid compared to a bacterium in the same medium. the way in which the fluid interacts with the external boundaries of the organism is important with locomotion through the fluid. for streamlined swimmers, the resistance or drag determines the performance of the organism. this drag or resistance can be seen in two distinct flow patterns: laminar flow, where the fluid is relatively uninterrupted after the organism moves through it, and turbulent flow, where the fluid moves roughly around an organism, creating vortices that absorb energy from the propulsion or momentum of the organism. scaling also affects locomotion through a fluid because of the energy needed to propel an organism and keep up velocity through momentum. the rate of oxygen consumption per gram body size decreases consistently with increasing body size. in general, smaller, more streamlined organisms create laminar flow (r < 0.5x106), whereas larger, less streamlined organisms produce turbulent flow (r > 2.0×106). also, increase in velocity (v) increases turbulence, which can be proved using the reynolds equation. in nature however, organisms such as a 6-foot-6-inch (1.98 m) dolphin moving at 15 knots does not have the appropriate reynolds numbers for laminar flow (r = 107), but exhibit it in nature. g. a. steven observed and documented dolphins moving at 15 knots alongside his ship leaving a single trail of light when phosphorescent activity in the sea was high. the factors that contribute are: the surface area of the organism and its effect on the fluid in which the organism lives. the velocity of an organism through fluid, which changes the dynamic of the flow around that organism – the shape of the organism becomes more important for laminar flow as velocity increases. the density and viscosity of the fluid. the length of the organism, as the surface area of just the front 2/3 of the organism has an effect on the drag. the resistance to the motion of an approximately stream-lined solid through a fluid can be expressed by the formula: cfρ(total surface)v2/2, where: v = velocity ρ = density of fluid cf = 1.33r − 1 (laminar flow) r = reynolds number the reynolds number r is given by r = vl/ν, where: v = velocity l = axial length of organism ν = kinematic viscosity (viscosity/density) notable reynolds numbers: r < 0.5×106 = laminar flow threshold r > 2.0×106 = turbulent flow threshold scaling also has an effect on the performance of organisms in fluid. this is extremely important for marine mammals and other marine organisms that rely on atmospheric oxygen for respiration and survival. this can affect how fast an organism can propel itself efficiently or how long and deep it can dive. heart mass and lung volume are important in determining how scaling can affect metabolic function and efficiency. aquatic mammals, like other mammals, have the same size heart proportional to their bodies. in general, mammals have hearts about 0.6% of their total body mass: heart weight = 0.006 m 1.0 =0.006^ , where m is the body mass of the individual. lung volume is also directly related to body mass in mammals (slope = 1.02). the lung has a volume of 63 ml for every kg of body mass, with the tidal volume at rest being 1/10 the lung volume. in addition, respiration costs with respect to oxygen consumption is scaled in the order of m 0.75 . this shows that mammals, regardless of size, have similarly scaled respiratory and cardiovascular systems and the same relative amount of blood: about 5.5% of body mass. this means that for similarly designed marine mammals, a larger individual can travel more efficiently, as it takes the same effort to move one body length. for example, large whales can migrate far distance in the oceans and not stop for rest. it is metabolically less expensive to be larger in body size. this goes for terrestrial and flying animals as well: smaller animals consume more oxygen per unit body mass than larger ones. the metabolic advantage in larger animals makes it possible for larger marine mammals to dive for longer durations of time than their smaller counterparts. that the heart rate is lower means that larger animals can carry more blood, which carries more oxygen. in conjuncture with the fact that mammals reparation costs scales in the order of m 0.75 , this shows having a larger body mass can be advantageous. more simply, a larger whale can hold more oxygen and at the same time demand less metabolically than a smaller whale. traveling long distances and deep dives are a combination of good stamina and also moving an efficient speed and in an efficient way to create laminar flow, reducing drag and turbulence. in sea water as the fluid, it traveling long distances in large mammals, such as whales, is facilitated by their neutral buoyancy and have their mass completely supported by the density of the sea water. on land, animals have to expend a portion of their energy during locomotion to fight the effects of gravity. flying organisms such as birds are also considered as moving through a fluid. in scaling birds of similar shape, it has also been seen that larger individuals have less metabolic costs per kg, as expected. birds also have a variance in wing beat frequency. beyond the compensation of larger wings per unit body mass, larger birds also have slower wing beat frequencies, allowing them to fly at higher altitudes, longer distances, and faster absolute speeds than smaller birds. because of the dynamics of lift-based locomotion and the fluid dynamics, birds have a u-shaped curve for metabolic cost and velocity. because flight, in air as the fluid, is metabolically more costly at the lowest and the highest velocities. on the other end, small organisms such as insects can make gain advantage from the viscosity of the fluid (air) that they are moving in. a wing-beat timed perfectly can effectively uptake energy from the previous stroke