Rodger Kram and Thomas Roberts discuss the impact of A. V. Hill's classic paper ‘The dimensions of animals and their muscular dynamics’, published in Science Progress in 1950.

Rodger Kram and Thomas Roberts discuss the impact of A. V. Hill's classic paper ‘The dimensions of animals and their muscular dynamics’, published in Science Progress in 1950.

It is probably unprecedented that an after-dinner speech for the general public is converted into a review article that, 65 years later, is still on an exponential growth curve of citations in the primary scientific literature. But then again, Archibald Vivian Hill was not your average after-dinner speaker. Hill shared a Nobel Prize in 1922 for his discoveries related to muscle heat production and efficiency, and his subsequent contributions were arguably even more important.

In November 1949, Hill presented ‘The Dimensions of Animals and their Muscular Dynamics’ as an Evening Discourse at the Royal Institution in London – a monthly lecture that continues to this day – and then published it in the journal Science Progress (Hill, 1950). Much of our understanding of the energetics and biomechanics of muscle is founded on Hill's lifetime of rigorous experimental work (Hill, 1970). But, in his 1950 article, Hill steps away from the bench and provides armchair speculation about how the properties of muscles, and simple scaling rules, shape the ways animals move. The topics covered in this paper range entertainingly from the contraction speed of kitten eye-blinking muscles, to treadmill-trotting terriers who like candy, to the heart beats of blue whales.

Straight from the horse's mouth, the paper begins with a brief review of fundamental muscle mechanics and energetics. Hill focuses on the observations that force, power and efficiency of a contraction depend on how quickly a muscle shortens. At fast shortening speeds, muscles are weak and inefficient. Hill notes that those muscle properties ultimately limit movement speed. The relationships between force, power, efficiency and speed of contraction are now central to the ‘Hill-type’ models that are implemented in modern simulations of muscle function during movement (e.g. Delp et al., 2007). However, much has been learned about how the designs of skeletal levers, muscle architectural features and elastic elements allow for muscle shortening speeds that are favorable for performance (Alexander, 2006).

Hill's article utilizes bicycles as a familiar illustration of muscle mechanics. Noting that normal cycling is most efficient at a cadence of ∼75 crank revolutions per minute, he predicts that power output would be maximized at nearly twice that cadence. Further, he projects that longer cranks would reduce the energetically optimal pedaling cadence. On the whole, those predictions have held true (e.g. McDaniel et al., 2002).

Hill's arguments build upon the idea that the force exerted by skeletal muscle per square centimeter of cross-sectional area (specific tension) is roughly constant between muscles and between different species. Today, we know that this is generally accurate among vertebrate skeletal muscles, but not for invertebrates. Hill's misassumption is understandable because the structural basis for variation in specific tension only arose in 1954 with the sliding filament theory for muscle force generation (Huxley and Niedergerke, 1954; Huxley and Hanson, 1954). Crab claw muscles, for example, can generate a force per cross-sectional area that is approximately five times that of vertebrate muscles, because their myosin filaments are  approximately five times as long (Taylor, 2000).

The crux thesis statement of Hill (1950) is that ‘dimensional reasoning … shows that similar animals of different size should be able to run or swim at the same linear speed and jump the same height’. He then devotes six pages to a recapitulation of the Gray's paradox (Gray, 1936) regarding dolphin swimming speed. Gray's calculations, based on anecdotal reports, seemed to violate what was known about hydrodynamics and physiological energy supply. In 1950, that idea still held water, but under modern scrutiny, the paradox has dissipated (Fish, 2005).

Regarding running, Hill provides likely reliable maximum running speeds for two dog breeds (whippets and greyhounds) and for racehorses, and notes that they differ by less than 25% despite the ∼50-fold difference in body mass. Hill then highlights the mechanical power needed to reciprocate the limbs during the swing phase. His dimensional reasoning leads to the conclusion that the mechanical work required to swing the limbs calculated per stride per kilogram body mass is independent of body size. He then considers the work required to overcome air resistance, comes to the same conclusion and seems to feel the issue is settled. However, our doctoral mentors, Dick Taylor and Tom McMahon, along with their colleagues, experimentally challenged Hill's reasoning in the 1980s. They found that neither the mechanical power for reciprocating the legs nor overcoming air resistance comprise a major portion of the overall mechanical power involved in running (Fedak et al., 1982). During each stride, lifting and decelerating/accelerating the center of mass of the whole body requires far more mechanical power (Heglund et al., 1982). A comprehensive collation of maximum speed reports for a wide variety of running animals (Garland, 1983) indicates that among mammals, maximum running speed scales with the 0.17 power of body mass, M0.17, but the most massive mammals (elephants) are not the fastest. A polynomial regression indicates that there is an optimum body size of ∼119 kg for a generic mammal, about twice the mass of cheetahs, the world's fastest terrestrial runners. Hill's observations about animal running speeds remain true, but neither his nor any ensuing explanation is entirely satisfying.

Hill's predictions that all animals should be able to jump to the same absolute maximum heights or horizontal distances regardless of body size have also not been supported by subsequent investigation. His calculations are well-reasoned, but are built on assumptions that have not held up. Hill assumes that: (1) maximum muscle shortening speed, measured in muscle lengths per second (L s−1), should scale with M−1/3, and (2) jumps are powered directly by muscle. We now know that maximum muscle speed (in L s−1) scales with ∼M−0.10 (Seow and Ford, 1991), though we still don't really know why. Further, many specialized jumpers escape the speed limits imposed by muscle properties using elastic catapult-like mechanisms. Flea jumps provide a good example. Hill credits their prodigious jumping performance to extremely high muscle shortening speeds. However, 15 years later, Bennet-Clark and Lucey (1967) showed that fleas actually jump by slowly contracting muscles, storing mechanical energy in elastic elements and then releasing the stored energy explosively to power the jump. This topic still ‘has legs’; the exact biomechanics of flea jumping were refined recently (Sutton and Burrows, 2011).

Specialized vertebrate jumpers also harness elastic mechanisms to escape muscle power limits during jumping. Bushbabies produce power outputs far in excess of their muscle capacity during jumping (Aerts, 1998), as do many frogs (Marsh and John-Alder, 1994). Among frog species, there are only small differences in the muscle properties between spectacular jumpers and the most pedestrian performers (Roberts et al., 2011). Elastic mechanisms weaken the link between muscle performance and locomotor performance in ways that undermine Hill's specific calculations, but arguably support his central thesis.

Both of our careers have been strongly influenced by this classic paper (Hill, 1950). Although the emphasis of Hill's paper was muscle shortening, he notes that generating isometric tension with muscle fibers that have faster intrinsic velocities is much more expensive. That is the basis for the idea that the cost of force generation is central to the whole-animal metabolic cost of running (Kram and Taylor, 1990). Later in his 1950 paper, Hill dismisses the importance of isometric muscle actions in animal movement: ‘maintained contractions are of special importance in connection with the statics, as contrasted with the dynamics, of animals’. That reasoning was sound in 1950 when tendons were thought akin to rigid struts. However, today we know that in vertebrate runners, tendons act as springs that allow muscles to act as nearly isometric force generators (Roberts et al., 1997).

Many scientists have cited Hill's 1950 paper because his statements are bold and likely also because the chance to point out where a Nobel laureate went wrong tempts the best of us. Regardless of the motivation, we like to think that A.V. Hill would have been pleased by the ensuing decades of scientific activity and critiques stimulated by his paper. Bernard Katz, Hill's PhD student, biographer and an outstanding scientist in his own right, recalled that Hill ‘…was unrelentingly critical and never ceased to probe for flaws in his own methods and deductions’ (Katz, 1978). Further, Katz noted that ‘…A.V. loved to make provocative statements and did not mind “sticking his neck out”, sometimes to inflict the punishment on himself later on! … he regarded errors in interpretation and the “built-in obsolescence” of scientific theories as an inevitable and indeed necessary part of progress, and the more provocative the statement, the better it was in jollying progress along’ (Katz, 1978). With regards to locomotion energetics and biomechanics, Hill (1950) has surely jollied many of us along.

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