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#2 Scaling law of locomotion-I

This was the second post in the Chinese version of 3min Biophysics. The suggested reading for this episode is at the end of the text.

To read the Chinese version of this post, please click the following links:
指數律_動物走路和跑步_上: https://goo.gl/9fegCW

Scaling laws are always the most fascinating yet difficult biophysical problems. For example, in our previous episode (#1 The physics of drinking water:https://goo.gl/oX904i), the authors found the lapping frequencies of felines are proportional to the body mass to the power -1/6, which matched the observed power β = -0.181. They have also mentioned [1] that the power is closed to another scaling law discovered by Heglund, Taylor & McMahon, and that is what we are going to talk about in these 2 episodes.

There are different states of locomotion varying with moving speed. For example, the transition from the “walk” state to the “run” state is not merely the change of speed but also involves the change in gait, which suggests that “walk” and “run” rely on different physical mechanisms. During the “walk” state, at least one leg directly contacts with ground at any time, while there may have “double float phase” during the “run” state. There are transitions among different locomotion states in animals, e.g. the transition from “walk” to “trot”, then to the “gallop” state.

Previous physicists postulated that the body size must have set a limit on our locomotion. However, they often focused on the effect of body size on the “maximum speed.” The problem of “maximum speed” is that we never know whether the observed animals “tried their best.” For this reason, a good theory explaining the relations between body size and locomotion was lacking.

Nonetheless, Heglund, Taylor & McMahon, the authors of our suggested reading, discovered another observable parameters in the 1970s. They found that the running speed and stride frequencies during the “walk-to-run” transition were fairly consistent among animals with similar body size, so they recorded the stride length, stride frequency and running speed of various animals during their “walk-to-run” transition states and analyzed how these parameters varied with body size. They discovered 3 scaling law: stride length M^(0.38)stride frequency M^(-0.14)running speed M^(0.24)

*The figure came from the Fig. 1 of the article: Matthis, J. S. & Fajen, B. R. (2013). Humans exploit the biomechanics of bipedal gait during visually guided walking over complex terrain. Proc. R. Soc. B. 280(1762): 20130700. The figure demonstrates the concept of inverse pendulum of human walking.

We have demonstrated how dimension analysis works in our previous episode. We will stop here and invite you to come up with a physical model to explain these 3 scaling laws. Here are some tips that may be useful. Previous scientist have found that during human walking, the change in center of mass resembles an inverted pendulum [2]. What kind of force is required for a pendulum motion? Where does it come from? We hope that you pause and think about it, and we will discuss how T. A. McMahon solved this problem.
*Suggested reading:
N. C. Heglund, C. R. Taylor, T. A. McMahon. Scaling stride frequency and gait to animal size: mice to horses. Science 186(4169), 1112 (1974).
[1] Reis, P. M., Jung, S., Aristoff, J. M., Stocker, R. (2010). How Cats Lap: Water Uptake by Felis catus. Science 330(6008): 1231-1234. 
[2] A. D. Kuo, J. M. Donelan, and A. Ruina. (2005). Energetic Consequences of Walking Like an Inverted Pendulum: Step-to-Step Transitions. Exercise and Sport Sciences Reviews. 33(2): 88-97.