1. Prefatory remarks


  2. Allometric scaling
    1. Physicists know it as dimensional analysis, in the form described in Percy Bridgeman's (1922) book.
      Examples:
    2. water waves: v µ (gl)½

    3. pendulum frequency: t µ ( l/g )½

    4. Impossibility of certain events

    5. Galileo, Two New Sciences (1638): there are no giants because
    6. Borelli, De Motu Animalium (1680-81): all animals
      jump the same height    

    7. Cautionary tales
      1. D'Arcy Thompson On Growth and Form (1917):
        1. motility is most efficient at pendulum frequency— Wrong!
        2. we increase our walking speed by increasing our stride length, or v  ­  s :     — Dead Wrong!

      2. Kleiber's Law B µM¾ (would have expected M2/3)


  3. R.M. Alexander: Froude number j =  v² / (gl) as scaling variable
    1. why should s/l scale as j ½ ?



    2. Alexander's empirical fit and what it means.


  4. Nature of muscle tissue:
    C.J. Pennycuick Newton Rules Biology (Oxford, 1992—out of print);
    D.J. Aidley, The Physiology of Excitable Cells (Cambridge, 1998)

    1. Striated muscle
      Each cross-bridge exerts 5.3´10-12 Nt over a distance of about 40 Å,
      giving DE » 0.13 eV per bond. This is about 5 kBT, as it better be!

    2. A.V. Hill's empirical relations for muscle:

              power vs. strain rate y = v / l


  5. Energy consumption in locomotion
    1. Theoretical considerations:
      There is energy loss µ v² at each walking step, and the
      frequency of stepping is µ v.

      Conversely, the force at each running step is constant, µ mg ; hence the specific power for each gait is
            
      where L is a constant.

    2. Do theoretical results make sense?
      • Energy per unit mass consumed in walking increases like v².     
        (But note effect of optimum strain rate y.)

      • Margaria, et al. (1963) measured O2 consumption of athletes on a
        treadmill. Since O2 and (food)-energy consumption are 1-1, we see
        that theory is verified by the data.

    3. Walk ® run transition is effected by need to economize the way power
      increases with speed. At some speed the linear increase in running beats the
      cubic increase in walking.

      The virtue of running is you increase your stride
      length s while keeping constant your stepping frequency f.


  6. Running out of time
    1. Why mice don't juggle: The time to get neural impulses to/from their muscles scales
      as l, whereas the falling time scales as l½. But the time brain centers require to process
      visual information is the same as for us. This means mouse effective reflex time is 50–75%
      of human, but available time is < 20% . They've run out of time.

    2. Empirics of stepping frequency vs. leg length:  f µ l–½

      • Is stepping frequency the same as pendulum frequency? (NOOO! —see table again)
        IMHO, the whole pendulum argument is Post hoc, ergo propter hoc.

      • In that case, why is  f µ l–½ ? IMO it is like Kleiber's Law, arising from a compromise
        —in this case, between strength/weight ratio and need for speed.

    3. Time in the air vs. stepping period t = 1/f:
             
      This means that at some size an animal will not have enough time to cycle its legs while it is
      off the ground. Max. air time is 0.4–0.7 seconds. This means a stepping frequency of at least
      1.4 Hz is needed. According to Pennycuick, then, the longest-legged galloping animal will
      have legs about 2 ±0.4 m. Thus we get an upper limit of 2.4 m for an animal to be able to run.

      Taller animals have also run out of time.


    4. Applications