The Physics of Frogs and Muscles

With our liberal arts education here at Colgate, I figured exploring the cross over between my Human Physiology class and our Physics class might be a little fun. In Human Physiology lab a little while ago, we performed an experiment on the gastrocnemius muscle of frogs (This more or less translates to the calf muscle in the back of the lower leg for both us and frogs). In our experiment, we prepped the isolated muscle and set it up to perform isotonic muscle twitches with varying weights. By finding the distance the muscle contracts and knowing the weight, we can easily calculate the work done by the muscle to lift the weights. Then, by plotting a graph of work done for a given weight lifted, I was able to obtain an optimal load at which the muscle does the greatest amount of work.

Accounting for the forces in play, we know that the force of gravity (mg) is determined by the varying weights we add to the muscle. Furthermore, we know that a force applied comes into play when the muscle contracts (and does work) to lift the weight. In order to simplify calculations, we ignored other forces besides the force of gravity on the weights and force applied. We ignored the weight of the muscle, and focused solely on the work done by the muscle on the weights. Our equation W = F * d helps us find the work done by the muscle on the weight.

My raw data for the lab was:

Weight (g)	Work	Height Lifted (mm)	Height (mv)
10	3.52725	0.352725	4.703
9	3.4803	0.3867	5.156
8	9.7782	1.222275	16.297
7	13.86368	1.980525	26.407
6	22.56345	3.760575	50.141
5	24.04088	4.808175	64.109
4	21.9609	5.490225	73.203
3	18.39974	6.133245	81.7766
2	13.95465	6.977325	93.031
1	7.6137	7.6137	101.516

Since we were stimulating the muscle at varying voltages, and similarly measuring the peak of the contraction in mV, I had to use a conversion factor to obtain the shortening distance (height lifted) for the muscle. This conversion factor was given as 75 mm / V. Also, to keep calculations simple in lab, we initially ignored gravity (as it multiples every mass by 9.8 equally), and considered the force of gravity = m (weight). This means that the force of contraction must overcome that downward force in order to have the weights move upward, and thus do work. To make this all a little more clear, I converted everything in SI units and corrected the force downward (and thus minimum upward) by multiplying gravity in.

Weight (kg)	Min. Force Exerted Upwards (N)	Distance (m)	Work Done (J)
0.01	0.098	3.53E-04	3.46E-05
0.009	0.0882	3.87E-04	3.41E-05
0.008	0.0784	1.22E-03	9.58E-05
0.007	0.0686	1.98E-03	1.36E-04
0.006	0.0588	3.76E-03	2.21E-04
0.005	0.049	4.81E-03	2.36E-04
0.004	0.0392	5.49E-03	2.15E-04
0.003	0.0294	6.13E-03	1.80E-04
0.002	0.0196	6.98E-03	1.37E-04
0.001	0.0098	7.61E-03	7.46E-05

Now, having everything in SI, I plotted the work vs. weight graph to find the optimal load where this muscle does the greatest amount of work using the numbers above, and found that this frog's gastrocnemius muscle's optimal load was 0.005kg, or 5g.


Let's take this a step further and calculate the power of this muscle.
Knowing power is the rate at which work is done, to find our average power of this muscle we can use the equation:
P = W / t.
For our isotonic muscle twitches, a contraction cycle lasted approx. 2 ms (variations occurred with changing weights, changing stimuli strength, how fatigued the muscle was, etc., but for simplicity's sake, 2 ms is a reasonable assumption).

At an optimal load of 5g (0.005kg), our muscle did 2.36E-4 Joules of work. Since 2 ms is 0.002 seconds, we simply divide to find that the frog's gastrocnemius muscle had an average power output of 0.118 W (J/s).

That's nice and all, but since I like to be self-centered now and then, how does that relate to us (and since the course is Human Physiology, how does it relate to humans?) Doing a little bit of research, I found the optimal power outputs of human gastrocnemius muscles at a walk and at a run in this article:
http://www.pnas.org/content/early/2012/01/04/1107972109.full.pdf

Granted they do some very interesting research and measure a variety of things, including changes in length for different parts of the muscle and power among different groupings, I used their data for muscle-tendon units to compare to my frog's gastrocnemius muscle power.

At a walk, the power output is around 37 W, and while running, our power output increases to around 48W.

Where frogs have a gastrocnemius muscle composed of fast twitch glycolytic muscle fibers that are useful for jumping quickly and moving only their small bodies, a power output of 0.118W makes sense. On the other hand, human gastrocnemius muscles contain a variety of fast twitch glyocolytic, fast twitch glycolytic oxidative, and slow twitch oxidative fibers. From an evolution standpoint, we need these variety of fibers to enable us to quickly run and jump (explosive movements ~ fast twitch fibers), but also maintain a standing posture throughout the day (long, gradual movements ~ slow twitch fibers). In order to maintain the work throughout the day, and to move our much larger masses, it makes sense that our power outputs are much higher than that of a frog's.







Farris DJ, Sawicki GS. Human medial gastrocnemius force-velocity behavior shifts
     with locomotion speed and gait. Proc Natl Acad Sci U S A. 2012;109:977-982. Found at  
     <http://www.pnas.org/content/early/2012/01/04/1107972109.full.pdf>.