Lifting While Small

Smaller is better

Because I work on a very tiny ant, Pheidole adrianoi, I have become very interested in how such a tiny creature experiences the world.  I have posted two essays on this subject, one on moving through air and one on drinking.  Now I want to deal with what it is like to lift things when one is that tiny (2 mm long, 40 micrograms weight).

I can't count the number of times I have mentioned that I work on ants, and my listener has remarked, it’s amazing how strong ants are!  They can lift xx times their own weight!  I usually respond by saying, yes, and if you were the size of the ant, you could do it too. And it’s true. And if you were an elephant, you would be very impressed with the strength of an ordinary human such as yourself.

The elephant (left) is weaker relative to her size than is the human (right).

It's not really hard to explain why, but there is some high school geometry and math involved. So here goes.  Muscles are built of bundles of contractile fibers, each fiber being more or less an elongated cell, but within the cell are bundles of rigid elongated protein fibers capable of sliding past one another.  Half of these fibers have many lateral projections that can grab the fixed, stiff neighboring fiber to pull itself along by a molecular distance as the projections contract.  This ratchet movement is repeated a few thousand times a second, contracting the muscle.  Upon relaxing again and sliding back to the starting position, it is ready for the next contraction. 

A schematic of the sliding filaments model of muscle contraction. The lateral extension on the central filaments ratchet themselves along the simple filaments. The movement is at the molecular level and requires energy.

All muscles throughout the animal kingdom have more or less the same construction.  They are just machines that can use energy to shorten upon the right stimulation, with each fiber exerting a pretty much fixed contractile force.  This being so, the strength of the whole muscle depends on how many contractile fibers are bundled together.  If you were to cut the muscle in cross-section and count the number of fibers you cut, that count would be tightly related to the amount of force the muscle could exert upon contraction.  This then is Point Number One--- the strength of a muscle is proportional to the cross-sectional area of the muscle (rather than the length).

The obvious corollary is that to make a stronger muscle, you must increase the cross-sectional area and therefore the number of fibers.  The muscles of larger animals, or larger muscles within the same animal, have generally greater cross-sectional areas.  So far, so good.  The cross-sectional area, and consequently the strength, increases with the square of the dimensions, but unfortunately and unavoidably, the mass of the muscle and the mass of the animal that it serves increases with the cube.  This means that the strength of the animal relative to its weight will decrease as it gets larger. There may be exceptional muscles that buck this trend, but taken all together, larger animals are relatively weaker than smaller ones (Point Number Two).

Let's illustrate with short pieces of aluminum rod, one with twice the length and diameter of the other.  Think of the rods as muscles that contract along their lengths. The strength of these imagined muscles would be proportional to their cross-sectional areas while the weight they are moving would be proportional to their volume.  The larger "muscle" thus weighs 8 times the smaller one but has only 4 times the cross-sectional area.   Therefore, adjusting for its size, it is only half as strong.

My imaginary muscles—- pieces of aluminum rod, the one on the left with double the dimensions of the one on the right. Because volume is proportional to the cube of the dimensions whereas cross-sectional area the square, the left “muscle” has eight times the weight but only four times the cross-sectional area.

Let’s do a fun little calculation.  Let’s say you are 2 meters tall and weigh 75 kg.  Because you have been going to the gym regularly, you can lift your own weight.   Pretty good.  Now let’s shrink you down to the size of a Pheidole adrianoi worker: 2 mm long, or 1000th of your original dimensions.  Assuming that your proportions remained the same, your weight (and the weight that your muscles must move) would now be one billionth of your original weight (1000 cubed; 0.075 mg) but the cross-sectional area and thus strength of your muscles would be one millionth of their original.  The ratio of strength to weight would now be 1000.  Because you could lift your own body weight at your original weight, you could now lift 1000 times your new body weight, or 75 mg.  The only thing that has changed is your size.  The rest is all geometry.

The same relationship would apply to our tiny ant, Pheidole adrianoi.  It weighs a bit less than your estimated new weight of 0.075 mg, namely 0.04 mg, but that could be the result of differences in body structure--- you are a human and it is an ant.  Nevertheless, we would predict that it could lift 1000 times its body weight, or 40 mg. I have not tested that it can actually do that, but the point is that at its small size, it is relatively much stronger than a large animal like you, and that your strength would be in the same range if you were as small as the ant. The image below shows as ant carrying a sand grain that is sixty times its own weight.  The ant did not appear to strain to do this, and no doubt can loft much heavier objects.

Left: A Pheidole adrianoi worker carrying a sand grain weighing 60 times her own weight. Right: the grain in the side view allowing measurement.

I smeared a 4 mg piece of brass foil with bullion paste to convince the ants that this object was worth collecting for food. Even though the foil-plus-bullion weighed well over 100 times as much as the forager, she hefted this weight, perhaps awkwardly, at a run and without much apparent effort.

What if we turned the calculation around?  If you were enlarged to the size of a blue whale, how strong would you be?  You would then be 15 times as long and weigh 100 metric tons (100,000 kg) or 1300 times your current weight. However, your strength would have increased only 225-fold, so now you would now be able to lift only 16% of your body weight. That's still 16,000 kg, not a paltry sum, but since you could previously lift your entire body weight, the whale would be pitifully weak compared to you, and astronomically weaker than my humble ant.

This simple example centered on strength reminds us that the world we live in always depends on scale.  The quantitative effects of scale on most aspects of life create worlds that are qualitatively different to such a degree that they are unrecognizable from one scale to the other. The whale cannot imagine what it is like to move the relatively huge objects we move, and we cannot imagine the relatively immense objects that ants heave around with ease. The rules of physics and chemistry that apply depend so strongly on size that they create vastly different experiences of the world.