Size and Scaling

Size and Scaling

Giant insects in print and on film

Many movie scripts and book plots have focused on the concept of giant insects roaming about the Earth, causing all sorts of havoc. For example, here is a picture from the web, apparently for a book supplement to a role-playing game,

Look here for a giant insect!

A particularly wonderful movie (tongue rolling around in cheek), apparently set in our very own South Florida, is "Empire of the Ants". In this movie, giant ants take over a sugar mill, in a location similar to the sugar producing region south of Lake Okeechobee, and enslave people to run the mill and produce lots of sugar for them to eat. This movie is very loosely based on the short story of the same name by H. G. Wells (1927), which attempted to use some elements of evolutionary theory to support the idea that someday ants might evolve to be the rulers of humankind. (You may remember that H. G. Wells is the author of The Time Machine and War of the Worlds.)

Let's scale up a hypothetical animal.

We can learn a lot about the physical limitations on size and structure of the body plan by looking at what would happen if we took a small animal, say a small marine invertebrate or an insect, and tried to scale it up to a much larger size. By scaling it up, we mean that we increase each of the three linear dimensions by the same factor, making a larger copy of the small animal. If the scale factor is two, then a leg will become twice as long and twice as wide. The eye will increase in size to twice the diameter of the original eye.

We can understand the basic problem involved by imagining a spherical animal which we scale up to twice its original size, keeping all the proportions the same:

Surface area to volume ratio affects heat gain and loss and nutrient and gas transport from the environment into body cells.

So, the spherical body of the larger animal has twice the radius of the smaller original. The volume of a sphere is (4 pi/3) r3 and the surface area of a sphere is (4 pi) r2. So the surface to volume ratio is 3/r - As the animal becomes larger, the surface to volume ratio decreases.. In calculating this ratio, we have learned something really important: The surface to volume ratio changes as the size of the animal changes; it does not remain constant! If their is not some compensating factor involved, this would have to have serious consequences for the physiology of the animal.

Many physiological phenomena depend up the amount of surface area or the amount of total volume. For example, imagine that our animal has no gut and absorbs nutrients across the external body wall. If we scale up the animal to twice the radius, we will have four times as much surface area over which to absorb nutrients. But, we will have eight times as much volume of cells which are using these nutrients. So, as the animal becomes larger in scale, each cubic millimeter of volume will have to share less and less of the nutrient material obtained from diffusion in through the body wall.

The conclusion we come to is that the body plan must change as the size of an organism increases. For example, we can help solve our nutrition problem by adding a long tubular gut with many branches, convolutions, or infoldings, thus increasing the surface area relative to volume. We can reduce volume of tissue, while keeping the same larger exterior dimensions, by creating an internal body cavity, the coelom. And, we can put muscles around the gut to force the food through faster than would occur by diffusion, thus making sure that a fresh supply of nutrients are always available for absorption. All of these adaptations appear early on in evolution of the animal body plan, as animals became more larger and more complex.

The same principles apply to absorption of oxygen and to removal of waste products, including carbon dioxide and nitrogenous waste such as urea or uric acid. And, finally, notice that a very small animal would have a large surface area relative to its volume. Thus heat would be lost at a greater rate and body temperature would be hard to regulate. Small homiotherms ("warm-blooded animals" - body temperature is kept constant) have a much higher metabolic rate and body temperature than large homiotherms, such as ourselves, because of this!

For a giant insect, we would expect that the animal would likely starve to death because it would not have the fine villi (invaginations in the gut wall) and complex foldings of the length of the intestine that give our intestines such a large surface area!

The giant insect would also suffocate from lack of absorption of oxygen from the air. Insects absorb air through spiracles, small openings on the surface of the abdomen which lead to tracheae, from which the oxygen is absorbed into the blood. Some arachnids - spiders - do have structures called book lungs, which consist of a folded structure. However, this is not as elaborate an infolding as found in the lungs of large animals such as ourselves. We have complex branched lungs, which provide a huge absorptive area. Additionally, we can compress and expand the lungs by muscles in the chest and thus force air in and out of the lungs.

During the Carboniferous, giant insects roamed the Earth. There were spiders the size of dinner plates and the flat five-foot-long Arthropleura, that crawled along the ground. Dragonflies, such as the Meganeuropsis permiana, had wingspans up to 2.5 feet. These and other organisms are described in a New York Times article, Tuesday, Feb. 3, 2004. According to this article, geologist Dr. Robert A. Berner, of Yale University, proposes that the reason that insects could be so large is that oxygen concentrations in the air during the Carboniferous might have been as high as 35%, as compared with 21% now. The higher oxygen concentration might have let the inefficient system of tracheae support a larger insect. When oxygen concentrations dropped, the large insects didn't fare so well and became extinct.

Area to volume considerations can be extended to the problem of support of an organism on its legs.

Suppose that we scaled up an insect by a factor of 100. The legs would have a diameter 100 times that of the original and a cross sectional area 100*100 = 10,000 times the original. But, the volume of the animal's body would be 100*100*100 = one million times the original volume. If the tissue and exoskeleton were exactly of the same composition, and thus density, the larger insect would weigh one million times as much as the original. But, the weight of the body would be supported on legs with only 10,000 times as much cross sectional area, and thus total strength, as the original. The weight supported per square millimeter of cross sectional area would have increased by 100 times! Force divided by area over which the force is applied is called pressure. The pressure on each millimeter of cross sectional area would have increased by 100 times! The result would likely be that the giant insect would crush its legs and collapse!

How can we solve this engineering problem? Well, like our nutrition problem, we can change construction methods and materials. Insects have an exoskeleton, often with a sclerotized (cross-linked) protein cuticle; vertebrates have mineralized internal skeletons made of bone. And, we can change proportions of the body plan, in this case by making the leg thicker relative to the body size as the animal becomes larger. In the classic work, On Growth and Form, D'Arcy Thompson gives measurements of the relative weight of bones of various mammals and their body weights: Bone is about 8% of the body weight of a mouse or a wren, 13% of a goose or a dog, and about 17 to 18% of a man's weight. As the animal becomes larger, the bone becomes a greater proportion of body weight because the bones are proportionally larger in diameter! Notice that for the smaller animals, D'Arcy Thompson pairs a mammal with a bird, showing that while the principle holds for two different groups of animals. It would have been nice if Thompson had given us data for a large bird, such as an ostrich or a very large mammal, such as an elephant, which has very large diameter legs, relative to its body size. ((Biography of D'Arcy Thompson)

All animals can jump to the same height (almost)!

Now we get to a common statement of wonder: "Isn't it amazing that a grasshopper can jump one hundred times (or something like that) its height while we cannot?". Let's analyze this situation. As an animal is scaled up in size, the volume increases proportionately to the cube of the length. So, the mass also increases as the cube of the length. Also, the mass of muscle, which will remain a constant fraction of body mass if we scale up the design of the animal keeping constant proportions, will increase as the cube of the length. It turns out that the energy developed by the muscle is directly proportional to the muscle mass. Twice the muscle mass gives twice the energy output. So, we conclude that the energy provided by the muscle and the total mass of the animal bothincrease proportionately to the cube of the length - exactly the same rule of increase. So, if we have twice the mass of animal (of the same body plan and design), we have twice as much energy to lift it up by jumping.

Now, you probably remember from physics that the height an object can be raised by a given amount of energy is directly proportional to the amount of energy: E = mgh, where E is the energy required to raise a body of mass m, a height h. (The constant g is the gravitational constant.) So, suppose we consider our spherical animal (with the little legs). If we double the radius, both energy provided by the muscles and total body mass will increase by 8 times. But the ratio of energy to mass, E/m, will remain constant. Therefore, the larger animal will jump to the same height as the smaller animal! The grasshopper, a mouse, man, and an elephant should then all be able to jump to the same height. Of course, this isn't exactly what we find. Some animals have different proportions of muscle. The muscles may be attached differently to the skeletal structure, giving different efficiencies for different types of motion. And, the biochemistry of the muscle proteins and enzymes involved in metabolism to produce ATP may be different. But, if you think about it carefully, you may come to the conclusion that a grasshopper and an elephant may jump to about the same height.

A giant grasshopper of science fiction fame would not be able to leap tall buildings in a single bound, but would be constrained to the same height jump as it always was - just a few feet, at most!

Walking and running speeds are determined by the mechanical properties of pendulums!

Walking is a really complex process. Imagine that our leg is a solid object which pivots at the hip - a pendulum! The natural rate of oscillation of a pendulum is proportional to the reciprocal of the square root of the length. A short pendulum oscillates faster and a long pendulum oscillates slower. (The principle is used in the design of a pendulum clock and a metronome, a device used by musicians for setting a rhythm.) A leg is an example of a "forced oscillator" - The energy lost by friction is replaced by "forcing" the movement with muscles. It turns out that the energy transfer in forcing the oscillation is most efficient when the oscillation is forced at the natural rate of oscillation - the rate that the pendulum will oscillate if just pulled back and let go.

So, it takes more energy to move a leg either faster or slower than the natural rate. It turns out that the walking speed for most people involves leg motion which is moving very close to the natural rate of oscillation.

But, our leg is not really a simple pendulum. It is a compound pendulum which is jointed at both the hip and knee. The rate of motion for walking is close to that predicted if the knee joint were rigid and the leg were a simple pendulum. When we change gait and run, we bend the knee a lot, and the leg becomes two short pendula, each one of which oscillates at a higher rate, as predicted by the theory of natural, unforced, oscillations.

How does this relate to our giant insect and the actual walking and running speeds of animals? Well, if one is larger, the leg is longer and oscillates slower. But each back and forth oscillatory movement takes a longer step because of the length of the leg. So, these two effects, the slower movement and the longer pace, tend to cancel out. So, it isn't clear that the giant insect would walk at much different a rate than a smaller one.

Eyes need to be different for small and large animals.

Our eye and the eye of the octopus both have a lens which produces an image on a retina which is comprised of an array of photoreceptive cells. This is an effective design for relatively large animals. But, if we tried to scale down this type of eye to fit onto an insect's body, we would have a big problem: The lens opening of the eye would be very small. Light passing through a small opening diffracts (spreads) due to the interference of the light waves. If the pupil is large, there is very little edge compared with the interior of the opening and there is little diffraction. (This in itself is a scaling problem similar to the surface to volume problem.) If we tried to scale down a vertebrate or mollusc eye to fit on an insect, diffraction would cause so much spreading that the image focused on the retina would be very blurry and essentially useless. So, insects make do with a much simpler type of eye. Insect compound eyes have many individual units, called ommatidia, each pointed in a different direction. These are simple detectors which signal how much light is coming from the direction into which the ommatidium is pointed.

The first of the following figures, both redrawn from Optima for Animals by R. McNeill Alexander (Edward Arnold), shows a diagrammatic representation of a compound eye with the individual ommatidia. Two tubular ommatidia are indicated in more detail, each pointing in a different direction as the result of the hemispherical shape of the compound eye. The insect can resolve two points (shown in blue and green and corresponding to the directions that two of the ommatidia are pointing) if these points are not separated by an angular distance, alpha, which is too small. The minimum resolvable distance is a function of two factors: 1) Optical diffraction causes "blurring" which becomes more serious as the diameter of the ommatidium decreases. So, for the least diffractive effect, the ommatidium must be as large as possible. 2) But, each ommatidium acts as a light pipe. All the light entering the pipe is averaged and an electrical signal is produced which indicates how much light enters that particular pipe. If the ommatidium is large in diameter, light will enter from a larger range of angles and the minimum alpha resolvable will increase. The optimum diameter for the ommatidium is where these two effects balance out. For a wavelength 0.5 micrometers (yellow-green), the optimum ommatidium diameter is 27 micrometers, a value very close to that measured in insects. For optimum resolution, the wavelength must be as short as possible. That is why many insects are most sensitive to light in the blue and near ultraviolet spectral regions.

If we tried to scale an insect up to human size, each ommatidium would view the same angle as in the smaller insect. The light pipe of the ommatidium would be longer and wider in the same proportion as for the smaller insect so the angle of view would be the same. But, diffraction would essentially disappear since the opening of each huge ommatidium would now be so large. So, the large insect might actually see a little better and would certainly have no trouble resolving two points which the smaller insect could resolve. Thus, our giant insect could still see but, like its smaller cousin, would not have anywhere near the fine resolving power of the lens-based eye of molluscs and vertebrates.

Hearing and flying are also affected by the scale of an animal.

There are many other scaling effects which could be considered. One is hearing. All types of hearing systems resonate, responding optimally to certain frequencies. Small structures resonate with higher frequencies. So, if we scaled our grasshopper up to human size or larger, we would expect that the sounds it could hear and the sounds it could make would be extremely low frequency.

This and other scaling problems indicate that physics puts very severe constraints on the types of structures that can be utilized at each level of scale and size. Small, less complex invertebrates use quite a different design plan in many ways that more complex vertebrates.

Recently, a news article pointed out problems of scaling ever so strikingly. Some engineers are trying to make micro-miniaturized aircraft, some no larger than an insect. These aircraft would be equipped with cameras, for example, and could fly into dangerous areas or provide surveillance. Naturally, the military is very interested in this. In order to get these tiny aircraft to fly, principles used by insects may have to be used - a conventional helicopter or wing design just doesn't work at small size. The tiny aircraft may have to use a rapidly beating wing like that of a bee!

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