PHYSICS AND SIZE IN BIOLOGICAL SYSTEMS

An Introduction and an Example

Suppose you wished to bake a cake large enough for four people but you only had the recipe for a cake that would feed eight people, such the the one below
 

A CAKE FOR EIGHT PEOPLE

4 cups flour 1 tsp. salt 2 tsp. baking powder 2 tsp. baking soda 2 cup sugar 2 cup cooking oil 4 eggs (added separately) Preheat oven to 300 degrees Fahrenheit. Mix well and pour into 9x13" non-stick pan and bake for 50-60 minutes.

For your smaller cake you would have to halve everything, so wouldn't the recipe for your four person cake read
 

A CAKE FOR FOUR PEOPLE

2 cups flour 1/2 tsp. salt 1 tsp. baking powder 1 tsp. baking soda 1 cup sugar 1 cup cooking oil 2 eggs (added separately) Preheat oven to 150 degrees Fahrenheit. Mix well and pour into 4 1/2 ' x 6 1/2" non-stick pan and bake for 25-30 minutes.

It had better not because that recipe produces not a cake for four people but a mess for four people. The amounts of flour, salt and so forth should be cut in half to produce a half sized cake but the temperature, length and width of the pan and the time the cake is cooked should not be. The question we will address here is WHY this is so.

The case of the half sized cake is an example of what is called a scaling effect. Scaling effects play vital roles in biological systems.

Another example, which we will examine more fully later, is that of the three Nevada game birds. Three quail-like game birds have been introduced into Nevada during the last two centuries. All three have similar physical characteristics and differ only in size. Yet one of them lives almost exclusively in the valleys, another lives in the foothills and the third only at higher elevations.

Do you know which one lives at the top of the mountain and the other in the valley and give the physics behind this? If not, read on.

The purposes of this page is (1) introduce the very interesting subject of biological scaling (allometry, Greek for 'difference measurement') to elementary physics teachers who may not already be familiar with it; (2) to offer those who do know about scaling more examples; and (3) to provide an in-depth bibliography on the subject.

[In the discussion below, the word size will generally refer to magnitude. Occasionally, however, it will assume the less familiar meaning of 'character, quality or status of a person or thing, especially with reference to importance, relative merit or correspondence to needs.']

This page will deal both with real situations (the game birds and elephants) and imaginary ones (Lilliput and King Kong) as the physics of that which we can only imagine is often as exciting as that we can experience. We must be careful entering the world of the imaginary because we cannot verify our theories and must not let our imagination carry us too far.

Elements of Scaling

Let us first look at a simple geometrical problem: If the dimensions of a bird are doubled, what happens to its volume and surface area? To make the problem simple, let us consider a 'cubic bird'. Happily for us, the results will be the same whether our birds are cubic, spherical, cylindrical or even bird shaped.
 

Figure 1

In Figure 1, the cube on the left (A) has sides of length 1 unit, so it has a volume of 1 cubic unit (=1 x 1 x 1). This cube has six sides, each with an area of 1 square unit (=1 x 1), so it a surface area of 6 x 1 = 6 square units. Therefore the ratio of the surface area to volume is 6/1 = 6.

For the cube on the right (B), each side has doubled in length to 2 units, so its volume is now 8 cubic units (=2 x 2 x 2). The surface area of each side is now 4 square units (=2 x 2) and the total surface area is 24 units (=4 per side x 6 sides). Therefore the ratio of the surface area to volume for cube B is 24/8 = 3.

So, by doubling the length of each side we have reduced the surface area to volume ratio by a factor of 2 (from 6 to 3). [You should convince yourself that if we had chosen a sphere as a demonstration, the surface area to volume ratio would reduce by a factor of 2 if we doubled the radius.]

For cubes (or spheres) whose length is 3 times that of a smaller one, the surface to volume ratio is 54/27 = 2. In general, the surface area for a cube is 6 l² and the volume is l³, so the surface to volume ratio is given by
 

Table 1 shows the result of extending these numbers.
 

What does this have to do with biological systems such as our game birds live? It has to do with the amount of heat they generate and how they can transfer it.

The amount of heat generated by animals which differ in size but are other wise alike (such as similar type birds or similar type reptiles but not birds and reptiles) depends on their masses. Since the mass is proportional to the density times the volume and the densities are the same (because the animals are alike), the amount of heat generated depends on the volume of the animal.

On the other hand, the amount of heat lost to the environment depends on the surface area, as heat is largely lost by radiation, which depends on the surface area.

So, if the surface to volume ratio is large (which means a small animal), that animal will lose heat too rapidly to survive in a cold climate (such as at the top of the mountain).

Therefore, small birds will live in the warmest climate, which are found in the valleys, while larger birds will live in the cooler climates of the higher elevations. By the way, the smallest, valley dwelling bird is the California quail, the one that is mid sized and lives in the foothills is the chukar, while the largest, mountain dwelling bird is the very appropriately named snow partridge.

This is not the only example of larger animals living in colder climates. The largest North American member of the rabbit family is the arctic hare. And please note again that when we make comparisons between two objects or animals based on their sizes, we are assuming that every other factor (availability of food, insulation of fur or feathers) is the same.

This scaling has some interesting and unexpected effects on human society as well. For example, the grain storage buildings constructed by the Inca civilization at low altitudes were fashioned from rocks that had been laboriously shaped by pounding with river rocks until they could be fitted tightly together. On the other hand, the grain storage buildings at high altitudes (up to 5.5 kilometers, or 18,000 feet!) were simply rough shelters covered with a roof. Why the difference? One reason was that small, warm blooded, grain eating animals such as rats and mice can not survive at high elevations because they lose body heat too fast. Therefore there was no need to construct buildings with closely fitting stones to keep these animals out.

A few simple, interesting and sometime surprising calculations can be made to examine what changes in the volume and the mass of animals occur when we change their dimensions. We assume that the mass of an animal is directly proportional to its volume. This is not exactly the case because very large animals must have bones larger in proportion to their size than do small ones. This will change the average density somewhat but we will overlook that for now.

What happens to height as we change mass? For example, if a 73-kg (160 lb.) person is 1.8 m (6 ft) tall, how tall is a 37-kg (80 lb.) person? Since we are assuming the density of the two persons is the same and since mass is proportional to the density times the volume and the volume is proportional to a length (or height) cubed, we have
 

We have the rather surprising result that a person with half the mass of another is not half as tall, but about 80% (=1.4/1.8 x 100%) as tall.

Now we can ask, what would be the mass,M₃, of a person who was half the height of our 73 kg six footer?
 

So a person with half the height of another has only about one eighth of the mass.

At this point you might wish to ask yourself some general questions such as:

• What factors in the lives of animals (including humans) depend on only the linear dimension (height, width, etc.) of the animal?
• What factors depend upon surface areas? Don't forget to include the inner organs such as the lungs and the intestines.
• What factors depend on the volume (or on the mass, which is proportional to the volume)?

Or ask yourself some more specific questions such as

• Why do we chew our food?
• Why do we use kindling wood to start a fire?

Now let's look at some consequences of the surface-to-volume ratio.

• Because they have such a large surface-to-volume ratio, mice eat 25 to 50% of their body weight every day in order to generate enough heat to stay warm.
• Hummingbirds eat more than their body weight each day.
• Pet birds such as canaries can die overnight if they do not have access to food and water after it gets dark.
• At the other end of the scale, elephants have difficulty, because of their small surface-to-volume ratio, eliminating their excess body heat. The ears of the elephant, which are quite large, contain many large, thin blood vessels, through which hot blood is circulated. The ears of the elephants act in the same fashion as the radiators of cars. In the same way, desert rabbits and foxes have much larger ears than the woodland varieties.

Terminal Velocities

Why can a mouse fall down a mine shaft without injury while a person cannot? We can get an answer from considering scaling effects.

When a object falls in a fluid (air or water, for example) there are two forces on it: the force of gravity downwards (=Mg,M being the mass, g the acceleration of gravity) and a drag force exerted upwards on the object by the air. This second force can be thought of as arising from the pressure (P) exerted on the object by the air molecules striking the area (A) of its lower surface. The pressure will increase as the object increases its velocity until, at some point,the force of gravity downwards and the drag force upwards will be equal and the object will no longer accelerate but move down with a constant terminal velocity,v_t.
 

The terminal velocity is proportional to the size (length or width or height, as long as we measure the same thing when we make our comparisons) of the animal. Since an elephant is about 100 times the size of a mouse, its terminal velocity is 100 times that of a mouse. When dropped from the same height, the mouse strikes with a much smaller velocity than the elephant and has a greater probability of surviving the impact.

Very small objects, such bacteria, fall very slowly indeed since their lengths are so small. A bacterium, for example, will fall in air only at about 2.8 x 10^-4 m/s (or about one half inch an hour). During the course of its fall, a bacteria will be buffeted about by air molecules and seem to undergo random, chaotic motion (Brownian motion). Nonetheless, they will settle to the ground, only it will be very, very slowly.

Gulliver's Physics

Winter, 1930 No longer in copyright.

In 1726 Jonathan Swift published Gulliver's Travel's, a satirical novel, in which the Gulliver travels to a number of fantastic new countries inhabited by strange new people. The most well known of these people are the tiny Lilliputians and the gigantic Brobdingnagians. The author was aware that scaling a 6-ft person linear dimensions down by a factor of 12 reduced his volume by 12³ = 1728 (the Lilliputians) or up by a factor of 10 increased his volume by a factor of 10³ = 1000 (the Brobdingnagians). He does not seem to be aware of some of the serious, perhaps even disastrous, consequences that would result from doing so. Let's look at the Lilliputians first.

Gulliver was 1.8 m (6-ft) high and weighed 720 N (160 lb.). A male Lilliputian was 0.15 m (6-in) high, and, using our formula from above relating weight (or mass) to height,would weigh only about 0.043 N (1.5 ounce). The brain of the Lilliputian man would be 1/1728 the size of Gulliver's as well. However, brain cells decrease in volume by about a factor of 8 from the largest mammals (elephants) to the smallest (mice), so let's give our Lilliputian the benefit and say that the number of brain cells reduces only by a factor of 200 compared the Gulliver's. This is about the size, in terms of the number of cells, of the brain of a kitten. Mr. Lilliput would not be very intelligent, compared to Gulliver, and it is doubtful he would have the power of speech.

But suppose Mr. Lilliput could speak, could Gulliver understand him? The vocal cords of animals vibrate very much like a stretched string for which the frequency f is related to the tension in the string T, its length l and its linear density (mass per unit length) µ. The relationship is: The mass per unit length is the density of the string, rho, times its volume and divided by its length. The volume is the cross sectional area times its length, so: and therefore the frequency is: Since the tension in the string ,T, and the density, rho, are constant, the frequency is proportional to: However, A is proportional to a length dimension square, so we have: Therefore, the frequency varies as l^-2.

The 1.8 m tall Gulliver, a healthy male, probably had a fundamental frequency, f_G, of about 150 Hz while Ms. Gulliver, back in England, had a fundamental frequency of about 250 Hz. Mr. Lilliput, who as you recall is 0.15 m tall, would therefore have a fundamental frequency, f_L, of:

That frequency is inaudible to most adult humans, so even if Mr. Lilliput had something to say to Gulliver, Gulliver could not have heard it. The fundamental frequency for Mrs. Lilliput would be 1.67 times that of her husband, or 37,000 Hz.

And it's worse in Brobdingnag

Rackhem, 1909 No longer in copyright.

The problems of the Brobdingnagians are also very serious. As they were 10 times taller (and 10 times deeper and 10 times wider) than Gulliver, they would have 1000 times the mass, so a male would weigh:

Likewise we can calculate the fundamental frequency of Mr. Brobdingnag's voice just as we did for Mr. Lilliput: Since the typical adult human can hear frequencies only in the range of 20 - 18,000 Hz, Mr. Brobdingnag would be speaking at a fundamental frequency much too low for him to be heard by Gulliver.

However, it still might be possible for Gulliver to hear some of the higher harmonics of the Brobdingnagian speech. In order for to have complete understanding of human speech, a minimum frequency range of 100 to 8,000 Hz is essential. Assuming that the numbers, frequencies and intensities of the harmonics in Brobdingnagian speech to be the same as for human speech with the frequencies scaled down for a factor of 100, the frequency range important to understanding Brobdingnagian speech would be 1 - 80 Hz (for Lilliputians it would be 8000 - 1,000,000 Hz). These three ranges would be essential for the Brobdingnagians, humans and Lilliputians to understand others of their own species. As the amplitudes of the harmonics outside these regions fall off very rapidly as do the response function of the ears at high and low frequencies, there could be no vocal communication among the three groups.

If you've got something to say, say it

There would be numerous other difficulties as well, but let us consider one more related to communications. Suppose Gulliver builds some sort of device that can detect the fundamental frequencies of the Lilliputians and Brobdingnagians and convert them into frequencies he can understand. What,exactly, does he hear. It takes about 1 second for a human to say the word 'Brobdingnagian', and speaking at 150 Hz, it means 150 wavelengths have entered our ears. When a Brobdingnagian says the same word, he says it at 1.5 Hz and it would take 100 times as long, or 100 seconds = 1 min 40 sec, for the same number of waves to enter our ears. That would try the patience of Job!

On the other hand, a Lilliputian would be speaking at a frequency 144 times ours, meaning a word that we humans said in one second, the Lilliputian would say in 1/144 s = 7 ms! If the Lilliputian were reading this web page out loud, it would take him less than 3 seconds to reach this point from the beginning. Do you think you would have understood anything he said?

The mental capacity of a person probably scales as the square of the characteristic length as part of the brain responsible for intelligence, the cerebal cortex, is only 2 - 3 mm thick but covers the brain like a skin and is therefore can be treated like an area. We therefore can get a good idea of the intelligence of the Lilliputians: The Lilliputian gentleman would likely have less than 1% of the intelligence of Gulliver, so even if Gulliver could detect his frequency and handle the rapidity of the words, he would find his tiny companion had essentially nothing to say. On the other hand, using the same formula as above but scaling for the Brobdingnagians, we would find them to be 100 times as intelligent as Gulliver! They would be saying extremely profound things, but in a very, very deep voice and extremely slowly.

If you can't talk the talk, can you walk the walk

As we have seen, it would be impossible for Gulliver to vocally communicate with either the Lilliputians or the Brobdingnagians directly, and that even if he could translate their speech into his frequency range and rate of communication, it would be unlikely that the different species would have anything to say to one another. So, let's leave that and consider another situation - could any two of the species walk along side each other?

Let us consider a very simple model for walking. In this model the foot of one leg is treated as a pivot and the other leg as a pendulum. The center of mass of the body is in front of the pivot and the pendulum leg swings outwards. In the animated gif to the right, the pendulum leg is shown as the darker line.

This is a quite stiff legged walk, but a reasonable first approximation. We can get an idea of the velocity, v, of this walk when we recall that where d is the distance traveled and t the time elapsed to travel that distance.

We can take the distance traveled to be the length of a single step and the time to be that of the period of that step. The length of the step, for similarly designed animals, will be proportional to the characteristic length of the animal. The period, since we are treating the leg as a pendulum, is proportional to the square root of the length of the leg, that is it is proportional to the square root of the characteristic length of the animal. Putting this into our formula: So, the velocity of walking goes as the square root of the characteristic length. Mr. Brobdingnag, being 10 times Gulliver's size, would walk about 3.2 times as fast as him, while Mr. Lilliput, being 1/12 the size of Gulliver, would be able to walk at only 29% of Gulliver's velocity.

Oddly enough, while our three companions could not walk at the same speed, they could run at the same speed. This is because running uses an entirely different mechanism to propel the animal in that, rather than using the force of gravity to move the leg, the leg is accelerated. We can use Newton's second law, to get an idea of how the acceleration scales with the characteristic length: where m is the mass of the leg and F the force accelerating the leg, which arises from the muscles in the shoulder of the animal. The mass of the leg must be proportional to the density of the leg times its volume, which in turn is proportional to the cube of the characteristic length. The force of the muscles would be proportional to the amount of muscle, which would be equal to the volume of the leg times the density of the muscle. In terms of formulas: Or, in words, the acceleration of the animal's leg is a constant. In running, the stationary leg is accelerated forward and outward and then planted on the ground and comes to a halt again. This acceleration, and hence the running velocity (as this process is repeated over and over) is a constant for animals of similar design.

One place you can see this is in the ugly English 'sport' of fox hunting. Humans, mounted on horses and accompanied by dogs, pursue a fox, which naturally runs away. The horse is probably about 10 times the size of the fox, the dogs are about 2 times the size of the fox and 1/5 that of the horse, and yet all three run at the same speed over considerable distances. Of course, if they were walking side by side, the horse would walk much faster.

It is interesting to think how one would design an animal so as to make it faster. Basically, you would want an animal with low mass legs, so the legs would be thin and with lots of muscle mass, that is big shoulders. And what type of animal is that? It is this type of animal. The next time you see a film of a cheetah in pursuit of a gazelle, check out the legs and shoulders of both. You'll see thin legs and massive shoulders on each, which means they are very fast.

There are quite a number of other physical characteristics that can be discussed using scaling. For example, the cross sectional area of the leg bones obviously scales as the square of the characteristic length while the mass of the body supported by the legs scales as the cube. Since the bone has some ultimate tensile strength in N/m², there would be some limit to the height of an animal, beyond which it would crumble under its own weight. You can also ask yourself, in the same fashion, what would be the tallest possible tree and the highest possible mountain?

Another question you might wish to address is heat loss and food intake. Because of their large surface to volume ratio, the Lilliputians would be losing heat at a tremendous rate and would have to be eating constantly to maintain the high body temperature of a mammal. On the other hand, the Brobdingnagians would be so massive and have such a relatively small surface area that they might have great difficulty cooling themselves off and avoiding heat stroke.

And why don't we see birds the size of 747's? And why do insects and other simple animals have simple eyes (and sometimes a very large number of them) rather than the compound eyes of humans, elephants and squids? And why, if dinosaurs are so much bigger than chickens aren't dinosaur eggs proportionaly bigger than chicken eggs?

These last three questions, and many others, are addressed in the primary resource for this page, the article Physics and Size in Biological Systems, which was written by Dr. George Barnes, professor emeritus of physics at the University of Nevada, Reno and which was published in the April 1989 issue of The Physics Teacher. I wish to thank Mrs. Barnes for granting permission to quote so extensively from that article; in fact, I basically translated the first half of the article into HTML. I have been using examples from this paper in my physics class for biology majors for years.

Another interesting paper is entitled Gulliver Was A Bad Biologist and was written by Francis Moog. It appeared in Scientific American in 1948.

A recent book, Air and Water - The Biology and Physics of Life's Media, by Mark Denny (Princeton University Press, 1993), discusses the effect of the scaling of the properties of the medium (air and water), such as its density, surface tension, viscosity, optical density, bulk modulus and so on, on the nature of the life that lives in that medium. It answers the question of if we have man made air balloons and nature made water balloon-like animals (jellyfish), why don't we have nature made air balloon-like animals?

The time of cooking would depend on the volume to surface area, which is for the small cake is twice what it is for the large cake. So, you would want to bake the cake for 25 - 30 minutes. Or until a straw inserted into the cake can be withdrawn cleanly.