Scaling and Life in Fluids

Main Points:

  1. Life operates at length scales spanning over 7 orders of magnitude and mass scales spanning 21 orders of magnitude.
  2. The size of organisms depends on their hierarchical organization.
  3. The sizes of organisms have profound consequences to movement in fluids such as water and air.
  4. The Reynolds number, expressing the ratio of the size of an organism and the velocity and density of the fluid around it to the viscosity of that fluid, is a key scaling relationship for life in fluids.
  5. Very small organisms live in a high viscosity world where gravity is unimportant (i. e. low Reynolds numbers).
  6. Large organisms living in a low viscosity world where gravity is extremely important (i. e. high Reynolds numbers).
  7. The differences in size have profound consequences on the structure of the organisms.
  8. Volume scales to the cube of length, but surface scales to the square of length: thus surface area of organisms has to increase in special ways to keep up with increased mass.
  9. Material transfer for small organisms is dominated by diffusion.
  10. Material transfer for large organisms is dominated by mass flow.
  11. Small organisms are at the mercy of the physical structure of their surroundings, although in aggregate they strongly modify the biogeochemistry of their environment.
  12. Large organisms grossly modify the physical structure of their surroundings and through that the distribution of smaller organisms.
  13. Constraints can be overcome by novel design though evolution.

1. Introduction:

One of the most remarkable things about life is the enormous range of sizes spanned by different kinds of individual organisms. From the smallest bacterium to the largest living individual animal, the blue whale, there are more than 7 orders of length and 21 one orders of mass as the following table shows.

Organism Length Mass
Mycoplasma <2x10-4 cm <0.1 pg <10-13 g
Bacterium 2x10-3 cm 0.1 ng 10-10 g
Ciliate eukaryote (Tetrahymena) 2x10-2 cm 0.1 g 10-7 g
Large amoeboid eukaryote 2x10-1 cm 0.1 mg 10-4 g
Bee 2 cm 100 mg 10-1 g
Hamster 2x101 cm 100 g 102 g
Human 2x102 cm 100 kg 105 g
Blue whale >2x103 cm >100 tons >108 g

It is important to note that while the total dimensions of these individuals are vastly different, all are composed as cells and at the cell scale life spans perhaps only 4 orders of magnitude in length and 9 orders of magnitude in mass. At the orgenelle/bacterium scale, the units from which eukaryote cells are made, life spans only 2 or 3 orders of magnitude in length and 6 orders of magnitude in mass.

Thus, in order for life to be larger, by orders of magnitude, the individual changes levels within the structural hierarchy.

Why does this need to be? Why cannot things just consist of giant cells in order to be larger?

2. Life in Fluids.

All life lives in fluids - air or water or some version of water. The order of magnitude size changes have some pretty remarkable implications for function within fluids.

First, we should recall what a fluid is:

A fluid is a substance that does not resist a shear stress.

It is very important to remember that a fluid can be a gas or a liquid, but never a solid. What matters for the deformation of a fluid is the rate at which it is deformed, not how much it is deformed. A nice analogy is given by Vogel (1994). The difference between a solid and a fluid is like the difference between a spring and a shock absorber. Both can be deformed, but if time is taken, a shock absorber can be put in any position, in which it stays one the force is removed, while a spring has a forced preference for a given length, and springs back. It is no accident that a shock absorber can do this because it consists of a loose fitting piston, in a sealed cylinder of fluid.

However, a fluid does require a force to deform, and for a given force, the deformation varies as function of a property called viscosity. For us, Vaseline is a very viscous fluid, but water is not.

Viscosity

Viscosity (or more properly dynamic velocity) is the relationship between the amount of deformation and the rate at which it occurs.

The formal meaning of dynamic viscosity) is probably easiest to understand by reference to the figure below:

In this drawing, dynamic viscosity is μ (mu), is the thickness of the fluid over which the deformation is occurring (z) times the force used to shear the fluid (F), divided by the velocity of the upper surface (U) times the area of that surface (S). In other words:

μ = Fz / US

The dimension are ML-1T-1, or mass divided by length times time.

The arrows of decreasing length downward represent a velocity gradient, with the velocity highest near the source of the fluid shear and lowest near the unmoving surface.

The ratio of dynamic viscosity to the fluid's density (ρ, rho) is the kinematic velocity, ν (nu) or:

ν = μ/ρ

Laminar vs. Turbulent flow

In the situation at right, the lines of flow, like the arrows representing the velocity gradient in the figure above, are parallel and smooth. The flow is said to be laminar.

Of course, fluids do not always behave like this. They often behave turbulently. In turbulent flow, the flow lines are chaotic, distributed in ever changing eddies (See figure at left).

Of particular interest to us is that the laminar or turbulent behavior of fluids is dependent on three things:

  1. The size of the object moving though the fluid, or the size of the vessel in which the fluid is moving.
  2. The velocity of the object, or the fluid relative to the vessel.
  3. The viscosity of the fluid.

The relationship between these variables is described by a scaling number, which is dimentionless, called the Reynolds number, Re.

Reynolds number

The Reynolds number is an indication of the tendency of flow to be laminar, if Re is very small or turbulent if Re is very large.

More precisely, Reynolds number is the ratio of the density of the fluid (ρ) times the size of the object (so called length factor, l) times the velocity (U), to the dynamic viscosity (μ).

Re = ρ l U / μ

It is hence also the ratio of the length factor times the velocity, and the kinematic viscosity.

Re = l U

For a given shape, Reynolds numbers tend to increase by orders of magnitude with increasing length factor, increasing relative velocity of the fluid, or decreasing velocity.

2000 and 200000 are large Reynolds numbers. 0.02 and 0.0002 are small Reynolds numbers.

We can say that at low Reynolds numbers, objects live in a world dominated by viscosity, while at high Reynolds numbers they live in a world dominated by gravity (i. e., interial forces).

A pollen grain can be suspended in air for years. A large rock simply drops.

After a 20 m whale gives a flap of its flukes, it glides on for many meters. When a 0.1 mm Daphnia moves its giant second antennae, it moves forward, but stops the moment the antennae stop. From the whale's perspective it lives in water, behaving like we would expect water to behave. But from the Daphnia's perspective, water behaves like our feeling for like Vaseline.

The flagellum of a sperm moves in much the same way as an eel. But the sperm is shoving its way through a viscous medium, while the eel gliding along a pressure wave in a low viscosity environment.

Consequences to the Structure of the Organisms

The different behavior of fluids at varying Reynolds numbers imposes constraints on organisms that pose challanges to design that in many circumstances have been met by natural selection.

Drag

A consequence of turbulence at high Reynolds numbers is that an object moving through a fluid experiences drag at its rear. This drag is a result of the dissipation of momentum in the turbulent wake rather than its transfer back to the object as a forward thrust as in laminar flow. For objects moving in a fluid at high Reynolds numbers, streamlining can prevent the separation of flow that occurs at the transition between laminar and turbulent flow along an object so that laminar flow prevails all along the organism surface at the same Reynolds number.

At left: Turbulence at a specific Reynolds number for a sphere.

At right: Flow at the same Reynolds number for a streamlined body.

For animals at high Reynolds numbers, adaptation though evolution has resulted in streamlined forms - spetactular examples are tuna and whales.

At lower Reynolds numbers, streamlining becomes less important, and the friction at the object's surface (skin friction) becomes relatively more important. This is certainly why very small animals tend to not be streamlined. The extra exposed surface is not worth it.

Because of the change in the behavior of fluid as the size of an object changes, it is mechanically extremely inefficent for the design that works well at one Reynolds number to be used at a very different Reynolds number to maintain the same velocity.

Self Streamlining

Self streamlining is a process where the organism changes its shape or orientation to a moving fluid in a way that maximizes the streamlining effect and minimizes drag.

Two examples of brown macroalgae orienting to flow are shown an the right. Flow is from the left to right. Genus on left is Nereocyctis and right is Postelsia (redrawn from Vogel, 1994).

Self streamlining is important because it allow organisms to adapt rapidly to changing flow directions while minimizing potentially damaging drag.

3. Length - Area - Volume relationships: Why be big?

Different sized organsisms face dramatically different physical constraints even if they are geometrically similar, not only because of the behavior of fluids around them, but also because of the consequences of Euclidian geometry. As you all know, the formulae for length, area, and volume are of a cube are: l, l2, and l3. As a consequence, as a cube gets bigger in length in equal increments its area and volume get bigger in different, larger increments. For all ojects changing shape while retaining similar geometry, we say that area increases with the square of its linear dimension, and the volume increases as the cube of its linear dimension.

Object Number of Dimensions Exponent Formula for Square Formula for Sphere
Length 1 1 l2 2 π r
Area 2 2 6(l2) 4 π r2
Volume 3 3 l3 (4/3) π r3

We can see the effects of these relationship by considering some graphs:

Thus, if we take a cube with a side of length 1 m (with area of 6 m2, and a volume of 1 m3) and double its length to 2 m, the area increases by a factor of 4 to 24 m2 (2 m x 2 m x 6 = 24 m2) but its volume increases by a factor of 8 to 8 m3 (2 m x 2 m x 2 m = 8 m3).

Consequences for "strength"

One important implication of the geometric relationship between length, area, and volume is that the "athletic" abilities of organisms does not scale linearly with size.

I am sure most of us have heard about fleas jumping many feet into the air, and if we were as strong. we could jump over skyscrapers. Well its not that fleas have especially strong muscles! Muscular strength increases by the cross sectional area of muscle tissue. But muscular effort acts on the mass of an animal. So as the size (length) of the flea was scaled up towards our size (without a change of shape) the strength would scale with the square of the length flea's muscles. Unfortunately, the flea's mass would be increasing with the cube of the length of the fleas muscles, and so as the flea get larger, it would get relatively weaker, compared to its mass. Once the flea reached our size, it would be no stronger that, say a kangaroo, of similar mass. It would jump higher than us, but probably only high enough to clear a large fence. It could do this only because a flea has proportionally larger leg muscles than us, as does a kangaroo.

The counterpart to such "athletic" abilities, is the load bearing ability of tissue, such as bones. The load bearing abilities of bones, again, scales to the cross-sectional area of the load bearing bone. But, the animal's mass scales to the cube of length.

As a land vertebrate gets larger, for it to be able to bear its weight, which is scaling to the cube of its length, the cross-sectional area must increase disproportionally with length. In other words the bone must CHANGE SHAPE as it increases in length.

The change in shape with size is called allometry.

As pointed out by Schmidt-Nielsen (1984) this relationship was pointed out by Galileo Galilei in 1637 using the following drawing.

Drawing from Galilei (1637). Here the larger bone is shown disproportionally thick compared to the smaller one. Thus, the shape changes with increasing size in this allometric relationship.

Allometry is a key characteristic of life and is one modification to get around constaints by modification of design.

However, as Schmidt-Nielsen (1984) notes, the bones of real animals do not scale with this relationship. Rather they scale so that they do not buckle by bending under a load. This is called scaling by elastic similarity. It predicts that bones get stouter than one would expect by scaling in proportion to the linear dimension, but not as stout as they would under the model of scaling to the compressive loads (mass). It is still an allometric relationship, but one that is not as dramatic.

Consequences for heat

The surface of organisms comprises a door for many metabolic processes. Critical among those is gaining and losing heat.

However, we have already seen that area and volume scale differently with increasing size. and that for objects of geometrically similar shapes area scales to 2/3 power of volume. Because mass scales proportionally with volume and the body mass is what gains loses or produces heat (through metabolic processes), we should expect that the ability to gain or lose heat decreases as body size increases.

Consequently, small organisms find it very difficult to maintain their inside temperature in the face of environmental changes, while very large animals have a large thermal inertia. This can be very useful in environments that experience fluctuations in temperature, or a problem if gaining heat or losing heat rapidly is necessary.

As you might expect, large animals often have to compensate for the surface area to volume relationship by changing their shape with increasing size. Hence, African Elephants have very large ears and the Early Permian age carnivorous synapsid Dimetrodon has a large sail on its back. However, we do not know if the sail of Dimetrodon was for gaining heat, say on a cold morning, or losing heat. Importantly, an herbivore, Edaphosaurus, also with a sail, lived in the same area, suggesting a pattern of "escalation", which will be explored later.

The sail-backed synapsids ("mammal-like reptiles") (A), Dimetrodon, a carnivore, and (B), Edaphosaurus, an herbivore.


Consequences for metabolism

Because the surface area of various tissues play a critical role in the metabolism of the mass of so many biological functions, one might expect that the Euclidean 2/3 power as seen in the surface area to volume relationship (a = v2/3) might be widespread among many biological functional relationships. In fact, even more widespread might by exponents that are multiples of 1/3.

This scaling relationship was expected by physiologists, but they found to their dismay that it was not the case. Instead, multiples of powers of 1/4 proved to be prevalent.

The relationship between mass and metabolic rate (above), long hoped to have an exponent of 2/3 (0.67) actually has an exponent of 3/4 (0.75) (from Schmidt-Nielsen, 1984; based on Benedict, 1938). Other examples of biological processes with exponents that are multiples of 1/4 include the diameters of tree trunks and aortas (3/8); rates of cellular metabolism and heartbeat (-1/4), and blood circulation and life span (1/4) (from West et al., 1999)

According to recent work by West et al. (1999) the exponents that are multiples of 1/4 are a consequence of the fractal properties of branching networks so prevalent in organisms and the natural selection for the minimal time of transport within these networks. At its most basic level, the exponents that are multiples of 1/4, occur because biological systems operate within 4 dimensions (length to the third power, plus time) rather than the 3 dimensions of Euclidean geometry.

West et al. (1999) believe that this scaling relationship (exponents that are multiples of 1/4) is an inherent and nearly universal property of life.

4. Molecular diffusion vs Eddy diffusion: Biogeochemical Consequences of Size

Perhaps one of the most important consequences of size relates to the changing properties of fluids with size. Very small organisms that live under conditions of low Reynolds numbers (i. e., viscosity-dominated) are slaves to the structure of their physical environment.

In specific, they tend to obtain their needs via molecular diffusion. As a consequence, if left to their own, organisms such as bacteria, tend to live as films or dispersed, in which their metabolic surface area as a population is passively maximized. This has profound implications for the structure of exclusively microbial communities and ecosystems, such as those that existed during most of the Precambrian (see next lecture).

Molectular diffusion tends to be very slow over distance orders of magnitude greater than the size of individual bacteria, and hence molecular diffusion of material through the environment is often a rate limiting process for microbes.

However, larger organisms, living at higher Reynolds numbers, can move fluids and materials en mass. They do not have to rely on molecular diffusion to bring them materials at length scales greater than themselves, and hence molecular diffusion is not a rate limiting process at their macroscale. For example, many polychaete worms can live in burrows in low-oxygen mud and actively pump fluids through their burrows, aerating them.

As a consequence of size, large organisms tend to dominate the physical structure of their environments, and very small organisms passively organize their communities to the larger structure.

Because the size of organisms is a function of their hierarchical organization, and since that has evolved dramatically through time, the biogeochemistry of ecosystems has changed enormously over the 3.5 billion years or so of life on Earth.

5. Key Evolutionary Innovations

Unlike purely physical processes, biological systems can overcome physical constraints by novel design. This is generally called adaptation. However, sometimes the innovation is of such great consequence that it opens the door to an entire universe of new possibilities. These we call key innovations. Over the next few lectures we will discuss the history of the appearance of major key innovations and their implications.

References

Galilei, G., 1637, Dialogues Concerning Two New Sciences (translated by H. Crew and A. De Salvio). New York, Macmillan, 1914, 300 pp.

Schmidt-Nielsen, K., 1984, Scaling: Why is Animal Size so Important. Cambridge, Cambridge University Press, 241 p.

Vogel, S., 1994, Life in moving fluids. Princeton, Princeton University Press, 467 p.

West. G. B., Brown, J. H., Enquist, B. J., 1999, The fourth dimension of life: fractal geometry and allometric scaling of organisms. Science, v. 284, p. 1677-1679.

Updated January 31, 2006
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