Scaling, Allometry, and fractal geometry

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<---- Is this is something we need to worry about?

To answer that and other questions, we examine issues of scaling and the constraints that scale imposes on organisms. Scaling is an issue in two distinct contexts:

Remember, ontogeny pertains to the individual, evolution to the lineage. (TV sci-fi plots notwithstanding.) As we will see in a later lecture, that doesn't mean they aren't connected in any way.

Issues of isometric scaling

What is the relationship between an object's size and the mechanical forces working upon it? Take the simple example of a cube, with a measure of 1 cm. in each linear dimension. That gives it:
  • a surface area of 6 cm2
  • a volume of 1 cm3
  • a cross-sectional area is 1 cm2 - equal to the area of its lower surface.
The cross-sectional area is significant because it must receive all of the compressive load of the weight of the material of which the cube is made. I.e., it is proportional to the object's strength, whereas the volume is proportional to its mass. Suppose the cube is made of a material weighing 1 g/cm3. In that case, each cm2 of cross-sectional area is supporting 1 g.

If we increase the size of the cube by a factor of 3 for each linear dimension (i.e. with geometric similarity), the volume and cross-sectional area scale up as inidcated below. Indeed, surfaces typically scale proportionally to the 2/3 power of volume.

If the cube were made of an infinitely strong material, we could scale it up infinitely. Real materials, however, have finite strength - mechanical limits at which they break. As we continue to scale our cube up, it will eventually break under its own weight as higher and higher loads are transmitted through the cross-sectional area.

Isometric Scaling: Scaling where all linear dimensions are scaled to the same factor, causing the scaled objects have the same shape.

In organisms, many physical constraints result from the mechanical limits of cross-sectional area. The obvious example is land vertebrate limbs, which must bear the entire creature's weight. In the case of Allometroraptor pusillius, the little dinosaur on the right, a cross-sectional area equal to a circular cross-section perpendicular to the long axis of either of its limbs must support the entire creature.

When we scale it up isometrically by a factor of two, its limbs are now dangerously overloaded.

To get around the limits of isometric scaling, we could have the cube change shape as it gets larger. In that way its cross-sectional area could keep up with the object's increasing weight. This is allometric scaling.

Thus, in Allometroraptor magnus the thickness of the legs scale allometrically, i.e. more than other measures.

NOTE: In this version, each parcel of bone in its legs now feels the same amount of loading as in Allometroraptor pusillius. This is called scaling with mechanical similarity

Allometroraptor is a contrived example, however allometric scaling with mechanical similarity is also evident in comparisons across real taxa. Small and large members of the same group such as:

Both are members of Artiodactyla, a group whose members typically impose similar mechanical stresses on their limbs. Thus, regardless of size, they move about in basically similar ways.

Scaling with mechanical similarity has limits, too. If the critter's legs scale up much faster than its heart, lungs, and the rest of the life-support system that powers them, those large limbs will be unable to function. Thus, large animals typically have markedly different behaviors than small ones to limit loading and conserve energy.

Scaling of surface area

The last example dealt with the scaling of volume and cross-sectional area. Simialr biomechanical problems pertain to the scaling of mass and any surface area, including overall surface area:
Consider Allometrosaurus parvus, a cold-blooded primitive synapsid. This creature must maintain an optimal body temperature by locating environmental heat sources and sinks. Fortunately, because it is small, it has a higher surface/volume ratio, enabling it to gain or shed heat quickly.

If we rashly scale A. parvus by a factor of two in each linear dimension, even if we remember to scale the limbs with mechanical similarity (as in the illustration on the left) the creature still has a problem. It's surface area is reduced per unit of volume. It must either spend more time warming up and cooling off, or be restricted to environments with stable temperatures.

Allometrosaurus ingens has solved the problem by changing its shape. The cooling fin on its back restores its SA/V ratio to that of Allometrosaurus parvus. Note that this structure can also be put to other uses, such as advertising species identity and breeding status.

This contrived example mimics trends observed in several fossil lineages during the Pennsylvanian and Permian, in which cold-blooded tetrapods including the synapsids Dimetrodon and Edaphosaurus, and the non-amniote tetrapod Platyhistrix independently evolved sail-backs. These structures must have conferred advantages sufficient to outweigh the obvious disadvantages of having to carry them around.

Indeed, since the evolution of more thermally competent predators during the Permian, sail-back have been very rare. If we look at the distribution of contemporary cold-blooded predators like lizards, we see large forms restricted to warm, relatively constant environments, whereas regions with hard winters and variable temperatures support only small-bodied forms.

Mathematics of Allometric Scaling

In the above example, the biomechanical issue was the diffusion of heat across the body surface, but similar issues pertain to the diffusion of gasses, water, and waste products. Thus, allometric scaling is crucially important to organisms, both across their ontogeny and across their taxonomic diversity. Thankfully, scaling across such diversity is easy to handle mathematically using the equation:



This is easily handled when log-transformed as

log y=log a+b(log x)

This yields a linear plot.

The slope of 2/3 characterizes isometric scaling. In allometric scaling, the parameters a and b can vary.
We aren't restricted to comparing surfaces and volumes. The comparison of the scaling of different volumes, such as the volume of body mass (gray) and skeletal mass (red), for instance, shows skeletal mass scaling up faster. Is there a skeletal safety factor?

Scaling and Evolution


Why bother getting big, if it entails problems with gas exchange, support, thermoregulation, etc. Likely advantages include:

In fact, Edward D. Cope (1840-1897) coined "Cope's Rule" - that members of evolving lineages tend to become larger through time. It is, indeed, a common trend, though many counter-examples can be cited (E.G.: the diminution of synapsids as they were eclipesd ecologically by archosaurs during the Triassic.)

The consequence of increased size is increased ecological specialization resulting from allometric scaling. We've already seen the example of the sail-backs. Dimetrodon was the dominant land predator of its environment, thanks to its relatively large size. It's hard, however, to imagine it climbing a tree or running under a low branch in pursuit of prey.

Scaling and Paleontological Reconstruction


It's good to be mindful of scaling issues in interpreting either the evolutionary processes that produced and organism or its behavior.
The largest deer ever, the Irish Elk (Megaloceros giganteus) from the Pleistocene (late Neogene) of Eurasia became extinct about 11,000 years ago. It was not a true elk - most closely related to the fallow deer, and has long been noted for its enormous antlers, up to 4 m. across. This has led to endless speculation - "just-so stories" on the evolutionary selective advantages of having them.

Plotted against the general diversity of cervid antler size, we see that they conform to the general scaling trend line. It is, thus, possible that they are a simple consequence of the deer's size. Indeed, the size of its scapulae and thoracic cavity also conform to the general allometric trend for cervids. Perhaps selective pressure was simply for large size in these animals or for consequences of large size such as offspring that were well-developed at birth.

Fractal Scaling

But this is all an oversimplification

A: Not all biological structures scale with simple allometry. Consider:

In living organisms, these are usually objects displaying fractal geometry. Such objects represent "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole." (Benoit Mandelbrot born 1924).
In the example of the "Koch curve," An equilateral triangle is inserted into the middle third of a line segment. This process is reiterated with the resulting line segments, and so forth. The result is a shape that displays self similarity at different scales.

Branching networks in biological systems behave similarly. Here is a contrived example in which the same branching pattern is reiterated at three scales.

And a real example - mammalian lungs. The dark outline on the left is an actual tracing of human bronchi, the schematic on the right is a computer-generated fractal representation. Measured across mammalian diversity, lung surface tends to scale to the 3/4 power of mass. Note: systems and networks that grow by iterative fractal branching exhibit developmental plasticity that allometrically scaling structures lack, and allow tissues and organs to respond adaptively to unusual circumstances.

Mathematics of Fractal Scaling

And yet even fractal scaling is mathematically simple. The fractal extent is given as



What we really care about is D, the fractal dimension, that allows us to characterize a fractal object in terms of its one, two, or three-dimensionality:

To arrive at this, we must also consider step length. In the Koch curve above, a 1 cm. scale is provided. If we measured each line only to the nearest cm, we would be able to distinguish the first and second iterations from one another, but not the third and fourth. In this case, the step length is 1 cm. To improve our approximation of the actual line lengths, we need a smaller step length and more steps. If we do this, the length that we measure in the third and fourth lines will be greater.

Indeed, the fact that an object's measurement increases with increasingly small step lengths is a clear indicator that is has fractal geometry. Consider:

Great Britain


Which has the more nearly fractal profile?

Fractal scaling is responsible for a fantastic range of features that we see in organisms, from tissue development to coloration and coat characteristics. In summation, scaling of structures depends on relative importance of allometric & fractal factors