ECOLOGY AND EVOLUTION LABORATORY
UNIT IV: MORPHOLOGY, GROWTH AND ALLOMETRY
WEEK ONE: April 15, 1998
This lab is meant to acquaint you with some of the challenges and methods associated with the analysis of biological form. It is composed of two main parts. In the first part, we introduce you to the concept of allometry, and examine the consequences of growth on biological shape. The second part gives you the opportunity to quantify the differences in skull shape that characterize different dog breeds, differences that your eye will readily see.
I. ALLOMETRY
Allometry is the study of the correlated growth of parts, and it is a useful analytic technique for understanding the relationships that exist between size, shape and function. In this lab, we will be paying particular attention to the consequences of increase in body weight on a variety of different morphological features: femur strength (cross-sectional area), canine length, cranial capacity, orbit diameter and palate (upper jaw) length. We will be looking at the relationships that exist between these variables both between and within species. Keep in mind that we can also look at the allometric consequences of increased size by measuring cranial capacity or canine length at different ontogenetic stages within a single species, or even a single individual. Today, however, we will look at the broader evolutionary picture that emerges from the study of different species within a closely related group. In addition, we will explore the intraspecific allometries of these measurements by focusing on variation within a human population.
A simple example of allometry:
Imagine a cube of edge 1. We could, if we had nothing better to do, ask the question "what happens to the perimeter of one face, the surface, and the volume of this cube if we double the edge length to 2?" Do these dimensions simply double? In the case of the perimeter of one face, it does indeed simply double, from 4 to 8. We say that perimeter is linearly related to edge length, increasing by the same factor as does edge length. Surface, however, does not scale linearly, increasing instead as the square of the edge. Volume, in turn, scales as the cube of the length, going from 1 to 8 in our example. How then might we express the relationship between surface and volume? It is here that we use an allometric relationship indicating that one variable increases as a power function of another-- the general form of allometric equations is Y= kXa, where a is referred to as the allometric coefficient. In the case of our growing cube, we can construct the allometric relationship as follows:
If S a l2
and V a l3 (where a means 'is proportional to')
then S3 a l6
and V2 a l6
thus S3 a V2
or S a V2/3 (in the case of a cube, S= kV2/3)
A function like the one we have just derived will appear as a curve when plotted on regular (arithmetic) graph paper. But scientists on the whole prefer to work with straight lines, rather than curves. How might we go about getting a straight line from an allometric equation? One possibility involves effecting a log transformation-- rather than plotting surface against volume, we plot the log of surface against the log of volume. This transformation can be carried out quite simply, and results in the following equation
log S = log k + 2/3 log V
This is, of course, the equation for a straight line of slope 2/3 and y-intercept k. We now have a convenient graphical method for finding the allometric coefficient: it is simply the slope of the line plot drawn on log-log paper.
The above example is one where shape does not change as size increases, since we are always dealing with a cube. In the case of organisms, it is often necessary to introduce functional considerations when we look at the consequences of size increase. The need for a given function to be maintained often means that organisms, or parts of organisms, must change shape as the organism grows. Nonetheless, it is still possible to carry out empirical studies to determine the allometric relationship that exists between two variables.
Here is an example that focuses on an interesting physiological variable, sleep, in relation to body size. If we measure the amount of sleep (average) that characterizes homeotherms as a function of body size, we obtain the following curious result:
You should note that the axes are both logarithmic-- in effect you are looking at the log of sleep duration as a function of the log body weight. Surprisingly, there is a reasonably tight fit between these two variables. In short, a constant relationship between the size of the animal and the amount of sleep it needs (or gets).
Note that the relationship is empirically derived -- someone measured these two quantities, and plotted them. The task at hand, of course is to explain this allometric correlation. In effect, there are three issues that need to be addressed:
1) why is there an allometric relationship (and how good is the fit)?
2) what does it mean that the particular slope of the allometric relationship (the allometric coefficient -0.14? What are the mechanisms that link size to sleep length (metabolism? activity? brain activity?)
3) what additional factors might explain outliers? Why, for instance are arctic ground squirrels so clearly above the line, and roe deer below it?
These sorts of questions suggest one of the main uses of these allometric or scaling curves-- they act as a baseline hypothesis, and they permit the identification of deviations that may require additional explanation. Saying that a mouse sleeps more than an elephant may not be a very interesting statement; this curve suggests that a mouse and an elephant sleep about as long as you would expect given their body sizes. We no longer have to give a separate explanation for the elephant's and mouse's sleep habits-- we now see a larger phenomenon, namely the scaling, that requires explanation.
In the data you will be collecting, there may also be some allometric trends, but you will have to think (oh no, horrors!) about how to plot and analyze the data. In the example shown below, the effects of plotting on different types of axes are clearly visible
As these plots show, the same data takes on a very different appearance when plotted on arithmetic, semi-log or logarithmic scales. In the exercises in this lab, you need to think and try different approaches in order to capture the fundamental trends in the data.
Lab Procedure:
Today you will be measuring various aspects of femurs and dog skulls. Please note:
1) These bones are very fragile, HANDLE THEM WITH CARE!
2) EACH MEASUREMENT IN THIS LAB WILL BE TAKEN AT LEAST TWICE -- ONCE BY YOU AND ONCE BY YOUR PARTNER. AVERAGE THE TWO MEASURES WHEN FILLING OUT THE MASTER CHART.
I) FEMUR SHAPES AND STRENGTHS
In this analysis, we are interested in the relationship between femur shape (and strength) and body weight.
1) Examine the femurs before you. Does shape appear to be maintained as size increases?
2) What would you expect, before you do any measuring, the relationship between femur cross-sectional area and body weight to be? What would you expect the relationship between femur length and body size to be?
3) Measure the bone circumference at its narrowest point. From this circumference, and assuming the bone has a cylindrical shape, calculate the cross-sectional area. Plot that area against body weight. Using your plot, can you determine the allometric coefficient, and give the formula relating femur c.s. area and body weight in the form y=kxa. Is there any other measurement that you might take that would better reflect femur strength?
4) Measure femur length, plot against body size, and determine the allometric coefficient.
II. MORPHOMETRICS
In this part of the lab, we ask you to think about the practice of morphometrics. More specifically, we want you to think about how we can quantitatively analyze biological shape. In this example, we are providing you with an amazing collection of dog skulls. As you know, there are radical differences in skull shape between different dog breeds, and you will see those differences immediately once you begin to look at these skulls. The challenge before you is to capture what your eye is clearly telling you with a small number of measurements. In other words, you need to decide what measurements you make on these skulls in order to quantify their variation. There are many possible answers to this question, and you should feel free to discuss this with your lab partners. The trick is to find linear measurements, or ratios, or angles, or whatever describe and quantify skull shape.
We will then ask you to think about how the measurements you have chosen are affected by body size. In other words, is a mastiff simply a scaled up Chihuahua? How can you approach this question empirically? (hint-- see Section I). Do all of these dog breed basically represent variations on a single theme, or do certain breeds suggest that a more drastic shape change has been brought about by artificial selection?
A COUPLE OF TECHNICAL POINTS:
1) WE WILL POST THE SPECIES IDENTITIES, BODY WEIGHTS AND COMPILED DATA ON THE WEB SOMETIME DURING THIS NEXT WEEK.
2) TRY TO LEARN HOW TO USE A BASIC GRAPHING PROGRAM SUCH AS EXCEL, JUMP, CRICKETGRAPH, ETC. THIS WILL HELP YOU GREATLY IN GRAPHING YOUR DATA.
WEEK TWO: April 22, 1998
Today's unit continues our exploration of the consequences of size, and of the relationship between size and shape in biology. We are now incorporating a number of primate skulls into the analysis.
As you did last week, we want you to examine and characterize primate femur size and shape, and to incorporate the measurements into last week's data. In addition, we will be examining a number of other features of the primate skull.
UPPER CANINE SIZE AND BODY WEIGHT
Simple logic tells us that a gorilla will have larger canines than a mouse lemur, but there is little to be gained by this observation, since the gorilla is so much larger an animal. What we might be interested in asking is whether a gorilla has proportionally larger teeth than a lemur, once body weight has been factored out. We can answer this question by looking at the relationship of canine size to body weight, and seeing if all our primates fall on the same line relating these two variables. Here then is another use for allometric analysis-- it allows you to identify those organisms which deviate from the expected canine size as predicted by the allometric analysis. Thus, individuals that fall way below the line linking canine size and body weight have disproportionately small canines for their size (Compared with the other primates in the study), individuals falling above the line have unusually large canines for their body size. Only after we have drawn the allometric plots can we identify those species whose canine size differs from our expectations and hence may require a special explanation. In this case, the allometric line acts as a baseline, enabling us to identify exceptional taxa.
1) Measure, using the calipers, the length of the upper canine in each specimen.
2) Plot, on log-log paper, the relationship between canine size and body weight. Plot each sex (where known) separately. Draw the line of best fit through your points.
3) Can you identify any species that deviate significantly from the line? What sorts of explanations might you suggest for this deviation? (Hint: what function might these canines be performing that would cause selection to favor unusually large canines in certain species? Is there a difference between the male and female allometric plots?
CRANIAL CAPACITY AND BODY WEIGHT
Once again, we are interested in the way in which cranial capacity scales with body weight in primates, both in order to establish the allometric coefficient linking the two variables and to be able to identify significant departures from the line. Ideally, the allometric coefficient might tell us something about the "design problem" being addressed by organisms as their size increases or about the necessary architectural consequences of the way an organism is constructed. In addition, by helping us to identify those species that do not display the cranial capacities we would expect from the allometric equation, the factors that affect cranial capacity can sometimes be identified.
1) Measure the cranial capacity indirectly by measuring cranial length (head length) and cranial width (head width) as indicated on the illustration. (NOTE: for a small number of the skulls, we can actually directly measure cranial capacity by filling the skulls with seed. We will point those skulls out to you). These linear measurements can be (roughly) transformed into cranial capacities using the following formula:
cranial capacity = [({HW+HL}/2)3 * 0.48]
2) Would you expect the relationship between cranial capacity and body weight to be linear or allometric? Why?
3) Plot cranial capacity against body weight, and give the allometric equation linking the two variables.
4) Based on this equation, or on your plot, what would you expect the cranial capacity of a 68 kg. human to be? Does your measure of human cranial capacity match your expectations? Why, or why not?
UPPER JAW LENGTH
Measure the upper jaw length as instructed, and plot as in previous sections. Is a general trend visible from the data? Is this a good proxy measure for changes in skull morphology that accompany hominid evolution?
ORBIT DIAMETER
Measure orbit height and width, as indicated. Average these two measurements to estimate orbit diameter. Plot as in previous sections. Does orbit size follow a general allometric trend? What additional biological or ecological factors (aside from size) might be required to understand orbit size in different primates?
INTRASPECIFIC VARIATION AND ALLOMETRY IN HOMO SAPIENS
You can measure head width and length and canine size on yourself and/or your partner. Plot these measurements against body weight (In Kgs.), again using both semi-log and log graph paper. Plot the two sexes separately, and estimate the line of best fit. Is the allometric slope the same for males and females? Why or why not? Do you find the same allometric relationships as in the previous analyses where we considered a range of primate species? Why or why not?