Compel your students to analyze evolution by involution of these intriguing cases.*
*in-vo-lu-tion. 1. a: the act or an instance of enfolding or entangling: involvement. b: complexity, intricacy. 2. exponentiation: the mathematical operation of raising a quantity to a power. 3. a: an inward curvature or penetration. (http://www.merriam-webster.com/dictionary/ involution)
Thomas Henry Huxley's reaction to Darwin's argument for natural selection is reported to have been "How truthfully simple. I should have thought of it." Each premise is truthfully simple, and the conclusion follows directly from them:
* Each and every species has a potential for geometric increase, yet rarely will all individuals in a population produce offspring according to this potential.
* This difference is not random: Some fare better than others because of heritable attributes that they gained from their parents and will most likely pass on to their offspring.
* The result is a natural form of selection within a variable population, leading to greater representation of heritable traits that confer reproductive and/or survival advantage, ergo "fitness."
Do our students really appreciate the simple power of this argument? Can they understand the nuances that have fostered lively discussion, theoretical progress, and clever investigations over the past 150 years? Hopefully, students have gained a qualitative understanding by the end of a unit on evolution. We should help them develop a quantitative understanding as well.
To that end, I use the case studies described here to estimate the power of selection mathematically. I challenge college freshmen in a biology majors course to develop confidence in quantitative methods, ask questions about the evolutionary process that would not have occurred to them otherwise, and apply critical-thinking skills. These case studies develop skill by repeatedly using a powerful tool and concept: the algebra of exponential growth.
While these may not be "case studies" in the traditional narrative sense, they are based on real situations described in outside readings that I assign to supplement the textbook (Raven et al., 2008). I begin the semester with a unit on evolution, introducing the Hardy-Weinberg equilibrium, mutation, selection, genetic drift, and the founder effect. For these "challenges," the students are encouraged to work together in finding solutions, but each is responsible for written discussion in his or her own words. For this article, I've abridged the examples, included blanks and tables that the students fill in, and inserted solutions for ABT readers in square brackets. First, I introduce the method with an intuitive example that raises issues relevant to our first case study, the growth of the world's human population.
* The Method: Exponential Population Growth
Let [w.sub.0] = [1.3 x [10.sup.8]] be the world's population at the beginning of year 1 AD and [w.sub.50] = [2.6 x [10.sup.8]] be the world's population 50 generations later, at 1000 AD. You can find [w.sub.0] and [w.sub.50] on page 1161 in your textbook. Use scientific notation. To find w use [w.sub.1] = [w.sub.0](1 + r) where r = the rate of population increase. To find w use [w.sub.2] = [w.sub.1] [(1 + r).sup.2] and, by substituting the definition above for [w.sub.1], find that [w.sub.2] = [w.sub.0](1 + r)(1 + r) = [w.sub.0] [(1 + r).sup.2]. This means that generally, after n generations, we can find any [w.sub.n] = [w.sub.0] [(1 + r).sup.n] if we know r, which is powerful in itself, but first we must find r. In this case, the world's population doubled in 1000 years (n = 50 generations, assuming 20 years/generation), so [w.sub.50]/[w.sub.0] = 2 = [(1 + r).sup.50]. To solve for r, we follow the steps and calculations in the first two columns of Table 1. Overall, the population was growing 1.4% in each generation. This growth rate is the birth rate minus the death rate. We are not saying that growth was steady; we only describe its averaged rate per generation. This is essentially how compound interest is calculated. The world's population doubled again from 1000 to 1650. Was this due to a higher growth rate or the greater initial population ([w.sub.50] above)? This means that the ratio [w.sub.n]/[w.sub.0] is 2 again, but this time spanning n = 32.5 generations (650 years), so [w.sub.n=32.5]/[w.sub.0] = 2 = [(1 + r).sup.32 5]. Follow the steps and fill in the blanks on the right-hand column of Table 1. What is your new value for r? [2.12%].
To segue into the first case study, I invite the class to hold a "BS" session (for "brain storming," of course) on the question "What does it mean that our rate of increase keeps increasing?" Historically, it could mean that death rates declined for a variety of reasons or that birth rates increased. Biologically, greater population density could beget faster population growth (e.g., the Allee effect), although it has also contributed to the spread of devastating epidemics and famine. Mathematically, the doubling time gets shorter and it seems that no single exponential rate is adequate for all times, but this is biology and conditions change. Our model still reflects the geometric-growth argument of Thomas Malthus that inspired Darwin's first premise.
* Case 1: Iron Man (or Woman)
1 assign the Scientific American article "Founder Mutations" (Drayna, 2005). Drayna discusses how haplotypes (consistent DNA regions) can be used not only to trace lineages but to estimate the age of origin of a mutation. A mutation in a gene called HFE (for a protein that normally modulates iron uptake) was traced to a single founder individual estimated (by comparing haplotypes in living descendants) to have lived around 800 AD. Like the sickle-cell allele, this mutant allele has different effects in heterozygotes and homozygotes. Drayna (2005) mentions a published estimate that 1 in 10 Caucasian Americans (22 million) are carriers. Homozygotes, especially males, can suffer from hemochromatosis, a condition of iron excess in the blood due to enhanced absorption of the element, and must maintain low-iron diets to avoid toxic levels. This sets the stage for a nuanced view of natural selection "for" a disease-causing mutation.
With our exponential model, we have a tool to ask how the growth of the population in general over the past 1200 years compares to a special subset of the population--descendants of our first "iron man" in whom the founder mutation occurred. I first ask: "Based on a Punnett Square, what are the odds that a carrier of a new mutation will leave carrier descendants? How likely are carriers to mate with other carriers?" Any new allele in a diploid species will be very rare initially and vulnerable to loss by genetic drift. Yet all 22 million American carriers are descended from one founder individual, most likely a northern European whose descendants survived the Black Death and untold hardships. This created great variation in Europe's population growth. To get a more accurate "baseline" estimate of r (using better population estimates) we begin with McEvedy and Jones's (1978) estimate of 65 million Europeans in 1500 and end with 297 million in 1900 (before the growth rate in Europe abated). I direct the students to follow the steps outlined in Table 1. So, we have 297/65, or a 4.56-fold increase in just 400 years (20 generations); thus, 4.56 = [(1 + r).sup.20], which yields r = 0.079, or 7.9%.
I instruct the students: "Now compare your result to the growth rate of HFE carriers in the population. The article says there are 22 x [10.sup.6] carriers of the allele in America, mostly of European descent. Around 800 AD, there was only one. Over the intervening 60 generations, find [r.sub.c], the growth rate of carriers (c). Use [c.sub.60]/[c.sub.0] = 22 x [10.sup.6]. Follow steps 2-6 in Table 1: log 22 x [10.sup.6] = 60 log (1 + [r.sub.c]). Solve for [r.sub.c] [0.325]."
Next it's time to initiate another BS session--this is a 4-fold increase over our 7.9% baseline estimate for Europeans in general. We didn't even consider all the carriers who stayed in Europe or who emigrated to other continents. Clearly something has favored the spread of this allele. But is it selection? I ask the students: "Could the ups and downs in Europe's population before 1500 have been as significant as (or more significant than) the period of steady growth that followed (see fig. 55.25 in Raven et al., 2008)? If the mutation results in enhanced iron absorption from food, when might that be an advantage? Can you relate this to another allele you learned about that is advantageous in the heterozygote under certain environmental conditions [e.g., the sickle-cell allele]? Could there have been a founder effect among immigrants to the New World? Did this allele help them to endure the hardships of a prolonged sea crossing or to establish settlements? If so, what difference would one expect in HFE frequency among Europeans who stayed behind? Why was bloodletting (often using leeches) a popular medical treatment in the Middle Ages? Could extra iron in the blood be protective against certain diseases or perhaps be advantageous in utero? Why are symptoms 3 times as common among males?"
Students can find more information about how this particular mutation results in hemochromatosis, and to what degree, at Online Mendelian Inheritance in Man (http://www.ncbi.nlm.nih.gov/entrez/ dispomim/cgi?id=235200). The Population Genetics section cites references to how a common HFE mutation called C282Y occurs at even higher frequencies in European populations of Celtic ancestry than among Americans and a probable selective advantage reported among other diasporas. Other causal links are still being investigated. The gene belongs to a class of HLA genes to which it is linked and with which it possibly shared selective success. Not all studies showed a significant increase in iron absorption by heterozygotes over noncarriers. Some have concluded that the mutation occurred within the past 2000 years in Northern Europe, spreading with Celtic or possibly Viking populations. Distante et al. (2004) suggested an earlier origin (circa 4000 BC) during the shift from hunting to agriculture in Europe.
* Case 2: Back from the Dark Side (May the Force Be with You)
Darwin discussed animal breeding in On the Origin of Species so that his readers would recognize what a "trait" was and realize that natural selection could act similarly: changing the frequency of traits by favoring the individual (or not). R. A. Fisher, J. B. S. Haldane, and Sewall Wright redefined evolution as the change in allele frequencies in a population. This distinction is worth exploring more deeply. We'll use one of the "textbook examples": the change in the melanic (dark) morph of the peppered moth (Biston betula). This also allows you, the teacher, an opportunity to present the scientific consensus that natural selection is implicated, in spite of the creationists' maligning of H. B. D. Kettlewell (Cook, 2000, 2003).
While Kettlewell was at Oxford studying the role of predators as the agent of selection, the ethologist Niko Tinbergen helped to film birds selecting moths against different bark backgrounds (Cook, 2000). Tinbergen famously distinguished proximate ("how") from ultimate ("why") questions in behavior. I ask the students to apply this dual concept of causation to natural selection: "Is it better to measure selection by a change in traits or by a change in alleles?" Both must be plural, because an increase in one thing (trait or allele) necessitates a decrease in another (as frequencies sum to one). Assign an ad hoc debate by dividing the class into an alleles camp and a traits camp. Alleles hold merit as an ultimate cause, whereas traits are the proximate features that selection can choose among and that observers can quantify more directly.
I assign (and recommend for all Biology majors) Sean Carroll's Making of the Fittest. In chapter 2 (Carroll, 2006: p. 52), he discusses the peppered moth story:
In just fifty years, from around 1848 to 1896, the dark form arose and evolved to as high as a 98 percent frequency in some areas [of Great Britian]. Haldane estimated from surveys of moth types taken over this period that the selection coefficient against light moths on dark trees was on the order of negative 0.2.
The students have been applying exponential growth, so I ask them to estimate the rate of increase in the dark morph with a "detectable" starting frequency of 1 in 500 in 1846 and ending with 96% in 1896. I set up the exponential model: 0.96 = 0.002 [(1 + s).sup.50] and, by following the steps of Table 1, we arrive at s = 0.13, or 13%. We use s instead of r in this case because we're describing selection strength. Students should iterate the function [m.sub.n+1], = [m.sub.n] (1.13) to make a table in Microsoft Excel[R] and graph it. The rise in melanic moths follows a J-shaped growth curve, but is this entirely realistic? Models of selection strength typically predict the increase of a favored allele to be sigmoid, like logistic population growth (Cook, 2003). The rise in the dark morph prior to 1900 was not as well documented as its fall after 1950, which resembles an inverted logistic growth curve (see fig. 2.2 in Carroll, 2006). If your students have learned about logistic population growth, ask them to discuss what factors could cause a deceleration phase in the dark morph's rise, particularly if predators were the agent of selection (e.g., through search image).
We will use Haldane's 1932 version of the selection coefficient: "... if in one generation the ratio of A to B (two alleles) changes from r to r(1 + k) we shall call k the coefficient of selection" (Haldane, 1990: p. 53.) Here, r denotes the ratio of alternative alleles and k the "rate" factor. The correspondence of k to s in our model is obvious when one expands to n generations and we modify it to use the ratio of alleles before and after: [p.sub.n]/[q.sub.n] = [p.sub.0]/[q.sub.0] [(1 + k).sup.n]. Note how Haldane used the ratio of alleles rather than of phenotypes to measure the strength of selection. The use of alleles allows us to contrast selection on dominant and recessive forms. So we can ask, "Is a rare, recessive mutant allele more or less likely to survive against genetic drift, and will it take longer to increase under similar selective intensity (k)?" I ask the students to plot an Excel[R] graph showing the increase in melanic moths from 1846-1896, then I have them add a graph showing Haldane's model for the melanism allele's increase. They use k = 0.18 from Table 2 and, beginning with [r.sub.1846] = 0.001/0.999 in the first cell (say, A1), they increment the next cell by the formula "= A1*(1 + 0.18)" and so on until [r.sub.1896]. In B1, they convert r to p by the formula "=A1/(1 + A1)" and copy that into the rest of column B. Then they graph column B. The graphs are compared in Figure 1. The phenotype shows the J-shaped curve, whereas the allele rises in an S-shaped (sigmoid) curve.
Before we contrast what happened to alleles for the dark morph (dominant) versus the wild type (recessive) in the peppered moth story, let us predict which should be more "responsive" to selection. Recessive alleles, even when they are disadvantageous, can initially increase in a population as a result of drift, because they are "under the radar" of natural selection in heterozygotes and, while they are rare, it is unlikely that random mating among the dominant phenotype will result in recessive phenotypes in the offspring. By contrast, a dominant allele's effect on fitness cannot be as covert. If it were advantageous, it should begin to increase in frequency more quickly, especially when rare (because fitness is always relative to the population as a whole.) The students draw population Punnett squares (after fig. 20.3 in Raven et al., 2008) with different values of p's and q's to map the relationship between allele and phenotype frequencies. If you apply the Hardy-Weinberg theorem to our situation in the 1950s, when dark moths constituted 96% of the population and 4% the recessive phenotype ([q.sup.2] = 0.04 and q = 0.2), you get the surprising result that one in five alleles is the disadvantageous recessive. (Technically, Hardy-Weinberg does not apply, because there is a selective disadvantage to one of the alleles. Ridley  provides an adjustment for such an allele [recessive phenotype frequency is [q.sup.2](1 - s)/(1 - [sq.sup.2])] and a simulation at http://www.blackwellpublishing. com/ridley/experiments/genefreq.asp.)
[FIGURE 1 OMITTED]
So I set up our problem: "Industrial melanism is a fascinating example--the same effect was measured in England and the northeastern United States in the same species as pollution increased, and then the trend reversed on both sides of the Atlantic following clean-air legislation. As light-colored moths rebounded from 1959 to 1995 (4% to 81% in Carroll's  fig. 2.2), what was the selection coefficient for the light-colored allele?"
If 19% is the 1995 frequency of dark moths, this corresponds to [p.sup.2] + 2pq in the Hardy-Weinburg formula, then 1 - 0.19 = [0.81] is [q.sup.2] so [q.sub.1995] = [0.9] and p1995 = [0.1]. The ratio [q.sub.1995]/[p.sub.1995] = . If 96% was the frequency of dark moths in 1959, then 4% or 0.04 is [q.sup.2], so [q.sub.1959] = [0.2] and [p.sub.1959] = [0.8]. The ratio [q.sub.1959]/[p.sub.1959] = [0.25]. We will use these allele ratios in our exponential-growth model, following Haldane: [q.sub.1995]/[p.sub.1995] = [q.sub.1959]/[p.sub.1959] [(1 + k).sup.36] Follow steps 2-5 in Table 2 to find k = [0.043 or 4.3%]. Make a graph in Excel[R] by incrementing the ratio (r) of q/p from 1959 to 1995 by iterating [r.sub.n+1], = [r.sub.n] (1 + k) in 36 cells of a column. Then, in the cells of the next column, convert r to q by the formula q = r/1 + r Then graph the rise in q over the 36-year period. Note the shape of the curve that results [gently sigmoid].
Table 2 illustrates how this model measures selection for and against the melanism allele in both periods (note that in the latter, k is negative).
If we consider 1 in 500 a likely frequency when dark moths were first detected (around 1846), these were almost certainly heterozygotes, so the initial value of p would be 1 in 1000 [(0.001] and q would be [0.999]. The steps and worked example show the estimation of k for the first period. Compare your result in the right-hand column to the k you obtained for the wild-type allele. Do you think that the recessive effect of the wild-type allele influenced the difference seen in its rate of rebound versus the dominant allele's earlier rate of increase? Why?
The rate of selective change depends on the rate of environmental change, and there's no evidence that the "sooting" and "unsooting" of tree trunks had similar rates. But we have shown that the rise of the dominant allele (per generation) was four times as fast as the rise of the recessive allele (18% vs. 4.3%), which is impressive considering that the dark allele was very rare to begin with, but perhaps not so surprising considering that a "fit" pale moth that mates with an "unfit" dark moth has, at best, one "fit" offspring in every two. But a fit dark moth can expect one fit offspring in every two if it mates with a wild-type moth and at least three out of four otherwise. The dominant allele rises more quickly than a recessive allele when rare (for a given selective advantage). The wild type rebounded quickly, but the allele had a higher covert frequency to begin with.
* Case 3: "... if it is to be of any service": Where Darwin Disagreed with Wallace
Surveys of what college freshmen understand about evolution often reveal a comfortably progressive or providential notion that we recognize as teleology. J. B. Lamarck is remembered today for his idea of the inheritance of acquired characters. But in Darwin's time, with the popularity of Paley's natural theology and an ignorance of genetics, Lamarck's was a popular and palatable view of evolution. And perhaps no topic brought these sentiments into focus more than the question of humanity and our oversized brain: ".and when Darwin received his copy of an article Wallace had written on this subject he was obviously shaken. It is recorded that he wrote in anguish across the paper, 'No!' and underlined the 'no' three times heavily in a rising furor of objection" (Eiseley, 1957: p. 79). I assign Eiseley's essay "The Real Secret of Piltdown" as background reading, along with Steadman et al. (2004), and use these to ask "How explosive was the growth of cranial capacity in hominids?" and "When and why did it begin?"
Wallace's ideas challenged the Great Chain of Being and Victorian notions of a racial hierarchy:
It is a somewhat curious fact that while all modern writers admit the great antiquity of man, most of them maintain the very recent development of his intellect, and will hardly contemplate the possibility of men, equal in mental capacity to ourselves, having existed in prehistoric times. (Wallace, 1876: p.114)
However, this was not Darwin's principal objection. That came earlier, when Wallace wrote that "Natural Selection could only have endowed the savage with a brain a little superior to that of an ape, whereas he actually possesses one very little inferior to that of a philosopher" (Wallace, 1870: p. 356). Wallace had been a more ardent selectionist than Darwin, yet here he is requiring some other force beyond selection to account for our brains, adding a too-familiar refrain: "... some higher intelligence may have directed the process by which the human race was developed" (Wallace, 1870: p. 359; the quotations above are given in Eiseley's  essay; Wallace's papers are available at http://www.wku. edu/~smithch/wallace/cited.htm).
Since Darwin and Wallace, we've learned much about the increase in cranial capacity among hominids over the past 4 million years. Though incomplete, the fossil record allows us to again apply our exponential model to brain growth. In the fifth edition of Raven and Johnson (1999), figure 22.17 nicely compares the cranial capacity of the hominids. Southwood (2003) provided the needed data, and students must complete the shaded cells in Table 3. I provide the answers for Homo habilis as a worked example and leave the Neanderthals blank for extra credit (and further discussion). To account for the allometry of brain size to body size, we use the scaling relationship for nonhuman primates provided by Schmidt-Nielsen (1984): brain mass = 0.03 body [mass.sup.0.7]. We then can find the excess cranial capacity (xcc) of fossil hominids.
Steadman et al. (2004) offered this clue: about 2.4 million years ago, a frame-shift mutation occurred in a gene for heavy-chain myosin that was normally expressed in the muscles of mastication derived from the first pharyngeal arch. This results in a smaller temporalis muscle that reaches from the vault of the cranium onto the jaw. This affects the prenatal growth of other skull regions. As the postnatal period of suckling became extended in hominids (and soft foods were offered thereafter), fusion of the sutures of the cranium (which ultimately limits cranial capacity) is delayed. This stands in contrast to Paranthropus, the lineage of robust hominids whose massive jaw muscles envelop a much smaller cranium, often with a sagittal crest for extra attachment surface.
We can now ask whether the growth rate of excess cranial capacity in hominids show a marked increase after 2.4 million years ago. The students calculate the rate of increase from Australopithicus afarensis ("Lucy") to H. habilis from 4 to 2 million years ago. We will express this rate per 1000 years, rather than per generation. The now-familiar formula becomes xccHh = xccAa [(1 + r).sup.2000]. From Table 3, we get 154 = 88 [(1 + r).sup.2000] and, solving as before, 154/88 = [(1 + r).sup.2000]. So log 1.75 = 2000 log (1 + r) and r = 2.8 x [10.sup.-4]. This is only 0.028% each millennium, which is hardly "explosive" compared with our other examples of the power of selection. Next, the students do the same operation from H. habilis to H. sapiens: xccHs = xccHh [(1 + r).sup.2000]. Substituting: 896 = 154 [(1 + r).sup.2000]. So, log 5.82 = 2000 log (1 + r) and r = 8.8 x [10-.sup.4] or 0.088%, a 3.14-fold increase in rate! Still, on a per-millennium basis, does 0.088% require us to invoke a supernatural intervention, as Wallace did? Even this rate would be statistically undetectable on anything but a "deep time" scale. There was nothing "sudden" about the expansion of our brains, but the rate of increase was greater over the past 2 million years. This is not to say that the increases over these vast periods were steady, nor that all these fossil hominids are in our direct lineage. One successful side branch, H. erectus, showed little change in cranial capacity over a much longer tenure than ours. A helpful chart and discussion of hominid cranial capacity over time are available at http://pandasthumb.org/archives/2006/09/fun-withhomini.html.
Students, even those who are open to "deep time" thinking, have been surprised by this. In their discussions, the students are invited to offer other potential forces acting on our ancestors, including neoteny. Our brain, even before the last Ice Age, had enough room for ideas like the Internet and the Human Genome Project and the capacity to recognize (and hopefully limit) the impact of our growing population on this planet. Does this mean that we have carried the burden of the brain's high metabolic cost and difficulty at birthing only in anticipation of today's genius? Wallace became somewhat of a mystic about this in his later years, whereas Darwin maintained a skeptical, guarded stance. He realized that our ancestors weren't involved in competition with apes so much as with each other. These struggles could well have accrued advantage to the prescient, the strategic, and the inventive ones.
Stephen J. Gould (2003) cautions us about the fallacy of equating current utility for reason of origin, and in the same book he repeats his favorite Darwin quotation: "How odd it is that anyone should not see that all observation must be for or against some view if it is to be of any service." This is exactly why we do this. Students may empathize with Wallace or Darwin, or wonder why "bad alleles" can be common. We explore these case studies not just to describe, or learn a new tool, but (hopefully) to inform our views. How piddling this growth of our most esteemed organ seems to be when compared with the rise of a disease-causing allele or the fickle fate of a moth. One can begin to address such comparisons by calculating the power of selection.
* Applications & Extensions
For each case study, I've raised many questions for student discussion. These are three of seven "Friday Challenges" I assign during the semester to generate opportunities for discourse, critical thinking, quantitative analysis, and written responses that show ability to integrate different sources and ideas. Were it not for space limitations, I would have included some of the more insightful student responses. Regardless, I feel it is important to challenge them in this way if they are to develop the reasoning skills to continue in biology.
Doubtless there are many other case studies of invasive species, epidemiology, or habitat fragmentation to which this method could be applied, and there are much more sophisticated, calculus-based models for the cases I use. But the value of these case studies is heuristic: it is vital that we begin with methods that students can comfortably master (repetition is important), and the value lies more in the new perspectives and questions than in the accuracy of any given model.
Carroll, S.B. (2006). The Making of the Fittest: DNA and the Ultimate Forensic Record of Evolution. NY: Norton.
Cook, L.M. (2000). Changing views on melanic moths. Biological Journal of the Linnean Society, 69, 431-441.
Cook, L.M. (2003). The rise and fall of the carbonaria form of the peppered moth. Quarterly Review of Biology, 78, 399-417.
Distante, S., Robson, K., Graham-Campbell, J., Arnaiz-Villena, A., Brissot, P. & Wor wood, M. (2004). The origin and spread of the HFE-C282Y haemochromatosis mutation. Human Genetics, 115, 269-279.
Drayna, D. (2005). Founder mutations. Scientific American (September), 78-85.
Eiseley, L. (1957). The Immense Journey: An Imaginative Naturalist Explores the Mysteries of Man and Nature. NY: Random House.
Gould, S.J. (2003). The Hedgehog, the Fox, and the Magister's Pox. NY: Three Rivers Press.
Haldane, J. (1990). The Causes of Evolution. [Reprint of 1932 edition.] Princeton, NJ: Princeton University Press.
McEvedy, C. & Jones, R. (1978). Atlas of World Population History. NY: Viking.
Raven, P. & Johnson, G. (1999). Biology,5th Ed. NY: WCB/McGraw-Hill.
Raven, P., Johnson, G., Losos, J., Mason, K. & Singer, S. (2008). Biology, 8th Ed. NY: McGraw-Hill.
Ridley, M. (2003). Evolution, 3rd Ed. Hoboken, NJ: Wiley-Blackwell.
Schmidt-Nielsen, K. (1984). Scaling: Why Is Animal Size So Important? Cambridge, UK: Cambridge University Press.
Southwood, R. (2003). The Story of Life. Oxford, UK: Oxford University Press.
Steadman, H.H., Kozyak, B.W., Nelson, A., Thesier, D.M., Su, L.T., Low, D.W., Bridges, C.R., Shrager, J.B., Minugh-Purvis, N. & Mitchell, M.A. (2004). Myosin gene mutation correlates with anatomical change in the human lineage. Nature, 428, 415-418.
Wallace, A.R. (1870). The limits of natural selection as applied to man. In The Action of Natural Selection on Man. New Haven, CT: C.C. Chatfield. (archived at: http://web2.wku.edu/~smithch/wallace/s165.htm).
Wallace, A.R. (1876). Rise and Progress of Modern Views as to the Antiquity and Origin of Man. In "Notices and Abstracts of Miscellaneous Communications to the Sections" portion of the Report of the British Association for the Advancement of Science 46 (1876) (John Murray, London, 1877): 100-119 (archived at http://web2.wku.edu/~smithch/ wallace/s257.htm).
WILLIAM BEACHLY is Professor in Biology at Hastings College in Hasting, NE 68901; e-mail: firstname.lastname@example.org.
Table 1. World population doubled from 1 AD to 1000 AD and again from 1000 AD to 1650 AD. The exponential model is solved for the first period. Students complete the shaded values in solving the exponential model for the second period. World Pop. 1-1000 AD The Exponential Model (50 generations) [w.sub.n] = [w.sub.o] 2.6 x [10.sup.8] = 1.3 x [(1 + r).sup.n] [10.sup.8] [(1 + r).sup.50] Step 1: [w.sub.n]/[w.sub.o] 2.6 x [10.sup.10]/1.3 x on the left [10.sup.10] = [(1 + r).sup.50] Step 2: Logarithm of both sides log 2 = 0.301 = 50 log(1 + r) Step 3: Divide by the power (n) 0.301/50 = 6.02 x [10.sup.-3] = log (1 + r) Step 4: Take the antilog [10.sup.(6.02 x 10-3) = 1.014 = ([10.sup.x]) 1 + r Step 5: Minus 1 1.014 - 1 = 0.014 = r Step 6: Find r as a % r = 0.014 x 100 = 1.4% World Pop. 1000-1650 AD The Exponential Model (32.5 generations) [w.sub.n] = [w.sub.o] 5.2 x [10.sup.8] = 2.6 x [(1 + r).sup.n] [10.sup.8] [(1 + r).sup.32.5] Step 1: [w.sub.n]/[w.sub.o] 5.2 x [10.sup.10]/2.6 x on the left [10.sup.8] = 2 = [(1 + r).sup.32.5] Step 2: Logarithm of both sides log 2 = 0.301 = 32.5 log (1 + r) Step 3: Divide by the power (n) 0.301/32.5 = 9.3 x [10.sup.-3] = log (1 + r) Step 4: Take the antilog [10.sup.9.3] x [10.sup.-3] = ([10.sup.x]) 1.021 = 1 + r Step 5: Minus 1 1.021 - 1 = 0.021 = r Step 6: Find r as a % r = 0.021 x 100 = 2.1% Table 2. Applying the exponential model to changes in the frequency of the melanism allele in the peppered moth over a period of increase (1846-1896) and one of decrease (1959-1995). Students complete the shaded values for the second period. Exponential Model Melanism Allele Increase (50 Generations) [p.sub.n]/[q.sub.n] = (0.8/0.2) = (0.001/0.999) [p.sup.0]/[q.sup.0] [(1 + k).sup.50] [(1 + k).sup.n] Step 1: Divide by [p.sub.o]/ 4/1x [10.sup.-3] = 3996 = [q.sub.0] [(1 + k).sup.50] Step 2: Logarithm of both sides log 3996 = 3.6 = 50 log (1 + k) Step 3: Divide by the power (n) 3.6/50 = 7.2 x [10.sup.-2] = log (1 + k) Step 4: Take the antilog (10x) 10 7.2 x [10.sup.-2] = 1.18 = 1 + k Step 5: Minus 1 1.18 - 1 = 0.18 = k Step 6: As a % k = 0.18 x 100 = 18% Exponential Model Melanism Allele Decrease (36 Generations) [p.sub.n]/[q.sub.n] = (0.1/0.9) = (0.8/0.2) [p.sup.0]/[q.sup.0] [(1 + k).sup.36] [(1 + k).sup.n] Step 1: Divide by [p.sub.o]/ 0.11/4 = 0.027 = [q.sub.0] [(1 + k).sup.36] Step 2: Logarithm of both sides log 0.027 = 1.56 = 36 log (1 + k) Step 3: Divide by the power (n) -1.56/36 = 4.32 x [10.sup.-2] = log (1 + k) 36 Step 4: Take the antilog (10x) [10.sup.-432 x [10.sup.-2] = 0.905 = 1 + k Step 5: Minus 1 0.905 - 1 = -0.095 = k Step 6: As a % k = -0.095 x 100 = -9.5% Table 3. Average brain volume and relation to probable body mass of hominids, showing steps to calculate the "excess" brain volume beyond that predicted by the allometric function [m.sub.brain] = 0.03 [m.sup.body] 0.7. Students brain body need to complete the shaded cells of columns 4-8 for the three species used in their exponential model. The values in columns 2 and 3 are from Southwood (2003). 1. 2. 3. Species Brain Brain volume/ volume body mass Australopithecus afarensis 457 [cm.sup.3] 12 [cm.sup.3]/kg Homo habilis 552 [cm.sup.3] 13 [cm.sup.3]/kg H. erectus 1016 [cm.sup.3] 18 [cm.sup.3]/kg H. sapiens 1355 [cm.sup.3] 26 [cm.sup.3]/kg Neanderthals 1512 [cm.sup.3] 20 [cm.sup.3]/kg 1. 4. 5. Species Body mass Brain mass (col. 2/col. 3) (col. 2) x 1.04 g/[cm.sup.3] Australopithecus afarensis 38.1 kg# 475.3 g# Homo habilis 42.5 kg# 574.1 g# H. erectus 56.4 kg 1056.6 g H. sapiens 52.1 kg# 1409.2 g Neanderthals 75.6 kg 1572.5 g 1. 6. 7. Species Predicted Expected brain brain mass volume 0.03 (col. 6) x 103 g/ [(col. 4).sup.0.7] kg/1.04 g/[cm.sup.3] Australopithecus afarensis 0.383 kg# 369 [cm.sup.3]# Homo habilis 0.414 kg# 398 [cm.sup.3]# H. erectus 0.505 kg 486 [cm.sup.3] H. sapiens 0.478 kg# 459 [cm.sup.3]# Neanderthals 0.620 kg 596 [cm.sup.3] 1. 8. Species Excess cranial capacity (xcc) (col. 2--col. 7) Australopithecus afarensis 88 [cm.sup.3]# Homo habilis 154 [cm.sup.3]# H. erectus 530 [cm.sup.3] H. sapiens 896 [cm.sup.3]# Neanderthals 916 [cm.sup.3] Students brain body need to complete the shaded cells of columns 4-8 for the three species used in their exponential model.…