Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  This was a major problem for biologists early in the twentieth century: How is genetic variation maintained in a sexually reproducing population?

  The solution to the problem involves a little careful analysis of the breeding process. The zygotes in the population with parents having allele frequencies as in equation (8.1) were shown to have allele pairs with frequencies according to equation (8.3). Now we ask: what will be the allele frequencies that we must use when this new, first generation becomes parents and produces the second generation? We can find that out by counting how the alleles occur in that new first generation and that is given by equation (8.3). The A allele occurs whenever the AA pair occurs, and in half the cases when the Aa pair occurs. Thus we find that the single chromosome put into the gamete has the A allele frequency

  Similarly, we find the a allele frequency in the new first generation to be

  Now amazingly, comparing equations (8.1) and (8.4), we see that the first new generation actually has allele frequencies f1 which are the same as those f for their parents.

  We now have the essential result: breeding has not changed the allele frequencies. So, when the new generation breeds to give the second generation, the same result must occur and the distribution of allele pairs in the new zygotes will again be as in equation (8.3).

  The dominant allele does not swamp out the genetic variation; in this way of reproducing, the genetic variation is stable. This result is known as the Hardy-Weinberg law, which was announced in 1908. It tells us that the blending problem worrying Darwin does not exist, and using Mendel's more particle-like concepts, the variation in a population is maintained.

  The message from the Hardy-Weinberg law is so significant that it is worth emphasizing: there is equilibrium in the genetic makeup of a sexually reproducing population of organisms so that variability is maintained. There are other ways to analyze the reproducing system described above, but they all come down to some sort of counting process and are basically simple calculations despite the attention to detail needed to follow them. This led the geneticist W. J. Ewens to conclude that “it does not often happen that the most important theorem in any subject is the easiest and most readily derived theorem for that subject.”11 You will hardly find it surprising that I am adding calculation 28, the Hardy-Weinberg law to my list of important calculations.

  8.3.3 Introducing Variability

  You may be saying: “but there really is variability, otherwise there would be no striking changes in populations and evolution.” The solution to this dilemma is to note that in deriving the Hardy-Weinberg law several assumptions have been made:

  Organisms are diploid and only sexual reproduction occurs.

  Mating is at random.

  Generations are nonoverlapping.

  Genders are evenly distributed among the three genotypes and all genotypes are equally fit.

  The population is very large (theoretically infinite).

  There is neither immigration or emigration.

  There is no mutation or artificial selection.

  If any of these conditions are violated, the Hardy-Weinberg law does not strictly hold in the form given above. In fact, checking the allele frequencies and genotypes is a way to check whether a population is in Hardy-Weinberg equilibrium and so to detect the influence of one of these conditions. Of course, the conditions will be violated in some ways and to certain degrees; mutations will occur, and hence there will be evolution as Darwin envisaged it.

  The analysis can be extended in various ways and new results produced. For example, the consequences of overlapping generations and multiple genes and alleles must be considered. There is now an enormous body of work in genetics and its mathematical treatment.

  8.3.4 The Remarkable Discoverer

  Hardy and Weinberg discovered their law in 1908. Wilhelm Weinberg was a German physician, and his work was not well known in the English- speaking world until 1943. For some time, the result was known as Hardy's law, and it was given in Hardy's 1908 Science paper. G. H. Hardy (1877–1947) was responding to claims that brachydactyly (a defect giving short fingers), through its genetic dominance, should be widespread and eventually afflict three quarters of the population. He constructed an argument using alleles A and a with frequencies p and q which led to the result given above He found what he termed “the stable distribution” and showed that the worries about widespread brachydactyly were unfounded since its occurrence is rare and will remain so whether the gene for it is dominant or recessive.

  The intriguing part of this story is that Hardy was one of the leading pure mathematicians of the first half of the twentieth century. His book on number theory remains a classic. In his final, wonderful book about his life and mathematics, A Mathematician's Apology, he expressed his disdain for applied mathematics and wrote that “I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.”12

  According to Hardy, the mathematics used in applications tends to be less demanding and innovative than that explored by the pure mathematician, and in his Science paper he writes of “the very simple point” that he wishes to make. Of course, it is often a simple mathematical argument that leads to a profound result in science, as several of the calculations discussed in this book illustrate, but that was not an appealing point for Hardy. It is ironic then, that Hardy the pure mathematician is probably little known compared with the Hardy in the Hardy-Weinberg law, which plays such an important and fundamental part in genetics.

  8.4 THE MATHEMATICS BEHIND COMPUTED TOMOGRAPHY (CT)

  In 1895, Wilhelm Röntgen discovered x-rays, which were soon used for medical diagnosis. For the first time, doctors could see inside the body and detect things like broken bones; the medical scene changed dramatically. However, the common x-ray gives only an overall picture produced by the total x-ray absorption profile of the whole body part through which the x-rays pass, and no details of the individual components involved are given. Early in the twentieth century, people began to think about “body-section radiography.” In 1937, Edward Wing Twining (1887–1939), the father of British neuroradiology, introduced the term “tomography”—from the Greek, tomos for slice, and graphein, to write. (For a history of tomography see the book by Webb and the article by Gordon, Herman, and Johnson.) The problem concerns how to use several x-rays transmitted through a section of a body or body part in order to deduce the details of the internal makeup of that section. This is an inverse problem much like that faced by seismologists using seismic waves to discover what is deep inside the earth, as explained in section 4.4. Perhaps the best way to explain the situation and how it may be tackled is to introduce a small, or “toy,” example.

  X-rays are partially absorbed as they pass through the body with different materials—muscles and bones, for example—absorbing different amounts. It is this differential absorption rate that gives the variable x-ray beam that is then used to create a picture on the x-ray plate. The problem is how to use these different absorption rates to build up a detailed picture of the inside of the body.

  8.4.1 A Simple Example Shows the Way

  Consider a box or grid split into nine equal parts as shown in figure 8.2. This could represent a square section of material with x-ray absorption coefficients as indicated—the top corner part has absorption set at unity, and the other parts have unknown absorptions denoted by xi. If x-rays are scanned through the section, can we determine those xi?

  Figure 8.2 indicates how eight scans may be made and gives some possible numerical outputs for measured absorptions. Thus the first scan tells us that x6 + x7 + x8 is equal to 13, but it gives no information about their individual values. They could be 3, 4, and 6, or 2, 9, and 2, for example. The same thing happens with each of the other seven scans; we get information about sums of absorptions, but not individual values.

  Figure 8.2. Scanning along the eight lines to give the indicated
sums. Figure created by Annabelle Boag.

  Notice that if the individual absorptions were given, it is a trivial matter to calculate the result of any scan by a simple addition. The important question is: Can we solve the inverse problem—can we use all the given scanning information to extract the individual absorption values? If we write out the measured information in mathematical form we get

  This is a set of eight linear equations for the eight unknowns, x1, x2,…x8, and there are standard, straightforward ways to solve them to get the values 2, 5, 3, 2, 4, 7, 1, and 5.

  This example illustrates the basic point that by employing a mathematical method, it is possible to use the data from several individual scans to calculate the internal structure of the sample being examined. An important note: while it is mathematics that allows the inverse problem to be solved, in practice, a computer is required to carry out the detailed calculations suggested by the mathematics. The set of equations given above could be solved without using a computer, but it is a tedious and time-consuming business.

  Notice that the practical requirement has changed because we now need a machine to record the numerical values of the absorptions; the output of the x-ray is now a numerical data set rather than a picture on a film.

  The processing of the data raises many questions related to accuracy and the influence of experimental error on the final output. There is also a fundamental question: Does the algorithm produced by the mathematical analysis of the general problem lead to a unique answer? In effect, can we really get the correct picture of what is inside the body section being examined? The solution to the set of linear equations used in the above example is known to be unique. It is possible to extend that method to produce a viable tomography technique, although there are problems in the detailed applications (see the article by Gordon, Herman, and Johnson, or chapter 7 in the book by Kak and Slaney).

  A little more of the relevant mathematics is given in the next section, but readers not interested in such matters may skip on to the actual historical development of tomography.

  8.4.2 The General Problem

  If an x-ray beam of initial intensity I0 travels along a line L through a material with absorption coefficient g(s), where s is distance along the line, then the emerging beam has intensity

  If we define fL by

  we have

  Thus the values of I give the values of fL. Does this lead to g(s), which is what we really want?

  Mathematically, we have the following problem: suppose that g exists everywhere for a finite region in a plane, and we may form the integrals along all possible lines L passing through that region to give all fL as in equation (8.6). The question is whether or not we can use all the fL data to find the values of g at all points in the region of interest.

  The answer to this question was actually given in 1917 by the Austrian mathematician Johann Radon, and the process of going from g to the set of line integrals fL is known as taking the Radon transform. Radon showed that the inversion problem could be uniquely solved; that is, given the line data, we can construct the function g over the region of interest. Thus the central mathematical question for tomography had already been answered in 1917, but most of the people working in the x-ray field half a century later were unaware of Radon's work and made the analysis again for themselves.

  It is important to understand that knowing that a problem has a solution in theory does not mean it will be simple to take it over for practical applications. In this case, there needs to be an enormous amount of work done to establish a viable inverse procedure and to convert the whole formalism to one in which discrete, rather than continuous, data sets may be used. Obviously, in practice, we cannot scan along all possible lines, although with the latest technology a very large number can be used. (There is now a large literature on this subject, and in the bibliography, I have given references to an early mathematical discussion by Shepp and Kruskal, and the books by Kak and Slaney, and Epstein.)

  8.4.3 Cormack's Contribution

  Allan Cormack published his paper “Representation of a Function by Its Line Integrals, with Some Radiological Applications” in 1963, and he published a second part of his work in 1964. In essence, Cormack set up the problem as explained above and then showed how it could be completely solved using a Fourier-series method and numerical analysis. The uniqueness of the solution was also examined.

  Cormack went further and proved the viability of his method using a disc specimen (often referred to as a phantom) and gamma rays rather than x-rays. His first specimen had circular symmetry as shown in figure 8.3 (a). It comprised a section of an aluminum cylinder of diameter 10 cm with a wooden surround of outer diameter 20 cm. The symmetry meant that lines at only one angle needed to be sampled, and Cormack scanned the beam as shown in figure 8.3 (b) with a line spacing of 5mm. The results given by Cormack's calculations proved to be accurate, so much so that the absorption coefficient of the aluminum section was shown to be slightly different in an inner and outer part. It turned out that when the specimen was manufactured, an inner part used pure aluminum, but then an alloy was used to complete the 10 cm diameter disc. In the next stage of his work, Cormack used a phantom lacking circular symmetry for which lines at different angles had to be scanned, see figure 8.3 (c) and (d). Again, he obtained impressive results.

  Allan Cormack received the Nobel Prize for this work, and his highly informative acceptance address reviewing his work is given in his 1980 Science article. He remarks that his work did not gain immediate recognition, but he did get a request from the Swiss Center for Avalanche Work asking about the relevance to objects buried in snow!

  Figure 8.3. A sketch of Cormack's scanning experiments. (a) and (c) show the type of specimens used, with shading to indicate the aluminum region. (b) and (d) indicate the scanning lines used. In the second case, scanning lines at twenty-five angles were used. Figure created by Annabelle Boag.

  8.4.4 Hounsfield's Contribution

  The Nobel Prize was shared by Godfrey Hounsfield (or Sir Godfrey, as he became in 1981). Hounsfield worked for EMI in England and produced the first viable scanners for medical purposes. His first patent was granted in 1972. An idea of the difficulties involved in operating the first CT scanners can be gained from the fact that an early test involved the recording of 28,000 measurements on paper tape; the data took nine days to collect, and the available computer took two and a half hours to analyze it. In one of the first clinical applications, a frontal-lobe tumor was detected in a 41-year-old patient. (See Webb for details and references.) Since then, CT scanning has gone from strength to strength as more sophisticated scanning machines and computers have been developed. Today, CT scans have revolutionized many areas of diagnostic medicine. (As an amusing aside, it is claimed that EMI could sponsor Hounsfield's research because it made large sums of money selling the Beatles’ records!)

  8.4.5 Should It Be CMT Scan?

  The invention of the CT scanner was a major step in medical science. Those early pioneering efforts by people like Cormack and Hounsfield have given rise to a wonderful tool making life easier for millions of people every day. The mathematics which is such an essential part of computed tomography is often forgotten, or certainly not fully appreciated. Without the mathematical basis there would be no CT scanner; equally, we must recognize that without significant computer power there would be no way to use that mathematical basis. Personally I would like to see this great tool referred to as “computed mathematical tomography” or CMT. It is probably too late for such a renaming, but at least I can add calculation 29, the mathematics behind the CT scan to my list of important calculations.

  8.5 LINKING ALL THE ANIMALS

  Darwin's ideas explain how different species can evolve and give us the great diversity of life that confronts us today. But this same theory also tells us that all life is somehow linked, and today we marvel at the similarities in the DNA of quite different creatures. Harvey showed how the blood of all animals is circulated wi
th the same mechanism involving a heart, veins, and arteries. Similarly, there is a nervous system carrying messages around inside animal bodies. These are general properties, and we know they follow from nature's blueprint for the growth and reproduction of life. But does this blueprint impose regularity beyond the mere similarity of the processes involved and the apparatus that grows to facilitate them? Can we find general principles that tell us how to describe these regularities and then how to explain them? This section is concerned with one approach to that question, and it relies on simple but powerful ideas about the scaling of physical objects. As an example, I ask what can be said about vertebrate animals as their size varies. As a measure of size, I take the mass M of the animal, which is the most simply defined and easiest quantity to measure consistently.

  8.5.1 The Basic Mathematics of Scaling

  We are used to seeing children's toys, such as cars or airplanes, as scaled-down versions of the real things, but probably few people naturally think of their pet dog as a scaled-up version of a mouse, or a scaled-down version of a horse or even an elephant. Does such a scaling idea make sense? Is it a useful idea? To explore these questions it is necessary to find a mathematical basis for the scaling concept.

  Consider a cube of material of side length L and density ρ. We can then find the following relationships:

  I have used the length L as the basic quantity, and the mass is proportional to its cube. I can also use the mass M as the basic quantity, and the volume is proportional to the mass. These ideas carry over to shapes other than cubes. For all quantities, I can write:

  In general, for any quantity y, I write:

  Equation (8.9) describes how some quantity y scales as the mass varies; b is called the scaling exponent, and a is the proportionality constant. For example, if we put y equal to the area, we use b = ⅔.