The situation in which allele frequencies in the gene pool of a population remain constant is called

The Hardy–Weinberg Principle

The Hardy–Weinberg principle relates allele frequencies to genotype frequencies in a randomly mating population. Imagine that you have a population with two alleles (A and B) that segregate at a single locus. The frequency of allele A is denoted by p and the frequency of allele B is denoted by q. The Hardy–Weinberg principle states that after one generation of random mating genotype frequencies will be p2, 2pq, and q2. In the absence of other evolutionary forces (such as natural selection), genotype frequencies are expected to remain constant and the population is said to be at Hardy–Weinberg equilibrium. The Hardy–Weinberg principle relies on a number of assumptions: (1) random mating (i.e, population structure is absent and matings occur in proportion to genotype frequencies), (2) the absence of natural selection, (3) a very large population size (i.e., genetic drift is negligible), (4) no gene flow or migration, (5) no mutation, and (6) the locus is autosomal. When these assumptions are violated, departures from Hardy–Weinberg proportions can result.

One useful way to think about the Hardy–Weinberg principle is to use the metaphor of a gene pool (Crow, 2001). Here, individuals contribute alleles to an infinitely large pool of gametes. In a randomly mating population without natural selection, offspring genotypes are found by randomly sampling two alleles from this gene pool (one from their mother and one from their father). Because the allele that an individual receives from their mother is independent of the allele they receive from their father, the probability of observing a particular genotype is found by multiplying maternal and paternal allele frequencies. Mathematically this involves the binomial expansion: (p + q)2 = p2 + 2pq + q2 (see the modified Punnett Square in Figure 1 for a graphical representation). Note that there are two ways that an individual can be an AB heterozygote: they can either inherit an A allele from their mother and a B allele from their father or they can inherit a B allele from their mother and an A allele from their father.

Additional insight can be found by considering an empirical example (Figure 2). Consider a population that initially contains 18 AA homozygotes, 4 AB heterozygotes, and 3 BB homozygotes. The alleles in the gene pool, 80% are A and 20% are B. After a single generation of random mating we observe Hardy–Weinberg proportions: 16 AA homozygotes, 8 AB heterozygotes, and 1 BB homozygote. Note that allele frequencies remain unchanged.

There are a number of evolutionary implications of the Hardy–Weinberg principle. Most importantly, genetic variation is conserved in large, randomly mating populations. A second implication is that the Hardy–Weinberg principle allows one to determine the proportion of individuals that are carriers for a recessive allele. Third, it is important to note that dominant alleles are not always the most common alleles in a population. Another implication of the Hardy–Weinberg principle is that rare alleles are more likely to be found in heterozygous individuals than in homozygous individuals. This occurs because q2 is much smaller than 2pq when q is close to zero.

The Hardy–Weinberg principle can be generalized to include polyploid organisms and genes that have more than two segregating alleles. Equilibrium genotype frequencies are found by expanding the multinomial (p1 + … + pk)n, where n is the number of sets of chromosomes in a cell and k is the number of segregating alleles. For example, tetraploid organisms (n = 4) with two segregating alleles (k = 2) are expected to have genotype frequencies of: p14 (AAAA), 4p13p2 (AAAB), 6p12p22 (AABB), 4p1p23 (ABBB), and p4 (BBBB). Similarly, diploid organisms (n = 2) with three segregating alleles (k = 3) are expected to have genotype frequencies of: p12 (AA), p22 (BB), p32 (CC), 2p1p2 (AB), 2p1p3 (AC), and 2p2p3 (BC). Genotype frequencies sum to one for each of the above scenarios. Although the Hardy–Weinberg principle can also be generalized to include genes located on sex chromosomes (e.g., X chromosomes in humans), it is important to note that it can take multiple generations for genotype frequencies at sex-linked loci to reach equilibrium values.

The Hardy–Weinberg Principle

The example given for theMN locus presents an ideal situation for gene frequency estimation because, owing to codominance, the three genotypes can easily be distinguished and counted. What happens when one of the homozygotes is indistinguishable from the heterozygote (i.e., when there is dominance)? Here the basic concepts of probability can be used to specify a predictable relationship between gene frequencies and genotype frequencies.

Imagine a locus that has two alleles, labeledA anda. Suppose that in a population we know the frequency of alleleA, which we will callp, and the frequency of allelea, which we will callq. From these data we wish to determine the expected population frequencies of each genotype,AA, Aa, andaa. We will assume that individuals in the population mate at random with regard to their genotype at this locus(random mating is also referred to aspanmixia). Thus the genotype has no effect on mate selection. If men and women mate at random, then the assumption of independence is fulfilled. This allows us to apply the addition and multiplication rules to estimate genotype frequencies.

Suppose that the frequency,p, of alleleA in our population is 0.7. This means that 70% of the sperm cells in the population must have alleleA, as must 70% of the egg cells. Because the sum of the frequenciesp andq must be 1, 30% of the egg and sperm cells must carry allelea (i.e.,q = 0.30). Under panmixia, the probability that a sperm cell carryingA will unite with an egg cell carryingA is given by the product of the gene frequencies:p ×p =p2 = 0.49 (multiplication rule). This is the probability of producing an offspring with theAA genotype. Using the same reasoning, the probability of producing an offspring with theaa genotype is given byq ×q =q2 = 0.09.

What about the frequency of heterozygotes in the population? There are two ways a heterozygote can be formed. Either a sperm cell carryingA can unite with an egg carryinga, or a sperm cell carryinga can unite with an egg carryingA. The probability of each of these two outcomes is given by the product of the gene frequencies,pq. Because we want to know the overall probability of obtaining a heterozygote (i.e., the first event or the second), we can apply the addition rule, adding the probabilities to obtain a heterozygote frequency of 2pq. These operations are summarized inFig. 3.30. The relationship between gene frequencies and genotype frequencies was established independently by Godfrey Hardy and Wilhelm Weinberg and is termed theHardy–Weinberg principle.

2.3.5 Statistical, Formal, and Population Genetics

A cornerstone of population genetics is the Hardy–Weinberg principle, named for Godfrey Harold Hardy (1877–1947), distinguished mathematician of Cambridge University, and Wilhelm Weinberg (1862–1937), physician of Stuttgart, Germany, each publishing it independently in 1908. Hardy [36] was stimulated to write a short paper to explain why a dominant gene would not, with the passage of generations, become inevitably and progressively more frequent. He published the paper in the American Journal of Science, perhaps because he considered it a trivial contribution and would be embarrassed to publish it in a British journal.

R.A. Fisher, J.B.S. Haldane (1892–1964), and Sewall Wright (1889–1988) were the great triumvirate of population genetics. Sewall Wright is noted for the concept and term “random genetic drift.” J.B.S. Haldane [37] (Fig. 1.9) made many contributions, including, with Julia Bell [38], the first attempt at the quantitation of linkage of two human traits: color blindness and hemophilia. Fisher proposed a multilocus, closely linked hypothesis for Rh blood groups and worked on methods for correcting for the bias of ascertainment affecting segregation analysis of autosomal recessive traits.

To test the recessive hypothesis for mode of inheritance in a given disorder in humans, the results of different types of matings must be observed as they are found, rather than being set up by design. In those families in which both parents are heterozygous carriers of a rare recessive trait, the presence of the recessive gene is often not recognizable unless a homozygote is included among the offspring. Thus, the ascertained families are a truncated sample of the whole. Furthermore, under the usual social circumstances, families with both parents heterozygous may be more likely to be ascertained if two, three, or four children are affected than they are if only one child is affected. Corrections for these so-called biases of ascertainment were devised by Weinberg (of the Hardy–Weinberg law), Bernstein (of ABO fame), and Fritz Lenz and Lancelot Hogben (whose names are combined in the Lenz–Hogben correction), as well as by Fisher, Norman Bailey, and Newton E. Morton. With the development of methods for identifying the presence of the recessive gene biochemically and ultimately by analysis of the DNA itself, such corrections became less often necessary.

Pre-1956 studies of genetic linkage in the human for the purpose of chromosome mapping are discussed later as part of a review of the history of that aspect of human genetics.

The Hardy-Weinberg Principle

Consider an “ideal” population in which there is an autosomal locus with two alleles, A and a, that have frequencies of p and q, respectively. These are the only alleles found at this locus, so that p + q=100%, or 1. The frequency of each genotype in the population can be determined by construction of a Punnett square, which shows how the different genes can combine (Fig. 7.1).

FromFig. 7.1, it can be seen that the frequencies of the different genotypes are:

If there is random mating of sperm and ova, the frequencies of the different genotypes in the first generation will be as shown. If these individuals mate with one another to produce a second generation, a Punnett square can again be used to show the different matings and their frequencies (Fig. 7.2).

FromFig. 7.2 the total frequency for each genotype in the second generation can be derived (Table 7.1). This shows that the relative frequency or proportion of each genotype is the same in the second generation as in the first. In fact, no matter how many generations are studied, the relative frequencies will remain constant. The actual numbers of individuals with each genotype will change as the population size increases or decreases, but their relative frequencies or proportions remain constant—the fundamental tenet of the Hardy-Weinberg principle. When epidemiological studies confirm that the relative proportions of each genotype remain constant withfrequencies of p2, 2pq and q2, then that population is said to be in Hardy-Weinberg equilibrium for that particular genotype.

12.3.1.1 Hardy-Weinberg Equilibrium

In 1908, two scientists—Godfrey H. Hardy, an English mathematician, and Wilhelm Weinberg, a German physician—independently worked out a mathematical relationship that related genotypes to allele frequencies called the Hardy-Weinberg principle, a crucial concept in population genetics. It predicts how gene frequencies will be inherited from generation to generation given a specific set of assumptions. When a population meets all the Hardy-Weinberg conditions, it is said to be in Hardy-Weinberg equilibrium (HWE). Human populations do not meet all the conditions of HWE exactly, and their allele frequencies will change from one generation to the next, so the population evolves. How far a population deviates from HWE can be measured using the “goodness-of-fit” or chi-squared test (χ2) (See Box 12.4).

Box 12.4

Hardy-Weinberg Equilibrium

The distribution of genotypes in a population in Hardy-Weinberg equilibrium can be graphically expressed as shown in the accompanying graph. The x-axis represents a range of possible relative frequencies of A or B alleles. The coordinates at each point on the three genotype lines show the expected proportion of each genotype at that particular starting frequency of A and B.

To check for HWE:

Consider a single biallelic locus with two alleles A and B with known frequencies (allele A = 0.6; allele B = 0.4) that add up to 1.

Possible genotypes: AA, AB and BB

Assume that alleles A and B enter eggs and sperm in proportion to their frequency in the population (i.e., 0.6 and 0.4)

Assume that the sperm and eggs meet at random (one large gene pool).

Calculate the genotype frequencies as follows:

The probability of producing an individual with an AA genotype is the probability that an egg with an A allele is fertilized by a sperm with an A allele, which is 0.6 × 0.6 or 0.36 (the probability that the sperm contains A times the probability that the egg contains A).

Similarly, the frequency of individuals with the BB genotype can be calculated (0.4 × 04 = 0.16).

The frequency of individuals with the AB genotype is calculated by the probability that the sperm contains the A allele (0.6) times the probability that the egg contains the B allele (0.4), and the probability that the sperm contains the B allele (0.6) times the probability that the egg contains the A allele. Thus, the probability of AB individuals is (2 × 0.4 × 0.6 = 0.48).

Genotypes of the next generation can be given as shown in the accompanying table.

The conclusions from HWE are follows:

Allele frequencies in a population do not change from one generation to the next only as the result of assortment of alleles and zygote formation.

If the allele frequencies in a gene pool with two alleles are given by p and q, the genotype frequencies is given by p2, 2pq, and q2.

The HWE principle identifies the forces that can cause evolution.

If a population is not in HWE, one or more of the five assumptions is being violated.

Thus, HWE is based on five assumptions:

Random selection: When individuals with certain genotypes survive better than others, allele frequencies may change from one generation to the next.

No mutation: If new alleles are produced by mutation or if alleles mutate at different rates, allele frequencies may change from one generation to the next.

No migration: Movement of individuals in or out of a population alters allele and genotype frequencies.

No chance events: Luck plays no role in HWE. Eggs and sperm collide at the same frequencies as the actual frequencies of p and q. When this assumption is violated and by chance some individuals contribute more alleles than others to the next generation, allele frequencies may change. This mechanism of allele change is called genetic drift.

Individuals select mates at random: If this assumption is violated, allele frequencies do change, but genotype frequencies may.

Evolutionary Dynamics

Unstable Equilibrium

At an equilibrium, the allele frequency does not change over time. An equilibrium is stable if small perturbations lead back to it. It is unstable if small perturbations lead away, typically toward other, stable equilibria. Heritable fitness differences are expected to lead to evolutionary change in a population over time, driven by natural selection. In the case of underdominance, heterozygotes are expected to produce fewer offspring in the following generation, corresponding to the fitness disadvantage. According to the Hardy–Weinberg principle (random pairing of alleles), alleles that are rare in a population (low starting frequency) are most often paired with alleles of another type, resulting in a heterozygous genotype. Thus, underdominance is expected to result in a disadvantage of rare alleles, which tend to be removed from the population by natural selection. However, the same alleles can proceed to fixation in a population if they occur as homozygotes sufficiently often, which requires a high starting frequency. There is an unstable equilibrium frequency that divides these two regimes. The direction of selection in underdominance is thus opposite of the one in overdominance, which is characterized by a stable polymorphic equilibrium frequency (see Figure 2).

Figure 2. Evolution of the frequency of allele A of a single-locus two-allele system with underdominance. For simplification, an infinitely large population with random mating is assumed. The fitness of AA homozygotes is 0.9 and the fitness of BB homozygotes is 1. Heterozygotes have a relative fitness disadvantage of 0.45 (as illustrated in the inset). Trajectories are shown for the five initial allele frequencies 0.2, 0.4, 0.55, 0.6, and 0.8. For the first two initial conditions, A goes extinct. For the last two initial conditions, A proceeds to fixation. In this example, 0.55 is exactly the unstable equilibrium allele frequency; small deviations, for example, caused by demographic noise, lead away from it.

Genetic Theory and Probabilities

The foundations of the genetic theory have been laid almost 150 years ago by Gregor Mendel. The field of application is limited to characteristics, or observation units (from classical traits such as color or form, to the outputs of technologically sophisticated methods such as electrophoresis or mass spectrometry) for which the population under study shows discontinuous variation (i.e., the individuals appear as grouped into discrete classes, called phenotypes). The theory assumes that for each of these characteristics, a pair of genetic information units exists in each individual (genotype), but only one is transmitted to each offspring at a time with equal probability (1/2). So, for nonhermaphroditic sexually reproducing populations, each member inherits one of these genetic factors (alleles) paternally and the other one maternally; in case of both alleles are of the same type, the individual is said to be a homozygote, and heterozygote in the case the alleles are distinct. The theory further assumes that for each of the observable units (or Mendelian characteristics), there is a genetic determination instance (a genetic locus; plural: loci) where the alleles take place and that the transmission of information belonging to different loci and governing, therefore, distinct characteristics is independent. It is now known that for some characteristics, the mode of transmission is more simple and that not every pair of loci is transmitted independently, but the hereditary rules outlined above apply to the vast majority of cases.

These rules allow us to predict the possible genotypes and their probabilities in the offspring knowing the parents’ genotypes or to infer parents’ genotypes given the offspring distributions. These predictions or inferences are not limited to cases where information on relatives is available. In fact, soon after the ‘rediscovery’ of Mendel’s work, a generalization of the theory from the familial to the population level was undertaken embodied in what is now known as the Hardy–Weinberg principle. This formalism states that if an ideal infinite population with random mating is assumed, and in the absence of mutation, selection, and migration, the squared summation of the allele frequencies equals the genotype distribution. That is, if at a certain locus, the frequencies of alleles A1 and A2 are f1 and f2, respectively, the expected frequency of the heterozygote A1A2 will be f1 × f2 + f2 × f1 = 2f1f2 (note that ‘A1A2’ and ‘A2A1’ are indistinguishable and are collectively represented by convention simply as A1A2); conversely, if the frequency of the homozygote for A1 is f1, the allele frequency would be the square root of this frequency (because the expected frequency of this genotype is f1 × f1).

In order to apply this theoretical framework to judicial matters, it must be clear that ‘forensics’ implies conflict, a difference of opinion, which formally translates into the existence of (at least) two alternative explanations for the same fact. In the simplest situation, the evidence is explained to the court as (1) being caused by the suspect (the prosecution hypothesis) or, alternatively, (2) resulting from the action of someone else, according to the defense.

In order to understand how genetic expertise can provide means to differently evaluate the evidence under these hypotheses, a brief digression into the mathematics and statistics involved is therefore required. The first essential concept to be defined is probability itself. The probability of a specific event is the frequency of that event, or in more formal terms, probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible. It is a convenient way to summarize quantitatively our previous experience on a specific case and allows us to forecast the likelihood of its future occurrence. But this is not the issue at stake when we move to the forensic scenario – the event has occurred (both litigants agree upon that) but there is a disagreement on the causes behind it, meaning that the same event can have different probabilities according to its causation.

Let us suppose that a biological sample (a hair, organic fluid, etc.) not belonging to the victim is found in a homicide scene. When typed for a specific locus, it shows the genotype ‘19’, as well as the suspect (provider of a ‘reference sample’). If allele 19 frequency in the population is 1/100, the probability of finding by chance such a genotype is thus 1/10 000. Therefore, under the prosecutor's hypothesis (the crime scene sample was left by the suspect), the probability of this type of observations (P|H1) is 1/10 000. While assuming the defense explanation (the crime scene sample was left by someone else), the probability of the same observations (P|H2) would be 1/10 000 × 1/10 000. In conclusion, the likelihood ratio takes the value of 10 000 (to 1), which means that the occurrence of such an event is 10 000 times more likely if both samples have originated from the same individual than resulting from two distinct persons (again provided the suspect does not have an identical twin). Note that this likelihood ratio is often referred as ‘probability of identity,’ although it is not a probability in the strict sense.

7 POPULATION GENETICS UPHOLDS DARWINISM

Mendel's hypothesis of inheritance of discrete factors that are not diluted should have resolved a major difficulty that Darwin encountered. Shortly after the publication of his Origin of Species, in 1867, Fleeming Jenkins showed that, adopting Darwin's theory of inheritance by mixing pangenes, would wash out any achievement of natural selection (see Hull [1973, 302-350]). Hugo de Vries and especially William Bateson, considered Mendel's Faktoren as indicated by his hypothesis of inheritance to provide a rational basis for the theory of evolution. Although as early as in 1902 Yule showed that, given small enough steps of variation, the Mendelian model reduces to the biometric claim [Yule, 1902], this was largely ignored in the bitter disputes between the Mendelians and the Biometricians [Provine, 1971], (see Tabery [2004]). Hardy's [1908] proof that in a large population, the proportion of heterozygotes to homozygotes will reach equilibrium after one generation of random mating (provided no mutation or selection interfered), developed in the same year by Weinberg [Stern, 1943], became the basic theorem of population genetics — the Hardy-Weinberg principle. It took, however, another decade for R. A. Fisher to convince that the continuous phenotypic biometric variation reduces to the Mendelian model of polygenes [Fisher, 1918]. Thus, finally the way was cleared to examine the Darwinian theory of natural evolution on the basis of Mendelian genetic analysis, not only in vivo but also in papyro. As formulated by Fisher in his fundamental theorem of natural selection: “The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time” [Fisher, 1930, 37].

Whereas Fisher examined primarily the effects of selection of alleles of single genes in indefinitely large population under the assumption of differences in genotypic fitness, J. B. S. Haldane concentrated on the impact of mutations on the rate and direction of evolution of one or few genes (and the influence of population size) [Haldane, 1990]. Sewall Wright in his models of the dynamics of populations wished to be more “realistic”, and stressed the influence of finite population size, the limited gene flow between subpopulations, and the heterogeneity of the habitats in which the population and its subpopulations lived [Wright, 1986].

Experimentally, the main British group, led by E. B. Ford adopted a strict Mendelian reductionist approach, emphasizing largely the effects of selection on single alleles of specific genes (the evolution of industrial melanism in moths, the evolution of mimicry in African moth species, the evolution of seasonal polymorphisms in snails, etc.). The American geneticists, especially Dobzhansky and his school, concentrated more on problems of whole genotypes, such as speciation (Sturtevant) and chromosomal polymorphisms (Dobzhansky) in Drosophila.

The triumph of reductionist Mendelism was at the 1940s with the emergence of the “New Synthesis” that defined natural populations and the forces that affect their evolution in terms of gene alleles’ frequencies [Huxley, 1943]. This notion dominated population genetics for the next decades. Attempts to emphasize the role of non-genetic constraints, such as the anatomical-physiological factors (e.g. by Goldschmidt [1940]), or the environmental (and evolutionary-historical) constraints (for example by Waddington [1957]) were largely overlooked.

The introduction of the analysis of electrophoretic polymorphisms [Hubby and Lewontin, 1966; Lewontin and Hubby, 1966] allowed a molecular analysis of allele variation that was also largely independent of the classical morphological and functional genetic markers (see also Lewontin [1991]). Although genes were still treated as algebraic point entities, inter-genic interacting system, such as “linkage disequilibrium” were considered [Lewontin and Kojima, 1960]. The New Synthesis was, however, seriously challenged when it was realized that a great deal of the variation at the molecular level was determined by stochastic processes, rather than because of differences in fitness [Kimura, 1968; King and Jukes, 1969].

This assault on the notion of the New Synthesis was intensified when, in 1972 Gould and Eldridge, two paleontologists, suggested a model of evolution by “punctuated equilibrium”, or long periods of little evolutionary change interspersed with (geologically) relatively short period of fast evolutionary change. Moreover, in the periods of (relatively) fast evolution large one-step “macromutational” changes were established [Eldredge and Gould, 1972]. Although it could be shown that analytically the claims of punctuated equilibrium could be reduced to those of classical population genetics [Charlesworth et al., 1982], these ideas demanded re-examination of the developmental conceptions that, as a rule, could not accept one-step major developmental changes since these called for disturbance in many systems and hence would have caused severe disturbances in developmental and reproductive coordination.

The need to reexamine the reductionist assumptions of genetic population analysis and to pay more consideration to constraints on the genetic determinations of intra- and extra-organismal factors coincided with the resurrection of developmental genetics. However, the major change in the analysis of evolution and development came from the molecular perspective. These allowed first of all detailed upward analysis, from the specific DNA sequences to the early products, rather than the analyses based on end-of-developmental pathway markers. Yet, arguably, the most significant development was the possibility of in-vitro DNA hybridization. This molecular extension of genetic analysis sensu stricto finally overcame the empirical impossibility to study (most) in vivo interspecific hybrids. The new methods of DNA hybridization had no taxonomic inhibitions whatsoever, and soon hybrid DNA molecules of, say mosquito, human and plant, were common subjects for research. Genetic engineering, which allowed direct genetic comparison between any species and the transfer of genes from one species to individuals of another, unrelated species, prompted the genetic analysis of the evolution of developmental process, or evo-devo.

Molecular genetic analysis of homeotic mutants, in which one organ is transformed into the likeness of another, usually a homologous one, revealed stretches of DNA that were nearly identical in other genes with homeotic effects (like the homeobox of some 180 nucleotides, that appear to be involved in when-and-where particular groups of genes are expressed along the embryo axis during development [McGinnis et al., 1984a; McGinnis et al., 1984b]). The method of determining homologies by comparing DNA sequences is nowadays done mainly in silico. As suggested many years ago [Ohno, 1970], the abundance of homologous sequences in the same species genome (paralogous sequence that do not necessarily share similar functions any more) or in different species (orthologous sequences that ‘usually’ have similar functions in different species), indicate that the system's structural and functional organization have been also causal factors rather than merely consequences in the history of the process of evolution.

3.2 Formalisation in Population Genetics

The phenomenon of heredity, although widely accepted since at least the Greco-Roman period, is extremely complex and an adequate theory proved allusive for several thousand years. Indeed, features of heredity seemed almost magical. Breeders from antiquity had a sophisticated understanding of the effects of selective breeding but even the most accomplished breeders found many aspects of heredity to be capricious. Even Darwin in the middle 19th century knew well the techniques of selective breeding (artificial selection) but did not have available a satisfactory theory of heredity when he published the Origin of Species [1859]. Although, he realized that his theory of evolution depended on heredity, he was unable to provide an account of it. Instead, he relied on the widely known effects of artificial selection and by analogy postulated the effects of natural selection in which the culling of breeders was replaced by forces of nature.

The first major advance came from the simple experiments and mathematical description of the dynamics of heredity by Gregor Mendel [1865]. Although Mendel's work went largely unnoticed until the beginning of the 20th century, its great strength lay in its mathematical description — elementary though that description was. Mendel performed a number of experiments which provided important data but it was his elementary mathematical description of the underlying dynamics that has had a lasting impact on genetics. His dynamics were uncomplicated. He postulated that a phenotypic characteristic (characteristic of organisms) is the result of the combination of two “factors” in the hereditary material of the organism. Different characteristics are caused by different combinations. Focusing on one characteristic at a time made the problem of heredity tractable. Factors could be dominant or recessive. If two dominant factors combined, the organism would manifest the characteristic controlled by that factor. If a dominant and a recessive factor combined, the organism would manifest the characteristic of the dominant factor (that is the sense in which it is dominant). If two recessive factors combine, the organism will manifest the characteristic of the recessive factor.

Mendel postulated two principles (often now referred to as Mendel's laws): a principle of segregation and a principal of independent assortment. The principle of segregation states that the factors in a combination will segregate (separate) in the production of gametes. That is gametes will contain only one factor from a combination. The principle of independent assortment states that the factors do not blend but remain distinct entities and there is no influence of one factor over the other in segregation. The central principle is the law of segregation. The law of independent assortment can be folded into the law of segregation as part of the definition of segregation. When gametes come together in a fertilised ovum (a zygote), a new combination is made.

Assume A is a dominant factor and a is a recessive factor. Three combinations are possible AA, Aa and aa. Mendel's experimental work involved breeding AA plants and aa plants. He then crossed the plants which produced only Aa plants. He then bred the Aa plants. What resulted was .25AA, .5Aa and .25aa. His dynamics explains this result. Since the factors A and a do not blend and they segregate in the gametes and combine again in the zygote, the results are fully explained. Crossing the AA plants with aa plants will yield only Aa plants:

Breeding only Aa plants will yield the .25:.5:.25 ratios:

Two of four cells yield Aa that is .5 of the possible combinations. Each of AA and aa occupy only one cell in four, that is, .25 of the possible combinations. In contemporary population genetics, Mendel's factors are called alleles. The location on the chromosome where a pair of alleles is located is called a locus. Sometime the term gene is used as a synonym for allele but this usage is far too loose. Subsequently, I will explore the confusion, complexity and controversy over the definition of “gene.” Mendel's dynamics assumed diallelic loci: two alleles per locus. His dynamics are easily extended to cases where each locus has many alleles any two of which could occupy the locus.

The basic features of Mendel's dynamics were modified and extended early in the 20th century. G. Udny Yule [1902] was among the first to explore the implications of Mendel's system for populations. In a verbal exchange between Yule and R. C. Punnett in 1908, Yule asserted that a novel dominant allele arising among a 100% recessive alleles would inexorably increase in frequency until it reach 50%. Punnett believing Yule to be wrong but unable to provide a proof, took the problem to G. H. Hardy. Hardy, a mathematician, quickly produced a proof by using variables where Yule had used specific allelic frequencies. In effect, he developed a simple mathematical model. He published his results in 1908. What emerged from the proof was a principle that became central to population genetics, namely, after the first generation, allelic frequencies would remain the same for all subsequent generations; an equilibrium would be reached after just one generation. Also in 1908, Wilhelm Weinberg published similar results and articulated the same principle (the original paper is in German, and English translation is in Boyer [1963]). Hence, the principle is known as the Hardy-Weinberg principle or the Hardy-Weinberg equilibrium.37 In parallel with these mathematical advances was a confirmation of the phenomenon of segregation and recombination in the new field of cytology.

Building on this early work, a sophisticated mathematical model of the complex dynamics of heredity emerged during the 1920s and 1930s, principally through the work of John Haldane [1924; 1931; 1932], Ronald Fisher [1930] and Sewall Wright [1931; 1932]. What has become modern population genetics began during this period. From that period, the dynamics of heredity in populations has been studied from within a mathematical framework.38

As previously indicated, one of the fundamental principles of the theory of population genetics, in the form of a mathematical model, is the Hardy-Weinberg Equilibrium. Like Newton's First Law, this principle of equilibrium states that after the first generation if nothing changes then allelic (gene) frequencies will remain constant. The presence of a principle(s) of equilibrium in the dynamics of a system is of fundamental importance. It defines the conditions under which nothing will change. All changes, therefore, require the identification of cause(s) of the change. Newton's dynamics of motion include an equilibrium principle that states that in absence of unbalanced forces an object will continue in uniform motion or at rest. Hence acceleration, deceleration, change of direction all require the presence of an unbalanced force. In population genetics, in the absence of some perturbing factor, allelic frequencies at a locus will not change. Factors such as selection, mutation, meiotic drive, and migration are all perturbing factors. Like many complex systems, population genetics also has a stochastic perturbing force, commonly call random genetic drift.

In what follows, the central features of the mathematical model of contemporary population genetic theory are set out. Quite naturally, the exposition begins with the Hardy-Weinberg Equilibrium. It is useful to begin with the exploration of a one locus, two-allele system. In anticipation, however, of multi allelic loci, we switch from A and a to ‘A1’ and ‘A2’. Hence, according to the Hardy-Weinberg Equilibrium, if there are two different alleles ‘A1’ and ‘A2’ at a locus and the ratio in generation 1 is A1:A2 = p: q, and if there are no perturbing factors, then in generation 2, and in all subsequent generations, the alleles will be distributed:

(p2) A1A1:(2pq)A1A2:(q2)A2A2.

The ratio of p: q is normalised by requiring that p + q = 1. Hence, q =1 — p and 1 — p can be substituted for q at all occurrences. The proof of this equilibrium is remarkably simply.

The boxes contain zygote frequencies. In the upper left box, the frequency of the zygote arising from the combination of an A1 sperm and A1 egg is p × p, or p2, since the initial frequency of A1 is p. In the upper right box, the frequency of the zygote arising from the combination of an A2 sperm and A1 egg is p × q, or pq, since the initial frequency of an A2 is q and the initial frequency of A1 is p.

Sperm

The lower left box also yields a pq frequency for an A1A2. Since the order doesn't matter, A2A1 is the same as A1A2 and hence the sum of frequencies is 2pq.

This proves that a population with A1: A2 = p:q in an initial generation will in the next generation have a frequency distribution: (p2)A1A1: (2pq)A1A2: (q2)A2A2. The second step is to prove that this distribution is an equilibrium in the absence of perturbing factors. Given the frequency distribution (p2)A1A1: (2pq)A1A2: (q2)A2A2, p2 of the alleles will be A1 and half of the A1A2 combination will be A1, that is pq. Hence, there will be p2 + (pq)A1 in this subsequent generation. Since q = (1 — p), we can substitute (1 — p) for q, yielding p2 + (p(1 – p)) = p2 + (p – p2) = p. Since the frequency of A1 in this generation is the same as in the initial generation (i.e., p), the same frequency distribution will occur in the following generation (i.e., (p2)A1A1: (2pq)A1A2: (q2)A2A2).

Consequently, if there are no perturbing factors, the frequency of alleles after the first generation will remain constant. But, of course, there are always perturbing factors. One central one for Darwinian evolution is selection. Selection can be added to the dynamics by introducing a coefficient of selection. For each genotype (combination of alleles at a locus39) a fitness value can be assigned. Abstractly, A1A1 has a fitness of W11, A1A2 has a fitness of W12, and A2A2 has a fitness of W22. Hence, the ratios after selection will be:

W11(p2)A1A1:W12(2 pq)A1A2:W22(q2)A2A2.

To calculate the ratio p: q after selection this ratio has to be normalised to make p + q =1. To do this, the average fitness, w, is calculated. The average fitness is the sum of the individual fitnesses.

w ¯=w11(p2)+w12(2pq)+w22(q2).

Then each factor in the ratio is divided by w, to yield:

((w11(p2))/w¯) A1A1:((w12(2pq)/w¯)A1A2:((w22( q2)/w¯)A2A2.))

Other factors such as meiotic drive can be added either as additional parameters in the Hardy-Weinberg equilibrium or as separate ratios or equations.

Against this background, a precise application of a X2-test of goodness of fit can be provided. The following example40 illustrates the determination the goodness of fit between observed data and the expected data based on the Hardy-Weinberg equilibrium. The human chemokine receptor41 gene CC-CKR-5codes for a major macrophage co-receptor for the human immunodeficiency virus HIV-1. CC-CKR-5 is part of the receptor structure that allows the entry of HIV-1 into macrophages and T-cells. In rare individuals, a 32-base-pair indel42 results in a non-functional variant of CC-CKR-5. This variant of CC-CKR-5 has a 32-base-pair deletion from the coding region. This results in a frame shift and truncation of the translated protein. The indel results when an individual is homozygous for the allele Δ3243. These individuals are strongly resistant to HIV-1; the variant CC-CKR-5 co-receptor blocks the entry of the virus into macrophages and T-cells.

In a sample of Parisians studied for non-deletion and deletion (+ and Δ32 respectively), Lucotte and Mercier (1998) found the following genotypes:

++: 224 + Δ32: 64 Δ32Δ32: 6

Dividing by the populations sample size yields the genotype frequencies:

++: 224/294 = 0.762 + Δ32: 64/294 = 0.218 Δ32Δ32: 6/994 = 0.20

Multiplying the number of homozygotes for an allele by 2 and adding the number of heterozygotes yields the number of that allele in the sample. Dividing that by the sample size times 2 (there are twice as many alleles as individuals) yields the allelic frequency of this sample. Hence:

The frequency of the + allele = 0.871

The frequency of the Δ32 allele = 0.129

What genotype numbers does the hardy-Weinberg equilibrium yield given these allelic frequencies?

(p2)++:(2pq)+Δ32:(q2)Δ32Δ32Yields(0.8712)++:(2(0.871X0.129))+Δ32:(0.1292)Δ32Δ32 =0.758641++:0.224718+Δ32:0.016641Δ32Δ32

Hence, in a population of 294 individuals, the Hardy-Weinberg equilibrium yields:

++:22.9+Δ32:66.2Δ32Δ32:4.9

As we would expect these add up to 294. A comparison of the values expected based on the Hardy-Weinberg equilibrium and those observed yields:

H-D expected: ++:222.9+Δ32:66.2Δ32Δ32:4.9 Observed:++:224+Δ32:64Δ32 Δ32:6

Now we can ask, how good is the fit between the H-D expected values based on the specified allelic frequencies and the observed values?

The X2-test is:

X2 = Σ (observed quantity – expected quantity)2/(expected quantity) There are three genotypes, hence:

To use this result to assess goodness of fit, it is necessary to determine the degrees of freedom for the test.

Degrees of Freedom (df) = (classes of data – 1) – the number of parameters estimated.

Since there are three genotypes, the classes of data is 3. Since p + q = 1(hence, q is a function of p; they are not independent parameters), there is only 1 parameter being estimated. Hence, the degrees of freedom for this test is:

Using the X2 result and 1 degree of freedom allows a probability value to be determined.

In this case, the relevant probability is 0.63. This is the probability that chance alone could have produced the discrepancy between the H-D expected values and the observed values. Since we are measuring the probability that chance alone could have produced the discrepancy (not to be confused with the similarity between the two44), the higher the probability, the more robust one's confidence that there are no factors other than chance causing the discrepancy and, hence, that there is a good fit between the values expected based on the model and the observed values45; any discrepancy is a function of chance alone.

The elementary framework sketched above has been expanded to include the Wright-Fisher model of Random Drift, mutations, inbreeding and other causes of non-random breeding, migration speciation, multiple alleles at a locus, multi-loci systems, phenotypic plasticity, etc. One important expansion relates to interdemic selection.

The account so far describes intrademic selection. That is, selection of individuals within an interbreeding population — a deme. However, the mathematical model also permits the exploration of interdemic selection (selection between genetically isolated populations) using adaptive landscapes. One outcome of such explorations is a sophisticated account of why and how populations reach sub-maximal, sub-optimal peaks of fitness. Richard Lewontin, building on concepts set out by Sewell Wright, provided the first mathematical description of this phenomenon.

Consider a population genetic system with two loci and two alleles (here for simplicity I revert to upper and lower case letter for alleles and for dominance and recessiveness). The possible combinations of alleles is:

There are 9 different combinations (genotypes). For each genotype a fitness co-efficient Wi can be assigned. In addition, for each genotype a frequency can be assigned based on p1 and q1, p2 and q2 (for locus 1 and locus 2 respectively). Let that frequency be Zi. The product of the frequency of a genotype and the fitness of that genotype is the contribution to the average fitness of the population w made by that genotype. The sum of the contributions of all the genotypes represented in the population is the average fitness w¯ of the population. Hence, the average fitness w¯ for a population

Consider the following calculation for a single population.

Since p1 + q1 = 1 and p2 + q2 = 1, the value of q can be determined from the value of p. Hence the value of p alone is sufficient to determine the genotype frequencies of the population.

In accordance with the Hardy-Weinberg equilibrium, the genotype frequencies can be calculated by multiplying the frequencies of the allelic combinations at each locus in the two loci pair. The resulting frequencies with assigned fitnesses, frequency-fitnesses, and the average fitness for the population is shown in the following table:

By plotting the average fitness w¯ of each possible population in a two loci system with the assigned fitness values Wi, an adaptive landscape for the system can be generated. This adaptive landscape is a three dimensional phase space (a system with a larger number of loci will have a correspondingly larger dimensionality):

The plotted point is the average fitness of the population described above. A complete adaptive landscape is a surface with adaptive peaks and valleys. An actual population under selection may climb a slope to an adaptive peak that is sub maximal (i.e., the average fitness of the population is less than the highest average fitness in the system). The only way to move to another slope which leads to a more maximal or maximal average fitness is to descend from the peak. This involves evolving in a direction of reduced average fitness that is opposed by stabilizing selection. Hence, the population is stuck on the peak at a sub maximal average fitness. When several populations are on different sub-maximal average fitness peaks, selection between populations (interdemic selection) can act.

This population genetic description has been used extensively to explain situations which cannot be explained in terms of intrademic selection. For example, body size which may have high individual fitness, and hence is selected for within a population, can reduce the fitness of the population by causing it to achieve a sub maximal average fitness and leave it open to interdemic selection.