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Hamilton's derivation of direct fitness from his 1970 paper

Hamilton's derivation of direct fitness from his 1970 paper

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In his 1970 paper "Selfish and Spiteful Behaviour in an Evolutionary Model", Hamilton uses Price's equation to derive his well-known rule $rb -c >0$. My question is about one of the steps in his derivation.

Hamilton considers a population of $n$ individuals. Let $s_{ij}$ be the effect of individual $i$ on the fitness of $j$. The fitness of an individual $j$ is defined as $w_j = 1 + sum_i s_{ij}$, $w=1/n sum_j w_j$ is the mean fitness in the population, and $q=1/n sum_j q_j$ is the average frequency of a certain allele. Using Price's equation, we get

$w Delta q = Cov (q_j, sum_i s_{ij})$.

So far so good. But Hamilton then says that this equation can be rewritten as

$wDelta q = sum_i 1/n sum_j (q_j - q)s_{ij}$

Based on the definition of covariance (i.e., $Cov (X,Y) = E((X-E(X))(Y-E(Y)))$, this seems to only be the case if $E(sum_i s_{ij}) = 1/n sum_j sum_i s_{ij} = 0$. But this would imply that the average fitness wouldn't change over time, which sounds odd for me. In sum, I don't understand this step in Hamilton's paper. What am I missing?

Actually the derivation is pretty straightforward. It's easier to use the fact that $Cov(X,Y) = E(XY) - E(X)E(Y)$ to derive this result. Suppose $x_{j} = sum_{i} s_{ij}$.
egin{align*} Cov (x_j, q_{j}) &= E (x_{j}q_{j}) - E (x_{j}) E (q_{j}) &= frac{1}{n}sum x_{j} q_{j} - frac{1}{n}sum x_{j} q &= frac{1}{n} sum x_{j} (q_{j} - q) end{align*}

Fisher, Medawar, Hamilton and the Evolution of Aging

THE idea that senescent decline in the performance of biological systems must have an evolutionary basis traces back almost to the beginnings of evolutionary biology (R ose 1991, Chap. 1), although Darwin does not seem to have discussed the problem. At first sight, the nearly universal existence of senescence in species of multicellular organisms is paradoxical, given that natural selection supposedly causes the evolution of increased, not decreased, fitness. As discussed by C omfort (1979), many biologists have, therefore, taken the view that senescence reflects an inevitable process of damage accumulation with age, and indeed an analog of senescence can be seen in complex machines such as cars (G avrilov and G avrilova 1991). But unicellular organisms, such as bacteria, which propagate simply by binary fission, and the germ lines of multicellular organisms, have been able to propagate themselves without senescence over billions of years, showing that biological systems are capable of ongoing repair and maintenance and so can avoid senescence at the cellular level. Senescence cannot, therefore, just be an unavoidable cumulative result of damage. The large amount of variation among different species in their rates of senescence also clearly indicates that aging is subject to variation and selection (C omfort 1979 F inch 1990 R ose 1991 W achter and F inch 1997). This conclusion is backed up by the existence of both quantitative genetic variation and major gene mutations affecting the rate of aging (F inch 1990 R ose 1991 W achter and F inch 1997).

Modern evolutionary theory has demonstrated that, in species with a clearcut distinction between parent and offspring, senescence is a virtually inevitable result of the fact that genes that affect survival or fecundity only early in life have a greater selective impact than genes whose effects are manifest only late in life. The purpose of this Perspectives article is to trace the history of this idea, with especial emphasis on William H amilton 's (1966) classic paper and its influence on subsequent work. This was motivated by Hamilton's untimely death earlier this year and the fact that his work on the evolution of senescence has probably received less attention than his other seminal contributions to evolutionary theory.

Within modern evolutionary genetics, the first discussion of the evolution of senescence was that of F isher (1930), in the context of his famous concepts of the “Malthusian parameter” and “reproductive value.” Fisher considered a sexually reproducing, age-structured population reproducing in continuous time, with a probability of survival to age x of l(x), and a rate of production m(x) of offspring of the same sex as their parent at age x. He pointed out that such a population reaches an asymptotic exponential rate of population increase, r, which is given by the (single) real root of the equation ∫ 0 ∞ e − r x l ( x ) m ( x ) d x = 1 . (1)

Fisher called r the Malthusian parameter of the population, in reference to Malthus's preoccupation with the supposedly inevitable exponential increase in numbers of the human species. Although not mentioned by Fisher, this result traces back to the work of E uler (1760) and S harpe and L otka (1911). r is now commonly referred to as the “intrinsic rate of increase” of the populations, and Equation 1 is often called the Euler-Lotka equation.

Fisher drew attention to the quantity defined by the relation v ( x ) = e r x l ( x ) ∫ x ∞ e − r y l ( y ) m ( y ) d y . (2)

This is the reproductive value of individuals of age x and measures their contribution to the future ancestry of a population growing at rate r, normalized to a value of unity at the time of conception. F isher (1930) stated (p. 27), “The direct action of Natural Selection must be proportional to this contribution.” He showed a curve of reproductive value for Australian women, which rises with advancing age during childhood, reaches a maximum around 19, and then declines steadily, reaching zero at close to 50, when most women have reached menopause. He commented (p. 29),

It is probably not without significance in this connexion that the death rate in Man takes a course generally inverse to the curve of the reproductive value. The minimum of the death rate curve is at twelve, certainly not far from the primitive maximum of the reproductive value it rises more steeply for infants, and less steeply for the elderly than the curve of reproductive value falls, points which qualitatively we should anticipate, if the incidence of natural death had been to a large extent moulded by the effects of differential survival.

These ideas greatly influenced M edawar (1946, 1952) when he was formulating the first explicit model of the evolution of aging. Medawar took it for granted that Fisher's reproductive value measures the relative effectiveness of selection at age x. Given this premise, a hypothetical mutant gene that increases survival over a small time interval at an age when reproductive value is high would thus have a higher net effect on fitness than a gene acting at an age when reproductive value is low. Since reproductive value declines over much of adult life, this leads to the expectation that selection will be more effective in improving performance early in adult life than late in life. He pointed out that this means that deleterious alleles with effects restricted to late stages of life would equilibrate at higher frequencies at mutation-selection balance than alleles that act earlier, the process now referred to as the “mutation-accumulation” theory of aging (R ose 1991, Chap. 4). Late-acting deleterious mutations are, of course, familiar to medical geneticists. H aldane (1941, pp. 192–194) had previously suggested that there would be selection for modifiers that postpone the age of onset of the effects of such mutations, and Medawar laid considerable stress on this possibility. However, the relevant selection pressure is of the order of the mutation rate to the trait in question, and so postponement is unlikely to be a major factor in the evolution of senescence (C harlesworth 1994, p. 200).

Medawar also pointed out that alleles with positive effects on performance early in life, but with negative effects because of physiological trade-offs later on, are more likely to be established by selection than alleles with the opposite pattern. This idea was more fully developed by W illiams (1957) and is now known as the “antagonistic pleiotropy” theory of aging (R ose 1991, Chap. 4). Both of these mechanisms could cause an initially nonsenescent life history, in which mortality rates are independent of age, to evolve gradually to a state in which death rates increase with age, the commonly used demographic criterion for senescence (C omfort 1979 F inch 1990). The relative importance of mutation accumulation vs. antagonistic pleiotropy in the evolution of aging is still an unsettled issue (R ose 1991 W achter and F inch 1997).

H amilton (1966) noted that it is fallacious to use reproductive value as a measure of the effectiveness of selection as a function of age, and that a different measure should be used. He started with the Fisherian assumption that r is an appropriate measure of net fitness, so that a mutant gene whose effect on survival or reproductive succcess increases the value of r over that for the current population will be favored by selection. He noted that it is possible to make explicit calculations of the effects on r of small changes in survival or fecundity at a given age, by the method of implicit partial differentiation of Equation 1. In the context of Fisher's continuous-time model, if the integral of the fecundity rate between ages x and x + δx is changed by a small amount δm(x), the associated change in r as δx approaches zero is given by δ r ≈ δ m ( x ) e − r x l ( x ) T , (3) where T is a measure of the generation time of the population, given by T = ∫ 0 ∞ x e − r x l ( x ) m ( x ) d x . (4)

A similar treatment can be applied to the age-specific mortality rate, defined as μ ( x ) = − d ln l ( x ) d x . (5) The change in r associated with a small change, δμ(x), in the integral of the mortality rate between ages x and x + δx is given by δ r ≈ − δ μ ( x ) ∫ x ∞ e − r y l ( y ) m ( y ) d y T . (6)

Neither of these formulae corresponds to reproductive value, as given by Equation 2, and they have rather different implications for the relation between the age of effect of a gene and its impact on fitness. If the population is stationary in size or growing, as must be the case in the long term if it is not doomed to extinction, Equation 3 implies that, all else being equal, there is always a greater selective premium on early rather than late reproduction, since l(x) declines with age. This is not predicted from the reproductive value curve, which increases during infancy, and it reflects the fact that a gene whose effect on fecundity occurs late in life may be removed by death of its carrier before this effect is expressed.

Similarly, Equation 6 implies that selection is indifferent to the timing of gene effects on age-specific mortality during infancy and that its intensity always decreases with age during adulthood. Again, this is quite different from the pattern predicted by reproductive value the difference arises from the fact that reproductive value is conditioned on an individual having survived to age x and discounts the amount of population growth that occurs over a time period x, whereas Equation 6 measures the expected fitness effect of a change in mortality at age x for individuals censused at conception. H amilton (1966) pointed out that these differences are non-trivial. For example, if fecundity increases exponentially with age during adulthood, reproductive value also increases exponentially, so that its use would lead to the conclusion that selection opposes senescence. In contrast, Equation 6 implies that there is always a selective premium on early survival, given the monotonic decrease in the magnitude of its right-hand side with age, although the rate of decline of the intensity of selection with age is greatly slowed if fecundity increases with age.

On the basis of these results, Hamilton proposed that the more rapid incorporation of favorable mutations with early effects on survival or fecundity than mutations with effects later in life would cause an initially nonsenescent life history to evolve in the direction of relatively high mortality rates and low fecundity late in life, without having to postulate any harmful mutations or trade-off effects. This explanation for the evolution of senescence has not been widely accepted, since (as Hamilton himself noted) it does not seem capable of explaining the evidently pathological aspects of many aspects of aging (R ose 1991, pp. 70–71). Instead, most applications of Hamilton's formulae to the evolution of senescence have applied the same basic results to the mutation-accumulation and antagonistic pleiotropy theories (C harlesworth 1994, Chap. 5).

Hamilton also pointed out that the oversimplified model of changes to mortality or fecundity at just one age can easily be extended, by calculating the net change in r owing to small changes in vital statistics at a whole range of ages. Functional relations among fecundity and mortality rates, reflecting resource allocation or physiological constraints, can also be included in such calculations, although he himself did not do this. The inclusion of such constraints has led to the development of elaborate models of life-history evolution, which attempt to predict optimal patterns of age-specific reproduction, growth, and survival and to relate comparative data on life histories to the predictions of these models (e.g., S tearns 1992 C harlesworth 1994, Chap. 5 M c N amara and H ouston 1996). H amilton 's (1966) method of calculating the sensitivity of r to age-specific changes in vital statistics, later applied to the equivalent matrix model of discrete-time populations (D emetrius 1969 G oodman 1971 C aswell 1989), is at the core of this enterprise. Somewhat ironically, reproductive value reappears in optimization models as a weighting function for the effect of a change in fecundity at a given age on mortality at that age (S chaffer 1974), and optimal life histories can be viewed as maximizing reproductive value at each age (S chaffer 1974 C harlesworth 1994, pp. 237–238).

Hamilton's analysis left one important gap, however. This concerns the validity of assuming that the Malthusian parameter r is indeed the correct measure of fitness for an age-structured population, in the sense that it accurately predicts the effect of selection on gene frequencies. No justification of this was provided by Fisher, who seems simply to have taken it for granted, as did most people who pioneered the theory of life-history evolution (e.g., L ewontin 1965) and many distinguished theoretical population geneticists such as Kimura (e.g., K imura 1958), for whom continuous-time models offered technical convenience.

It is easy to define r for a particular genotype, as the solution to Equation 1 for a (hypothetical) population consisting entirely of individuals with the set of l(x) and m(x) values characteristic of the genotype in question this is presumably what Fisher had in mind as the Malthusian parameter of a given genotype. It is also easy to see that, with competition among clonally reproducing genotypes, the genotype with the highest r will outcompete the rest, since this situation is simply equivalent to a set of populations growing at different rates. It is less easy to see how to model a sexually reproducing diploid population in which each parent produces a mixture of genotypes, especially as changes in genotype frequencies induced by selection must cause continual changes in age structure (M oran 1962, p. 90 C harlesworth 1970).

There is, thus, no obvious guarantee that the use of r as a fitness measure gives correct results. In fact, although Fisher characteristically made no reference to their work, H aldane (1927) and N orton (1928) had made great progress toward solving this problem. H aldane (1927) derived integral equations to represent the effects of selection on genotype frequencies in an age-structured population. He analyzed them by making the simplifying assumption that selection was weak and population growth was slow. Using some rather cumbersome algebra, he was able to obtain an approximate expression for the rate of change of gene frequency, in terms of the differences in lifetime expectations of offspring among genotypes. He returned to this problem at the end of his life (H aldane 1962).

N orton 's (1928) paper is one of the most profound papers in both demography and population genetics. Harry Norton was a mathematician at Trinity College, Cambridge (UK), and a member of the Bloomsbury group of British intellectuals. Eminent Victorians was dedicated to him by Lytton Strachey. He is mentioned in Strachey's biography as the only person in the group who could hold his own with Bertrand Russell and John Maynard Keynes (H olroyd 1971). He had earlier anticipated H aldane 's (1924) paper on the rate of change of gene frequency under selection, in a set of calculations published as an appendix to P unnett 's (1915) book on mimicry. Using integral equations similar to Haldane's, which Norton had derived independently in 1910 (H aldane 1927), he examined the asymptotic properties of a diploid, randomly mating population segregating for a single locus with two alleles. Under some simplifying assumptions, notably random mating with respect to age and genotype and no sex differences in vital statistics, the genotype with the highest r value will supplant the others if there is directional selection. With heterozygote advantage in r, a polymorphism is maintained with heterozygote disadvantage, polymorphism is eliminated. He also showed that, with heterozygote advantage, the population ultimately approaches the neighborhood of a fixed gene frequency.

Later work, reviewed by C harlesworth (1994, Chaps. 3 and 4), has extended these pioneering analyses of age-structured populations. Sex differences in vital statistics, nonrandom mating, density-dependent modification of mortality and fecundity rates, and the effects of spatially and temporally fluctuating environments have all been included in the models. In the simplest case described above, when these complications can be neglected, we now know that the initial rate of increase of a nonrecessive rare mutant allele introduced into a large population is governed by its effect on the intrinsic rate of increase, even if its effect is arbitrarily large. In general, however, differences among genotypes in intrinsic rates of increase can be used only as approximate predictors of the rate of change of allele frequency under selection, although the approximation is very good if selection is weak.

Equilibrium frequencies of genotypes under the standard scenarios of population genetic models can, however, be calculated from equations that are of exactly the same form as those of the familiar discrete-generation models of deterministic population genetics (C row and K imura 1970), such that the Wrightian fitness weight for a genotype, wi, is replaced by the expression w i = ∫ 0 ∞ e − r x l i ( x ) m i ( x ) d x , (7) where r is the growth rate of the population as a whole, and li(x) and mi(x) are the vital statistics for individuals of genotype i. If selection is weak, differences in the wi among genotypes are approximately proportional to differences in the corresponding genotypic intrinsic rates, where the constant of proportionality is equal to the value of T for some standard genotype (C harlesworth 1994, p. 178).

It is interesting to note that the equilibrium gene frequency predicted from Equation 7 in the case of heterozygote advantage at a single locus with two alleles corresponds to the value that N orton (1928) showed is approached asymptotically. An exact analysis of K imura 's (1958) use of Malthusian parameters also enables Equation 7 to be recovered (C harlesworth 1970). Bill Hamilton would probably have regarded these results as minor technicalities, but it is satisfying that the assumptions that underly the use of r in explaining the evolution of senescence and life-history patterns can be made explicit and analyzed in population genetic terms.

Our understanding of the evolution of senescence is, at one level, very complete we know that senescence is an evolutionary response to the diminishing effectiveness of selection with age and that this explains many aspects of the comparative biology of senescence (W illiams 1957 R ose 1991 C harlesworth 1994 R icklefs 1998). On the other hand, it is at present hard to be sure which of the two most likely important mechanisms by which this property of selection influences senescence (accumulation of late-acting deleterious mutations or fixation of mutations with favorable early effects and deleterious late effects) plays the more important role, especially as these are not mutually exclusive possibilities. If senescence does not take its toll, perhaps a future Perspectives by this author will provide an update on this problem.


The old physical anthropology was primarily a technique. The common core of the science was measurement of external form with calipers. The new physical anthropology is primarily an area of interest, the desire to understand the process of primate evolution and human variation by the most efficient techniques available (Washburn, 1951 , p. 298).

This year, 2018, marks the centennial of the founding of the American Journal of Physical Anthropology, a notable milestone for the journal, for the American Association of Physical Anthropologists which it represents, and for the larger field of inquiry chronicled in its pages. To celebrate this milestone, I invited my predecessor Editors-in-Chief of the AJPA to join me in assembling a special, centennial issue. We solicited contributions from a number of our colleagues, representing, if not entirely encompassing, the diversity of subfields within physical anthropology. We asked the contributors to reflect on the changes that have occurred since the journal was founded and its role in those changes. In inviting these special contributions, called “Centennial Perspectives,” we granted the authors a great deal of scope as to content and style. The space available is obviously insufficient for comprehensive reviews. Instead, we encouraged the authors to provide their personal perspectives, to identify what seemed to them important and to draw out the themes that they found striking.

The result is a wonderful collection of essays that not only look backward on the development of the discipline of physical anthropology, but forward to its future. We expect that they will be of broad interest, both to newer students and to seasoned researchers, and that they may even be of interest to future generations of students and researchers as a penetrating exercise in self-assessment conducted early in the 21st century.

Many of the Centennial Perspectives make note of Washburn's call for a “new physical anthropology,” and virtually all implicitly or explicitly evoke the relationship of changes in the area they are covering to developments in evolutionary theory. However, none of them directly consider those changes as a central axis running through the history of our discipline. As an introduction to the entire issue, therefore, it may be useful to briefly consider what has been an extremely dynamic century for the larger field of evolutionary biology and its impact on physical anthropology.

Social evolution theory

Before discussing possible points of confusion, it is useful to provide a basic summary of relevant theory. As stated above, the problem of cooperation is why should an individual carry out a cooperative behaviour that appears costly to perform, but benefits other individuals ( Hamilton, 1963, 1964 )? Theoretical explanations for the evolution of cooperation (or any behaviour) can be broadly classified into two categories: direct fitness benefits or indirect fitness benefits (Fig. 1 Hamilton, 1964 Brown & Brown, 1981 Grafen, 1984 Taylor, 1996 Lehmann & Keller, 2006 West et al., 2006b ). This follows from Hamilton's insight that individuals gain inclusive fitness through their impact on the reproduction of related individuals (indirect fitness effects) as well as directly through their impact on their own reproduction (direct fitness effects) ( Hamilton, 1964 Grafen, 1984 ). The terms direct and indirect fitness were introduced by Brown & Brown (1981) , although Fisher (1930 , chapter 2) discussed indirect effects in a similar context.

A classification of the explanations for cooperation. Direct benefits explain mutually beneficial cooperation, whereas indirect benefits explain altruistic cooperation. Within these two fundamental categories, the different mechanisms can be classified in various ways – here we follow West et al. (2006 see also Sachs et al., 2004 Lehmann & Keller, 2006 ). These possibilities are not mutually exclusive, for example a single act of cooperation could have both direct and indirect fitness benefits. We have listed some of the many different terms that have been used to describe the mechanisms for enforcing cooperation to emphasize that reciprocity (reciprocal altruism) is only one of many ways to obtain direct fitness benefits through cooperation. These enforcement mechanisms can also alter the indirect benefits of a behaviour ( Lehmann & Keller, 2006 ), and determining the relationships between these terms remains an important task. Kin selection has been used to refer to (i) just those indirect benefits involving coancestry (i.e. limited dispersal and kin discrimination), or (ii) all indirect benefits (i.e. also including greenbeard effects).

The first class of explanations for cooperation is that it may provide a direct fitness benefit to the individual that performs the behaviour, which outweighs the cost of performing the behaviour ( Sachs et al., 2004 ). One possibility is that individuals have a shared interest in cooperation. For example, in many cooperative breeding species, larger group size may provide a benefit to all the members of the group through factors such as greater survival or higher foraging success – in this case, individuals can be selected to help rear offspring that are not their own, in order to increase group size ( Kokko et al., 2001 ). Another possibility is that there is some mechanism for enforcing cooperation, by rewarding cooperators or punishing cheaters ( Trivers, 1971 Frank, 2003 ). This could happen in a variety of ways, which have been termed punishment, policing, sanctions, reciprocal altruism, indirect (reputation based) reciprocity and strong reciprocity (see below).

The second class of explanations for cooperation is that it provides an indirect benefit because it is directed towards other individuals who carry the cooperative gene ( Hamilton, 1964, 1970, 1975 ). The easiest and most common way in which this could occur is if genes are identical by descent – by helping a close relative reproduce, an individual is still passing on its own genes to the next generation, albeit indirectly. Hamilton (1964) pointed out that this could occur via two mechanisms: (i) kin discrimination, when cooperation is preferentially directed towards relatives (ii) limited dispersal (population viscosity) keeping relatives together, allowing cooperation to be directed indiscriminately towards all neighbours (this will be favoured as those neighbours tend to be relatives). The second way to obtain an indirect fitness benefit is if cooperation is directed towards nonrelatives who share the same cooperative gene. This assortment or ‘greenbeard’ mechanism requires a single gene (or a number of tightly linked genes) that both causes the cooperative behaviour and can be recognized by other individuals due to a distinctive phenotypic marker, such as a green beard ( Hamilton, 1964, 1975 Dawkins, 1976 Jansen & van Baalen, 2006 ). An alternative is to conceptualize greenbeards as a form of kin discrimination.

Replicator and vehicles

Upon publication, The Selfish Gene ( Dawkins 1976) received both enthusiastic praise (e.g., Hamilton 1977) and fierce criticism (e.g., Lewontin 1977). A common theme among critics was that the gene cannot be the unit of selection because selection cannot act on them directly, only via their effects on individual organisms ( Gould 1977). The distinction between replicator and vehicles ( Dawkins 1982a, 1982b also known as interactors, Hull 1980) was introduced partly to address this issue. Under this model of evolution, natural selection requires 2 different units playing different roles in the evolutionary process ( Godfrey-Smith 2000). Replicators are entities that faithfully produce copies of themselves that are transmitted across generations. In biological evolution, as far as we know, genes play this role. A vehicle is an entity that interacts with the environment, and whose phenotype has evolved to preserve the replicator that it carries. Since it is the differential survival and reproduction of vehicles that lead to the spread of replicators, selection can be said to act on replicators via their effects on the vehicles that house them. However, since individual organisms and groups are transient occurrences, vehicles cannot be a unit of selection. Genes, on the other hand, are units of selection because they are “potentially immortal” ( Dawkins 1982a, p. 97 Bourke 2011).

To see how selfish genetic elements fit into the replicator/vehicle distinction, it is first worth noticing that Dawkins himself changed his mind slightly about the implications of the distinction ( Sterelny and Kitcher 1988 Okasha 2008b). In The Selfish Gene ( Dawkins 1976), he argues that the gene level offers a uniquely correct representation of the causal processes underlying evolutionary change. In The Extended Phenotype ( Dawkins 1982a), however, he presents a weaker argument. Here, Dawkins argues that the gene’s-eye view and the traditional individual centered view as 2 different, equivalent perspectives of evolution—2 orientations of a Necker Cube, as he puts it. Whereas selfish genetic elements are easily accommodated by the first, stronger, argument, the equivalence of the individual and gene’s-eye view is more problematic. Selfish genetic elements are the textbook example of a phenomenon not explainable by the traditional individual-centered perspective. A way around this, as has been suggested multiple times ( Sober and Wilson 1998 Reeve and Keller 1999 Okasha 2008b Lloyd 2012), is to treat replicators that are selfish genetic elements also as vehicles. Thus, whereas all genes are replicators, and can only improve their chances of transmission by contributing to the fitness of the vehicle that houses them, selfish genetic elements play a dual role.

Press release

for the discovery of “restriction enzymes and their application to problems of molecular genetics”.

Restriction enzymes provide the “chemical knives” to cut genes (= DNA) into defined fragments. These may then be used (1) to determine the order of genes on chromosomes, (2) to analyse the chemical structure of genes and of regions of DNA which regulate the function of genes, and (3) to create new combinations of genes. These techniques open up new avenues to study the organisation and expression of genes of higher animals and to solve basic problems in developmental biology. In medicine, increased knowledge in this area should help in the prevention and treatment of malformations, hereditary diseases and cancer.

Arber discovered restriction enzymes. He postulated that these enzymes bind to DNA at specific sites containing recurring structural elements made up of specific basepair sequences.

Smith verified Arber’s hypothesis with a purified bacterial restriction enzyme and showed that this enzyme cuts DNA in the middle of a specific symmetrical sequence. Other restriction enzymes have similar properties, but different enzymes recognize different sequences.

Nathans pioneered the application of restriction enzymes to genetics. He demonstrated their use for the construction of genetic maps and developed and applied new methodology involving restriction enzymes to solve various problems in genetics.

This year’s Nobel Prize in medicine or physiology is awarded for discoveries with far reaching consequences for genetics. The task of genetics is to describe and explain how genes are organized and expressed in cells and in living organisms. The discovery of restriction enzymes provided new tools for the detailed chemical analysis of the mechanism of gene action. Even though these enzymes have been available only during a few years their application to genetics has already led to new and far reaching results, in particular concerning the organisation and expression of genes (= DNA) of higher animals. All work in this area carried out by many research groups all over the world, is based on the discoveries made by the three laureates.

Restriction enzymes are used as tools to dissect DNA into smaller defined fragments. These can be used to determine the order of genes on chromosomes, to analyse the chemical structure of genes and to recombine genes by chemical means. Most important restriction enzymes are used to analyse the function of regions of DNA which regulate gene expression. This opens up new areas of research to study the connection between heredity and function. We can now begin to answer questions of central biological importance in developmental biology: how do genes direct the evolution of a single fertilized egg to a complete individual with many different organs? What determines that the cells within one organ normally retain their specialized functions? Different diseases are expressions of disturbances in normal functions and increased knowledge in molecular genetics should aid in preventing and treating malformations, hereditary diseases and cancer.

Werner Arber started this field of research in Geneva during the 1960’s. He discovered restriction enzymes. Arber was studying an earlier known phenomenon, “host controlled restriction of bacteriophages”, and found that this process involved changes in the DNA of the virus. The process apparently served to form a barrier against foreign genetic material. Arber showed that the phenomenon could be divided into two components: restriction and modification. Restriction involved a breakdown of DNA, modification was a change (= methylation) of DNA which prevented restriction. Arber postulated that both processes are catalyzed by specific restriction and modification enzymes. He proposed that DNA molecules contain specific sites with the capacity to bind both types of enzymes. These sites are created by recurring structural elements formed from specific basepair sequences. The enzymes act at these sites either by cleaving the molecule (= restriction) or by methylating it (= modification).

Hamilton Smith verified Arber’s hypothesis. He is a biochemist and worked independently of Arber in Baltimore. In 1970 he published two classical papers which described the discovery of a restriction enzyme from the bacterium Heamophilus influenzae and characterized in detail the mechanism of its action. Other scientists before Smith had unsuccessfully tried similar experiments. The restriction enzyme from Haemophilus influenzae degrades foreign DNA to large fragments, about 1000 basepairs in size, but does not touch the DNA of the host bacterium. Most important, Smith showed that all fragments at their beginning and end had the same three basepairs showing that the enzyme had cleaved DNA wherever a specific sequence of 6 basepairs was present. This sequence was internally symmetric and was cleaved in the middle. Many other restriction enzymes have by now been characterized by others using the methodology worked out by Smith. More than 100 such enzymes are known and in most cases the same pattern is observed: a restriction enzyme recognizes certain symmetrical basepair sequences and cleaves DNA wherever these sequences occur. Different enzymes recognize different sequences and by now a battery of enzymes is available which can be used to cleave DNA at different sites in order to produce a multitude of defined fragments.

Dan Nathans pioneered the application of restriction enzymes to problems of genetics. He works in Baltimore at the same university as Smith. All his contributions in this area of research were made during the 1970’s. Nathans uses in his experiments the small DNA from a simian virus, called SV40, but his results are of general significance. In his first communication from 1971 he showed that the restriction enzyme discovered by Smith cleaves SV40 DNA into 11 well defined fragments. In this communication Nathans also discussed other possible applications of restriction enzymes in genetics and in a brilliant way predicted much of the later development. Nathan’s publication from 1971 no doubt served as a major source of inspiration for scientists who subsequently started to use restriction enzymes. Two years later he described the cleavage patterns of SV40 DNA obtained with two additional restriction enzymes. He could then piece together the fragments obtained from the three cleavages and construct the complete genetic map of SV40 DNA, the first obtained by a chemical method. The general approach designed by Nathans for SV40 was later used by other scientists for mapping increasingly complex DNA structures. The map of SV40 DNA was further refined by other scientists. Today we know the complete nucleotide sequence of the molecule and thus can write the complete chemical formula for all the genes of an animal virus. Nathans himself continuously contributed new ideas and developed new methods for the application of restriction enzymes to genetic problems and has continuously been a main source of inspiration in this field of research.

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Twelve laureates were awarded a Nobel Prize in 2020, for achievements that have conferred the greatest benefit to humankind.

Their work and discoveries range from the formation of black holes and genetic scissors to efforts to combat hunger and develop new auction formats.

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America's Founding Fathers — including George Washington, John Adams, Thomas Jefferson, James Madison, Alexander Hamilton, James Monroe and Benjamin Franklin — together with several other key players of their time, structured the democratic government of the United States and left a legacy that has shaped the world.

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For over a century, biologists have been describing the remarkable ability of amphibious fishes to survive out of water. The ability to emerse and breathe air allowed ancestral fishes to exploit a new O2-rich habitat and avoid poor aquatic conditions and aquatic predators. However, there are many negative consequences to breathing air (Table 1) thus, emersion must confer a significant survival advantage. Here, we have provided the first list of extant amphibious fish species, revealing a remarkable evolutionary diversity of form and function across >200 species. These amphibious fishes have met the challenges of life out of water by expressing a variety of plastic traits and relying on fixed adaptations that enhance survival on land. There is a tremendous diversity of life histories, from species that are highly active out of water but return to an aquatic refuge multiple times each day to other species that are quiescent on land and survive there for entire seasons. We understand little about the full extent of phenotypic flexibility and developmental plasticity in amphibious fishes and the underlying regulatory mechanisms. Early rearing in air has been shown to profoundly alter the phenotype of later life stages in Polypterus (Standen et al., 2014), but further exploration of developmental plasticity in other species may provide some insights into genetic assimilation and the evolution of terrestriality. Finally, there is inherent value in uncovering the physiological mechanisms (plastic or otherwise) used by amphibious fishes with different life histories and evolutionary origins to cope with emersion, especially given that we only have data on a few taxonomic groups.

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