‘Hardy’s Law’ and genomics in Anaesthesia
2008; Wiley; Volume: 63; Issue: 12 Linguagem: Inglês
10.1111/j.1365-2044.2008.05736.x
ISSN1365-2044
Autores Tópico(s)Genomics and Rare Diseases
ResumoIn this issue of the journal, Huang et al. link pressure pain sensitivity in women to certain polymorphisms in opioid receptor genes [1]. As handling editor, I had asked the authors whether their reported allele frequencies ‘were in line with proportions predicted by Hardy’s Law’. Many readers will be familiar with this question and understand the authors’ response. Others may, however, welcome some discussion of the underlying science. GH Hardy (1877–1947) was an extraordinary mathematician. Fourth Wrangler in Tripos, Fellow of Trinity College Cambridge, Professor at Oxford (and then at Cambridge), Fellow of the Royal Society and recipient of its Copley Medal, he was friend and contemporary of (amongst others) CP Snow, the philosophers Bertrand Russell and Ludwig Wittgenstein, the poets Rupert Brooke and DH Lawrence (who apparently thought Hardy the only person worth talking to in Cambridge), economist J Maynard Keynes and pharmacologist Henry Dale. Hardy is a protagonist in a novel [2] and the subject of two films in progress (see: http://news.bbc.co.uk/2/hi/south_asia/4811920.stm and http://ia.rediff.com/movies/2006/mar/31ram.htm). By Hardy’s own admission, though, his greatest achievement was the discovery of the even more extraordinary mathematician Srinivasa Ramanujan (1887–1920, the first Indian elected to a fellowship at Trinity or the Royal Society) [3]. To explain the Law, I will use as focus a question I have often asked at entrance interview for the past c. 15 years as admissions tutor at an Oxford college. First posed by Udny Yule in 1908 [4] and phrased variously, the gist is: ‘How is it that dominant alleles do not inevitably over time predominate and squeeze recessive alleles out of existence?’ or: ‘If the brown-eyed allele is dominant, how is it that we are not all brown-eyed?’ or ‘Why are dominant alleles or phenotypes not more common?’ etc. Designed to elicit interview discussion, a specific answer exists – provided by Hardy in a beautifully succinct letter to Science [5]. Later it was discovered that Weinberg had independently offered a similar explanation [6] and ‘Hardy’s Law’ is more commonly taught to generations of school biology students as the ‘Hardy-Weinberg Principle’. A gene is a single inheritable piece of information transmitted through generations in deoxyribonucleic acid (DNA). Usually by coding for proteins (but sometimes coding to influence the transcription/translation of other parts of the genome) it broadly determines a certain characteristic (e.g. eye or fur colour). An allele is a version of the gene (e.g. specifically determining blue or brown eye colour; or red or grey fur). Hardy’s insight was to realise that alleles must be considered as if they were freely able to combine in a ‘pool’, created by the random mating of species within a population [5]. He moved away from considering individuals, couples or families to considering the population as a whole. Therefore, if we assume that two alleles, F (dominant) and f (recessive) have respective frequencies in the population denoted by p and q, Hardy realised that there were two necessary conditions that must always be fulfilled, regardless of the biological function of alleles F and f: p + q = 1, i.e. the proportions of alleles F and f must always sum to unity; if allele F constitutes 10% of the population, then allele f must form 90%. The ratios of homozygote dominant FF: heterozygote Ff: homozygote recessive ff is always p2 : 2pq : q2. Less intuitive than point (1), this is readily understood by use of a Punnett square (Appendix). Well-known to school biologists and preclinical students to help estimate the risk of offspring or siblings inheriting certain alleles, this tool is here it is applied to a whole population [4]. Incidentally, it was during Punnett’s lecture at the Royal Society that Yule had asked his question. Unable to respond at the time, Punnett later sought Hardy’s help after their regular game of tennis. Hardy’s equations help answer Yule’s question. Let us suppose that, in one reproductive generation of a population, 80% of the alleles in the fur gene are dominant F (grey fur); thus 20% are recessive f (red fur) as per point (1) above. Punnett’s square shows the resulting ratios of homozygotes FF: heterozygotes Ff: homozygotes ff produced in the offspring of that generation are 64 : 32 : 4 – clearly, a huge dominance of grey furred animals. But Punnett’s square also shows that the proportion of dominant alleles F in the population of offspring is, as before, exactly 80% and the proportion of recessive alleles is still 20%. When time comes for the offspring generation to mate, there is no change in the Punnett square: the overall gene pool is the same. In other words, while the ‘dominance’ of an allele makes it likely to be expressed in the phenotype, its ‘dominance’ does not make it any more likely that it will be inherited. Hardy’s Law makes this last statement mathematically explicit and also enables calculation of allele frequencies without even being able to measure them directly; all we need do is identify the incidence of a dominantly or recessively inherited characteristic. If the incidence of a recessive disease like cystic fibrosis is, say, c. 1/2000 or 0.0005 (0.05%), we immediately know (Hardy’s equation (2), above) that the recessive allele frequency (q) in the population must be √0.0005 = c. 0.02 (c. 2%). Thus the normal or dominant allele frequency (p) must be [by formula (1)] 0.98 (c. 98%). Therefore, homozygote normals must constitute p2 or 0.96 (96%), and heterozygotes constitute 2pq or c. 0.02 (2%). Hardy’s equations can be plotted graphically (Fig. 1) to show all possible combinations of heterozygote:homozygote proportions for the full range of possible allele frequencies. This plot is universal: it applies to any gene, genetic disease or species. Hardy’s equations plotted graphically. The x-axis consists of two parallel axes, one representing the full range of proportions of a dominant allele F in the population (denoted p, from 0 to 100%) and one axis representing the corresponding proportion of a recessive allele f (denoted q, from 100 to 0%). The y-axis indicates the relative frequencies of each possible combination of these alleles in the population. These are: recessive homozygotes ff (given by Hardy’s equations as equal to q2), dominant homozygotes FF (given by Hardy’s equations as equal to p2), and heterozygotes Ff (given by Hardy’s equations as equal to 2pq). The arrow shows the plot of the example for fur colour used in the text and in the Punnett square in the Appendix, for the case of a population with 80%F and 20%f: the points where the arrow crosses each of the lines gives the frequency of resulting homozygotes ff (4%), heterozygotes Ff (32%) and homozygotes FF (64%). The same method can be performed for any combination of two alleles. Hardy himself noted some limitations and caveats to his ‘law’ [5]. It applies to a stable and large population. Ongoing ‘selection pressure’ will cause proportions to change, especially in small populations (if a predator arrives that only eats red animals, red fur may soon evolve out of a small population). The human population is assumed both relatively large and stable, but allele frequencies may differ across ethnic groups (a point made by Huang et al. [1]) [7]. Hardy’s Law also assumes near-random mating of roughly equal males and females within a population, so enforced mating of animals or selectively altering the male:female ratio will cause allele proportions to deviate from those predicted by Hardy’s Law. Other alleles may actually make it less or more likely for the carrier to breed with a specific partner, and not randomly. Random genetic mutations (an inherent part of evolution and adaptation to selection pressure) will also cause deviations from the Law, but normal rates are extremely low (c. 10−11 per DNA base pair per replication) [8], so the natural effect of this in the absence of selection pressure is small. A chi-squared test can be used to assess the significance of any deviation from Hardy’s proportions. It is, incidentally, possible to calculate proportions for genetic traits which have more than two alleles or which are sex-linked: the mathematics becomes quite complex and we will not discuss this, or some other trivial deviations from the Law [9]. All this may be very (un)interesting to readers; but what relevance is it for anaesthesia? Well, when an anaesthetist performs any intervention, be it laryngoscopy, pre-oxygenation, administers drugs, etc, the patient’s response is in part (or wholly) determined by their genetic make-up: differences in alleles (gene polymorphisms) may explain these outcomes [10-12]. This is a highly clinically relevant notion. Many of the 20th century’s interventions were ‘empirical’, i.e. management that worked, but it was not really known why. One example of this is the efficacy of magnesium for eclampsia: it works [13], but we do not know (yet) why it should work better than other treatments. Empiricism cannot be relied upon in the long-term and only knowledge of fundamental processes, especially at molecular or cellular level, enables refinements in treatment – tailored to the disease process – to be made. In many studies, we are generally used to measuring the ‘average’ treatment effect in a relatively large, randomly selected sample. In reality, this ‘average’ effect may consist of a discrete mix of responses of subgroups of patients with different allele composition. Quite apart from the self-evident genetic disorders that have a very direct influence on the conduct of anaesthesia (e.g. malignant hyperthermia [14] or pseudocholinesterase deficiency [15]), anaesthetists are applying this line of thinking to study of the possible role of specific alleles in cardiovascular [16] and cerebral [17] outcomes after surgery, clotting postsurgery [18], renal function [19] and even less ‘dramatic’ issues such as the effectiveness of topical anaesthesia creams [20]. Whether a response to an anaesthetic intervention has a ‘genetic’ basis can be investigated in the following general steps. First the study identifies (from previous work, expert opinion or ‘best guesses’) certain candidate genes whose function is likely associated with the phenomenon being investigated (let us suppose postoperative myocardial infarction, MI, is being studied). For many candidate genes (e.g. one that codes for a proinflammatory cytokine) publicly available information of the human genome (e.g. http://snp.cshl.org/thehapmap.html.en) records the relevant alleles (usually two alternative alleles). In a ‘cross-sectional’ study design, the presence or absence of these alleles is assessed in a random sample from the population of patients, for both patients who suffer MI and those who do not (i.e. ‘genotyping’). In a ‘case–control’ design, a specified number of affected cases (those who suffered MI) is compared with a matched number of control cases (who did not suffer MI), selected at random from the population. For each study design, it is possible to estimate the degree of ‘risk’ (e.g. odds ratio for MI) conferred by the possession of one allele versus another, using statistical techniques such as logistic regression analysis [21]. For both study designs, non-random samples make it inappropriate to extrapolate results to the wider population. A strength of gene studies is that fewer assumptions of randomness (or ‘representativeness’) of the sample need to be made: we can actually test for this using Hardy’s Law. In a properly randomly-selected group, the frequencies of the alleles should follow the proportions indicated by Hardy’s Law, calculated from their known frequencies in the general population. If a study sample exhibits instead a different proportion of allele frequencies, then it may not have been selected sufficiently randomly, or may not be sufficiently representative of the population as a whole. This may lead to bias, skewed data, etc. Even worse, if the allele frequency is well-established to follow Hardy’s Law in the general population but does not do so in the study sample, then genotyping error may have occurred during the technical aspects of the experiment, which might invalidate the entire results of the study. Authors who have investigated the influence of opiate receptor genotype on postoperative morphine consumption [22] and the influence of β-receptor genotype on cardiovascular outcome [23] and who failed to check if their allele samples followed Hardy’s Law have been reminded of this omission by distinguished correspondents [24, 25]. Such omissions are not confined to the anaesthetic literature. Studying polymorphisms for genes that code for the enzyme heme-oxygenase-1 as a risk factor for arteriovenous fistula impotency in patients with renal failure, Lin et al. [26] were criticised for their oversight in not reporting adherence of their measured alleles to Hardy’s Law [27]. Indeed, general guidance issued on the proper conduct and reporting of genotype–phenotype association studies confirms Hardy’s Law as an essential step in the quality control process [28]. So what of Weinberg? A Stuttgart physician with a thriving private practice, he was apparently a difficult man, a loner who had no immediate colleagues or students [29]. Although he published >160 papers, many were vitriolic pieces, abusive to competitors and his writing style was not easy to understand. Perhaps this is why his similar work was only rediscovered c. 40 years after Hardy’s Law was established [5]. To his credit, he apparently had a strong sense of social justice and did much charitable work [29]. Many feel that the importance of Hardy’s contribution to genetics ‘cannot be overstated’ [8], but Hardy himself thought his letter was trivial and never commented on genetics again. Instead he wrote a thoughtful piece on the philosophy of mathematics that is itself an education in how to use the English language [30], and published numerous papers with Ramanujan that are as close to fine art as mathematics can get [31]. Hardy’s little letter [4] offers insight into the data presented by Huang et al. [1]. Such studies will soon – properly – become commonplace in our clinical anaesthetic literature. In the meantime, I will need to find a new interview question. A simple Punnett square. The columns show two alleles F (dominant) and f (recessive) in the ‘male’ gametes in the population pool. The rows show the same two alleles in the ‘female’ gametes of the population pool. For each allele, p and q represent their respective proportions in the population. The Punnett square is like a multiplication table where the each box is the product of the relevant row and column, and so it shows both the genotype that results and their proportions when gametes fuse. The Punnett square above is now replaced with specific values for the example used in the text: F is the dominant allele for grey fur, f the recessive allele for red fur. Note that the final proportions of the allele pool in the population is unchanged in the offspring despite overwhelming preponderance of the dominant gene
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