Inbreeding and the Genetic Complexity of Human Hypertension

Corresponding author: Harry Campbell, University of Edinburgh Medical School, Teviot Pl., Edinburgh EH8 9AG, Scotland, UK., harry.campbell{at}ed.ac.uk (E-mail)

Communicating editor: D. CHARLESWORTH

	ABSTRACT

TOP ABSTRACT SUBJECTS AND METHODS RESULTS DISCUSSION APPENDIX LITERATURE CITED

Considerable uncertainty exists regarding the genetic architecture underlying common late-onset human diseases. In particular, the contribution of deleterious recessive alleles has been predicted to be greater for late-onset than for early-onset traits. We have investigated the contribution of recessive alleles to human hypertension by examining the effects of inbreeding on blood pressure (BP) as a quantitative trait in 2760 adult individuals from 25 villages within Croatian island isolates. We found a strong linear relationship between the inbreeding coefficient (F) and both systolic and diastolic BP, indicating that recessive or partially recessive quantitative trait locus (QTL) alleles account for 10–15% of the total variation in BP in this population. An increase in F of 0.01 corresponded to an increase of 3 mm Hg in systolic and 2 mm Hg in diastolic BP. Regression of F on BP indicated that at least several hundred (300–600) recessive QTL contribute to BP variability. A model of the distribution of locus effects suggests that the 8–16 QTL of largest effect together account for a maximum of 25% of the dominance variation, while the remaining 75% of the variation is mediated by QTL of very small effect, unlikely to be detectable using current technologies and sample sizes. We infer that recent inbreeding accounts for 36% of all hypertension in this population. The global impact of inbreeding on hypertension may be substantial since, although inbreeding is declining in Western societies, an estimated 1 billion people globally show rates of consanguineous marriages >20%.

THE extensive literature on the health effects of inbreeding has largely focused on its impact on reproduction, childhood mortality, and Mendelian disorders (BITTLES et al. 1991 ; BITTLES and NEEL 1994 ). Remarkably little has been published on the effects of inbreeding on genetically complex late-onset disorders that account for most of the public health burden of disease. This is despite the observation in other species that the deleterious effects of inbreeding may increase with age, suggesting greater sensitivity of homeostatic mechanisms to inbreeding in later life (CHARLESWORTH and HUGHES 1996 ; CHARLESWORTH and CHARLESWORTH 1999 ).

We postulated that the quantitative trait, blood pressure (BP), and the related late-onset disorder, essential hypertension, might be mediated by recessive and partially recessive quantitative trait locus (QTL) alleles, which would be influenced by the increased homozygosity found in inbred individuals. In support of this hypothesis, several studies of small inbred communities worldwide have reported an increased prevalence of hypertension (KRIEGER 1968 ; MARTIN et al. 1973 ; HURWICH et al. 1982 ; THOMAS et al. 1987 ; WAHID SAEED et al. 1996 ; HALBERSTEIN 1999 ). In addition, analogous observations have come from experiments in inbred ("spontaneous") and engineered animal models of hypertension (STOLL et al. 2000 ). To investigate the relationship between inbreeding and BP we studied a large population sample from well-characterized genetic isolates from the Dalmatian islands, Croatia (Fig 1).

View larger version (67K): In this window In a new window Download PPT slide   Figure 1. Map of middle Dalmatia, Croatia, showing villages and islands.

	SUBJECTS AND METHODS

TOP ABSTRACT SUBJECTS AND METHODS RESULTS DISCUSSION APPENDIX LITERATURE CITED

Study population:
The village populations of three neighboring islands in the eastern Adriatic, Middle Dalmatia, Croatia (Brac, Hvar, and Korcula—see Fig 1) represent well-characterized genetic isolates. Over 100 publications describe the ethnohistory, migration patterns, genealogical reconstruction, biological trait measurements, disease prevalence, and environmental and sociocultural characteristics of this population (RUDAN et al. 1987 , RUDAN et al. 1992 , RUDAN et al. 1999 ; WADDLE et al. 1998 ). Population genetic characteristics of the study population based on a number of serogenetic polymorphisms were reported by ROBERTS et al. 1992 and JANICIJEVIC et al. 1994 . Subsequent analyses of variable number of tandem repeat and short tandem repeat DNA polymorphisms and mtDNA characterized genetic variation in specific islands (MARTINOVIC et al. 1998 , MARTINOVIC et al. 1999 ; KLARIC et al. 2001A , KLARIC et al. 2001B ; TOLK et al. 2001 ). The results indicated that village populations in these islands have preserved separate characteristics over the course of history to the present day. Measures of genetic kinship and genetic distances revealed isolation of individual villages or village groups from each other and from the mainland. Specific village clustering was noted on Brac and Hvar islands, which coincided with known historic processes. An appreciable degree of genetic homogeneity within the studied villages has been noted, which is especially true for the most geographically isolated villages. The 25 villages chosen for this study were founded during one of three periods: the BC era (by admixture of Illyrians, Greeks, and succeeding Romans), the 7th century AD (by Croats who immigrated from Asia), and the 16th to 18th centuries AD (by Croats who left the Balkan peninsula fearing Ottoman expansion). The subsequent tendency toward inbreeding in each village has been influenced by geographic isolation, political ("Pastrovic") privileges given to residents of certain communities, and by sociocultural factors (RUDAN et al. 1992 ). These island populations present a range of inbreeding patterns at both individual and subpopulation levels, as documented in previous studies reporting endogamy, isonymy, mating choice, genealogical information, and genetic marker distributions. High inbreeding levels have been implicated in at least three Mendelian disorders characterized in neighboring island populations: Mal de Meleda in Mljet (FISCHER et al. 2001 ), hereditary dwarfism in Krk (KOPAJTIC et al. 1995 ), and hereditary mental retardation in Susak (BOHACEK 1964 ). Measures of diet and lifestyle factors show restricted variation in this population, suggesting its suitability for genetic studies of hypertension (RUDAN et al. 1992 ).

Blood pressure and other measurements:
We measured blood pressure, height, and weight between 1979 and 1981 in 2760 adult individuals selected at random from voting lists from 25 isolate villages on three islands (Brac, Hvar, and Korcula) in middle Dalmatia, Croatia (representing a 20% sample of the village populations). In addition, we collected data on body mass index, diet, education level, occupation, and smoking status. This was carried out with the informed consent of participants by the Institute for Anthropological Research in Zagreb, Croatia, in collaboration with the Smithsonian Institute in Washington, DC. None of the examinees had ever received antihypertensive treatment. Blood pressure was measured by a single observer in local health centers and dispensaries between 6 AM and 12 noon following standard procedures as described by Weiner (WEINER and LOURIE 1969 ). BP values were adjusted for the major determinants of BP (age, height, weight, and smoking status in the analyses) and were reported separately in males and females. Hypertension was defined as systolic BP 160 or diastolic BP 95 mm Hg.

Computation of individual inbreeding coefficients:
A single researcher (I. Rudan) computed individual inbreeding coefficients independently and blind to BP status for each study participant on the basis of pedigree information on four ancestral generations (five generations where these occurred over a similar time frame) recorded during the initial field work and supplemented by study of parish registries stored in local churches during 1997–2000. The individual inbreeding coefficients (F) were then computed according to Wright's path method,

(WRIGHT 1922 ), where m_i and n_i refer to the number of paths from the ith common ancestor, and c refers to the number of common ancestors. The genealogical inbreeding coefficient for each village was then computed as the average of all individual F values. To further support these estimates, F was calculated from isonymy as proposed by CROW 1980 ,

where S = p_kq_k, p_k, q_k are the frequencies of the surname k in males and females, respectively, P is the proportion of marriages between spouses carrying the same surname among all marriages, and the summation is over all surnames. We calculated average inbreeding measures for each of the 25 villages on the basis of isonymy, which provides an upper bound (ROGULJIC et al. 1997 ) (Table 1).

Statistical analysis and modeling:
Comparisons of BP among villages were based on systolic and diastolic BP measurements adjusted for age, body mass index (weight/ height²), and smoking status. A step-down multiple regression analysis was performed using MINITAB 12.21 software to investigate the correlation between individual BP measurements and inbreeding coefficients. The model explored the relationship between systolic and diastolic blood pressure (as dependent variables) and a number of explanatory variables: individual inbreeding coefficient (F), island and village of residence, smoking status, and the major known risk factors for hypertension—age, sex, (log-transformed) height, and (log-transformed) weight. Variables that made the least contribution to the explained variation were dropped one at a time until all the remaining variables were statistically significant (defined as P < 0.05 for main effects and P < 0.01 for higher-order effects; Table 2). A model was developed from quantitative genetic theory to derive a lower bound, n_L, for the number of genetic loci of equivalent effect contributing to the dominance variance in BP, as

(1)

where D_T is the overall slope of the regression on F, V_G is the total genetic variance, V_P is the total phenotypic variance, and H² is the broad-sense heritability (see the Appendix). This extends to multiallelic loci the result given by CHARLESWORTH and HUGHES 1999 for the biallelic case and is valid except in the unlikely case of strong overdominance at all, or most, loci. To correct for possible unobserved background inbreeding preceding the earliest generation of which we had knowledge, we inflated F values by a factor equal to the ratio of the mean of village inbreeding levels based on isonymy methods to the mean based on pedigree methods.

Population-attributable fraction:
The population-attributable fraction (PAF) for hypertension (defined as either systolic >160 mm Hg or diastolic >90 mm Hg) was calculated by multiple logistic regression allowing for individual differences in the variables: village, sex, age, height, weight, and smoking. We determined the appropriate regression as a function of all associated variables (including F) and then noted each individual's probability of being hypertensive if their F was set equal to 0. The sum of all such probabilities, P_sum, is an estimate of the number affected in the absence of inbreeding, but with other variables remaining unaltered. Then PAF = 1 - P_sum/N_aff, where N_aff is the actual number affected.

Modeling the effects of individual QTL loci:
For biallelic loci, the relation between the true number, n say, of recessive QTL loci affecting a trait and n_L (see above) is n = n_L(1 + ²), where denotes the coefficient of variation of the frequency distribution of locus effects. Following ZENG 1992 , we modeled this as gamma with parameter L 1; i.e., f(x) = x^L-1e^-x/(L). This family of distributions has ² = L^-1. Since the contribution of a biallelic locus with nonadditive effect, x (or D_j in the notation of the Appendix), to the dominance variance is just x², the distribution of such contributions is also gamma, but with parameter L + 2. Hence, for given L, we can compute the minimum proportion of loci contributing any specified proportion of the overall variance. Finally for given n_L, we obtain an estimate of the actual minimum number of loci by multiplying this proportion by n_L(1 + L^-1). As shown by ZENG 1992 this number is relatively insensitive to L in the range ¹/₁₆ L 1.

	RESULTS

TOP ABSTRACT SUBJECTS AND METHODS RESULTS DISCUSSION APPENDIX LITERATURE CITED

Measurements recorded during a survey in 1979–1981 in an untreated population permitted analysis of BP as a quantitative trait. Body mass index, diet, education level, occupation, smoking status, and inbreeding values among study participants are shown in Table 1 by village of residence. The prevalence of hypertension among individuals with no known inbreeding in their recent ancestry in the study population was 20%, and the mean ages of those males and females were, respectively, 45.9 (SD 13.9) and 47.0 (SD 13.9) years. Average inbreeding measures for each of the 25 villages based on Wright's path method and isonymy gave a consistent pattern of ranking of villages by level of inbreeding. This supports the use of F values as a means of ranking individuals and villages by inbreeding coefficient (Table 1).

We found a highly significant linear correlation between mean inbreeding coefficient of study individuals in each village and the prevalence of hypertension (Fig 2). To explore this further, we performed multiple regression analysis of systolic and diastolic BP on individual inbreeding coefficients (F), controlling for the main recognized determinants of BP (age, sex, height, and weight), village of residence, and smoking status. We found a strong linear correlation between F and adjusted systolic and diastolic BP in both males and females (Fig 3). Both systolic and diastolic BP levels correlated positively with age, weight, and individual inbreeding coefficients and negatively with height and smoking status in both males and females. The regression model explained 35–50% of the phenotypic variance in BP. The strongest effect was clearly individual inbreeding coefficients, which alone explained 15% of the variance in males and 10% in females in both systolic and diastolic levels (Table 2). An increase in F of 0.01 corresponded to an increase of 3 mm Hg in systolic and 2 mm Hg in diastolic BP in both sexes.

View larger version (14K): In this window In a new window Download PPT slide   Figure 2. Relationship between average inbreeding coefficients (F) computed from genealogical information and prevalence of hypertension in 25 study villages.

View larger version (33K): In this window In a new window Download PPT slide   Figure 3. Multiple regression analysis: plot of residuals against inbreeding for (a) systolic BP and (b) diastolic BP after adjusting for effects of village, sex, age, height, weight, and smoking and (c) height after adjusting for village, sex, and age.

The effect of inbreeding (F) on BP depends on the number and dominance properties of QTL alleles, their frequencies, and average effects on the trait (MUKAI et al. 1974 ; FALCONER and MACKAY 1996 ). Using result (1) and taking the total phenotypic variance as an upper limit for the value of V_D, we found the genetic component of blood pressure variability in this population to be influenced by not less than several hundred recessive QTL, with 405 and 306 loci for systolic BP and 615 and 375 loci for diastolic BP in males and females, respectively.

The distribution of recessive QTL effects can be approximated as gamma-type with mode at zero and parameter L < 1 (ZENG 1992 ). From our data, if the estimated minimum QTL number is 400, and L is between ¹/₁₆ and 1, then the minimum numbers contributing the upper 25th and 50th percentiles of the distribution are, respectively, 8–16 and 30–55 (Fig 4).

View larger version (38K): In this window In a new window Download PPT slide   Figure 4. The effect of L on the minimum numbers of loci contributing specified proportions of the overall dominance variance, assuming nL = 400.

Height was analyzed in a similar fashion since in many populations it shows additive variance but no major dominance component (KRIEGER 1968 ; TAMBS et al. 1992 ). The results showed that the slope of the regression of F on height did not differ significantly from zero, as predicted (Fig 3), supporting our interpretation of the data.

	DISCUSSION

TOP ABSTRACT SUBJECTS AND METHODS RESULTS DISCUSSION APPENDIX LITERATURE CITED

It is widely recognized that essential hypertension is under considerable genetic influence. However, apart from isolated successes in mapping rare monogenic loci, which account for <5% of hypertension, no major progress has been made in defining the genetic basis of essential hypertension (LIFTON et al. 2001 ). A common, often implicit, assumption in mapping studies of such complex traits is that relatively few genetic loci of moderate to large effect account for a large component of the underlying genetic variance despite the paucity of empirical data to support this. We have demonstrated that the effects of recessive QTL on BP are widespread, accounting for 10–15% of the total variation in BP in this population. These effects are attributable to a very large number of loci (at least 300–600), which will almost certainly show a range of effects on BP.

The model makes several assumptions that may influence these estimates. First, the inbreeding coefficient is based on measures of recent inbreeding (over four to five generations). We therefore calculated isonymy estimates for each village (Table 1) and found that their mean value exceeded the median F value by a factor of 1.35. Since isonymy is widely recognized to overestimate inbreeding (TAY and YIP 1984 ), this represents an upper bound to the inbreeding estimate. Inflating the F values by 135% in the model reduces the above estimates of minimum QTL numbers by a factor of 1.35² (=1.83). On the other hand, if, as seems likely, V_D/V_P is nearer 33% than the 100% assumed here, the effect would be to triple the estimates. CAVALLI-SFORZA and BODMER 1971 , for example, estimated V_D/V_P to be 0.38 and 0.33 for systolic and diastolic blood pressure, respectively, using the data of MIALL and OLDHAM 1963 . Second, as in many genetic models, all loci were assumed to have equal effects, whereas both theory (BRINK 1967 ) and empirical data in animals (MACKAY 2001 ) show that the QTL effects vary widely and may even be of opposite sign. Ignoring this again results in underestimating the true number of QTL. Third, if a substantial proportion of the phenotypic variance is due to epistatic effects, additive and dominance variances may be upwardly biased and lead to an underestimation of the number of QTL. However, many studies suggest that epistatic QTL effects are uncommon (MACKAY 2001 ). Finally, we assume that recombination between adjacent loci is sufficiently frequent that the identity-by-descent (IBD) status of any locus can be considered independent of its neighbors. In effect, the method treats tightly linked loci as a single "superlocus" (FLINT and MOTT 2001 ), leading to further underestimation of the true number of loci.

The magnitude of the inbreeding effect on BP is large (equivalent to a rise in systolic BP of 20 mm Hg and diastolic of 12 mm Hg in offspring of first-cousin marriages; F = 0.0625) but very similar to the only other two published estimates we could identify in other isolate populations. KRIEGER 1968 found a 35 mm Hg increase in diastolic BP associated with a 0.1 increase in F in a study of 3465 children in Brazil and MARTIN et al. 1973 reported a 7–28 mm Hg increase in systolic BP in adult Hutterites associated with an increase in F of 0.0625. This may be because inbreeding has a greater influence on late-onset traits than on traits that are subject to early selection (CHARLESWORTH and HUGHES 1996 ). It is also possible that low environmental variation, or underestimation of F due to individuals being related through multiple lines of descent, contributes to the size of inbreeding effect in these isolate populations (KRIEGER 1968 ; MARTIN et al. 1973 ; HALBERSTEIN 1999 ; ABNEY et al. 2001 ). Thus the observed effect size may be less in more environmentally diverse or outbred populations. The unidirectionality of the effect is also striking and consistent with a linear unidirectional effect seen in an S-Leut isolate population (MARTIN et al. 1973 ), but the mechanism is unclear. A change in BP with inbreeding is predicted as a consequence of recessive or partially recessive variants with the direction of change toward the value of the more recessive alleles. Physiological homeostasis may also act to support a directional change in BP, for example, through selection against variants tending to reduce BP to maintain circulatory viability. Directional dominance may also occur in late-onset traits when environmental factors are directional (e.g., increase in adult blood pressure due to dietary salt) and when selective constraints are weak compared with blood pressure maintenance in early life.

The estimate of several hundred recessive QTL relevant to human hypertension is realistic and indeed may be conservatively low. It is consistent with a complex and genetically highly variable (HALUSHKA et al. 1999 ) system of blood pressure control mediated by cardiac output, blood vessel architecture, renal function, and central nervous system integration and requiring the interaction of homeostatic systems, including baroreceptors, natriuretic peptides, renin-angiotensin-aldosterone, kinin-kallikrein, adrenergic receptors, and local vasodilator mechanisms (LIFTON et al. 2001 ). Furthermore, published work from animal models of hypertension supports a polygenic rather than oligogenic basis for hypertension (LIFTON et al. 2001 ) and yet these models probably underestimate the genetic complexity, since they are typically bred to achieve fixation of a small subset of the diversity found in wild populations (FLINT and MOTT 2001 ). The greater genetic complexity of a diverse and outbred human population would seem to be self-evident, despite the fact that humans show less haplotype and polymorphic diversity than several other species, including other primates (REICH et al. 2001 ).

Our minimum estimates of the number of recessive QTL influencing blood pressure control do not in themselves reveal the relative magnitudes of locus effects. There is, however, good evidence for an L-shaped (leptokurtotic) distribution of allelic-effect sizes (SHRIMPTON and ROBERTSON 1988 ; TANKSLEY 1993 ; BOST et al. 2001 ; HAYES and GODDARD 2001 ; MACKAY 2001 ; BARTON and KEIGHTLEY 2002 ). In addition, as shown by ZENG 1992 , their distribution can be approximated as gamma-type with mode at zero (i.e., with parameter L < 1), implying that most loci contribute little to the overall genetic variation and that the number contributing a large proportion is both small and relatively insensitive to L. The model developed from our data predicts the minimum QTL numbers contributing the upper 25th and 50th percentiles of the distribution are, respectively, 8–16 and 30–55 (Fig 4). Thus, the QTL with the largest effect account individually for a small proportion of the total dominance variation and 50–75% of the variation is mediated by many QTL of very small effect, which are probably undetectable using current methods (TERWILLIGER and GORING 2000 ).

This study demonstrates an important effect of inbreeding on the genetically complex late-onset disorder, hypertension, which appears to be mediated by a large number of recessive QTL alleles as a result of increased homozygosity. Several factors support the validity of the data and reinforce the conclusions: first, the standard measurement procedures that were adopted and the exclusion of known confounding factors; second, the consistency of findings in diverse populations (KRIEGER 1968 ; MARTIN et al. 1973 ; HALBERSTEIN 1999 ); third, the linear increase in BP with increasing F (prevalence of hypertension rises by 10% for every increment in F of 0.01 up to F = 0.06); fourth, the overall strength of the effect; fifth, the existence of biologically plausible mechanisms, all of which point to a causal relationship between inbreeding and hypertension. Moreover, the consistency of the observation in a random sample of individuals across 25 villages is not explicable by a kinship effect. In terms of health impact, the results show that 36% of hypertension incidence in this population can be attributed to inbreeding (population-attributable fraction). The population prevalence of hypertension among individuals with no known inbreeding in their recent ancestry is 20%, similar to most outbred populations, but it increases steeply among 50-year-olds as the inbreeding coefficient rises (Fig 5).

View larger version (18K): In this window In a new window Download PPT slide   Figure 5. Prevalence of hypertension as a function of inbreeding. The data are shown for nonsmokers of age 50 years, height 1.7 m, and weight 75 kg and with mean village effect, after adjusting for the effects of these variables by binary logistic regression. The dashed lines are 95% confidence limits.

Inbreeding is generally decreasing among nonimmigrant Western societies but it is highly prevalent globally. Consanguineous marriages, defined as a union between individuals related as second cousins or closer (equivalent to F 0.0156 in their progeny), has been conservatively estimated to occur at 1–10% prevalence among 2.811 billion and at 20–50% prevalence among 911 million people globally (BITTLES 1988 ; BITTLES et al. 2001 ). In addition, the extent of homozygosity by descent in outbred populations may have been underestimated (BROMAN and WEBER 1999 ). The global impact of inbreeding on hypertension (and stroke) could therefore be significant in health economic terms. In addition, the results provide new insights into the genetic architecture of a common disorder, which should inform and improve the design of QTL-mapping studies and explain some of the observed differences in trait distributions among different populations.

	ACKNOWLEDGMENTS

The authors thank Professor Bill Hill, Professor Brian Charlesworth, and Dr. Peter Visscher for helpful discussions, comments, and suggestions. This work was supported by the Wellcome Trust (IRDA) grant to H.C. and I.R., the Croatian Ministry of Science and Technology (CMST) grant 01960101 to P.R., 0196005 to P.R., 0196001 to N.S.N., and 0108330 to I.R., and the joint British Council and CMST grant ALIS 054 to H.C. and I.R. I.R. was supported by funds from the UK Medical Research Council, the University of Edinburgh, and the Overseas Research Scheme.

Manuscript received June 28, 2002; Accepted for publication November 19, 2002.

	APPENDIX

TOP ABSTRACT SUBJECTS AND METHODS RESULTS DISCUSSION APPENDIX LITERATURE CITED

THE EFFECT OF INBREEDING ON A MULTILOCUS PHENOTYPE
Model, notation, and assumptions:
The phenotype of the ith individual is modeled by

(A1)

where x_ij denotes the contribution to the phenotype of the genotype at the jth locus (j = 1, ... , n), and _i is a random "environmental" contribution, uncorrelated between individuals and with mean 0 and variance ². Each locus is assumed to have K_j alleles, A_jk, with frequencies p_jk (k = 1, ... , K_j), and loci are assumed to act additively and independently. Individuals with genotypes A_jkA_jk and A_jkA_jk' (k k') are distributed around mean phenotypic values of 2a_jk and a_jk + a_jk' + d_jkk', respectively. Thus, d_jkk' = 0 represents additivity, and d_jkk' = a_jk - a_jk' complete dominance of A_jk over A_jk'.

The effect of inbreeding:
Assuming Hardy-Weinberg equilibrium (HWE),

(A2)

where

(A3)

(A4)

and _k denotes summation from k = 1, ... , K_j, and likewise for _k'. (Note that for mathematical conformity, we assume d_jkk = 0, k.) If the two alleles at locus j are IBD, then

(A5)

Hence, if the level of inbreeding, P(IBD), of the ith individual is F_i, we have

(A6)

where _j denotes summation from j = 1, ... , n. Thus, a plot of y_i against F_i is linear with slope -_jD_j.

The components of genetic variance:
A1 shows the steps needed to compute the additive (V_Aj) and dominance (V_Dj) components of total genetic variance (V_Gj) at locus j, defining

(A7)

and

(A8)

This is a generalization of Falconer's treatment for the biallelic case (FALCONER 1964 ). By adding the squared deviations weighted by their frequencies and simplifying, we find

(A9)

and

(A10)

where

(A11)

Since loci are assumed to be independent the overall components of variance are derived by summing over all j. Hence,

(A12)

where

(A13)

Lower limit for the number of loci, n:
We make use of the mathematical result that, for any two sets of real numbers {z_i, i = 1, ... , n} and {w_i, i = 1, ... , n}, if z_i² w_i (i = 1, ... , n) then

(A14)

This is an application of Cauchy's inequality (see, e.g., HARDY 1960 ). In the present context, if we set z_j = D_j and w_j = V_j (where V_j can denote any component of variance, V_Aj, V_Dj, or V_Gj, as required), and provided that we can show that D_j² V_j for all j, then we have a sufficient, but not necessary, condition that the quantity

(A15)

is a lower bound for n. Here, D_T = -_jD_j and V_T = _jV_j denote the overall slope and variance, respectively, the additivity of both relationships being a consequence of assuming that different loci act independently.

Special cases:
The condition D²_j V_j does not hold in all circumstances. However, consideration of special cases suggests that the circumstances under which it breaks down are rather exceptional. In the biallelic case (model BA), the condition always holds since D²_j is identical to V_Dj. For the multiallelic case we consider two models (MA1 and MA2), in both of which all alleles at every locus have equal frequency and successive homozygotes are evenly spaced—that is, p_jk = 1/K_j and a_jk = a_j[k - (K_j + 1)/2] (k = 1, ... , K_j; j = 1, ... , n). In model MA1, the dominance effects are assumed to be equal in absolute magnitude, i.e., d_jkk' = a_jd_j ( j and k k'), whereas in model MA2 they are assumed to be proportional to the interhomozygote distances; i.e., d_jkk' = a_jj|k - k'| ( j, k, k'). With this definition, |_j| = 1 corresponds to full dominance of one allele in each possible pair, and |_j| = 0 to complete absence of dominance. By analogy, it seems logical in model MA1 to scale d_jkk' by half the mean interhomozygote distance, i.e., a_j(K_j + 1)/3.

By straightforward though tedious algebra it can be shown that the condition D_j² V_Gj is satisfied under all models unless the level of dominance exceeds (model MA1) or (model MA2). These asymptotic limits are bounded from above as K_j . Since such levels imply a considerable degree of overdominance in the same direction and between all pairs of alleles, it is unlikely to apply in most situations. For example, in sickle-cell anemia, in perhaps the most widely quoted and extreme case of overdominance in human genetics, FALCONER 1964 quotes relative fitness values of 0.80, 1.00, and 0.25 for the "normal" homozygote, the heterozygote, and the sickling-trait homozygote, respectively. The implied level of overdominance () is then 1.73.

Extreme overdominance:
In the most extreme form of overdominance, all homozygotes have one assigned value (0, say), and all heterozygotes have another (d, say). Then at a single locus, and dropping the suffix j, we have

(A16)

(A17)

(A18)

and

(A19)

where

(A20)

Hence,

(A21)

since R₂ K^-1. On summing over all loci and applying Cauchy's inequality, we obtain

(A22)

Note that this depends in turn on the easily proved result that, for A_j > 0,

(A23)

Conclusions:
The models explored here suggest that a sufficient condition for n_L (with V_T = V_GT = _jV_Gj) to be a lower bound for n is likely to be satisfied in most practical circumstances and will fail only in situations of extreme overdominance. In the most extreme such situation, when all homozygotes have the same genetic value and all heterozygotes have a different one, Cauchy's inequality leads to the result that

(A24)

and hence, if denotes the mean number of alleles per locus, that

(A25)

On the other hand, because the condition is sufficient but not necessary it will in practice be more widely applicable than the above models suggest. For example, in the multiallelic models, the requirement that the absolute dominance is less than a certain limit may allow dominance to be much greater for some pairs of alleles than for others and even of opposite sign, so long as the average dominance remains within the required limit.

Finally, it should be borne in mind that the present method reveals nothing about the relative magnitude of the dominance effects at different loci or of course about the presence of additive effects.

	LITERATURE CITED

TOP ABSTRACT SUBJECTS AND METHODS RESULTS DISCUSSION APPENDIX LITERATURE CITED

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