Ionizing radiation and genetic risks VIII. The concept of mutation component and its use in risk estimation for multifactorial diseases

Mutation Research 405 Ž1998. 57–79

Ionizing radiation and genetic risks VIII. The concept of mutation component and its use in risk estimation for multifactorial diseases C. Denniston a , R. Chakraborty b, K. Sankaranarayanan

c,)

a Laboratory of Genetics, UniÕersity of Wisconsin–Madison, Madison, WI 53706, USA Human Genetics Center, UniÕersity of Texas School of Public Health, P.O. Box 20334, Houston, TX 77225, USA Department of Radiation Genetics and Chemical Mutagenesis, SylÕius Laboratories, Leiden UniÕersity Medical Centre, Wassenaarseweg 72, 2333 AL Leiden, Netherlands b

c

Accepted 27 May 1998

Abstract Multifactorial diseases, which include the common congenital abnormalities Žincidence: 6%. and chronic diseases with onset predominantly in adults Žpopulation prevalence: 65%., contribute substantially to human morbidity and mortality. Their transmission patterns do not conform to Mendelian expectations. The model most frequently used to explain their inheritance and to estimate risks to relatives is a Multifactorial Threshold Model ŽMTM. of disease liability. The MTM assumes that: Ži. the disease is due to the joint action of a large number of genetic and environmental factors, each of which contributing a small amount of liability, Žii. the distribution of liability in the population is Gaussian and Žiii. individuals whose liability exceeds a certain threshold value are affected by the disease. For most of these diseases, the number of genes involved or the environmental factors are not fully known. In the context of radiation exposures of the population, the question of the extent to which induced mutations will cause an increase in the frequencies of these diseases has remained unanswered. In this paper, we address this problem by using a modified version of MTM which incorporates mutation and selection as two additional parameters. The model assumes a finite number of gene loci and threshold of liability Žhence, the designation, Finite-Locus Threshold Model or FLTM.. The FLTM permits one to examine the relationship between broad-sense heritability of disease liability and mutation component ŽMC., the responsiveness of the disease to a change in mutation rate. Through the use of a computer program Žin which mutation rate, selection, threshold, recombination rate and environmental variance are input parameters and MC and heritability of liability are output estimates., we studied the MC-heritability relationship for Ži. a permanent increase in mutation rate Že.g., when the population sustains radiation exposure in every generation. and Žii. a one-time increase in mutation rate. Our investigation shows that, for a permanent increase in mutation rate of 15%, MC in the first few generations is of the order of 1–2%. This conclusion holds over a broad range of heritability values above about 30%. At equilibrium, however, MC reaches 100%. For a one-time increase in mutation rate, MC reaches its maximum value Žof 1–2%. in the first generation, followed by a decline to zero in subsequent generations. These conclusions hold for so many combinations of parameter values Ži.e., threshold, selection coefficient,

)

Corresponding author. Tel.: q31-71-5276155; Fax: q31-71-5221615

0027-5107r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 7 - 5 1 0 7 Ž 9 8 . 0 0 1 4 6 - 8

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C. Denniston et al.r Mutation Research 405 (1998) 57–79

number of loci, environmental variance, spontaneous mutation rate, increases in mutation rate, levels of ‘interaction’ between genes and recombination rates. that it can be considered to be relatively robust. We also investigated the biological validity of the FLTM in terms of the minimum number of loci, their mutation rates and selection coefficients needed to explain the incidence of multifactorial diseases using the theory of genetic loads. We argue that for common multifactorial diseases, selection coefficients are small in present-day human populations. Consequently, with mutation rates of the order known for Mendelian genes, the FLTM with a few loci and weak selection provides a good approximation for studying the responsiveness of multifactorial diseases to radiation exposures. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Genetic risk of radiation; Multifactorial disease; Radiation risk; Mutation component for multifactorial disease

1. Introduction The genetic risks of exposure of human populations to ionizing radiation are conventionally expressed as the expected number of cases of genetic diseases in addition to those occurring ‘naturally’ as a result of spontaneous mutations. Since the relationship between mutation and disease is not straightforward or the same for all classes of genetic diseases, the assessment of the extent to which the different classes of genetic disease will respond to an increase in the mutation rate caused by radiation is of paramount importance. The 1972 BEIR report of the U.S. National Academy of Sciences w1x introduced the concept of the mutation component ŽMC. to deal with this problem for all classes of genetic diseases with particular focus on multifactorial diseases. Subsequently, Crow and Denniston w2,3x and Denniston w4x discussed this concept in more detail and showed how, in the absence of any detailed knowledge on the genetic basis of multifactorial diseases, one can still address the question of their responsiveness to increases in mutation rate. One important conclusion that emerged from their studies is that MC is related to ‘heritability’ Ži.e., the proportion of the total variability of a trait that can be attributed to genetic causes of variation. and that heritability can provide a meaningful guidance on the magnitude of MC. In the preceding paper of this series w5x, we discussed the MC concept and its application to estimate the responsiveness of Mendelian diseases to increases in mutation rate. In this paper, we extend the study to the assessment of MC for multifactorial diseases. The model developed for this purpose incorporates some elements of the ‘standard’ Multifactorial Threshold Model ŽMTM. of disease liability Žthe latter, a latent variable defining the disease risk. and the equilibrium theory Ži.e., the concept of a balance between mutation and selection. and is called the Finite-Locus Threshold Model ŽFLTM.. The FLTM developed here relates the change in the frequency of a multifactorial disease more directly to an increase in mutation rate and with the simultaneous effect of selection Žagainst affected individuals. than the one discussed by Crow and Denniston w2,3x and Denniston w4x. However, the formulation of the model is such that MC cannot be expressed in the form of a single equation as a function of the mutation and selection parameters. Therefore, we developed a computer algorithm to characterize the model properties and show how it can be used to evaluate MC in any generation following an increase in mutation rate. In order to provide a background for the discussions, we first summarize some essential aspects of multifactorial diseases and the standard MTM. From this, we then rationalize the concept of a finite-locus model for multifactorial diseases ŽFLTM., and develop the algebraic formulations for MC. For this purpose, we consider the presence of fitness differential of individuals Žfor evaluating the opposing effects of mutation and selection. and sporadic occurrences of diseases Žwhich do not respond to any change in the mutation rate, but contribute to disease frequency and the average fitness of the population due to selection against the affected individuals.. In this context, the genetic load theory is also discussed in so far as it is relevant for predicting a biologically meaningful range of parameter values for the FLTM, applicable for common multifactorial diseases.

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We show that under the assumptions and parameter values that were used, the MC for multifactorial diseases in the early generations following either a one-time or a permanent increase in the mutation rate is quite small Ži.e., of the order of 1–2%., which in turn is much smaller than that for Mendelian diseases w5x.

2. Multifactorial diseases 2.1. General aspects The term ‘multifactorial’ is a general designation assigned to a disease known to have a genetic component in its aetiology, but whose transmission pattern cannot be described as simple Mendelian. The common congenital abnormalities Že.g., neural tube defects, cleft lip with or without cleft palate, etc.., with an estimated birth prevalence of about 6% w6x and common chronic, degenerative diseases of adults Že.g., coronary heart disease, essential hypertension, diabetes, etc.., with an estimated population prevalence of 65% Žfor about 25 clinical diseases. w7x are examples of phenotypes that conform to the description of multifactorial traits. These diseases are interpreted as resulting from a large number of causes, both genetic and environmental, the nature of which can vary between individuals, families and populations. The fact that genetic factors are involved is evident from observations of familial aggregation, i.e., these diseases ‘run’ in families, but the risks to relatives do not conform to their respective expectations for Mendelian diseases. For most multifactorial diseases, the risk to first-degree relatives Že.g., sibs. is generally less than one in four, while the sibling risk for simple dominant and recessive traits are one in two and one in four, respectively. This subject has been reviewed in earlier papers of this series w8,9x. 2.2. Multifactorial Threshold Model (MTM) and disease liability To explain the patterns of transmission and to be able to predict risks to relatives of affected individuals—in the near-absence of knowledge of the genetic and of environmental factors that underlie multifactorial diseases —Falconer w10x first proposed what is referred to here as the standard Multifactorial Threshold Model ŽMTM. of disease liability. The MTM itself is an extension of the concepts developed for the study of quantitative characters to qualitative, i.e., all-or-none traits. In its simplest form, the model assumes contribution of numerous genetic and environmental factors, each conferring a small amount of what is called ‘liability’ to disease. The distribution of liability in the population is, therefore, normal or Gaussian. Individuals beyond a certain threshold value of liability are those who are affected. The properties of the normal distribution, the concept of threshold and the use of statistical techniques developed for the study of quantitative characters Žespecially for the estimation of heritability and of correlation of liabilities between relatives. enable prediction of recurrence risks to relatives of affected individuals from population frequencies of the disease. In most cases, these predictions are consistent with empirical observations and are still used in genetic counseling. 2.3. Heritability Since the concept of heritability is an important one from the standpoint of our model, it merits some comments here. In the broad sense, it is the degree to which a trait is genetically determined, calculated as the ratio of genetic variance to the total phenotypic variance Žwhich includes environmental effects. and is symbolized as h 2B . In the narrow sense, heritability is the degree to which a trait is transmitted from parent to offspring, estimated as the ratio of additive genetic variance to the total phenotypic variance and is symbolized by h2N . For most multifactorial diseases of relevance in the present context, the available estimates of h2B are in the range of 30% to 90% Žsee Table 1 in Ref. w7x and Table 8 in Ref. w8x.. Further, as discussed in Appendix A,

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C. Denniston et al.r Mutation Research 405 (1998) 57–79

Table 1 Notations and descriptions of parameters of the Finite-Locus Threshold Model Symbol

Description

n g e x fŽ g.

Number of loci Number of mutant genes in a genotype A normally distributed random variable with a mean of zero and variance Ve f Ž g .q e, the liability g qw g Ž g y1.r2xsyn, where syn measures the intensity of epistasis and dominance; when syn s 0, the model is additive at the level of liability The threshold such that if x GT, the individual is affected the ‘trait’, an indicator variable that takes the value of zero if the individual is unaffected, and one when affected The selection coefficient, such that affected individuals have fitness of 1y s in relation to the fitness of one for unaffected individuals The mutation rate at locus j The parameter indicating an increase of mutation rate; e.g., from m to mŽ1q k . The frequency of gametes with haploid genotype whose binary representation is equal to j For example, for a five-locus model, with normal alleles represented by capital letter alphabets and the mutants by small letter ones, the gamete ABcdE has the binary representation 00110 which is equal to decimal 5, so its frequency is denoted by y5 . The frequencies of gametes carrying j mutant alleles; for example, for a two-locus model, q0 s y 0 for the gamete AB; q1 s y1 q y 2 for gametes Ab, aB and q2 s y 3 for gamete ab and for a three-locus model, q0 s y 0 for the gamete ABC; q1 s y1 q y 2 q y4 for gametes ABc, AbC, aBC; q2 s y 3 q y5 q y6 for gametes Abc, aBc, abC and q3 s y 7 for gamete abc. g Ý js0 q j q g y j s the frequency of Ždiploid. genotypes carrying g mutant alleles; note that q j s 0 for j- 0 and j) n. The probability that a genotype with g mutant alleles has a liability that exceeds the threshold T n Ý2gs 0 Pg Q g s the population frequency of individuals above the threshold T The frequency of the sporadics in the population aqŽ1y a. P s the total frequency of affected individuals in the population including sporadics 1y sQ g s the fitness of a genotype with g mutant alleles; g s1,2, . . . ,2 n n Ž . Ý2gs 0 Pg f g s the mean liability in the population n Ý2gs0 Pg Q g2 y P 2 s the genetic variance of the trait VG qVe sVariance of liability in the population P, the population frequency of the trait P Ž1y P . s the variance of the trait The broad-sense heritability of liability The broad-sense heritability of the trait

T f s mj k yj

qj

Pg

Qg P a PT Wg EŽ x . VG VŽ x. EŽ f . VŽf . h 2x hf2

heritability of liability and heritability of trait are different quantities; the latter, which can be calculated from the former, is generally much smaller than heritability of liability. 3. The mutation component (MC) concept The MC concept has been discussed earlier w2–5x and is used with the doubling dose method of risk estimation. With this method, one first estimates what is referred to as the doubling dose ŽDD., which is the

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dose of radiation that is required to produce as many mutations as those occurring spontaneously in a generation. It is estimated as the ratio of the average spontaneous mutation rate of a set of gene loci to the average rate of induction of mutations by radiation at the same set of loci. The reciprocal of DD Ži.e., 1rDD. is the relative mutation risk per unit dose. Consequently, the risk is estimated as the product of three quantities, namely, the natural incidence of diseases in the population Ž P ., the relative mutation risk Ž1rDD. and the mutation component ŽMC.: Risk per unit dose s P = Ž 1rDD. = MC,

Ž 1.

where MC s w D PrP x r w D mrm x ,

Ž 2.

in which D P is the change in disease incidence relative to P, and D m is the increase in mutation rate relative to m, the spontaneous rate. Therefore, MC is the relative change in disease incidence per unit relative change in mutation rate. It implies that when the mutation rate is increased from m to mŽ1 q k ., the disease incidence increases from P to P Ž1 q k MC.. It should be stressed that: Ži. the MC concept is applicable only when there is a change in mutation rate; Žii. MC is not the same as the genetic component of a disease; rather, MC quantifies the responsiveness of the genetic component of the disease to increases in mutation rate; Žiii. if a disease is only partly genetic, since only the genetic component will respond to an increase in mutation rate, MC for such a disease will be lower than that for a fully genetic disease and Živ. if the disease is entirely environmental in origin, the MC concept does not apply. In estimating risk in the manner discussed above, it is assumed that: Ži. the population is in equilibrium between mutation and selection Ži.e., P represents the disease incidence at mutation–selection equilibrium. and Žii. radiation exposure Žeither in one generation only or generation after generation. will cause an increase in mutation rate and, over a number of generations, the population will attain a new equilibrium between mutation and selection. Therefore, the risk estimate that is obtained pertains to that at the new equilibrium after the increase in mutation rate. Traditionally, the increase Ž D P . is calculated from the old to the new equilibrium although it can be calculated over any other time interval of interest, given P, 1rDD and MC for the specified generation. The theory and procedures for estimating MC for Mendelian diseases are relatively straightforward w5x, but the situation is more involved for multifactorial diseases and is dependent on the model used and its assumptions.

4. The Finite-Locus Threshold Model (FLTM) 4.1. Rationale The MTM mentioned earlier is a descriptive model and is not designed to answer questions on the impact of induced mutations on disease incidence which are the ones of interest in radiation risk estimation. The Finite-Locus Threshold Model ŽFLTM. developed for this purpose Ži. assumes a finite number of loci, Žii. uses the concept of threshold of liability Žwhich is identical to the one used in MTM. and that of mutation–selection equilibrium and Žiii. enables one to examine the relationship between broad-sense heritability of liability Ž h 2x . and MC. The use of such a model was dictated by the following considerations. First, for most well-studied multifactorial diseases Že.g., coronary heart disease, essential hypertension, diabetes, etc.., the standard MTM with the assumption of virtually infinite number of loci cannot be sustained at present: the biometrical and molecular studies with these diseases lend credence to the view that the number of loci underlying these diseases is probably small, each with moderate effects and that genes with major effects at

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C. Denniston et al.r Mutation Research 405 (1998) 57–79

the population level seem infrequent w9x. Furthermore, with the assumption of a finite number of loci, FLTM allows a direct incorporation of effects of mutation and selection in the analysis, while for MTM, any assessment of the joint impact of mutation and selection is troublesome to make. Second, in the absence of precise knowledge on the genetic basis of most multifactorial diseases, a finite-locus model provides a useful starting point because with such a model, the meaning of parameters reflecting mutation and selection can be quantitatively assessed in terms of those of single-gene defects. Even if the number of loci is large, the predominant contribution to a clinical phenotype is generally from large effects of mutant alleles at a few loci. Third, the concept of threshold is implicit in clinical medicine. For example, dietary therapies are recommended for those individuals whose cholesterol concentrations Ža risk for coronary heart disease. are in the range of 5.2 to 6.5 mMrl, more stringent dietary recommendations and some drug therapy for those with levels in the range of 6.5 to 7.8 mMrl and aggressive individualized therapies for those with levels exceeding 7.8 mMrl Žand low density lipoprotein cholesterol levels exceeding 4.9 mMrl.. A similar situation is true also for essential hypertension. In addition, it has been shown w11x that a strict truncation is a good approximation even when the truncation is more gradual, e.g., follows a sigmoid curve. Finally, initial studies incorporating mutation and selection into the standard MTM revealed that the model does not yield reproducible results owing to the fact that there were several parameter values, some of which, the ones of the mutation rate and selection coefficients in particular, were not uniquely determined by the others. 4.2. Assumptions and specifications of the FLTM The FLTM assumes that the genetic contribution to liability is discrete and that of the environmental factors, continuous. Consider a finite number Ž n. of autosomal loci, each with two alleles Žor two groups of alleles., a normal and a mutant one. The discreteness of genetic contribution to liability is determined by the total number of mutant genes Ždefined as a random variable, g, the number of mutant genes in a genotype. at these n loci. The continuous effect of environmental factors is represented by a random variable e, which has a Gaussian distribution with a mean of zero and variance Ve . The total liability, thus, consists of two components: Ža. a function w f Ž g .x of the number of mutant genes in the n-locus genotype of an individual and Žb. the normally distributed environmental effect Ž e .. The threshold characteristic of the model is described by assuming that the individuals with liability exceeding T are phenotypically affected and have a fitness of 1 y s and those below it, are normal with fitness equal to one. These, as well as all other parameters and their descriptions, are given in Table 1 for ready reference.

5. Genetic load, sporadics and mutation component for the FLTM In the FLTM formulated above, the detrimental effect of a multifactorial disease is measured by selection against the affected individuals Žfitness of 1 y s relative to the fitness of one for unaffected individuals.. Two concepts, therefore, become immediately relevant. First, since some individuals have reduced fitness in comparison to others, the average fitness in the population will be smaller than the maximum possible. To evaluate the potential burden of genetic variation on such a reduction in fitness, Muller w12x introduced the concept of genetic load defined as the reduction in the fitness in a population from the maximum possible because of genetic variation in the population relative to the maximum fitness. With a maximum fitness of one for some individuals Že.g., unaffected ones., the genetic load Ž L. may then be represented by L s 1 y W, where W can be computed for the population considering sources that can produce affected individuals in a population either due to genetic causation Žmaintained by opposing effects of mutation and selection. or by non-genetic factors.

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While the theory of genetic load has been discussed mainly in the context of single gene Mendelian diseases, for the FLTM, genetic load can be evaluated in terms of the genetic make-up of the affected individuals. This allows an apportionment of the contributions of different sources to MC and to genetic load Ž L., since some individuals may be affected irrespective of their genotype. Further, they may have liability exceeding the threshold because of a large value of e, the environmental contribution, despite being mutant free at any of the underlying n loci governing the trait. The second concept is that of sporadics. For instance, an individual may develop diabetes, coronary heart disease, or hypertension irrespective of hisrher genotype at the n loci, i.e., a certain proportion of diseases may be environmental or stress-induced, independent of genetic predisposition. We let a be the frequency of sporadics in the population. With P being the frequency of individuals having liability Ž x . above the threshold ŽT ., as before, the total frequency of affected individuals in the presence of sporadics, PT , becomes PT s a q Ž1 y a. P. Therefore, under the FLTM, a reduction in fitness Ži.e., load. may occur from three different sources: LS s the load due to the presence of sporadics in the population Žalthough such individuals have the trait irrespective of their genotype, they are assumed to suffer a reduction of fitness of the same magnitude as those whose liability exceeds the threshold.; LT s the load due to individuals whose liability exceeds the threshold although they carry no mutant allele at any of the n loci governing the trait and LU s the load due to mutant alleles at one or more of the n loci. In Table 2, the loads due to the combined effects of the above three sources are represented by LST , LT U and LST U , respectively. Of these three components of genetic load, the responsiveness to an increase in mutation rate applies only to the loads evaluated through LU , LT U and LST U because only they involve mutations. The maximum fitness Žequal to one. used in this context implies a zero mutation rate. In other words, the load calculations shown in Table 2 are done in reference to individuals who are free of mutations at all loci underlying the trait, which makes our analysis less prone to the usual criticisms of the genetic load theory Žsee e.g., Refs. w12,13x. in which the designation of an ‘ideal’ phenotype of maximum fitness is implicit. By measuring fitness in relative terms of that of the mutation-free phenotypes, we do not imply that the individuals with no mutation necessarily have the maximum absolute fitness in the entire population. The three mutation components MC U , MC T U and MC ST U become relevant in the context of FLTM, which are represented in the last row of Table 2. In these, the quantity P X y P denotes the change in the frequency of individuals above the threshold resulting from an increase of mutation rate from m to mŽ1 q k ., with k measuring the relative increase in the mutation rate. From these computations, the relationships among the three different sources of MCs can be established as follows. First, recall that Q 0 represents the probability that an individual may have liability) T in spite of having no mutant allele at any of the n loci. Thus, the MC due to the mutant genes, MC U , becomes MC U s

PX y P 1 P y Q0 k

.

Ž 3.

The MC due to the combined effects of mutant genes and the presence of individuals whose liability exceeds the threshold although they carry no mutant genes, MC T U , is given by

ž

MC T U s 1 y

Q0 P

/

MC U ,

Ž 4.

which equals MC P , ascribed to disease incidence. Finally, the MC due to the combined effects of all sources that contribute to the load LST U is

ž

MC ST U s 1 y

a PT

/

MC T U ,

Ž 5.

64

Source of decreased fitness

Maximum fitness

Average fitness

Loads

Sporadics

1

1y as

Sporadics and threshold

1y as

1yw asqŽ1y a. sQ0 x

Sporadics, threshold and mutation

1yw asqŽ1y a. sQ0 x

1yw asqŽ1y a. sP x

LS s as Ž 1y a . sQ0 LT s 1y as Ž 1y a . s Ž P y Q0 . LU s 1y asy Ž 1y a . sQ0

LST s asqŽ1y a. sQ0 LT U s

Ž 1y a . sP

X

Corresponding mutation components

MC U s

P yP 1 P y Q0 k

1y as X

MC T U s

P yP 1 P k

LST U s asqŽ1y a. sP X

P MC ST U s Ž1y PaT . P y P

1 k

C. Denniston et al.r Mutation Research 405 (1998) 57–79

Table 2 Mutation components due to different sources and their corresponding fitness and genetic loads

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so that the MC of total load ŽMC ST U . may be written as

ž

MC ST U s 1 y

a PT

/ž

1y

Q0 P

/

MC U .

Ž 6.

Thus, the presence of sporadics and individuals above the threshold, but without mutant alleles at any of the n underlying loci both decrease MC, as expected. When the heritability is high, and therefore, Q 0 is small, MC T U , MC U ; in addition, if there are no sporadics, MC ST U s MC T U , MC U . The above formulations indicate that in order to compute the MCs from the different sources, the quantities PT , P, P X , and Q0 are to be evaluated under the FLTM as functions of selection coefficient Ž s ., mutation rate Ž m., threshold ŽT . and number of loci Ž n., for which no explicit formulae can be derived. We, therefore, used a computer simulation, a description of which is given in Appendix B. The criteria for choosing parameter values for simulations, in turn, are dictated by the extent of genetic load that can be maintained by mutation–selection balance under the FLTM. Using the different components of genetic load described earlier, the formulation of King w14x can be used to predict the number of gene loci underlying a threshold trait for any given combination of incidence Ž P ., mutation rate Ž urlocusrgeneration. and selection coefficient Ž s ., as discussed in Section 6.

6. Mutation load and King’s formula for FLTM and prediction of the number of gene loci Consider first a single autosomal locus with genotypes, frequencies and fitnesses represented by: Genotype

AA 2

Aa

aa

2 pq

q2

Frequency

p

Fitness

w Ž 1 y hs .

w

wŽ1 ys. ,

so that the average fitness becomes W s w Ž1 y 2 hspq y sq 2 . and the load is, therefore, L s 2 hspq q sq 2 . Further assume that the allele A mutates to ‘a’ at a rate of u per generation, so that in the next generation, the new gene frequency Ž qX . becomes w q y hspq y sq 2 q u Ž 1 y q .

qX s

W

.

Ž 7.

Let X s 2 q or X X s 2 qX be defined as the number of ‘a’ alleles per genotype. Correspondingly, the quantity 2 hspq q 2 sq 2 may be written as nAa LAa q n aa Laa , the sum of the components of the genetic load contributed by the genotypes, each weighted by the number of ‘a’ alleles in that genotype. In turn, this may be written as NL s Ž nAa LAa q n aa Laa .rŽ LAa q Laa ., where N s the average number of ‘a’ alleles eliminated per genetic death Ži.e., the mean number of mutant alleles in those individuals removed by selection.. Finally, 2 uŽ1 y q . represents the number of new ‘a’ alleles created by mutation. We, then, have XXs

w X y NL q 2 u Ž 1 y q . W

.

Ž 8.

Since Wrw s 1 y L, we get at equilibrium Ž X X s X . between mutation and selection Ls

2 uŽ 1 y q . NyX

.

Ž 9.

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When there are multiple interacting loci, this formula can be extended to give Ls

2U Ž 1 y q . NyX

,

Ž 10 .

where now, X s 2Ýqi , the average number of mutant alleles per zygote; U s Ýu i , the genomic mutation rate; and q s Ýu i qirÝu i , the weighted mean mutant allele frequency Žweighted by the locus-specific mutation rates.. This equation, first derived by King w14x, implies that the mutation load approximately equates to twice the gametic mutation rate divided by the difference between the mean number of mutants in those individuals removed by selection and the mean number before selection. More exactly, the mutation load is given by L s L xrŽ1 q L x . Žsee Ref. w15x., where L x refers to the King’s formula of Eq. Ž10. above. In the FLTM, assuming a balance of mutation and directional selection, the load is equal to sP, where s measures the selection against those above the threshold and P is the frequency of affected individuals. This approximation of load for FLTM holds unless the heritability of the trait is very low and further, since genetic load is reduced in the presence of partly non-genetic causation of disease Žimplicit in the formulation of FLTM., we have the inequality sP F

2U Ž 1 y q . NyX

,

Ž 11 .

or, PF

1 2U Ž 1 y q . s

2U F

NyX

s

.

Ž 12 .

In the context of this work, the above inequality is relevant in that it enables one to determine the minimum number of loci needed to explain a disease incidence Ž P . for any given combination of mutation rate Ž u. and selection coefficient Ž s .. In other words, this inequality offers guidance on the combination of parameter values Ž s, u, P and n. for which the FLTM can represent a biologically valid mathematical model for a multifactorial disease. Noting that 2U s 2 nu, where u is the mutation rate per locus and n is the number of loci, the inequality Ž12. predicts that a large number of loci would be needed to explain threshold traits whose incidence is of the order of 10y3 , when the selection against the phenotype is strong. For example, for a congenital abnormality to have an incidence of the order of 10y3 , assuming a mutation rate of about 10y6 and selection coefficients of s s 0.5 and 0.2, the number of loci predicted by the inequality are: 250 and 100, respectively, but when s s 0.05, this number is only 25. Further numerical illustrations of applications of this inequality are shown in Table 3. It is clear that the FLTM can explain threshold diseases with appreciable incidences only when either the mutation rate is very high or the selection coefficient is much smaller than the ones used in Table 3, or some

Table 3 Minimum number of loci needed to explain a specified prevalence Ž P . for various mutation rates Ž u rlocusrgeneration. and selection coefficients Ž s . for affected individuals P

us10y5 rlocus ss 0.05

0.001 0.001 0.01 0.1 a

a

1 2 25 250

us10y6 rlocus ss 0.20

ss 0.50

ss 0.05

ss 0.20

ss 0.50

1 10 500 5000

2 25 250 2500

2 25 250 2500

10 100 1000 10 000

25 250 2500 25 000

For sPr2 u-1, a value of one is used.

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combinations of the two. There is no a priori reason to assume that the mutation rates of genes underlying threshold traits are higher than those of loci for simple Mendelian traits. It is likely that selection coefficients for most threshold traits are smaller than those shown in Table 3.

7. Computer simulations: general procedure and results on the relationship between heritability and MC All results discussed in this section are for free recombination; five loci were used in the simulations, at each of which the rate of mutation is equal, and the change of mutation rate considered is from 10y6 to 1.15 = 10y6 . Each simulation started with equal gametic frequencies in the initial generation. Since heritability is a composite outcome of input parameters used, various values of s, Ve and T were used in order to examine the relationship between MCs and heritability. Note that heritability, in this context, refers to the broad-sense heritability of liability. 7.1. Effects of a permanent increase of mutation rate on MC at equilibrium when there is no gene interaction and no sporadics When the gene effects are additive Ži.e., there is no epistatic interaction of the number of mutant alleles at different loci., the FLTM results can be numerically evaluated with the genotypic value, f Ž g ., defined as the number of mutant alleles in the n-locus genotype of an individual. This case corresponds to the parameter

Fig. 1. Relationship between MCs and heritability of liability Ž h2x . at equilibrium under conditions of a permanent increase in mutation rate from 10y6 rlocusrgeneration to 1.15=10y6 rlocusrgeneration for a five-locus model with no gene interactions Ži.e., syns 0.. Ža. MC U is the mutation component due to mutant genes and Žb. MC T U is the mutation component due to mutant genes under the FLTM.

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syn s 0 Žsee Table 1.. First, we discuss the results for this case under the absence of any sporadic occurrence of the disease. Fig. 1 depicts the relationship between the heritability of liability Ž h 2x . and the equilibrium mutation components, MC U Žin panel a. and MC T U Žin panel b., for a five-locus model, following a permanent 15% increase in the mutation rate Žfrom 10y6 rlocusrgeneration to 1.15 = 10y6rlocusrgeneration. and selection coefficients of s s 0.2 to 0.8. Recall that MC U is a measure of the impact of the change in mutation rate on the mutation load and MC T U is the corresponding impact on the disease incidence ŽTable 2.. Although MC U does not, by itself, depict the impact of mutational increase on disease incidence, the numerical results for MC U are included to examine the effect of elimination of mutants due to selection; this is because the mutation load is approximately twice the gametic mutation rate, adjusted for the number of mutations among individuals who survive the effect of selection ŽEq. Ž9... The results shown in Fig. 1 indicate that for heritability higher than about 10%, MC T U is above 0.8, and for h2x greater than 40%, MC T U is essentially 1.0 at equilibrium. MC T U is substantially smaller Žsay, 0.6 to 0.7. when h 2x is very small Žsay, less than 10%.. For example, when h2x is about 10%, a 15% permanent increase in mutation rate will result only in a 12% Žs 0.8 = 15%. increase in disease incidence at equilibrium. For h2x G 40%, however, the increase in disease incidence will be equivalent to the relative increase in mutation rate Ži.e., MC T U s 1.. An increase in mutation rate will result in new mutations in individuals, but since s ) 0, not all of them will contribute to an increase in disease frequency. Thus, at the new equilibrium, the average number of mutant genes in individuals who survive the effects of selection, will increase; the magnitude of this increase can be predicted from MC U . The MC U values shown in Fig. 1 Žpanel a. are essentially one except for h2x - 10%, when it drops to about 0.8. In other words, unless heritability is quite low Žsay, - 10%., almost all the induced mutations survive the effect of selection Ži.e., MC U s 1., leading to the conclusion that a 15% increase in mutation yields a 15% increase in disease frequency at equilibrium. Although the estimates discussed are for the five-locus model Ž n s 5., these conclusions remain qualitatively unaltered for other values of n. For example, for n s 3, 4 or 6 Ždata not shown., the relationships between h 2x and the two MCs are essentially similar to those shown in the two panels of Fig. 1. 7.2. Effect of a permanent increase of mutation rate on MC in the first few generations when there is no gene interaction and no sporadics Fig. 2 shows the values of MC U Žtop row panels. and MC T U Žbottom row panels. in the first few generations Žat generations 1, 5 and 10. after a permanent increase of 15% in mutation rate. These results show that in early generations, both MC U and MC U are always very small Žnote the scale of the Y-axis in this figure in contrast to that in Fig. 1., except when the heritability, h 2x , is extremely small Žsay, - 5%.. As mentioned earlier, the broad-sense heritability of liability, h2x , is not a direct input of the algorithm used. In contrast, the h2x values at any generation Žand even at mutation–selection equilibrium. are determined by the combinations of parameter values n, m, s, Ve and T, so that the spread of h 2x values shown in the different panels of Fig. 2 are obtained by varying these parameter values. We shall return to the dependence of h2x on these parameters and the consequent effect on the mutation component ŽMC T U . later. However, one way to have a small h 2x is to have the threshold, T, shifted far to the left of the distribution. This is so, because when T is shifted to the left of the liability distribution, it does not take many deleterious genes to cause the disease. When T is small, although generally the heritability will be small, depending upon the other parameters, the response to mutation is quite unstable, since the approach to the new equilibrium may be quite fast. For example, in the most extreme case when T s 1 Žall genotypes are lethal except the one completely free of any deleterious allele., the equilibrium MC U s 1 is reached in one generation. Therefore, the trends of dependence of MC U and MC T U on h 2x shown in all panels of Fig. 2 are in fact composite pictures of different individuals trends, each of which shows that both MCs decrease with increasing values of heritability of liability.

(

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Fig. 2. Relationship between MCs and heritability of liability Ž h2x . in early generations Ž1, 5 and 10. under conditions of a permanent increase in mutation rate from 10y6 rlocusrgeneration to 1.15 = 10y6 rlocusrgeneration for a five-locus model with no gene interactions Ži.e., syn s 0.. The top row represents the MC U -values and the bottom row represents the MC T U -values.

7.3. Dependence of heritability of liability, h 2x , on parameter Õalues in the first few generations after a permanent increase in mutation rate The results of how the different parameter values affect h 2x and consequently, how MC T U is affected, are presented in Fig. 3. For a clearer display of the trends of dependence of h2x on the individual parameters Žwith others held constant., in these numerical computations, we deliberately introduced a doubling of mutation rate Žfrom 10y6 to 2 = 10y6 . and labelled the individual data points for their respective values of s Žthe selection

Fig. 3. Relationship between MC T U and heritability of liability Ž h 2x . in generations 1, 2, 5 and 10 under conditions of a permanent increase in mutation rate from 10y6 rlocusrgeneration to 2 = 10y6 rlocusrgeneration for a five-locus model with no gene interactions Ži.e., syn s 0., plotted for different values of s, T and se.

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(

coefficient., T Žthe threshold. and se Žs Ve , the environmental standard deviation.. The results plotted are taken from outputs of the computer program at different generations after the permanent doubling of mutation rate, and these are identified by different symbols. These computations reveal that as the environmental variance, Ve , increases, h2x decreases, as expected. With other parameters held constant, generally with increasing values of Ve , the MC T U decreases. However, as T decreases, while h 2x decreases, MC T U is increased with the other parameters held constant. Finally, a decrease of s tends to increase h 2x resulting in a corresponding decrease in MC T U . These calculations, thus, show that heritability estimates are very sensitive to changes of individual parameter values. However, for the purpose of studying the impact of an increase in mutation rate on disease incidence, this ‘fine-tuned’ behaviour of h 2x does not affect the general result, that is, that MC is a decreasing function of heritability, and unless the latter is very small, in the immediate generations following a permanent increase in mutation rate, MC is quite small. 7.4. Effect of gene interaction (epistasis) on MC in the absence of sporadics With epistatic gene interaction Ži.e., syn / 0., the above results remain basically unaltered. To arrive at this conclusion, we conducted simulations by altering the genotypic value, the f Ž g . component of liability x Žsee Table 1., by assuming syn s 1. This represents the case of ‘synergistic’ gene action, in which the genotypic value, f Ž g ., becomes g Ž g q 1.r2, with g denoting the number of mutant alleles in the n-locus genotype of an individual ŽTable 1.. As before, we considered no sporadics and a 15% increase in mutation rate Žfrom 10y6 to

Fig. 4. Relationship between MC and heritability of liability Ž h 2x . at equilibrium under conditions of a permanent increase in mutation rate from 10y6 rlocusrgeneration to 1.15=10y6 rlocusrgeneration for a five-locus model with gene interactions Ži.e., syns1.. Ža. MC U is the mutation component due to mutant genes and Žb. MC T U is the mutation component due to mutant genes under the FLTM.

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Fig. 5. Relationship between MCs and heritability of liability Ž h2x . in early generations Ž1, 5 and 10. under conditions of a permanent increase in mutation rate from 10y6 rlocusrgeneration to 1.15 = 10y6rlocusrgeneration for a five-locus model with gene interactions Ži.e., syn s 1.. The top row represents the MC U -values and the bottom row represents the MC T U -values.

Fig. 6. Comparison of the relationship between MC T U and heritability of liability Ž h 2x . at equilibrium Žclosed circles. and at generation 10 Žopen circles. under conditions of a permanent increase in mutation rate from 10y6 rlocusrgeneration to 1.15 = 10y6 rlocusrgeneration for a five-locus model with no gene interactions Ži.e., syn s 0.. The shaded areas of heritability Žrange: 0.3–0.8. are those of interest for risk estimation.

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1.15 = 10y6 .. As shown in Fig. 4 ŽMC U in panel Ža. and MC T U in panel Žb.., the equilibrium MC values are again close to one. Likewise, in the first few generations following a permanent increase in mutation rate, both MC values are below 2% even for relatively small values of h2x . These are shown in Fig. 5 for generations 1, 5 and 10, following the permanent increase in mutation rate. The results shown in Figs. 1–5 are essentially unaltered for three-, four- and six-locus models as well Ždata not shown.. Finally, Fig. 6 provides a comparison of the MC T U values at equilibrium and in generation 10, following a permanent increase in mutation rate. The shaded areas in this figure are the ones of interest for risk estimation and represent the range of heritability between 30% and 80%. These calculations re-emphasize the main qualitative result, that is, for a 15% increase in mutation rate, the equilibrium MC T U is close to one, whereas even in generation 10, the corresponding values of MC T U are quite small. 7.5. Effect of a one-time increase in mutation rate The simulation was also used to examine the consequences of a one-time increase in mutation rate Ži.e., the mutation rate was increased for one generation and then brought back to the original value for all subsequent generations.. As expected, the first generation MC values are identical to the ones shown in Figs. 2 and 5 Žfor syn s 0 and syn s 1, respectively.. This is followed by a gradual decline of MC back to zero even in absence of sporadics. Given the relatively small values of both MC U and MC T U in the first generation, details of these computations do not provide any new biological insight, hence, they are not shown graphically. 8. Limitations of the model and other considerations The numerical results discussed in the Section 7 were obtained for an arbitrary 15% increase in mutation rate Žexcept that for Fig. 3. at each of the five underlying loci, using syn s 0 and 1 to study the effect of gene interaction. The other parameter values Ž s, T and Ve . were varied to obtain different heritabilities, so that the predictions on MC could be made based on h 2x values alone. Further, our selection model corresponds to a truncation selection in which selection is against individuals having liability exceeding the threshold, T, and the increase in mutation rate was imposed on a population at mutation–selection equilibrium. It is instructive to inquire how the results might differ when mutational increase occurs Ži. in a population which has not reached a mutation–selection equilibrium, Žii. under balancing selection as opposed to truncation selection, Žiii. when there are interactions between genes that are different from a complete synergistic effect Ži.e., when syn / 0 does not necessarily imply that syn s 1. and Živ. when sporadics are present. 8.1. Effect of mutational increase in non-equilibrium populations In our model, we assumed that at the time the increase in mutation occurs, the population is at mutation–selection equilibrium. However, with the relaxation of selection pressure due to advances in medicine and public health during the past two or three generations, it is almost certain that no present-day population is at mutation–selection equilibrium. The issue of interest for radiation risk, therefore, is how the prediction on MC will differ, if mutational increase occurs in a population ‘on its way’ to a new equilibrium after a relaxation of selection. For a simple autosomal dominant trait with no sporadics, the underestimation of MC may be substantial. This is so, because with relaxed selection, a population on its way to a new equilibrium will experience occurrences and survival of further affected individuals which will happen even in the absence of radiation-induced increase of mutation rates. However, for threshold traits, MC is probably not seriously underestimated. This is especially true for chronic diseases in which the selection against affected individuals Žmeasured as pre-reproductive death or reduced fitness. before and after the relaxation of selection has been and remains small.

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Further, when the genotypic value of liability is mainly dictated by the number of mutant alleles in the multi-locus genotype of the individual, the effect of relaxed selection accumulates predominantly in the first few generations w5x. Hence, unless radiation-induced increase in mutation rate occurs exactly at the same generation when a relaxed selection takes place, the underestimation of MC due to our mutation–selection equilibrium assumption is not expected to be large. 8.2. Impact of mutational changes under truncation Õs. balancing selection So far, we have assumed that mutations are unidirectional and that selection operates against individuals above a threshold. However, it is possible that selection operates at both ends of the liability distribution. For simple genetic traits under balancing selection Ži.e., heterozygotes that are more fit than either homozygote., the mutant gene frequency Žand consequently, the disease frequency. will be determined primarily by selection, with mutation having only a secondary influence. For multifactorial traits under centripetal selection Ži.e., intermediate optimum., Bulmer w16x showed that the trait frequency will be determined by the genetic component of liability; the upper bound for this is determined by the product of the intensity of selection and mutation rate. As a consequence, the general result that MC s 1 at equilibrium still holds, and the impact of mutational increases in early generations will still be dictated by the strength of selection. The assumption that selection works at only ‘one tail’ of the liability distribution amounts to overemphasizing MC. Furthermore, when both tails of the liability distribution result in impaired health, the phenotypes at these tails may be different. Therefore, for computing phenotypic frequencies in a population that responds to mutational increase, still, only one tail of the liability distribution is relevant. 8.3. Impact of mutational increase when gene interactions are present The effect of gene interaction at the liability level for multifactorial diseases is varied and does not readily lend itself to modeling. However, in the results presented in this paper, some effects of gene interaction were illustrated by introducing the parameter syn. With syn s 1, the genotypic value is represented by g Ž g q 1.r2, instead of g. Thus, in a sense, the numerical calculations shown in Figs. 4 and 5 are upper bounds for the effects of gene interactions. The results show that even under this scenario, MC T U values in early generations are trivial for a wide range of heritability values Ž h 2x ) 0.10.. 8.4. Effect of the presence of sporadics As mentioned earlier, throughout all of the numerical computations discussed above, we considered the absence of sporadic occurrences of the disease. The effect of sporadics is simply to decrease our estimate of MC by a factor of Ž1 y PaT . Žsee Eq. Ž5... Likewise, correlations between relatives are also reduced in the presence of sporadics. The extent to which correlation between relatives under the FLTM is reduced in the presence of sporadics is derived in Appendix C. 8.5. Other considerations In the formulation of the FLTM, we assumed only two segregating alleles per locus. This oversimplification is probably not unrealistic because of the fact that our model is effectively equivalent to two groups of alleles, first of which consists of a number of normal isoalleles and the other represents the group of mutant alleles, the average of which is considered explicitly in the model. The latter assumption is justified by the fact that most recessive alleles show intermediate dominance with each other, while pooling the dominant mutations in a single class Žin relation to the normal isoalleles. is even more legitimate.

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Further, in the model that we used, a specified increase in mutation rate at all the genes underlying the disease has been assumed. Radiobiological wisdom dictates that such a simultaneous increase in mutation rate for several specific genes is unlikely to occur under low-dose radiation conditions. This means that the estimates of MC, as presented in this paper, are probably overestimates of the responsiveness of multifactorial diseases to radiation-induced mutations. We have not attempted to fit the FLTM to empirical data on multifactorial diseases for the following reasons: such model fitting involves: Ži. the estimation Žs reconstruction. of mutation rates and selection coefficients that should have operated in the past to result in the present assumed equilibrium prevalences for the various multifactorial diseases and risk to relatives and Žii. use of these estimates as a starting point to examine the consequences on an increase in mutation rates with different selection coefficients. Based on the results obtained in this study, it was felt that such efforts are unlikely to add more precision to the conclusions reached and therefore, were not undertaken.

9. Use of MC for risk estimation for multifactorial diseases As noted in Eq. Ž1., risk is estimated as a product of three quantities, namely: P Ždisease incidence., 1rDD Žrelative mutation risk. and MC. In this paper, we provided the necessary formulations and algorithm for estimating MC in any generation of interest following radiation. At present, there are no compelling reasons to alter the P estimate of 65% for multifactorial diseases w7x. However, the applicability of the currently used doubling dose of 1 Gy Žfor low LET, low-dose, radiation conditions. has been questioned w17x and is under investigation ŽK. Sankaranarayanan, unpublished results.. In view of this, no actual risk estimates are presented in this paper. This issue will be considered in a future paper of this series.

10. General conclusions The Finite-locus Threshold Model and its numerical evaluations presented in this paper permit some general predictions regarding the effect of induced mutations on the incidence of multifactorial diseases. First, in general, MC is a function of mutation rate, selection, threshold value, environmental variance and recombination. The FLTM developed here does not explicitly yield any direct functional form of this dependence, but the value of MC at equilibrium, as well as in the first few generations following any increase in mutation rate can be roughly predicted from knowledge of the heritability Žof liability. alone. Second, in the absence of sporadics, under conditions of a permanent increase in mutation rate, the equilibrium MC is close to one, unless the heritability of liability is small Žsay, h 2x - 10%., a situation which probably does not apply for most multifactorial diseases. Third, MC in early generations Žsay, within the first five or 10 generations. following a permanent increase in mutation rate is generally less than 2%, except when the threshold value is very far to the left of the liability distribution. In fact, the numerical results indicate that MC T U in early generations following the increase in mutation rate seems always less than 2% for values of T higher than n, the number of loci. Fourth, the results described in this paper used values of the selection coefficient, s s 0.2–0.8, that are perhaps too high for most multifactorial diseases. Decreasing s, holding all other parameters constant, appears to have the general effect of increasing heritability and consequently, decreasing MC in early generations ŽFig. 3.. The impact of weaker selection on the equilibrium MC is negligible. In other words, even when the selection is weaker, the equilibrium MC remains close to one. Fifth, Eq. Ž9., along with the results of Table 2, suggests that for s s 0.2–0.8, the Finite-locus Threshold Model requires a large number of loci to explain common multifactorial diseases, unless the mutation rates are large. For common diseases Ži.e., incidence, P, larger than the mutation rate, u., it seems reasonable to assume

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that most are maintained in the population through weak selection. High mutation rates for the underlying loci seem unreasonable, since such a situation would predict that new mutants should be abundantly found in the population Ždetected through pedigrees with a single member affected.; this does not appear to be the case. Thus, for most multifactorial diseases, s is probably much weaker, leading to even lower MC values under conditions of an increase in mutation rate. Sixth, the presence of sporadics would make MC lower both at equilibrium and in the early generations following an increase in mutation rate. As shown in Appendix C, the factor by which the numerical results are to be changed is roughly Ž1 y arPT ., the complement of the relative proportion of sporadics among the total number of affected individuals in the population. Finally, while these predictions on MC are based on a truncation selection model in which the increase in mutation rate occurs in a population under mutation–selection balance, other forms of selection Že.g., balancing selection. and the induced mutations occurring in a non-equilibrium population do not seem to increase MC drastically. In contrast, assumptions used in the numerical evaluations of the FLTM may have resulted in MC that may be regarded as overestimates of actual MC expected for most common multifactorial diseases.

Acknowledgements This work was carried out within the framework of a Task Group of the International Commission on Radiological Protection ŽICRP. and we are grateful to ICRP for the encouragement received. We thank Prof. W.J. Schull ŽHouston. and Prof. J.F. Crow ŽMadison. for their valuable advice throughout the course of this study and for their comments on an earlier version of this paper. In part, this work was supported by the Atomic Energy Control Board of Canada ŽAECB Project number 7.219.1 to R.C.. and by the European Union ŽContract number F14PCT-96-0041 to K.S...

Appendix A. Relationship between the heritabilities of liability and trait for threshold traits With affection status defined as having liability above the threshold ŽT ., the trait is a binary variable Ž f . that takes a value of one for affected individuals and zero for unaffected ones. In contrast, under the FLTM, the liability Ž x . is a composite variable Ž x s f Ž g . q e ., one component w f Ž g .x of which is discrete Žand is dependent on the number, g, of mutant genes in the genotype., while the other Ž e, the environmental effect. is a continuous variable, assumed to be normally distributed with a mean of zero and variance Ve . Consequently, these make the heritability of trait different from that of liability. Using the parameter definitions shown in Table 1, a relationship of these two heritability coefficients can be determined as follows. First, the expectation w EŽ f .x and variance w V Ž f .x of f are determined by P, the population frequency of individuals having liability above the threshold T, namely: EŽ f . s P ,

Ž A1.

and V Ž f . sP Ž1yP . .

Ž A2.

With the definition of parameters of the FLTM ŽTable 1., the quantity P can be evaluated in two steps. The probability that a genotype with g mutant alleles has a liability that exceeds the threshold T is given by `

Qg s

1

HT (2p V

e

exp y

Ž xyf Ž g . . 2Ve

2

d x,

Ž A3.

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and since Pg is the probability of such genotypes in the population ŽTable 1., the disease incidence, P and EŽ f . can be written as 2n

Ps

1

`

Pg

Ý gs0

exp y

HT (2p V

Ž xyf Ž g . . 2Ve

e

2

d x.

Ž A4.

With the broad-sense heritability of liability, h2x , defined as

h 2x s

V fŽ g. V f Ž g . q Ve

,

Ž A5.

the Eq. ŽA4. can be re-written in terms of h2x as 2n

Ps

Ý

1

`

Pg

gs0

Ht (2p 1 y h Ž . 2 x

exp y

Ž xyf Ž g . .

2

2 Ž 1 y h 2x .

d x,

Ž A6.

(

in which the threshold T is re-scaled as t s Tr V Ž f Ž g . . q Ve . To compute the broad-sense heritability, h 2B Ž f ., we now evaluate the genetic variance of the trait, which becomes

`

VG s

1

Hy` (2p h

2 x

exp y

x2 2 h2x

¶ 1 Ž yyx. ~°H exp y d y• d x y P , ¢ (2p Ž1 y h . ß 2 Ž1 y h . 2

`

t

2

2

2 x

2 x

Ž A7.

so that the broad-sense heritability of trait f is evaluated as

h2B Ž f . s

VG P Ž1yP .

.

Ž A8.

The additive genetic variance of the trait f is approximated by VA s z 2 h2x , where z is the standard normal deviate corresponding to the threshold T w18x. Thus, the narrow-sense heritability of the trait f is given by

h2N Ž f . s

z 2 h2x P Ž1yP .

.

Ž A9.

In Table A1, we present some numerical evaluations of these two heritability coefficients for a threshold trait for various values of incidence and heritability of liability Ž h 2x ., in which the latter is defined in the broad sense only.

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Table A1. Broad and narrow sense heritabilities of a threshold trait for various values of trait incidence Ž P . and heritability of liability Ž h 2x . Ps Ts zs h2x 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.001 3.090 0.003 h 2N Ž f . 0.011 0.010 0.009 0.008 0.007 0.006 0.004 0.003 0.002 0.001

0.005 2.576 0.014 h 2B Ž f . 1.000 0.440 0.263 0.159 0.094 0.053 0.029 0.014 0.006 0.002

h 2N Ž f . 0.042 0.038 0.034 0.029 0.025 0.021 0.017 0.013 0.008 0.004

0.01 2.326 0.027 h2B Ž f . 1.000 0.506 0.334 0.224 0.148 0.095 0.058 0.033 0.016 0.006

h 2N Ž f . 0.072 0.065 0.057 0.050 0.043 0.036 0.029 0.022 0.014 0.007

0.05 1.645 0.103 h2B Ž f . 1.000 0.537 0.371 0.259 0.179 0.121 0.077 0.046 0.024 0.009

h2N Ž f . 0.224 0.202 0.179 0.157 0.134 0.112 0.090 0.067 0.045 0.022

h2B Ž f . 1.000 0.618 0.469 0.360 0.274 0.204 0.146 0.098 0.058 0.026

h2N Ž f . s The narrow-sense heritability of the threshold trait. h2B Ž f . s The broad-sense heritability of the threshold trait.

Appendix B. An outline of the computer program for the numerical evaluation of the FLTM With n loci, there are 2 n possible gametes and 2 n possible recombinational outcomes. Each gamete may be represented as a binary number. For example, for three loci, the eight gametes are represented by the binary numbers zero to seven as follows. ABC 000

ABc 001

AbC 010

Abc 011

aBC 100

aBc 101

abC 110

abc 111

Likewise, the eight recombinational events are also represented by the binary numbers zero to seven, in which M and P refer to alleles of maternal and paternal origin, respectively. MMM 000

MMP 001

MPM 010

MPP 011

PMM 100

PMP 101

PPM 110

PPP 111

Each such recombinational event, r , has a probability P Ž r ., that is a function of the recombination frequencies and measure of interference. Let yi be the frequency of the ith Žby the binary designation. multi-locus gamete Gi . Then, the change of gamete frequencies in two successive generations due to selection can be expressed by the recurrence equation yXi s Ý yj y k wjk P Ž r . rW ,

Ž A10.

for i s 0, 1, . . . 2 n y 1, where wjk is the fitness of the multi-locus genotype Gj G k , W is the average population fitness and the summation is taken over all combinations of j, k and r , such that the gamete Gi results from all genotypes of the form Gj G k through recombinations at one or both of the gametes Gj and G k . Expressed mathematically, the summation is over all combinations of j, k and r , such that i s Ž j F r .DŽ k F r c ., in which r c denotes the complement of r .

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Further changes in gamete frequencies occurring in successive generations through mutation can be represented by yYi s Ý yXjQ Ž i , j . , Ž A11. k . Ž . dŽ3, k . for i, j s 0, 1, . . . ,2 n y 1, where Q Ž i, j . s P Ž1 y m k . dŽ1, k . mdŽ2, 0 , for i s 1,2, . . . ,n, and k dŽ1,k . s 1 if Gi and Gj both carry normal alleles at locus k; 0, if otherwise dŽ2,k . s 1 if Gi carries a mutant allele and Gj carries a normal allele at locus k; 0, if otherwise dŽ3,k . s 1 if Gi carries a normal allele and Gj carries a mutant allele at locus k m k s the mutation rate at the k th locus. To calculate the mutation component following an increase in the mutation rates, the population is run to equilibrium Žresulting in mutation–selection balance. for a particular set of parameters specified by n s the number of loci; s s the selection coefficient; T s the threshold point; se s Ve s the environmental standard deviation; m k s the mutation rate at locus k; syn s the intensity of epistasis; ™ r s the array of generalized recombination frequencies and ™ y s the starting array of gametic frequencies. Once the population reaches equilibrium, the mutation rates are increased and the iterative run is resumed with the new mutation rates. The output of the first 10 generations after the increase in mutation is printed out and the run is terminated when the population reaches a new equilibrium due to mutation–selection balance. The output variables for which records are kept are: ™ y s the array of gamete frequencies; ™ q s the array of gene 2 2 frequencies; h x s the heritability of liability; hf s the Žbroad-sense. heritability of the trait; L s the load; P s the frequency of the trait; MC U s the mutation component due to mutation alone; MC T U s the mutation component of the threshold model and K s King’s formula for the load Žfrom Eq. Ž10... The program codes are so designed that the user may specify the generations for which the output variables are saved. Also, one can specify in the program whether a permanent increase in mutation or a ‘burst’ is desired. The program can be run either manually, i.e., watching the screen and stopping a run when the gamete frequencies and other output results remain virtually stable Žsay, up to seven or eight decimal places. for several generations, or automatically, in which several criteria for an equilibrium have been met; for example, the first two gamete frequencies, y 0 and y 1 , both change less than 1 = 10y1 1 over two successive generations. Finally, the fitness parameters in the threshold model used in the program are according to a multi-locus mutation–selection model without any sporadics, having the fitness matrix structure 1 y sQ i for any genotype that carries exactly a total of i deleterious alleles in its n-locus genotype, irrespective of the number of deleterious alleles in individual gametes. For example, for a three-locus model, the genotypes AbcrABC and aBCrAbC will have the same fitness of 1 y sQ2 , with Q g representing the probability that a genotype with g mutant alleles has a liability exceeding the threshold T Žas defined in Table 1.. Note that under such a mutation–selection model, no general analytically closed form of the equilibrium is available.

(

Appendix C. Sporadics and the correlation between relatives under the FLTM For traits with a dichotomous classification Žaffected and unaffected., heritability estimates are generally obtained from correlation between relatives. Thus, it is relevant to investigate the expected correlation between relatives under the FLTM and examine how it is influenced by the presence of sporadic occurrences of diseases. Let a s the proportion of sporadics Žaffected for reasons unrelated to the genotype., P s the proportion of the general population with x G T, PT s the proportion of affected individuals in the general populations a q Ž1 y a. P, R s the coefficient of relationship between relatives of degree R, PR s the proportion of R relatives of individuals with x G T who are also above T and q s the proportion of affected individuals among R relatives of affected individuals. Then, the trait Ž f . correlation between R relatives is rs

PT q y PT2 PT Ž 1 y PT .

s

q y PT 1 y PT

.

Ž A12.

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Now, if the affected proband is a sporadic, with probability arPT , then, the probability the R relative is also affected is just PT ; if the proband is affected because he is above the threshold, with probability Ž1 y a. PrPT , then, the probability the relative is also affected is a q Ž1 y a. PR . So, the term PT q becomes aPT q Ž1 y a. P w a q Ž1 y a. PR x. Thus, the correlation between relatives can be expressed as rs

aPT q Ž 1 y a . P a q Ž 1 y a . PR y PT2 PT Ž 1 y PT .

2

Ž 1 y a . Ž PPR y P 2 . s . PT Ž 1 y PT .

Ž A13.

When a s 0, with PT being equal to P, the correlation between relatives reduces to Ž PR y P .rŽ1 y P ., as it should. Now, PR y P s Rh2Q Ž1 y P ., where h2Q is the heritability of the trait in the absence of sporadics; this is true only for uni-lineal relatives. The simplest case is that of a completely recessive which can be effectively thought of as a threshold trait Žone needs two ‘mutant genes’ to be affected.. In that case, letting p s the recessive gene frequency, PR s 2 Rp q Ž1 y 2 R .p 2 and P s p 2 . The heritability is 2prŽ1 q p .. So, we have 2 Rp q Ž1 y 2 R .p 2 y p 2 s 2 Rp Ž1 y p 2 .rŽ1 q p ., which agrees with the well-known formula 2 Rp Ž1 y p .. Also, r s 2 . PRh Q Ž 1 y P . Rhf2 , where hf2 is the heritability of the f trait Žwith sporadics.. So, r s Ž1 y aP and since 1 y P Ž T. T a a 2 2 2 2 P s Ž PT y a.rŽ1 y a., r s Ž1 y PT . Rh Q s Rhf . Hence, hf s Ž1 y PT . h Q . The computer simulations used in this work do not take sporadics into account. However, in view of the above, sporadics simply reduce MC by a factor of arPT throughout. 2

References w1x NAS, The effects on populations of exposure to low levels of ionizing radiation, National Academy of Sciences, National Research Council, Washington, DC, 1972. w2x J.F. Crow, C. Denniston, The mutation component of genetic disease, Science 212 Ž1981. 888–893. w3x J.F. Crow, C. Denniston, Mutation in human populations, Adv. Hum. Genet. 14 Ž1985. 59–123. w4x C. Denniston, Are human studies possible? Some thoughts on the mutation component and population monitoring, Environ. Health Perspect. 52 Ž1983. 41–44. w5x R. Chakraborty, N. Yasuda, C. Denniston, K. Sankaranarayanan, Ionizing radiation and genetic risks. VII: The concept of mutation component and its use in risk estimation for Mendelian diseases, Mutat. Res., 1998, in press. w6x A. Czeizel, K. Sankaranarayanan, The load of genetic and partially genetic disorders in man. I: Congenital anomalies: estimates of detriment in terms of years of life lost and years of impaired life, Mutat. Res. 128 Ž1984. 73–103. w7x A. Czeizel, K. Sankaranarayanan, A. Losonci, T. Rudas, M. Keresztes, The load of genetic and partially genetic disorders in man. II: Some selected common multifactorial diseases: estimates of population prevalence and of detriment in terms of years of life lost and years of impaired life, Mutat. Res. 196 Ž1988. 259–292. w8x K. Sankaranarayanan, N. Yasuda, R. Chakraborty, G. Tusnady, A.E. Czeizel, Ionizing radiation and genetic risks. V: Multifactorial diseases: a review of epidemiological and genetical aspects of congenital abnormalities in man and models on maintenance of quantitative traits in populations, Mutat. Res. 317 Ž1994. 1–23. w9x K. Sankaranarayanan, R. Chakraborty, E. Boerwinkle, Ionizing radiation and genetic risks. VI: Chronic multifactorial diseases: a review of epidemiological and genetical aspects of coronary heart disease, essential hypertension and diabetes mellitus, Mutat. Res., 1998, submitted. w10x D.S. Falconer, The inheritance of liability to certain diseases, estimated from the incidence among relatives, Ann. Hum. Genet. 29 Ž1965. 51–76. w11x J.F. Crow, M. Kimura, Efficiency of truncation selection, Proc. Natl. Acad. Sci. U.S.A. 76 Ž1979. 396–399. w12x H.J. Muller, Our load of mutations, Am. J. Hum. Genet. 2 Ž1950. 111–176. w13x W.J. Ewens, Mathematical Population Genetics, Springer-Verlag, Berlin, 1979. w14x J.L. King, The gene interaction component of the genetic load, Genetics 53 Ž1966. 403–413. w15x A.S. Kondrashov, J.F. Crow, King’s formula for the mutation load with epistasis, Genetics 120 Ž1988. 853–856. w16x M.G. Bulmer, Maintenance of genetic variability by mutation–selection balance: a child’s guide through the jungle, Genome 31 Ž1989. 761–767. w17x K. Sankaranarayanan, Ionizing radiation, genetic risk estimation and molecular biology: impact and inferences, Trends Genet. 9 Ž1993. 79–84. w18x E.R. Dempster, I.M. Lerner, Heritability of threshold characters, Genetics 35 Ž1950. 212–236.