Hum Genet (1982) 60:271-273

© Springer-Verlag 1982

Segregation Analysis of o~-L-Fueosidase Activity

L. Iselius l, J. R. Playfer 2, and D. A. Price Evans 2

1 Population Genetics Laboratory, University of Hawaii, Honolulu, Hawaii, USA, and Department of Clinical Genetics, Karolinska Hospital, Box 60500, S-10401 Stockholm, Sweden

2Department of Medicine, University of Liverpool, Liverpool L69 2BX, Great Britain

Summary. Complex segregation analysis of plasma c~-L-fucosid- ase in 45 British families provides evidence for an additive major gene causing low activities of fucosidase. There was no significant evidence of polygenic heritability or common family environment.

Introduction

The activity of the enzyme c~-L-fucosidase in plasma has been shown to be under genetic control. Low activity, which is present in about 10% of the population, has been thought to be caused by an autosomal recessive gene (Ng et al. 1976; Playfer and Price Evans 1976; Gatti et al. 1979). The low activity is present in plasma but not in leucocytes, suggesting at least two different loci coding for fucosidase. The frequency distribution of plasma fucosidase activity in the populat ion shows bimodality (Fig. 1) with one group consisting of individuals with low activity, thought to be due to homozygosity for a recessive gene, and a second group with high activity consisting of heterozygotes and normal homozygotes. Subdivision of the second group has proved to be difficult, although attempts have been made to define a threshold between heterozygotes and normal homo- zygotes (Gatti et al. 1979). This threshold was partly defined by assuming additivity at the major locus and calculating the expected proportions using the Hardy-Weinberg equilibrium. Using the threshold, an analysis of six families was performed to elucidate the mode of inheritance of plasma fucosidase defi- c i e n c y - w i t h its element of circularity (Gatti et al. 1979). In addition, no attempt was made to allow for polygenes or family environment, which also might be of importance for the variation of fucosidase activity.

The only method available incorporating incomplete dominance, polygenic inheritance, and common family environ- ment is the mixed model for complex segregation analysis (Morton and MacLean 1974) which allows for quantitative variables, without resorting to a more or less arbitrary threshold. Dichotomizing or trichotomizing quantitative data can only decrease the power to distinguish between different hypotheses 0VIacLean et al. 1975). The aim of the present study was to perform complex segregation analysis of the 45 British families described by Playfer and Price Evans (1976) to look for a major locus in the presence of polygenes and common family environ- ment.

Offprint requests to." L. Iselius, Stockholm (address above)

Material and Methods

The 45 British white two-generation families are described elsewhere (Playfer and Price Evans 1976) and are given in Table 2. The plasma c~-L-fucosidase activity was measured using an automated enzyme assay with p-nitrophenol c~-L- fucosidase as substrate. The actual activity measurements were used for segregation analysis. No zero values were observed. The fucosidase activities exhibited positive skewness. Since skewness can simulate a major locus (MacLean et al. 1975), the data (x)

were standardized (to mean zero and variance one) and skewness was removed using the power transform y = rp -~ [(xr -~ + 1) p- 1] (MacLean et al. 1976), where r is a scale parameter introduced to ensure that every xr ~ + 1 is positive in the sample, r was taken to be equal to 6. Maximum likelihood estimation yielded a value of P = 0.355 with no residual skewness.

The mixed model for segregation analysis (Morton and MacLean 1974) allows for a major locus with two alleles, a continuous variable representing polygenic and/or cultural inheritance, common family environment or other sources of sibling resemblance, and continuous random error. The pheno- type, for a quantitative trait, has mean U and variance V in the general population. The quantitative trait x can be parti t ioned as x = g + c + e , where g is the effect due to a major locus, c is the breeding value to a large number of additive genetic factors, and e is the environmental contribution. The three sources are acting independently. The environmental component is partit ioned

32

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E

~ members

~ d donors

20 60 100 140 180 220 260 300 340 380 420

o~-L-Fucosidase activity (tumol PNP/I) Fig. 1. Frequency histograms for ~-L-fucosidase activities

0340-6717/82/0060/0271/$ 01.00

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Table 1. Tests of hypotheses and estimates for the mixed model (non-iterated parameters within parenthesis)

Hypothesis - 2/n + C H B q t/l/Y- d

q = d= t = H= 0 323.86 (0) 0.397 (0) (0) (0) No major locus, no polygenic heritability

q= d= t = 0 288.12 0.813 0.000 (0) (0) (0) No major locus

H= 0 280.58 (0) (0) 0.691 2.80 0.531 Generalized single locus

General 278.64 0.042 (0) 0.622 2.68 0.545

d=H= 0 306.76 (0) (0) 0.611 1.49 (0) Recessive major locus, no polygenic heritability

d= 0.5, H= 0 281.21 (0) (0) 0.687 2.72 (0.5) Additive major locus, no polygenic heritability

d= 1, H= 0 310.95 (0) (0) 0.150 1.47 (1) Dominant major locus, no polygenic heritability

Table 2. c~-L~Fucosidase activities in 45 British families. The values for female offspring are underlined

Family Father Mother Children number

1 100 210 124 136 260 130 120

2 288 180 290 196 180

3 280 100 244 110 160

4 120 115 124 120 115 140

5 100 260 120 250 100 140

6 118 168 196 13 28

7 140 276 216 226 174

8 158 310 120 226 245 154 110

9 340 104 125 145

10 200 196 256 234 264

11 264 102 165 118 198 79 166

12 128 82 102 126 200

13 162 130 212 140 104 196

14 225 325 316 248 240

15 10l 246 122 259

16 82 165 104 104

17 180 260 265 264

18 280 340 308 240 210 160

19 184 114 170 190

20 350 146 150 185 226 106

21 140 140 144 6 160

22 102 84 76 200

23 244 178 240 144

24 80 194 94 120

25 174 196 324 216 26 248 84 156 264 196

27 200 150 92 110 166

28 165 226 165 218

29 302 152 258 262

30 104 200 104 116 144

31 110 140 160 130 12 130

32 270 170 130 130 33 162 80 94 644

34 360 180 320 202 35 180 108 250 84 36 110 218 110 180 210

210 192

Table 2 (continued)

Family Father Mother Children number

37 160 220 120 100

38 140 206 110 140

39 248 6 112 108

40 80 16 144 28

41 374 50 98 196

42 132 10 16 18

43 10 276 20 88

44 16 340 196

45 14 280 162 78

90 200 120 105

into two components ; environment common within a sibship

and random environment. A locus is major if it causes a displacement of more than one phenotypic s tandard deviation

between "normal" and "abnormal" genotypes. At the major locus, q is the gene frequency of the allele with a positive effect

and t is the displacement between the two homozygous geno- type means [measured in s tandard deviation units (V V)]. The degree of dofninance (d) is the position of the heterozygous mean relative to the means of the two homozygous classes, d = 0 corresponds to a recessive gene, d = 0.5 to an additive gene, and d= 1 to a dominant gene. The polygenic heritability is called H,

and B is the relative variance due to common environment within sibships.The different parameters of the model are estimated by maximum likelihood using Newton-Raphson iteration. Likeli- hoods of the set of children's phenotypes are taken as conditional

on parental phenotypes. The general model has seven parameters. Different subhypo-

theses can be tested by fixing certain parameters, in terms of the likelihood ratio criterion. For each hypothesis -2 ln L + c was

calculated, where In L is the log-likelihood of the sample and c a constant. If - 21n L~ + c is the value when m + k parameters are estimated and -21n L2 + c when only m of the m + k parameters are estimated, then (-21n L2 + c) - (-21n LI n u C ) = 21n (L1/L2) follows a X 2 distribution with k d f testing a null hypothesis on these k parameters. Since ascertainment is through the parent

generation selection is complete.

Results

Frequency his tograms for e-L-fucosidase activities for the 218 family members are shown in Fig. 1. There was no significant evidence of a sex difference of the activities. Likel ihoods and estimates of the pa ramete r s under different subhypotheses of the mixed model are given in Table 1. There was no evidence for envi ronment c o m m o n to sibs(B) and it was therefore excluded in the remain ing par t of the analysis. There was significant evidence for a ma jo r gene 0(3 = 288.12 - 278.64 = 9.48; 0.01 < P < 0.025). Polygenic her i tabi l i ty was not significant (y~ = 280 .58 - 278.64 = 1.94). An addit ive ma jo r gene wi thout polygenic her i tabi l i ty is consistent with the data. The estimate of the gene frequency for the gene causing high activities, q = 0.691, is similar to the f inding of Playfer and Price Evans (1976) for the British popu la t ion (q = 0.706).

Discussion

The classical evidence tha t the low activity phenotype is an au tosomal recessive character , rests upon classification of individuals into two or three genotypes using the appa ren t ant i - modes as thresholds. The thresholds are no t easy to define due to considerable overlap. In addi t ion, assuming a par t icular mode of inheri tance before analysis clearly makes any r igorous a t t empt to elucidate the mode of inher i tance impossible.

The mixed model of segregation analysis does not require that the data be d ichotomized, nor tha t classification errors be

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negligible. Using this me thod of analysis we have shown tha t a

simple addit ive major gene, wi thout polygenes and c o m m o n family env i ronment , can explain the inher i tance of p lasma a-L-

fucosidase activity. The addi t ional var ia t ion is p robab ly due to r andom envi ronment , e.g. measurement error.

Acknowledgements. PGL paper no. 253. This work was supported by grant 3681 from the Swedish Medical Research Council and grant GM 23498 from the U.S. National Institute of Health.

References

Gatti R, Cavalieri S, Romeo G (1979) Relationship between a-L- fucosidase deficiency in plasma and e-g-fucosidase activity in leucocytes. Hum Genet 48 : 23-30

MacLean C J, Morton NE, Lew R (1975) Analysis of family resemblance. IV. Operational characteristics of segregation analysis. Am J Hum Genet 27 : 365-384

MacLean CJ, Morton NE, Elston RC, Yee S (1976) Skewness in com- mingled distributions. Biometrics 32:695-699

Morton NE, MacLean CJ (1974) Analysis of family resemblance. III. Complex segregation of quantitative traits. Am J Hum Genet 26: 489-503

Ng WG, Donnell GH, Koch R, Berggren WR (1976) Biochemical and genetic studies of plasma and leukocyte c~-L-fucosidase. Am J Hum Genet 28 : 42-50

Playfer JR, Price Evans DA (1976) Enzyme activity in fucosidosis. Lancet ii: 1415-1416

Received January 12, 1981 / Revised February 11, 1982