THEORETICAL POPULATION GENETICS OF REPEATED GENES FORMING A MULTIGENE FAMILY * TOMOKO OHTA National Institute of Genetics, Mishima, 411 Japan Manuscript received September 23, 1977 Revised copy received December 19, 1977 ABSTRACT The evolution of repeated genes forming a multigene family in a finite population is studied with special reference to the probability of gene identity, i.e., the identity probability of two gene units chosen from the gene family. This quantity is called clonality and is defined as the sum of squares of the frequencies of gene lineages in the family. The multigene family is under- going continuous unequal somatic crossing over, ordinary interchromosomal crossing over, mutation and random frequency drift. Two measures of clonality are used: clonality within one chromosome and that between two different chromosomes. The equilibrium properties of the means, the variances and the covariance of the two measures of clonality are investigated by using the diffusion equation method under the assumption of constant number of gene units in the multigene family. Some models of natural selection based on clonality are considered. The possible significance of the variance and covari- ance of clonality among the chromosomes on the adaptive differentiation of gene families such as those producing antibodies is discussed. INrecent years, much attention has been paid to the problem of evolution of repeated genes. Although how repeated genes emerged and what role they play in evolution are far from being elucidated, their universal existence in the genome of higher organisms is well demonstrated (see, Proc. Cold Spring Harbor Symp. 1974). HOOD,CAMPBELL and ELGIN (1975) have recently classified repeated genes into three classes: simple-sequence multigene families (highly repeated sequences), multiplicational multigene families (genes for ribosomal RNA, transfer RNA and histone) and informational multigene families (genes for immunoglobulins). They reviewed and discussed the possible evolutionary mechanism and significance of these classes of repeated gene families, and they particularly emphasized the possible importance of the last class for organizing enormous amounts of genetic information to cope with the complexity of higher organisms. There are two remarkable characteristics of multigene families (HOOD, CAMPBELLand ELGIN1975; TARTOF1975). They exhibit contraction-expansion of gene number in a family, as well as coincidental evolution during the course of evolution. The latter represents the process in which parallel or simultaneous * Contribution No. 1182 from the National Institute of Genetics, Mishiia, 411, Japan. Genetics 88: 845-861 April, 1978. 846 T. OHTA mutant fixation occurs in many members of the gene units in the same family of a species. The model of unequal crossing over provides one of the most satis- factory explanations for this process (SMITH1974; BLACKand GIBSON1974). Here it is assumed that unequal crossing over is continually occurring in a germ- cell line, which is responsible for the horizontal expansion or contraction of mutant gene number within a chromosome. To put it in another way, unequal crossing over increases the homogeneity among the members of gene units in a family through duplication and deletion, similar to random fixation of mutant alleles by random drift. I have investigated the population consequences of a multigene family by incorporating the above knowledge (OHTA1977,1978). In other words, I studied the evolutionary mechanism of repeated genes at the population level. Partic- ularly in view of their possible relationship with “polygenes” (WRIGHT1977, p. 374), it is desirable to investigate ihe evolutionary mechanism of repeated genes by taking into account the above recent findings. In my study, I started from a simple situation in which the number of gene units in a family is kept constant. Then the probability of gene identity was studied (OHTA1977, 1978). This is the quantity that is used to represent the level of homogeneity among members of the family. Following SMITH(1974), I shall call this probability “clonality.” It is defined as the sum of squares of the frequencies of gene lineages within the multigene family. This quantity is equivalent to homozygosity in population genetics. In the present paper, further analyses on the clonality of repeated genes will be reported. In particular, the derivation of the variance and covariance of clonality among chromosomes in a finite population will be pre- sented. The possible significance 01 such quantities on the adaptive differentia- tion of repeated genes will be discussed. BASIC THEORY Let us assume a random-mating population with effective size Ne.A multigene family is located on a particular chromosome and continuously undergoes un- equal somatic crossing over, ordinary interchromosomal crossing over, mutation and random genetic drift. Let n be the number of gene units in the family and let it be constant throughout. Let U be the mutation rate per gene unit per generation, so that there are nu mutations per family per generation. We assume here that whenever a mutation appears, it represents a new type of gene; i.e., the so-called infinite-allele model of KIMURAand CROW(1964) is used. After unequal crossing over, the duplication-deletion of gene units in a family occurs. One cycle of unequal crossing over is defined as two successive unequal crossovers of duplication and deletion of one gene unit (OHTA1976; PERELSON and BELL1977). I assume that the continuous process of unequal crossing over in unit time may be equated to an effectiue number of cycles of unequal cross- overs. This assumption seems to be satisfied under fairly general situations in which the number of duplicated or deleted gene units is not restricted to one, but is some kind of random variable ( OHTA1978). POPULATION GENETICS OF A MULTIGENE FAMILY 847 Let K be the effective number of cycles of unequal crossovers per family per generation. Further, let p be the rate of interchromosomal crossover per family per generation. It is assumed that the interchromosomal crossing over is not accompanied by the duplication or deletion of gene units. The clonality of a gene family may be defined at two different levels; the clonality of a gene family on one chromosome and that between two homologous families on different chromosomes randomly chosen from the population. Let Xz,k be the frequency of the k-th gene lineage in the i-th gene family type. Here a gene lineage means a group of genes genetically identical and a family type also means a genetically identical gene family. Then the two measures of clonality may be written as follows an ' where c, is the clonality of the i-th family type, czjis the clonality between the i-th and the j-th family types and summation is over all k. In the following, I shall present the equilibrium properties of clonality of the family under the pressure of mutation, random sampling drift and crossing over. In my previous report ( OHTA 1978), the mean value of clonality was investi- gated. In the present paper, the variance and the covariance of clonality are studied by using the diffusion-equation method. (i) Unequal crossing ouer It was shown that the process of unequal crossing over is a stochastic one anal- ogous to that of random sampling of gametes in finite populations and that one cycle of unequal crossing over is equivalent to n2/4Negenerations of random gene frequency drift, where Neis the effective population size (OHTA 1976; PERELSON and BELL 1977). Therefore, the theory of the diffusion model for the moments of gene frequencies in finite populations may be applied. OHTA and KIMURA (1971) have shown that the following equation holds for the rate of change of the moments of allelic frequencies d --{f(xi, xs, . =E[L{f(xi, . xn)}I dt xn)} xz) (2) where E denotes the expectation, x, is the frequency of the ith allele in the popu- lation, f is a continuous function of x, values and L is the differential operator of the Kolmogorov backward equation in the following form in which M, V and W designate, respectively, the mean, the variance and the covariance in the rate of change of xi values that appear as subscripts. Equation (2) is equivalent to the one used by LI and NEI (1975) for getting the variance of heterozygosity and genetic, distance. 848 T. OHTA In om case, the mean and the variance of change of xi,TCby one cycle of unequal crossing over are given in Table 1 of OHTA (1976). Let Sxi,k be the amount of change of xi,k per K cycles of unequal crossing over (per one generation). Then, the covariance of Sxi,k and Sxi,Z may be similarly obtained. The mean, the vari- ance and the covariance of Sxi,k values become, respectively, after K cycles of unequal crossing over M6xi,k = 0 VSXi$ = cY1i.k (1 -xi,k) (4) and in which CY = 2x/n2. By using formulas (2) and (4),the change of moments of xi,k in one generation is calculated. The details of the calculation are not given here; only the resulting equations will be presented later in matrix notation. (ii) Random sampling of gametes The above formula (2) can again be applied for evaluating the effect of ran- dom drift. This time, the dynamics of the frequencies of the various types of a gene family within the population must be analyzed. Let pi be the frequency of the ith type of the gene family in the population. Then the mean clonality of a chromosome (CO)and that between the two chromosomes (CI) in the population become, respectively cl?= + cipi and (5) where summation is over all i and 1. In applying equation (2) for generating the moments of clonality, the follolwing standard formulas in population genetics for the mean, the variance and the colvariance of the change of pi values in one generation are used (e.g., CROWand KIMURA1970, p.