Recombination and the Evolution of Satellite DNA

Genet. Res., Comb. (1986), 47, pp. 167-174 With 1 text-figure Printed in Great Britain 167

Recombination and the evolution of satellite DNA

WOLFGANG STEPHAN* Institute of Animal Genetics, University of Edinburgh, West Mains Road, Edinburgh EH9 3JN, U.K. (Received 11 September 1985 and in revised form 6 December 1985)

Summary In eukaryotic chromosomes, large blocks of satellite DNA are associated with regions of reduced meiotic recombination. No function of highly repeated, tandemly arranged DNA sequences has been identified so far at the cellular level, though the structural properties of satellite DNA are relatively well known. In studying the joint action of meiotic recombination, genetic drift and natural selection on the copy number of a family of highly repeated DNA (HRDNA), this paper looks at the structure-function debate for satellite DNA from the standpoint of molecular population genetics. It is shown that (i) HRDNA accumulates most probably in regions of near zero crossing over (heterochromatin), and that (ii), due to random genetic drift the effect of unequal crossover on copy numbers is stronger, the smaller the population size. As a consequence, highly repeated sequences are likely to persist longest (over evolutionary times) in small populations. The results are based on a fairly general class of models of unequal crossing over and natural selection which have been treated both analytically and by computer simulation.

1. Introduction lem of the function of these DNAs has been the sub- Eukaryotic chromosomes contain nucleotide se- ject of major controversies during the past two quences of lengths from about 10 to several hundred decades. So far no function of highly repeated, tan- base pairs that are repeated thousands to millions of demly arranged DNA sequences has been clearly times per haploid genome. HRDNA is arranged identified. It is largely the association between largely in long tandem arrays which are associated heterochromatin and satellite DNA that gave rise to with the heterochromatic regions of chromosomes. speculations about the function of HRDNA. Numer- Satellite DNA is associated with the bulk of ous attempts have been made to resolve the question constitutive heterochromatin but it does not need to of function from a more and more detailed analysis of parallel the distribution of heterochromatin (John & structure and changes in structure. But it is an assump- Miklos, 1979; Brutlag, 1980). Thus in humans four tion that important answers will be found exclusively satellites which have been identified make up about 3 % at the cellular level (Miklos, 1982). of the genome, whereas the C-banding material It was proposed in an earlier paper (Charlesworth, amounts to approximately 20%. Similar situations are Langley & Stephan, 1986) that the distribution of reported from rye, Secale cereale (Peacock et al. 1977; HRDNA along the chromosome is a consequence of Flavell, 1982), and from the Chinese hamster, an evolutionary equilibrium between genetic drift, Cricetulus griseus (see John & Miklos, 1979). Hetero- natural section and mutation pressure (amplification) chromatin and HRDNAs are distributed along the in regions of restricted recombination and is not a chromosome arms. Though heterochromatic segments property of HRDNAper se. The first step in the develop- are mostly found in centromeres and telomeres, it is ment of this hypothesis is to understand why certain now apparent that chromosomes in many organisms regions of the chromosome should have intrinsically also show interstitial C-bands which are highly low recombination rates. The treatment of this variable in their locations (reviewed by John & Miklos, question is the subject of our previous paper 1979). (Charlesworth et al. 1986). Several observations Whereas very much is known about the structure, reviewed there suggest that the suppression of variability and location of satellite DNA, the prob- crossing over in regions such as centromeres and telomeres is not a direct physical property of HRDNA, but Present address: Physikalische Chemie I, Technische Hochschule Darmstadt, Petersenstr. 20, D-6100 Darmstadt, F.R.G. is a consequence of the long-range effects of centro-

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meric and telomeric factors. A second important step histone or ribosomal RNA genes which presumably is to study the association between satellite DNA se- have optimal copy numbers of repeats. On the other quences and regions of restricted crossing over. Thus hand, an upper limit to copy number seems likely, since I will present here mathematical models which may cells with large amounts of H RDNA must have substan- explain why HRDNA accumulates preferentially in tially altered properties such as long division times chromosomal regions where virtually no meiotic (John & Miklos, 1979). A possible form of wt is given recombination occurs (heterochromatin), but is hardly by found in euchromatin.

which is mostly used in the following and referred to 2. The model as the additive selection model. It is clear from data on species comparisons (John & Unequal exchange. Little is known at present about Miklos, 1979, and references therein) that changes in the distribution of equal and unequal meiotic ex- heterochromatin and satellite content are common- changes in HRDNA. However, large changes in the place during evolution, and it is widely recognized that amount of heterochromatin in specific chromosomal the mechanisms for their changes are unequal regions have been observed in humans from one genera- crossing over and saltatory amplification. Whereas tion to the next (Craig-Holmes, Moore & Shaw, 1975; HRDNA is formed by saltatory events, unequal ex- Seabright, Gregson & Mould, 1976). This suggests that change is considered as a secondary randomization unequal exchanges can involve a large number of process which leads to homogenization of the se- repeats. Various models of unequal exchange have quences within an HRDNA array or to variation of been proposed (Kriiger & Vogel, 1975; Perelson & cluster size, but is not used as a direct means of Bell, 1977; Ohta & Kimura, 1981). The present amplification (see the definition of unequal exchange, analysis is based on the model of Takahata (1981), below). Before outlining details of these processes, let which seems most general. It only assumes that the me briefly describe the biological background which exchange is symmetric, i.e. the probability that an the model is based on. For simplicity, I assume a sexual exchange between two chromosomes with copy haploid species, in which transient diploid zygotes are numbers j and k results in a daughter chromosome formed by random mating. Meiosis follows immedi- with copy number / (1 ^i 1. Given the threshold, fl. This is because I am not considering above selection scheme, it can be shown multigene families with specific functions such as (Charlesworth et al. 1986) that, having initially been

accumulated, HRDNA will ultimately be lost from the boundary Q. This may be realistic for HRDNA since population in the absence of (further) amplification or selection is very weak and thus Q = s"1 must be very migration. In the following I give some asymptotic large; hence, formulae for the mean time to loss of HRDNA. The effect of recombination on the rate of loss is of -j)- (5) particular interest. Given equations (4) or (5), the Markov chain (3) is fully 3. Asymptotic theory determined. To calculate the expected time until HRDNA is lost from a given population, it is imprac- (i) Mean time to loss for small Ny ticable to use the method of our previous paper by I assume here that population size 2N and recom- considering the eigenvalues of the matrix {ptj}. bination rate y are sufficiently small that the popu- Instead, we resort to a diffusion approximation of the lation is usually fixed for a single gamete type. In the Markov chain model (Ewens, 1979; chapter 4). absence of selection, this requires 2Ny <^ 1, so that no Let Y(z) indicate the state of the Markov chain at new type is produced by unequal crossing over while time T. The expected change in Y between z and T+1 a given one is on its way to fixation (a process which can easily be calculated using equations (2) and (5): takes in the neutral case an average of AN generations while the time between successive recombination E{Y(z+l)- Y(z) I Y(z) = 1} = (7-O events amounts to approximately l/Ny generations; since, assuming neutrality, only \/2N recombinant gametes get fixed,th e present model requires 2Ny <^ 1). A similar approach has been adopted by Walsh (1985) According to the above assumption I neglected in describing the evolution of multigene families under boundary effects here too, so that 2i < Q always holds. gene conversion. It should be noted that the following 2* The term S Qut (J— Ovanishesbecauseitisasumovera formal treatment is also valid for sister-strand exchange. product of a symmetric and an asymmetric function. It is convenient to study the process first on the time Similarly, scale of successive fixation events. Since I assumed an upper limit, Q = s"1, to copy numbers, the system E{(Y(z +1)- Y(z)f I Y(z) = i} * ^ I Qm{j-i?. occupies Q discrete states Et(i = 1, ..., Q), numbered (7) after the copy numbers of the individuals, and its Since 5 is assumed to be very small this is approxi- dynamics can be described by a finite Markov chain. mately equal to the variance in the change of Y Let pi} be the probability of transition from state Et, between z and z+1. The sums on the right-hand sides to Ej between times r and T+1 on the scale of fixation of equations (6) and (7) can elementarily be evaluated events, p^ is related to the unequal crossing-over to be transition probabilities by O« = J(?-1). (8)

Pu= (3) In order to obtain the operator for the diffusion J equation, space is rescaled by introducing 1 where ri} is the probability of fixationo f a variant with X(i) = (Y(t)-l)/n and time as / = (NyY *. Since in j copies in a population originally fixed for i copies. the additive selection model s = Or1, the mean a(x) The calculation of the probability of fixation, ri}, is and the variance b(x) in the rate of change in X(t) = x difficult unless we assume that a variant chromosome per generation are then obtained as withy copies (say7 < 0 has an overall selective advantage over all other individuals of the population (note a(x) x - \x(x + 2s), (9 a) that apart from a chromosome withy copies, a variant with 2/—y copies is also produced by unequal crossing (9 b) over between two individuals both carrying i copies). In the additive selection model the overall advantage As shown in our previous paper, the process starting is given by s(i—j). This can be inserted into the result at a certain copy number will ultimately get absorbed by Moran (1962, chapter 5) for haploid populations to in state Eu where each chromosome carries only a obtain an approximate expression for the probability single copy. If the initial copy number of the repeated of fixation at state Ep coming from Et: sequences is i0, the expected time to absorption at copy number 1 is in the diffusion approximation given by (4) 4 dx + e Nx x(x + 2s) — Xo A simpler expression for ri} can be obtained by + 2s)i assuming that the system is always far away from the with ;cn = (10)

If population size is small, such that N < Q//o and if current mean, I, only and will be approximated by a \

(ii) Mean time to loss for large Ny Clearly, these equations would still contain frequency In this case population size 2N and recombination rate terms without some further but minor approxi- y are supposed to be sufficiently large for the follow- mations. I approximated in equations (lib) and (17c) ing argument to be applied. Since selection is weak relative to recombination, the mean copy number changes slowly, relative to the rapid establishment of j+k odd a quasi-equilibrium distribution of copy number, after by I, and similarly, a recombination event had occurred. (Sampling drift a can be neglected, because N is assumed to be large.) i,k-l The dynamics of the process is thus a function of the j+k odd

by \i. Some insight into the validity of these approxi- moments. It follows then from equation (19) that the mations can be obtained by considering the following variance is given by two extreme cases. If copy numbers are equally distributed, the first term is given by \—^ and the second i- D2} (21) by |f— ^j. On the other hand, if the system is in the s neighbourhood of the absorption state, both sums will 48-H-25 Y vanish, as they should do. provided / is sufficiently small. However, since selec- Since the equation for a given moment always tion is presumably very weak, say s< 10~10—10~8, depends on the next higher moment, the moment expan- and recombination rate is assumed to be rather high, sion does not stop, unless one breaks it off at a certain conditions (20) do not imply any greater restraints on point. This is done after the third moment because the the present analysis. Note that the dynamics becomes term containing the fourth moment in equation (17 c) independent of recombination, if f is sufficiently small, can be neglected in the kind of approximation I pursue say I < 0-05 y/s (see (21)). (see below). To solve the moment equations I em- Finite population. The deterministic model, presen- ployed an adiabatic approximation procedure (Haken, ted above, studies the change in copy number under 1977, chapter 7), noting that unequal crossing over unequal crossing over and selection only. In finite popu- does not affect mean copy number in the absence of lations sampling has to be taken into account. This selection, and using the argument from the outset of occurs in a given generation after selection to form new this section. For our case of very weak selection it is zygotes. Let {Xf} again be the frequency distribution of thus possible to conclude that the variance and third copy number in a given generation after selection. A moment change rapidly on the time scale of the very possibly different frequency distribution {xt} is pro- slowly moving mean, so that, after a short transient duced by N male and N female gametes when taken as phase, one has random samples from {xt}. The expected change of the mean, E{Ai}, per generation of the distributions {x't} and {xt} can be calculated from standard multinomial -03}«0. j (18) formulae (Crow & Kimura, 1970, p. 330): In dealing with the fourth moment, it is realistic to E{AT} = Af (22 a) assume that it is of the order of E{(i—I)2}2 or smaller. (This is fulfilled for most of the known distributions.) and the variance of the change of the mean is given by In the following, I consider only the case E{(i—z)4} < 6E{(i—j)2}2. (Relaxing this condition a bit does not E{(Ai-E{Ai})*} = ±jE{(i-r (22b) change the main result (equation (21)) and has only slight influence on the domain of validity of this result.) where E{...} indicates that expectation has to be taken Under these circumstances, 13E{(i- i)2}2 - E{(i- J)4} | over the distributions {xj and {x't}, respectively. (see equation (17c)) can be estimated by 3E{(i—f)2}2. Formulae (22) constitute the relationship between Using this approximation and equation (18), the deterministic model and finite populations. They allow one to employ a diffusion approximation, with equations (lib) and (17c) can be combined to give the the mean copy number as the diffusing variable. The following equation, which is quadratic in the second combination of equations (17a) and (22a) and, simi- moment: larly, of equations (21) and (22b) yields the mean, a(x), and the variance, b(x), in the change in x per generation: (?-l)«fl. (19) a(x) x -s (23 a) The first term of equation (19), which is proportional 48-X + 25 to the square of the second moment and which is Y basically an upper bound of the fourth moment can, in turn, be neglected if (23b) y (20a) If the initial mean copy number is To, the expected time to absorption at copy number 1 is approximately given by: or, to be more specific and requiring the left-hand side 48-(x+l) + 25 of (20a) to be smaller than 01, if (\-e~ !~' *** 10;t(;c+2) T*0-nl (20b) Under these conditions one is allowed to approximate — (e*Nsx0_] L e-i Nsx system (14) of equations by their three lowest order S 10X(JC+2) ' (24)

where z is the upper boundary determined by conditions (20). For evaluating the integrals, z can be replaced by oo. The integrals can approximately be carried out if l0 is sufficiently large. The mean time to loss is then given by

In £ (25a) and

F(fo) In (256) These results may again be compared with those of the corresponding neutral case (conditional on the absorption state Ex having been reached). In this case the mean copy number does not change, so that AT = 0, whereas equation (21) has to be replaced by 2 E{(i — T) } « §(F— 1). (26) io3 - The expected time to loss of HRDNA, conditional on E1 having been reached, is then given by

/(io)« lOWln^, (27) in2 provided the process started with a high copy number

Fig. 1. Mean time to loss of satellite DNA vs inverse recombination rate. The simulations were done with the 4. The simulations following sets of parameter values: selection coefficient, s = 00001; initial copy number, i0 = 50; and population To obtain the expected time to loss of HRDNA from sizes, IN = 10 (A), 40 (D), 100 (•) and 400 (O). The a given population for intermediate values of the recom- theoretical curves are represented by solid lines, with the corresponding population size by the curve. bination rate y and to examine the validity of the analytic results, a simulation study was performed. A haploid species of population size 2N was considered. generations had almost no effect on the estimates of In each generation, the following processes were the times to loss since, for the given set of parameter allowed to modify copy numbers: sampling and selec- values, the boundary has hardly been reached. tion of gametes, and recombination among the However, a slight artifact has arisen through the way sampled and survived individuals. Sampling and selec- in which the simulations were realized on the com- tion were done simultaneously. Individuals were puter. Since each batch job could run for only a sampled at random from the population, with limited amount of time, longer runs remained replacement. Their survival was determined unfinished (and could not be counted) more often than according to the additive selection scheme of shorter ones. The times to loss are therefore slightly equation (1). This process was repeated until underestimated, in particular, for the larger popu- population size became 2N. Recombination was simulations (2N = 100 and 400) and small ys. lated as follows. Two gametes were chosen randomly In Fig. 1 the simulation results are compared with from the pool of the sampled and survived individuals those predicted by the model, as obtained by numeri- and were paired up according to the probability distri- cal integration of equations (10) and (24), respect- bution (equation (2)) to generate recombinants. The ively. The asymptotic results agree reasonably well number of crossovers between the parental chromo- with the simulations for large ys. The agreement is the somes had a Poisson distribution with mean Ny. better the larger the recombination rates and popu- The simulations were started with an initial copy lation sizes are. The deviations for very low popu- number i0 = 50 (for each gamete) and were run for lation sizes, 2N = 10, are due to the semi-deterministic 5 x IO6 generations, or until absorption at copy approach used in the analysis. On the other hand, the number 1 occurred. For populations of sizes 10 and 40, adiabatic approximation technique leading to the simulations were repeated 50 times, for sizes 100 equation (18) is responsible for the deviations of the and 400 25 times. The upper limit to copy number was theoretical curve for smaller ys (y < 1). In this approxi- Q, = io4. The results of the simulations are sum- mation, I is nearly independent of y since, apart from marized in Fig. 1. Terminating runs at 5xlO6 the change in the mean (see equation (17a)), the vari-

ance E{(i—f)2} is hardly affected by unequal crossing euchromatin (Szauter, 1984, and references cited there- over, due to the weakness of selection (see equation in). Apart from the fact that we do not know (21)). On the whole, the comparison between theoreti- whether small amounts of satellite DNA have gone cal and simulation results suggests that for realistic undetected at other regions throughout the lengths of population sizes the mean time to loss of HRDNA chromosomes, there is a strong association between from a population is correctly described by the model heterochromatin and satellite DNA. As a contri- over a wide range of recombination rates including bution to the structure-function debate about those of euchromatic chromosomal regions. HRDNA, the possibility of this association has been More important for the evolution of HRDNA and investigated in this paper on the basis of quantitative heterochromatin where virtually no meiotic exchange modelling. occurs (Szauter, 1984) is the other asymptotic case To this end, the time to loss of HRDNA from a 2Ny <£ 1. The model predicted an inverse linear given population has been calculated, both analyti- relationship between / and recombination rate for cally and by computer simulations. To understand small ys (see equation (10)). This character of tis found why HRDNAs are likely to persist longest in regions in the simulations over an even wider range of y than of suppressed recombination, it was not necessary to required by the condition 2Ny <^ 1, a fact which introduce an explicit model of amplification. The evolu- suggests that the model could also be used as an approxi- tionary dynamics of HRDNA can also be understood mation in circumstances in which copy number is in the following way. At each time, when new known to vary from individual to individual within a repeated sequences are regenerated by an amplifi- given population (for instance, deca-satellite in the cation event the (unidirectional) process of loss of African green monkey genome; Maresca, Singer & HRDNA is restarted. Since the rate of producing new Lee, 1984). The values obtained from equation (10) are copies of DNA sequences is presumably very low slightly higher than those found by the simulations. To (Schimke, 1984), interaction between selection (operat- some extent, this is due to the time limitations of the ing through unequal exchange) and amplification thus batch jobs, mentioned above. leads to a mutation-selection balance, i.e. a steady- Two remarkable consequences of the model, which state probability distribution of copy numbers in evolu- have been obtained analytically, are confirmed by the tionary times. Sufficiently strong amplification could simulations. First, the expected times to loss of certainly generate limit cycles or chaotic behaviour, HRDNA are greater in small populations. This result but this possibility seems somewhat remote. Given the is not intuitively obvious and is in contrast to the large results presented above, the steady-state distribution Afy-case. It follows from this effect, which is a conse- of copy numbers will be such that mean copy number quence of sampling drift, that the amount of HRDNA is highest in segments of chromosomes with low values should be higher in small populations. Fig. 1 indicates of y. In fact, the greatest effect of meiotic recom- that the effect is not large. But a final answer cannot bination on HRDNA is in the neighbourhood of zero be given without including the amplification process crossing over (see Fig. 1). explicitly. Secondly, the simulations confirmed the pos- Thus the present analysis questions the often-used sibility of the accumulation of HRDNA in chromo- argument that the localization of H RDNAs along chro- somal regions where recombination is suppressed. In mosomes gave hints to understanding the functional such regions the mean time to loss of HRDNA in- aspects of satellite DNA. Many arguments concern- creases by several orders of magnitude relative to seg- ing satellite DNA function are based on the centric ments of high recombination rates, i.e. Ny ^ 1. The or telomeric localization of these DNAs, but it is now interpretation of this result is as follows. Frequent apparent that heterochromatin is distributed through- unequal exchanges tend to spread out the distribution out the entire chromosome arms (see Introduction) of copy number, so allowing selection to be very effec- and may persist everywhere in the genome where recom- tive in the elimination of individuals with high copy bination is restricted. Under these circumstances our numbers. Selection itself, being not supported by earlier paper (Charlesworth et al. 1986) attempted to unequal crossing over, appears to be rather weak in answer the question why recombination is suppressed preventing the spread of tandemly repeated DNA in some chromosomal regions separately and to treat sequences. the accumulation of HRDNAs as a pure consequence of such restriction. Two (independent) observations, the non-uniform distribution of meiotic exchanges 5. Conclusions along the chromosomes and the distribution of satel- Meiotic exchanges occur non-uniformly along the chro- lites, are thus brought together to constitute a causal mosomes of most higher eukaryotes. There are at least relationship. We presented evidence that this relation- two aspects of this. First, there is virtually no crossing ship is essentially established by long-range effects of over in heterochromatin. Secondly, there are fewer centromeric and telomeric factors in controlling cross-overs per unit of physical length in regions near recombination. the centric, telomeric and apparently also near the A more complete picture of the accumulation of interstitial heterochromatin than in the remainder of satellite DNA can be obtained, considering possible

12 GRH 47

counter-forces of the mechanisms controlling recom- Flavell, R. B. (1982). Sequence amplification, deletion and bination. Recent data on minisatellites (Jeffreys, rearrangement: major sources of variation during species Wilson & Thein, 1985) show that repeated, non-tran- divergence. In Genome Evolution (eds. G. A. Dover and R. B. Flavell). London: Academic Press. scribed DNA sequences are not inherently incapable Haken, H. (1977). Synergetics, an Introduction. Berlin: of recombination. These observations suggest, that, in Springer. order to behave as recombinationally inert, satellite Jeffreys, A. J., Wilson, V. & Thein, S. L. (1985). Hyper- sequences must obey certain constraints with respect variable ' minisatellite' regions in human DNA. Nature to their DNA structure. Further evidence pointing in 314, 67-73. John, B. & Miklos, G. L. G. (1979). Functional aspects of this direction has been reported by Strauss & satellite DNA and heterochromatin. International Review Varshavsky (1984). They found a non-histone protein of Cytology 58, 1-114. that binds to a-satellites from African green monkeys Kruger, J. & Vogel, F. (1975). Population genetics of at three specific sites per repeat. Due to their apparent unequal crossing over. Journal of Molecular Evolution 4, nucleosome-positioning activity, such proteins may 201-247. Maresca, A., Singer, M. F. & Lee, T. N. H. (1984). Con- underlie the highly regular nucleosome arrangements tinuous reorganization leads to extensive polymorphism in regions of repetitive DNA and so stabilize repeated in a monkey centromeric satellite. Journal of Molecular DNA sequences against strand breakage and subse- Biology 179, 629-649. quent recombination events. Accordingly, we ad- Miklos, G. L. G. (1982). Sequencing and manipulating vanced the following hypothesis on the evolutionary highly repeated DNA. In Genome Evolution (eds. G. A. Dover and R. B. Flavell). London: Academic Press. accumulation of satellite DNA. There is a preferential Moran, P. A. P. (1962). The Statistical Processes of Evolu- accumulation of repeated sequences in the neighbour- tionary Theory. Oxford: Clarendon Press. hood of centromeres and telomeres, due to these Ohta, T. (1983). Theoretical study on the accumulation of regions having been selected for restricted rates of selfish DNA. Genetical Research 41, 1-15. crossing over, and the HRDNA sequences which Ohta, T. & Kimura, M. (1981). Some calculations on the amount of selfish DNA. Proceedings National Academy of accumulate are those with low rates of exchange. Sciences, USA 78, 1129-1132. Peacock, W. J., Lohe, A. R., Gerlach, W. L., Dunsmuir, P., This research has been carried out at the Universities of Dennis, E. S. & Appels, R. (1977). Fine structure and Sussex and Edinburgh. I am considerably indebted to Brian evolution of DNA in heterochromatin. Cold Spring Charlesworth and Charles H. 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