J. theor. Biol. (1979) 78, 425-438

Diploidy, JULIAN

Department The Middlesex Hospital (Received 25 October

Evolution

LEWIS?

and Sex

AND LEWIS WOLPERT

qf Biology as Applied to Medicine, Medical School, London WlP 6DB. England

1978, and in revised,form

7 February

1979)

A new gene for a new purpose may be created by mutation of a pre-existing gene. But if that original gene is still required for its original purpose, and is to be retained side by side with the new, a spare copy is needed initially as raw material for the innovation. Thus in haploids the original gene must be duplicated before it is modified. But in diploids a spare copy of every gene is always available, and a mutant allele serving a new purpose can be easily established and maintained by heterosis in parallel with the old allele. Subsequent gene duplication will lead, via crossing-over, to insertion of the new gene in tandem with the old, as a permanent addition to the genome. Calculations show that dipioids can thus enlarge their genomes with new genes for new purposes much more readily than haploids ; in particular, they can more easily evolve the complex gene control systems characteristic of differentiated multicellular organisms. Sexual reproduction preserves diploidy, and so can be seen as the basis of these richer possibilities for evolutionary innovation. 1. Introduction

New genes are not conjured out of thin air : they must be adapted from preexisting genetic material. If that pre-existing material itself has a function, then a mutation that fits it for a new task will in all likelihood unfit it for the old. If the old task remains essential, and there is no other gene to do it, the evolution of a gene for the new task will be hindered ; it can occur only through those very rare mutations which create a single dual-purpose gene endowed with the new capability as well as the old. Thus innovation is difficult for a haploid organism. In a diploid organism, however, there is a spare copy of every gene, free to mutate to serve a new purpose, while the original allele remains to Serve the old. This makes innovation easy. But such innovation uses up the spare gene copies: the two homologous parts of the genome diverge, and the diploid state is gradually lost. Sex, however, checks this trend: there is a limit to the degree of heterozygosity compatible with sexual reproduction. Again we encounter a barrier to innovation. That t Addressfor reprints:Dept.of Anatomy,King’sCollege,Strand,LondonWC2,England. 425

0022-5193/79/l

10425+

14 $02,00/O

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barrier can be surmounted, both in haploid and in diploid species, by enlarging the genome by gene duplication. But the haploid must duplicate first and mutate after, whereas the diploid can mutate first and duplicate after, maintaining the mutation in the interim as a balanced polymorphism. Hence, as we shall argue in detail below, an additional gene for a new purpose can become fixed in a haploid population only after a happy coincidence of two rare chances ; whereas in a diploid species the new gene can become fixed by a two-step process, in which the two rare chances have merely to occur in sequence: The diploid genome can thus be enlarged and enriched with new genes for new purposes much more easily than the haploid. More generally, mutations which are forbidden as lethals in the haploid may be tolerated as recessive lethals in the diploid; and these mutations may constitute an important pathway of evolutionary change. The diploid species has a greater freedom to experiment with its genes. Evolution by gene duplication is no new idea. It was suggested by Bridges in 1918 and has been discussed by many others since, especially by Ohno (1970) in his book on the subject. Sequence homologies within many groups of related proteins provide overwhelming evidence of their origin by some chain of events involving gene duplication and divergence. Furthermore, Fincham (1966) and Partridge & Giles (1963) have already remarked how in a diploid species a mutant gene may first become established by heterosis, and then become incorporated in series with the old gene through a subsequent duplication. The theory of the process has been studied in some detail by Spofford (1969). In this paper we show that heterosis followed by duplication is not only a possible way to add new genes to the genome, but that it is also an especially rapid and important way; and we point out the peculiar advantage consequently enjoyed by sexually-reproducing organisms over those that are purely vegetative. For sex keeps a species diploid ; and through being diploid, as we shall argue, it is able to evolve the complex genome that generates a differentiated multicellular organism, instead of a mere crowd of microbes. 2. How are New Genes Added to the Genome?

Suppose that there is a certain gene G from which another gene G* may be derived by mutation ; and that an individual endowed with copies both of G and of G* has a selective advantage over individuals having only the gene G, or only the gene G*. If at first all individuals in a population have only the gene G, selection will favour enlargement of the genome to include the gene G* also. The mechanism for this step in evolution will be different for haploids and diploids. We want to compare the two.

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Consider the haploid first. The initial genotype is then G; the final genotype is GG* ; and this must evolve via an intermediate form GG, in which the gene G is duplicated : in symbols G -+ GG -+ CC*. Since the duplication is a precondition for the mutation, the rate of occurrence of the process as a whole will be proportional to the frequency of duplications GG multiplied by the frequency of mutations G + G*. To be specific, let there be N individuals in the initial population. Let r be the relative frequency of individuals with the duplication GG, so that there are on average Nr such individuals in existence at any time. Let p be the probability per generation per pair of copies of the gene G of a mutation G -+ G*. Then there is a probability Nrp per generation that an individual will crop up with the favoured genotype CC*. (We assume Nrp < 1.) Let p be the probability that such an individual will survive and propagate so that the genotype GG* becomes fixed in the population and supersedes the primitive genotype G. The time taken in such a case for the genotype CC*, once it has been introduced, to spread until it predominates in a population of size N will then be of the order of 1 t spreadIs ln p _ ln j In N generations wherefandf* are the fitnesses of individuals with the genotypes G and G* respectively. Assuming that the fitnesses are not drastically different, so that If*-fl $f, we can put roughly 1 lnf*-lnf

f* * f*-f’

Thus the total number of generations which must on average elapse until the genotype GG* arises by duplication followed by mutation and spreads through the population will be of the order of --+

1

f* ---1nN. Nvv f*-f If we define the selective advantage of the GG* genotype as ‘hapbid

this can be written 1 ‘hapbid

-

~

Nrpp

+ilnN. s

Finally, we can insert a value for p, the probability

of fixation of the mutant

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gene :

The derivation

is given by Crow & Kimura ---

‘hapbid

(1970). Thus

1 +IlnN. 2Nrps s

Consider now the diploid species. The initial genotype, in which each parent contributes only one copy of the gene G, may be represented OG/OG. We want to see how this may evolve to give a genotype GG*/GG*, in which each parent contributes both variants of the gene G. The sequence of evolutionary events must again include a duplication and a mutation. If the duplication precedes the mutation, the calculation is essentially the same as for the haploid, and the same formula gives the average time which must elapse before the GG*/GG* genotype arises and becomes common: tdm

-

1 +I1nN. 2Nrps s

We want to consider by contrast the course of events in which the mutation precedes the duplication : OGIOG -+ OGfOG* + GG*/GG*. The crucial feature of this process is that selection may favour the intermediate genotype OG/OG*, so that it becomes common, and serves as an easy stepping-stone to the final genotype GG*/GG*. As before, let the number of individuals in the population be N, let p be the probability per generation per pair of copies of the gene G of a mutation G + G*, and let p be the probability that a OG/OG* individual will survive and propagate so that the gene G* can become fixed in the population. Once this has happened, the gene G* will rapidly spread through the population thanks to heterozygote advantage, until the relative frequencies of the genes G and G* reach the equilibrium proportion off* -f** :f* -f, where f, f* and f** are respectively the fitnesses of the zygotes OGIOG, OG/OG* and OG*/OG*. (See, for example, Crow & Kimura, 1970.) The time taken for the G* gene, once introduced, to spread through the population will again be of the order of

f* f*-flnN generations.

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Thus the total number of generations which must on average elapse until the allele G* arises and spreads through the diploid population will be of the order of t hetero

Putting

-

j&

+

+j *

In

N.

again

s - .f* -f .f* we thus have t hetero

1 -

m

+IlnN. s

Let us now suppose, as before, that there is a certain small probability r per gamete of duplication of the gene G, so that, in the absence of the allele G*, there are approximately 2Nr individuals with the genotype GG/OG. If the allele G* is introduced and becomes common, recombination will yield some GG* gametes; and since these confer the advantage possessed by OGjOG* heterozygotes on every zygote to which they contribute, they will tend to increase in frequency until the whole population consists of GG*/GG* individuals. The equations which describe this process and an estimate of the time it takes, tdup, are derived in the Appendix. The general formulas are rather complicated. To make the important points clear, we present here only a simple limiting case; it is easy to check, by numerical computations, that the behaviour in many other cases is roughly similar (see Spofford, 1969). The simple limiting case is defined as follows : (I) All zygotes with at least one copy of the gene G plus at least one copy of the gene G* have the same superior fitnessf*; (II) all other zygotes, having copies only of G or only of G*, have the same inferior fitness f; (III) the original G locus and its duplicate are closely linked. We measure the linkage by a parameter y, the recombination fraction, that is, the relative frequency of crossing-over between the original G locus and its duplicate. Close linkage means small y. Then the number of generations required for the population to evolve from a state of balanced OG/OG* polymorphism to a state in which more than half the gametes produced have the favoured genotype CC*, with the new gene and the old in tandem, is roughly

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AND

L. WOLPERl

where

as before. Thus the total number of generations required on average for the gene G* to arise and spread by duplication until a large proportion of the population is GG*/GG* is of the order of ’ md =

thctero

+ ‘dup

_

l..--

+

!

2Nps

]n

N

+

1 ln

s

s

$,

i’

or somewhat less, allowing for the fact that GG* gametes will start to be engendered even before the population has attained a state of balanced OGIOG* polymorphism. We can now draw the comparison between the rates of the two modes of genome enlargement, the first by duplication followed by mutation (possible for haploids as well as diploids) the second by mutation followed by duplication (possible only for diploids) :

fdm -h t md

1 __2lvpr

+!lnN s

1 P+~lnN+~ln~ 2Np.5

s

s

2r;

1 l+ZNprlnN =~‘1+2N~[lnN+2ln(~’ On any plausible estimates of N, s, r and y (that is, say, for lo2 < N < lOlo, 10-j < s < 1, lop9 < r < 1O-2 and 10e4 < y < 0.5) we shall have 5 < In N 5 25 5 ,< In N +2 In (s/Zry) ,< 80. A middling roughly

estimate might be N - 106, s - lo-‘,

r - 10w6, 7 - 10e2, giving

1nN - 15 In N+2 In (s/2ry) - 45. For the order-of-magnitude arguments which follow, it does not much matter which of these values we use. Using the middling estimate, we have 1 1+30Npr tdm -Ad-* tmd r 1+9ONp

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Then, assuming r 4 1, we have

if 90Nn >> 1 b 30Npr 1 if 9ONi1 < 1. $1 I- I In words, the evolutionary process involving duplication followed by mutation will be much slower than the peculiarly diploid process involving mutation followed by duplication, if there is only a small probability, for the population as a whole in any generation, of having the favourable mutation occur de novo in an individual who already chances to carry the duplication. In the unlikely case where this condition is not satisfied, the two rates will be of the same order of magnitude. For higher animals, the mutation rate per gamete per locus is of the order of 1O- ’ or 1Oe6, and the rate of occurrence of a specific favourable mutation might well be one or two orders of magnitude less than this (Cavalli-Sforza & Bodmer, 1971): so let us say p - 10m7. It is hard to find estimates of r, the probability that a given gamete will have a given gene in duplicate. In bacteria, genes occur in duplicate with quite a high relative frequency, of the order of 10d4 (Anderson & Roth, 1976). In eukaryotes, the frequence of duplication is probably much less at most loci: in one well-studied case, it was found that dihydrofolate reductase genes occurred in duplicate in mouse cells with a frequency of the order of, or less than, one in a million (Ah et al., 1978). Thus if we suppose. rather arbitrarily, that r - 10d6, and put N 5 106, we get -tdm - 105. t md

Even if r were as large as 1O-*, we should still have t&,/f& - 10. Thus we see that sexually-reproducing diploid species do indeed have a peculiarly rapid way of enriching their genome by the addition of favourable new genes. The important process, in all likelihood, is not one of duplication followed by mutation of the duplicate gene, as has so often been suggested (Bridges, 1918, quoted by Sinnott, Dunn & Dobzhansky, 1958; Lewis, 1951; Ohno, 1970); it is rather one of mutation followed by duplication on the lines proposed by Fincham (1966) and Spofford (1969). That is, mutations preserved in balanced polymorphism act as a driving force for enlargement of the genome; through recombination, they lend their heterotic selective

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advantage to any duplication that may arise, and tend to carry it to fixation : and thereby they establish themselves as permanent additions to the genome. 3. Multicellular

Organisms

Could this explain why sexual, diploid organisms have evolved in ways so much more rich and complex than haploids? Could it explain in particular why they have evolved into differentiated multicellular animals and plants, while the prokaryotes in general have not‘? We would answer yes; but the argument is not quite straightforward. A bacterium may have a generation time of 20 minutes, while a large plant or animal may have a generation time of several years; thus even if an evolutionary process required 10’ more generations for the bacterium than for the sexual dipioid, the rate of that process might be the same for the two types of organism if measured in terms of years rather than generations. In any case, as Williams (1975) has argued, there is little reason to suppose that the rate of mutation is the factor that limits the rate of evolution. We have to show, not that the haploids follow slowly in the footsteps of the diploids, but that they tend to take an alternative evolutionary pathway. It is difficult to devise a general proof: that would require a general analysis of the possible evolutionary pathways. We might, however, reasonably expect that the balance between, for example, tendencies to innovation and tendencies to economy would be struck differently in the two types of organism. It is perhaps best to point the contrast by considering a particular example, of special importance in the evolution of plants and animals. To form a differentiated multicellular organism, the constituent cells must adjust their behaviour according to their position in the whole, as indicated presumably by chemical cues of some sort. To be specific, let us consider differentiation mediated by the control of gene transcription. Suppose that a certain gene H is to be silenced under certain conditions C, but transcribed otherwise; and suppose that this control is to be exerted by means of a repressor protein P which is present in an active conformation only under the conditions C, and then acts by binding to a control locus L adjacent to the gene H. How might this control system evolve, given a primitive organism in which H was transcribed indiscriminately? In particular, how might the specific binding of a protein P to a control locus L evolve, if primitively they had no special mutual affinity? It is here that we find a crucial difference between haploid and diploid organisms. There are two basic strategies: (i) the original locus L, adjacent to the gene H may mutate so as to bind the protein P; or {ii) the original protein P, may mutate so as to bind to the locus L. The air of symmetry between these alternatives is deceptive.

DIPLOIDY.

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Let us first take case (i). Under this heading we should consider not only (a) the possibility of point mutations in the DNA adjacent to H, but also (b) the possibility of translocations carrying H into the domain of a preestablished control locus elsewhere in the genome (Hegeman & Rosenberg, 1970). Both types of change are entirely feasible. In the haploid organism, either of these modifications would stand a good chance of being carried to fixation by selection favouring control of transcription of the gene H. But in the diploid, both types of mutation would probably be recessive, since heterozygotes would still have a copy of H that was not subject to control; thusmutations of these types would stand a poor chance of being carried to fixation, even though selection for control of H might favour homozygous mutants. Now consider case (ii), in which the protein P, mutates to a form P which binds to the control locus L. The protein P, as a rule will presumably have some prior function : it may bind to some other control locus in the genome. or have some enzymatic or structural role. Mutations which give it a specific affinity for the control locus L are likely to spoil it for these other purposes, and so on balance to be deleterious in the haploid organism. In the diploid, by contrast, there will remain a copy of the old gene for P,, and heterozygotes carrying also the new variant P will be at an advantage: the control exerted by the mutant protein P that binds to L will be dominant. It follows that haploids and diploids will tend to evolve differently in response to selection favouring the development of this sort of control over transcription. Haploids will tend to modify the locus L adjacent to the gene H that is to be controlled. Diploids will tend instead to evolve a new type of controlling protein P. These two courses of evolution open up quite different vistas. In the haploid there is no change in the number or variety of genes that code for protein; instead the control locus L is adapted so that the original protein P, can serve a dual purpose-on the one hand to control transcription of H, and on the other hand to perform also its previous catalytic, structural or controlling task. The transcription of H is thus firmly tied to the other, original, functions of PO. The arrangement is economical, but leaves little scope for further elaboration. In the diploid, the contrary is true. A new allele coding for a new type of protein is established. By the mechanism analysed in the first part of this paper, this allele will eventually become incorporated in the genome as an additional gene. The protein P encoded by the new gene serves just one purpose, the control of transcription of H ; and so it is free to mutate to give the optimum control, independently of any requirements that must be satisfied by the ancestral protein P,,. The tendency in the diploid is towards a

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division of labour, towards enlargement of the genome by insertion of a new gene making a protein specialized for a new purpose. New systems of control can thus be easily established, and elaborate networks can be built up to do the complex tasks involved in the development of a differentiated multicellular organism (Wolpert & Lewis, 1975). It would be rash to maintain that this was the whole explanation of the differences of size and sophistication between prokaryotes and eukaryotes. But our argument does at least illustrate the way in which diploid organisms may have an advantage over haploids in the matter of evolutionary innovation. 4. The Importance

of Sex

This advantage can, moreover, be viewed as a benefit conferred by sex. Without sexual recombination, the diploid state would not be preserved. The two copies of each gene would diverge until there was, instead of a diploid genome, a haploid genome of inflated size and complexity. Sexual recombination prevents the two copies of the genome from diverging too far. The general argument can be easily stated. Suppose that there are 1 independently segregating loci at which recessive lethals are maintained by heterosis at a high frequency of the order of 1, say. Then the fraction of zygotes that are viable, in that they are not homozygous for a recessive lethal mutation at any locus, will be proportional to (1 -AZ)1 - e-“‘. This rapidly becomes desperately small if I becomes large. In other words, in a sexual species there is a large price to pay for the accumulation of a large number of heterotic recessive lethals. The same will be true, in only slightly less degree, for heterotic recessively deleterious mutations. This is indeed precisely the selection pressure that tends to bring about duplications of the type we have analysed above, and is relieved by them. If there is indeed an evolutionary advantage in being diploid, we have here an answer to the question “What use is sex?” (Maynard Smith, 1971). That difficult and much-disputed problem can, however, be posed in several different senses (Williams, 1975); we do not claim, for example, to have discovered a selective mechanism that maintains the sexual mode of reproduction in preference to the asexual in a given species. But we would argue that sex brings with it richer possibilities of evolutionary change: by keeping a species diploid, it endows it with genetic material to spare for experiment and innovation. Thus it is that the most complex among living creatures are the products of sex. We thank the Medical Research Council for support.

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REFERENCES ALT, F. W., KELLEMS, R. E., BERTINO, J. R. & SCHIMKE. R. T. (1978). J. Biol. Chem. 253, 1357. ANDERSON, R. P. & ROTH, J. R. (1977). Ann. Rev. Microbial. 31, 473. CAVALLI-SFORZA, L. L. & BODMER. W. F. (1971). The Genetic.\ of’ Human Population.!. pp. 102 -110. San Francisco: Freeman. CROW, J. F. & KIMLIRA, M. (1970). An Introdwtion to Population Genetiu Theory, pp. 418-422. New York: Harper & Row. EWENS, W. J. (1969). Population Genetics. pp. 106-108. London: Methuen. FINCHAM, J. R. S. (1966). Genetic Complementation. New York: Benjamin. HEGEMAN, G. D. & ROSENBERG, S. L. (1970). Ann. Rev. Microbial. 24, 429. LEWIS, E. B. (1951). Cold Spring Harb. Symp. Quant. Biol. 16, 159. MAYNARD, Surm, J. (1971). J. theor. Biol. 30, 319. OHNO, S. (1970). Evolution by Gene Duplication. London : Allen & Unwin. PARTRIDGE, C. W. H. & GILES, N. H. (1963). Nature 199, 304. SINNOTT. E. W., DUNN. L. C. & DOBZHANSKY, T. (1958). Princ~iples ofGenetics. 5th edn. p. 386. New York : McGraw-Hill. SFQFFORD, J. B. (1969). Am. Nat. 103, 407. WILLIAMS, G. C. (1975). Se-r and Evolution. p. 146. Princeton: University Press.

APPENDIX Our problem is an instance of the “two-locus problem” (see Crow & Kimura, 1970). We have four possible types of gamete, OG, OG*, GG and GG*, with relative frequencies a,, a2, a3 and a4 respectively, where a, +a, +a, +a, = 1. We assume that mating is random. Let the zygote formed of the ith and jth types of gamete have fitness fij. Let y be the probability of recombination between the original G locus and its duplicate. The relative numbers of gametes produced by each type of zygote are then as shown in Table 1. TABLE

Zygote

OG/OG OG/OG* OGIGG OGJGG* OG*/OG* OG*/GG OG*/GG+ GGIGG GG/GG* GG*/GG*

Frequency

a? 2aIa2 ha3 la, a4 4 %a3 ha4 4 ha4 ai

1

Fitness

2: i:

Gametes

produced

OG

OG*

GG

GG’

2 1 1 l-;

0 1 0 ;’ 2 l-; 1 0 0 0

0 0 1 ;’ 0 1-y 0 2 1 0

0 0 0 1-1 0 1’ 1 0 1 2

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Thus the frequencies of the four types of gamete in the next generation are

4 = ~I.f;ra:+fi2zlna+/lia,zi+f;air,cla+~(.12saIli-tlaala4)1

where W is a factor fixed by the condition = 1;

a;+a;+a;+ak

that is, W = 2 1 fijaiaj. ij

There is no known general analytic solution of this system of equations for the change of gene frequencies with time. The behaviour can, however, be easily computed numerically for particular values of the parameters. It is also possible to work out analytically the initial behaviour, during the period when a3 and a4 are much smaller than a1 and a2 (Ewens, 1969). Thus we can get an analytical estimate of the time it takes for the duplication GG* to reach significant proportions in the population. For this purpose, suppose that aI and a2 are initially in the equilibrium proportions a1 -=

f,2-f-22 f, 2 - f, I

a2

appropriate to a population in a state of balanced polymorphism with a3 = a4 = 0 ; and suppose that a3 and a4 are small compared with a1 and al. Then to lowest order in the small quantities a3 and a4 we can approximate (with a little rearrangement): -1

1 w

=*12-

( f12-fil

+ fi2

a, = constant = 2.h

./;2lf2:

i

-f22

2 -f22

-“f-i

1

fl2--fil

a2 = constant = vi

2 -f2

2 -fi

1

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a =~C.fi~a~+(l-y)f~~a~l;

b = -k jtf;4al

1 c = -y&al;

d = w CC1-y)f14al +f;4a21.

1

W

To find out how a3 and a0 change over a large number of generations, we express (a,, ad) as a linear combination of eigenvectors (II,, L’+ ) and (U~, I’ ) of the eigenvalue equation

The eigenvalues are A. f = +[a+dfJ(d-a)*+4bc]

The corresponding

eigenvectors satisfy %-

b

L‘*

I.,--a

i,-d c

.

Initially, the frequency of the duplicate gametes GG is assumed to be I, and that of the GG* gametes zero, so that (~~>‘“‘=(~)=.+,r’-,~,.+(::)+u~l~~~u+v-

(1:).

Thus after t generations we have

=

rL’+ v U,D_-U-V,

U+ 0 v,

+

ri’- v + u-v,

-u,v-

u( VP 1.

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Substituting for u*/u+_ we thus find that the frequency of the GG* gamete after t generations is

for t 9 (In 1+/J.-)-‘. We want to know the time it takes for the gamete GG* to spread through the population, that is, the time, tdup, required for CL, to grow to a value approaching unity-let us say, to be specific, to a value of ), (This was the criterion used elsewhere in this paper in estimating the corresponding time for spread of the gamete OG* through the population). The above formula for at’ has been derived on the assumption that c@ 4 1. For the purposes of an order-of-magnitude estimate, however, we may nevertheless apply the formula for a:’ = 3 and so find 1 ln A+ -;I‘dw - K

2rc

We can now substitute for i+, 13. _ and c in terms of y and the fij. But to get a formula simple enough to make sense of, we need to make some assumptions about the fitnesses fij. Specifically, let us suppose that all types of zygote that contain at least one G gene plus at least one G* gene have the same superior fitnessf*, and that all other zygotes, having only the one type of gene or only the other, have the same inferior fitnessf. That is,

fil =f22 =f,3 =f33 =f fl2 =f23 =f14 =f*4 =f34 =f44 = f*. We define

f*-f S=f"' If we assume also that the original G locus and its duplicate are closely linked, so that y 6 1, we find eventually that

for y 6 s.