A simulation model for the evolutionary advantage of sex

Giacinto Libertini

 www.r-site.org/ageing www.programmed-aging.org

 giacinto.libertini@tin.it

The term “sex”, synonymous of “mixis”, means the recombination of genes between two individuals.

The term “recombination” is not a synonymous of “sex”, because there are species that recombine genes within a single individual and without

mating (e.g., autogamy, Bell 1982)

The “classic” theory (Weismann 1889; Guenther 1906; Fisher 1930; Muller 1932, 1958 and 1964; Crow & Kimura 1965) hypothesizes that sexual reproduction is advantageous because it allows the combination of new advantageous alleles while for asexual reproduction this is not directly possible.

Maynard Smith (1968) criticised the “classic” hypothesis with the following argument:

In a haploid species, we have two genes (a, b), with alleles (A, B, respectively) having an advantage (sA, sB) on a and b, respectively. The frequency of a combination xy at the generation n+1 (P’xy) is in function of combination frequency at generation n (Pxy):

P’ab = Pab / T; P’Ab = PAb(1 + sA) / T

P’aB = PaB(1 + sB) / T; P’AB = PAB(1 + s’) / T

where:s’ = [(1 + sA)(1 + sB) - 1] k;k = interaction (epistasis) between the fitnesses;T = the sum of numerators.

If, at generation zero, there is no linkage disequilibrium, namely: Pab PAB = PAb PaB

and no epistasis (k = 1), in the next generation it will be always: P’ab P’AB = P’Ab P’aB

with or without recombination, which can only halve linkage disequilibrium at each generation. Therefore, with these conditions sex is not advantageous.

With negative linkage disequilibrium (D = Pab PAB - PAb PaB < 0) sex would be advantageous, while with positive linkage disequilibrium sex would be disadvantageous.

With positive epistasis (k > 1) between sA and sB, sex results disadvantageous because it breaks the more advantageous combination AB. The opposite happens if there is negative epistasis (k< 1).

Maynard Smith tried to overcome his argument observing that it was valid for infinite populations but that “linkage disequilibrium is bound to arise by chance in a finite population” and in conditions of negative linkage disequilibrium sex would be advantageous (Maynard Smith 1976), as previously observed by Felsenstein (1974).

Many scholars did not accept the arguments of Maynard Smith (e.g., Crow & Kimura 1969, Williams 1975) and, however, why conditions of negative linkage disequilibrium, favorable for sex, should prevail over positive occurrences?

The doubts about the validity of the “classic” explanation of sex caused the flourishing of alternative hypotheses:

- Muller’s Ratchet (Muller 1964; Felsenstein 1974)- Best-Man hypothesis (Williams 1966; Emlen 1973; Treisman 1976)- Hitch-hiker hypothesis (Hill & Robertson 1966; Felsenstein 1974)- Tangled Bank (Ghiselin 1974; etc.)- Red Queen (Van Valen 1973; Glesener & Tilman 1978; Bell 1982; Ridley 1993, etc.)- Sex is advantageous because it slows down evolution and excessive specialization (Williams 1975; Stanley 1976)- Historical hypothesis (Williams 1975)

The crisis of the “classic” theory and the absence of a valid and undisputed alternative require the development of a reliable simulation model capable to confirm or falsify the classic theory:

- confirming the valid theoretical argument of Maynard Smith about the absence of advantage of sex in populations in linkage equilibrium;

- showing that sex is advantageous in finite populations;

- avoiding any bias on the prevalence of sex favoring conditions as negative epistasis (k < 1) or negative linkage disequilibrium

- avoiding any hypothesis about advantages of sex for the species or for successive generations.

The simulation model

We hypothesize two (a, b) or three (a, b, c) genes present in all the haploid individuals of a population composed by N individuals.From these genes, more advantageous alleles (A, B, C, respectively) originate with frequency ux. The frequency of inverse mutation is wX:

ua ub uc a A b B c C wA wB wC

The advantages of A, B, C on a, b, c are sA, sB, sC, respectively.

For simplicity, we suppose:

u = ua= ub = uc

w = wA = wB = wC

s = sA = sB = sC

In the case of only two genes (a, b), the transformation of a combination to another is illustrated by the following schemes:

From To Prob.

aB AB u

Ab AB u

ab aB u-u2

ab Ab u-u2

ab AB u2

aB ab w

Ab ab w

AB aB w-w2

AB Ab w-w2

AB ab w2

In the case of three genes (a, b, c), the transformation of a combination to another is illustrated by the following schemes:

From To Prob.

ABc ABC u

aBC ABC u

AbC ABC u

Abc ABc u-u2

Abc AbC u-u2

aBc ABc u-u2

aBc aBC u-u2

abC aBC u-u2

abC AbC u-u2

Abc ABC u2

aBc ABC u2

abC ABC u2

abc Abc u-2u2

abc aBc u-2u2

abc abC u-2u2

abc ABc u2-u3

abc aBC u2-u3

abc AbC u2-u3

abc ABC u3

From To Prob.

Abc abc w

aBc abc w

abC abc w

ABc Abc w-w2

ABc aBc w-w2

aBC aBc w-w2

aBC abC w-w2

AbC Abc w-w2

AbC abC w-w2

ABc abc w2

aBC abc w2

AbC abc w2

ABC ABc w-2w2

ABC aBC w-2w2

ABC AbC w-2w2

ABC Abc w2-w3

ABC aBc w2-w3

ABC abC w2-w3

ABC abc w3

The advantage (s’) for a combination with 2 more advantageous alleles is given by the formula:

s’ = [(1 + s)2 –1] k

where k is the interaction (epistasis) between the two advantages.

By Maynard Smith, sex is predicted to be advantaged with positive epistasis (k > 1) and with positive linkage disequilibrium: D = Pab PAB - PAb PaB > 0and disadvantaged with opposite conditions.

The advantage (s”) for the combination with all the three more advantageous alleles (ABC) is given by the formula

s” = [(1 + s)3 –1] k2

In the population there are two alleles:R- -> not allowing recombinationR+ -> allowing recombination with other R+ individuals.

In the model, any disadvantage of sex is disregarded.

At zero generation, the frequencies of R+ and R- (R+0, R-0) are always 0.5

The simulation model should show whether after a certain number of generations (e.g., 250) the frequency of R+ (R+250) is greater than R- frequency (= sex advantageous) or less (= sex disadvantageous).

With these conditions and using the above said formulas and others that simulate recombination for R+ individuals, a first simulation model has been created. No expedient has been used to simulate finiteness of population. The results are very simple and confirm Maynard Smith’s observations.

With k=1 (no epistasis), sex is neutral with any value of u, w or s

With varying values of s (in the simulations, s oscillates from –0.1 to +0.1 each 150 generations), sex is neutral too

This means that Red Queen hypothesis, which explains sex as due to oscillating values of advantages caused by continuous interactions with other species, is insufficient to justify sex.

three genes

three genes

two genes

two genes

If k>1 (positive epistasis), sex is disadvantageous, and, on the contrary, if k<1 (negative epistasis), sex is advantageous, as predicted by Maynard Smith (k = 1.03 in the left figure; k = 0.97 in the right figure):

With positive linkage disequilibrium sex is disadvantaged, while with negative linkage disequilibrium sex is advantaged, as predicted by Maynard Smith (D = +0.04 in the left figure; D = -0.04 in the right figure):

However, it is unlikely and undocumented to justify sex as caused by prevailing conditions of negative epistasis or of negative linkage disequilibrium

k > 1 k < 1

D > 0 D < 0

As a matter of fact, it is indispensable a modification of the model, because real population are finite and discrete, namely not composed by fractions of individuals but by integer and finite numbers of individuals:

if for mutation, advantage, recombination, genetic drift or other, the frequency of a combination passes from a frequency XYn at generation n to a frequency XYn+1 in the next generation with an increment ΔXY, this increment must be always an integer number.

In the model, each of these integer numbers is obtained emulating the function "rbinom" of the package R of The R Foundation for Statistical Computing© (http://www.r-project.org/), which generates integer random deviates.

This method of “rbinom emulation” is used in the program to simulate the variations of frequencies due to:- Mutations (e.g., a -> A)- Inverse mutations (e.g., A-> a)- Advantage- Recombination- Genetic drift- Diffusion of combinations among demes (when the population is composed by more than one deme)

At each generation, rbinom emulation is used many times as described in the following table, which illustrates simulation times too.

* on a 2.13 GHz PC

Calls of rbinom emulation

program routine

Simulation times* for 23 groups (with log10N = 1 to 12, step 0.5) of 1,000 simulations for each group

2 genes case 3 genes case 2 genes case 3 genes case

1 deme 38 calls 114 calls 7 minutes 20 minutes

10 demes 460 “ 1,300 “ 52 “ 166 “

100 demes 4,600 “ 13,000 “ 491 “ 1,618 “

(RESULTS)Examples of single simulations(with u = w = 0.00001; s = 0.1; k = 1; and log10N = 3, 5, 6, 9 respectively)

log10N = 3

log10N = 6

log10N = 5

log10N = 9

(RESULTS)Series of simulations with log10N varying from 1 to 12 step 0.5 (means and S.D. of 1,000 simulations for each point) and with u = w = 0.00001; s = 0.1; k = 1

* = p < 0.001

(RESULTS)Series of simulationsSame conditions of the first series, but varying the value of u

In logaritmic unities, an {u} variation of 1 shifts the left side of the curve of 1and the right side of 2 (two genes) or 3 (three genes)

(RESULTS)Series of simulationsSame conditions of the first series, but varying the value of s

Note: with s = 0.01, simulations have been extended to 2,000 generations.

(RESULTS)Series of simulationsSame conditions of the first series, but with the population (now metapopulation) divided in d demes each with N individual and with an interdemic interchange of individuals equal to 0.1 for generation

In logarithmic unities, a {d} variation of 1 shifts the curve of 1.Note: d demes, each with N individuals, for the advantage of sex are practically equivalent to a single deme with N d individuals.

(RESULTS)Series of simulationsSame conditions of the previous simulations, but the value of u is 0.0001 instead of 0.00001

Note: if d = 100, even with log10N = 1 sex is advantaged.

So, in the simulations sex is advantageous practically for any size of the metapopulation.However, to predict the diffusion of sex among the various species, in alternative to asexual reproduction, or to predict the occurrence of sexual phases for species alternating sexual and asexual phases, it is necessary to consider the disadvantage deriving from sexual reproduction:

The prediction of “classic” theory is very simple and immediate:when disadvantage of sex is greater, asexual reproduction is favoured by natural

selection, and vice versa.

Advantage of sex(recombination of

genes)

Disadvantage of sex(time and energy to

find a partner and to mate)

A theory is valid or not valid whether its predictions are confirmed or falsified by empirical data.

Therefore, it is necessary to verify if predictions of the “classic” hypothesis of sex evolutionary advantage are

confirmed or falsified by data from natural observation.

It is useful to do the same for two other theories (Best Man hypothesis, Tangled Bank) that make precise predictions about

the diffusion of sex (Bell 1982).

PREDICTIONS OF

Best-Man hypothesis

Tangled-Bank

hypothesis

Red Queen hypothesis

Classic hypothesis

Data from natural

observation

PART 1: INTERSPECIFIC COMPARISON

Correlation with different habitats (for Best-Man and Tangled-Bank hypothesis’ predictions and for Data from Natural Observation, see Bell, pp. 359-65)

Freshwater, Higher latitudes, Severely disturbed environments, r-selection, Ecological periphery of a species range, Novel habitats, Recently glaciated areas, Xeric environments

Sexual Asexual Asexual Asexual Asexual

Ocean, Lower latitudes, Constant environment, K-selection, Ecological center of a species range, Ancient habitats, Unglaciated areas, Non-xeric environments

Asexual Sexual Sexual Sexual Sexual

Legenda: red = wrong prediction; green = right prediction

Best-Man hypothesis

Tangled-Bank

hypothesis

Red Queen

hypothesis

Classic hypothesis

Data from natural

observation

Other conditions (see Bell, pp. 378-83 and 364)

Parasitism The same as

observed in nature

The same as

observed in nature but thelitoky is expected not rare

The same as

observed in nature

The same as observed

in nature

Sexual whenever possible. Thelitoky

extremely rare, more common in free-living

form

Very small size of soma Sexual - Asexual Asexual Asexual

Large size of soma Asexual - Sexual Sexual Sexual

Recombination (see Bell, pp. 411-35)

Correlation between achiasmy and Ocean, Lower latitudes, constant environment, K-selection, etc.

Expected negative

Expected positive

Expected positive

Not

expected

Not found

Correlation between chromosome number and proclivity for vegetative reproduction

Expected negative

Expected positive

Expected positive

Not

expected

Not found

Correlation between crossing over freq. and proclivity for vegetative reproduction

Expected negative

Expected positive

Expected positive

Not

expected

Not found

Best-Man hypothesis

Tangled-Bank hyp.

Red Queen hypothesis

Classic hypothesis

Data from nat. observ.

PART 2: INTRASPECIFIC COMPARISON

Intermittent sexuality (see Bell, pp. 365-70)

During growing season (exponential growth of population)

Asexual - - Asexual Asexual

Before climatic changes Sexual - - - -

At times of high population density Asexual Sexual Sexual Sexual Sexual

At times of minimal population density Sexual Asexual Asexual Asexual Asexual

Elicitation of sex in laboratory (see Bell, pp. 370-71)

Signals of a change in environment Sex elicited - - - -

Crowding and starvation - Sex elicited Sex elicited Sex elicited Sex elicited

Dispersal and dormancy (see Bell, pp. 371-77)

Actively dispersing stage Sexual Sexual (with some

reservation)

- Sexual Sexual

Dormant stage Sexual (for most Best-

Man models)

Sexual / Asexual

- Sexual (Asexual if

the change of environment conditions is

abrupt)

Sexual / Asexual

CONCLUSIONMatt Ridley wrote (The Red Queen, 1993):I asked John Maynard Smith, one of the first people to pose the question ‘Why sex?’, whether he still thought some new explanation was needed. ‘No. We have the answers. We cannot agree on them, that is all.’Now, to this statement we can reply with:The advantage of sex is rationally explained by “classic” theory and, considering the disadvantage of sex too, it is possible to formulate predictions about its diffusion in nature that are confirmed by data from natural observation.No other theory is justified by sound theoretical arguments and/or confirmed by empirical data.

SUMMARY

Best-Man hypothesis

Tangled-Bank hypothesis

Red Queen hypothesis

Classic hypothesis

Differences

Concordances

No prediction

12

3

1

4

7

5

3

8

5

0

14

2

DISPROVED DISPROVED DISPROVED CONFIRMED

Thanksfor your attention

giacinto.libertini@tin.it www.r-site.org/ageing