Recapitulating Kirkpatrick and Jenkins
We confirm the KJ effect [8] by computer simulation (see Methods section for the way in which the genetic algorithm is set up). In the simplest case, where the parasite is a uniform population that is held fixed, the simulation provides a 17-point numerical scale to measure the adaptive value of an individual host allele. Diploid genotypes have a 33-point scale because of the averaging of allelic adaptations. Figure 1(a) shows the response of a uniform diploid population that is maladapted by one mutation per allele in a static environment. The starting adaptation score of the diploid genotype is 1.0 - 0.0625 = 0.9375. The sexual population moves to an average adaptation score of 1.0 after adoption of one beneficial mutation, whereas the clonal population moves to an average adaptation score of 0.96875 after the first mutation, and 1.0 after the second. Figure 1(a) is the same as Figure 1 in [8] but generated computationally in real time using our simulation. Adoption of beneficial mutations in the simulation is stochastic. Figure 1(b) shows data for time to adoption plotted on a lognormal scale. The times to adoption, in generations, for the sexual population (measured at population adaptation = 0.953) are in red. Times to adoption of first mutation in a clonal population (measured at population adaptation = 0.953) are in blue; time to adoption of the second clonal mutation (measured at 0.984) in green. Lognormal distributions were Gaussian for all three curves as tested by the method of Burmaster and Hull [31]. The time to adoption of the second mutation is 2.5-3 times that of the first. This ratio is expected since the target size for the second mutation is half the size of that for the first mutation and the clock starts for the second mutation as the first mutation is adopted. There is evidence of slight differences in the effects of drift on homozygote and heterozygote populations. The slower adoption times of the second clonal mutation is general across a range of mutation rates, μ, and population sizes, N. (N = 300 – 10,000; μ = 10-6 – 10-3).
The KJ effect with multiple loci
Kirkpatrick and Jenkins [8] adopted an analytical approach to the calculation of population fitness for sexual and clonal populations over the medium term, and made a number of simplifying assumptions. These included: (i) loci were numerous; (ii) loci were binary, i.e. 0 or 1; and (iii) loci fully adopted mutations before another locus proceeded to adoption. These, and other assumptions, are restrictive and have been noted as such [10]. We avoid the restrictions by setting mutation rates independently for hosts and parasites and allowing for stochastic adoption of mutations. We then compute all adaptation scores on-the-fly for each allele for each generation, placing no limitations on the order of mutation adoption. We also use 16-bit loci to increase allele space, and study single loci and small-to-moderate numbers of loci where the parasite environment evolves continuously to reduce the adaptation of the host through NFDS. A key relationship at steady state (defined in Methods) is the ratio of mutation rate in the parasite, μ
p
, to that in the host, μ
h
, since a function of this ratio gives the lag load, LL (i.e. LL = f (μ
p
/μ
h
)). As noted in the Introduction, a lag load provides opportunities for continuous adoption of advantageous mutations by the host [3, 9].
Data are shown in Figures 2(a-d) for a range of lag loads generated in two populations (N = 100, 3000) as parasite mutation rates increase at fixed host mutation rates (Figures 2(a,c), μ
h
= 10-6 bits/allele/generation; Figures 2(b,d), μ
h
= 10-3 bits/allele/generation). Adaptation scores decline as parasite mutation rates increase relative to those of the host (i.e. μ
p
/μ
h
increases). The average adaptation scores of the sexual populations (black curves) are always higher than those of the clonal populations (red curves) under equivalent conditions.
The long-run steady state values of population adaptation that arise at low host mutation rates (10-5 or less) are largely independent of the absolute values of the mutation rates μ
p
and μ
h
, but are a function of the ratio, μ
p
/μ
h
. When μ
p
/μ
h
~ 1, both sexual and clonal populations are essentially monomorphic (effective allele number = ~ 1.0), but as μ
p
/μ
h
approaches 100 or more, sexual populations remain close to monomorphic, whereas clones become dimorphic (effective allele number = ~2.0). Supporting data is shown in Figure 3. What is seen in Figures 2(a) and (c), therefore, is a straight Kirkpatrick and Jenkins (KJ) rate advantage to sex emerging as lag loads increase (that is, ratio of effective allele number, clonal/sexual tends to 2). Large lag loads are needed because these loads indicate maladaptation, and that, in turn, increases the target size for adoption of advantageous mutations at a particular locus. This dilutes the chances that an advantageous mutation will convert a clonally heterozygote position to homozygosity. The progressive loss of lag load sees a reduction in the size of the mutation target and an increase in the chance of heterozygotes converting to homozygotes (reflected in trend of effective allele number to 1). When host matching of the environment reaches a maximum, and lag loads are very small, the lower fecundity of sexual reproduction greatly reduces its fitness relative to clones. The data in Figure 2(c) show an adaptation advantage to sex of ~2.3 when μ
h
= 10-6 and μ
p
= 10-4, which attenuates as μ
p
progressively declines. Figures 2(b) and (d) show a second effect emerging as host mutation rates are increased. For a host mutation rate in Figure 2(d) of 10-3, the parasite mutation rates needed to produce the same lag loads as μ
p
at 10-4 in Figure 2(c) are 10-2 for sex and ~3 x 10-3 for clones, an ~10-fold reduction in μ
p
/μ
h
for sex and ~35-fold reduction of μ
p
/μ
h
in clones. At the same time, the number of alleles rises to 18.9 for the sexual population and 20.2 for the clonal, from 1.00 and 2.01 respectively when μ
h
= 10-6 under similar lag loads. This large increase in polymorphism with increase in host mutation rate markedly interferes with clearance of maladapted alleles in clonal populations. We return to this point in presenting the data that underpins Figures 4(a-d).
So far, we have shown data for adaptation scores. Fitness, in this simulation, is a multiplicative function of adaptation and fecundity, and the two-fold cost of males in sexual reproduction is reflected as a two-fold fecundity difference between clonal and sexual populations. The green curves in Figures 2(b) and (d) represent the fitness of sexual populations relative to clonal populations obtained by halving the adaptation scores of sexual populations; clonal fitness is represented by the solid adaptation curve in red, since the fecundity factor is 1.0 relative to 0.5 for sexual reproduction. These data show that sexual populations are less fit than clonal ones at small lag loads, but a transition point exists past which sexual fitness exceeds clonal fitness. This transition point represents a parasite mutation rate, μ
p
, at which, for a given host mutation rate, μ
h
, the parasite environment causes an equal loss of fitness in both populations; above this parasite mutation rate, clonal populations are less fit than sexual ones. Importantly, this cross-over point sits within a not-implausible range of lag loads. When the same approach is adopted for data in Figure 2(a), there is effectively no cross-over point, and for data in Figure 2(c), it applies at relatively high lag loads. We conclude from these data that the KJ effect at low mutation rates, both μ
p
and μ
h
, allows sexual populations to go a considerable way in matching the fitness advantage of clonal populations, subject to the existence of adequate lag loads.
Emerging adaptation differences: understanding the mechanism
In this section, we provide data that show how, in part, the effects described previously arise. First, we extract information from the simulation about allele lifetimes and allele frequencies during the course of a simulation. For technical reasons related to the tractability of computation, these data are derived from diploid genotypes with three loci, each of 2 × 16 bits. Figures 5(a) and (b) show typical profiles for sexual and clonal populations, respectively, under middle-range lag loads (adaptation scores for individual loci: sex, 0.707; clone, 0.548) using mutation rates, μ
h
, per locus of 10-4 bits/allele/generation for hosts, and 10-2 bits/allele/generation for parasites; N = 3000. Sexual reproduction at these mutation settings is characterized by a series of short-lived allele sweeps that mostly move towards temporary homozygosity (effective allele number = ~2). Occasionally, the population supports two alleles simultaneously at any given locus. Lowering mutation rates without shifting the ratio, μ
p
/μ
h
, between parasite and host extends allele lifetime and effective allele number tends to 1.0. In all cases, alleles come into existence and then die, rather than cycle. This behaviour reflects the size of the allele landscape being used. The non-cycling behaviour of these alleles is in contrast to those cited in [20].
Clonal populations show significantly different allele behaviour under the same conditions. Here, maximum allele frequencies are capped at 0.5 as lag load develops and effective allele number tends to 9.0. As with sexual reproduction, lowering mutation rates without shifting the ratio, μ
p
/μ
h
, extends allele lifetime without lifting the frequency cap. We argue the explanation for the emergence of the cap is as follows. An average clonal adaptation score of 0.55 per locus (see previous paragraph) represents a mismatch of approximately 14 bits for a diploid locus of 2 × 16 = 32 bits. If the one-bit components of the locus are homozygous (either 0,0 or 1,1) and adopt an advantageous mutation, they move to heterozygosity (0,1 or 1,0). The next advantageous mutation to undergo adoption by the locus has no better than a 1-in-14 chance of occurring at the heterozygotic bit in question. The chance of adoption is actually lower than that, as we show shortly, because alleles start losing adaptation as soon as they have been adopted. Taken together, the low chance of adopting a second mutation at the heterozygote locus and the deterioration in the adaptation of its carrying allele make its chances of moving to homozygosity effectively zero, and this is borne out by the data. The cap on the 0.5 frequency rises as the parasite mutation rate is progressively lowered (through 10-3, 10-4, 10-5, etc.) relative to the host (that is, as the value of μ
p
/μ
h
falls, because the average adaptation score of the clonal population increases, as does the chance of adopting a heterozygote-converting advantageous mutation.
Histograms of allele frequency distribution are shown in Figure 3. At the low μ
p
of 10-6 bits/allele/generation, with μ
h
= 10-4 bits/allele/generation, the allele distributions of sexual and clonal populations are essentially identical, and both populations are almost entirely homozygotic (that is, effective allele number is close to 1.0). This reflects the effect of stabilizing selection in a near-constant environment. When the environment changes more rapidly, the average adaptation of the population falls. The precise level of the fall depends on the nature of the environmental change. A random walk has almost no effect, whereas NFDS imposes a lag load whose size depends on the ratio μ
p
/μ
h
and the mode of reproduction (sexual or clonal). Where significant lag loads emerge (using μ
p
of 10-2 in this particular case), there is capping of the highest clonal frequency. This result is an inevitable consequence of the KJ effect under significant lag load. It has the important effect of enforcing greater polymorphism (strictly, a higher effective allele number) on the clonal population than the sexual population, thereby creating a more substantial opportunity for negative interactions between co-alleles at single loci. The effect is independent of absolute mutation rates, but rests on a suitable value of μ
p
/μ
h
.
Allele lifetimes and declines in adaptation scores
Here we present the data that provide evidence of greater relative interference with selection in clones as the number of loci increase. These data fall into two sections. First, we compare average allele lifetimes in sexual and clonal populations, where there are either one or three loci in contention (Figures 6(a-d). The data for the sexual populations show effectively no difference between the distribution of allele lifetimes at a single locus (Figure 6(a)) and three loci (Figure 6(b)), nor rates of decline of adaptation scores. Figure 6(c) shows the average allele lifetime behaviour for a single clonal locus under the same conditions. The maximum allele frequency is much less, due to the capping effect at significant lag loads, and there is already evidence of interference with selection, as evidenced by the long tail for allele frequency against time. The rate of decline of the adaptation score is also slower. We know from the data on single loci in sexual reproduction that declines in adaptation score can be much faster. We attribute the slower decline in clonal reproduction to increasing polymorphism that, in turn, spreads the effects of NFDS across a larger number of different alleles. This reduces differences in selection pressure on different clonal genotypes and slows elimination of individual allele species. The argument is supported by the lower adaptation score in clonal alleles on starting, which indicates that the parasite population already has wider surveillance of the host population than its equivalent in the sexual populations. As a consequence, selection pressure on individual alleles is reduced and the effect of polymorphism is enhanced further. The slower timeline of the decline in adaptation score also increases the likelihood that a novel advantageous mutation will occur in an allele of declining adaptation, with a co-allele whose adaptation is also declining. These data support the existence of interference between alleles at single clonal loci.
Figure 6(d) shows the effects of coupling in a clonal three-locus genotype. This extends further the average lifetime of alleles and increases the effective allele number. The longer tail when compared with Figure 6(c) is due to a combination of intra-locus and inter-locus interference with selection, and greater spreading of the clonal targets facing parasites deploying NFDS. The starting adaptation score is now lower than in Figure 6(c) because inter-locus interference has increased effective allele number still further. The obligatory coupling that occurs in clones increases the probability that a beneficial mutation will occur in a poorly adapted background. The effect increases with the number of loci that are coupled, because overall effective allele number increases as a power function of the effective allele number at single loci, as well as contributing additional interference. The existence of multiple loci has relatively little effect in sexual reproduction if the loci segregate independently. This underpins Fisher’s explanation for the relative advantage of sex [12].
The data also show that the average initial rate at which clonal alleles sweep is essentially the same as sexual alleles for those alleles that survive drift. This addresses and eliminates another possible source of difference between sexual and clonal reproduction.
Optimizing host mutation rates and the effects of polymorphism and locus number
This section describes the outcome when data, used in Figure 2, on the interaction between host and parasite mutation rates, are redrawn in Figure 4 to show the response of host populations to fixed parasite mutation rates. We extend these data to include examples of the effect of locus number.
All four graphs in Figure 4 show that host populations have higher fitness when their mutation rates approach those set by the parasite. These data confirm that parasites that mutate rapidly require high rates of mutation in host defences if host populations are to optimize their own fitness. Figure 4(a) shows that clonal and sexual populations progressively approach an adaptation score that is close to unity for both populations as their mutation rates rise. There is a small mutational load at these mutation rates under the conditions of the simulation, and this mutational load can be increased as selection is relaxed. The difference in adaptation scores between clonal and sexual populations at large lag loads is attributable to the Kirkpatrick and Jenkins (KJ) effect [8], covered in previous sections of these Results.
Figure 4(b) shows the effect of increasing parasite mutation rate by 10-fold. The host now shows adaptation optima at higher host mutation rates, as expected. The increased gap that emerges between clonal and sexual population fitness can be attributed, in the first instance, to higher polymorphism and an increase in clonal interference. However, as already discussed, the parasite NFDS response becomes spread over a number of genotypes as polyclonality increases, and this spread reduces differences in selective pressure on genotypes and, indirectly, any given allele. It slows the ability of the clonal population to respond to high parasite mutation rates.
When the number of host loci is increased at the lower parasite mutation rate used in Figure 4 (10-3), there is a small difference in maximum adaptation scores between clonal and sexual populations that is consistent. Intermediate parts of the curves show a greater disparity in adaptation between sexual and clonal populations than in Figure 4(a). In both cases, the gap is likely to reflect the emergence of clonal interference and its consequences for NFDS.
Figure 4(d) illustrates the outcome with a higher mutation rate (10-2) and a larger number of loci (20). There is a marked collapse in the adaptation scores of the clonal population as effective allele number rises. The KJ effect seems likely to be a significant contributor at large lag loads, where maladaptation is substantial. The picture is likely to be more complex where host populations are better adapted because of higher host mutation rates. These higher rates, in clones, would be expected to produce greater polymorphism, higher levels of heterozygosity, and weaker selection on different genotypes. Whatever the reason, sexual reproduction is now fitter over the entire range of host mutation rates.
Testing Maynard Smith
Maynard Smith illustrated the two-fold cost of males in sexual reproduction using an example in which a sexual mother was converted to a clonal mother by mutation [3]. The effect of the conversion is that the clonal mother, immediately after conversion, has the adaptation she enjoyed as a sexual mother, but now has twice the fecundity, and therefore twice the fitness. Left to her own devices, she and her descendants will rapidly outbreed the sexual population and drive it to extinction. We have shown in simulations of separate sexual and clonal populations that a clonal population can experience a much greater loss of adaptation than a sexual population under the same conditions, and this disparity provides a mechanism by which sex can, in principle, prevent invasion by clones. In our computational model, the clone is chosen from the pool of best-adapted sexual mothers. This gives the clone an adaptation score that, at steady state, is far higher than a clonal mother would normally enjoy. The outcome of inserting a clone is an empirical matter, however, and rests on the rate at which the parasite environment causes loss of adaptation in the host. Too slow a response from the parasite and the clonal population expands rapidly at little cost. Figure 7 shows data for three populations (N = 102,103,104), where the numerical value of μ
p
/μ
h
(= 1.0) is sufficient to give sex a fitness advantage under group selection at steady state. At least two predictions follow from this brief outline. First, the parasite mutation rate, μ
p
, needs to be above some threshold that causes rapid deterioration in the environment, and second, the outcome should show some sensitivity to population size, since larger populations provide for a larger number of generations for NFDS to extinguish the descendants of a clone. Because the sexual female chosen for conversion to a clone is the best adapted, it inevitably means she has just been introduced into the simulation; she is rare and without a negative frequency-dependent response from the parasite. This frequency-dependent response has to develop quickly if the clone and its descendants are to undergo elimination. Larger host populations provide a greater number of generations within which to generate this negative frequency-dependent response. Both these predictions are borne out by the data in Figure 7.