Fixation probability of a beneficial mutation conferring decreased generation time in changing environments

Background

One central building block of population genetics is the fixation probability. It is a probabilistic understanding of the eventual fate of new mutations. Moreover, the fixation probability of new beneficial mutations plays an important effect on the adaptation of populations to environmental challenges. Great progress has been made in the study of the beneficial mutations that increases offspring number. However, the fixation probability of beneficial mutations with a shorter generation time under various genetic and ecological conditions has not been explored.

Results

Here we extend the classical result of the fixation probability of beneficial mutations obtained by Haldane, and estimate the fixation probability of a beneficial mutation with a reduced generation time in a changing environment. Assuming that the selective advantage is very small, we concentrate all the changing factors of environment on a single quantity: effective selective advantage. Using a time-dependent branching process, we get the analytic approximation for the fixation probability of beneficial mutations that decrease the generation time. Then, we apply this approximation to four interesting biological cases.

Conclusions

In these instances, we show the comparison of the approximation with the accurate values. We find that they are consistent, demonstrating the effectiveness of our result for the fixation probability of beneficial mutations conferring a reduced replication time.

     

Fixation probability and time in subdivided populations

New alleles arising in a population by mutation ultimately are either fixed or lost. Either is possible, for both beneficial and deleterious alleles, because of stochastic changes in allele frequency due to genetic drift. Spatially structured populations differ from unstructured populations in the probability of fixation and the time that this fixation takes. Previous results have generally made many assumptions: that all demes contribute to the next generation in exact proportion to their current sizes, that new mutations are beneficial, and that new alleles have additive effects. In this article these assumptions are relaxed, allowing for an arbitrary distribution among demes of reproductive success, both beneficial and deleterious effects, and arbitrary dominance. The effects of population structure can be expressed with two summary statistics: the effective population size and a variant of Wright's F(ST). In general, the probability of fixation is strongly affected by population structure, as is the expected time to fixation or loss. Population structure changes the effective size of the species, often strongly downward; smaller effective size increases the probability of fixing deleterious alleles and decreases the probability of fixing beneficial alleles. On the other hand, population structure causes an increase in the homozygosity of alleles, which increases the probability of fixing beneficial alleles but somewhat decreases the probability of fixing deleterious alleles. The probability of fixing new beneficial alleles can be simply described by 2hs(1 - F(ST))N(e)/N(tot), where hs is the change in fitness of heterozygotes relative to the ancestral homozygote, F(ST) is a weighted version of Wright's measure of population subdivision, and N(e) and N(tot) are the effective and census sizes, respectively. These results are verified by simulation for a broad range of population structures, including the island model, the stepping-stone model, and a model with extinction and recolonization.     

Fixation probabilities depend on life history: fecundity, generation time and survival in a burst-death model

The burst-death model has been developed to describe the life history of organisms with variable generation times and a burst of a fixed number of offspring. The model also includes an optional constant clearance rate, such as washout from a chemostat, and the possibility of sustained periods of population growth followed by severe bottlenecks, as in serial passaging. In this model, a beneficial mutation can either increase the burst rate or the burst size, or reduce the clearance rate, thus increasing survival. In this article we examine the effects of these three possible mechanisms on both the Malthusian fitness and the fixation probability of the lineage. We find that equivalent relative increases in the burst rate or burst size confer equivalent increases in the Malthusian fitness of a lineage, whereas increasing survival typically has a more moderate effect on Malthusian fitness. In contrast, for beneficial mutations that confer the same increase in fitness, mutations that increase survival are the most likely to fix, followed by mutations that increase the burst rate. Mutations that increase the burst size are the least likely to fix. These results imply that mutant lineages with the highest Malthusian fitness are not, in many cases, the most likely to escape extinction.     

Choosing probability distributions for modelling generation time variability

This paper explores the variation in generation time of bacterial systems growing at suboptimal temperatures. Generation time generally has a distribution with a long right-hand tail, suggesting a model with variance proportional to the second or third power of its mean. Suitable non-normal probability distributions include the 'gamma' and 'inverse Gaussian', with the modelling being carried out by 'generalized linear regression'. The procedure is illustrated with replicated data on Pseudomonas fluorescens obtained using a gradient temperature incubator with nutrient broth as the growth medium. The results show that the 'gamma' distribution is a suitable stochastic assumption when modelling generation time. This enables one to predict, for example, a mean generation time of 615 min at 2·4°C, and that 0·1% of the observed values will fall below 471 min and one in a million below 405 min. Use of an unreplicated set of data gave less conclusive results but favoured the 'inverse Gaussian' distribution as the stochastic model.     

Metapopulation structure favors plasticity over local adaptation

We describe a model for the evolutionary consequences of plasticity in an environmentally heterogeneous metapopulation in which specialists for each of two alternative environments and one plastic type are initially present. The model is similar to that proposed by Moran (1992) but extends her work to two sites. We show that with migration between sites the plastic type is favored over local specialists across a broad range of parameter space. The plastic type may dominate or be fixed even in an environmentally uniform site, and even if the plasticity has imperfect accuracy or bears some cost such that a local specialist has higher fitness in that site, as long as there is some migration between sites with different distributions of environmental states. These results suggest that differences among taxa in dispersal and hence realized migration rates may play a heretofore unrecognized role in their patterns of adaptive population differentiation. Migration relaxes the thresholds for both environmental heterogeneity and accuracy of plastic response above which plasticity is favored. Furthermore, small changes in response accuracy can dramatically and abruptly alter the evolutionary outcome in the metapopulation. A fitness cost to plasticity will substantially reduce the range of conditions in which the plastic type will prevail only if the cost is both large and global rather than environment specific.     

High herbivore pressure favors constitutive over induced defense

Theoretical and empirical studies show that, when past or current herbivory is a reliable cue of future attack and defenses are costly, defenses can be induced only when needed and thereby permit investment in other functions such as growth or reproduction. Theory also states that, in environments where herbivory is constantly high, constitutive defenses should be favored. Here, we present data to support the second aspect of the induced resistance hypothesis. We examined herbivore‐induced responses for four species of Inga (Fabaceae), a common canopy tree in Neotropical forests. We quantified chemical defenses of expanding leaves, including phenolic, saponin and toxic amino acids, in experimental field treatments with and without caterpillars. Because young leaves lack fiber and are higher in protein than mature leaves, they typically lose >25% of their leaf area during the few weeks of expansion. We predicted that the high rates of attack would select for investment in constitutive defenses over induction. Our data show that chemical defenses were quite unresponsive to herbivory. We demonstrated that expanding leaves showed no or only small increases in investment in secondary metabolites, and no qualitative changes in the phenolic compound profile in response to herbivory. The proteinogenic amino acid tyrosine, which can be toxic at high concentrations, showed the greatest levels of induction. Synthesis: These results provide some of the first support for theoretical predictions that the evolution of induced vs. constitutive defenses depends on the risk of herbivory. In habitats with constant and high potential losses to herbivores, such as tropical rainforests, high investments in constitutive defenses are favored over induction.     

Fixation probability in spatially changing environments.

The fixation probability of a mutant in a subdivided population with spatially varying environments is investigated using a finite island model. This probability is different from that in a panmictic population if selection is intermediate to strong and migration is weak. An approximation is used to compute the fixation probability when migration among subpopulations is very weak. By numerically solving the two-dimensional partial differential equation for the fixation probability in the two subpopulation case, the approximation was shown to give fairly accurate values. With this approximation, we show in the case of two subpopulations that the fixation probability in subdivided populations is greater than that in panmictic populations mostly. The increase is most pronounced when the mutant is selected for in one subpopulation and is selected against in the other subpopulation. Also it is shown that when there are two types of environments, further subdivision of subpopulations does not cause much change of the fixation probability in the no dominance case unless the product of the selection coefficient and the local population size is less than one. With dominance, the effect of subdivision becomes more complex.     

Short communication. Determination of generation time and asexual fecundity of doliolids (Tunicata, Thaliacea)

The goal of this study was to provide temporal information on the generation time of Dolioletta gegenbauri and some of the life cycle's components. At 20C and 90 g C l^-1 of ingestible phytoplankton. D.gegenbauri's life cycle is completed in 20.5 days. Phorozooids 5 mm produce on average 11.0 gonozooids day^-1 over a period of 8-18 days. Utilizing field data on the abundance and size distribution of an assemblage of phorozooids and nurses, in conjunction with experimentally obtained rates, indicates that asexual production per cubic meter by phorozooids with that of nurses should result in rapid colonization of a wide shelf by doliolids, as observed during July and August 1981 on the southeastern continental shelf of the USA.      

Sexual conflict over mating in a spider: increased fecundity does not compensate for the costs of polyandry

Female multiple mating (polyandry) is a widespread but costly behavior that remains poorly understood. Polyandry may arise when whatever benefits females accrue from multiple mating outweigh the costs, or males manipulate females against the females' best interests. In a polyandrous spider Stegodyphus lineatus females may mate with up to five males, but behave aggressively toward additional males after the first mating. Female aggressiveness may act to select for better quality males. Alternatively, females may try to avoid superfluous matings. To test these alternatives, we allocated females into single-mating (SM) and double-mating treatments. Double-mated females either accepted (DM) or rejected (RE) the second male. DM females laid more eggs, but did not produce more offspring than SM and RE females. Offspring of DM females were smaller at dispersal than offspring of SM and RE females. Also, nest failure was significantly more common in DM females. Paternal variables did not influence female reproductive success, whereas maternal body condition explained much of the variation. We show that polyandry is costly for females despite the production of larger clutches and suggest that multiple mating results from male manipulation of female remating behavior.     

Fixation probability for lytic viruses: the attachment-lysis model

The fixation probability of a beneficial mutation is extremely sensitive to assumptions regarding the organism's life history. In this article we compute the fixation probability using a life-history model for lytic viruses, a key model organism in experimental studies of adaptation. The model assumes that attachment times are exponentially distributed, but that the lysis time, the time between attachment and host cell lysis, is constant. We assume that the growth of the wild-type viral population is controlled by periodic sampling (population bottlenecks) and also include the possibility that clearance may occur at a constant rate, for example, through washout in a chemostat. We then compute the fixation probability for mutations that increase the attachment rate, decrease the lysis time, increase the burst size, or reduce the probability of clearance. The fixation probability of these four types of beneficial mutations can be vastly different and depends critically on the time between population bottlenecks. We also explore mutations that affect lysis time, assuming that the burst size is constrained by the lysis time, for experimental protocols that sample either free phage or free phage and artificially lysed infected cells. In all cases we predict that the fixation probability of beneficial alleles is remarkably sensitive to the time between population bottlenecks.     

Fixation probability of rare nonmutator and evolution of mutation rates

Although mutations drive the evolutionary process, the rates at which the mutations occur are themselves subject to evolutionary forces. Our purpose here is to understand the role of selection and random genetic drift in the evolution of mutation rates, and we address this question in asexual populations at mutation‐selection equilibrium neglecting selective sweeps. Using a multitype branching process, we calculate the fixation probability of a rare nonmutator in a large asexual population of mutators and find that a nonmutator is more likely to fix when the deleterious mutation rate of the mutator population is high. Compensatory mutations in the mutator population are found to decrease the fixation probability of a nonmutator when the selection coefficient is large. But, surprisingly, the fixation probability changes nonmonotonically with increasing compensatory mutation rate when the selection is mild. Using these results for the fixation probability and a drift‐barrier argument, we find a novel relationship between the mutation rates and the population size. We also discuss the time to fix the nonmutator in an adapted population of asexual mutators, and compare our results with experiments.     

Paternity costs from polyandry compensated by increased fecundity in the hide beetle

Polyandry-induced sperm competition is assumed to impose costson males through reduced per capita paternity success. In contrast,studies focusing on the consequences of polyandry for femalesreport increased oviposition rates and fertility. For thesespecies, there is potential for the increased female fecundityassociated with polyandry to offset the costs to males of sharedpaternity. We tested this hypothesis by comparing the proportionand number of offspring sired by males mated with monandrousand polyandrous females in the hide beetle, Dermestes maculates,both for males mating with different females and for males rematingwith the same female. In 4 mating treatments, monandrous femalesmated either once or twice with the same male and polyandrousfemales mated either twice with 2 different males or thricewith 2 males (where 1 male mated twice). Polyandrous and twice-matingmonandrous females displayed greater fecundity and fertilitythan singly mating monandrous females. Moreover, males rematedto the same female had greater paternity regardless of whetherthat female mated with another male. In both polyandrous treatments,male mating order did not affect paternity success. Finally,although the proportion of eggs sired decreased if a male matedwith a polyandrous female, multiply mating females or femalesthat remated with a previous mate laid significantly more eggsand thus the actual number of eggs sired was comparable. Thus,males do not necessarily accrue a net fitness loss when matingwith polyandrous females. This may explain the absence of anyobvious defensive paternity-protection traits in hide beetlesand other species.     

Fixation probability in a two-locus model by the ancestral recombination-selection graph

We use the ancestral influence graph (AIG) for a two-locus, two-allele selection model in the limit of a large population size to obtain an analytic approximation for the probability of ultimate fixation of a single mutant allele A. We assume that this new mutant is introduced at a given locus into a finite population in which a previous mutant allele B is already segregating with a wild type at another linked locus. We deduce that the fixation probability increases as the recombination rate increases if allele A is either in positive epistatic interaction with B and allele B is beneficial or in no epistatic interaction with B and then allele A itself is beneficial. This holds at least as long as the recombination fraction and the selection intensity are small enough and the population size is large enough. In particular this confirms the Hill-Robertson effect, which predicts that recombination renders more likely the ultimate fixation of beneficial mutants at different loci in a population in the presence of random genetic drift even in the absence of epistasis. More importantly, we show that this is true from weak negative epistasis to positive epistasis, at least under weak selection. In the case of deleterious mutants, the fixation probability decreases as the recombination rate increases. This supports Muller's ratchet mechanism to explain the accumulation of deleterious mutants in a population lacking recombination.     

Shorter life and reduced fecundity can increase colony fitness in virtual Caenorhabditis elegans

In the nematode Caenorhabditis elegans, loss of function of many genes leads to increases in lifespan, sometimes of a very large magnitude. Could this reflect the occurrence of programmed death that, like apoptosis of cells, promotes fitness? The notion that programmed death evolves as a mechanism to remove worn out, old individuals in order to increase food availability for kin is not supported by classic evolutionary theory for most species. However, it may apply in organisms with colonies of closely related individuals such as C. elegans in which largely clonal populations subsist on spatially limited food patches. Here, we ask whether food competition between nonreproductive adults and their clonal progeny could favor programmed death by using an in silico model of C. elegans. Colony fitness was estimated as yield of dauer larva propagules from a limited food patch. Simulations showed that not only shorter lifespan but also shorter reproductive span and reduced adult feeding rate can increase colony fitness, potentially by reducing futile food consumption. Early adult death was particularly beneficial when adult food consumption rate was high. These results imply that programmed, adaptive death could promote colony fitness in C. elegans through a consumer sacrifice mechanism. Thus, C. elegans lifespan may be limited not by aging in the usual sense but rather by apoptosis‐like programmed death.     

The increased flexibility of CDR loops generated in antibodies by Congo red complexation favors antigen binding

The dye Congo red and related self-assembling compounds were found to stabilize immune complexes by binding to antibodies currently engaged in complexation to antigen. In our simulations, it was shown that the site that becomes accessible for binding the supramolecular dye ligand is located in the V domain, and is normally occupied by the N-terminal polypeptide chain fragment. The binding of the ligand disrupts the beta-structure in the domain, increasing the plasticity of the antigen-binding site. The higher fluctuation of CDR-bearing loops enhances antigen binding, and allows even low-affinity antibodies to be engaged in immune complexes. Experimental observations of the enhancement effect were supported by theoretical studies using L lambda chain (4BJL-PDB identification) and the L chain from the complex of IgM-rheumatoid factor bound to the CH3 domain of the Fc fragment (1ADQ-PDB identification) as the initial structures for theoretical studies of dye-induced changes. Commercial IgM-type rheumatoid factor (human) and sheep red blood cells with coupled IgG (human) were used for experimental tests aimed to reveal the dye-enhancement effect in this system. The specificity of antigen-antibody interaction enhanced by dye binding was studied using rabbit anti-sheep red cell antibodies to agglutinate red cells of different species. Red blood cells of hoofed mammals (horse, goat) showed weak enhancement of agglutination in the presence of Congo red. Neither agglutination nor enhancement were observed in the case of human red cells. The dye-enhancement capability in the SRBC-antiSRBC system was lost after pepsin-digestion of antibodies producing (Fab)2 fragments still agglutinating red cells. Monoclonal (myeloma) IgG, L lambda chain and ovoalbumin failed to agglutinate red cells, as expected, and showed no enhancement effect. This indicates that the enhancement effect is specific.