How do lethal alleles remain in a population

Interestingly, the balance between positive and negative selection would explain a rapid increase in the population under strong selection for growth traits, as has been the case for pig breeding lines at least since the introduction of modern breeding techniques.

This increase in growth rate with the most significant effect in the test period from 25— Kg and feed intake, likely results from a heterozygous loss-of-function of the BBS9 gene.

Loss-of-function mutations in human BBS9 and other members of the BBSome cause Bardet-Biedl syndrome, associated with a series of clinical features including obesity, renal anomalies, and retinopathy, with the obese phenotype as one of the key features of Bardet-Biedl syndrome patients. Studies have hypothesized that cilia defects are likely to affect feeding and satiety, causing an increased appetite and lack of satiation [ 29 ].

In addition, heterozygous carriers of a mutant BBS allele in humans show increased levels of obesity, without showing any of the other Bardet-Biedl syndrome features [ 31 ], analogous to the observed phenotype in carriers of the BBS9 deletion.

Mouse null-mutants in genes that form the BBSome complex have been associated with similar phenotypic features including obesity, lower birth weights, and partial embryonic lethality [ 32 , 33 ], again supporting the BBS9 role in increased growth rate, and the lower birth weight. We cannot completely exclude, however, that other genomic factors, in high LD with the kb deletion, contribute to the observed phenotype as well.

The question remains which gene or regulatory element is causal for the early death of homozygous individuals. Enhancers are important drivers of transcription and loss of enhancer elements can lead to decreased expression of the associated gene.

Naturally occurring knock-outs of the BBS9 gene does not result in fetal lethality in human [ 18 ] and is therefore not likely to be causal for the lethal phenotype.

The lower expression of the BMPER gene is supported by allele specific expression for the non-deletion haplotype in carriers, not observed in individuals only carrying wild-type haplotypes. BBS4 [ 33 ] and BBS7 [ 36 ] , we can therefore not exclude the possibility that a complete loss of a functional BBS9 protein contributes to the early lethality as well. This work describes a striking example of balancing selection in pigs, maintaining a recessive lethal allele that shows pleiotropic effects on fertility and growth traits at moderate frequency in the population.

Other examples in pigs include the Porcine Stress and Pale Soft Exudative Meat Syndrome, caused by a homozygous missense mutation in the RYR1 gene, while heterozygotes show increased muscle mass [ 37 ]. Moreover, several instances of balancing selection have been described in domestic cattle breeds among which a kb deletion causing embryonic lethality in homozygotes, while having increased milk yield in heterozygotes [ 5 , 15 ].

Identifying balancing selection on lethal alleles can be challenging, as the only consequence observed is a somewhat lower fertility in the parental animals, lacking affected liveborn individuals.

We expect that this type of balancing selection might be more prevalent within pig populations than previously thought, especially for the somewhat higher frequency lethal alleles, which are less likely to be purely the result of drift effects. Moreover, the relatively subtle effects found in this study could only be made apparent because phenotypic data derived from a very large number of pigs was available. In this study we report a kb deletion with antagonistic effects on fertility and growth.

We show that homozygotes for the deletion die around mid- to late-gestation, becoming mummified. Compared to other lethal alleles identified in this population, the deletion seems to be maintained at moderate frequency This moderate carrier frequency is likely not a result of random drift effects, as heterozygotes for the deletion-haplotype show, despite a lower birth weight, increased growth rate, and feed intake, important traits in the breeding goal.

The balancing scenario observed, most likely, is a consequence of pleiotropic effects of the deletion on two different genes affecting fertility BMPER and growth BBS9. The large amount of genotype data accumulating in modern breeding schemes applying genomic selection in combination with the large amount of phenotypic data deliver a powerful tool to monitor and control deleterious alleles much more efficiently.

Samples collected for DNA extraction were only used for routine diagnostic purpose of the breeding programs, and not specifically for the purpose of this project. Therefore, approval of an ethics committee was not mandatory. Sample collection and data recording were conducted strictly according to the Dutch law on animal protection and welfare Gezondheids- en welzijnswet voor dieren.

The dataset consists of 23, purebred Large White animals. We discard markers that did not meet following filtering criteria: A minimum call rate of 0. Moreover, markers with unknown location on the Sscrofa All steps were performed in Plink v1. We performed haplotype phasing and imputation of missing sites in Beagle4.

Reference and test phased VCF files were merged using bcftools 1. We tested the SS18 haplotype for the expected number of homozygotes using both parents haplotype information sire, and dam with the formula described in Fritz et al.

An exact binomial test was applied to test the number of observed homozygotes with the number of expected homozygotes. Next, we applied logistic regression to distinguish carrier from non-carrier animals using the sci-kit learn Python library [ 44 ] S6 Fig.

The dataset consists of 73 whole genome sequenced Large White individuals with a total volume of 1. The data was sequenced on Illumina Hiseq We used sickle software for quality trimming of raw reads. Next we aligned the sequences to the Sscrofa Samtools dedup function was used to remove PCR duplicates [ 41 ].

Variant calling was performed with Freebayes v1. Variants were annotated using the Ensembl variant effect predictor VEP, release 90 [ 17 ]. The impact of missense variants was predicted using SIFT [ 48 ]. The sequenced population was phased using Beagle4. Analysis on structural variation SV was performed using Lumpy with default settings [ 49 ], taking the aligned BAM files as input.

Alignments and SV events were visualized using the JBrowse genome viewer version 1. In addition, we analyzed two other pigs from Duroc, and Pietrain genetic background on five different tissues. RNA-seq reads were aligned to the Sscrofa We applied the following steps to examine allele specific expression: First, samtools [ 41 ] was used to extract uniquely mapped reads from the BAM alignment files.

Next, WASP [ 53 ] was used to reduce the mapping reference sequence bias. Visual examination of the alignments and transcripts was performed in JBrowse [ 50 ]. RT-qPCR was started with: 3. Reaction was performed as follows:. All primers and results are listed in S12 and S13 Tables.

We analyzed the frequency of the SSC18 haplotype per half-year starting from jul We assessed the frequency based on total population live animals on each time point by looking at the proportion of carrier and non-carrier animals in the population.

The number of animals per time point are provided in S8 Table. In this study, we evaluated 16 traits used in the Large White breeding program. Deregressed estimated breeding values DEBV were used as a response variable for each trait under study. The estimated breeding value EBV was separately deregressed for each trait using the methodology described by Garrick et al [ 54 ]. The EBV of each animal was obtained from the routine genetic evaluation by Topigs Norsvin using an animal model.

The heritabilities used for the deregression were also extracted from the routine genetic evaluation. Parent average effects were also removed as part of the deregression process to obtain more accurate estimates of the genetic merit of each individual. Finally, weighting factors based on the estimated reliability of the DEBV were also estimated according to Garrick et al [ 54 ] using a value of 0. We simulated changes in allele frequency across multiple populations under the model of Wright [ 57 ].

Each allele is associated with a fitness, and we set the fitness to zero for homozygotes for lethal recessive allele and fitness to 1 no negative fitness effect for carriers and non-carriers. We assume constant population size through time, and matings are simulated randomly at each generation. We first calculated the average TSI, and estimated breeding values for six important traits in the breeding line S11 Table. Next, we used the ratio of carrier TSI over non-carrier TSI to estimate the selective advantage in the breeding program.

Next, we simulated the long-term allele frequency change assuming random matings based on the selective advantage, and the loss of homozygous animals using the Hardy-Weinberg principle. Similar analysis was performed using the selective advantage on growth exclusively.

C: X-ray of mummified piglet status unknown. The authors thank Meenu Bhiati for useful input on the allele specific expression analysis. We thank Michel Georges and Carole Charlier for useful comments on this work, Remco Pieters for generating the X-rays, and Richard Crooijmans for library preparation and lab work. This research was funded by the STW-Breed4Food Partnership, project number From sequence to phenotype: detecting deleterious variation by prediction of functionality.

The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. National Center for Biotechnology Information , U. PLoS Genet. Published online Sep Martijn F. Marcos S. Martien A. Tosso Leeb, Editor.

Author information Article notes Copyright and License information Disclaimer. All authors declare that the results are presented in full and as such present no conflict of interest. Received Apr 24; Accepted Aug This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. This article has been cited by other articles in PMC.

S2 Fig: Screen capture of the deletion boundary in the JBrowse genome browser. S3 Fig: LRR signal intensities within the haplotype region for two "fresh born" homozygotes of the kb deletion. S6 Fig: Logistic regression to distinguish carrier from non-carrier animals farm 2 litters. S8 Fig: Genetic progress for growth daily gain in the Large White breed. S10 Fig: Simulation of the SSC18 carrier frequency if selection would be exclusively applied on growth.

S3 Table: Markers and genomic positions. S4 Table: Haplotypes from the four tracked carrier-by-carrier matings, including parent animals. S8 Table: Deletion carrier frequency and the number of genotyped animals per time point from — Abstract Livestock populations can be used to study recessive defects caused by deleterious alleles. Author summary We report a large deletion within the BBS9 gene that induces late fetal mortality in homozygous affected animals in a commercial pig population.

Introduction Domesticated animals are excellent models to study the effect of inbreeding on fitness, and the role of selection in inbreeding depression.

Results A haplotype inducing foetal lethality segregates at moderate frequency in a Large White pig population Genomic loci that harbour recessive lethal alleles can be identified by searching for haplotypes showing reduced or missing homozygosity.

Table 1 SSC18 haplotype characteristics and phenotypic effects. Position, Mb SSC Open in a separate window. Genotyping the offspring of carrier-by-carrier matings confirms early lethality of homozygous animals We tracked five recent CxC matings. Carriers exhibit a kb deletion affecting the BBS9 gene To identify candidate causal mutations, we analysed whole genome sequence data from 73 individuals from the same Large White population and identified 10 carrier animals for the SSC18 haplotype S5 Table.

Fig 1. Fig 2. Tracing the origin of the deletion To investigate the origin of the deletion, we analysed the frequency of the deletion over the last decades. Fig 3. SSC18 carrier frequency from — In humans, existing estimates of these quantities are based on comparisons between consanguineous and nonconsanguineous couples, an approach that confounds socioeconomic and genetic effects of inbreeding.

To overcome this limitation, we focused on a founder population that practices a communal lifestyle, for which there is almost complete Mendelian disease ascertainment and a known pedigree. Focusing on recessive lethal diseases and simulating allele transmissions, we estimated that each haploid set of human autosomes carries on average 0.

Comparison to existing estimates in humans suggests that a substantial fraction of the total burden imposed by recessive deleterious variants is due to single mutations that lead to sterility or death between birth and reproductive age. In turn, comparison to estimates from other eukaryotes points to a surprising constancy of the average number of recessive lethal mutations across organisms with markedly different genome sizes.

IN diploid organisms such as humans, the efficacy of selection on a deleterious mutation depends both on the fitness of homozygotes and on the fitness of heterozygotes, which reflects dominance relationships among alleles. Since recently introduced mutations are mostly present in heterozygotes, they will be purged less effectively by selection when recessive and segregate at higher frequencies compared to dominant or semidominant alleles that cause a similar fitness reduction in homozygotes.

For this reason, recessive alleles are expected to constitute a large fraction of strongly deleterious alleles segregating in diploid populations and in particular of Mendelian disease mutations in humans. One context in which the effects of recessive mutations are unmasked is in the presence of inbreeding, which leads to an excess of homozygotes compared to Hardy—Weinberg expectation.

Because closely related individuals may co-inherit alleles from one or more common ancestors, the genomes of offspring of consanguineous couples are more likely to be identical by descent, revealing recessive, deleterious traits.

If there are many recessive or partially recessive deleterious mutations i. Estimating the burden of recessive deleterious mutations in humans is therefore key to predicting adverse outcomes of consanguineous unions due to genetic factors Morton et al. Two main methods have been developed to these ends: both aim to quantify the burden by comparing the health states of offspring of nonconsanguineous and consanguineous matings.

The first considers couples with variable degrees of relatedness and regresses the viabilities of their offspring on their inbreeding coefficients, F Morton et al. When applied to humans, this method suffers from a number of limitations.

For one, the estimate relies heavily on accurate assessments of degrees of relatedness, and yet the F values estimated from recent pedigrees do not capture inbreeding among more distant ancestors. This will bias the results if, as seems plausible, consanguineous marriages tend to occur in families with a tradition of close-kin unions Hussain and Bittles ; Hamamy et al.

Even if F is calculated based on deeper pedigrees, it represents only the expected proportion of the genome that is identical by descent, whereas the realized proportion could vary tremendously across individuals.

In practice, this variation, combined with sampling variance, can lead to considerable uncertainty in the estimated effects of recessive alleles Bittles and Makov , Moreover, due to the restricted range of F and the small number of data points, the estimate of the combined effect of recessive deleterious mutations is highly sensitive to the choice of the regression model Makov and Bittles Perhaps most importantly, consanguineous and nonconsanguineous groups differ with respect to socioeconomic factors, in ways that influence the mortality and morbidity of the progeny Schull and Neel ; Neel et al.

How estimates of genetic effects will be affected is unclear, as the strength of the correlation between socioeconomic status and inbreeding—and even the direction of the correlation—varies across societies e. Thus, this approach could either overestimate or underestimate the genetic effects of consanguinity on health outcomes.

To minimize these concerns, a second approach focuses specifically on the comparison between offspring of first-cousin marriages and of nonconsanguineous marriages in a large number of populations Bittles and Makov ; Bittles and Neel ; Bittles and Black b. Even in this approach, however, genetic effects may be confounded by socioeconomic conditions that differ between consanguineous and nonconsanguineous groups within a population Bittles and Neel ; Hussain and Bittles Here, we introduce an approach that is not confounded by environmental effects, considering a founder population that practices a communal lifestyle, with a known pedigree and close to complete disease ascertainment over the past few generations Hostetler Founder populations have contributed greatly to the identification of Mendelian disease mutations, because the founding bottleneck and subsequent inbreeding increase the chance of recent identity-by-descent and thus the incidence of a number of otherwise rare, recessive diseases Boycott et al.

With a known pedigree, we can estimate the probability that an autosomal recessive, deleterious founder mutation manifests itself i. From this estimate and the number of recessive diseases observed in the pedigree, we can obtain an estimate of the total number of deleterious mutations carried by the founders. Since the number of founders is known, this estimate translates into the average number of recessive lethal alleles in each haploid set of autosomes. An advantage of our approach is that, by directly utilizing the pedigree information, there is no need to calculate an inbreeding coefficient or to compare among groups that are potentially subject to different socioeconomic conditions.

A difficulty, however, is that the transmission probability of a recessive deleterious allele depends on its selection coefficient in homozygotes s , which is in general very hard to quantify. Since recessive lethal mutations are only a subset of all deleterious mutations, our estimate provides a lower bound on the burden of recessive deleterious mutations, as well as information on the tail of the distribution of fitness effects of deleterious mutations e.

To assess the probability of a recessive lethal mutation manifesting itself after , we ran two sets of gene-dropping simulations. When using this pedigree, we assumed no transmission distortion Meyer et al. If reproductive compensation is incomplete, our simulations are expected to lead to an underestimate of the fraction of recessive lethal alleles purged by selection since the founding and an overestimate of the probability of manifestation.

In each replicate, we assigned a mutation to one founder and simulated the genotypes of other individuals generation by generation. For all other situations, we simulated the offspring genotype based on the parental genotypes according to Mendelian inheritance rules. This simulation scheme relies on the assumption of complete reproductive compensation, but is robust to missing data in the pedigree.

After generating genotypes of all individuals in the pedigree, we examined the numbers of heterozygotes and homozygotes for that mutation among people born after Because individuals in the pedigree are related to each other, the incidences of a disease in the pedigree are not independent, so we considered the number of unique diseases instead of the number of affected individuals.

We performed , gene-dropping simulations replicates for each founder. The mutation was lost before in , We also considered all individuals born after as the cohort. Among the , simulations performed, the mutation was lost before in , We focus on this simulation scheme in the main text.

Individuals who fell into one of the following three groups were included in this pedigree: 1 before the separation of the three leuts, individuals who had descendants in the Schmiedeleut S-leut ; 2 all S-leut individuals who were born since the separation of the three leuts through ; and 3 S-leut individuals who were born between and and participated in our ongoing studies Chong et al.

When using this larger pedigree, we assumed no reproductive compensation and no transmission distortion Meyer et al. Specifically, in each replicate, we assigned a mutation to one founder and then simulated the genotypes of all other individuals according to Mendelian inheritance rules. Because individuals homozygous for a recessive lethal mutation cannot reproduce, any individual who has offspring cannot be a homozygote recessive. To model this property of recessive lethals, we retained only replicates that were consistent with this condition, i.

This simulation scheme would be exact on a complete pedigree, in which differences in family size are entirely attributable to recessive lethals and stochasticity.

However, it is sensitive to incompleteness of the pedigree and other nonrandom factors that affect family size, in ways that are expected to lead to an overestimate of the fraction of recessive lethal alleles purged by selection since the founding and hence an underestimate of the probability of manifestation.

We performed 78, gene-dropping simulations on the larger pedigree replicates for each founder. Among the 67, replicates retained, the mutation was lost before in 46, The proportion of recessive lethal mutations lost is higher than that of neutral variants We also ran 39, replicates where we considered individuals born after as the cohort replicates for each founder.

Among the 34, replicates retained, the mutation was lost in 23, For both scenarios, we also considered the situation where more than one copy of the same mutation was present in the founders. In each replicate, we randomly sampled two or three founders to be the carriers and simulated the genotypes of other individuals as described above.

A total of , simulations were run with the minimum pedigree with two or three carriers, and the mutation was manifested in A total of 10, simulations were performed with the larger pedigree with two or three carriers, and the mutation was manifested in These findings indicate that the probability of manifestation is approximately proportional to the number of carriers among the founders, enabling us to estimate the total number of recessive lethal alleles some of which could be copies of the same mutation carried by the founders by dividing the number of distinct diseases by the probability of manifestation with one carrier.

Most Mendelian diseases reported in Hutterites were summarized in a review by Boycott et al. To incorporate newly identified diseases, we searched PubMed for genetic diseases in Hutterites that were reported since We included infertility due to biological reasons e.

To restrict the number of recessive lethal diseases to the pedigree under study, we required there to be affected individuals in Hutterites in South Dakota. This narrowed down the list to four diseases cystic fibrosis, nonsyndromic mental retardation, restrictive dermopathy, and myopathy with movement disorder and intellectual disability. We excluded restrictive dermopathy when considering the minimum pedigree, because the only reported patient in S-leut was not included in the minimum pedigree although the parents were included and confirmed to be carriers by genotyping Chong et al.

For the other three diseases, genotype data of the extant individuals confirmed the presence of an individual s homozygous for the disease allele in the pedigree Chong et al. Both alleles lead to severe phenotypes such that homozygous or compound heterozygous males are completely sterile and homozygous or compound heterozygous females cannot survive to reproductive age in absence of treatment. We therefore treat them as two copies of the same recessive lethal mutation.

We further note that although the p. Fdel mutation is common in Europeans, it is present on a single haplotype in Hutterites, suggesting that it was introduced into the population by only one founder Chong et al. The p. MK mutation is rare in Europeans but was identified on two haplotypes in Hutterites Zielenski et al.

The two haplotypes differ at multiple sites on both sides of the mutation, indicating either that at least two recombination events have occurred or that the p.

MK mutation was introduced by two founders Chong et al. Therefore, it is likely that two or three carriers of these two CFTR mutations were present in the founders. Given that the probability of manifesting a mutation is approximately proportional to the number of carriers in the founders, we can treat it as introduced by only one founder. We used a Bayesian approach to estimate the credible interval for mean number of recessive lethal alleles carried by each haploid set of human autosomes, R.

Given that D recessive lethal diseases have been observed, the posterior probability of R is. Let X i be the number of unique recessive lethal mutations carried by the i th founder, among which Y i mutations manifested themselves in the pedigree.

We assume that each X i is independently Poisson distributed:. For simplicity, we assume all mutations carried by the founders are unique. If the transmission of each mutation is independent and each of the X i mutations carried by the i th founder has a manifestation probability of p i , then Y i follows a binomial distribution:. Because of the thinning property of Poisson distribution, conditional on R , Y i follows a Poisson distribution:.

While both are improper priors, the resulting posterior distributions are proper gamma distributions:. A genomic parameter of interest is the total number of functionally important sites that, if mutated, would give rise to recessive lethal alleles.

Using a diffusion model and a low mutation rate approximation, Li and Nei derived the expectation for the total number of heterozygotes that carry a recessive deleterious mutation in a finite population [designated by n 1 p ],.

For simplicity, we assume that there are m autosomal genes in the genome that each can lead to complete sterility or lethality between birth and reproductive age and that gene i has l i sites at which mutations will give rise to recessive lethal alleles. The expected frequency of heterozygotes carrying a recessive lethal allele at this gene i is.

Assuming that the founders of S-leut Hutterites were drawn from a random-mating population at equilibrium, each of them should have carried twice that number of recessive lethals. Therefore, at equilibrium, an individual in a population with larger N e will have a greater expected number of recessive lethal mutations. The recent population growth experienced by humans represents a transition from small N e to large N e , which will therefore lead to an increase in the average number of recessive lethal mutations per individual.

As a result, the estimated target size after recent growth will be smaller than that estimated from the long-term N e. As in the completely recessive case, we assume that there are m autosomal genes in the genome that can mutate to deleterious alleles with these properties and that gene i has l i such sites. Under these assumptions, the expected total number of heterozygotes affected by a mutation in a finite population at gene i can be approximated as.

So the average frequency of those partially recessive lethal mutations is. Therefore, the total number of such partially recessive lethal mutations carried by a random haploid genome is.

We focused on the Hutterites, a group ideally suited for the new method we propose. The Hutterian Brethren is an isolated founder population, which originated in the Tyrolean Alps in the s and eventually settled in North America on three communal farms in the s after a series of migrations throughout Europe. The three colonies thrived and shortly thereafter gave rise to three major subdivisions, referred to as the Schmiedeleut S-leut , Lehrerleut L-leut and Dariusleut D-leut , with most marriages since taking place among individuals within the same leut.

Each leut practices a communal lifestyle, with no private ownership and hence no socioeconomic differences among members Hostetler Additionally, the time to approach equilibrium depends on the pre- and post-jump values of h , and each curve takes different times to approach equilibrium.

A measure of such times to equilibrium is given in Table 2. To make comparisons, it is useful to have a measure of the time to equilibrium. It is, however, hard to provide a precise definition of the time to equilibrium. For intermediate times the half time to equilibrium can be approximated by. We can obtain results for the Wright-Fisher model described by Equation 7 , which is a discrete time Markov chain where some exact results are known Haigh, and for which many results can be numerically calculated.

Before we give results based on numerical calculations, we note that analytical insight into the Wright-Fisher model, and the phenomena occurring in a finite population, can be gained using a diffusion approximation of this model Kimura, The diffusion approximation treats both the allele frequency and time as continuous quantities, and replaces the random frequency X t of the Wright-Fisher model by a continuous function of continuous time, which we write as X t.

The probability density obeys the diffusion equation. Some intuition about the phenomena occurring in a finite population with a lethal genotype can be gained from the form of the coefficients V x and F x that appear in the diffusion equation Equation On neglecting small terms of order x 2 , xu and u 2 we find that for small x. We assume that h is not equal to 1, so the disease-causing allele is not fully dominant.

This is unlike the standard i. This is positive, and hence has the tendency to push the frequency to positive values, as we would expect mutation to do. Under the Wright-Fisher model, the fraction of a very long period of time spent by the population at a particular frequency is given by the value of the stationary distribution at this frequency Gillespie, The expected value of the frequency in this distribution is the finite population analog of the equilibrium frequency in an infinite population.

In Table 1 we illustrate the dependence of E stat [ X ] on the parameters u , N and h. The results of Table 1 suggest that E stat [ X ] is: i an increasing function of N , ii a decreasing function of h , iii an increasing function of u. Table 1.

For a finite population, we have so far considered the stationary distribution of the disease allele's frequency. Let us now try to get some insight into the transient behavior that also occurs in a finite population. We assume the same scenario of changes of the dominance coefficient h as before see section 3. A basic characterization of this problem is in terms of the expected value of the frequency. We note that compared with the infinite population result, we now have an additional parameter in the problem, namely the population size, N , and results will depend on the value adopted for this parameter.

In Figure 3 the logarithm of the mean allele frequency, log 10 E [ X t ] , is plotted in colored curves against the time, t. In Figure 3 the behavior of the finite population results for E [ X t ] can be seen to be qualitatively similar to those of an infinite population, but quantitatively different.

Mean frequencies in a finite population are, from the figure, smaller than the corresponding equilibrium frequencies of an infinite population. In this work we have provided an analysis of the implications of lethal mutations in both effectively infinite and finite populations.

For an effectively infinite population, we have given the general form for the deterministic evolutionary force which acts in such a system, along with the equilibrium frequency. We have also provided some illustrations and a characterization of the transient behavior of the frequency when the fitness of the heterozygote discontinuously changes.

For a finite population, we have provided the appropriate i. We have presented properties of a finite population, such as the stationary distribution and its transient behavior. For populations of finite size, Wright-Fisher models and their diffusion approximations have often been employed in describing the evolution of a focal allele see e.

One assumption that is typically made when taking this approach is that selection is a weak evolutionary force, in the sense that selection coefficients are small compared with 1. However, the assumption of weak selection becomes untenable for lethal mutations; lethality represents the strongest level of selection against one genotype. Thus an important consideration with lethality, is the explicit need to treat the action of selection on genotypes, rather than on alleles.

This would appear to make the analysis of lethal mutations significantly more complicated than when selection acts weakly on all genotypes in which case a description in terms of single allele frequency suffices. However, perhaps surprisingly, lethal selection has no more complexity than weak selection. This arises for a single locus with two alleles since a description of the population is generally required in terms of three genotype frequencies, but the three genotype frequencies add to unity so just two are independent, and when there is also lethality of one homozygote, this allows elimination of one of the two independent genotype frequencies, with the substantial simplification that just a single frequency is required to describe the population.

Thus, lethality of one genotype has the effect of simplifying the model. The multinomial distribution that is required to relate genotype frequencies in adjacent generations under more general schemes of selection Nagylaki, collapses to a binomial distribution, thereby making the problem mathematically no more complex than a weak selection problem, which is also described by just a single frequency and also involves a binomial distribution.

Thus, while it might be viewed as improbable , but not impossible, that a lethal mutation can rise to a frequency above 1 2 , the analysis presented in this work indicates that this can never be the case. In Table 1 we gave expected values of the frequency of the disease-causing allele, in a finite population, in the stationary distribution, E stat [ X ].

We infer that with weak overdominance, the lethal allele can, in larger populations, reach higher frequencies that are more in line with some lethal disorders.

In a recent study on lethal mutations by Amorim et al. In particular, for three of the four mutation types, the observed frequency was significantly higher than the theoretical expectation. Here we have found, for a large effectively infinite population size, that as the mutation rate decreases, the sensitivity of the equilibrium allele frequency to overdominance increases Figure 2.

Consequently, for lethal alleles with a low mutation rate, even very weak overdominance can result in highly inflated equilibrium frequencies, that largely escape mutation limitation.

It may be of some relevance that when considering the transient behavior of the mutation frequency Figure 3 , the time for the mutation to approach equilibrium subsequent to a period of overdominance can be considerable: e.

Realistically, the proportion of lethal recessive disorders found to be at unusually high incidences because of periodic overdominance is likely to be a small subset, the majority being more likely due to an ascertainment bias in identification Amorim et al. The results we have established in this work relate to the subset of Mendelian disorders corresponding to a lethal disease homozygote. Although the majority of lethal disorders are autosomal recessive conditions, such as cystic fibrosis and Tay-Sachs, it should be noted that the treatment outlined in this work can also be applied to rare dominant lethal conditions, such as achondroplasia, where individuals homozygous for the mutation are unlikely to survive infancy, unlike the non-lethal heterozygous state Pauli et al.

Thus, despite involving strong selection, such diseases are susceptible to a detailed analysis. To summarize, we believe the results presented in this work shed new light on the possible behaviors that can occur in well-characterized genetic systems involving lethal alleles. DW and AO conceived the paper. DW carried out the analyses. DW and AO co-wrote the paper. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Amorim, C. The population genetics of human disease: the case of recessive, lethal mutations. PLoS Genet. Ewens, W. Mathematical Population Genetics: I. Theoretical Introduction. Google Scholar. Fisher, R. The Genetical Theory of Natural Selection. Oxford: Oxford University Press.