Mendelian recessive disease allele set

We relied on two datasets, one that describes 173 autosomal recessive diseases [19] and another from a genetic testing laboratory (Counsyl [20]; <https://www.counsyl.com/>) that includes 110 recessive diseases of clinical interest. From these lists, we obtained a set of 44 “recessive lethal” diseases associated with 45 genes (S1 Table), requiring that at least one of the following conditions is met: (i) in the absence of treatment, the affected individuals die of the disease before reproductive age, (ii) reproduction is completely impaired in patients of both sexes, (iii) the phenotype includes severe mental retardation that in practice precludes reproduction, or (iv) the phenotype includes severely compromised physical development, again precluding reproduction.

Based on clinical genetics datasets and the medical literature (see Methods for details), we were able to confirm that 417 Single Nucleotide Variants (SNVs) in 32 (of the 44) genes had been reported with compelling evidence of association to the severe form of the corresponding disease and an early-onset, as well as no indication of effects in heterozygote carriers (S2 Table). By this approach, we obtained a set of mutations for which, at least in principle, there is no heterozygote effect, i.e., for which the dominance coefficient h = 0 in a model with relative fitness of 1 for the homozygote for the reference allele, 1-hs for the heterozygote, and 1-s for the homozygote for the deleterious allele, and the selective coefficient s is 1.

A large subset of these mutations (29.3%) consists of transitions at CpG sites (henceforth CpGti), which occur at a highly elevated rates (~17-fold higher on average) compared to other mutation types, namely CpG transversions, and non-CpG transitions and transversions [18]. This proportion is in agreement with previous estimates for a smaller set of disease genes [21] and for DMD [22].

Empirical distribution of allele frequencies of disease mutations in Europe

Allele frequency data for the 417 variants were obtained from the Exome Aggregation Consortium (ExAC) for 60,706 individuals, of whom 33,370 are non-Finnish Europeans [23]. Out of the 417 variants associated with putative recessive lethal diseases, three were found homozygous in at least one individual in this dataset (rs35269064, p.Arg108Leu in ASS1; rs28933375, p.Asn252Ser in PRF1; and rs113857788, p.Gln1352His in CFTR). Available data quality information for these variants does not suggest genotype calling artifacts (S2 Table). Since these diseases have severe symptoms that lead to early death without treatment and these ExAC individuals are healthy (i.e., do not manifest severe Mendelian diseases) [23], the reported mutations are likely errors in pathogenicity classification or cases of incomplete penetrance (see a similar observation for CFTR and DHCR7 in [24]). We therefore excluded them from our analyses. In addition to the mutations present in homozygotes, we also filtered out sites that had lower coverage in ExAC (see Methods), resulting in a final dataset of 385 variants in 32 genes (S2 Table).

Genotypes for a subset (91) of these mutations were also available for a larger sample size (76,314 individuals with self-reported European ancestry) generated by the company Counsyl (S3 Table). A comparison of the allele frequencies in this larger dataset to that of ExAC suggests that the allele frequencies for individual variants are concordant between the two datasets (Pearson’s correlation coefficient of 0.79, S1 Fig) and that the overall distributions do not differ appreciably (Kolmogorov–Smirnov test, p-value = 0.23). Thus, both data sets appear to reflect the general distribution of these disease alleles in Europeans. In what follows, we focused on ExAC, which includes a greater number of disease mutations.

Models of mutation-selection balance

To generate expectations for the frequencies of these disease mutations under mutation-selection balance, we considered models of infinite and finite populations of constant size [3] and conducted forward simulations using a plausible demographic model for African and European populations [25] (see Methods for details). In all these models, there is a wild-type allele (A) and a deleterious allele (a, which could also represent a class of distinct deleterious alleles with the same fitness effect) at each site, such that the relative fitness of individuals of genotypes AA, Aa, or aa is given respectively by:

• w_AA = 1;
• w_Aa = 1-hs;
• w_aa = 1-s;

The mutation rate from A to a is u; we assume that there are no back mutations.

For a constant population of infinite size, Wright [26] showed that under these conditions, there exists a stable equilibrium between mutation and selection, when the selection pressure is sufficiently strong (s>>u). In particular, when the deleterious effect of allele a is completely recessive (h = 0), its equilibrium frequency q is given by: (1)

For a finite population of constant size, Nei [3] derived the mean (Eq 2) and variance (Eq 3) of the frequency of a fully recessive deleterious mutation (h = 0) based on a diffusion model, leading to: (2) (3) where N is the diploid population size and Γ is the gamma function (see Simons et al. [1] for a similar approximation).

In a finite population, the mean frequency, , therefore depends on assumptions about the population mutation rate (2Nu). If the population mutation rate is high, such that 2Nu>>1, is approximated by (4) which is independent of the population size and equal to the equilibrium frequency in an infinite population, i.e., the right hand side of Eq (1). The important difference between the two models above is that in a finite population, there is a distribution of frequencies q (because of genetic drift), whose variance is given in Eq (3), rather than a single value, as in an infinite population.

In contrast, when the finite population has a low population mutation rate (2Nu<<1), the mean allele frequency, , is approximated by: (5) which depends on the population size [3].

We note that Nei [3] assumed a Wright-Fisher model, in which there is no distinction between census and the effective population sizes. However, when the two differ, it is the effective population size that governs the dynamics of deleterious alleles, so the N in the analytical results in fact represents the effective population size. In humans, the mutation rate at each bp is very small (on the order of 10⁻⁸ [18]) and the effective population size not that large, even recently [27,28], so the second approximation should apply when considering each single site independently.

The expectation and variance of the frequency of a fatal, fully recessive allele (i.e., s = 1, h = 0) are then given by: (6) and (7) This single site model implicitly ignores the existence of compound heterozygosity in modeling the strength of selection acting on an individual site.

Comparing mutation-selection balance models

Although an infinite population size has often been assumed when modeling deleterious allele frequencies (e.g. [5,29–32]), predictions under this assumption can differ markedly from what is expected from models of finite population sizes, assuming plausible parameter values for humans. For example, the long-term estimate of the effective population size from total polymorphism levels is ~20,000 individuals (assuming a mutation rate of 1.2 x 10⁻⁸ per bp per generation [18] and diversity levels of 0.1% [33]). In this case and considering a mutation rate of 1.5 x 10⁻⁸ for exons (which have a higher mutation rate than the rest of the genome, because of their base composition [34]), the average deleterious allele frequency in the model of finite population size is ~23-fold lower than that in the infinite population model (Fig 1).

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Fig 1. Expected allele frequencies in a population, according to mutation-selection balance models.

The blue bar denotes the expected allele frequency under an infinite population size, the green bar the mean under a finite constant population, and the red bar the mean under a plausible demographic model for European populations; for this last case, the entire distribution across 100,000 simulations is shown in the grey histogram. All models assume s = 1 and h = 0, i.e., fully recessive, lethal mutations. For the finite constant population size model, we present the mean frequency for a population size of 20,000 (see S2A Fig for other choices). Population allele frequencies (q) were transformed to log10(q), and those q = 0 were set to 10⁻⁷ for visual purposes, but indicated as “0” on the X-axis. The density of the distribution is plotted on a log-scale on the Y-axis. The mutation rate u was set to 1.5 x 10⁻⁸ per bp per generation in all models.

https://doi.org/10.1371/journal.pgen.1006915.g001

Because the human population size has not been constant and changes in the population size can affect the frequencies of deleterious alleles in the population (e.g. [2,35]), we also simulated the population dynamics of disease alleles under a plausible demographic model for European populations based on Tennessen et al. [25]. The original model assumes a genome-wide mutation rate of 2.36 x 10⁻⁸ per bp per generation, when current, more direct estimates are approximately two-fold smaller [18,34,36]. We therefore rescaled the demographic parameters of the Tennessen et al. model, based on a mutation rate of 1.2 x 10⁻⁸ [18] (see Methods). Assuming a mutation rate of 1.5 x 10⁻⁸ per bp (as recently estimated for exons [34]), the mean allele frequency of a lethal, recessive disease allele obtained from this model was 7.10 x 10⁻⁶, ~1.33-fold higher than expected for a constant population size model with N_e = 20,000 (Fig 1). The mean frequency seen in simulations instead matches the expectation for a constant population size of 35,651 individuals (see Methods and S2A Fig). Increasing the effective population size in a constant size model is not enough to capture the dynamics of disease alleles appropriately, however. For example, if simulation results obtained under the Tennessen et al. [25] demographic model are compared to those for simulations of a constant population size of N_e = 35,651, the mean allele frequencies match, but the distributions of allele frequencies are significantly different (Kolmogorov-Smirnov test, p-value < 10⁻¹⁵; S2B and S2C Fig). These findings thus confirm the importance of incorporating demographic history into models for understanding the population dynamics of disease alleles [5,37,38]. In what follows, we therefore tested the fit of the more realistic demographic model [25] (and variants of it) to the observed allele frequencies.

Comparing empirical and expected distributions of disease allele frequencies

The mutation rate from wild-type allele to disease allele, u, is a critical parameter in predicting the frequencies of a deleterious allele [4,39]. To model disease alleles, we considered four mutation types separately, with the goal of capturing most of the fine-scale heterogeneity in mutation rates [27,36,40,41]: transitions in methylated CpG sites (CpGti) and three less mutable types, namely transversions in CpG sites (CpGtv) and transitions and transversions outside a CpG site (nonCpGti and nonCpGtv, respectively). In order to control for the methylation status of CpG sites, we excluded 12 CpGti that occurred in CpG islands, which tend not to be methylated and thus are likely to have a lower mutation rate [36] (following Moorjani et al. [42]). To allow for heterogeneity in mutation rates within each one of these four classes considered, we modeled the within-class variation in mutation rates according to a lognormal distribution (see details in Methods and [27]).

For each mutation type, we then compared the mean allele frequency obtained from simulations to what is observed in ExAC, running 100,000 replicates. To this end, we matched simulations to the empirical data with regard to the number of individuals sampled and number of mutations observed of each mutation type and focused the analysis on the largest sample of the same common ancestry, namely Non-Finnish Europeans (n = 33,370) (Fig 2A). We found significant differences between empirical and expected mean frequencies for nonCpGtv (30-fold higher on average; two-tailed p-value < 1 x 10⁻⁴; see Methods for details) and nonCpGti (15-fold higher on average, two-tailed p-value < 1 x 10⁻⁴), but only marginally so for CpGtv (5-fold higher on average, two-tailed p-value = 0.08). The mean frequency for CpGti is also somewhat higher than expected, but insignificantly so (1.17-fold higher on average, two-tailed p-value = 0.59). Intriguingly, the discrepancy between observed and expected frequencies becomes smaller as the mutation rate increases (Fig 2B).

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Fig 2. Comparison between expected and observed mean allele frequencies of recessive, lethal disease mutations.

(A) Shown are the expected and observed mean sample frequencies of disease mutations for four different mutation types. The title of the panel indicates the mutation type, followed by K, the total number of mutations of that type considered in this study, with the p-value for the difference between observed and expected mean frequencies given below. Distributions in grey are the mean sample allele frequencies across K mutations based on 100,000 simulations, and rely on a plausible demographic model for European populations [25] (see Methods). Blue bars represent the observed mean frequencies of the four mutation types, estimated from 33,370 individuals of European ancestry from ExAC. (B) Fold increase in the observed mean allele frequency in relation to the expected, as a function of the mutation rate u (on a log-scale), for each of the four mutation classes.

https://doi.org/10.1371/journal.pgen.1006915.g002

Two additional factors that we have not included in our model should further decrease the predicted frequencies of disease alleles. Given that frequencies in ExAC are already unexpectedly high, these factors would only exacerbate the discrepancy between observed and expected frequencies of deleterious alleles. First, we have ignored the effects of compound heterozygosity, the case in which combinations of two distinct pathogenic alleles in the same gene lead to lethality. This phenomenon is known to be common [43], and indeed, in the 320 cases in which we were able to obtain this information, 58.44% were initially identified in compound heterozygotes. In the presence of compound heterozygosity, each deleterious mutation will be selected against not only when present in two copies within the same individual, but also in the presence of lethal mutations at other sites. Since the purging effect of selection against compound heterozygotes was not modeled in simulations, we would predict the frequency of a deleterious mutation to be even lower than shown (e.g., in Fig 2A).

In order to model the effect of compound heterozygosity in our simulations, we re-ran our simulations, but focusing on a gene rather than a single site and so considering the sum of frequencies of all known recessive lethal alleles within a gene. In these simulations, we used the same set-up as in the site level analysis, except for the mutation rate, U, which is now the sum of the mutation rates u_j at each site j that is known to cause a severe and early onset form of the disease [8] (S2 Table; see Methods for details). This approach does not consider the contribution of other mutations in the genes that cause the mild and/or late onset forms of the disease, and implicitly assumes that all combinations of known recessive lethal alleles of the same gene have the same fitness effect as homozygotes. Comparing observed frequencies of disease alleles for each gene to predictions generated by simulation, about a fourth of the 27 genes for which we implemented the gene-level analysis (see Methods) differ from the expected distribution at the 5% level, with a clear overall trend for observed frequencies to be above expectation (S4 Table; Fig 3; Fisher’s combined probability test p-value = 6 x 10⁻⁸).

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Fig 3. Sample allele frequencies at the gene level.

The expectation (grey) is based on 1000 simulations, assuming no fitness decrease in heterozygotes, but allowing for compound heterozygosity (see Methods for details). The sum of allele frequencies of known recessive lethal disease mutations in each gene (purple bars) was obtained from ExAC considering 33,370 European individuals. Genes are ordered according to the two-tailed p-value (S4 Table; see Methods). Genes are bolded when they differ significantly from expectation (at the 5% level). Violin plots show the distribution of the log₁₀ combined allele frequency of all segregating alleles obtained from simulations and boxes represent the fraction of simulations in which no deleterious allele was observed in the simulated sample at present time.

https://doi.org/10.1371/journal.pgen.1006915.g003

This finding is even more surprising than it may seem, because we are far from knowing the complete mutation target for each gene, i.e., all the sites at which mutations could cause the disease. If there are additional, undiscovered sites in the gene at which mutations are fatal when carried in combination with a known recessive lethal mutation, the purging effect of purifying selection on the known mutations will be under-estimated in our simulations, leading us to over-estimate the expected frequencies of the known mutations in simulations. Therefore, our predictions are, if anything, an over-estimate of the expected allele frequency, and the discrepancy between predicted and the observed is likely even larger than what we found.

The other factor that we did not consider in simulations but would reduce the expected allele frequencies is a subtle fitness decrease in heterozygotes, as has been documented in Drosophila for example [44]. To evaluate potential fitness effects in heterozygotes when none had been documented in humans, we considered the phenotypic consequences of orthologous gene knockouts in mouse. We were able to retrieve information on phenotypes for both homozygote and heterozygote mice for only eight out of the 32 genes, namely ASS1, CFTR, DHCR7, NPC1, POLG, PRF1, SLC22A5, and SMPD1. For all eight, homozygote knockout mice presented similar phenotypes as affected humans, and heterozygotes showed a milder but detectable phenotype (S5 Table). The magnitude of the heterozygote effect of these mutations in humans is unclear, but the finding with knockout mice makes it plausible that there exists a very small fitness decrease in heterozygotes in humans as well, potentially not enough to have been recognized in clinical investigations but enough to have a marked impact on the allele frequencies of the disease mutations. Indeed, even if the fitness effect in heterozygotes were as small as h = 1%, a 79% decrease in the mean allele frequency of the disease allele is expected relative to the case with complete recessivity (h = 0) (S3 Fig).

Disease allele set

In order to identify single nucleotide variants within the 42 genes associated with lethal, recessive Mendelian diseases (S1 Table), we initially relied on the ClinVar dataset [56] (accessed on June 3^rd, 2015). We filtered out any variant that is an indel or a more complex copy number variant or that is ever classified as benign or likely benign in ClinVar (whether or not it is also classified as pathogenic or likely pathogenic). By this approach, we obtained 769 SNVs described as pathogenic or likely pathogenic. For each one of these variants, we searched the literature for evidence that it is exclusively associated to the lethal and early onset form of the disease and was never reported as causing the mild and/or late-onset form of the disease. We considered effects in the absence of medical treatment, as we were interested in the selection pressures acting on the alleles over evolutionary time scales rather than in the last one or two generations, i.e., the period over which treatment became available for some of diseases considered. To evaluate the impact of treatment, we decreased s from 1 to 0 (i.e., we assumed a complete absence of selective effects due to treatment) in the last three generations and compared the mean allele frequencies across 100,000 simulations implemented with or without this readjustment in selection coefficient. Because of the stochastic nature of the simulations, we repeated this pairwise comparison 10 times in order to get a range of expected increase in allele frequencies. We observed only a minor increase in the mean allele frequency (2.6% at most) across the 10 replicates. This simulation procedure corresponds to a scenario in which there is an extremely effective treatment for all diseases for the past three generations, which is an overestimate of the effect and length of treatment for the disease set considered.

Variants with mention of incomplete penetrance (i.e. for which homozygotes were not always affected) or with known effects in heterozygote carriers were removed from the analysis. This process yielded 417 SNVs in 32 genes associated with distinct Mendelian recessive lethal disorders (S2 Table). Although these mutations were purportedly associated with completely recessive diseases, we sought to examine whether there would be possible, unreported effects in heterozygous carriers. To this end, we used the Mouse Genome Database (MGD) [57] (accessed July 29^th, 2015) and were able to retrieve information for both homozygote and heterozygote mice for eight out of the 32 genes (all of which with a homologue in mice) (S5 Table).

In addition to the information provided by ClinVar for each one of these variants, we considered the immediate sequence context of each SNV, to tailor the mutation rate estimate accordingly [18]. To do so, we used an in-house Python script and the human genome reference sequence hg19 from UCSC (<http://hgdownload.soe.ucsc.edu/goldenPath/hg19/chromosomes/>).

Genetic datasets

The Exome Aggregation Consortium (ExAC) [23] was accessed on August 9^th, 2016. The data consist of genotype frequencies for 60,706 individuals, assigned after Principal Component Analysis to one of seven population labels: African (n = 5,203), East Asian (n = 4,327), Finnish (n = 3,307), Latino (n = 5,789), Non-Finnish European (n = 33,370), South Asian (n = 8,256) and “other” (n = 454) [23]. We focused our analyses on those individuals of Non-Finnish European descent, because they constitute the largest sample size from a single ancestry group. We note that, some diseases mutations, for instance, those in ASPA, HEXA and SMPD1, are known to be especially prevalent in Ashkenazi Jewish populations, which could potentially bias our results if Ashkenazi Jewish individuals constituted a great portion of the sample we considered. However, this sample includes only ~2,000 (~6%) Ashkenazi individuals (Dr. Daniel MacArthur, personal communication).

From the initial 417 mutations, we filtered out three that were homozygous in at least one individual in ExAC and 29 that had lower coverage, i.e., fewer than 80% of the individuals were sequenced to at least 15x. This approach left us with a set of 385 mutations with a minimum coverage of 27x per sample and an average coverage of 69x per sample (S2 Table). For 248 sites with non-zero sample frequencies, ExAC reported the number of non-Finnish European individuals that were sequenced, which was on average 32,881 individuals [23]. For the remaining 137 sites, we did not have this information. Nonetheless, the coverage across all samples is reported and does not differ significantly between the two sets of sites (by a Kolmogorov-Smirnov test, p-value = 0.90; S7 Fig). We therefore assumed that mean number of individuals covered for all sites was 32,881 and used this number to obtain sample frequencies from simulations, as explained below.

A second genetic dataset was obtained from Counsyl (<https://www.counsyl.com/>). Counsyl is a commercial genetic screening laboratory that offers, among other products, the “Family Prep Screen”, a genetic screening test intended to detect carrier status for up to 110 recessive Mendelian diseases in couples that are planning to have a child [20]. A subset of 294,000 of its customers was surveyed by genotyping or sequencing for “routine carrier screening”. This subset excludes individuals with indications for testing because of known personal or family history of Mendelian diseases, infertility, and consanguinity. It therefore represents a more random (with regard to the presence of disease alleles), population-based survey. For these individuals, we had details on self-reported ancestry (14 distinct ethnic/ancestry/geographic groups) and the allele frequencies for 98 mutations that match those that passed our variant selection criteria described above, of which 91 are also sequenced to high coverage in the ExAC database (S2 Table). We focused our analysis of this dataset on the 76,314 individuals of self-reported Northern or Southern European ancestry.

Simulating the evolution of disease alleles with population size change

We modeled the frequency of a deleterious allele in human populations by forward simulations based on a crude but plausible demographic model for human populations from Africa and Europe, inferred from exome data for African-Americans and European-Americans [25]. To this end, we used a program described in [1]. In brief, the demographic scenario consists of an Out-of-Africa demographic model, with changes in population size throughout the population history, including a severe bottleneck in Europeans following the split from the African population and a rapid, recent population growth in both populations [25]. As in Simons et al. [1], we simulated genetic drift and two-way gene flow between Africans and Europeans in recent history.

The original demographic model was inferred using a mutation rate u of 2.36 x 10⁻⁸ per bp per generation [25,58]. More recent estimates, based on direct resequencing of human pedigrees, instead point to mutation rates about 50% smaller than that [18,34,36]. To incorporate what is believed to be a more accurate mutation rate estimate, we rescaled all demographic and time parameters in the original Tennessen et al. [25] model by a factor of 1.97, based on the difference between the mutation rate considered in the original study and that of Kong et al. [18] (which is similar to that found in other studies [48]). We refer to this model as the rescaled Tennessen model and rely on it throughout.

Negative selection acting on a single bi-allelic site was modeled as in the analytic models. Allele frequencies follow a Wright-Fisher sampling scheme in each generation according to these viabilities, with migration rate and population sizes varying according to the demographic scenario considered. Whenever a demographic event (e.g., growth) altered the number of individuals and the resulting number was not an integer, we rounded it to the nearest integer, as in Simons et al. [1]. A burn-in period of 10Ne generations with constant population size Ne = 14,328 individuals was implemented in order to ensure an equilibrium distribution of segregating alleles at the onset of demographic changes in Africa, 11,643 generations ago.

In contrast to Simons et al. [1], our simulations always start with the ancestral allele A fixed and mutation occurs exclusively from this allele to the deleterious one (a), i.e., a mutation occurs with mean probability u per gamete, per generation, and there is no back-mutation. However, recurrent mutations at a site are allowed, as in Simons et al. [1].

When implementing the simulations, we considered a mean mutation rate u of 1.5x10^-8 per bp, per generation, as has been estimated for exons [34], as well as mutation rates for four distinct mutation types (CpGti = 1.12 x 10⁻⁷; CpGtv = 9.59 x 10⁻⁹; nonCpGti = 6.18 x 10⁻⁹; and nonCpGtv = 3.76 x 10⁻⁹) estimated from a large human pedigree study [18]. While these four categories capture much of the variation in germline mutation rates across sites, a number of other factors (e.g., the larger sequence context or the replication timing) also influence mutation rates, introducing heterogeneity in the mutation rate within each class considered [27,36,40,59]. To allow for this heterogeneity as well as for uncertainty in the point mutation rates estimates, in each simulation, when compared to human exome data (Figs 1, 2, S4 and S5), instead of using a fixed rate u, we drew the mutation rate M from a lognormal distribution with the following parameters: (8) such that that E[M] = u. σ was set to 0.57 (following [27]).

For each mutation type, we then proceeded as follows:

1. We ran two million simulations, thus obtaining the distribution of deleterious allele frequencies expected for the European population.
2. We sampled K allele frequencies from the two million simulations implemented for each mutation type, where K is the number of identified mutations of that type. Sample allele frequencies were simulated from these population frequencies by Poisson sampling, so to match ExAC’s number of chromosomes.
3. We repeated step (2) 100,000 times, thus obtaining a distribution for the mean allele frequency across K mutations.

To assess the significance of the deviation between observed and expected mean, we obtained a two-tailed p-value, defined as 2 x (r+1)/(100000+1), where r is the number of simulated allele frequencies that were greater or equal to that of the empirical mean [60], for each mutation type separately.

A well-known source of heterogeneity in mutation rate within the CpGti class is methylation status, with a high transition rate seen only at methylated CpGs [21]. In our analyses, we tried to control for the methylation status of CpG sites by excluding sites located in CpG islands (CGIs), which tend to not be methylated [42]. The CGI annotation for hg19 was obtained from UCSC Genome Browser (track “Unmasked CpG”; <http://hgdownload.soe.ucsc.edu/goldenPath/hg19/database/cpgIslandExtUnmasked.txt.gz>, accessed in June 6th, 2016). BEDTools [61] was used to exclude those CpG sites located in CGIs. We note that the CpGti estimate from [18] includes CGIs, and in that sense the average mutation rate that we are using for CpGti may be a very slight underestimate of the mean rate for transitions at methylated CpG sites.

Unless otherwise noted, the expectation assumes fully recessive, lethal alleles with complete penetrance. Notably, by calculating the expected frequency one site at a time, we are ignoring possible interaction between genes (i.e., effects of the genetic background) and among different mutations within a gene (i.e., compound heterozygotes). These assumptions are relaxed in two ways. In one analysis (S3 Fig), we considered a very low selective effect in heterozygous individuals (h = 1%), reasoning that such an effect could plausibly go undetected in medical examinations and yet would nonetheless impact the frequency of the disease allele. Second, when considering the gene-level analysis (Fig 3), we implicitly allowed for compound heterozygosity between any pair of known lethal mutations [8]. For this analysis, we ran 1000 simulations for a total mutation rate U per gene that was calculated accounting for the heterogeneity and uncertainty in the mutation rates estimates as follows: (i) For j sites in a gene known to cause a recessive lethal disease and that passed our filtering criteria (S2 Table), we drew a mutation rate u_j from the lognormal distribution, as described above; (ii) We then took the sum of u_j as the total mutation rate U; (iii) We then ran one replicate with U as the mutation parameter, and other parameters as specified for site level analysis. Because the mutational target size considered in simulations is only comprised of those sites at which mutations are known to cause a lethal recessive disease, it is almost certainly an underestimate of the true mutation rate—potentially by a lot. We note further that by this approach, we are assuming that compound heterozygotes formed by any two lethal alleles have fitness zero, i.e., that they are identical in their effects to homozygotes for any of the lethal alleles. Moreover, we are implicitly ignoring the possibility of complementation, which is (somewhat) justified by our focus on mutations with severe effects and complete penetrance (but see Discussion). Since we were interested in understanding the effect of compound heterozygosity, for this analysis, we did not consider the five genes in which only one mutation passed our filters (BCS1L, FKTN, LAMA3, PLA3G6, and TCIRG1).

Modeling the effect of the ascertainment bias in disease discovery

To calculate the probability of ascertaining a recessive, lethal mutation, we assumed that all currently known disease mutations were identified in a putative ascertainment study of sample size n_a in a population with an inbreeding coefficient of F_a. Under this model, we can estimate P_asc, the probability of ascertaining a disease mutation, as following:

For a disease allele (denoted as a) at frequency q in the present population, if we randomly sample an individual with inbreeding coefficient of F_a, the probabilities of the three genotypes are: (9) (10) (11)

Thus, if n_a unrelated individual are surveyed, the probability of not seeing any homozygote for the disease allele (which is the same as the probability of not being ascertained in this set) is: (12) Therefore, the probability of ascertainment is (13) which is an increasing function with regard to q (the population allele frequency), F_a (the inbreeding coefficient of the population under study) as well as n_a (the sample size of the putative ascertainment study) (S6 Fig).

We also demonstrate the relationship between the probability of ascertainment and mutation rate using simulations of ascertainment bias implemented according to the following steps:

1. For each of the four mutation types considered, we generated 10⁶ allele frequencies q from the results of the simulations based on a realistic demographic model [25].
2. We generated n_a independent diploid genotypes, given the allele frequencies from step 1 and an inbreeding coefficient F_a. We ran this step for a range of n_a and F_a values (Table 1).
3. With a given combination of n_a and F_a values, we identified the cases (out of the 10⁶ observations from step 1) where at least one homozygote individual was observed in step 2. These cases correspond to disease mutations that were ascertained; the reasoning being that given complete penetrance, a recessive disease mutation can only be identified if there is at least one affected individual in the studied population. With this step, we calculated the probability of ascertainment by taking the fraction of cases that satisfy the criteria above.
4. Finally, for each one of the 10⁶ simulations from step 1, we generated a sample allele frequency of the disease mutation, matching ExAC’s sample size (i.e., considering 2n = 65,762 chromosomes). We can then compare q_u, the unbiased average allele frequency of all disease mutations, to q_a, the mean frequency of the subset of cases ascertained in step 3, i.e., those cases for which at least one homozygote individual is observed.

These simulations were meant to illustrate the likely impact of ascertainment bias, rather than to precisely describe the disease mutation identification process or to quantify the expected effect. Notably, we performed these simulations for single sites, so the criteria for ascertainment in step 3 did not include the possibility of compound heterozygotes, despite the fact that an estimated 58.4% of the disease mutations included in our study were initially identified in compound heterozygotes. However, this simulation framework could readily be extended in this direction and it would not change our qualitative conclusion.