Introduction

In diploid organisms, the fitness effect of an allele, or a group of alleles, can be categorized as additive, dominant or recessive, or as part of a more general epistatic network. A large body of existing work is devoted to the development of statistical methods for the detection and quantification of selection using DNA sequencing data, including comparative genomics and the sequencing of population samples [1–3]. However, much less progress has been made toward developing methods to identify the mode of selection as additive, recessive or dominant. Substantial experimental work in the last 50 years has been devoted to identifying the average dominance coefficient in model organisms, often with disagreement between different studies and techniques [4, 5]. These studies, in an attempt to identify the relationship between dominance coefficients and selective effects, largely focus on mutation accumulation experiments and subsequent laboratory propagation, determining dominance coefficients from the viability of crosses [4, 6]. At least one study attempts to determine the relationship between dominance coefficient and selective effect from natural populations, propagating crosses directly from wild-type samples, however the methodology relies on the often inapplicable assumption of mutation-selection balance [7]. A particularly useful overview of various techniques and studies can be found in [8], with some more modern techniques described in [9]. Additionally, more recent work taking advantage of a large amount of yeast knockout data has made progress towards quantifying the distribution of dominance effects (restricted to the discussion of nonsense mutations), with emphasis on the variance and skew of this distribution [10, 11].

Despite these substantial steps forward, all of the methods employed rely on the ability to rapidly breed laboratory-friendly organisms, either for the purposes of mutation accumulation or production of homozygotes and heterozygotes through crosses. Unfortunately, such techniques are infeasible when dealing with long-lived macroscopic organisms, particularly in the case of humans. In the present work, we hope to provide steps towards the development of techniques applicable to natural populations of such organisms by making use of naturally occurring demographic events and describing the dynamic response of populations to such events.

The genetics of model organisms and of human disease provide plenty of anecdotal evidence in favor of the general importance of dominance [12]. Although genome-wide association studies suggest that alleles of small effects involved in human complex traits frequently act additively, estimation of genetic variance components from large pedigrees suggests a substantial role for dominance in a number of human quantitative traits; LDL cholesterol levels, for example, have a substantial dominance component, as shown in [13]. Alleles of large effects involved in human Mendelian diseases often behave similarly to large effect (and even lethal) spontaneous and induced mutations in model organisms, such as mouse, zebrafish, or flies, that are frequently recessive [4, 14]. In spite of these observations, the role of dominance in population genetic variation and evolution remains largely unexplored in the majority of diploid species and no formal statistical framework is currently available to identify dominance coefficients in natural populations deviating from mutation-selection balance.

A number of theoretical studies suggested that demographic processes associated with the increase in variance of allele frequency distribution result in a more efficient removal of recessive deleterious alleles [15–18]. Such demographic scenarios include population bottlenecks, population subdivision, range expansion, and inbreeding. Increase in the variance of allele frequency distribution during a bottleneck can be characterized by inbreeding coefficient (even in case of a panmictic population). For structured populations, the increase in variance is characterized by F_ST. Substantial theoretical work and associated experimental studies explored the removal of recessive variants due to increased inbreeding coefficient during sustained population bottlenecks [19–22]. Additionally, several studies note that bottlenecks have a strong effect on nonadditive variation, specifically loci with epistatic interactions [19, 23–30]. To complement these analyses, we focus on genetic variation in panmictic populations that experienced a population bottleneck and subsequent re-expansion, similar to the scenario recently analyzed in [30]. Using a combination of theoretical analysis and computer simulations, we demonstrate that recessive selection can be qualitatively distinguished from additive selection in populations that recently recovered from a temporary bottleneck, and detail the dynamics of the average number of mutations per haploid.

An important study by Kirkpatrick and Jarne [31] qualitatively described how, perhaps counterintuitively, the number of deleterious recessive alleles per haploid genome is transiently reduced after re-expansion following a population bottleneck, while the number of additively or dominantly acting alleles is increased. We focus on this insight and quantitatively extend the analysis of these dynamics to show that, in spite of a well-documented increase in the frequency of some recessively acting variants in founder populations, the average number of deleterious recessive alleles (with dominance coefficient h ≪ 0.5) carried by an individual is reduced as a consequence of the bottleneck. With the growing availability of DNA sequencing data in multiple populations, these results demonstrate the potential to directly evaluate the role of dominance, either on a whole genome level, or in specific categories of genes.

Population bottlenecks are a common feature in the history of many human populations. For example, the “Out of Africa” bottleneck involved the ancestors of many present-day human populations. Numerous recent bottlenecks affected, among others, the well studied populations of Finland and Iceland. More generally, bottlenecks followed by expansions are standard features in the recent evolution of most domesticated organisms, including an analogous “Out of Africa” event in Drosophila melanogaster [32], highlighting the ubiquity of these events in natural populations. We suggest that complex demographic history may assist rather than complicate statistical inference of selection in population genetics.

Here we focus on a comparison between two populations that recently split, after which their demographic histories diverged, one exhibiting a founder’s event (a population bottleneck followed by subsequent re-expansion), and the other maintaining a fixed population size. We analyze their accumulated differences to shed light on the type of selection dominating the dynamics of deleterious alleles, and show that the average number of mutations per individual, 〈x〉, is dependent on the mode of selection characterized by the average dominance coefficient, h. We introduce a measure B_R (the “burden ratio” defined below) that is the ratio of per-haploid deleterious allele accumulation in the two populations. This potentially allows for the qualitative distinction between predominantly additive selection (h ≈ 0.5), where mutations accumulate due to relaxed selection during a bottleneck, resulting in B_R < 1, and predominantly recessive selection (h ≪ 0.5), where homozygous deleterious mutations are purged from the population after re-expansion from the bottleneck, resulting in B_R > 1, as shown in Fig 1.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Response of the B_R statistic for additive and recessive variation.

A schematic representation of two populations is presented above (A). Initially a single population prior to the bottleneck event, the populations split and have distinct demographic profiles. The equilibrium population maintains a constant size for easy comparison to the founded population. The latter drastically reduces its population size to N_B for a short time T_B during the founder’s event. Our statistical comparison between populations is represented here for cases of purely additive (B) and purely recessive (C) variation. The statistic B_R > 1 for recessive variation (dominance coefficient h = 0) and B_R < 1 for additive variation (h = 1/2), providing a simple indicator for the primary mode of selection of polymorphic alleles in the populations.

https://doi.org/10.1371/journal.pgen.1005436.g001

For qualitative demonstration and development of intuition, the analysis assumes strictly additive and strictly recessive selection with a highly idealized demography. However, this behavior is not restricted to the simplified demographic model presented in this paper, but rather suggests a quite generic qualitative signature for the presence of recessive (or near-recessive) selection in comparison between two populations, one of which experienced a bottleneck event. Additionally, our simulations suggest the potential to distinguish between partially recessive and additive alleles, as the change in the qualitative behavior of B_R occurs at intermediate values of the dominance coefficient, h. The temporal dependence of the “critical dominance coefficient”, h_c, describing the boundary between B_R > 1 and B_R < 1, as well as the sensitivity to partial recessivity, is discussed in the S1 Text.

To ask whether the behavior of the B_R statistic is consistent with the dynamics of recessive selection in natural populations, we perform a statistical analysis of genes annotated in the literature as causing autosomal recessive (AR) disease. We use the “Out of Africa” event to differentiate between variation in African and European populations, potentially allowing for the identification of recessive selection in natural human populations. We find that sets of AR disease genes show a statistically significant deviation from neutrality, with B_R > 1. This suggests that at least some disease-associated genes with autosomal recessive mode of inheritance may be under recessive selection. Although this observation is not surprising, it is nontrivial, as disease genes could be neutral, highly pleiotropic, or contain variants with different modes of inheritance. This analysis demonstrates the potential to use our methodology to identify sets of genes under predominantly recessive selection.

Results

Population dynamics

As noted above, a qualitative insight on the effect of the bottleneck on recessive variation was previously obtained by noting that the expected change in frequency of recessive allele is accelerated due to the increased variance of allele frequencies (inbreeding coefficient). We offer a different approach and attempt to quantitatively describe the difference in dynamics between additive and recessive variation.

We follow the expected number of mutations per chromosome in the population, noting that it is simply the first moment of SFS. (2)

When multiplied by s, this is the effective “mutation load” of each individual in the additive case, but in the case of purely recessive selection this is not proportional to the fitness, as selection acts only on homozygotes. We refer to this statistic generally as the “mutation burden” to avoid assumption of any given mode of selection. As described below, comparison between the mutation burden in the equilibrium and founded populations in the form of the “burden ratio”, B_R, may prove useful in the identification of sets of alleles under recessive selection. (3)

To gain intuition for this qualitative difference, we work to quantitatively understand the population dynamics in a simple demography, first for purely additive selection, and then for purely recessive selection for comparison.

Additive selection and response to a bottleneck.

The initial site frequency spectrum for purely additive alleles is given by Eq (1) with h = 1/2. (4)

Here θ₀ = 4N₀U_d. In the deterministic limit, when 2N₀s ≫ 1, the SFS rapidly decays as x → 1 simplifying the functional form [34]. We approximately compute the initial mutation burden as follows. (5)

This describes the deterministic mutation-selection balance for mutations under strong selection. Now we deviate from equilibrium by reducing the population size to 2N_B chromosomes, representing a population bottleneck. The effect that a bottleneck has on the site frequency spectrum is twofold: a fraction of alleles are removed from the population due to increased random drift, and the mean of the remaining alleles occurs at higher frequency. The dynamics of the distribution ϕ(x, t) during such a change in demography can be computed from Kolmogorov’s forward equation, as detailed in the S1 Text. The first moment of the distribution, the mutation burden, follows the temporal dynamics derived from summing the Kolmogorov equation over all alleles in the genome, and takes the following form. (6)

As discussed in [35, 36], the burden of additive mutations is not directly affected by drift, as the drift term vanishes from the dynamics of the first moment, however the dependence on the second moment introduces an indirect dependence on drift. In the strong selection regime, in the limit where 〈x²〉 ≪ 〈x〉, extinction of some alleles is exactly compensated for by an increase in the frequency of other alleles. This is true in the equilibrium distribution prior to the bottleneck when N₀s ≫ 1, where and , as can be computed directly from Eq (4). During the bottleneck the mutation burden 〈x〉 monotonically increases; the second moment 〈x²〉 increases, as well, reaching a maximum value in the case of a long bottleneck where it has re-equilibrated and scales as . Provided N_Bs ≫ 1, the second moment is guaranteed to be subdominant to the first moment, such that Eq (6) is well approximated by in the strong selection limit with the well known solution . For a finite duration bottleneck of T_B generations, the population immediately recovers and remains in mutation-selection balance throughout the bottleneck with final burden . After instantaneous re-expansion to the initial population size, the dynamics of the distribution ϕ(x) are completely analogous to those inside the bottleneck in this limit, such that the mutation burden never deviates during the demographic perturbation.

In the opposite limit of completely relaxed selection during the bottleneck, the dynamics of the mutation burden are completely driven by the influx of new mutations. For a bottleneck with duration T_B generations, the net effect of mutation accumulation due to relaxed selection is given simply by the following expression. (7)

Additionally, one can show that the second non-central moment gains an analogous contribution in addition to the net effect of drift. (8)

Here we have expressed the second moment as a function of the bottleneck intensity . Immediately after re-expansion from the bottleneck, selection is again efficient, so that the dynamics are completely described by Eq (6). Although the second moment is increased due to relaxed selection during the bottleneck, we find that this increase is negligible in comparison to the direct accumulation of the first moment provided that I_B ≪ 1. As a result, the primary effect of the bottleneck in this limit is to accrue new mutations that are subsequently purged when selection is again efficient in the re-expanded population. The dynamics for the two limiting cases can be summarized as follows. (9)

Here T_B represents the duration of the bottleneck, and t represents the time after re-expansion from the bottleneck. The top result is for the deterministic strong selection limit, and the bottom result is for the case of completely relaxed selection such that during the bottleneck the dynamics are effectively neutral. We note that 〈x〉_founded ≥ 〈x〉_eq at all times in both limiting cases, and asymptotically decays to the equilibrium frequency on a timescale given by the strength of selection of the accumulated deleterious mutations. In the case of a single-generation bottleneck, we find that the mutation burden is only slightly shifted even if selection is fully relaxed, resulting in effectively no observable change at either limit. Our statistical measure, the burden ratio B_R, in the additive case can be written approximately as follows. (10)

We note that the mutation burden in each population is proportional to the mutation rate, such that mutation rates cancel as long as they are the same in both populations leaving B_R independent of mutation rate.

As we will see in the following sections, recessive selection results in a depleted mutation burden with corresponding values B_R > 1, proving a contrast to the additive scenario and is thus a signature of recessivity.

Recessive selection and dynamics of the mutation burden.

Prior to the bottleneck, the initial site frequency spectrum for alleles under recessive selection is given by the h = 0 limit of Eq (1). (11)

At low frequencies with the spectrum decays more slowly than in the additive case, representing alleles protected from recessive selection by existing primarily in heterozygous form. In contrast, at high frequencies the spectrum decays faster than the additive exponential decay, falling off as e^−2N₀sx2.

Single-generation population bottlenecks.

First, we restrict our analysis to a single-generation bottleneck with intensity I_B = 1/2N_B, as this provides insight into the non-equilibrium response of the frequency spectrum to a downsampling event. Later, we extend our analysis to finite bottlenecks that persist for T_B generations, with total intensity I_B = T_B/2N_B. We represent the increase in drift due to a single-generation bottleneck by downsampling. During this time step, N_B diploid individuals are chosen at random from the initial larger population of N₀ individuals. (12)

Binomial sampling gives the distribution ϕ_B of deleterious alleles with frequency x = k/2N_B. There is a loss of allelic variation due to the bottleneck, corresponding to the k = 0 term in Eq (12).

Re-expansion is modeled as up-sampling the distribution ϕ_B(x) from N_B to N₀ diploid individuals, which has a negligible effect on the first and second moments of the distribution. As a result of drift to higher frequencies during the bottleneck, much of the existing variation appears in homozygous form immediately after the increase in population size. These individuals are rapidly selected out of the population, driving high frequency alleles to lower frequencies on a very short time scale, as was initially described in [31]. Since drift is once again suppressed, selection becomes far more efficient, particularly for alleles of large selective effect.

The time evolution of ϕ after the bottleneck is given by the forward Kolmogorov equation for recessive selection (see S1 Text). The mutation burden follows the time dependence, (13)

Here we suppress a selection term proportional to 〈x³〉 of in analogy to the additive case. Since recessive selection depends quadratically, rather than linearly, on the allele frequency, the increased variance of the distribution drives the motion of the mutation burden. Alleles with frequency appear in homozygous form and are rapidly pushed down to lower frequencies. This happens on a time scale of order s^−1/2 and effectively reduces the variance, slowing the decrease in the mutation burden 〈x〉. New mutations introduced during this period slowly drift to appreciable frequencies, replacing those lost in the bottleneck. This process is drift controlled, rather than selection controlled, and thus occurs on a time scale of generations. As a result, the mutation burden quickly decreases due to selection immediately after the bottleneck until it slows to a stop, and then gradually increases as the population accumulates new mutations and re-equilibrates.

A minimum in the mutation burden 〈x(t)〉_founded occurs when the time derivative vanishes. This corresponds to a characteristic time scale associated with the selective effect s, where our statistical measure is maximized. Since this time scale is shorter than the time scale of drift, we can imagine rescaling time by the effective population size 2N₀ and then working in the perturbative regime t/2N₀ ≪ 1. This allows us to Taylor expand near the re-expansion time t = 0 to understand the motion of the mutation burden at times soon after the bottleneck. (14)

To understand the time dependence of 〈x²〉, specifically the time derivative, we analyze the higher moments in the same fashion as employed for the first moment in Eq (13). All relevant moments are computed in the S1 Text and we note sufficient convergence to validate this expansion. This allows for the re-expression of Eq (14) to second order in t in terms of the first three moments of the site frequency spectrum immediately after re-expansion. The moments of the post-bottleneck initial distribution can be written in terms of the initial equilibrium distribution using the integral form given in Eq (12). Details of this calculation appear in the S1 Text. In the strong selection limit 2N₀s ≫ 1 these initial equilibrium moments are readily approximated by standard convolutions of a polynomial with a Gaussian. Suppressing subdominant contributions in the limit corresponding to a low intensity bottleneck, we find the following approximation to the trajectory of the mutation burden immediately after the bottleneck re-expands. (15)

Concentrating on this second order expansion in t, the time after re-expansion from the bottleneck, we find that the curve first drops from its initial value , quickly reaches a minimum, and is then brought back up by the positive second order term. The location of the minimum is easily found to have the following parameter dependence. (16)

The second derivative is positive at this extremum, implying a local minimum. Plugging t_min into our expression for 〈x(t)〉 in the limit N₀s ≫ 1, we find the following minimum value for the average number of recessive deleterious mutations per genome following a bottleneck. (17)

We note that is the approximate mutation burden for the equilibrium distribution in the deterministic 2N₀s ≫ 1 limit, allowing us to simply write the extreme value of the B_R statistic as follows. (18)

The burden ratio is again independent of mutation rate due to cancellation, as discussed above. We find the following dependence on time in immediate response to a population bottleneck. (19)

This expansion is only valid in the small time limit where the quadratic term is subdominant, such that all values are positive. As seen in simulations described in the following section, for recessive deleterious mutations, the burden ratio remains positive at all times.

This precise result applies strictly in the limit of a strong, single-generation bottleneck, where N₀ ≫ N_B. Additionally, the technique used to compute integral expressions required the strong selection limit 2N₀s ≫ 1. Analysis of higher order contributions to the trajectory are made substantially easier by restricting to the low bottleneck intensity limit , which may be biologically reasonable in human populations, for example, where many identified founding events are relatively short and on the order of N₀ ∼ 10⁴ or N_B ∼ 10³, with the notable exception of the Out of Africa event (see further discussion in S1 Text on general dominance coefficients). Despite these analytic restrictions in parameter space, our simulations described below indicate that the signature of B_R > 1 is ubiquitous for populations under predominantly recessive selection.

Extended population bottlenecks.

We argue that for the case of relatively low intensity bottlenecks, where intensity is defined as I_B ≡ T_B/2N_B ≪ 1, we can approximately express the magnitude of B_R using a simple substitution (2N_B)⁻¹ → I_B. This is equivalent to the claim that for low intensity bottlenecks, the B_R statistic depends only on the ratio of the bottleneck time to the bottleneck population size, and any explicit dependence on T_B occurs in subdominant contributions. This intuition is confirmed by simulations described in below, where we show that the accuracy of our analytic approximation breaks down as I_B → 1. For short bottlenecks with I_B < 1/10, the approximation of a single-generation sampling event remains sufficiently accurate, even for strong selective coefficients s ∼ 0.1. Under this trivially extended single-generation approximation, B_R(t) can be written in terms of the intensity of a short bottleneck in the low intensity limit as follows. (20)

The B_R of maximum effect, has a magnitude given approximately by, (21)

For illustration of the behavior described in the above analytics we present a time series of recessive simulations with curves representing various selection coefficients in Fig 2. The time dependence of the B_R statistic is plotted to demonstrate the simulated population’s response to a founder’s event. Crucially, we find that the peak B_R values vary in both magnitude and time as a function of s, as is consistent with our analytic understanding and intuition.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Time dependence of the B_R statistic after re-expansion.

The time dependence of B_R(t) after a population bottleneck is shown for for alleles under recessive selection (h = 0) for various selection strengths. Peak B_R values vary in both magnitude and time as a function of s. The founded population was simulated with 2N₀ = 20000, 2N_B = 2000, and T_B = 200 and plotted for 5000 generations after re-expansion.

https://doi.org/10.1371/journal.pgen.1005436.g002

Transient response and time of observation determine detectable selection coefficients.

Thus far, we have detailed the dynamic dependence of a set of alleles in a population, all with selective effect s, in response to demographic perturbation in the form of a bottleneck. Notably, for recessive selection, a peak response occurs in the B_R statistic at some time t_min after re-expansion. In general, both the magnitude of B_R(t_min) and the time of the peak itself depend sensitively on the selection coefficient. In general, a distribution of mutations with different selective effects will be present, many of which may be simultaneously polymorphic in a given population. Since alleles of different selective effect respond to the bottleneck on different time scales, one can ask what selective effect is most likely to be observed at a given time. For example, very strong selection has the tendency to peak and subsequently re-equilibrate immediately after the bottleneck, such that observation of alleles with large s is substantially more difficult at later times. On the other hand, alleles under relatively weak selection have a peak effect at very late times, such that at the time of data collection a statistically significant response may not yet have occurred.

We would like to understand the transient behavior of the burden ratio B_R(t), as well as the value of the selection coefficient s for which B_R is largest at a given time. When comparing theory to population data, one has little control over the demographic history, and thus it becomes important to understand the selective coefficient that dominates at the time of observation. According to the time dependent expression in Eq (20), we expect the effect to decrease quite rapidly for very large s. However, the peak occurs quite early in the case of larger s values, allowing the mutation burden to equilibrate over a longer period of time between the peak and observation to return to mutation burden values close to B_R ∼ 1. This tells us that the equilibration process is what reduces the magnitude of B_R for large s. In the case of very recent bottlenecks, the large s values dominate, but for later times of observation, this signal has partially equilibrated, potentially allowing a smaller s value to dominate the statistic. At a given time of observation t_obs, one can represent B_R(s, t_obs) as a function of various selection coefficients s. Fig 3 represents B_R(s) for a fixed t_obs for various dominance coefficients h. We concentrate here on recessive variation with h = 0, but note that a critical value occurs at some h_c where additive and recessive effects offset each other in the B_R statistic, the dynamics of which are detailed in S1 Text and illustrated in S1 Fig). Based on our analytics, we expect the peak to shift from extreme high s values at early times to extreme low s values at late times, eventually dissolving into neutrality. We take the s derivative of Eq (20) to find the maximum at t_obs. (22) (23)

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. The B_R statistic at the time of observation.

ABOVE: At the time of observation t_obs, the value of B_R(t_obs) is plotted as a function of the average strength of selection s and dominance coefficient h. Dominance coefficients appear as solid lines with fully recessive selection (h = 0) at the top and purely additive selection () at the bottom. For strong selection B_R → 1 due to the rapid transient response. For weak selection B_R → 1 due to the nearly neutral insensitivity to the bottleneck. For some intermediate dominance coefficient h_c, a critical value occurs (h_c ∼ 0.25 in the example shown, but explored more generally in S1 Text) where additive and recessive effects cancel, yielding B_R(h_c) ∼ 1. A low intensity bottleneck (I_B = 0.05) is shown, with parameters 2N₀ = 20000, 2N_B = 2000, T_B = 100, and t_obs = 1000. BELOW: The same range of parameters is plotted for a realistic demographic model of the Out of Africa event comparing Africans and Europeans [48], where B_R = 〈x〉_African/〈x〉_European. The European bottleneck has estimated intensity I_B ∼ 𝒪(0.5), an order of magnitude stronger than the simple bottleneck above, allowing for potentially observable deviations from B_R ∼ 1 if a large fraction of analyzed variants act recessively with h < h_c ∼ 0.25.

https://doi.org/10.1371/journal.pgen.1005436.g003

One can easily show that the second derivative evaluated at this point is negative, confirming that this is a maximum. This result matches our intuition: maximum s values of B_R(s, t) are found at high s for early times, s_max(t → 0) ≫ 1, and at low s for late times, s_max(t → ∞) ≪ 1. This is qualitatively observed in our simulations by comparing the relative values of B_R(s) as a function of time.

As the effect is transient, we can define a relaxation time t_relax corresponding to the vanishing of any response to the bottleneck. This is given by determining when s_max is dominated by effectively neutral variation at roughly s_max ∼ 1/2N₀. After this time, B_R(s, t) cannot be differentiated from one for any s. (24)

We note that the return to equilibrium happens on a time scale faster than random drift, even for the weakest selective effects, thus validating our perturbative approximations using t/2N₀ ≪ 1. Higher order time dependence in Eq (20) may substantially correct this estimate, but we feel that the presentation of this methodology is conceptually important and provides a greater understanding of the transient dynamics of population response to bottlenecks. As it is relevant to human populations, we note that if both populations expand exponentially after the bottleneck, the effect may persist long beyond t_relax. This is explored analytically in the S1 Text.

Comparison of analytic results to simulations

We checked our analytic results using a forward time population simulator, described in detail in the S1 Text. Given the ubiquity and analytic simplicity of the exponential decay in the additive scenario, we focus here on our predictions for recessive variation. We compare analytic expressions of B_R(t_min) at the peak response given in Eq (21) for various selection coefficients. We simulated a wide range of bottleneck parameters to probe the limitations of our theoretical understanding. In Fig 4, we demonstrate the accuracy of our analytic results, by plotting the ratio of the simulated values of B_R(t_max, s, I_B) to our analytic predictions B_R(t_max, s, I_B) as presented in Eq (21). We arrange our simulated data by bottleneck intensity I_B, as we expect the single-generation bottleneck approximation to break down as intensity is increased due to longer bottleneck duration T_B ≫ 1. As plotted, complete agreement between simulated data and analytic predictions is represented by a flat line at . As expected, we find deviations as we approach the limitations of our perturbative approximation, roughly around T_b ∼ 2N_B/10 when I_B ∼ 0.1. Below these higher intensities, we find quite good agreement for all parameter sets well below 10% error, even at I_B = 0.05. Further comparison between simulation and analytic results is presented in S1 Text and illustrated in S2 Fig.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Comparisons of analytic and simulation results.

Maximum response values of the burden ratio B_R(t_min) are plotted for recessive selection as a function of bottleneck intensity. A wide range of parameter sets is plotted with all combinations of 2N_B = {2000,1000,400,200,100}, s = {0.1,0.02,0.01,0.001}, T_B = {200,100,50,20,10}, each simulated for 10⁸ nucleotide sites. For relatively low intensity bottlenecks we note excellent agreement over the parameter ranges plotted. Intensities with I_B = T_B/2N_B > 0.1 are excluded, as the single-generation bottleneck scaling breaks down in favor of a long bottleneck scaling. The approximation necessarily weakens for simulations that represent longer bottlenecks, and only for strong selective coefficients, as expected. This quantifies the limitations of the single-generation bottleneck approximation, as we observe substantial deviation only around I_B = 0.1 and with selection strength s = 0.1.

https://doi.org/10.1371/journal.pgen.1005436.g004

Empirical detection of recessive selection

The B_R statistic provides a qualitative indication of recessive selection (h ≪ 0.5), in that values over one theoretically correspond to recessivity. This corresponds to a reduction in the average number of deleterious alleles per haploid locus in a founder population relative to a non-bottlenecked population. To test whether the statistic is sensitive to recessive selection, we analyze human exome data from the Exome Sequencing Project (ESP) [37]. We compare European Americans (EA), known to have undergone a relatively intense bottleneck during the “Out of Africa” event, to African Americans (AA), who have substantial African ancestry that did not experience this founder’s event. We aggregate a set of genes and compute the per-haploid mutation burdens, 〈x〉_AA and 〈x〉_EA for each gene set by summing the frequencies of all variants occurring in those genes within the AA and EA populations separately, such that and . This provides a group burden ratio score for the entire gene set ranging from predicted additive (or dominant) with B_R < 1 to predicted recessive with B_R > 1. While this strategy could in principle be applied directly to a single gene, substantial statistical fluctuations tend to make this measure unreliable on the individual gene level.

We assemble sets of genes associated with known autosomal recessive (AR) diseases, some of which are potentially under recessive selection, and compute a corresponding B_R score. In the absence of pleiotropy and the presence of purifying selection against these disease phenotypes, we naively expect these genes to act under partial (h < 0.5) or total recessive selection (h ≈ 0). We check for significant deviation from B_R = 1 in several gene sets: 44 genes associated with diseases with “autosomal recessive” in the name of the disease with at least 5 annotated variants in the Human Gene Mutation Database (HGMD), 37 genes associated with congenital hearing loss (HL) and found only with AR mode of inheritance in a clinical genetics lab, and 1348 genes with Clinical Genomic Database (CGD) AR annotations [38–40]. Additionally, we aggregate non-overlapping HGMD and HL genes into a larger combined list of 72 genes.

To compute B_R gene scores, we assume that derived variants at a given locus are deleterious, and include derived alleles of all frequencies, including those fixed in one or both of the populations. We restrict our analysis to nonsense variants and non-synonymous variants predicted to be damaging using a human-free version of PolyPhen2 [36] developed to remove bias due the ancestry of the human reference. Derived alleles fixed in one of the two populations are included in the analysis of the burden, as they contribute to the weighted mean 〈x〉.

We estimate significance using bootstrapped standard errors, as described in detail in the S2 Text. First, we compute the burden ratio for all genes in the genome, and find no statistical deviation from one, replicating previously published results [35, 36]. Analysis of the CGD gene set again shows no statistically significant deviation from one. Given the whole genome result, this is not unexpected, as this set of over 1000 genes is plausibly large enough to representatively sample the set of all genes. It is likely that many of these genes have only one or a few variants under recessive selection, with the rest being neutral or even dominantly acting. In contrast, we find statistically significant B_R > 1 values in the potentially more reliable HGMD and HL gene sets, despite their small size, as well as in the combined set. We partially replicate our results from ESP using an independent dataset, from the 1000 Genomes Project (1KG), again finding statistical significance in the HGMD disease gene set [41]. A detailed discussion of the data sets and statistical analyses used is provided in S2 Text and detailed in S1 Table (with full gene lists included in a supplemental spreadsheet).

We find statistical significance for two separately obtained disease gene sets, as well as in the combined set. The HGMD gene set is significant in both ESP and 1KG. Additionally, we find null results in nearly all controls presented in S2 Text and detailed in S2 Table. Together, the empirical analysis provides suggestive evidence that genes associated with autosomal recessive disease and thus potentially under recessive selection can show significant burden ratio values B_R > 1. The resulting analysis is summarized in Table 1. In light of these findings, we believe we have demonstrated the potential usefulness of this method for identifying sets of genes under recessive selection.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Empirical results for autosomal recessive disease gene sets.

https://doi.org/10.1371/journal.pgen.1005436.t001

Given the significant observed values of B_R > 1 in these gene sets, one can gauge the degree of recessivity for a given set. Specifically, we can readily estimate the average dominance coefficient for damaging and nonsense mutations within a set of genes under the assumption that these mutations all act with a single average dominance coefficient and an average selection strength . We caution that estimates using a single h and s pair of values for all derived mutations may be inappropriate if there is substantial variance in either or both of these parameters. In the absence of information about the variance in dominance coefficients, we believe this approximation may still be informative (if only as a rough guide) in gene sets that clearly deviate from neutrality. Given the details of the Out of Africa demography, the data for the HGMD gene set are consistent with an average dominance coefficient (with 95% confidence), however, this bound is conservative over all possible values of the average strength of selection in this gene set. For average selective strengths of in damaging and nonsense variants, we find that the corresponding allowed average dominance coefficients are (with 95% confidence), respectively. Note that the non-monotonicity in these values is a consequence of the behavior shown for the Out of Africa demography in Fig 3. Additionally, all average dominance coefficients for HGMD are inconsistent with weak average selective strengths below roughly . Complementary population data from distinct founder’s events may provide stricter bounds on both the average dominance coefficients and average selective strengths for a given gene set.

Discussion

The increase in prevalence of recessive phenotypes following population bottlenecks has attracted the interest of geneticists for a long time [19, 42]. Theoretical analysis of allele frequency dynamics in a population expanding after a bottleneck suggested that frequency of an individual allele may rise due to increased drift [42–44]. Here, we focus on a more general question of the collective dynamics of recessively acting genetic variation. In line with the qualitative description found in [31], our analysis suggests that the number of recessively acting variants per haploid genome is reduced in response to a bottleneck and subsequent re-expansion. Generally, we have demonstrated that features of the derived allele spectrum of recessive deleterious polymorphisms behave distinctly from additively acting variation following a population bottleneck and subsequent re-expansion. The response of additive variation depends crucially on the average number of deleterious alleles, and on the number of generations for which selection is relaxed during the bottleneck. In contrast, the dynamics of recessive variation crucially depend on the variance of the site frequency spectrum, rather than the average number of mutations per individual, such that the accumulation of deleterious mutations can respond strongly even to a single-generation bottleneck. Importantly, the temporal dynamics of the accumulation of deleterious alleles depends qualitatively on dominance coefficient and quantitatively on selection coefficient. The qualitative dependence on dominance coefficient suggests that one can learn about recessivity from analysis of the population dynamics in response to a founder‘s event. If the variation is additive, the number of deleterious variants per a haploid genome is larger in a bottlenecked population than in a corresponding equilibrium population. If the variation acts recessively, this number is smaller. The selection coefficient determines the timing of response to a bottleneck.

By explicitly analyzing the non-equilibrium response to a bottleneck, we suggest that naively confounding demographic features may actually shed light on underlying population genetic forces. In realistic populations, for example in modern humans, substantial work has been done to identify and understand the recent demographic history of geographically disparate populations [37, 45–54]. In a recent paper, Simons, et al. [35] use the B_R statistic on the whole genome level to empirically compare the accumulation of mutations in European Americans and African Americans. The authors find no statistically significant differences in the whole genome mutation burden of these populations, a result that was extended to all two-point comparisons between a diverse set of humans by Do, et al. [36]. To explain this observation, Simons, et al. derive a complementary theoretical treatment of the dynamics of segregating alleles using branching process techniques and extensive simulations, providing results that are consistent with those presented here.

In the case of the “Out of Africa” event, a historically substantiated and believable demographic model can be used to understand the difference between African and European populations since their divergence. The comparison between populations that have and have not undergone a bottleneck can be used to elucidate plausible selection and dominance coefficients by making use of a simulated version of this demography. As shown in Fig 3 for the comparison between Africans and Europeans, a realistic demographic model can be used to bound the selection and dominance coefficients in modern populations based on a single observation, such as those detailed in [35, 36].

Although the net number of recessive deleterious mutations is reduced as a consequence of a founder‘s event and subsequent re-expansion, the fitness of individuals carrying these alleles is not necessarily increased, as the number of homozygotes is known to increase after a population bottleneck. However, the number of heterozygous deleterious sites, or the average carrier frequency for associated alleles, is suppressed, such that the mating of individuals from disparate bottlenecked populations may result in a decreased incidence of recessive phenotypes in such mixed lineages. In studies of model organisms, this may have applications when comparing laboratory populations founded from a few wild type individuals to their corresponding natural populations.

We have demonstrated that analysis of the B_R statistic on the gene set level shows significant deviations above one in genes known to be responsible for autosomal recessive human disease. In principle, the results of this study can be extended to the analysis of any specific groups of genes beyond those with a known mode of inheritance. Sufficiently large subsets of alleles that are medically relevant may be analyzed in humans to identify the mode of selection for candidate variants of potentially recessive diseases.

In sum, the non-equilibrium dynamics induced by demographic events is an essential, and indeed insightful, feature of most realistic populations. Population bottlenecks, abundant in laboratory populations and in natural species, have the potential to provide a novel perspective on the role of dominance in genetic variation.

Methods

Simulation details. We performed analysis using a forward time population simulator, custom written in C, available at http://genetics.bwh.harvard.edu/wiki/sunyaevlab/dbalick. For computational speed, the simulator only keeps track of allele frequencies in a freely recombining diploid system, rather than containing full genome information. We use an infinite sites model with a mutation rate of 2 × 10⁻⁸ per generation per site. Allele counts in the current generation are sampled based on the frequencies in the previous generation x_old, the selection coefficient s, and the dominance coefficient h. We calculate the expected frequency x_current in the current generation as: (25)

The simulator has arguments for per base mutation rate U_d, selection coefficient s, and dominance coefficient h, with a default burn-in of 300,000 generations where sampling occurs every 100 generations in sped-up mode before transitioning to sampling every 1 generation at 1000 generations before time t = 0.

The code was designed to allow for flexible demographic histories, in order to accurately represent events such as the “Out of Africa” migratory event in human population genetic history. For the purposes of comparison to our analytic results, we ran simulations for a simple, square bottleneck of varying population sizes for both the equilibrium population with size 2N₀ = 2 × 10⁴ and bottlenecked populations with temporarily reduced sizes of 2N_B = {2000,1000,400,200,100} for a duration of T_B = {200,100,50,20,10} generations. These simulations were performed under both purely additive (h = 0.5) and purely recessive (h = 0) selection, for a wide range of selection coefficients s = {1,0.1,0.02,0.01,0.001}. For simulations of a range of selective effects and dominance coefficients shown in Fig 3, we used a square bottleneck with parameter 2N₀ = 20000, 2N_B = 2000, T_B = 100, and t_obs = 1000 and a realistic Out of Africa demography detailed in Tennessen, et al. [48].

Human polymorphism data. We analyze exome data from the Exome Sequencing Project (ESP) and validate some of our findings using exome data from the 1000 Genomes Project (1KG)[37, 41]. We use available frequency information for polymorphic variants to compute an average per haploid mutation burden per gene for all genes in ESP in 1088 European Americans(EA) with largely European ancestry and 1351 African Americans (AA) with substantial African ancestry. In 1KG, we compare 85 Northern Europeans from Utah (CEU) to 88 Yorubans (YRI) by computing the same statistic. We sum these mutation burdens over genes of interest to compute an aggregate B_R score for a given gene set.

Human-free Polyphen2. To compute mutation burden gene scores for putatively deleterious mutations, we restrict our analysis to non-synonymous nonsense variants and variants predicted to be damaging using a human-free version of PolyPhen2 [36]. This software was developed to remove bias due to the mixed ancestry of the human reference sequence, and annotates derived alleles based on chimpanzee orthologs.

Disease gene sets. We use several lists of genes associated with AR diseases that we naively expect to act under partial or total recessive selection. First we compile a set of genes from the Human Gene Mutation Database (HGMD) only associated with diseases with “autosomal recessive” in the disease name [38]. We restrict this set to genes with at least 5 disease-associated variants to guarantee sufficient polymorphism and reduce noise in the B_R statistic. This set contains 38 genes that appear in the list of ESP scored genes (44 in 1KG) and is referred to as “HGMD”. We use Congenital Hearing Loss as an example of a polygenic, largely recessive disease. We obtained an annotated gene list of AR genes associated with hearing loss from the Laboratory for Molecular Medicine (LMM) [39]. This list contains 30 genes in ESP (37 in 1KG) and is referred to as “Hearing Loss”. Notably, this list excludes connexin 26 (GJB2), among other genes, which has additional association with AD hearing loss. Additionally, we assemble a combined list of all genes from HGMD and Hearing Loss, with a total of 60 genes in ESP (72 in 1KG) after removing overlap, referred to as “Combined”. To assemble a larger, though noisier gene set, we use all annotated AR genes in the Clinical Genomic Database, referred to as “CGD”, which contains 1268 genes in ESP and 1348 genes in 1KG [40].

Supporting Information

Acknowledgments

The authors would like to thank Ivan Adzhubei, Benjamin Good, Daniel Jordan, Alexey Kondrashov, Nick Patterson, Konstantin Popadin, Jonathan Pritchard, and Guy Sella for particularly useful discussions.