Explaining the timescale dependence of d_N/d_S through an exclusively purifying selection model poses several challenges. Firstly, fitting observed data with purifying selection requires a preponderance of mutations with extraordinarily small effects on fitness (selective coefficients, s), which are challenging to eliminate effectively (Haigh, 1978). Secondly, the occurrence of an adaptive event during the extensive time required to purge weakly deleterious mutations interrupts the purification of such mutations. Lastly, neutral bottlenecking processes, such as those observed during host-to-host transmission, exacerbate the accumulation of deleterious mutations. For most of this section, we will disregard these last two complications and focus on the problem of small s. To provide clarity, we first detail the classic purifying selection model.

Mutations can be divided into three classes, the first two of which accumulate at a constant rate per unit of time: synonymous mutations (S), neutral nonsynonymous mutations (Nneut), and non-neutral, transient, nonsynonymous mutations (Ntransient). We restate the timescale dependence of d_N/d_S as the observation that, in a population starting from a single wild type (WT) cell, the average number of non-neutral nonsynonymous mutations per cell in the population (N¯transient) increases and then asymptotes. Assuming an infinitely large population size and an infinite genome size (to circumvent saturation of mutations), the exclusive purifying selection model (Garud et al., 2019; Rocha et al., 2006) can thus be written as(1)dNdS=N¯neut+N¯transient3S¯

and(2)dN¯transient=UNdt−sN¯transientdt.

Here, UN is the non-neutral mutation rate per core genome per generation, s is the selective disadvantage of a non-neutral nonsynonymous mutation (or the harmonic mean of such mutations; see Appendix 1, Section 1.1), and t is the number of generations. The 3 in the denominator of Equation 1 accounts for the discrepancy in the number of potential nonsynonymous and synonymous sites (Yang and Nielsen, 2000). We solve for N¯transient by assuming N¯transient(t=0)=0 to obtain:(3)N¯transient(t)=UN(1−e−st)s.

We further simplify and combine these equations to create an equation for d_N/d_S with only two parameters as previously done (Garud et al., 2019). First, since d_N/d_S plateaus with time (Figure 1a), we have limt→∞N¯neut+N¯transient3S¯=N¯neut3S¯=α. Conveniently, α represents both the asymptote of d_N/d_S and the proportion of nonsynonymous mutations that are neutral. This allows us to leave only s as the other free parameter, obtaining (see Appendix 1, Section 1.1)(4)dNdS=α+(1−α)1−e−stst.

As sequence analysis is not privy to the actual number of generations, we approximate t assuming that synonymous mutations accumulate according to a molecular clock (t=dS2(1/4)μ), where μ is the mutation rate per generation per base pair, ¼ represents the proportion of random mutations that are synonymous (Yang and Nielsen, 2000), and 2 accounts for the fact that divergence is a measure between a pair of genomes. As selection and mutation are both in units per time, any change in μ results in a corresponding change in s. Both model fits and consequences are largely dependent on the ratio of these two variables (more on this below), and thus are not sensitive to the choice of μ. We use a relatively high mutation rate of 10^-9 per base pair per generation (Drake, 1991; Barrick and Lenski, 2013) as lower rate would imply even weaker purifying selection.

Fitting the data from Garud et al., 2019, we infer median values of α ≈ 0.10 (0.09–0.14) and s ≈ 3.5 × 10^-5 (2.6 × 10^-5-6.5 × 10^–5) across all species (‘Methods’, Figure 1a). Aggregating all of the data at once results in a similar optimal fit of α ≈ 0.11 and s ≈ 2.8 × 10^–5. The similarity across the 10 species is perhaps not surprising, given that all are human gut residents of the order Bacteroidales; these values are also in line with the values obtained previously from aggregating across all species (Garud et al., 2019). These values indicate a model in which only ~10% of nonsynonymous mutations are neutral and the remaining ~90% are so weakly deleterious that they are beyond the limit of detection of any experimental method to date (s ≳ 10^–3) (Gallet et al., 2012). Higher values of s that better reflect experimental observations (Kibota and Lynch, 1996; Trindade et al., 2010; Robert et al., 2018) result in poor fits to the data (Figure 1—figure supplement 3). While the implied proportion of deleterious mutations may seem high, deep mutational scanning experiments have revealed that most amino acid-changing mutations in essential genes are deleterious enough to be measured in the lab (Kelsic et al., 2016; Dewachter et al., 2023); complex real-world environments are expected to constrain an even larger fraction of the genome.

In finite populations, the presence of so many weakly deleterious mutations becomes quickly problematic. When s is smaller than UN, organisms without any deleterious mutations (or with the fewest number of deleterious mutations, the ‘least-loaded class’; Haigh, 1978) can be easily lost from a finite population before they outcompete less fit organisms and fitness decay begins to occur. The likelihood of loss depends on the population size and mutation-selection balance (UN/s), a parameter that estimates the average number of deleterious mutations per cell relative to the least-loaded class. Given a core genome of L = 1.5 ×10⁶ bp that can acquire deleterious mutations, we then expect 0.001 new deleterious mutations per genome per generation (UN=34(1-α)Lμ). Thus, the value of UN/s for the above fits is ~29, indicating that most cells in the population contain dozens of deleterious mutations (see Appendix 1, Section 1.2). With this value of the mutation-selection balance parameter, the frequency of mutation-free organisms in a population is extremely small, even for a population that starts without any deleterious mutations (<10^–12 after 100,000 generations). If the flexible genome also contains deleterious mutations, the least-loaded class is pushed down even further. Simulations substantiate this prediction of mutation accumulation and decrease in frequency of the wild type (Figure 2a, ‘Methods’).

The time until the least-loaded class is completely lost from the population depends on the strength of genetic drift. The strength of genetic drift is inversely proportional to population size in well-mixed populations (Gillespie, 2004), and in less well-mixed or otherwise nonideal populations, is inversely proportional to a smaller parameter, the effective population size, N_e. N_e is often estimated by assessing polymorphisms in a population (Gillespie, 2004) but is hard to estimate from data following a recent bottleneck. Because each individual’s gut microbiome is thought to be well mixed (census size = 10¹³) (Sender et al., 2016), it has been recently argued that N_e ≈ 10¹¹ reflects drift processes for dominant gut species (Ghosh and Good, 2022; Labavić et al., 2022). On the other hand, lower values of N_e ≈ 10⁹ or less have been estimated for global populations of bacteria (Bobay and Ochman, 2018) because of the slow rates of bacterial transmission across people. While this decrease in N_e when increasing scales may seem paradoxical, we note this use of N_e only reflects the magnitude of the force of drift; for other calculations in nonideal populations, census population size or other parameters should be used.

Without extremely large values of N_e, the least-loaded class will be lost recurrently, rapidly lowering the fitness of the population (i.e., Muller’s ratchet; Haigh, 1978). Assuming s is small and thus approximately additive, this recurrent process of fitness decay occurs roughly when the following inequality is satisfied (see Appendix 1, Section 1.2; Neher and Shraiman, 2012):(5)2sNee−UNs<<1.

Given UN/s = 29 as derived above, N_e > 10¹⁵ is required to avoid continual deleterious mutation accumulation and fitness decline (Figure 2b). Thus, the purifying model requires levels of drift unrealistic at the within-person or across-globe scales. Simulations confirm that deleterious mutations accumulate and compromise the ability of the purifying model to explain empirical d_N/d_S decay in reasonably finite populations (Figure 2c). Moreover, continuous accumulation of mutations in such populations decreases fitness so much that the average genome contains a sizable fraction (~10%) of deleterious alleles after 1 million years (Figure 2c), assuming N_e = 10⁹ and one generation a day (Korem et al., 2015). Even if this decreased fitness was biologically maintainable, the accumulation of so many deleterious mutations would lead to many potential adaptive back mutations, complicating the efficiency of purifying selection. Consequently, this value of UN/s is simply incompatible with a model where a vast majority of alleles are already optimal.

Lastly, the intolerance of the purifying model to adaptation and transmission is particularly problematic. Within-host adaptive sweeps have been observed in Bacteroides fragilis (Zhao et al., 2019) and other Bacteroides (Garud et al., 2019). Such adaptation interferes with inefficient purifying selection; deleterious mutations are likely to hitchhike (Desai et al., 2013) to fixation on the genomic background of adaptive mutations. Any given weakly deleterious mutation with s = 3.5 × 10^–5 cannot be purged from a within-host population on the timescale of human lifetime (assuming ~1 generation per day), and thus if any adaptive sweep occurred within that host, it would either hitchhike to fixation or be completely removed from the population. Similarly, deleterious mutations can also hitchhike to fixation during neutral transmission bottlenecks, thereby raising the average number of deleterious mutations per cell in the population, furthering mutation accumulation, and hampering the efficiency of purifying selection. Simulations confirm that even infrequent adaptive sweeps and bottlenecks have tangible impacts on d_N/d_S, including raising the asymptote (Figure 2—figure supplement 3).

Homologous recombination, which occurs at detectable rates within human gut microbiomes and within the Bacteroidales order (Liu and Good, 2024), cannot rescue a population from Muller’s ratchet when such weakly deleterious mutations are so frequent. If we assume a generously high rate of recombination, such that a mutated nucleotide is 500 times more likely to be reverted via recombination than mutation (r/m = 500) (Torrance et al., 2024; Liu and Good, 2024) and brings along a single linked synonymous mutation during each recombination event, the decay of d_N/d_S still cannot be recreated in a population of size 10⁹ and fitness will still decay (Figure 2—figure supplement 4). The inability of recombination to suppress mutation accumulation in this regime arises because the selective advantages themselves are still too small to sweep faster than the rate at which mutations accumulate. While recombination does allow d_N/d_S to eventually decay, the rate of decay is much slower than observed, resulting in a poor fit to the data (Figure 2—figure supplement 4). While higher values of N_e or a higher recombination rate could theoretically approximate the absence of linkage and escape of Muller’s ratchet, we note that the maximum r/m across bacteria is estimated to be <50 (Torrance et al., 2024) and our simulations are therefore conservative.

Our presentation so far has implicitly assumed that weak purifying selection has been acting continuously and that values of s are constant for any given allele over time. However, apparently weak purifying selection might theoretically emerge from mutations that spend periods under neutral selection (or even local positive selection) and larger periods under strong negative selection, with the estimated value of s reflecting the harmonic mean (Culyba and Van Tyne, 2021; Loo et al., 2020). However, such models will have a hard time overcoming mutation accumulation. For example, a model in which purifying selection acts only during transmission still cannot prevent mutation accumulation without unrealistic assumptions. In particular, the selection-at-transmission model would still require ~29 non-neutral mutations in the average adult population, which implies a very low frequency of the least-loaded class. Assuming each host’s population gets replaced once every 10,000 bacterial generations (~26 years), such a model would require the least-loaded class to be 6000× more likely to colonize them than the average genotype in the population ((1+10,000s)29). The presence of rare cells with strong selective advantages would suggest super-spreading across human microbiomes, which has yet to be reported in the human microbiome (Faith et al., 2013). More importantly, Muller’s ratchet would still click because of the low frequency of this least-loaded class.

If purifying selection cannot explain the decay in neutral mutations, what can? One particularly attractive process that removes nonsynonymous mutations over time is strong adaptive mutation and subsequent strong adaptive reversion of the same nucleotide when conditions change. Such reversions are likely to sweep in large populations when mutations are adaptive locally but deleterious in other environments (Ascensao et al., 2023). In the gut microbiome, these alternative environments could represent different hosts (Figure 3a) or environmental changes within a single host (e.g., diet, medication, other microbes). As an illustrative example, the presence of a bacteriophage in one gut microbiome might select for a loss-of-function mutation (premature stop codon or otherwise) in a phage receptor, driving this mutation to fixation in its host, but reverting to the wild-type receptor when transmitted to a phage-free host. Reversions are most likely when compensatory mutations that counteract a mutation’s deleterious effects are either scarce or not as beneficial as direct reversion (Levin et al., 2000) (i.e., provided a premature stop codon); we discuss models that include compensatory mutations later in this section.

Adaptive nonsense mutations have been observed to emerge frequently within individual people in both pathogens (Culyba and Van Tyne, 2021; Lieberman et al., 2011; Key et al., 2023; Shopsin et al., 2008) and commensals (Zhao et al., 2019; Barreto et al., 2023). Identifying reversions in vivo requires both high temporal resolution and deep surveillance such that the probability of persistence of ancestral genotype is removed (Snitkin et al., 2013) despite this difficulty, reversions of stop codons have been observed in mouse models (Sousa et al., 2017) and during an outbreak of a pathogen infecting the lungs of people with cystic fibrosis (Poret et al., 2024). While direct reversion has not yet been observed in gut microbiomes, premature stop codons are frequently observed. Among the 325 observed nonsynonymous de novo mutations in a study of within-host B. fragilis adaptation (Zhao et al., 2019), 28 were premature stop codons. This frequency is significantly higher than expected by chance (p=0.015; ‘Methods’). Moreover, 4 of the 44 mutations in 16 genes shown to be under adaptive evolution on this short timescale were stop codons. These same 16 genes show a signature of purifying selection on long timescales (Figure 1a).

Traditionally, mutational reversions of stop codons and other mutations have been considered exceedingly unlikely and have been ignored in population genetics (Tajima, 1996), with a few exceptions (Charlesworth and Eyre-Walker, 2007). However, for a bacterial population within a human gut microbiome, the likelihood of a mutational reversion is quite high. A single species within the gut microbiome can have a census population size of 10¹³, with generation rates ranging from 1 to 10 per day (Sender et al., 2016; Korem et al., 2015). Taking a conservative estimate of one generation per day and a within-person N_e of 10¹⁰ (e.g., bacteria at the end of the colon may not contribute much to the next generation; Labavić et al., 2022), reversions become highly probable (Figure 3b; see Appendix 1, Section 2.2). Given a mutation rate of 10^–9 per site per generation, we anticipate 10 mutants at any given site each generation. In the large population sizes relevant for the gut microbiome, a beneficial mutation will then take substantially longer to sweep the population than occur, with values of s_ben > 1% generally sweeping within 10 years (Figure 3b). Consequently, if selection strongly benefits a reverting mutation, a genotype with a beneficial mutation is essentially guaranteed to emerge within days to weeks and replace its ancestors within the host within months to years.

Given its plausibility, we now consider if the reversion model can explain the observed decay of d_N/d_S. The dynamics of the reversion model can be given by. (6)dN¯transientdt=1τflip(nloci−N¯transient)−1τflipN¯transient

With the corresponding solution for N¯transient being (see Appendix 1, Section 3.1)(7)N¯transient(t)=nloci2(1−e−2tτflip).

Here, nloci denotes the number of loci that experience distinct sources of fluctuating selection. The parameter τflip represents the average number of generations required for the sign of selection at a chosen locus to flip and determines the key point in the d_N/d_S decay curve where N¯transient(t) begins to drop. We note that a locus here could be a nucleotide, gene, or gene set – any contiguous or noncontiguous stretch of DNA in which two knockout mutations would be just as beneficial or harmful as one mutation. We again use α to represent the proportion of nonsynonymous mutations that are neutral. Using Equation 7, we obtain a formula for d_N/d_S that has only three free parameters when a single value for μS is chosen:(8)dNdS=α+nloci6μSt(1−e−2tτflip).

When fitting the d_N/d_S curve, the values obtained are reasonable in the context of bacterial genomics, with median best-fit values across species of τflip=46,000 bacterial generations (range 25,000–105,000) and nloci=55 (range 34–80). Given daily bacterial generations, this value of τflip suggests the sign of selection on a given allele would flip approximately every 110 years. The average time for any pressure to flip would thus be approximately every 2 years, or less frequently if adaptive events occur in bursts (e.g., upon transmission to a new host). While 55 loci under distinct selective pressures may seem high, Bacteroidetes genomes are known to have dozens of invertible promoters (up to 47 in B. fragilis; Jiang et al., 2019). Invertible promoters are restricted out of the genome and re-ligated in the opposite direction to turn gene expression on or off. The number of invertible promoters in a given genome approximates a lower bound on the number of fluctuating selective pressures that these genomes frequently experience. Interestingly, adaptive loss-of-function mutations reported in B. fragilis affect the same genes regulated by invertible promoters (Zhao et al., 2019). The plausibility of these fit parameters lends support to a model in which d_N/d_S decays solely based on strong and recurrent local adaptations.

To ensure a reversion model is robust to finite populations, we performed simulations using fit parameters. These simulations capture the dynamics of a single population evolving as it transmits across a series of hosts through random bottlenecks (Figure 4a; ‘Methods’); these simulations allow for clonal interference between adaptive mutations. We allow new pressures to arise independent of bottlenecks as new selective forces (phage migration [Koskella and Brockhurst, 2014]; immune pressures [Barroso-Batista et al., 2015]; dietary changes [Carmody et al., 2019]) can emerge throughout the lifespan and independent of migration; forcing transmission and bottlenecks to coincide gives similar results (Figure 4—figure supplement 1). As in the purifying selection simulations, the per base pair mutation rate is 10^–9, and 90% of nonsynonymous substitutions are deleterious, but this time they have a larger s of 0.003 (Robert et al., 2018) and are thus purged more quickly from the population. Notably, while some of these deleterious mutations hitchhike to fixation during bottlenecks and adaptive sweeps, fitness does not decay because these mutations are subsequently reverted with adaptive sweeps (Figure 4—figure supplement 2). If deleterious mutations had significantly smaller s, they would be unable to be reverted due to the long time needed to reach fixation, even if bottlenecks and adaptive events are less frequent (Figure 2—figure supplement 3).

We note that other complex models that include reversion and other processes are also possible. For example, a model with a very large number of loci with selective tradeoffs and pressures that act only transiently (nonfluctuating) could potentially fit the data. However, the agreement between nloci and the number of invertible promoters, and the finding of parallel evolution in vivo, suggests the fluctuating selection model is more realistic than a very many-sites model.

So far, we have assumed that only exact reversions are selected upon when the sign of selection returns to its original state. However, the reversion model can also accommodate compensatory mutations that exclude any selective advantage for reversion; these compensatory mutations can also be subject to reversion themselves. We conceptualize this as a random walk, in which a locus at a nonancestral state acquires a compensatory mutation with probability p or obtains a true reversion with probability 1 – p (see Appendix 1, Section 3.2). As long as p≤0.5, d_N/d_S will decay to the same asymptote despite adaptive dynamics occurring. While compensatory mutations shift the timing of d_N/d_S decay to the right, it can be shifted backward by decreasing nloci (Figure 4—figure supplement 3a). The condition p≤0.5 is easily met when sben = 0.03, until excluding compensatory mutations are 10 times more likely than true reversion and provide 95% of the selective advantage of the true reversion (Figure 4—figure supplement 3b). If selective pressures are stronger (as they might be in the presence of phage), true reversions will outcompete compensatory mutations even if the supply of compensatory mutations is greater or such mutations provide better relative compensation.

A critical consequence of the reversion model is that apparent and actual d_N/d_S values diverge quickly. Even when the true genome-wide d_N/d_S exceeds 1 – meaning that adaptive sweeps have been a dominant force in shaping genomes – the observed value can be close to 0.1 on long timescales. This disparity complicates the interpretation of d_N/d_S as it becomes challenging to determine whether a genome or gene lacks nonsynonymous mutations due to reversions or negative selection. We confirmed the inability to detect adaptive selection on a gene when reversion is rampant by simulating protein phylogenies; even the advanced software PAML (Phylogenetic Analysis by Maximum Likelihood) (Yang, 2007) significantly underestimates actual d_N/d_S (Figure 4b; ‘Methods’). Without sufficient temporal sampling, no software can realistically estimate these hidden, adaptively driven nonsynonymous mutations.

Lastly, we sought to find evidence of past reversions of stop codons in certain genes by analyzing codon usage. Both leucine and serine have the property that they can be encoded by six codons, only one of which is highly stop codon adjacent (TTA for leucine and TCA for serine). Across the B. fragilis genome, these codons are depleted overall (13.48% usage rather than the neutral expectation of 16.67%). However, specific Clusters of Orthologous Genes (COG) categories are enriched in TTA and TCA codons relative to this baseline, including genes associated with transcription and cell envelope biogenesis (Figure 4c, ‘Methods’, Supplementary file 1). Further, when functionally annotated genes are further categorized by cellular localization, more gene categories exhibit enrichment (Figure 4c, ‘Methods’, Supplementary file 1), most notably genes involved in inorganic ion transport and metabolism that are localized to the outer membrane. Genes implicated in within-host B. fragilis adaptation (Zhao et al., 2019) are also found disproportionately in this category of outer membrane transporters (p=1.22 × 10^–4; ‘Methods’, Supplementary file 1). Both the cell envelope and membrane-bound transporters are known to mediate interactions with the immune system and phage (Sukhithasri et al., 2013; Ongenae et al., 2022), and are therefore expected to experience fluctuating selective pressures. The enrichment of stop-codon adjacent codons in pathways associated with environment-dependent costs further supports a model in which adaptive mutational reversions are frequent.

Data was obtained from Shoemaker et al., 2022 and was initially generated by Garud et al., 2019. Pairwise d_N/d_S values can be found in the GitHub repository. The parameters are estimated using scipy.optimize.curve_fit. The fit minimizes the RMSD of the logarithmic d_N/d_S. If we fit the data by minimizing just d_N/d_S on a linear scale, we get s ≈ 2.0 × 10^–5, which suggests an even weaker purifying selection (Figure 1—figure supplement 1). We analyzed the 10 species with the most data points reflecting short divergence times (d_S < 0.0005), which is critical for data fit.

The majority of the computational simulations performed are built upon the idea of the Wright–Fisher model with selection (Tataru et al., 2017) that population generations can be determined from a multinomial distribution. However, we have made some changes to generalize this model for our purposes.

First, the simulations do not necessarily assume a constant population size but rather assume the population grows via a logistic growth model with a capacity to allow for the implementation of bottlenecks. Specifically, if P[t] is the population on generation t and K is the population capacity, thenP[t+1]¯=P[t]+P[t](1−P[t]K)

AndP[t+1]=Poiss(P[t+1]¯).

The population size is a Poisson random variable as we choose to determine the offspring of individual genetic classes as a Poisson random variable. We note that except for the very first few generations and after bottlenecks, the population size only has small fluctuations around a fixed capacity.

To speed the simulation up and enable the simulation of very large population sizes, we implemented a variety of genotype classes, rather than tracking each genotype individually. Genotype classes are similar to the practice of simulating fitness classes (Desai and Fisher, 2007), though we manage the number of unique classes via Poisson merging and splitting.

For all simulations, we start from a single organism that begins with 500,000 neutral alleles, representing a core genome size of this many codons that has yet to receive any mutations. When a mutation occurs, one allele may change types or stay the same, depending on the mutation received and the state of the randomly chosen codon. For example, a deleterious mutation occurring at a codon already in a deleterious state does not change the genotype class of the organism.

The specific implementations and additional parameters used for this model are provided in the following sections. Here, we outline the theory that ensures that genotype classes accurately represent such a population and enable the calculation of fitness. Consider the total population of size, P[t], at generation t, as a composite of multiple different classes. The number of individuals in class j on generation t will be Aj[t]. We haveP[t]=∑j≥0Aj[t].

Within class j, we store several variables that provide information about the genotype of members of Aj[t]. Specifically, we store a number jk that specifies the number of alleles of type k in the class j. Examples of potential types that are used in our work include deleterious alleles, adaptive alleles, and alleles that result from reversion. Each type of allele is associated with a specific selective advantage sk. We can now write a formula to calculate the absolute fitness Fj of class j:Fj=∏k≥0(1+sk)jk.

From the absolute fitness Fj, we calculate the average absolute fitness of the population on generation t viaE[F]=∑j≥0FjAj[t]P[t].

We now calculate the relative fitness of class j on generation t asfj=FjE[F].

Next, we calculate the expected size of class j in the next generation withAj[t+1]=fj(Aj[t]+Aj[t](1−P[t]K)).

Note thatP[t+1]=∑j≥0Aj[t+1]=∑j≥0fj(Aj[t]+Aj[t](1-P[t]K))=∑j≥0FjE[F](Aj[t]+Aj[t](1−P[t]K))=P[t]+P[t](1−P[t]K).

This allows us to use a logistic model of growth to represent population size rather than being constrained to fixing it, which is useful for simulating bottlenecks.

To account for genetic drift through random fluctuations, we rewrite the above equations to beAj[t+1]=Poiss(fj(Aj[t]+Aj[t](1−P[t]K))).

which also impliesP[t+1]=Poiss(P[t]+P[t](1−P[t]K)).

Note that this simulation still has equivalent dynamics of the frequencies of classes as a Wright–Fisher model with selection in the case of a fixed population size due to the ability to split Poisson processes, that is,Pr(Aj[t+1]=X|P[t+1])=Pr(Bin(P[t+1],fjAj[t+1]P[t])=X).

Mutations are added in every generation depending on the mutation rate. Only single mutants are generated per generation, and an organism cannot get more than one mutation per generation. The number of new mutants is determined by the binomial distribution. New mutants are added then to their appropriate class j. For example, if a deleterious mutation is gained in a class with 10 deleterious alleles (and nothing else), this new mutant will increase the population size of the class with 11 deleterious alleles (and nothing else) while decreasing the population size of the class with 10 deleterious alleles (and nothing else).

By grouping individuals in classes rather than by genotype, computational costs can be greatly cut down. Grouping individuals does not affect the dynamics of the simulation because the merging of Poisson processes is still Poisson. The downside to this approach is information loss, though by designing custom alleles, we can track specific mutational histories like reversions.

The purifying selection simulations (Figure 2) utilized the base framework as mentioned above. Effective population sizes (N_e) varied from 10⁶ to 10¹⁸ depending on the simulation. The simulation begins with an initial organism with 500,000 neutral alleles (representing a WT core genome). The population quickly grows to the carrying capacity (N_e) and follows logistic growth (see ‘Population simulations overview’). The deleterious nonsynonymous mutation rate per genome per generation is 1.01 × 10^–3. These deleterious mutations have a selective disadvantage of s ≈ 3.5 × 10^–5. Synonymous and neutral nonsynonymous mutations are not simulated directly as they are neutral and are instead assumed to accumulate in the population with an average rate of 3.75 × 10^–4 and 1.12 × 10^–4 per genome per generation, respectively.

We estimate the average d_N/d_S of the population by taking the average number of codon differences between two individuals in the population to be twice the average number of mutations in the population. This approximation is valid due to the lack of selective sweeps, bottlenecks, and large effective population size, which results in expected coalescent time between random individuals being 10⁶–10¹⁸ generations (far longer than our simulations).

Variations of this basic purifying selection model are performed as described in the article, including increasing the mutation rate to 1.13 × 10^–3 mutations per genome per generation (Figure 2—figure supplement 2) and the simulation of recombination (Figure 2—figure supplement 3). For simulations of recombination, we assume that transitions to the ancestral state (purging the deleterious allele) occur at a rate of 2.5 × 10^–7 per codon per genome per generation and bring along a synonymous mutation (tracked via the number of recombinations to the ancestral state). This procedure does not allow for recombination to purge multiple deleterious alleles at a time; such events are unlikely given that deleterious alleles are rare and randomly distributed. We also include a model in which synonymous mutations are not included in this reversion event.

The simulation of purifying selection through bottlenecks and infrequent adaptive sweeps was performed as in the modified version of the reversion model (see ‘Reversion simulations’), though with less frequent adaptation and larger and less frequent bottlenecks.

Table S7 in Zhao et al., 2019 provides an Excel sheet detailing all observed mutations. There were 325 observed nonsynonymous mutations of which 28 were stop codons. Under a null model, there are 415 possible permutations of initial codon and codon one mutation (see ‘Code availability’) away that result in a nonsynonymous substitution of which 23 lead to a stop codon. Assuming no preference for specific mutation or initial codon, we would expect roughly 18 stop codons in this data. Under a null binomial distribution, the p-value for obtaining 28 or more is 0.015.

We simulated gut bacterial populations using a modified Wright–Fisher model (see ‘Population simulations overview’) to monitor mutation acquisition over time compared to an ancestor. Like all simulations, we begin with a single organism with 500,000 neutral alleles to represent the WT core genome. The population can grow to a capacity of 10¹⁰ via a logistic growth model. Environmental changes occur with a probability of nlociτflip per generation, triggering an average of one selective pressure per environmental change, modeled by a Poisson distribution. Population bottlenecks to 10 individuals occur independently of environmental changes with a probability of 10^–4 per generation (see Figure 4—figure supplement 1 for an alternative in which bottlenecks and environmental change are correlated).

Both adaptive selective pressures and adaptive mutations are categorized into two allele types: forward and reverse. These classes are designed to enable tracking of complete mutational history and therefore recorded relative to the ancestral state rather than the current state. Thus, actual d_N/d_S is calculated as the sum of these mutations and observed d_N/d_S using their difference (plus asymptomatic d_N/d_S). When releasing beneficial selective pressures, their classification as forward or reverse is based on the balance of previously released selective pressures: the probability of an adaptive pressure being classified as a reverse adaptation increases as the number of forward pressures increases and is equal to the difference between the forward pressures previously released (qF) and reverse pressures previously released (qR) divided by the number of loci (i.e., qF−qRnloci). All beneficial mutations have a selective advantage of s_ben = 0.03 (for forward or reverse). The rate at which mutations occur given an available pressure depends on whether the mutation is adapting to a forward or reverse pressure: the reversion rate is set at one-fifth the rate of the nonsynonymous per codon mutation rate (4.5 × 10^–10 per generation per cell), while forward mutations are set at a rate five times higher than the nonsynonymous mutation rate because they can happen at multiple sites (25× the reversion rate; 1.1 × 10^–8 per generation per cell).

This simulation treats each adaptive mutation as occupying a unique codon in the core genome for simplicity. This assumes that the ancestral allele at a given locus has been purged before the next environmental change affecting that locus (or selective pressure); as theory suggests that a beneficial mutation takes 768 generations to fix (see Appendix 1, Section 2.2), compared to 46,200 generations for pressure shifts at any locus we believe this assumption is reasonable. To confirm this theory still holds in the presence of bottlenecks and clonal interference, we tracked the average number of beneficial mutations in the population relative to the number of selective pressures released at any generation in simulations; we found only a 2% deviation between the average and expected total beneficial mutations over 2 × 10⁶ generations.

Throughout the simulation, deleterious mutations occur at a rate of 1.01 × 10^–3 mutations per genome per generation, with a selective disadvantage of s = 0.003, and can be reverted to a deleterious reversion allele class (separate from the adaptive reversion allele class).

To calculate d_N/d_S, we assume the simulated population could be compared to an equivalent population but with distinct mutations, allowing us to calculate the d_N/d_S as using double the current observed substitutions.

We assume during the reversion simulations that the ancestor has no initial transient mutations. We make this assumption for computational simplicity but the theoretical curve is equivalent whether starting from no revertible mutations or the equilibrium where half of the loci currently have forward mutations (assuming forward and reverse mutations occur at equal rates; see Appendix 1, Section 3.3).

We simulated gene sequences with selective pressures acting at specific sites for Figure 4b. For each genomic distance investigated (every 500,000 bacterial generations), we ran 10 simulations as described below, with each simulation resulting in 10 sequences derived from a branching process. In the permanent adaptations simulation (blue), adaptive mutations in the phylogeny are acquired simply and permanently. In the transient adaptations simulation (red), only more recent mutations in the same phylogeny will be visible while older mutations are obscured by reversion. Both sets of sequences were then fed to PAML v4.8 (Yang, 2007) for the estimation of d_N/d_S values. PAML uses maximum likelihood analysis to estimate the rate of substitution that best explains a given phylogenetic tree.

For each simulation, we generated a random 1500 bp open-reading frame and designated 10% of codons as neutral, 10% under positive selection, and 80% under purifying selection. We introduced mutations and branches across several cycles, with each cycle representing 100,000 generations. For each cycle, we assigned mutations at random according to the following probabilities: 67.5% that a nonsynonymous mutation occurred at a codon under positive selection, 12.5% for synonymous mutation at any codon, 3.75% for nonsynonymous mutation at a nonselected site (neutral), and 16.25% for no mutation. These values were selected to give a d_N/d_S of around 2 and to match the general ratios in the reversion model.

Two phylogenies were constructed from each simulation: both received identical mutations, but they differed in how nonsynonymous mutations at selective codon sites were visible at the end of the simulation. In the transient adaptations version of the phylogeny, nonsynonymous mutations at selective codon sites were reverted at the end of the simulation, except those that occurred within the last 500,000 generations. Reverted sites were converted to a synonymous substitution at a frequency based on the codon table (assuming an equal probability of all nucleotide mutations). Both versions of the sequences underwent multiple sequence alignment and neighbor-joining tree construction (Biopython; Cock et al., 2009). We calculated treewide d_N/d_S ratios using PAML v4.8’s codeML feature (Álvarez-Carretero et al., 2023), employing the M2a model to analyze site-specific selection.

To evaluate possible enrichment for stop codon adjacency, we focused on TTA and TCA codons. TTA and TCA are ideal for measuring the likelihood of nonsense mutations because each has two point mutations that yield a stop codon, unlike the five other redundant codons encoding for the same amino acids (for both leucine and serine, one codon is singly stop codon adjacent and the last four are not stop codon adjacent). In B. fragilis, these codons have a codon usage rate of 13% for leucine and 14% for serine.

We annotated the reference genome NCTC_9343 with Bakta v1.9 (Schwengers et al., 2021) and obtained COG categories for each gene using eggNOG v5.0 (Cantalapiedra et al., 2021). Genes that did not have a functional COG category (35%) were removed. To control for unusual outlier genes skewing results, only the 15 COG groups that had at least 50 genes were considered for enrichment analyses. For each COG category, we calculated a null codon usage proportion based on the proportion of leucine and serine codons and compared this to the actual proportion using a one-proportion Z test. To address the fact that genes in the same functional category but localized to different parts of the cell may be under different selective pressures, we analyzed cellular location classifications from PSORTb v3.02 (Yu et al., 2010) and categorized genes by the combination of function and localization. We analyzed the 15 function-location combinations with more than 50 genes. After identifying those categories that were significantly enriched, we cross-referenced which categories of genes shown to be under adaptive within-person evolution in a previous study of B. fragilis within-person evolution were in Zhao et al., 2019. Of the 16 genes reported in that paper, 8 were assigned a functional COG category/cellular location and in the reference genome (NCTC_9343). Four of these were in outer membrane inorganic ion transport and metabolism, a significant enrichment (p=1.22×10−4; binomial test) (Supplementary file 1, Supplementary file 2).

Code and simulation results are available at https://github.com/PaulTorrillo/Microbiome_Reversions (copy archived at Torrillo, 2023).