2.1 Experimental data on fitness bottlenecks

Poelwijk et al. [27] analyzed a family of variants of the naturally occurring wild type Entacmea Quadricolor red fluorescent protein eqFP611 [53]. More specifically, from a natural variant of eqFP611, called eqFP578 (with 76% amino acid sequence homology), the TagRFP mutant was engineered via random mutagenesis. TagRFP is 21 mutations away from eqFP578 [54, see S1 Text] and has enhanced fluorescence and stability. From TagRFP, two engineered variants have been derived to have distinct fluorescence (See https://www.fpbase.org/protein/eqfp578 for the complete tree of variants.):

• mTagBFP2, which is fluorescent in the blue and has 13 mutations and 6 insertions relative to TagRFP [55];
• mKate2, which is fluorescent in the deep red and has 8 mutations and one insertion relative to TagRFP [56].

The sequences of mTagBFP2 and of mKate2 differ by 13 mutations (12 of which are shown in Fig 2A reproduced from Ref [27]), thus resulting in a total of different combinations of intermediate amino acid substitutions. The experiment of Poelwijk et al. [27] measured the fluorescence of all these intermediates, both in the red and in the blue. Note that there are also 5 insertions in mTagBFP2 with respect to mKate2 that were neglected in the experiment (details were not found in Ref [27]).

We focus on these published data as they characterize a combinatorial landscape bridging two reference genotypes with distinct functionalities, within which a functional bottleneck was explicitly identified. A more recent study, employing a similar experimental design but investigating a different protein, reported the absence of such a bottleneck [50]. This discrepancy highlights the topological variability of fitness landscapes and motivates our search for the minimal conditions that govern the emergence of these constraints.

We downloaded the original data from Ref [27] and represented the genotypes with binary vectors of 13 variables, with being the blue mKate2 and being the red mTagBFP2. In the experiment, E.Coli cells expressing the mutant genotypes were sorted by a microfluidic device able to push them into separate channels according to their fluorescence. From the raw experimental measurement, we assign to each genotype a red fitness, , and a blue fitness, , based on the enrichment in the two channels, each normalized to its corresponding reference value such that (see the S1 Text for details). This normalization provides a consistent reference scale, with each reference variant serving as the baseline functional state for its phenotype, enabling a more direct quantitative comparison between the red and blue fitness values.

Note that a specific measurable trait – fluorescence intensity – is here employed as a proxy for fitness, despite its lack of direct coupling to reproductive success. Instead, fluorescence quantifies the protein‘s functional efficiency. Under the assumption that enhanced biophysical performance may confer a selective advantage within the relevant environment, fluorescence serves as a proxy for fitness. This approach is standard in many empirical studies of fitness landscapes, such as Refs [27,50].

For each measured phenotype (we will use a suffix ‘R’ for red and ‘B’ for blue fluorescence), we can define an underlying trait and a non-linear mapping to fitness . Inverting the non-linear function, we can thus derive the underlying trait from the measured fitness,

(2)

The optimal methodology for inferring the function remains a subject of active debate [24,32,37,40–42]; for instance, it has been suggested that the nonlinearity be inferred directly through rank-based statistics [42]. While acknowledging that results can be sensitive to the specific choice of ϕ, for our illustrative purposes we keep the same power-law form utilized in Ref [27] for both phenotypes. Alternative specifications, such as the sigmoid function shown in Fig 2A, may yield quantitatively different results, but the qualitative topology of the landscape remains robust.

We note that when there are two phenotypes only, for simplicity, we can follow Alqatari and Nagel [38] and encode them both in a single trait

(3)

which is positive for the ‘blue’ phenotype and negative for the ‘red’ phenotype. This is only possible when the two phenotypes are exclusive, as in Ref [27], which may not be true in other cases [32,50–52]. We also define the reference trait values as

(4)

The structure of the two proteins and the 13 substitutions are illustrated in Fig 2A, together with a schematic illustration of the non-linear mapping between and the two fitnesses.

Fig 2B shows the histogram of across all genotypes, with red and blue lines corresponding to the two reference variants. Note that the reference values are and . Their absolute value slightly differs from one due to the contribution from the other phenotype. As observed in Ref [27], most genotypes are non-functional with . There is a rather sharp peak of blue genotypes around , while red genotypes exhibit a much broader distribution, ranging from down to large negative values of E. This indicates that some intermediates are significantly more fluorescent in the red than the red reference variant itself.

Having defined , we can analyze the distribution of SMEs. For each of the backgrounds , we consider all 13 single mutations and we compute the SME,

(5)

The histogram of the absolute value of the SMEs is reported in Fig 2C. (Note that the SMEs include each mutation and its reverse, hence the histogram would be symmetric by construction, which is why we focus here on .) Although we have limited data, we clearly observe that the distribution is fat-tailed. A good fit is achieved by a Pareto distribution,

(6)

here with . The fit is just an indication, in particular due to the limited amount of data and the limited range of the experimental fitness, which both cutoff the distribution at large . While other distributions could also fit the data, the distribution tails are quite broad, indicating substantial heterogeneity among SMEs (in the S1 text we show similar results for another, independent dataset of SMEs [37]).

Finally, in Fig 2D, we examine the topology of the resulting functional bottleneck. To perform this analysis, we need to first establish a formal definition of a ‘functional’ genotype. As noted in Ref [27] and illustrated in Fig 2B, the threshold for functionality is inherently context-dependent; shifting this threshold can substantially alter the perceived landscape topology [38]. Indeed, viability is not an intrinsic property of a genotype alone but is contingent upon the environment – specifically the selection pressure exerted on the relevant trait – which varies across different experimental conditions. In this work, we consider that the fitness function is growing fast around a given threshold (either on the positive or negative side, see Fig 2A). Hence, genotypes with are considered dysfunctional and evolutionarily disallowed. Because the reference variants are functional, they must of course be located above the functionality threshold, i.e., . However, Fig 2D shows that no path of single mutations exists that connects the two reference variants while always maintaining , hence we need to consider in order to preserve the connection between the reference sequences. (Note that a different result was obtained in Ref [27] when the two phenotypes were not normalized to the reference values. We discuss this point in the S1 Text.) What is, then, the largest value of the functionality threshold such that at least one viable mutational path exists between the two reference variants?

In order to precisely answer this question, following Ref [38], we characterize the space of the paths connecting the two reference variants upon increasing the threshold . We consider the subset of paths that (i) are made of single mutations, (ii) connect the red and blue reference sequences, and (iii) are such that all along the path. We start from , and gradually increase to the maximum value such that at least one path remains. The resulting value, which we call , quantifies how ‘hard’ it is for evolution to find a path that realizes the functionality switch. If , then one can find a path of single mutations, such that all intermediates are equally functional to the reference variants. If instead , in order to connect the two reference variants one has to accept a loss of fitness with respect to the reference fitness, which makes the transition less likely, but still possible.

For the experimental data of Ref [27], this analysis yields a value of approximately , or . The resulting space of paths, depicted in Fig 2D, exhibits the characteristic bottleneck shape also reported in Ref [38], where all paths traverse a single ‘jumper’ genotype with . In Fig 2D, the jumper genotype has a blue phenotype, after which a single mutation brings to the red phenotype. This functional switch is driven by a mutation located roughly at the midpoint. The number of functional intermediates determines the number of evolutionary paths that can reach the critical ‘jumper’ genotype. More paths imply greater evolutionary accessibility, making it easier for evolution to find the key mutation; while fewer paths indicate a more constrained transition requiring a specific mutational sequence. Note that the analysis in Ref [27] found a less pronounced bottleneck, with multiple mutational paths surviving – a scenario that can be reproduced here by adopting a less stringent definition of which allows for multiple paths, see Ref [38].

2.2 Stylized model of global epistasis

The goal of this paper is to determine whether a bottleneck structure like the one observed in Sec. requires network epistasis, or if it can also be explained by a simpler model of global epistasis.

To formulate a simple stylized model, we begin by an important observation. The reference proteins analyzed in Ref [27] are not the product of natural evolution, but rather highly engineered sequences. In engineered proteins, derived either by random mutagenesis as in Sec. or by directed evolution, one typically observes that the fitness improvement is achieved by just a few highly beneficial mutations, see, e.g., [37,57,58]. While this might seem, in principle, a highly specific feature of engineered proteins, a similar scenario has been observed when a natural protein needs to quickly adapt to a novel environment and acquire a new function. A notable example is the SARS-COV-2 Spike protein that displayed a few highly beneficial mutations just after the virus jumped to human hosts (see, e.g., [59]), but similar dynamics has been observed in other viral proteins [60], and in the immune system [32]. In each of these cases, just a few advantageous mutations allow the protein to acquire the new desired function, whether the selection pressure is applied artificially or emerges from a change in the natural environment. A very different dynamics might be at play in the neutral evolution of optimized initial proteins [61–64], which might explain why the combinatorial landscape of Ref [50] does not display a bottleneck.

To capture a few minimal ingredients inspired by the results of Sec. and the above discussion, we introduce a stylized model featuring only global epistasis. The model is defined as follows:

• The genotype is a binary sequence (i.e., ) of length L.
• An underlying additive trait has random SMEs that are identically and independently distributed according to a symmetric input distribution . After having drawn the values independently, we impose that by shifting their mean, i.e., .
• The two fitness functions that will be associated with the two ‘colors’ are non-linear functions of the underlying trait, and , as illustrated in Fig 2A. If then the genotype is functional ‘blue’, if it is functional ‘red’, and if , the genotype is non-functional.

The fitness functions grow sharply around , in a way controlled by the parameter β. The choice we will make implicitly, following Ref [38], is to consider the limit , in which the fitness is close to a threshold (or Heaviside) function at (see Fig 2A, dashed curves). Note that the values can be multiplied by an arbitrary constant that can be absorbed into and β, which allows us to fix the overall scale (e.g., the variance) of without loss of generality.

Given the above definitions, both genotypes and have and are thus non-functional. We stress that, contrarily to the analysis of experimental data reported above, these two genotypes will not be the reference sequences in the stylized model. Following Ref [38], we thus introduce a ‘tuning procedure’ to generate two reference variants, the ‘red’ with and the ‘blue’ with , for a chosen ‘tuning’ value .

The procedure starts from the genotype , which is thus considered as a ‘common ancestor’ or ‘wild type’ (for example, in the experiment described in Sect II A, it would correspond to the eqFP578 protein). One then sequentially introduces mutations as follows.

• With probability p, we perform a ‘greedy’ step. We scan all the values that have not been introduced yet (those for which ), and choose the one that has the maximum impact towards the desired goal, i.e., the largest positive for the blue reference variant or the smallest negative for the red reference variant. We introduce the corresponding mutation by switching .
• With probability , we perform a ‘random’ step. We choose at random one site such that , and mutate it to .

The tuning procedure is performed independently for the two colors starting from , and continues until for the first time to construct the red reference variant, or to construct the blue reference variant.

This procedure is not designed to be a fully realistic model of molecular evolution. Nonetheless, it is grounded in experimental observations. As noted at the beginning of this section, directed evolution experiments often yield final protein variants carrying a small number of beneficial mutations together with a few neutral ones. More generally, when a protein is challenged to acquire a new function, adaptation often involves selecting a handful of advantageous substitutions on top of incidental neutral changes. Our approach for constructing the reference variants is simply the most straightforward way to reproduce this pattern, without any claim of population-genetic realism. We also believe that this procedure mimics the tuning of elastic networks performed in Ref [38], in which the network is tuned based on a local proxy for the global fitness; we hereby assume that this local proxy can indeed provide the best choice (with probability p) or miss it (with probability ).

The parameters that define the model are the sequence length L, the a priori distribution of SMEs , the probability p and the tuning parameter that controls the reference sequence construction. We note that may be interpreted as describing the a priori distribution of mutational effects – arising, for instance, from underlying biophysical constraints – which represents the potential ‘pool’ of mutations accessible to evolution. We found that various choices of lead to similar outcomes, provided the model is appropriately calibrated. For the sake of parsimony, and recognizing that these are stylized representations, we restrict our analysis to two distributions that capture distinct mutational behaviors.

1. Gaussian: A Gaussian distribution with unit variance.
2. Pareto cutoff: Motivated by the analysis of experimental data in Fig 2C, a Pareto distribution, i.e., Eq (6) with , where we fix and . We also include an upper cutoff as in the data, i.e., we set for and adjust the normalization constant accordingly.

For a given choice of parameters, an ‘instance’ of our random model is thus defined by the a priori pool of the L SMEs, , and by the two reference variants and . For a given instance, we can define the number of mutations M separating and (i.e., the number of that differ in the two sequences), and we can then investigate the directed paths that connect them, obtained by introducing these mutations in all possible orders. Following Ref [38] and as discussed above, we can find the maximum value of such that at least one single-mutational path connecting the two reference variants, with all intermediate states having , survives, and we call this value . The closer is to , the easier it is to perform the functional switch.

Given and , we wish to characterize the statistical properties of the quantities M, , and , which characterize the overall topology of the fitness landscape. Another interesting quantity is the distribution of SMEs selected by the tuning procedure to generate the two reference variants (i.e., those that correspond to in each reference genotype), which may be interpreted as the distribution of mutations that are ‘fixed’ by evolution. We will also define . In the following, we will use brackets, , to denote the statistical average of an observable A over an ensemble of random instances generated by the model. Unless otherwise specified, the average is taken over 20000 independent instances of the model for each value of the parameters.

Our model is similar in spirit to Fisher’s geometric model (FGM) as discussed in Refs [2,3,5,19,43], from which it however differs in two important aspects. First, the choice of non-linearity, which reflects here the existence of two distinct fitness functions associated to each color, see also Ref [43] for a similar choice. Second, the fact that we choose two reference variants using the stochastic procedure described above, and we then restrict to the space of intermediates between the selected variants. As we show in the following, these two ingredients, together with a proper calibration of the model, give rise to the desired bottleneck structure.

Before discussing the details of the calibration procedure, a typical example of the space of mutational paths obtained from a calibrated model is shown in Fig 1. The qualitative similarity with Fig 2D, modulo the (irrelevant) change of scale of E, is striking.

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Fig 1. Experimental data from Ref [27].

(A) Schematic shape of the fitness functions for the red and blue phenotypes as a function of the underlying trait E. Sigmoid functions have been used for illustrations. The two reference structures with the corresponding mutations are also shown, from Ref [27]. (B) Histogram of the value of E obtained from Eqs (2), (3) for each of the 2¹³ variants. The red and blue lines correspond respectively to the reference values and . (C) Histogram of the absolute value of single mutational effects (SMEs) from Eq (5) for all single mutations that can be obtained from the dataset, shown in log-log (main panel) and lin-log (inset) scales. The black curve is the fit obtained with the Pareto distribution in Eq (6). (D) Topology of the space of paths obtained keeping only genotypes with , and finding the largest possible value of (dashed line) such that the red and blue reference sequences remain connected. The lower panel reports the values of for each functional genotype (blue dots for and red dots for ) as a function of the number of mutations from the blue reference, with the gray lines connecting pairs of genotypes that differ by a single mutation. The upper panel shows the resulting graph of connections.

https://doi.org/10.1371/journal.pcbi.1014000.g001

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Fig 2. Same representation as in. Fig 2D, using an instance of the calibrated model with Gaussian P(h) instead of the experimental data.

The lower panel reports the value of for each functional variant as a function of the distance from the blue reference variant (here with and ). The threshold value for which at least a mutational path remains is (dashed lines) and when normalized with respect to the largest of the two reference genotypes it reads .

https://doi.org/10.1371/journal.pcbi.1014000.g002

2.3 Calibration of the model

The first step is to calibrate the parameters in such a way that our stylized model reproduces the basic phenomenology observed in Refs [27,38]. The requirements are the following.

• The phenotype of the two reference sequences should be comparable to , i.e., , in order for to be a meaningful fitness scale.
• We want to reproduce a ‘bottlenecked’ structure of the space of evolutionary paths connecting the two reference variants, as shown in Figs 2D and 1 and in Ref [38]. This is characterized by a single ‘jumper’ genotype through which all paths must pass. The mutation responsible for the functionality switch occurs just after (or just before) this genotype. We note that the jumper genotype can be connected by a single mutation to only one, or more than one, genotypes carrying the other phenotype.
• The ‘jumper’ genotype should occur approximately at half distance along the evolutionary trajectory, as it is observed in experimental data (Fig 2D). Furthermore, we want the largest possible number of functional intermediate variants to survive before and after the jump, thus maximizing the number of functional evolutionary paths connecting the two reference variants.
• The fitness variation produced by the ‘jumper’ mutation(s) should be as large as possible, in such a way that is as close as possible to . If this is the case, a path can go in a single jump from having fitness close to the red reference, to having fitness close to the blue reference. Otherwise, one would necessarily have to tolerate a decrease in fitness to perform the functional switch. This makes the resulting mutational paths more viable for evolution.
• Finally, for practical reasons, the number of mutations M connecting the two reference variants should be small enough, otherwise enumerating all the paths becomes computationally very expensive. To avoid this, we impose , a value close to that chosen in Ref [38].

An important remark is in order about the expected value of . Consider for illustration the case in which the tuning procedure reaches the target with a single greedy step for both colors. We can thus expect and . In absence of network epistasis, the largest jumper mutation that can happen along a path has , but this mutation must bring the system from to , hence . Based on this simple argument, the largest possible value of we can expect is such that , as we observe numerically (Fig 3C).

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Fig 3. Calibration of the two models with different , a Gaussian distribution or a Pareto distribution with cutoff.

The black symbols correspond to the choice of p after calibration. (A) Value of for which as a function of p. Dashed lines are linear fits. (B) Average value as a function of p. (C) Average of as a function of p. (D) Probability of having a single jumper at , or equivalently that , as a function of p.

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To fix the model parameters we proceed as follows. First of all, we fix as the length of a typical protein. We checked that the results are very weakly dependent on L (see the S1 Text). Next, for each p, we progressively increase until the average number of mutations separating the two reference variants is . The resulting is plotted as a function of p for both choices of in Fig 3A, and we find that it is approximately linear in p. The linear fit allows us to fix for all p.

The next step is to calibrate p. For this, we consider the ratio for both colors, which we want to be close to one. The evolution of its statistical mean with p is shown in Fig 3B. For small p, we mostly perform random steps in the training, and the requirement that M is small forces to be small as well (Fig 3A). As a result, it is very likely to ‘overshoot’ the target and end up with . Hence, in order to have , we need a large enough p.

Next, we consider the connectivity threshold . We recall that is obtained, for each instance, in such a way that at least one single-mutational path connecting the two reference variants exists, with all intermediate states having . Because both reference sequences must always be included in the space of allowed genotypes, increasing the value of beyond is not meaningful [38]. This gives an upper bound for the largest value of for each instance. We show in Fig 3C its average value as a function of p, where the average is restricted to instances such that . When p is too large, the tuning procedure requires too many greedy steps and as a consequence tends to decrease, contrarily to what we want in order to have bottleneck topologies with large . This suggests to lower the value of p as much as possible.

However, in Fig 3D we report the probability of having a single jumper at , which we estimate for numerical convenience by the probability that , as a function of p. When p is too small, this probability drops quickly below one. This is because starts increasing, the reference variants are obtained by many random steps, and the bottleneck structure is lost.

Taken together, these results suggest to choose the lowest value of p such that the probability of a single jumper is still very close to one, which also gives both and . The value of is fixed by the requirement that . For both choices of , this optimal value is around , although slightly larger values could be considered as well.

The crucial insight from the calibration is that one needs to tune p in order to achieve a good balance between greedy and random steps. Correspondingly, as we will discuss later, the distribution of the SMEs that are selected by the tuning procedure that generates the reference sequences (that may be interpreted as those ‘fixed by evolution’) is very heterogeneous, featuring a coexistence of large SMEs (generated by greedy steps) and small SMEs (generated by random steps).

2.4 Results for the calibrated model

In the following, we choose and the parameters corresponding to the black dots in Fig 3 as the final calibrated parameters for which we report more detailed results: Gaussian with and for the tuning procedure, and Pareto cutoff with and . Keeping these two models fixed, we generated many () random instances of the model and studied the distribution of the relevant quantities that characterize the topology of the fitness landscape. An example of a ‘good’ topology, closely resembling the one found experimentally, has already been given in Fig 1. We want to quantify the probability of generating such a topology in the stylized model.

We first focus on the properties of the two reference variants, shown in Fig 4 for both models. More specifically, Fig 4A shows the histograms of M, the total number of mutations separating the two reference sequences. We note that, by construction, , and we fixed . We see that the distribution is quite broad, leading to a significant fraction of instances with M as large as 20, and resembles that shown for elastic networks in Ref [38]. However, when we investigate the mutational paths, we restrict the analysis to instances with for computational tractability, as in Ref [38]. Fig 4B shows the histograms of for the blue reference variant (similar results are obtained for the red reference variant due to the inherent symmetry of the model). We recall that, by construction of the training procedure, . We observe that the histogram is peaked slightly above the minimal value, which confirms that the tuning procedure produces and . Fig 4C shows the histograms of the number of greedy steps in the tuning procedure, which is peaked around one, as expected. Finally, Fig 4D shows the histograms of the M SMEs that are selected by the tuning procedure. These mutations characterize a posteriori (after tuning) the space of intermediates between the reference variants, and can be thought of as being those ‘fixed’ by evolution. We observe a strongly bimodal distribution for both input distributions , which reflects the choice of the tuning procedure: mutations selected by the random steps tend to be small, while mutations selected by the greedy steps tend to be much larger (in a way that can be quantified by extreme value statistics, see the S1 Text).

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Fig 4. Distribution of some relevant quantities for two different calibrated models, the Gaussian model and the Pareto cutoff model.

In the former, SMEs are extracted from a Gaussian distribution with unit variance and the tuning procedure has , ; in the latter, SMEs are extracted from a distribution with Pareto tails decaying with and a cutoff , and the tuning procedure has , . In each plot the dashed and dot-dashed vertical lines represent the mean and the median of the data, respectively. (A) Distribution of the number of mutations. (B) Distribution of the value of the positive (blue) reference phenotype divided by . (C) Distribution of the number of greedy steps needed to reach the target value when generating the two reference variants. (D) Distribution of SMEs selected by the tuning procedure to generate the two reference variants.

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Next, we analyze the statistical properties of the space of paths connecting the reference variants (Fig 5). We recall that in the calibrated models we find with probability one. The space of paths thus contains a single genotype through which all paths must pass: raising would remove this genotype from the set, and disconnect the reference variants. A bottleneck is thus observed. We note that the jumper genotype, in most cases, is connected by a single mutation to only one genotype carrying the other phenotype, as in Fig 1. However, in a few cases, we also observe situations where the jumper genotype is connected to a few genotypes carrying the other phenotype. In Fig 5A and 5D we show the histograms of , separately for instances with given M. We observe that the histogram is almost independent of M (with more variability observed for the Pareto cutoff model), and peaked around as discussed above. Fig 5B, 5E shows the histogram of the location of the jumper, called j, along the mutational path. We observe that the jumper is often located mid-way between the reference variants, leading to a peaked histogram around . The peak (slowly) becomes sharper upon increasing M. The qualitative shape of the possible topologies produced by the model is indicated in Fig 5G. Finally, we can compute the number of paths that remain viable at . The average of the logarithm of this number is shown in Fig 5C, 5F as a function of M. We find that it grows almost linearly at large M, indicating that the number of paths grows exponentially in M. This suggests that there are exponentially many ways to reach the bottleneck, which could thus maintain a good evolutionary accessibility from both reference sequences [38].

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Fig 5. Statistical properties of the space of paths for two different calibrated model, the Gaussian model (A-C) and the Pareto cutoff model (D-F), with the same parameters as in Fig 4.

(A), (D) Distribution of values of , separately for several values of M. The mean and the median (dashed and dot-dashed vertical lines, respectively) are computed with respect to the full distribution (with all M values). (B), (E) Probability distribution of the jumper position j along the path, for different values of M. (C), (F) Average of logarithm of the number of paths that remain viable at , as a function of M. (G) Some examples of topologies, each associated to a schematic representation used in panel b.

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We conclude that our stylized model, which features a global non-linearity on top of an underlying additive phenotype, with properly calibrated parameters, generates topologies that are very close to those observed experimentally in Ref [27] and Sec. An example is shown in Fig 1, to be compared with the result obtained re-analyzing the experimental data from Ref [27], shown in Fig 2D. The model generates an ensemble of random fitness functions [5], whose topology is likely (with probability one) characterized by a bottleneck, i.e., by a genotype through which all paths connecting the two reference variants must pass. Of course, the bottleneck can be made less pronounced if one allows for less stringent selection, i.e., reduces [38]. The bottleneck is more likely found mid-way along the path. These features are essentially independent of M, i.e., the number of mutations separating the two reference variants, at least in the limited range investigated here.

We note that there is quite a lot of flexibility in the choice of parameters. The initial size L can be varied in a broad interval without affecting much the results (see S1 Text). The tuning parameter has been chosen here in such a way to obtain the desired average of M, but other choices are possible. The value of p, i.e., the probability of making a greedy step when constructing the reference variants, can be varied in a relatively wide range, provided it is neither too small (otherwise is too small and the reference phenotypes end up overshooting the fitness threshold) nor too large (otherwise decreases). Finally, the a priori distribution that characterizes the input distribution of SMEs can be varied essentially at will in a broad range (see also the S1 Text for other choices of the input distribution). The tuning procedure used to construct the reference variants ensures that the SMEs selected a posteriori are broadly distributed, featuring the right balance between almost neutral SMEs and strongly beneficial or deleterious SMEs (Fig 4D), which is a necessary condition for the emergence of a bottleneck. We thus believe that our results are robust and do not depend crucially on the specific choices made during calibration.

4.1 Genotype representation and underlying additive phenotype

The first step consists in choosing a value of L and considering a genotype as a binary sequence of length L, where each site i can take values . The genotype-dependent underlying additive trait is given by:

(7)

where the single-mutation effects (SMEs) are independent and identically distributed according to the a priori distribution . The two choices we used are a Gaussian distribution with zero mean and unit variance, and a Pareto distribution with cutoff:

(8)

with and the constant A determined by normalization. After sampling, the SMEs are shifted to ensure . These choices fix an additive constant and an overall multiplicative factor in , which can be absorbed by β and in the fitness function, without loss of generality:

4.2 Fitness function and phenotypic classes

The global epistatic fitness function is a non-linear transformation of , defined separately for the blue (B) and red (R) phenotypes:

(9)

where without loss of generality, is the functionality threshold parameter, and controls the sharpness of the fitness transition. Accordingly, genotypes with are functional for the “blue” phenotype, those with are functional for the “red” phenotype, and those with are non-functional.

4.3 Reference variants construction

The red and blue reference variants, and , are independently generated starting from the ancestral genotype by sequentially introducing mutations. At each step, with probability p, the mutation with the largest contribution to E in the desired direction is chosen (greedy step), and with probability , a random site is mutated (random step). The process is iterated until and .

4.4 Mutational paths and bottleneck characterization

The mutational distance M between and is defined as the number of differing sites. All possible evolutionary paths between the reference variants are considered, and a value is determined such that at least one path of single mutations exists, maintaining at all intermediate steps. The distribution of and the likelihood of a single critical ‘jumper’ mutation define the presence and severity of the bottleneck.