Estimating fitness of evolutionary-relevant genotypes

We studied His3, a gene coding for imidazoleglycerol-phosphate dehydratase (IGPD, His3p), an enzyme essential for histidine synthesis. In a multiple alignment of His3 orthologues from 21 yeast species we identified 686 extant amino acid states (S1 Supporting Information), which were evenly distributed across the His3p structure (Fig 1D). These 686 states, which represent the end product of ~400 million years of evolution [33] (Fig 1B and 1C) correspond to ~10⁸³ sequences, even a tiny fraction of which would be too many to survey. Thus, we sectioned His3 into 12 independent segments such that the full combinatorial set of amino acid states that occurred in His3 during yeast evolution comprised 10,000-100,000 genotypes per segment (see Methods and S1A Fig). The 12 segments were of similar length, constrained by the molecular methods employed for library construction (see Methods), and covered a diverse range of secondary structures and functional elements (S1C Fig). For each of the 12 segments of His3 we performed an independent experiment surveying its fitness landscape. For each segment we used degenerate oligonucleotides to construct genotypes consisting of combinations of amino acids present in extant His3 sequences, and determined the fitness conferred by these genotypes by expressing them in a Δhis3 strain of S. cerevisiae and measuring the rate of growth (S1B Fig). This way for each segment we assayed the fitness landscape of the genotype space that was traversed over the course of the last ~400 million years of evolution [33].

The segmentation of the His3 protein into 12 segments at first glance appears not to be relevant to understanding the fitness landscape of the entire protein. Indeed, it would have been more informative to survey combinations of extant amino acid states across the entire His3 sequence. Our choice to study independent segments was dictated by our preference for depth over width, in other words, we were more curious to consider combinations of extant amino acid states from distant orthologues rather than consider combinations of extant amino acids states from a few closely related species across the entire gene. Our choice of selecting random segments across the protein is justified in the same way as surveys of individual genes; segments may interact to form a functional protein while genes interact to contribute to the overall organismal fitness. Indeed, the lessons learned from fitness landscapes of random protein segments are likely to be scalable to the level of the entire protein to a greater extent than how fitness landscapes of an entire protein can be scaled to understand the fitness landscape of a genome.

Across 11 experiments, we measured fitness for a total of 4,018,105 genotypes (875,151 unique amino acid sequences) with high accuracy (S2 Fig). Of these, 422,717 consist solely of combinations of extant amino acid states from His3 orthologues, while the remaining genotypes incorporate other amino acid changes (S2 Supporting Information and Methods). Throughout the study, we considered the logarithm of fitness to allow for the sum of the impact of individual amino acid replacements on fitness to represent the fitness function. For one segment, 9, the accuracy of our experiment was low, and it was not used in cumulative analyses. For each segment we measured fitness for 60% - 99.8% of all possible genotypes from the combinatorial set of selected extant amino acid states found in 21 yeast species and a smaller fraction of combinations found across all domains of life (S2 Supporting Information), characterizing the evolutionary relevant fitness landscape (Fig 1B). For segment 3 for instance, 11 out of 17 amino acid sites had more than one extant amino acid state: L145 = 2, L147 = 2, Q148 = 3, K151 = 2, V152 = 2, D154 = 3, L164 = 3, E165 = 4, A168 = 2, E169 = 4, A170 = 4, with the full yeast combinatorial set consisting of 2*2*3*2*2*3*3*4*2*4*4 = 55,296 genotypes out of which we determined the fitness for 48,198, or 87% of the possible yeast extant states combinations in our library. Our experimental design not only generated combinations of extant amino acids, but also allowed us to distinguish real variants from less abundant aberrations originating from sequencing errors (S2 Fig). We further reduced genotyping errors by sequencing every variant twice through paired-end sequencing (S2A Fig), and by using an error-correction algorithm (see Sequencing error rate section).

Most of the genotypes that can be created by the combination of extant amino acid states are not ancestral His3 protein segments. In a high-dimensional sequence space, the number of actual ancestral sequences is exponentially smaller than the number of combinations of extant states (Fig 1A). Out of the 48,198 extant amino acid state combinations in segment 3 only seven sequences actually match the reconstructed ancestral states of the entire segment. The combinations of extant amino acid states represent the evolutionary-relevant segment of the fitness landscape because they represent all possible alternative evolutionary trajectories in sequence space that consist of the same substitutions that have occurred in evolution of the extant protein sequences under consideration. Furthermore, some of these seven sequences were likely found in a different context of the entire protein, so they do not represent the ancestral states of the entire His3 protein but rather the ancestral states found in segment 3 across different genetic context of the rest of the His3 protein.

A substantial proportion of combinations of extant amino acid states led to genotypes with low fitness (Fig 2, Fig 3A and 3B, S3 Fig), an observation that takes into account the false discovery rate in our data (S2 Supporting Information). This observation could be explained by i) some extant amino acids having a universally deleterious effect, ii) some amino acid states exerting a negative effect on fitness because of intergenic interactions with other S. cerevisiae genes, or iii) by epistatic interactions between the extant amino acid states within His3 [34]. We exclude the possibility that some extant amino acid states had a universally lethal effect in S. cerevisiae background because no extant amino acid states were present only in unfit genetic backgrounds, genotypes conferring a fitness of zero. Indeed, no extant amino acid states had a universally strong deleterious effect, a decline of fitness by more than 0.4 (Fig 3D), and the smallest number of fit backgrounds in which any extant state was found was approximately 300 (Fig 3C). We use the 0.4 threshold because it corresponds to ~1% false discovery rate (see Methods). We exclude the possibility that some extant amino acid states disrupt intergenic interactions because the complete His3 coding sequences from extant species fully complemented a His3 deletion in S. cerevisiae (S4C Fig). Thus, the observed genotypes with low fitness can only be explained by epistatic interactions among extant amino acid states within the His3 gene in the same or different segments. Remarkably, 85% (330/389) of replacements between extant amino acid states had substantially different effects on fitness in different backgrounds (Fig 3D). By contrast, only 15% of amino acid replacements are truly neutral, in the sense that they do not exert strong influence on fitness in any genetic background. Three quarters of the universally neutral amino acid replacements were observed in the disordered region of the protein (44/59). Taken together, the His3 fitness landscape across the 11 segments with high accuracy was strongly influenced by epistasis on a macroevolutionary scale, i.e. the impact of an extant amino acid state on fitness often depends on the background in which it occurs [34–37]. An epistatic fitness landscape is rugged in the sense that evolving genotypes must avoid fitness valleys that emerge through deleterious combinations of amino acid states that may also be found in fit genotypes [19,20,37–38]. Characterizing the ruggedness and the mechanisms that determine the underlying epistasis becomes the primary challenge in understanding the fitness landscape of His3.

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Fig 2. Visual representations of the fitness landscape.

a, The fitness landscape for all assayed genotypes in segment 7. Nodes represent unique amino acid sequences with edges connecting those separated by a single amino acid replacements. Colour saturation represents the minimum fitness of the two connected nodes. b, For segment 7, fitness of ancestral and extant nodes and genotypes one amino acid replacement away from the nodes in the background of S. cerevisiae gene on the yeast phylogeny (black lines), are shown in colour ranging from grey (lowest fitness) to blue (highest fitness). The abbreviations represent the following species: Scer: Saccharomyces cerevisiae, Soct: Schizosaccharomyces octosporus, Sbay: Saccharomyces bayanus, Cgla: Candida glabrata, Scas: Saccharomyces (Naumovozyma) castellii, Kwal: Kluyveromyces waltii, Klac: Kluyveromyces lactis, Sklu: Saccharomyces (Lachancea) kluyveri, Agos: Ashbya gossypii, Clus: Clavispora (Candida) lusitaniae, Dhan: Debaryomyces hansenii (Candida famata), Cgui: Candida (Pichia) guilliermondii, Ctro: Candida tropicalis, Calb: Candida albicans, Cpar: Candida parapsilosis, Lelo: Lodderomyces elongisporus, Ylip: Yarrowia lipolytica, Anid: Aspergillus nidulans, Ncra: Neurospora crassa, Sjap: Schizosaccharomyces japonicus, Spom: Schizosaccharomyces pombe.

https://doi.org/10.1371/journal.pgen.1008079.g002

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Fig 3. Fitness distributions.

a, the distribution of fitness for genotypes composed of combination of extant amino acid states (green) and non-extant amino acid states (purple) at the same positions. b, The fraction of unfit genotypes per segment among genotypes consisting entirely from extant amino acid states (green) and those incorporating non-extant amino acid states (purple). c, Amino acid states can be found in a different number of fit backgrounds. In this figure, we show the number of amino acid states that were found in the lowest number of fit genetic backgrounds. Some non-extant amino acid states were found in just a few genotypes with high fitness. The most infrequent extant amino acid state was found in at least 300 different genotypes that had high fitness. d, The percent of backgrounds in which a specific amino acid replacement is neutral (white), beneficial (dark blue) or deleterious (dark grey). The region marked in green shows amino acid replacements that never have large effects (> 0.4) on fitness. Beneficial and deleterious effects are shown only if the frequency for a given amino acid replacement was higher than the false discovery rate (S2 Supporting Information). Data from segment 9 were excluded for this figure.

https://doi.org/10.1371/journal.pgen.1008079.g003

Unidimensional epistasis of the His3 fitness landscape

The ruggedness of the fitness landscape can be characterized by different measures of complexity of the underlying epistatic interactions. In the simplest case, epistasis may be unidimensional, in the sense that the fitness landscape can be described as a function of an intermediate variable, the fitness potential [40–42]. The fitness landscape is a function from the space of genotypes to fitness. In analogy with a scalar field, we can characterize the ruggedness of this function with standard measures of complexity if genotypes are arranged in a linear space. The simplest case is that of a linear predictor called the fitness potential: p = c₁x₁ + c₂x₂ + … + c_nx_n, where c_i is a coefficient and x_i is a binary variable that signifies the presence (1) or absence (0) of a given amino acid at a given position. By definition, e^p describes a non-epistatic fitness landscape because the effect of every amino acid replacement is multiplicative, and it depends only on the associated c. Any other function of p leads to epistasis. If the f(p) function is “simple”, meaning that it has a small number of local extrema, such as a bimodal function, the epistasis is called unidimensional [42]. The limitation of simplicity of f(p) is necessary because any function f₀(x₁, …, x_n) can be represented by a function f’(p) and choosing appropriate coefficients c₁, …, c_n in p. Thus, a simple f(p) leads to unidimensional epistasis because the entire genotype space can be reduced to a single dimension [42].

To quantitatively determine how well fitness differences between genotypes can be explained by unidimensional epistasis we used a deep learning approach to estimate the coefficients c for each allele x in the fitness potential and determine the unidimensional function f(p) that best approximated the fitness landscape. We used a dense neural network architecture composed of three layers. Each neuron in the architecture performed a linear transformation of its input and then applied a nonlinear (sigmoid) function. The single neuron of the first layer computes the fitness potential, which is then mapped to a fitness value obtained from f(p), the function of the fitness potential which is found by the three layers of the neural network architecture (see Methods; Fig 4). For ten protein segments, f(p) converged to the same shape: a threshold function in which organismal fitness remains constant with decreasing fitness potential and then is rapidly reduced to lethal after a certain threshold (Fig 5A). This analysis, coupled with previous observations of thresholds in fitness landscapes [7,10, 43] suggests that macroevolutionary fitness landscapes may in fact represent instances of truncation selection. Remarkably, no fitness functions showed a defined optimum, implying a lack of stabilizing selection on His3 protein function and that it was maintained at an optimized state throughout its evolution. The ability of the cliff-like threshold fitness function [44] to predict fitness from genotype varied between the His3 segments from near perfect (r² = 0.97) in segment 7, to relatively poor (r² = 0.44) in segment 5 (S5 Fig). Thus, while the fitness landscape of His3 is approximately unidimensional for some segments, it has a higher degree of complexity for others.

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Fig 4. Schematic representation of a deep learning approach able to fit any arbitrary fitness function.

a, each genotype was encoded as a binary vector (x). During training, each of the amino acid replacements was assigned a coefficient (c_i), comprising a vector of coefficients (c). The multiplication of these two vectors is the fitness potential of the genotype. After going through three layers, each with a sigmoid activation function, the predicted fitness is obtained. b, The fit of a mock fitness function (yellow) and the fit achieved by our neutral network (black). The mock fitness function was created by generating a set of amino acid states with defined coefficients (effects on fitness potential), which were then combined to generate genotypes across a range of fitness potential values.

https://doi.org/10.1371/journal.pgen.1008079.g004

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Fig 5. Epistasis and the His3 fitness landscape for segments 2, 5 and 7.

a, Fitness as a function of a single fitness potential (black curve, the fitness of individual genotypes is orange). b, A network depiction of sign epistasis between amino acid replacements. Colour coded sites with reciprocal sign epistasis (black lines) and unidirectional interactions (grey arrows) are shown. c, Genotypes containing replacements with a higher number of sign epistatic interactions are less likely to be fit by the threshold function of the fitness potential. d, Increasing the number of neurons in the first layers of the neural network, which is equivalent to increasing the number of underlying fitness potentials, leads to more accurate models for segments with detected sign epistasis. Each dot corresponds to an independent optimization of model parameters. e, Fitness as a function of two fitness potentials (black dots, measured fitness is depicted in orange).

https://doi.org/10.1371/journal.pgen.1008079.g005

Ruggedness and multidimensional epistasis of the His3 fitness landscape

Ruggedness is a general property of fitness landscapes that quantifies the accessible paths of high fitness that connect fit genotypes [45–47]. A path between highly fit genotypes is inaccessible when one of the intermediate genotypes has low fitness [6,21–23,45,48] (e.g., for genotypes AB and ab, the intermediate are aB and Ab). Such instances also manifest in sign epistasis on the fitness landscape, that the same amino acid replacement may be beneficial or deleterious when occurring in a different genetic background [48,49]. To quantify the ruggedness of the His3 fitness landscape we identified instances of sign epistasis: replacements between extant amino acid states that were strongly beneficial in some backgrounds (increasing fitness by at least 0.4 in absolute fitness) or strongly deleterious (decreasing fitness by at least 0.4 in absolute fitness) in other backgrounds [48]. Some of these instances may be due to miscalled fitness of very few genotypes. Therefore, we considered a pair of extant amino acid states to be under sign epistasis only when sign epistasis was observed in a statistically significant number of different genetic backgrounds (see Methods section Quantifying sign epistasis).

An example of sign epistasis is the C141S replacement in the second segment that had an opposite effect on fitness depending on amino acid at site 143 (I, V or T). The I143T replacement in turn exhibits sign epistasis depending on the amino acid at site 163 (F, I, V or L) (Fig 6A). These epistatic interactions can be represented by a graph in which nodes represent a pair of extant amino acid states at a specific site and nodes are connected by edges if strong sign epistasis has been detected between them (C141S - I143T - I163F) (Fig 5B). We found that 86 out of 128 (67%) sites in our library exhibit sign epistasis and 46% (59/128) exhibit reciprocal sign epistasis with 8% (968/11597) of all pairs of sites exhibiting sign epistasis (see S2 Supporting Information Table 5). Most sites showed a sign epistatic interaction with multiple other sites (Fig 6C, S6 Fig) demonstrating that, although sign epistasis affects few genotypes, it leads to a fitness landscape that requires the interaction of multiple sites for proper characterization.

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Fig 6. Sign epistasis.

a, Amino acid replacement C->S at site 141 in segment 2 more frequently has a positive effect on fitness in the background of T at site 143, a negative effect in the background of 143I and is equally likely to be strongly deleterious or strongly beneficial in the background of 143V. b, Predicted change in folding free energy using Rosetta [64] following a C141S replacement in all genetic backgrounds with an I or T at 143 and that are closer than four mutations away from S. cerevisiae. c, the fraction of genotypes in which the amino acid replacement under sign epistasis has the less frequent effect on fitness.

https://doi.org/10.1371/journal.pgen.1008079.g006

Sign epistasis can appear when fitness is described by a unidimensional function of the fitness potential, for example, when the fitness landscape is a unimodal function with an optimum in an intermediate range of the fitness potential [45,49]. However, sign epistasis may also be a sign of multidimensional epistasis, when a unidimensional function of the fitness potential cannot fully describe genotype fitness [42]. Many genotypes were predicted poorly by a unidimensional function of the fitness potential (S5A Fig). Two lines of evidence suggest that such genotypes reveal the presence of multidimensional epistasis. First, genotypes with a higher number of amino acid replacements influenced by sign epistasis were less well-predicted by a unidimensional fitness function (Fig 5C and S7B Fig). Second, we explain a larger fraction of genotypes by using a more complex neural network architecture accommodating multiple fitness potentials instead of one. We found that increasing the number of neurons in the first layer of the neural network architecture, which is equivalent to increasing the number of independent fitness potentials, gradually improves the prediction power of the obtained models for most of the segments (Fig 5D). Thus, adding dimensions to the function of fitness potential increases the prediction power of the model. For example, for a two-dimensional case fitness was described by f₁(p₁, p₂) with p₁ = a₁x₁ + a₂x₂ + … + a_nx_n and p₂ = b₁x₁ + b₂x₂ + … + b_nx_n. For several His3 segments, a fitness function with multiple underlying fitness potentials described the fitness landscape more accurately than a simple unidimensional function of a single fitness potential (S7A Fig). For instance, for these segments, fitness function of two fitness potentials described the shape with a higher degree of accuracy than a function of a single fitness potential (Fig 5D and 5E). By contrast, epistasis in segment 7 is entirely unidimensional (Fig 5D and 5E and S3 Supporting Information); we do not see any improvement in the model’s predictive power when adding extra dimensions.

Evolutionary trajectories on the His3 fitness landscape

On a smooth fitness landscape, evolution can proceed along any of the evolutionary paths connecting two fit genotypes, as none of the intermediate genotypes confer low fitness (see Box 2 in [50]). Alternatively, the fitness landscape is rugged when it contains non-connected fitness peaks, such that there are no viable paths between some pairs of genotypes that confer high fitness [4,5]. In other words, the presence of deleterious intermediate genotypes between highly fit ones leads to inaccessibility of some evolutionary trajectories between extant or ancestral sequences [6,21–23,48]. The simplest explanation for the substantial ruggedness of the landscape observed in many of the His3 segments lies in the unidimensional threshold fitness function (Fig 7A). On a threshold function, a combination of amino acid replacements that are all neutral in some genetic backgrounds can take a genotype beyond the fitness threshold through their additive effect on fitness potential, making some genotypes inaccessible for evolution (Fig 7A). Between any two fit genotypes, the fraction of intermediate genotypes that are unfit depends on the fitness potential of the two parental genotypes (Fig 7B). Evolution between two fit genotypes with high fitness potential can proceed unhindered because all intermediate genotypes also have high fitness potential and, consequently, high fitness. Conversely, when both fit genotypes are located close to the threshold, many of the intermediate genotypes between them have low fitness and many evolutionary paths between them are inaccessible (Fig 7C). Thus, the cliff-like threshold fitness function is the major determinant of the observation that not all paths between two fit genotypes are accessible to evolution (Fig 7B). We find that unfit intermediate genotypes are in genetic proximity with each other and are on a limited number of paths; the fraction of inaccessible paths is smaller than if the same number of unfit genotypes were distributed randomly in genotype space (Fig 7D and 7E).

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Fig 7. Analysis of evolutionary pathway accessibility.

a, A threshold fitness potential function can lead to some paths being inaccessible between two genotypes of high fitness (abe, ABE) if the joint contribution of several alleles to the fitness potential (abE, aBE) leads to the fitness potential below the threshold. b, The fraction of unfit intermediate genotypes between two fit genotypes as a function of their average fitness potential. c, The grey area represents all genotypes in segment 7. When two fit genotypes (red dots) have high fitness potential, many paths between them will be accessible because many intermediate genotypes will also have high fitness potential and fitness (blue dots). d, Two fit genotypes in our dataset with intermediate genotypes between them. Intermediate genotypes are those that can be found between the two genotypes when making amino acid replacements from one genotype to the other. By this definition, the intermediate genotypes are located on the shortest paths connecting the two genotypes, with shortest paths are those that include only forward replacements. Accessible paths are those that incorporate only fit genotypes. The fraction of accessible shortest paths between two fit genotypes from data (orange) shows a modest decline as a function of Hamming distance between the two genotypes. The fraction of accessible shortest paths between two fit genotypes declines more rapidly when the same number of unfit genotypes as observed in real data are randomly drawn from intermediate genotypes (grey), demonstrating that in real data unfit genotypes are clustered in sequence space. Error bars are standard deviation. e, Genotypes can be represented by a graph with edges connecting genotypes if they can be connected by one amino acid replacement. We calculate the degree of connectivity (number of edges for each node) when all genotypes one replacement away are connected (blue), only unfit (fitness = 0) genotypes are connected (orange) and when unfit genotypes are connected with unfit genotypes selected at random keeping the same number of unfit genotypes as in our data (grey). With the exception of segment 9, in all segments the connectivity of unfit genotypes is higher than random, confirming that unfit genotypes are clustered in sequence space.

https://doi.org/10.1371/journal.pgen.1008079.g007

The effect of synergistic epistasis dominates the His3 fitness landscape (Fig 5A), affecting over 85% of amino acid replacements from our library that occurred in His3 evolution (Fig 3D). This synergistic epistasis may reflect the folding free energy of the protein [10,51,52], as evidenced by a weak to modest correlation between the fitness potential and the impact of amino acid replacements on the folding free energy of His3p (S8 Fig). Similarly, instances of sign epistasis may also be explained by changes in protein stability; for example, in the 143T background C141S increased fitness and also had a positive effect on stability (Fig 6B). Consistent with protein stability contributing to the observed sign epistasis we find that sites that exhibited reciprocal sign epistasis are close together in the His3p structure (S8 Fig). Only a relatively distant His3 structure to S. cerevisiae was used in this analysis, with about 30% divergence, likely reducing the accuracy of the free energy estimates. An additive contribution to free energy can lead only to a unidimensional fitness function [51], indicating that other non-additive mechanisms, such as catalytic activity or inter-subunit interactions, or non-additive contribution to protein stability must be responsible for the multidimensionality of the His3 fitness landscape.

Inference of inter-segmental epistatic interactions in the His3 gene sequence

Epistasis may be caused by interaction among positions within a segment (intra-segmental epistasis) or by interaction of the segment with the rest of the S. cerevisiae His3 sequence (inter-segmental epistasis). In other words, a specific genotype of one segment may be associated with low fitness because of interaction of amino acid states on that segment with amino acid states at other segments. To some extent, the contribution of inter- versus intra-segmental interactions can be decoupled. Given two fit genotypes (e.g., ABC & abc in one His3 segment), any unfit intermediate states (e.g., aBc in the same His3 segment) must be due to intra-segmental epistasis because the rest of the protein remains constant. For each segment, we took as a measurement of intra-segmental epistasis all pairs of fit genotypes and calculated the proportion of unfit intermediate genotypes as a function of the Hamming distance between the two fit genotypes. We then compared this proportion with the total proportion of all unfit genotypes as a function of Hamming distance from S. cerevisiae, a measurement that includes both inter- and intra-segmental epistasis. We found three times more inter-segmental than intra-segmental epistasis (S9 Fig), likely because a single segment provides a much smaller target space for interactions than the entire His3 protein. On the other hand, inter-segmental epistasis was just three times more frequent, rather than ten times more frequent, possibly due to higher probability of interaction of amino acid residues that are proximal in primary structure [53]. The proportion of sites under epistatic interactions increased exponentially with Hamming distance (S9 Fig), analogous to Orr’s snowball, the accumulation of genetic incompatibilities in the course of speciation [34,54].

Data access

The raw and processed data have been submitted to the NCBI Gene Expression Omnibus (GEO;http://www.ncbi.nlm.nih.gov/geo/) under accession number GSE99990. A virtual machine containing a running version of the data processing pipeline is available as a Docker image https://hub.docker.com/r/gui11aume/epi/. The scripts to reproduce the figures are on Github at https://github.com/Lcarey/HIS3InterspeciesEpistasis.

Study design

The His3 gene was selected for three principal reasons, it is short, conditionally essential and has not been known to be involved in protein-protein interactions. Studying 20²²⁰ variants of His3 is impossible, thus, we have chosen an approach to survey the fitness landscape in a manner that would elucidate the area most relevant to His3 evolution while managing the technical limitations of our experimental design. We considered amino acid states found in extant species, focusing on yeast species, which translated into a full combinatorial set of ~10⁸³ unique genotypes. Technically, it is feasible to measure fitness on the order of 100,000 unique genotypes in a single growth experiment. Therefore, we split the His3 gene into 12 independent segments such that the full combinatorial set of extant amino acid states from 21 yeast species in each segment was 10,000 – 100,000 genotypes. We then considered the combinatorial library for each segment in an independent growth experiment, which allowed us to study a tractable section of the sequence space while considering trajectories across a vast part of the space connecting extant species (Fig 1A). We constructed these combinations in 12 plasmid libraries and transformed them into a haploid His3 knockout S. cerevisiae strain. Growth rate (fitness) of yeast carrying different mutations in His3 was measured using serial batch culture in the absence of histidine.

We split the His3 gene sequence into segments in a manner agnostic to the structure of the His3 protein (S1A Fig). For technical reasons, a segment consisted of two variable regions with a constant region between them (S1B and S2A Fig). All growth experiments were performed independently for each segment, with the exception of one experiment on a limited group of genotypes from each segment which was done for the normalization of fitness values across different segments (S4 Fig).

As a control, we measured the rate of growth of S. cerevisiae whose entire His3 gene sequence came from another distant species. We found that the replacement of an entire gene sequence of His3 leads to wild-type rates of growth of S. cerevisiae even when the His3 sequence comes from very distant yeasts, as far as S. pombe (S4 Fig). Therefore, His3 appears to be an independent unit of the fitness landscape and is a good model for the study of fitness landscapes of an isolated gene.

Measuring fitness

Growth rates of isolated strains

We isolated 197 strains from all segment libraries of extant amino acid combinations (9-26 strains per segment) and used Sanger sequencing to determine the sequence. For each strain we performed 6 repeats of growth assay and calculated the average growth rate. Fitness values from competition and growth rates are highly correlated (r = 0.82, p = 10^-48). Correlation was significant and greater than 0.6 for all segments except segment 9, where all selected genotypes appeared to be neutral (S4 Fig).

Initial data filtering

The individual sequences of the variants were recovered from pair-end reads with the following steps: the constant region between the two variable regions was identified by inexact matching allowing up to 20% errors using the Seeq library version 1.1.2 (https://github.com/ezorita/seeq). The reads are not oriented because the Illumina sequencing adapters were added by ligation, so the constant regions were searched on both reads. Forward and reverse reads were swapped when a match was found on the reverse read. This ensured that all of the sequences are in the same orientation. For multiplexing purposes, the sample identity was encoded in the left and right primers used to PCR-amplify the variants. To demultiplex the reads, we used inexact matching with the candidate primers, allowing up to 20% errors. To merge the reads, the sequence of the reverse reads was reverse complemented and the constant region was searched by inexact matching allowing up to 20% errors. The position of the constant part in each read indicated how they must be stitched together. This approach was faster and less error-prone than using FLASH [56]. In the region of overlap, the consensus sequence was determined by picking the nucleotide with highest quality as indicated in the quality line of the fastq files. If 'N' persisted in the final sequence, the reads were discarded. The PCR primers were trimmed so that all the sequences of the same competition would start and end at the same location.

Reads that did not have the constant region, that could not be oriented or that could not be demultiplexed were discarded. The remaining errors in the reads were corrected by sequence clustering. We used Starcode version 1.0 [57] with default parameters and allowing up to two errors. The corrected reads were translated using the genetic code. Variants encoding the same proteins were not merged; they were kept separate for downstream analyses. A running Docker virtual machine with commented scripts to replay the whole the process is available for download at https://hub.docker.com/r/gui11aume/epi/.

DNA sequence variant frequency calculation and data filtering

The total number of reads for 12 segments, 3 time points and 4 replicas are shown in S2 Supporting Information. Genotypes frequencies are defined as the number of reads for a given genotype divided by the total number of reads in that replicate. Mean frequency was calculated over 4 replicas to be used in further analysis. However, to eliminate influence of outliers the median was taken instead of mean if absolute difference between mean and median was greater than the median value. Only genotypes present in both technical replicas of both biological replicas with at least ten reads (summed across all time points) in each of them were kept.

Sequencing error estimate

The length of the segment was designed so that each variable region was read twice by pair-end reads (S2A Fig). This strategy led to a substantial reduction in the sequencing error rate because mismatches between the two reads were corrected to the nucleotide with the higher quality call.

The raw Illumina sequencing error rate was estimated by measuring the frequency of mismatches between forward and reverse reads. The rationale of this estimate is that each mismatched nucleotide must be a sequencing error for at least one of the reads. The variant calling error rate was estimated by collecting groups of reads that differ by only one nucleotide in the constant region (S2A Fig). Since the variable regions are identical, such reads come from the same variant and the different nucleotide in the constant region must originate from a calling error (mutations in S. cerevisiae are negligible because they occur at a much lower rate). The frequency of such reads was used to approximate the per-nucleotide variant calling error rate. The raw Illumina error rate was computed by custom Python scripts and reads differing by one nucleotide were collected using the “sphere” clustering option of Starcode with a maximum distance of 1 (see the Data access section for the code repositories). The results are summarized in S2 Supporting Information.

The design strategy and the low error rate allow us to distinguish variants incorporated in the library from random sequencing errors. Each library contained on the order of 10⁵ individual sequence variants, so that each library variant would be found at a frequency several orders of magnitude higher than variants created by sequencing errors. For example, for segment 7, the number of nucleotide variants was 176,879 with 31,815,448 possible single mutants of these variants. The error rate in segment 7 was 0.04% per nucleotide, which translates into 2.4% of reads being miscalled. Thus, the expected frequency of a particular miscalled variant is 0.024 / 31,815,448 ≈ 8x10^-10, considerably smaller than the expected frequency of real variants 0.976 / 176,879 ≈ 6x10^-6. The estimated frequencies of library variants for all segments of His3 are reported in S2 Supporting Information.

The final variant calling error rate was smaller than the numbers shown in S2 Supporting Information because errors were further corrected by sequence clustering using Starcode (see Initial data filtering section). In line with the rationale above, low frequency erroneous reads are converted to the closest high frequency variant at a maximum Levenshtein distance of 2.

PCR recombination rate estimate

It is known that PCR can create new genotypes by template switching [58]. To test the magnitude of this effect, we took advantage of the two-block design of the variants and estimated the frequency of recombination between the left and the right variable regions. The structure of reads can be represented like this: AAAAAA-------BBBBBB. In this example, “A” represents the left part of the variable region of the segment, “-“ represents the invariable region and “B” represents the right part of the variable region. Insertions of two or more nucleotides are rare events caused by errors during the library synthesis; so, if the same insertion in a left half of the variant is associated to several variants on the right half, it is likely an occurrence of template switching (the same holds for insertions in the right half of the variant). For example, if the region with A*AAA*AA-------BBBBBB, where “*” represents a deletion, is found with several different variants of the B segment then such a situation likely represents template switching. Focusing on the initial time point to avert the effect of selection, we counted a total of 11,454 variants with two or more insertions on either the left side or the right side. Among those, 76 had the same insertion as another variant. This means that > 98.6% of the variants were free of template switching. Extrapolating to the rest of the dataset, this means that the “leakage” of reads between variants is substantially lower than the magnitude of the observed epistasis and we can rule out this artefact as a potential explanation for our results. In either case, all errors in our experimental pipeline, including template switching, are taken into account when we calculate the false discovery rate.

Noise estimation

The major factors causing noise in genotype frequency measurements are sampling errors, PCR amplification errors and genetic drift during the competition. For all of these factors, the amount of error depends on the genotype frequency. Therefore, we estimated measurement errors as the function of genotype frequency.

For a given segment, time point and a pair of biological or technical replicas for each genotype we calculated the mean frequency and the squared difference of frequencies from these two replicas. We sorted genotypes by mean frequency and grouped them such that each bin contains 5000 genotypes. We calculated the average frequency and the average squared difference in each bin. Additionally, squared error for frequency 0 was set equal to , where N_i and N_j are total read numbers in replicas i and j. Finally, by linear interpolation we obtained dependencies of squared differences as a function of frequency, , where i and j are different replicas.

Using squared differences from pairwise comparison of replicas we can estimate variance of mean frequency over four replicas. Let numerate replicas 1, 2, 3, 4 where 1, 2 are technical replicas of the first biological repeat and 3, 4 are the technical replicas of the second biological repeat. Errors coming from the competition (e.g.: genetic drift) are shared for replicas 1, 2 and for replicas 3, 4. Let’s call them and and their variances and , respectively. Technical errors of sampling from the population and from PCR are unique for each replica. Let’s call them and their variances , respectively. All variances are function of frequency and when writing we assume .

In the introduced notations the mean frequency over 4 replicas is: where f* is the true frequency. Applying basic properties of variance, the variance of mean frequency:

To estimate we used squared differences from pairwise comparison of replicas calculated above :

Therefore, the variance of mean frequency f can be found as: Recalling that variance and squared differences are a function of frequency: For each segment and time point we calculated the numerical function σ²(f). Then for each genotype having mean frequency f_x we estimated its variance as σ²(f_x)

Merging amino acid genotypes

We merged nucleotide genotypes that corresponded to the same amino acid sequence and summed their frequencies and variances. We filtered out all genotypes x that had any of following patterns:

or or . Fraction of such genotypes were <0.5% for all segments except S9, for which it was 4.5%

For further analysis, this amino acid dataset was used except when specified.

Fitness estimation

Number of cells in a pool with particular genotype x after time interval t increases exponentially where s_x is absolute fitness. Frequency of genotype x as well depends exponentially on absolute fitness with an additional multiplicative factor: where N^t and N⁰ are total cell numbers in a pool at time points 0 and t. Factor reflects the total growth of population, it changes with time but is the same for all genotypes. Therefore, we can rewrite genotype frequency at time t as: where

In the measured dataset for each genotype x we have 3 measurements of frequency and their errors . To estimate genotype fitness, we minimized relative squared errors of exponential fit as function of fitness s_x and initial frequency : (1)

This formula contains four parameters common for all genotypes from one segment: . Further we will perform additional shifting and scaling of fitness values (see next section), therefore, without loss of generality we could set and t₁ = 1. Ideally, t₂/t₁ should equal 14; however, we noticed that this ratio does not hold for many segments and fitted k = t₂/t₁ from data instead of using value 14.

To find specific and k for each segment we selected genotypes with high frequencies at t₀ (t₀>25∙10⁻⁶) that corresponds to ~500-1000 reads per technical replicate. Each segment contains 10³-10⁴ genotypes that meet this criterion. We minimized eq. (1) for selected genotypes trying all possible combinations of () from a grid where with step 0.01 and kϵ[1,14] with step 0.1 and choose () which gives minimal (*).

Finally, given () for each segment we found s_x for each genotype. Errors for fitness values, , were estimated as standard error of best-fit parameter.

For genotypes with frequencies pattern fit of Eq (1) cannot be obtained. Therefore, we defined upper boundary for their fitness value as , where are total read numbers at time point t₁ in i-th replica.

Fitness rescaling

We scaled fitness such that lethal genotypes have fitness 0 and neutral genotypes have fitness 1. We assumed that genotypes with a stop codon or frame shift are lethal. Thus, for each segment we linearly rescaled the fitness distribution so that 95% of genotypes with nonsense mutations have a fitness of 0 and so that the local maximum of the fitness distribution of genotypes with extant amino acids is 1. The scaling around the local maximum led to the shift of fitness values of less than +/- 0.025 in each of the 12 segments compared to the measured wildtype strains and did not affect our results (for scale, we called an amino acid change non-neutral if its effect on fitness was > 0.4). All fitness values that became smaller than 0 were set to 0.

Quality control and comparison of synonymous sequences

We used nucleotide synonymous sequences as an internal control. The error rate for a measurement of fitness of an amino acid sequence depends on the number of synonymous sequences, n, that were used to estimate it. Therefore, we estimated the false discovery rates separately for categories with n = 1,..10 variants. For each amino acid genotype with more than n synonymous variants we merged random combination of n of its nucleotide genotypes and estimated fitness. We then calculated the difference between this fitness and the fitness of the corresponding amino acid sequence. We classified case as “false unfit” if difference was <-0.4 and as “false fit” if difference was >0.4. The fraction of such cases gives us false discovery rates for genotypes having n synonymous variants. To get total false discovery rates for each segment we averaged “false unfit” and “false fit” rates for different n with weights equal to the fractions of genotypes in amino acid dataset which have n synonymous variant (S2 Supporting Information). The high correlation between biological replicas (S2 Supporting Information) confirms high accuracy of our high-throughput experiments, with the exception of segment 9.

Analysis of amino acid replacement effects in different backgrounds

For each amino acid replacement, we calculated its fitness effect in different backgrounds. We estimated the fraction of backgrounds in which a replacement exhibits deleterious, beneficial and neutral effects. To get the fraction of backgrounds with neutral effects we utilized the approach of mixture distribution analysis from Sarkisyan et al. [10]. We assume that neutral replacements have the same distribution as the fitness effects of synonymous replacements and are caused by measurement noise. We then calculated the fraction of backgrounds in which mutations have a neutral effect as the overlap between the distribution of synonymous mutations and distribution of fitness effects across different backgrounds. In remaining backgrounds, the amino acid replacement was called to have a non-neutral effect. Among them we counted those with strong fitness effects, including deleterious mutations when the fitness effect was < -0.4 and beneficial when the fitness effect was > 0.4. We concluded that a particular amino acid replacement exhibits a strong deleterious or beneficial effect in some background if the fraction of such backgrounds exceeded the false discovery rate.

Predicting fitness using deep learning

To predict the unidimensional fitness function based on additive contribution of extant amino acid states we used deep learning, a machine learning technique capable of constructing virtually any function, even with a simple neural network architecture [59] (Fig 4B). To convert amino acid sequences into a binary feature matrix we used one-hot encoding strategy, in which each feature (column in the matrix) indicates the presence or absence of a particular amino acid state. For neural network implementation the TensorFlow library was used (www.tensorflow.org/about/bib).

To optimise the accuracy/overfitting ratio, we tested different combinations of neural network architectures and parameters. As a starting point, we selected a number of complex architectures, which describe our data but are prone to overfitting due to their large number of parameters. We then gradually reduced the number of layers and neurons to reduce the overfitting, while empirically controlling for accuracy.

Our final architecture consists of three layers and 22 neurons in total (Fig 4). Each neuron performs a linear transformation of the input and subsequently applies a non-linear sigmoid activation function to the result. The output of the first layer is a single sigmoid of a linear transformation of the feature vector, i.e. where x is the feature vector, c₁ is the vector of coefficients, b₁ is the bias and σ(t) = (1+e^−t)⁻¹. Looking forward, observe that is a fitness potential of the genotype x (see main text). The second layer decompresses the hidden nonlinear representation into 20 sigmoids, the combination of which is further linearly transformed with the only neuron of the third layer and wrapped into another sigmoid function: In the formula above, c_2,i is the coefficient of the i-th neuron in the n-th layer, and b_2,i is the bias of the i-th neuron in the second layer (the biases of the only neurons of the first and third layers are b₁ and b₃, correspondingly).

The key idea of our approach is that the number of neurons in the first layer of the neural network determines the number of linear combinations of mutations (or fitness potentials) used in order to predict the fitness of the variant. In other words, each neuron in the first layer assigns a single unique weight to every amino acid state in the dataset (Fig 4). Thus, the number of neurons in the first layer of the architecture is the dimensionality of epistasis in the model (i.e. one in this case). The fitness potentials are then transformed by a nonlinear phase shift function constructed by the 22 neurons of the neural network.

The simplicity of the architecture minimizes overfitting, which was further prevented by keeping 10% of the data as a test set (in order to see how well the model performs on a fraction of the data it has never seen) and early stopping (training was stopped if the test accuracy did not improve for 10 epochs). The loss function that is being optimised is not convex, which leads to a high probability of getting stuck in different local minima. To ensure reproducibility, each of our models was constructed ten independent times using random train-test splits. The accuracies of the 10 constructed models varied by at most 2%.

Each model was trained for under 100 epochs using mean squared error as the loss function. An adaptive learning rate method proposed by Geoffrey Hinton, RMSProp, was used as the optimiser [60]. This algorithm is a version of a mini-batch stochastic gradient descent, utilising the gradient magnitude of the recent gradients in order to normalise the current ones. All the weights were initialised using Xavier normal initialiser [61].

Paths between pairs of fit genotypes

For analysis in Fig 7D, we first choose two fit “parental” genotypes, one randomly chosen genotype (eg: ABE) and the other parental genotype that is either S. cerevisiae wildtype genotype (inter-segmental) or another random fit genotype in the data (intra-segmental) (eg: abe). The two genotypes in this example are Hamming Distance 3 apart (HD = 3). We next compute all (2^HD-2) intermediate genotypes (eg: AbC, aBc, et cetera) and retain the subset that were experimentally measured. We represent the two parental genotypes and all measured intermediate genotypes as an undirected graph in which each genotype is a vertex. All genotypes one amino acid replacement apart are connected by an unweighted edge. The shortest possible path for a given pair of genotypes is of length HD. We find all shortest paths between the two parental genotypes using a breadth-first search. We next remove all vertices (genotypes) that are unfit, and recompute the number of shortest between the two parental genotypes. For example, in Fig 7A, there are six paths of length three if you take into account all genotypes, but only three paths of length three if you take into account only fit genotypes.

Clustering of unfit genotypes in sequence space

For the analysis in Fig 7E, we first represent the two parental genotypes and all measured intermediate genotypes as an undirected graph in which each genotype is a vertex. All genotypes one amino acid replacement apart are connected by an unweighted edge. We can then compute the degree (number of genotypes of distance one) for each vertex (genotype). We do so randomly drawing from all measured genotypes and using only unfit genotypes or using the same number but randomly chosen genotypes. For the randomly chosen genotypes, the value is the average over 1000 runs.

Quantifying sign epistasis

For each amino acid replacement (eg: C -> S at position 141), we considered only those that exhibit a large fitness effect (abs. difference > 0.4) comprising a set of amino acid replacements with large effects. For each amino acid replacement, we divided the genetic backgrounds into two categories: those in which the replacement caused a > 0.4 increase in fitness, and those backgrounds in which the replacement caused > 0.4 decrease in fitness. A single amino acid replacement can cause a large increase in fitness in some backgrounds and a large decrease in others due to two possible reasons: sign epistasis or experimental error. To differentiate the two cases, we identified secondary amino acid replacements that significantly alter the ratio of large increases to large decreases in fitness (Fisher’s exact test, Bonferroni corrected p-value < 0.05). We only consider a site to be under sign epistasis if there is a second site that alters the frequency of sign epistasis in a statistically significant manner, i.e. more frequently than expected by chance alone.

Ancestral state reconstruction

We reconstructed ancestral amino acid states using maximum likelihood approach implemented in CODEML program of PAML 4 [62].

Structural analysis