Statistical models of repeat-protein families

We start by building statistical models for the three repeat protein families presented in Fig 1 (ANK, TPR, LRR). These models give the probability P(σ) to find in the family of interest a particular sequence σ = (σ₁, …, σ_2L) for two consecutive repeats of size L. The model is designed to be as random as possible, while agreeing with key statistics of variation and co-variation in a multiple sequence alignment of the protein family. Specifically, P(σ) is obtained as the distribution of maximum entropy [28] which has the same amino-acid frequencies at each position as in the alignment, as well as the same joint frequencies of amino acid usage in each pair of positions. Additionally, repeat proteins share many amino acids between consecutive repeats, both due to sharing a common ancestor and to evolutionary selection acting on the protein. To account for this special property of repeat proteins, we require that the model reproduces the distribution of overlaps between consecutive repeats. Using the technique of Lagrange multipliers, the distribution can be shown to take the form [17]: (1) with (2) where h_i(σ), J_ij(σ_i, σ_j), and {λ_ID}, ID = 0, 1, …, L, are adjustable Lagrange multipliers that are fit to the data to reproduce the experimentally observed site-dependent amino-acid frequencies f_i(σ_i), joint probabilities between two positions, f_ij(σ_i, σ_j), and the distribution of Hamming distances between consecutive repeats P(ID(σ)), which is equivalent to maximize the likelihood of the data under the model. We fit these parameters using a gradient ascent algorithm: we start from an initial guess of the parameters, then generate sequences via Monte-Carlo simulations and update the parameters proportionally to the difference between the empirical and model generated observables , and P(ID(σ)) − P(ID(σ))^model. We repeat the previous steps until the model reproduces the empirical observables defined above, with a target precision motivated according to the finite size of our original dataset, as in Ref. [17]. See Methods for more details. We tested the convergence of the model learning by synthetically generating datasets and relearning the model (see Methods).

By analogy with Boltmzan’s law, we call E(σ) a statistical energy, which is in general distinct from any physical energy. The particular form of the energy (2) resembles that of a disordered Potts model. This mathematical equivalence allows for the possibility to study effects that are characteristic of disordered systems, such as frustration or the existence of an energy landscape with multiple valleys, as we will discuss in the next sections.

Eq 2 is the most constrained form of the model, which we will denote by E_full(σ). One can explore the impact of each constraint on the energy landscape by removing them from the model. For instance, to study the role of inter-repeat sequence similarity due to a common evolutionary origin, one can fit the model without the constraint on repeat overlap ID, i.e. without the λ_ID term in Eq 2. We call the corresponding energy function E₂. One can further remove constraints on pairwise positions that are not part of the same repeat, making the two consecutive repeats statistically independent and imposing h_i = h_i+L (E_ir), or only linked through phylogenic conservation through λ_ID (E_ir,λ). Finally one can remove all interaction constraints to make all positions independent of each other (E₁), or even remove all constraints (E_rand ≡ 0).

Statistical energy vs unfolding energy

The evolutionary information contained in multiple sequence alignments of protein families is summarized in our model by the energy function E(σ). Since this information is often much easier to access than structural or functional information, there is great interest in extracting functional or structural properties from multiple sequence alignments, provided that there exists a clear quantitative relationship between statistical energy and physical energy.

Such a relationship was determined experimentally for repeat proteins by using E(σ) to predict the effect of point mutations on the folding stability measured by the free energy difference between the folded and unfolded states, ΔG, called the unfolding energy [17, 20]. Synthetic sequences with low E(σ) have also been shown to reproduce the fold and function of natural sequences [29]. Here, extending an argument already developed in previous work [30–33], we show how this correspondance between statistical likelihood and folding stability arises in a simple model of evolution.

Evolutionary theory predicts that the prevalence of a particular genotype σ, i.e. the probability of finding it in a population, is related to its fitness F(σ). In the limit where mutations affecting the protein are rare compared to the time it takes for mutations to spread through the population, Kimura [34] showed that the probability of a mutation giving a fitness advantage (or disadvantage depending on the sign) ΔF over its ancestor will fix in the population with probability 2ΔF/(1 − e^−2NΔF), where N is the effective population size. The dynamics of successful substitution satisfies detailed balance [35], with the steady state probability (3)

Again, one may recognize a formal analogy with Boltzmann’s distribution, where F plays the role of a negative energy, and N an inverse temperature. If we now assume that fitness is determined by the unfolding free energy ΔG, F(σ) = f(ΔG(σ)), then the distribution of genotypes we expect to observe in a population is (4)

Note that a similar relation should hold even if we relax the hypotheses of the evolutionary model. While in more general contexts (e.g. high mutation rate, recombination), the relation between ln P(σ) and F(σ) may not be linear, such nonlinearities could be subsumed into the function f.

Identifying terms in the two expressions (1) and (3), we obtain a relation between the statistical energy E, and the unfolding free energy ΔG: (5)

For instance, if we assume a linear relation between fitness and ΔG, f(ΔG) = A + BΔG, then we get a linear relationship between the statistical energy and ΔG, as was found empirically for repeat proteins [17].

Strikingly, the relationship f does not have to be linear or even smooth for this correspondance to work. Imagine a more stringent selection model, where f(ΔG) is a threshold function, f(ΔG) = 0 for ΔG > ΔG_sel and −∞ otherwise (lethal). In that case the probability distribution is P(σ) = (1/Z)Θ(ΔG − ΔG_sel), where Θ(x) is Heaviside’s function. Using a saddle-point approximation, one can show that in the thermodynamic limit (long proteins, or large L) the distribution concentrates at the border ΔG_sel, and is equivalent to a “canonical” description [30, 31, 33]: (6) where the “temperature” T_sel is set to match the mean ΔG between the two descriptions: (7)

This correspondance is mathematically similar to the equivalence between the micro-canonical and canonical ensembles in statistical mechanics.

Statistical energy and unfolding free energy are linearly related by equating (Eq 1) and (Eq 6): (8) despite f being nonlinear. Eq 8 is in fact very general and should hold for any f in the thermodynamic limit in the vicinity of 〈E〉.

Equivalence between two definitions of entropies

There are several ways to define the diversity of a protein family. The most intuitive one, followed by [26], is to count the total number of amino acid sequences that have an unfolding free energy ΔG_sel above a threshold ΔG_sel [2]. This number naturally defines a Boltzmann entropy, (9)

Alternatively, starting from a statistical model P(σ), one can calculate its Shannon entropy, defined as (10) as was done in Ref. [27]. What is the relation between these two definitions?

By the same saddle-point approximation as in the previous section, the two are identical in the thermodyamic limit (large L), provided that the condition (Eq 7) is satisfied. We can thus reconcile the two definitions of the entropy in that limit.

To calculate the Boltzmann entropy (Eq 9), one needs to first evaluate the threshold E_sel in terms of statistical energy. This threshold is given by E_sel = E₀ − ΔG_sel/T_sel, where E₀ and T_sel can be obtained directly by fitting (Eq 8) to single-mutant experiments. E_sel can also be obtained as a discrimination threshold separating sequences that are known to fold properly versus sequences that do not [26]. In that case, assuming that the linear relationship (Eq 8) was evaluated empirically using single mutants, this relationship can be inverted to get ΔG_sel in physical units.

Calculating the Shannon entropy Eq (10), on the other hand, does not require to define any threshold. However, the threshold in the equivalent Boltzmann entropy can be obtained using Eqs 7 and 8, i.e. E_sel = 〈E〉, where the average is performed using the distribution defined in Eqs 1 and 2.

Entropy of repeat protein families

To compare how the different elements of the energy function affect diversity, we calculate the entropy of ensembles built of two consecutive repeats from a given protein family for the different kinds of models described earlier, from the least constrained to the most constrained: E_rand, E₁, E_ir, E_ir,λ, E₂, E_full. In the case of models with interactions, calculating the entropy directly from the definition Eq (10) is impossible due to the large sums. A previous study of entropies of protein families used an approximate mean-field algorithm, called the Adaptive Cluster Expansion [27], for both parameter fitting and entropy estimation. Here we estimated the entropies using thermodynamic integration of Monte-Carlo simulations, as detailed in Methods. This method is expected to be asymptotically unbiased and accurate in the limit of large Monte-Carlo samples.

The resulting entropies and their differences are reported in Table 1 and Fig 2. All three considered families (ankyrins (ANK), leucine-rich repeats (LRR), and tetratricopeptides (TPR)) show a large reduction in entropy (∼40 − 50%) compared to random polypeptide string models of the same length 2L (of entropy S_rand = 2L ln(21)). Interactions and phylogenic similarity between repeats generally have a noticeable effect on family diversity, although the magnitude of this effect depends on the family: (S₁ − S_full)/S_full = 7% for ANK, versus, 13% for LRR, and 16% for TPR. Thus, although interactions are essential in correctly predicting the folding properties, they seem to only have a modest effect on constraining the space of accessible proteins compared to that of single amino-acid frequencies. However, when converted to numbers of sequences, this reduction is substantial, from to for ANK, from 10³⁹ to 10³⁴ for LRR, and from 7 ⋅ 10⁵⁰ to 4 ⋅ 10⁴² for TPR.

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Table 1. Entropies (in bits, i.e. units of ln(2)) of sequences made of two consecutive repeats, for the three protein families shown in Fig 1.

Entropies are calculated for models of different complexity: model of random amino acids (S_rand = 2L ln(21), divided by ln(2) when expressed in bits); independent-site model (S₁), pairwise interaction model (S₂); pairwise interaction model with constraints due to repeat similarity λ_ID (S_full); pairwise interaction model of two non-interacting repeats learned without (S_ir) and with (S_ir,λ) constraints on repeat similarity. Fig 2 shows graphically some of the information contained in this table.

https://doi.org/10.1371/journal.pcbi.1007282.t001

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Fig 2. Contributions of within-repeat interactions (S₁ − S_ir green), repeat-repeat interactions (S_ir,λ − S_full, purple), and phylogenic bias between consecutive repeats (S_ir − S_ir,λ, blue), to the entropy reduction from an independent-site model.

All three contributions are comparable, but with a larger effect of within-repeat interactions and phylogenic bias in TPR. The fourth bar (orange) quantifies the redundancy between two constraints with overlapping scopes: the constraint on consecutive-repeat similary, and the constraint on repeat-repeat correlations. This redundancy is naturally measured within information theory by the difference of impact (i.e. entropy reduction) of a constraint depending on whether or not the other constraint is already enforced.

https://doi.org/10.1371/journal.pcbi.1007282.g002

By considering models with more and more constraints, and thus with lower and lower entropy, we can examine more finely the contribution of each type of correlation to the entropy reduction, going from E₁ to E_ir to E_ir,λ to E_full. This division allows us to quantify the relative importance of phylogenic similarity between consecutive repeats (λ_ID) relative to the impact of functional interactions (J_ij), as well as the relative weights of repeat-repeat versus within-repeat interactions (Fig 2). We find that phylogenic similarity contributes substantially to the entropy reduction, as measured by S_ir − S_ir,λ = 4.5 bits for ANK, 4.3 bits for LRR, and 10.7 bits for TPR. The contribution of repeat-repeat interactions (S_ir,λ − S_full ∼ 5 bits for all three families) is comparable or of the same order of magnitude as that of within-repeat interactions (S₁ − S_ir = 4.3 bits for ANK, 6.9 bits for LRR, and 11.4 bits for TPR). This result emphasizes the importance of physical interactions between neighboring repeats in the whole protein.

On a technical note, we also find that pairwise interactions encode constraints that are largely redundant with the constraint of phylogenic similarity between consecutive repeats, as can be measured by the double difference S_ir − S_ir,λ − S₂ + S_full > 0 (Fig 2, orange bars). This redundancy comes from the fact that, in absence of an explicit constaint on P(ID) in E₂, the interaction couplings J_i,i+L(σ, σ) between homologous positions in the two repeats is expected to favor pairs of identical residues to mimic the effect of λ_ID. This redundancy motivates the need to correct for this phylogenic bias before estimating repeat-repeat interactions.

Comparing the three families, ANK has little phylogenic bias between consecutive repeats, and relatively weak interactions. By contrast, TPR has a strong phylogenic bias and strong within-repeat interactions.

Effect of interaction range

We wondered whether interactions constraining the space of accessible proteins had a characteristic lengthscale. To answer this question, for each protein family in Fig 1, we learn a sequence of models of the form Eq 2, in which J_ij was allowed to be non-zero only within a certain interaction range d(i, j) ≤ W, where the distance d(i, j) between sites i and j can be defined in two different ways: either the linear distance |i − j| expressed in number of amino-acid sites, or the three-dimensional distance between the closest heavy atoms in the reference structure of the residues. Details about the learning procedure and error estimation are given in the Methods; see also S1 Fig for an alternative error estimate.

The entropy of all families decreases with interaction range W, both in linear and three-dimensional distance, as more constraints are added to reduce diversity (Fig 3 for ANK, and S2 Fig for LRR and TPR). The initial drop as a function of linear distance (Fig 3A) is explained by the many local interactions between nearby residues in the sequence. The entropy then plateaus until interactions between same-position residues in consecutive repeats are included in the W range, which leads to a sharp entropy drop at W = L. This suggests that long range interactions along the sequence generally do not constrain the protein ensemble diversity, except for interactions at exactly the scale of the repeat. This result suggests that the repeat structure is an important constraint limiting protein sequence exploration. These observations hold for all three repeat protein families. The importance of 3D structure in reducing the entropy can also be appreciated in the entropy decay as a function of physical distance (Fig 3B for ANK) where most of the entropy drop happens within the first 10 angstroms, indicating that above this characteristic distance interactions are not crucial in constraining the space of accessible sequences.

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Fig 3. Entropy reduction as a function of the range of interactions between residue sites.

A) Entropy of two consecutive ANK repeats, as a function of the maximum allowed interaction distance W along the linear sequence. The entropy of the model decreases as more interactions are added and they constrain the space of possible sequences. After a sharp initial decrease at short ranges, the entropy plateaus until interactions between complementary sites in neighbouring repeats lead to a secondary sharp decrease at W = L − 1 = 32 (dashed line), due to structural interactions between consecutive repeats. B) Entropy of two consecutive ANK repeats as a function of the maximum allowed three-dimensional interaction range. The entropy decreases rapidly until ∼10 Angstrom, after which decay becomes slower. In both panels entropies are averaged over 10 realizations of fitting the model; See Methods for details of the learning and entropy estimation procedure. Error bars are estimated from fitting errors between the data and the model; see Methods and S1 Fig for error bars calculated as standard deviations over 10 realizations of model fitting.

https://doi.org/10.1371/journal.pcbi.1007282.g003

Multi-basin structure of the energy landscape

The energy function of Eq (2) takes the same mathematical form as a disordered Potts model. These models, in particular in cases where σ_i can only take two values, have been extensively studied in the context of spin glasses [36]. In these systems, the interaction terms −J_ij(σ_i, σ_j) imply contradictory energy requirements, meaning that not all of these terms can be minimized at the same time—a phenomenon called frustration. Because of frustration, natural dynamics aimed at minimizing the energy are expected to get stuck into local, non-global energy minima (Fig 4), significantly slowing down thermalization. This phenomenon is similar to what happens in structural glasses in physics, where the energy landscape is “rugged” with many local minima that hinder the dynamics. Incidentally, concepts from glasses and spin glasses have been very important for understanding protein folding dynamics [37].

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Fig 4. A rugged energy landscape is characterized by the presence of local minima, where proteins sequences can get stuck during the evolutionary process.

The set of sequences that evolve to a given local minimum defines the basin of attraction of that mimimum.

https://doi.org/10.1371/journal.pcbi.1007282.g004

We asked whether the energy landscape of Eq (2) was rugged with multiple minima, and investigated its structure. To find local minima, we performed a local energy minimization of E_full (learned with all constraints including on P(ID), but taken with λ_ID = 0 to focus on functional energy terms). By analogy with glasses, such a minimization is sometimes called a zero-temperature Monte-Carlo simulation or a “quench”. The minimization procedure was started from many initial conditions corresponding to naturally occuring sequences of consecutive repeat pairs. At each step of the minimization, a random beneficial (energy decreasing) single mutation is picked; double mutations are allowed if they correspond to twice the same single mutation on each of the two repeats. Minimization stops when there are no more beneficial mutations. This stopping condition defines a local energy minimum, for which any mutation increases the energy. The set of sequences which, when chosen as initial conditions, lead to a given local minimum defines the basin of attraction of that energy mimimum (Fig 4). The size of a basin corresponds to the number of natural proteins belonging to that basin.

Performing this procedure on natural sequences of consecutive repeat pairs from all three families yielded a large number of local minima (Fig 5). To control for the phylogenetic bias that links natural sequences, we repeated this analysis on sequences synthetically generated from the model (E_full), and obtained very similar results (see S6 Fig for ANK). When ranked from largest to smallest, the distribution of basin sizes follows a power law (Fig 5A for ANK and S3A and S4A Figs for LRR and TPR). The energy of the minimum of each basin generally increases with the rank, meaning that largest basins are also often the lowest. Despite this multiplicity of local minima, the Monte-Carlo dynamics that we used in previous sections for learning the model parameters and for estimating the entropy did not get stuck in these minima, suggesting relatively low energy barriers between them.

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Fig 5. Interactions within and between repeats sculpt a rugged energy landscape with many local minima.

Local minima were obtained by performing a zero-temperature Monte-Carlo simulation with the energy function in Eq (2), starting from initial conditions corresponding to naturally occurring sequences of pairs of consecutive ANK repeats. A, bottom) Rank-frequency plot of basin sizes, where basins are defined by the set of sequences falling into a particular minimum. A, top) energy of local minima vs the size-rank of their basin, showing that larger basins often also have lowest energy. Gray line indicates the energy of the consensus sequence, for comparison. B) Pairwise distance between the minima with the largest basins (comprising 90% of natural sequences), organised by hierarchical clustering. The panel right above the matrix shows the size of the basins relative to the minima corresponding to the entries of the distance matrix. A clear block structure emerges, separating different groups of basins with distinct sequences. C-D) Same as A) and B) but for single repeats. Since single repeats are shorter than pairs (length L instead of 2L), they have fewer local energy minima, yet still show a rich multi-basin structure. Equivalent analyses for LRR and TPR are shown in S3 and S4 Figs.

https://doi.org/10.1371/journal.pcbi.1007282.g005

The partition of sequences into basins allows for the definition of a new kind of entropy S_conf = −∑_b P(b) ln P(b) called configurational entropy, based on the distribution of basin sizes, P(b) = ∑_σ∈b P(σ), where σ ∈ b means that energy minimization starting with sequence σ leads to basin b. This configurational entropy measures the effective diversity of basins, and is thus much lower than the sequence entropy S_full, while the difference S_full − S_conf measures the average diversity of sequences within each basin. We find S_conf = 5.1 bits for ANK, 6.0 bits for LRR, and 10.4 bits for TPR. As each basin corresponds to a distinct sub-family within each family [32], this entropy quantifies the effective number of these subgroups.

While basins are very numerous, they are also not independent of each other. An analysis of pairwise distances (measured as the Hamming distance between the local minima) between the largest basins reveals that they can be organised into clusters (panels B of Fig 5, S3 and S4 Figs), suggesting a hierarchical structure of basins, as is common in spin glasses [36].

The impact of repeat-repeat interactions on the multi-basin structure can be assessed by repeating the analysis on the model of non-interacting repeats, E_ir. In that model the two repeats are independent, so it suffices to study local energy minima of single repeats—local minima of pairs of repeats follow simply from the combinatorial pairing of local minima in each repeat. The analyses of basin size distributions, energy minima, and pairwise distances in single repeats are shown in panels C and D of Fig 5, S3, and S4 Figs. We still find a substantial number of unrelated energy minima, suggesting again several distinct subfamilies even at the single-repeat level. For comparison, the configurational entropy of pairs of independent repeats is 6.9 bits for ANK, 6.7 for LRR, and 7.6 for TPR. While for ANK and LRR repeat-repeat interactions decrease the configurational entropy, as they do for the conventional entropy, they in fact increase entropy for TPR, making the energy landscape even more frustrated and rugged.

Note that the independent sites model E₁ defines a convex energy landscape with a single local minimum—the consensus sequence—as all constraints h_i can be optimized independently. To address how the interactions contribute in shaping the sequence space, going from a convex to a rugged landscape, we repeated the analysis with a limited linear interaction range W of 3 and 10 (models of Fig 3A). We find that the more interactions we add, the more local minima we find (S5A and S5B Fig for ANK with W = 3, and C and D for W = 10). The minima cluster into clearer sub-blocks structure as the interaction range is increased, consistent with the entropy reduction observed in Fig 3A.

In summary, the analysis of the energy landscape reveals a rich structure, with many local minima ranging many different scales, and with a hierarchical structure between them.

Distance between repeat families

Lastly, we compared the statistical energy landscapes of different repeat families. Specifically, we calculated the Kullback-Leibler divergence between the probability distributions P(σ) (given by Eqs 1 and 2) of two different families, after aligning them together in a single multiple sequence alignment (see Methods).

We find essentially no similarity between ANK and TPR, despite them having similar lengths: D_KL(ANK||TPR) = 227.6 bits, and D_KL(TPR||ANK) = 214.1 bits. These values are larger than the Kullback-Leibler divergence between the full models for these families and a random polypetide, D_KL(ANK||rand) = 122.8 bits, and D_KL(TPR||rand) = 157.6 bits. LRR is not comparable to ANK or TPR as it is much shorter, and a common alignment is impractical. These large divergences between families of repeat proteins show that different families impose quantifiably different constraints, which have forced them to diverge into different troughs of non-overlapping energy landscapes. This lack of overlap makes it impossible to find intermediates between the two families that could evolve into proteins belonging to both families.

Data curation

We use a previously curated alignment of pairs of repeats for each family [17]: ANK (PFAM id PF00023 with a final alignment of 20513 sequences of L = 66 residues each), LRR (PFAM id PF13516 with a final alignment of 18839 sequences of L = 48 residues each) and TPR (PFAM id PF00515 with a final alignment of 10020 sequences of L = 68 residues each). Those multiple sequence alignments of repeats were obtained from PFAM 27.0 [6, 7]. In order to improve the data obtained from the PFAM database, we used original full protein sequences available in UniProt database [43] to add available information using the headers of the original alignement. Firstly, to decrease the number of gaps positions, misdetected initial and final amino acids in repeats were completed with residues from full sequences. Secondly, individual repeats which appeared consecutively in natural proteins were joined into pairs. Finally, positions with more than 80% of gaps along the alignment were removed, eliminating in this way insertions.

From the multiple sequence alignement of each family, they were calculated the observables that we use to constrain our statistical model. Particularly, we calculated the marginal frequency f_i(σ_i) of an amino acid σ_i at position i and the joint frequency f_ij(σ_i, σ_j) of two amino acids σ_i and σ_j at two different positions i and j. These quantities were calculated using only sequences selected by clustering at 90% of identity computed with CD-HIT [44] and then normalizing by the amount of sequences. In this way, the occurrences of residues in every position are not biased by overrepresentation of proteins in the database. Furthermore, to take into account the repeated nature of the protein families that we are considering, an additional observable was calculated, the distribution of sequence overlap between two consecutive repeats, P(ID(σ)), with .

Model fitting

In order to obtain a model that reproduces the experimentally observed site-dependent amino-acid frequencies, f_i(σ_i), correlations between two positions, f_ij(σ_i, σ_j), and the distribution of Hamming distances between consecutive repeats, P(ID(σ)), we apply a likelihood gradient ascent procedure, starting from an initial guess of the h_i(σ_i), J_ij(σ_i, σ_j) and λ_ID(σ) parameters.

At each step, we generate 80000 sequences of length 2L through a Metropolis-Hastings Monte-Carlo sampling procedure. We start from a random amino-acid sequence and we produce many point mutations in any position, one at a time. If a mutation decreases the energy (2) we accept it. If not, we accept the mutation with probability e^−ΔE, where ΔE is the difference of energy between the original and the mutated sequence. We add one sequence to our final ensemble every 1000 steps. Once we generated the sequence ensemble, we measure its marginals and , as well as P^model(ID(σ)), and update the parameters of Eq 2 following the gradient of the likelihood. The local field and λ_ID(σ) are updated along the gradient of the per-sequence log-likelihood, equal to the difference between model and data averages: (11) (12)

As the number of parameters for the interaction terms J_ij is large (= 21² L²), we force to 0 those that are not contributing significantly to the model frequencies through a L₁ regularisation γ∑_{ij, σ, τ}|J_ij(σ, τ)| added to the likelihood. This leads to the following rules of maximization: If J_ij(σ_i, σ_j)^t = 0 and (13)

If J_ij(σ_i, σ_j)^t = 0 and (14)

If (15)

If (16)

The optimization parameters were set to: ϵ_m = 0.1, ϵ_j = 0.05, ϵ_ID = 10, and γ = 0.001.

To estimate the model error, we compute and . We also calculate the difference of generated and natural repeat similarity distribution for all the possible repeats Hamming distances, penalized by a factor 5 to better learn the parameter λ_ID: 5(P(ID(σ)) − P(ID(σ))^model). We repeat the procedure above until the maximum of all errors, , and 5|P(ID(σ)) − P(ID(σ))^model|, goes below 0.02, as in Ref. [17].

Models with different sets of constraints

Using this procedure we can calculate the model defined in Eq 2 with different interaction ranges used in the entropy estimation in Fig 3A. We start from the independent model h_i(σ_i) = log f_i(σ_i). We first learn the model in Eq 2 with J = 0. We then re-learn models with interactions between sites i, j along the linear sequence such that |i − j| ≤ W, in a seeded way starting from the previous model. The first and last point of Fig 3 correspond to the independent site model with λ_ID and the full model in Eq 2

The entropy in Fig 3B is calculated in the same way as in Fig 3, but now interactions are turned on progressively according to physical distance in the 3D structure rather than the linear sequence distance. In order to obtain the physical distance between residues we use as a reference structure the first two repeats of a consensus designed ankyrin protein 1n0r [45, 46], which have exactly 66 amino-acids. We define the 3D separation between two residues as the minimum distance between their heavy atoms in the reference structure.

To learn the Potts model without λ_ID (E₂) we remove λ_ID from Eq 2 and re-learn the Potts field using the full model parameters as initial contition.

To learn the single repeat models with and without λ (E_ir and E_ir,λ, we take as initial condition the model with interactions below the length of a repeat (W = L − 1, dashed vertical line in Fig 3), and then learn a model removing all the J_ij terms between different repeats. We also impose that the h_i fields and intra-repeats J_ij terms are the same in each repeat, and the experimental amino-acid frequencies to be reproduced by the model are the average over the two repeats of the 1- and 2-points intra-repeats frequencies f_i(σ_i) and f_ij(σ_i, σ_j), such that (17) and (18) if i and j represent sites within the same repeat. In this way we obtain a model for a single repeat that can be extended to both the repeats in the original set of sequences of our dataset.

Entropy estimation

In practice to calculate the entropy S of the protein families we relate it to the internal energy E = −log p(σ) and the free energy F = −log Z: (19)

We generate sequences according to the energy function in Eq 2 and use them to numerically compute 〈E〉. To calculate the free energy we use the auxilliary energy function: (20) where the interaction strength across different sites can be tuned through a parameter α that is changed from 0 to 1. We generate protein sequence ensembles with different values of α and use them to calculate F as a function of α, : (21) where the average over α is taken over the sequences generated with a certain value of α, characterized by the ensemble with probability . F(0) is the free energy for an independent sites model: (22) where the first sum is taken over protein sites and the second over all possible amino-acids at a given site. Eqs 22 and 19 result in the thermodynamic sampling approximation for calculating the entropy [47]: (23)

We generate 80000 sequences using Monte Carlo sampling for the energy in Eq 20 with 50 different α values, equally spaced between 0 and 1 at a distance of 0.02, and then numerically compute the integral in Eq 23 using the Simpson rule.

Entropy error

The entropy estimate is subject to three sources of uncertainty: the finite-size of the dataset, convergence of parameter learning, and the noise in the thermodynamic integration. We estimate the contribution of each of these errors using the independent sites model. In the independent sites model each site i is simply described by a multinomial distribution with weights given by the observed amino-acid frequencies in the datasets. The variance in the estimation of the frequencies from a finite size sample is Var(f_i(σ_i)) = (p_i(σ_i)(1 − p_i(σ_i)))/N_s and the covariance between the frequencies of different amino-acids σ and σ′ at the same site i is where N_s is the sample size and p_i(σ_i) are the weights of the true multinomial distribution sampled. Through error propagation from these quantities we calculate the variance in the entropy of the independent sites model, to first order in 1/N_s: (24)

Evaluating this equation using the empirical frequencies p = f assuming they are sampled from an underlying multinomial distribution, gives an estimate of the standard deviation of 0.05. We assume that the interaction terms do not change the order of magnitude of this estimation. Also the standard deviation in the averages in Eq (23) scales as with N_s = 80000.

The parameter inference is affected not only by noise, but also by a systematic bias depending on the parameters of the gradient ascent described above and the initial condition that we chose to start learning from. S1 Fig shows the average entropy of 10 realizations of the learning and thermodynamic integration procedure for the ANK family and its standard deviation as error bars. If we learn the models with an increasing W window progressively we get a different profile than learning each point starting from the independent model, and above L these two profiles are more distant than the magnitude of the standard deviation, signalling a systematic bias. S1 Fig also shows that progressively learning the model results in a better parameters convergence to values that give lower entropy values.

In order to estimate how this bias is reflected in the entropy estimation we take the single-site amino-acid frequencies produced by the inferred energy function in the last Monte-Carlo phase of the learning procedure and calculate the corresponding entropy for this independent-sites model. We compute the absolute value of the difference between this estimate of the entropy and the independent-sites entropy calculated from the dataset. Again in doing this we assume that neglecting the interaction terms does not change the order of magnitude of this error. These procedure results in the errorbars shown in Figs 3 and 2, Table 1, S2 Fig.

We repeat 10 realizations of both the parameter inference procedure and the entropy estimation, and in Fig 3 we show the average entropy of these 10 numerical experiments for the ANK family where error bars are estimated as explained above to sketch the order of magnitude of the error coming from systematic bias in the parameters learning. S1 Fig shows the mean entropy of ANK as in Fig 3A with the standard deviations of the realizations entropy as error bars, to give an idea of the combined noise in the thermodynamic integration and in the gradient descent, starting from the same initial conditions and with the same update parameters (see Section). The combined noise is smaller than the entropy decrease at 33 residues, showing the decrease is real.

To further check the robustness of the entropy estimation procedure, we generate two synthetic ANK datasets, one with an independent sites model, the other with a model of two non-interacting repeats obtained as explained above, and relearn the model from the synthetic datasets. Repeating the learning and entropy estimation procedure on each on the synthetic protein families gives results that are consistent with the model used for the dataset generation. The entropy of the model learned taking an independent sites dataset does not decrease with the interaction range W and the entropy of the model learned taking a non-interacting repeats dataset does not show any drop around the repeat length.

We repeat the procedure described for the LRR and TPR repeat-proteins families reaching similar conclusions (S2 Fig).

Calculating the basins of attraction of the energy landscape

In order to characterize the ruggedness of the inferred energy landscapes and the sequence identity of the local minima, we start from all the sequences in the natural dataset as initial conditions and for each of them we perform a T = 0 quenched Monte-Carlo procedure. Repeating this analysis on sequences synthetically generated from E_full yields very similar results (see S6 Fig for ANK)

We perform this energy landscape exploration learning the parameters of the Hamiltonian in Eq 2 (refer to Section for the learning procedure), and then set λ_ID = 0 in the energy function because we want to investigate the shape of the energy landscape due to selection rather than the phylogenic dependence.

We scan all the possible mutations that decrease the sequence energy and then draw one of them from a uniform random distribution. The possible mutations are all single point mutations. If the same amino-acid is present in the same relative position in the two repeats we allow for double mutations that mutate those two positions to a new amino-acid, that is identical in both repeats, at the same time. We do this so that the phylogenetic biases that are still partially present in the parameters of the model do not result in spurious local minima biasing the quenching results. The Monte-Carlo procedure ends when every proposed move results in a sequence with an increased energy, and the identified sequence is a local minimum of the energy landscape.

To explore how turning on interactions makes the energy landscape more rugged, we perform the same procedure with the Hamiltonian corresponding to two intermediate interaction ranges in Fig 3A. That is Eq 2, in which J_ij was allowed to be non-zero only within a certain interaction range W. We picked W = 3 and W = 10.

In order to assess what is the role of the inter-repeat interactions we repeat this T = 0 quenched Monte-Carlo procedure on single repeats, with all the unique repeats in the natural dataset as initial condition. The learning procedure of the Hamiltonian for a single repeat is explained in Section. In this single repeat case the possible mutations are just the single point mutations.

Once we have the local minima of the energy landscape, we obtain the coarse-grained minima using the Python Scipy hierarchical clustering algorithm. In this hierarchical clustering the distance between two clusters is calculated as the average Hamming distance between all the possible pairs of sequences belonging each to one cluster. As a result we plot the clustered distance matrix, the clustering dendogram and the basin size corresponding to the distance matrix entries.

In the end we can repeat the quenching procedure described above for LRR and TPR families. The result are sketched in S3 and S4 Figs and lead to similar conclusions as for the ANK family.

Kullback-Leibler divergence

The Kullback-Leibler divergence between two families A and B is defined as D_KL(A||B) = ∑_σ p_A(σ) log₂ p_B(σ)/p_A(σ). We can substitute the sequence ensembles for ANK and TPR in the definition of the probabilities obtaining: (25) (26) where the notation 〈〉_ANK means that the average is calculated over sequences drawn from the ANK ensemble: . Therefore 〈E_TPR〉_ANK is the average TPR energy function evaluated, via the structural alignment between the two families, on 80000 sequences generated through a Monte Carlo sampling of the ANK model 2 (and analogously for 〈E_ANK〉_TPR). The terms F_ANK and F_TPR are calculated in the same way as when estimating the entropy through Eqs 21 and 22, as explained in Section.

For the control against a random polypeptide of length L we use D_KL(FAM||rand) = log Λ − S(FAM), where Λ = 21^L is the total number of possible sequences of length L.