 Method
 Open access
 Published:
Sensitive inference of alignmentsafe intervals from biodiverse protein sequence clusters using EMERALD
Genome Biology volume 24, Article number: 168 (2023)
Abstract
Sequence alignments are the foundations of life science research, but most innovation so far focuses on optimal alignments, while information derived from suboptimal solutions is ignored. We argue that one optimal alignment per pairwise sequence comparison is a reasonable approximation when dealing with very similar sequences but is insufficient when exploring the biodiversity of the protein universe at treeoflife scale. To overcome this limitation, we introduce pairwise alignmentsafety to uncover the amino acid positions robustly shared across all suboptimal solutions. We implement EMERALD, a software library for alignmentsafety inference, and apply it to 400k sequences from the SwissProt database.
Background
When exploring the diversity of life, we tend to either reduce observations according to similar principles and patterns shared across lineages (comparative method) or aim to deduce the individual mechanistic function with a causeandeffectrevealing experimental design (functional assessment). In genomics, such attempts translate into comparing genetic sequences according to the similarity of their DNA or protein composition (comparative genomics) or mechanistic analysis of threedimensional structural conformations of proteins (functional genomics). Recent breakthroughs in protein structure prediction from primary sequence alone [1, 2] uncovered that integrating comparative and functional genomics into a predictive model can yield groundbreaking insights useful enough to guide mechanistic studies in molecular life sciences. Building on this integrative foundation of sequence comparison and protein structural prediction, we explore how the sequence diversity across the tree of life can be compressed into alignmentrobust sequence regions while minimizing the loss of protein structural information.
To compare two biological sequences according to a predefined scoringscheme, pairwise alignment methodologies have proven useful for various practical applications [3, 4]. When constructing a pairwise alignment, a combinatorial space of possible alignment configurations is explored and a single optimal alignment setting is selected (based on the predefined scoringscheme) and reported to equip experimenters with one plausible solution rather than overwhelming them with a wide range of possible solutions. While sufficient for many applications, including protein sequence similarity search, the reduction of comparison to one solution (even when optimal) can cause enormous information loss about biologically relevant, but suboptimal, alignment configurations, thereby systematically biasing the comparative method when applied at treeoflife scale. One could argue that handling only optimal alignments is the most parsimonious approach to dealing with complexities when scaling to millions or even billions of pairwise sequence comparisons when organizing a diverse sequence space according to their pairwise identities. However, analogous to the concept of point estimates and confidence intervals in statistical parameter inference [5], neglecting the goodness of fit for any application may result in unrealistic technical optima rather than focusing on quantifying the biological relevance (e.g., functional protein configuration) of reported alignment solutions.
It seems therefore surprising that the experimental community has grown accustomed to interpret algorithmically derived optimal alignment solutions as biologically most relevant configuration of pairing similar proteins, although theory clearly states that such approximation may only be reasonable when comparing very similar sequences [6] and not when dealing with distant homologs. We argue that by exploring the space of suboptimal alignment configurations using a quantification method able to capture stable positions across possible alignment paths (alignmentsafe intervals), novel insights with greater biological relevance can be unveiled and quantified with particular relevance for protein structure evolution at treeoflife scale.
Fortunately, previously collected evidence suggests that sufficiently screening the suboptimal alignment space for particular configurations that are biologically more relevant can in fact be achieved [7, 8]. Previous work, such as [9,10,11], has shown that there is a significant connection between the suboptimal alignment space and the structural alignment space and used this link to improve the accuracy of pairwise sequence alignments. It has also been shown that suboptimal alignments are often more accurate than strictly optimal ones and that they contain a high number of correct amino acid residue pairs [7], indicating their use in protein structure prediction. Vingron M and Argos P. [12] showed that socalled reliable regions, defined in terms of a robustness measure of individual aligned amino acids can identify conserved and functionally relevant regions among two protein sequences. They demonstrate this functional relevance by validating that these conserved regions also correspond to aligned regions of their respective tertiary structures. In detail, Chao KM et al. [13] introduce a robustness measure for a single pair of aligned symbols between two sequences to assess the difference between the optimal alignment score of the compared sequences without restrictions and the optimal alignment score not containing that aligned pair. A related approach suggested by Naor D and Brutlag DL [6] notes that the space of suboptimal alignments (whose score is within a difference \(\Delta\) to the optimal solution) can reveal conserved regions when manually inspecting the “graphic representation” of these possible alignments. While this initial suggestion to explore the suboptimal alignment space yielded promising visual insights, no solution was given on how to automate or scale this approach to millions or billions of pairwise comparisons. Currently, most applications favor multiple sequence alignments (MSAs), hidden Markov modelbased approaches (HMMs), or protein language models [14] when interested in sensitively locating conserved regions inside a set of sequences for largescale phylogenetic applications (i.e., those positions or regions that remain unchanged in the phylogenetic tree). While providing reliable results in a molecular evolution context, calculating optimal MSAs or constructing relevant HMMs for deep homology searches scales exponentially with the number of sequences, which is computationally expensive and not suitable for treeoflife applications. In addition, MSAs are designed to process highly similar sequences and often perform poorly when comparing divergent sequences [15, 16].
Here, we overcome these limitations by introducing the application of solution safety [17], (the “Methods” section―Definition 1), for pairwise protein sequence alignments. With alignment safety, we can explore the space of optimal and suboptimal alignment configurations (i.e., possible pairings of amino acids) and find entire intervals that are common to all or to a given proportion of alignment solutions. We implement this approach in a command line tool, EMERALD [18]. Instead of forcing two (possibly very diverse) sequences into a single optimal alignment configuration, EMERALD embraces the diversity of possible alignment solutions, by revealing alignmentsafe intervals of the two sequences which appear as conserved (and not even necessarily identical) in the entire space of optimal and suboptimal alignments (Fig. 1). To demonstrate the effectiveness of this procedure for sizeable protein comparisons, we first cluster all protein sequences stored in the SwissProt database [19, 20] using DIAMOND DeepClust [21] (the “Results” and “Methods” sections) and apply EMERALD to each nonsingleton cluster. With this comprehensive analysis, we aim to explore whether our method can provide a competitive solution to project alignmentsafe primarysequence intervals onto the structural conformation of proteins and thereby accelerate the biologically relevant exploration of sequence evolution and their corresponding structural divergence across the tree of life.
Results
The main purpose of this study is to notify the genomics community about the advantages of exploring the suboptimal alignment space when dealing with comparisons of vastly divergent sequences where subsequent structural information will be used to make claims about protein sequence and fold evolution at treeoflife scale. EMERALD allows to achieve this task by equipping users with an automated and scalable software solution to infer alignmentsafe subsequences extracted from the suboptimal alignment neighborhood (Fig. 1), which we demonstrate experimentally to correspond to conserved regions of the underlying protein structure (such as alphahelix, etc.).
Defining alignmentsafety for pairwise protein sequence alignments
Initially developed in the context of genome assembly [17], and later extended to other applications admitting multiple solutions, such as flow decompositions for RNA transcript assembly [22], and RNA folding [23], solution safety identifies a set of partial solutions (e.g., an interval of an alignment) to infer safe positions or windows present in all optimal and \(\Delta\)suboptimal configurations to the problem. Using this definition, previous work by [6, 12, 13] can be reintroduced in the context of studying maximally safe partial alignments, whereby maximal denotes the property that such partial alignment cannot be extended left or right without losing the characteristic of being safe.
To achieve this property at scale and with biological relevance, we introduce novel computational aspects of the alignmentsafety concept and implement these into the C++ command line tool EMERALD (the “Methods" section) [24]. First, we generalize the notion of safety by defining a partial solution to be \(\alpha\)safe (\(\alpha \in (0,1]\)) if it is present in at least a proportion \(\alpha\) of all possible solutions. We further generalize \(\alpha\)safety to the space of \(\Delta\)suboptimal solutions from [6] that are at most \(\Delta\) away from the optimal solution, by defining a jointparameter \((\alpha ,\Delta )\) which captures partial solutions that appear in at least a proportion \(\alpha\) of all \(\Delta\)suboptimal solutions. In other words, while \(\Delta\) captures the boundaries of the suboptimal alignment space that a user wishes to explore, \(\alpha\) regulates the quantilerange over all possible alignment solutions. The jointparameter \((\alpha ,\Delta )\) then allows users to specify the suboptimal alignment space within a particular quantile range over all solutions that shall be explored. This approach allows us to address the fact that an \((\alpha ,\Delta )\)safe partial solution is an interval of arbitrary alignment length (not only a single pair of aligned symbols, as is the case for the robustness measure from [12]). Together, we denote all such \((\alpha ,\Delta )\)safe intervals as the collection of alignmentsafe protein sequence intervals or in short safety windows. Intuitively, \(\Delta\) allows sufficient exploration across the suboptimal alignment space within an \(\alpha\) range, thus enlarging the solution space, and leading to shorter safetywindows, while \(\alpha\) relaxes the safety requirement by enforcing that only a \(\alpha\)fraction of the \(\Delta\)suboptimal alignmentconfigurations need to have the same amino acid to extend safety windows. We show that optimizing the \(\alpha\) and \(\Delta\) configuration can be a powerful tool for regulating the biological relevance safetywindows can capture in diverse protein sequences and their respective threedimensional structures.
Additionally, we explore the biological signatures that can be captured when exploring the suboptimal alignment space using our alignmentsafety approach. In order to annotate which biological features alignmentsafe residues may encode in a particular threshold and substitutionmatrix configuration, we use the Stride [25] annotation such as alphahelices and loops to exemplify how users can biologically assess the output of EMERALD. However, we would like to point out that users need to pay close attention to what suffices as valid “ground truth” for their benchmarking, since EMERALD is agnostic to particular biological applications and the choice of substitution matrix may induce a well characterized amino acid bias for evolutionary more stable residues [26,27,28]. Recent advancements in protein structure prediction can also provide users with a more structurally oriented validation dataset [28]. While it is well understood that secondary structure elements align better than nonsecondary structure elements or disordered parts of proteins, our methodology can reveal the concrete regions of a protein sequence (alignmentsafety windows) that are the same across all (considered) suboptimal alignment solutions. This feature of EMERALD allows users to explore how robust certain sequence regions are to alternative alignment configurations and different substitution matrices. For this purpose, we extracted the secondary structure from the SwissProt database for each protein sequence using the command line tool Stride [25]. For each residue in a sequence, Stride assigns a secondary structure type, in our case: “AlphaHelix,” “310Helix,” “PiHelix,” “Strand,” “Bridge,” “Coil,” and “Turn.” We further distinguish secondary structure types by placing them into two distinct categories: stable and not stable. Coil, Strand, and Turn were labeled as not stable while the rest of the secondary structure types were labeled as stable. While this classification is naive in the first instance (for example, because there are wellknown cases where Coils, Strands, and Turns can be stable to fix certain protein conformations and viceversa, alphahelices can be fairly flexible and unstable [16, 29]), we use this distinction only to exemplify how users can categorize known proteinstructural features into distinctive structural feature classes to benchmark the suboptimal alignmentspace for their domainspecific application and quantification of biological relevance. The main motivation behind such categorization is to test which regions of a protein structure are robustly encoded in alignmentsafe intervals. We envision that users make extensive use of this benchmarking setup to explore, quantify and test the biological relevance of harnessing the suboptimal alignmentspace for encoding structural features and protein evolution of their (divergent) sequences of interest.
In silico experimental design to investigate the biological relevance of alternative suboptimal alignment configurations
The concept of solutionsafety [17] guarantees the following property. If the true alignmentconfiguration is inside the full solution space of the dynamic programming matrix (i.e., optimal and \(\Delta\)suboptimal alignmentconfigurations), then windows that are the same across all such alignmentconfigurations (safety windows) are also part of the true alignment (which cannot be observed in reality). In other words, analogous to sampling theory in statistics where the ground truth of a data universe cannot be measured, we assume that a representative sample drawn from this data universe allows us to infer rules and principles about the data universe itself. Alignmentsafety windows can thus be seen as analogy of a representative sample drawn from the true (unobserved) alignment. The biological relevance of this approach can then be studied by annotating these alignmentsafe windows/positions in regard to their overlap with protein secondary structural features, their contribution to protein selection pressures such as synonymous vs nonsynonymous substitution rates (dN/dS), and mappings onto AlphaFold2 structures or their underlying multiplesequence alignments (MSAs). The focus of our experimental design is, therefore, to test whether exploring the suboptimal alignment space when dealing with millions or billions of pairwise alignments spanning a comprehensive sequence diversity across the tree of life yields sufficiently more information (compared to a single optimal alignment configuration) to increase biologically relevant inference power when dealing with protein structure prediction and structural evolution tasks. To achieve this, we retrieved the SwissProt database from [20] (June 2021), which represents a manually curated subset of the UniProtKB [19] database. In the version of June 2021, SwissProt contains approximately 560k protein sequences which can be retrieved as a FASTAfile. After data retrieval, we filter out protein sequences that did not have corresponding AlphaFold2 predicted threedimensional structures [1, 30]. We then clustered this dataset with DIAMOND DeepClust, a new sensitive deepsequenceclustering method implemented into DIAMOND since version 2.1.0 [21]. We filter out cluster members with no stable bases and clusters of size 1, resulting in 15934 clusters and 396k sequences in total. To achieve a comprehensive overview of deephomology associations between proteins across the full diversity intrinsic to the 396k SwissProt sequences, clustering was carried out using a percentidentity threshold of 20% and 75% length coverage. This threshold was motivated mainly by two factors: (1) conformity with the twilight zone of protein evolution where sequence identity greater than 20–35% can still be reliably associated with structural similarity above this zone, while this association is “breaking” otherwise [31] and (2) to maximize the diversity of the protein sequence space across the tree of life. It is important to note that although the minimum pairwise identity threshold ensures that distant similarities are detected, some clusters can yield pairwise sequence compositions that have significantly higher identityrelations than the chosen threshold of 20%, since DIAMOND uses no upper identity threshold when clustering.
Next, we run EMERALD to calculate safety windows of all sequence and representative pairs of each cluster. To explore the influence of the suboptimal alignment space depth on capturing biologically relevant features, we test various parametercombinations performed using three different \(\alpha\) parameters (0.51, 0.75, 1) and seven \(\Delta\) parameters (0, 2, 4, 6, 8, 10, 15), values that are in the same order of magnitude as the BLOSUM62 metric [32], resulting in 21 safety window calculations per cluster (Figs. 2 to 3). As cluster representative, we selected the cluster centroids reported by DIAMOND DeepClust’s greedy set cover method. Finally, we benchmarked the CPU runtime and maximal memory consumption of these runs (Fig. 4).
Benchmarking and analysis of stable versus unstable protein structural features that are encoded by alignmentsafe windows
For quantification and comparison of safe intervals in the context of stable versus unstable bases, we utilize several benchmarking metrics: safety coverage, stable coverage, stable structure overlap, and stable structure retention. Safety coverage is defined by the portion (%) of a protein sequence which is reported by EMERALD as (alignment) safe, while stable coverage is the portion (%) of a sequence that is considered stable according to the distinct proteinstructural features classification (userdefined) described in the previous section. We further classify a sequence position as true positive (TP) if it is both safe and stable, false negative (FN), if it is not safe but stable, and false positive (FP) if it is safe but not stable. Based on these distinctions, we compute the metrics stable structure retention (recall) = TP/(TP+FN), indicating the percentage of stable positions captured by safety windows, stable structure overlap (precision) = TP/(TP+FP), indicating the proportion of safety windows that is also stable, and their combination, F1score, which is the harmonic mean of the stable structure retention and overlap. This analytics tool allows users to benchmark according to their own distinction of structural classes when determining how much information is gained when incorporating the suboptimal alignment space to the sequence comparison task.
We represent the sequence identity ranges as intervals of real numbers with notation [i, j] where i defines the lower identity boundary and j the upper identity boundary. Figure 2 illustrates the stable structure retention and safety coverage benchmark for our 396k sequence dataset. Naively, we expect the stable structure retention and the safety coverage to be similar, as we are likely to cover, e.g., 60% of all stable elements if 60% of the sequence is covered. Similarly, if all amino acids are alignmentsafe, the stable structure retention is equal to 100% meaning that no further biological information can be obtained. Thus, we are usually interested in an increased gap between the stable structure retention and the safety coverage. Ideally, the stable structure retention would always be close to 100% with all varying safety coverages, meaning that safety corresponds perfectly to stable elements. Figure 2 further shows that as we consider sequences that are more dissimilar, safety coverage decreases, since we increase the space of optimal and suboptimal solutions. Figure 2b shows the stable structure retention that is restricted to all sequences with a safety coverage of at most 70%, resulting in a decrease of the stable structure retention. In both Fig. 2 a and b, the average values of the stable structure retention exceeds the average value of the corresponding safety coverage, and in the identity range \([40\%, 70\%]\), the average stable structure retention is at 80% even when we restrict the sequences to have no more than 80% safety coverage. This result shows that the proportion of stable positions inside the safety windows is larger than the proportion of stable positions out of all the sequence (i.e., \((\text {stable} \cap \text {safe})/{\text {safe}} > \text {stable}/\text {sequence length}\)). In addition to Fig. 2, we also assessed the safety coverage of each individual amino acid structure type in (Additional file 1: Fig. S1), which shows that unstable amino acids have a smaller safety coverage than stable ones. “Coil” has the smallest coverage followed by “Turn,” with an exception of “Strand” having a higher coverage than “AlphaHelix,” which we defined as stable. EMERALD sufficiently captures amino acids of type “Bridge” and “310Helix,” despite STRIDE only assigning these types to \(0.8\%\) and \(3\%\) of all amino acids, respectively.
Figure 2a illustrates that the best tradeoff between safety coverage and stable structure retention is the identity range \([40\%, 70\%]\) sequence identity. This range setting introduces the biggest gap between medians while ensuring that the stable structure retention does not drop below 70%. In Fig. 2b, the median of the stable structure retention in the identity range of 40–70% is at 80%, despite the safety coverage being restricted to be at most 80%. Furthermore, we analyzed the stable structure retention over all combinations of parameters (Additional file 2: Fig. S2; Additional file 3: Fig. S3; Additional file 4: Fig. S4). Overall, the stable structure retention is always slightly larger than the safety coverage. For \(\Delta = 0\) (i.e., when we consider only optimal alignment parts) and all three \(\alpha\) values 0.51, 0.75 and 1.0, the safety coverage is close to 100% for identities of at least 20%, which implies that the substitution matrix is very good at finding a unique optimal alignment, which motivates exploring the suboptimal alignment space. As we predicted, decreasing \(\alpha\) corresponds to increasing the stable structure retention and safety coverage, while increasing \(\Delta\) corresponds to decreasing them.
The choice of parameters also has an impact on the contiguity of safety window lengths and counts. For example, when selecting very conservative parameter settings, a larger proportion of shorter safety windows are generated. To buffer this effect, we added the additional constraint as a new EMERALD parameter m (–windowmerge) which determines whether short safety windows that appear in close proximity (i.e., intersecting or adjacent to each other), are subsequently joined into one large safety window. We quantify this parameter selection behavior in (Additional file 5: Table S1), which now provides the average sequence and safety window lengths (summary statistic of contiguous stable intervals). As expected, lowering the pairwise sequence identity results in shorter and overall less contiguous safety windows. Additionally, a substantial amount of the safety windows of sequence alignments in the \([70\%,100\%]\) identity range have a length exceeding 50% of their full sequence length. Over the whole dataset and all combinations of parameters, the stable structure overlap stays constant (Additional file 6: Fig. S5) with the median at around 43%.
Figure 3 further explores the relationship between stable structure retention and safety coverage. It restricts the sequences in the xaxis to those with stable structure retention of at least x, plotting the safety coverage (on the left) and the proportion of considered sequences (on the right). Strikingly, in Fig. 3a, we can see that for about 20% of all sequences, or of all sequences in the identity range \([40\%, 70\%]\) (dark red, and purple curves, respectively on the right for \(x = 1.00\)), EMERALD retains all their stable positions, with a safety coverage of only 15%. This illustrates that, in contrast to optimal alignment approaches, for lower identity bounds (dissimilar sequences) EMERALD manages to reveal structurally conserved intervals. Similarly, EMERALD achieves a stable structure retention of 80% for around 75% of all the sequences with a safety coverage of around 65%. In other words, EMERALD declares 65% of the sequence as safe, capturing 80% of all stable amino acids. This result further shows that by relaxing the stable structure retention criteria from 100% down to 80%, EMERALD is able to reduce the sequences to their safety intervals from full length to 65%. In Fig. 3b, we analogously restrict the sequences only to those of safety coverage of at most 80%. Here, for example, the purple curve shows that around 10% of the sequences in the identity range 40–70% can be reduced to a safety coverage of nearly 5%, without losing the stable structural retention constraint of 100%. This suggests that EMERALD embraces the sequence diversity to narrow down the structurally conserved context of a sequence.
(Additional file 7: Fig. S6) analyses the safety coverage and the F1 score, and it shows that, as we consider clusters with smaller postcomputed identity values, F1score has only a minor decrease, but safety coverage has a marked decrease. This indicates additionally that safety windows have a better ability to capture stable structural elements in clusters with smaller identity values (Additional file 8: Fig. S7).
Finally, we benchmarked EMERALD’s run time and memory consumption (Fig. 4) on a computing server with 32 cores (2x hyperthreading, 64 threads) and 512GB of RAM using the clustered 396k sequences from the SwissProt database. Overall, none of our clusters ran more than 17 min, and, except for two clusters, they did not use more than 6.5GB of memory. While the time increases quadratically in the lengths of sequences, trivially, it increases only linearly with the number of sequences in the cluster. This result illustrates that EMERALD can scale to millions of sequences and thousands of clusters to explore the protein universe across the tree of life.
The pseudocode denoting the algorithmic procedure underlying the pairwise alignmentsafety inference implemented into EMERALD is illustrated in (Additional file 9: Algorithm 1).
Discussion
For the past decades, pairwise sequence alignments have served silently and reliably as foundation of comparative and functional genomics applications. Algorithmic innovation focused on computing ever faster heuristics for retrieving optimal alignment solutions at scale or extending comparisons to multiple sequence alignments or Hidden Markov Model based statistical alignment approaches. However, only little innovation occurred in the pairwise alignment field on quantifying the biologically most relevant alignment configuration from a collection of up to millions of possible (suboptimal) alignment solutions. While biological meaning in the context of alignment optimization is a vague concept, in the early days of comparative and functional genomics the ability to encode structural information of proteins was among the main applications underlying the benchmarking of biologically meaningful alignment configurations [5, 7]. The consensus then was that focusing on optimal alignments is a reasonable heuristic when dealing with highly similar sequences, since the optimal alignment solution can indeed encode a good representation of conserved protein structure such as alphahelix [33]. Still applied as a main assumption in functional genomics today when employing pairwise alignment searches, protein sequences are usually screened for high similarity across distant species or strains to test whether this retained sequence identity translates into structural conservation and potential functional similarity.
In this study, we aimed to determine whether quantifying suboptimal alignment configurations in pairwise sequence alignments can significantly improve the sensitivity of identifying relationships between features of protein structure across the tree of life when analyzing a large diversity of protein sequence space and making comparisons between hundreds of thousands of species [21]. To approach this quest, we designed an in silico experiment to annotate all alignmentsafe positions of the SwissProt database. Using this annotation approach, we investigate how the quantification of the suboptimal alignment space can refine biologically relevant interpretations such as conserved protein structural features. We asked how information about suboptimal alignment configurations at scale can be harnessed to predict protein structural change when the proportion of alignmentsafe positions in distant sequence alignments is reduced. Finally, we introduced alignmentsafety as a new methodology to approach such questions and the command line tool EMERALD to implement our methodology at scale.
While previous work focused on assessing the quality or statistical robustness of optimal alignment configurations in comparison with a set of suboptimal alignment solutions (optimal alignment neighborhood) [34,35,36] in the context of protein homology modeling or threading, our work significantly extends these concepts to associate protein sequence evolution with their respective structural change while scaling to millions of pairwise comparisons and trillions of suboptimal solutions at treeoflife scale. We achieve this by inferring the robust amino acid positions across a range of suboptimal alignmentconfigurations. As a result, we learned that when attempting to relate the sequence biodiversity of the protein universe to the evolution of protein structure, a detailed inference and exploration of how alignmentsafe positions are retained or lost throughout the tree of life can serve as robust proxy for how conserved structural features change over evolutionary time. We interpret this result such that alignmentsafe positions can be associated with the stably folded backbone of a protein structure and that further research is required to unveil the causal associations between alignmentsafe positions and positions giving predictive signatures when inferring structural conservation above the twilight zone of protein evolution [31].
While our main purpose is to raise awareness and stimulate further discussion on how to scale sampling from the true alignmentconfiguration when dealing with the sequence diversity across the tree of life, a current limitation of exploring the suboptimal alignment solution space using our alignmentsafety methodology is that inferring alignmentsafety windows with EMERALD is sensitive to the choice of the parameters \(\Delta\) and \(\alpha\). To further soften this limitation, we plan to establish an analogous approach to topological persistence [37], where we extend EMERALD by adding a procedure we refer to as inference of persistent safetywindows. In this approach, the \(\alpha\) parameter is fixed and a safetywindow is only referred to as \(\Delta\)persistent if it is also \((\alpha , \Delta )\)safe. By iteratively increasing \(\Delta\) starting from \(\Delta = 0\), we can then extend the suboptimal alignment graph for the new \(\Delta\) parameter and update the corresponding safetywindows. We envision that the persistence of a safetywindow (i.e., the range of \(\Delta\) parameters for which we can guarantee safety) will not only eliminate ambiguity in choosing the right EMERALD parameters, but will also inform us in greater detail about the protein features such alignmentsafety windows encode.
Conclusions
We envision that studies exploring the suboptimal alignment space when comparing protein sequences pairwise across a biodiverse protein universe will stimulate further research attempting to address the remaining shortcomings of fold predictions such as dealing with disordered parts of proteins and incorporating diverse mutation patterns into fold evolution predictions. Our alignmentsafety methodology and EMERALD software are designed to assist these efforts of associating new information gained from suboptimal alignments with biologically relevant phenotypes of proteins in an evolutionary context. We designed our approach to be sufficiently fast and sensitive to scale to millions of sequences with various degrees of divergence to supply the dataintensive demands of the biosphere genomics era.
Methods
Conceptually, we build on the approach introduced by Naor and Brutlag [6] and vastly extend its methodological depth and computational scalability for treeoflife scale applications (Fig. 5). In detail, the approach introduced by [6] is restricted to consider only aligned symbols that are part of all suboptimal alignments, ignoring pairs which are part of the optimal alignment itself. For example, suppose that an aligned pair is common to all the 1000 optimal alignments of score q, there is a single suboptimal alignment of score \(q1\) not containing the pair, and the next suboptimal alignments not containing the pair have score \(q100\). In this case, the single suboptimal alignment of score \(q1\) makes the pair “not conserved” under the approach of [6] and drastically decreases the robustness of the pair (from 100 to just 1), since the robustness measure cannot quantify the proportion of (sub)alignments containing the pair. In addition, they define robustness independently for each aligned aminoacid pair thereby excluding the opportunity to quantify robust regions, which are often observed in natural settings. As a result, although [6] provide a first theoretical template to study suboptimal alignment spaces, they fail to deliver scalable algorithmic solutions, software tools, and biological validation to investigate the protein sequence diversity space in the context of robust  alignment safe regions for treeoflife scale applications. In fact, their methodology does not exceed a manual analysis of graphical alignment representations with little potential for automation and efficient scaling.
Inference of alignmentsafe protein sequence windows
We formally introduce the calculation of alignmentsafe intervals through the thresholded exploration of the suboptimal pairwise alignment space. Let A and B be two strings over an alphabet \(\Sigma\) of length n and m, respectively. In this study, we refer to an optimal alignment between A and B, when a particular alignment maximizes a scoring function based on an externally provided match/mismatch cost matrix. First, we consider scores provided by a amino acid substitution matrix (such as BLOSUM62 [32]) and solely focus on optimal alignments, before we introduce affinelinear gap costs in the Introducing gap penalties section [38] and the suboptimal alignment space to explain the necessary changes in our approach.
In sequence bioinformatics, it is established that such optimal global alignments between A and B can be computed via dynamic programming [3]. A common result is that alignments of maximum score are in bijection with maximumweight (optimal) paths in the directed acyclic graph (DAG) which corresponds to this dynamic programming table. More formally, we can define the alignment DAG of A and B as \(G(A,B) = (V,E)\), with \(V = \{0,\dots ,n\} \times \{0,\dots ,m\}\), with each node (x, y) denoting three outgoing edges in E to \((x+1,y), (x, y+1)\) and \((x+1,y+1)\) (and thus, the DAG has unique source \(s = (0,0)\) and unique sink \(t = (n,m)\)). We denote paths in a graph by the notation \(P = (v_1,v_2,\dots ,v_k) \in V^k\) and their restriction by \(P[v_L..v_R] = (v_L,v_{L+1},\dots ,v_R)\). The first two of the edges correspond to a gap and the third edge corresponds to an alignment of the symbols A[x] and B[y]. If we assign the scores of the substitution matrix as well as the gap score to the corresponding edges as weights, then finding an optimal alignment corresponds to finding a maximumweight path from s to t (st path) in G. We define such score maximizing paths as optimal. We are then interested in discovering those safe partial alignments that are common to all optimal alignments.
We can further generalize the notion of safety by also considering a parameter \(\alpha \in [0,1]\) and analogously explore \(\alpha\)safe partial alignments that appear in at least the proportion \(\alpha\) of all optimal alignments. For \(\alpha =1\), 1safety coincides with safety and considering \(\alpha < 1\) allows us to have potentially longer \(\alpha\)safe paths. We can define these notions formally based on the graphcentric definition of an optimal global alignment.
Definition 1
Let \(\alpha \in [0,1]\), let A and B be two strings, and let G(A, B) be the global alignment DAG connecting A and B. We denote that a path P in G is

safe, if P is a subpath of all optimal st paths of G;

\(\alpha\)safe, if P is a subpath of at least an \(\alpha\) proportion of all optimal st paths of G.

maximally \(\alpha\)safe, if it is not a subpath of a longer \(\alpha\)safe path.
We define the set of all maximal \(\alpha\)safe paths as \(\alpha\)safety windows. We will omit \(\alpha\) when it is clear from the context.
Given the alignment DAG G(A, B) that connects A and B, we define \(G_0(A,B) = (V_0,E_0)\) as the unweighted minimal subgraph of G(A, B) to include all optimal st paths in G(A, B) (thus s and t are nodes in both G(A, B) and \(G_0(A,B)\)). Backtracking an optimal solution in G(A, B) can be done by taking any incoming edges of any node (x, y) that is part of an optimal path. Due to the bijection property of \(G_0\), we can ignore edge weights and focus on exploring only the set of all its st paths, since these are bijectively linked with the optimal alignments between A and B. We note that \(G_0(A,B)\) is weakly connected (i.e., there is an undirected path between any two pairs of nodes, since by definition every node appears in some st path of \(G_0(A,B)\)). It is easy to notice that the safe edges of G(A, B) are exactly the edges that when removed from \(G_0(A,B)\) leaves \(G_0(A,B)\) no longer weakly connected. Such edges are also referred to as bridges of the undirected graph underlying \(G_0(A,B)\), and they can be computed in linear time and in proportion to the size of \(G_0(A,B)\) [39].
Here, we propose a refined approach to compute \(\alpha\)safe paths based on counting st paths in \(G_0(A,B)\). For each node v, let d(v) be the number of paths from v to t. We can calculate these numbers with the following recurrence:
where \(\mathcal {N}^+(v) = \{ u \in V_0 ~~ (v,u) \in E_0\}\). Since \(G_0\) is a DAG, the recurrence is welldefined, and it stores the total number of st paths in d(s). Analogously, we can also compute the number \(d_r(u)\) of su paths for any \(u \in V_0\). Given these two counts for any edge \(e = (u,v) \in E_0\), we can define
as the proportion of all st paths of \(G_0\) that e is part of. We can also understand p(e) as the probability of e appearing in an arbitrary st path of \(G_0\). Likewise, given a path \(P = (v_1,\dots ,v_k)\) of nodes \(v_i \in V_0\), the proportion of st paths that P is part of is given by
Thus, P is \(\alpha\)safe if and only if \(p(P) \ge \alpha\).
The following lemma shows that if \(\alpha > 0.5\), then there exists an st path \(P^*\), such that any \(\alpha\)safe path is a subpath of \(P^*\). This not only simplifies the algorithm, but it also makes it computationally efficient since it guarantees that the number of maximal \(\alpha\)safe paths is proportional to the size of this path (i.e., \(\mathcal {O}(n+m)\)). Moreover, since \(P^*\) is an st path in \(G_0\), it corresponds to an optimal alignment between A and B, and thus all safety windows can be reported as intervals of this alignment.
Lemma 1
Let A and B be two strings based on an amino acid alphabet \(\Sigma\). If \(\alpha \in (0.5, 1]\), then there is an st path \(P^*\) which contains all the \(\alpha\)safe paths of G(A, B).
Proof
As defined above, for an edge e let \(p(e) \in [0,1]\) denote the proportion of optimal paths it is part of. It is clear that
for all nodes v, where \(\delta ^+(v)\) is the set of all outgoing edges of v.
We assume the opposite, meaning that there are two \(\alpha\)safe edges \(e_1\) and \(e_2\) for which no common st path exists. Since by definition of \(\alpha\)safe edges, \(p(e_1) > 0.5\) and \(p(e_2) > 0.5\), we obtain the contradiction
as the paths that cross \(e_1\) are distinct of the paths that cross \(e_2\) but all start at node s. Finally, since each pair of \(\alpha\)safe edges share a path, there must exist a single path \(P^*\) containing all \(\alpha\)safe edges. \(\square\)
Next, we give an approach on how to find such a path \(P^*\).
Lemma 2
Let \(G = (V,E)\) be a DAG with source \(s\) and sink \(t\). Given a set of edges \(A = \{ e_1, e_2, \dots , e_k \}\), we can find an \(st\) path that either contains all the edges in \(A\) or return the information that such path does not exist, in time \(\mathcal {O}(V + E)\).
Proof
Let T be a topological order of all nodes in V and let A be sorted topologically by the tails of the edges. Assume we have a path from s to an edge \(e_i = (u_i, v_i)\). The task is to determine a path to the edge \(e_{i+1} = (u_{i+1}, v_{i+1})\). For any path \(P_{e_i}\) from \(v_i\) to \(u_{i+1}\), it must hold that \(T[x] \in [T[v_i], T[u_{i+1}]] {=:} I~(\subseteq V_0)\) for all nodes x contained in \(P_{e_i}\). To achieve this, we perform a graph traversal (for example, depthfirst search) in I and visit every node at most once. If \(u_{i+1}\) is not found after the graph search, it follows that the path does not exist. At the end of this procedure, if no edge in A is left, we just connect the path to t.
In (Additional file 9: Algorithm 1), we provide the pseudocode to infer safetywindows. In lines 1 to 6 we construct the subgraph \(G_0(A,B)\) of optimal paths, calculate the ratios p(e) for all \(e \in E\) and find a path \(P^*\) according to Lemma 1. Next, we calculate the safetywindows with a twopointer algorithm on \(P^*\). The iterator variable here is R (right pointer), and we use a second left pointer L with \(L \le R\). The idea is to keep the value L as small as possible, to maintain the safetywindows left maximal, while moving L up until the interval [L, R] of \(P^*\) becomes \(\alpha\)safe. On line 13 we add the interval into the list of safetywindows if it is also rightmaximal (line 12).
Theorem 1
Let \(\alpha \in (0.5,1]\). Given two strings \(A\) and \(B\) of length n and m, respectively, (Additional file 9: Algorithm 1) computes the safety windows of A and B in time \(\mathcal {O}(n\cdot m)\), assuming constanttime arithmetic operations.
Proof
We analyze the runtime line by line. First, constructing \(G_0(A,B)\) can be performed in \(\mathcal {O}(n\cdot m)\) by the standard dynamic programming approach. As shown earlier, d(v) overall can be calculated in linear time (here \(\mathcal {O}(n\cdot m)\)) for DAGs. Line 5 runs in constant time, so this forloop also runs in linear time.
Next, in line 6, we find an st path \(P^*\) containing all \(\alpha\)safe edges, using the approach given in Lemma 2. Note that the path exists by Lemma 1.
Finally, we calculate the safety windows by iterating over \(P^*\) and store them in the set W. Both variables R and L require at most \(\mathcal {O}(n+m)\) iterations and such W will at most contain \(\mathcal {O}(n+m)\) safety windows.
Let \([\ell ,r) \in W\). According to the choice of \(\ell\) in line 10, such an interval is \(\alpha\)safe. It is maximal and thus a safety window, since neither \([\ell 1,r) \in W\) nor \([\ell , r+1) \in W\).
Next, we show that all safety windows are in W. Let \([\ell , r)\) be a safety window. According to the iteration in line 9, R will eventually be equal to r. Consider the start of this iteration. If \(L \le \ell\), the iteration in line 10 will ensure that L will be equal to \(\ell\) by definition of safety windows and we insert \([\ell , r) = [L, R)\) into W. We now assume \(L > \ell\). In previous iterations \(R \le r\), we must have increased L to make it larger than \(\ell\). But since \([\ell , r)\) is \(\alpha\)safe, all intervals \([\ell , R)\) for \(R \le r\) are \(\alpha\)safe, and we never increase L if \(L = \ell\) and \(R \le r\). \(\square\)
Note that we assume that arithmetic operations run in constant time. If \(G_0(A,B)\) contained all vertices in G(A, B), then the number of paths will be equal to the exponentially growing Delannoy numbers [40] (i.e., if \(G_0(A,B) = G(A,B)\), then \(d(v) \in \mathcal {O}(5.8^n/\sqrt{n})\), as this value is iteratively the sum of three previous values). Though, even if \(G_0(A,B)\) only consists of nodes close to the diagonal line, the d values can grow up to \(\mathcal {O}(2^n)\).
Introducing gap penalties
So far, we only considered scores given by a substitution matrix \(M[1..\sigma ][1..\sigma ]\) for \(\sigma\) characters as well as by gaps through insertions and deletions when determining (sub)optimal global alignments between two protein sequences. While accounting for gaps in a global pairwise alignment is necessary to retrieve biologically relevant alignment configurations, finding the right balance between gap introduction and similarity optimization is an important consideration to make.
To achieve this, we generalize the gap scoring function to be affinelinear. This means that when creating a gap of length \(\ell\) in the alignment, we obtain the score \(\text {gap}_{p,g}(\ell ) = p + \ell \cdot g\). The value \(p \in \mathbb {Z}\) is the gap penalty, chosen in order to minimize the number of gaps in an optimal alignment. Before having introduced a gap score, we have set \(p = 0\). To accommodate gap scores in our previously defined DAG G, we can slightly modify the graph to address the known issue that paths are then not in 1:1 correspondence to alignments anymore. Following [38], we replace each node (i, j) by three nodes C(i, j), D(i, j) and I(i, j). Nodes D and I correspond to being inside a gap, erasing the characters in A and B respectively, while node C is the default node, from which we can decide to either match two characters or to introduce a new gap. We thus create the following edges for C: \(C(i,j) \rightarrow C(i+1,j+1)\) to match \(A_i\) with \(B_j\), \(C(i,j) \rightarrow D(i+1,j)\) to align \(A_i\) with a gap and \(C(i,j) \rightarrow I(i,j+1)\) to align \(B_j\) with a gap. The last two of the edges also define the start of a gap. For D (and analogously for I), we create the following edges: \(D(i,j) \rightarrow C(i,j)\) to close a gap and \(D(i,j) \rightarrow D(i+1,j)\) to extend a gap. The new graph is an st DAG of size \(\mathcal {O}(n\cdot m)\) which retains all properties to calculate safetywindows as previously described. Additionally to the substitution matrix and the gap score, we assign edge weights for closing and opening a gap, commonly chosen as 0 for closing and some negative integer for opening in order to punish creating too many gaps.
Extending our DAG approach to incorporate suboptimal alignments
We previously stated that the graph \(G_0(A,B)\) has the property that all its st paths are optimal paths in G(A, B). This gives us a compact data structure able to store exponentially many optimal paths. We now generalize the subgraph \(G_0\) to contain also suboptimal paths. This is achieved by introducing a critical threshold \(\Delta\), which users can specify to either extend or narrow down the suboptimal alignment space. For a given \(\Delta \ge 0\), an st path in G(A, B) is said to be \(\Delta\)suboptimal if it is of weight at least \(OPT  \Delta\), where OPT denotes the weight of an optimal path. We denote paths that appear in at least an \(\alpha\)proportion of all \(\Delta\)suboptimal paths in G(A, B) as \((\alpha ,\Delta )\)safe and define maximal \((\alpha ,\Delta )\)safe paths as \((\alpha ,\Delta )\)safety windows.
Following the notion of [6], we define \(G_\Delta (A,B) = (V_\Delta , E_\Delta )\) to be the subgraph of G(A, B) which is induced by the set of edges \(e \in E\) such that there is a \(\Delta\)suboptimal st path in G(A, B) crossing e. By definition, \(G_\Delta (A,B)\) contains all \(\Delta\)suboptimal st paths in G(A, B). Additionally, however, it may contain st paths that are not \(\Delta\)suboptimal. This implies that some paths of \(G_\Delta (A,B)\) appear in at least a proportion \(\alpha\) of st paths which are not necessarily \((\alpha ,\Delta )\)safe paths (in G(A, B)).
Naor D and Brutlag DL [6] have shown that \(G_\Delta (A,B)\) is the smallest subgraph of G(A, B) that contains all \(\Delta\)suboptimal st paths. In fact, our data spanning treeoflife scale protein sequence diversity does not indicate that these spurious nonsuboptimal paths have any effect in practice. Naor D and Brutlag DL [6] argue that the only possible solution to capture only suboptimal paths would be to naively enumerate over all suboptimal paths, which is unfeasible, since in the worst case there can be exponentially many suboptimal paths of size G(A, B). We address this limitation by approximating \((\alpha ,\Delta )\)safe paths with those paths of \(G_\Delta (A,B)\) appearing in at least a proportion \(\alpha\) within its set of st paths. We compute these paths according to (Additional file 9: Algorithm 1), with the only difference that we start with \(G_\Delta (A,B)\) in line 1 instead of \(G_0(A,B)\).
To compute \(G_\Delta (A,B)\), we proceed as in [6]: for all nodes \(v \in V\), we find the weight w(v) of an optimal vt path of size G(A, B) in linear time by traversing the nodes in reverse topological order, such that the weights of all outneighbors of v are computed when we reach v. Similarly, we compute the weight \(w^r(v)\) of optimal sv paths for all \(v \in V\). An edge \(e = (u,v) \in E\) is now part of \(G_\Delta (A,B)\) only if \(w^r(u) + w(v) + w(e) \ge OPT  \Delta\), where w(e) denotes the weight of the edge e. We can thus construct \(G_\Delta (A,B)\) in linear time and proportional to the size of G(A, B).
Implementing alignmentsafety into the scientific software EMERALD
We designed EMERALD to efficiently implement our theoretical methodology to explore the suboptimal alignment space by inferring alignmentsafe intervals when performing pairwise protein sequence comparisons. In detail, EMERALD accepts a protein sequence cluster in FASTA format of k sequences as input and returns \(k  1\) safetywindow sets in a custom safety window format. Our tool aligns all sequences in the cluster against a userselected representative sequence of the cluster, which by default is the first sequence in the FASTA file, thereby maximizing the score of the alignment. EMERALD by default uses BLOSUM62 as substitution matrix; however, any affinelinear gap cost function can be used via the command line parameters. This makes EMERALD extendable to sequence alignments beyond protein sequences. For a cluster containing k sequences, the goal is to be able to compare the safetywindows of all \(k1\) pairs with each other. Thanks to Lemma 1, we can project the safety windows to the sequences. Node indices in the alignment graph are of the form (i, j) with \(0 \le i \le n\) and \(0 \le j \le m\), and such safety windows can be written as \([(i_1, j_1), (i_2, j_2)]\). For example, given the strings S \(=\) “AB” and T \(=\) “BC” and the alignment “\(AB\)” and “\(BC\),” a safety window of the first gap would be of the form [(0, 0), (1, 0)]. Even though by usual convention \(T[0] = B\), this position is not part of the safety window. EMERALD then returns the interval [0, 1] for string S and [0, 0] for string T. In other words, the tuple indices of the nodes are between the string characters. This coincides with the fact that the node set is the product \(\{0,\dots ,n\} \times \{0,\dots ,m\}\) (i.e., \((n+1)\cdot (m+1)\) nodes) and with the fact that we want to be able to include gaps in safety windows.
Availability of data and materials
Implementation
EMERALD is available on GitHub [24] and on Zenodo [18]. The source code is released under the GNU Public License v3.0. The EMERALD GitHub repository [24] contains all the details on how to install and run the software on Linux operating systems through the command line.
Datasets
All data preprocessing and data generation is implemented as a reproducible and parameterised Snakemake pipeline [41] and computationally reproducible scripts can be found on GitHub [42]. All datasets, including the 396k protein sequences, DIAMOND DeepClust clusters, EMERALDoutput files, and annotation files can be retrieved from Figshare [43].
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Acknowledgements
We thank V. Alva, H. Ashkenazy, C. Behr, A. Dobbelstein, F. Gabler, L. Maischak, K. K. Ullrich and D. Weigel for their interest and helpful discussions. H.G.D. and B.B. thank D. Weigel for ongoing support and sponsorship of their work. H.G.D. thanks the BMBFfunded de.NBI Cloud within the German Network for Bioinformatics Infrastructure (de.NBI) (031A532B, 031A533A, 031A533B, 031A534A, 031A535A, 031A537A, 031A537B, 031A537C, 031A537D, 031A538A) for support. Finally, H.G.D. and B.B. acknowledge that their work was supported by the Max Planck Society.
Review history
The review history is available as Additional file 10.
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Veronique van den Berghe was the primary editor of this article and managed its editorial process and peer review in collaboration with the rest of the editorial team.
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Open Access funding enabled and organized by Projekt DEAL. This work was partially funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 851093, SAFEBIO), and partially by the Academy of Finland (grants No. 322595, 328877, 352821) as well as by the BMBFfunded de.NBI Cloud and by the Max Planck Society.
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A.T. and H.G.D. designed and led this study. An.G. implemented EMERALD with feedback from Ar.G., F.D., B.B., A.T., and H.G.D. Data analysis was performed by An.G., Ar.G., F.D., B.B. Data interpretation was performed by A.T., H.G.D., An.G., Ar.G., F.D., and B.B. The manuscript was written by H.G.D., An.G., and A.T., with contributions from Ar.G., F.D., B.B.
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Twitter handles: @HajkDrost (HajkGeorg Drost); @AlexTomescu0 (Alexandru I. Tomescu).
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Supplementary information
Additional file 1: Figure S1.
Safety coverage restricted on each structural type as assigned by STRIDE (i.e., for every structural type A (number of amino acids of type A that are safe) / (total number of amino acids of type A)).
Additional file 2: Figure S2.
Stable structure retention compared to safety coverage for various EMERALD parameter configurations and several identity ranges.
Additional file 3: Figure S3.
Stable structure retention compared to safety coverage for α = 0.75, Δ = 0,2,4,6,8,10,15 and several identity ranges.
Additional file 4: Figure S4.
Stable structure retention compared to safety coverage for α = 0.51, Δ = 0,2,4,6,8,10,15 and several identity ranges.
Additional file 5: Table S1.
Summary statistics of pairwise sequence assessments in regard to their parameterspecific safetywindow length construction.
Additional file 6: Figure S5.
Stable structure overlap for each pair of α = 0.51, 0.75, 1 and Δ = 0, 2, 4, 6, 8, 10, 15 on the full dataset covering all 396k SwissProt sequences.
Additional file 7: Figure S6.
F1score for α = 0.75, Δ = 8, for sequences in clusters of varying identity ranges.
Additional file 8: Figure S7.
Direct comparison of sequence logos between multiple sequence alignments (MSAs) and onevsall (cluster members against cluster representative) EMERALD alignments derived from one DIAMOND DeepClust clusters with various identity ranges (90% identity, 40% identity, 20% identity).
Additional file 9: Algorithm 1.
Pseudocode denoting the algorithmic procedure underlying EMERALD for inferring alignmentsafety windows from pairwise protein sequence alignments.
Additional file 10.
Review history.
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Grigorjew, A., Gynter, A., Dias, F.H. et al. Sensitive inference of alignmentsafe intervals from biodiverse protein sequence clusters using EMERALD. Genome Biol 24, 168 (2023). https://doi.org/10.1186/s13059023030086
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DOI: https://doi.org/10.1186/s13059023030086