DCAlign v1.0: aligning biological sequences using co-evolution models and informed priors

Abstract

Summary

DCAlign is a new alignment method able to cope with the conservation and the co-evolution signals that characterize the columns of multiple sequence alignments of homologous sequences. However, the pre-processing steps required to align a candidate sequence are computationally demanding. We show in v1.0 how to dramatically reduce the overall computing time by including an empirical prior over an informative set of variables mirroring the presence of insertions and deletions.

Availability and implementation

DCAlign v1.0 is implemented in Julia and it is fully available at https://github.com/infernet-h2020/DCAlign.

1 Introduction

A common task in Bioinformatics is to cast evolutionary-related biological sequences into a multiple sequence alignment (MSA). The objective of this task is to identify and align conserved regions of the sequences by maximizing the similarity among the columns of the MSA. State-of-the-art alignment methods, like HMMER for proteins (Eddy 2011) and Infernal (Nawrocki and Eddy 2013) for RNAs, use hand-curated MSAs of small representative subsets of sequences to be aligned (the so-called seed alignments). Whereas for proteins, HMMER builds the Hidden Markov Model (HMM) by using only the seed alignment, Infernal needs also secondary structure information to generate a Covariance Model (CM). In both cases, HMM (for proteins) or CM (for RNAs) are used to align query sequences. However, homologous sequences show signals of correlated mutations (epistasis) undetected by profile models.

Conservation and co-evolution signals are at the basis of Direct Coupling Analysis (DCA)-based statistical models (Morcos et al. 2011, Cocco et al. 2018). Recently, these models have been used to align biological sequences (Muntoni et al. 2020) and perform remote homology search (Wilburn and Eddy 2020) by alignment of the sequences to a seed model, or by pairwise alignments of seed models (Talibart and Coste 2021). The method in Muntoni et al. (2020), viz. DCAlign, returns the ordered sub-sequence of a query unaligned sequence which maximizes an objective function related to the DCA model of the seed. In this latter case, standard DCA models fail to adequately describe the statistics of insertions and gaps. To alleviate this limitation, we added to the objective function gap and insertion penalties learned from the seed alignment. While for the insertions, the computational complexity is negligible, inferring gap penalties is a time-consuming problem [see (Muntoni et al. 2020) and Supplementary text]. Here, we treat penalties in terms of informed priors computed from the seed sequences. The parameters for gaps and insertions, extracted from the seed alignment, are determined in an unsupervised manner. Finally, to further speed up the learning of the seed-based objective function, we obtain the parameters of the DCA model using pseudo-likelihood maximization (Ekeberg et al. 2013) instead of Boltzmann Machine Learning (Figliuzzi et al. 2018, Muntoni et al. 2021). DCAlign v1.0, is a computational pipeline that allows for the computation of the seed-model parameters in a few minutes, contrary to its original implementation which required at least a day of computation in the best scenario. The alignment problem is then solved approximately through a message-passing algorithm (see Supplementary text).

2 Methods

Our alignment algorithm estimates the optimal ordered sub-sequence compatible with a DCA model and empirical knowledge of insertions and gaps of the seed. Let $\mathit{A}$ be an unaligned sequence of length N, and $\mathit{S}$ be its aligned counterpart of length L (which is the length of the seed MSA). We only consider the $L\le N$ case. At each $i=1,\dots ,L$, we define a Boolean variable $x_i\in \{0,1\}$ and a pointer $n_i\in \{0,\dots ,N+1\}$. The variable $x_i$ indicates whether the position i is a gap ‘-’ ($x_i=0$) or a match, i.e. a symbol in $\mathit{A}$. When i is a match, $n_i$ identifies where $S_i$ matches $\mathit{A}$, i.e. $S_i=A_{n_i}$; instead, for $x_i=0$, the value of $n_i$ is used for keeping track of the last matched symbol in $\mathit{A}$. Let us define a pointer-difference variable as $\Delta n_{i,j}=n_j-n_i$ for $i=1,\dots ,L$ and $j>i$. Each auxiliary variable $\Delta n_{i,j}$ quantifies how many symbols of the unaligned sequence $\mathit{A}$ are present between two i, j positions of the aligned counterpart $\mathit{S}$. If a configuration of the $\mathit{n}$ is given, the full set of the pointer differences reveal the presence of insertions and gaps between any columns i and j of the alignment (see Supplementary text).

2.1 Seed modeling

Together with a DCA model of the aligned seed (see Fig. 1, central panel), for every site i (in red), we compute the $\Delta n_{i,j}$ for $j>i$ for all the seed sequences, and we learn an empirical probability $P_{i,j}\left(\Delta n_{i,j}\right)$ as shown in the bottom central panel of Fig. 1 (this procedure is computationally very fast). The color gradient is associated with the value of j, the lighter the color, the larger is j. In Fig. 1 (bottom central panel), we consider as an example three sequences differing in the nature of the $\Delta n_{i,i+1}$.

Figure 1.

Schematic representation of the DCAlign v1.0 pipeline. From a (given) hand-curated alignment (the seed, shown in the left panel), our algorithm learns (i) a DCA model $\mathcal{H}$ exploiting the one-site and two-site statistics of the seed (upper central box), and (ii) the gap and insertion penalties by means of the empirical distribution of the pointer differences $P\left(\Delta n_{ij}\right)$ for $i=1,\dots ,L$, and $j>i$ (bottom central box). The three sequences represent the three scenarios that can occur between position i and $j=i+1$: some insertion can appear, no insertion and no gap is present, or $i+1$ contain a gap, so $\Delta n_{i,i+1}=0$. For $j>i+1$ (gradient shaded region on the left end of the sequence), both insertions and matched symbols contribute to the computation of the $\Delta n_{i,j}$, while gaps do not carry any contribution (see Supplementary Fig. S1 for a more detailed example). The alignment problem is then mapped into a constrained optimization problem over the $\left(\mathit{x},\mathit{n}\right)$ variables. The constraints on the variables and an example of alignment are shown in the right panel.

2.2 Alignment procedure

We can express the alignment problem in terms of the following optimization problem: $\mathit{x},\mathit{n}={\text{argmax}}_{\overline{\mathit{x}},\overline{\mathit{n}}}\frac{e^{-\beta \mathcal{H}\left(\overline{\mathit{x}},\overline{\mathit{n}}\right)}}{Z(\beta )}\prod _{i,j}{P}_{ij}^{\beta }\left(\overline{\mathit{n}}\right)$, where $\mathcal{H}$ is the DCA model describing the seed (see Fig. 1, top central panel), Z is a normalization factor, and $\beta$ is a free parameter whose relevance will be discussed below. The maximization only runs over the feasible assignment of the variables, i.e. we impose that $n_{i+1}>n_i$ for every column i. The informed prior will guide the optimization process toward solutions that, among those that maximize the Boltzmann distribution associated with $\mathcal{H}$, reproduce the statistics of the seed pointer differences. Unfortunately, the problem thus stated is unfeasible as the normalization function Z cannot be efficiently computed. Similarly to the first DCAlign version, we use an approximate message-passing algorithm coupled with an annealing scheme over $\beta$ (i.e. we iteratively increase $\beta$) to get the best alignment for the query sequence $\mathit{A}$ (see Supplementary text and Supplementary Fig. S2).

3 Results

We can classify the type of tests performed to assess the performance of our computational strategy into three different categories:

• Comparison with the previous implementation: As in Muntoni et al. (2020), we compared our results against HMMER, Infernal (the last algorithm only for RNA sequences) on four Pfam (PF00035, PF00677, PF00684, PF00763), and Rfam (RF00059, RF00162, RF00167, RF01734) families. A detailed description of the dataset is contained in Supplementary Tables S2 and S3. We utilized the following comparison metrics: (i) the positive predictive value (PPV) of the DCA-based contact prediction (Morcos et al. 2011, Cocco et al. 2018), (ii) the proximity measures between the generated and the seed MSAs. As far as the contact map prediction is concerned, we observe either a mild improvement or a similar performance. With respect to the proximity measures, we notice a negligible increase in the average distance between seed sequences and generated alignments (see Supplementary Figs S3–S6 and Supplementary Tables S7–S10).

• Leave-one-out experiment: As a stress test for DCAlign v1.0 we also compared our results to 25 ground-truth MSAs either extracted from benchmark sets (Bahr et al. 2001, Thompson et al. 2005, Freyhult et al. 2007) or built from structural alignments (Akdel et al., 2020) (see Supplementary Tables S2, S4, and S5). The numerical experiments consist of iteratively excluding one of the sequences of the reference alignment and training HMM, CM, or DCAlign using the remaining sequences. The excluded sequence is then aligned and quantitatively compared to the ground truth (viz. the structural alignment, or the benchmark sets). The emerging picture depends on the data type considered: for benchmark sets all computational strategies seem to perform reasonably well. In particular, HMMER (resp. Infernal) and our algorithm provide similar outcomes for protein (resp. RNA) domains (see Supplementary Figs S7–S10 and Supplementary Tables S11 and S12). However, when we consider structural alignments as our reference ground truth, our method significantly outperforms HMMER as shown in Supplementary Figs S11 and S12 and Supplementary Tables S13 and S14.

• Divergent sequence alignment: Finally, to assess our algorithm’s remote homology detection performance, we considered three RNA benchmark sets (the seed of Rfam RF00162 (Kalvari et al. 2020), Twister type P1 (Roth et al. 2014), tRNA (Sprinzl et al. 1998), see Supplementary Table S6) from (Wilburn and Eddy 2020). Results suggest that Infernal is the best-performing method on two of the three datasets, while our method achieves the best alignment for the tRNA case. Note that Infernal is trained using secondary structure information that our algorithm does not use. All results are presented in Supplementary Fig. S13 and Supplementary Table S15.

From a computational efficiency point of view, the time needed to train the algorithm is significantly smaller than both our old implementation and CM-Infernal (see Supplementary text and Supplementary Fig. S14). However, the time necessary to align a sequence is equivalent compared to DCAlign, and probably to other computational strategies taking into account epistasis (Wilburn and Eddy 2020, Talibart and Coste 2021).

4 Conclusion

DCAlign v1.0 is a new implementation of the DCA-based alignment technique, DCAlign, which conversely to the first implementation, allows for a fast parametrization of the seed alignment. The new modeling significantly drops the pre-processing time and guarantees a qualitatively equivalent alignment of a set of target sequences.

Supplementary data

Supplementary data are available at Bioinformatics online.

Conflict of interest

None declared.

Funding

A.P.M. and A.P. acknowledge financial support from Marie Skłodowska-Curie, grant agreement no. 734439 (INFERNET).

Data availability

We created a Zenodo link associated with the GitHub repository https://zenodo.org/badge/latestdoi/269696171.