Singular value decomposition of protein sequences as a method to visualize sequence and residue space

Abstract

Singular value decomposition (SVD) of multiple sequence alignments (MSAs) is an important and rigorous method to identify subgroups of sequences within the MSA, and to extract consensus and covariance sequence features that define the alignment and distinguish the subgroups. This information can be correlated to structure, function, stability, and taxonomy. However, the mathematics of SVD is unfamiliar to many in the field of protein science. Here, we attempt to present an intuitive yet comprehensive description of SVD analysis of MSAs. We begin by describing the underlying mathematics of SVD in a way that is both rigorous and accessible. Next, we use SVD to analyze sequences generated with a simplified model in which the extent of sequence conservation and covariance between different positions is controlled, to show how conservation and covariance produce features in the decomposed coordinate system. We then use SVD to analyze alignments of two protein families, the homeodomain and the Ras superfamilies. Both families show clear evidence of sequence clustering when projected into singular value space. We use k‐means clustering to group MSA sequences into specific clusters, show how the residues that distinguish these clusters can be identified, and show how these clusters can be related to taxonomy and function. We end by providing a description a set of Python scripts that can be used for SVD analysis of MSAs, displaying results, and identifying and analyzing sequence clusters. These scripts are freely available on GitHub.

The importance of input sequence set to consensus‐derived proteins and their relationship to reconstructed ancestral proteins

“…This allows the correlations among the sequences to be concentrated into the first few dimensions of SVD space, significantly reducing the dimensions of sequence space in which sequences can be visually compared. One feature of the SVD algorithm is that it numbers the dimensions in order of decreasing information content; as such, the first dimension of the SVD, which contains the most variation, corresponds to the consensus sequence (see Baxter‐Koenigs et al (2022)). One can compare MSA sequences, both visually and quantitatively, by determining the extent to which they spread out and/or cluster with one another along the first few axes of the SVD.…”

“…Four hundred and nine aligned RNase H sequences from Hart et al (2014). were represented as a binary matrix, which was transformed into an orthogonal sequence and residue matrices using singular value decomposition (Baxter‐Koenigs et al, 2022). Plots show sequences (small points) plotted in the first three dimensions of SVD space (left), and in dimensions 2 and 3 (right).…”

“…One can compare MSA sequences, both visually and quantitatively, by determining the extent to which they spread out and/or cluster with one another along the first few axes of the SVD. Moreover, by alignment to the MSA, sequences not in the original MSA (such as ancestors and consensus sequences) can be projected into SVD space (Baxter‐Koenigs et al, 2022).…”

“…SVD is an approach for “dimensionality reduction” in the same vein as the more familiar Principal Component Analysis (Baxter‐Koenigs et al, 2022). SVD reduces each sequence to a point in a high‐dimensional space.…”

“…SVD was carried out as detailed previously (Baxter‐Koenigs et al, 2022). Briefly, the 409 sequence‐MSA used for ancestral reconstruction was encoded as a binary matrix in which each position is represented using 20 variables reflecting the 20 amino acids, and the variable corresponding to the residue at a particular position of a sequence is assigned a one, and the other 19 variables are assigned a zero.…”