[Paper Review] Compositional representation of protein sequences and the number of Eulerian loops

Bailin Hao, Huimin Xie+1 more|ArXiv.org|Mar 10, 2001

Genomics and Phylogenetic Studies2 references11 citations

TL;DR

This paper proposes a compositional representation of protein sequences using overlapping K-strings and models the reconstruction problem as counting Eulerian loops in a directed Euler graph. Using a modified BEST theorem to account for parallel arcs and self-loops, the study shows that for K ≥ 5, the vast majority of proteins in the pdb.seq database have unique reconstructions, indicating that K-strings provide a nearly unique representation of protein sequences.

ABSTRACT

An amino acid sequence of a protein may be decomposed into consecutive overlapping strings of length K. How unique is the converse, i.e., reconstruction of amino acid sequences using the set of K-strings obtained in the decomposition? This problem may be transformed into the problem of counting the number of Eulerian loops in an Euler graph, though the well-known formula must be modified. By exhaustive enumeration and by using the modified formula we show that the reconstruction is unique at K equal or greater than 5 for an overwhelming majority of the proteins in the PDB.seq database. The corresponding Euler graphs provide a means to study the structure of repeated segments in protein sequences.

Motivation & Objective

• To investigate whether protein sequences can be uniquely reconstructed from their overlapping K-strings, a compositional representation method.
• To model the reconstruction problem as counting Eulerian loops in a directed Euler graph derived from K-string frequency data.
• To develop and apply a modified version of the BEST theorem that accounts for parallel arcs and self-loops in the graph representation.
• To assess the uniqueness of reconstruction across real protein sequences in the pdb.seq database for varying K values.
• To identify proteins with highly non-unique reconstructions, which may indicate complex repeated structural motifs.

Proposed method

• Decompose each protein sequence into overlapping K-strings of length K, forming a multiset of K-strings with their frequencies.
• Construct a directed Euler graph where nodes represent (K-1)-mers and edges represent K-strings, with edge multiplicities reflecting frequency.
• Apply a modified BEST theorem formula: R = Δ × ∏(di−1)! / ∏(aij!) to count the number of Eulerian loops, where Δ is the Kirchhoff matrix minor, di is the degree of node i, and aij is the multiplicity of edge (i,j).
• Use the modified formula to compute the number of distinct sequences that can be reconstructed from the same K-string multiset.
• Perform exhaustive enumeration and database screening on 2820 proteins from pdb.seq to validate the formula and assess reconstruction uniqueness.
• Coarse-grain the amino acid alphabet in future work to enhance biological relevance and detect higher-order correlations.

Experimental results

Research questions

• RQ1 To what extent can a protein sequence be uniquely reconstructed from the multiset of its overlapping K-strings?
• RQ2 How does the number of possible reconstructions (R) change with increasing K, and at what K value does uniqueness become typical?
• RQ3 Which proteins exhibit a high number of possible reconstructions, and what structural features might explain this behavior?
• RQ4 Can the modified BEST theorem accurately predict the number of Eulerian loops in graphs derived from real protein sequences?
• RQ5 What biological insights can be drawn from proteins with highly non-unique reconstructions, particularly regarding repeated segments and evolutionary mechanisms?

Key findings

• For K ≥ 5, 76.7% of the 2820 proteins in the pdb.seq database have unique reconstructions, increasing to 99.0% at K = 10.
• The number of Eulerian loops R is computed using a modified BEST formula that accounts for parallel edges and self-loops in the Euler graph representation.
• Proteins with extremely large R values (e.g., 491,166,720 at K=5) are rare but identifiable, such as CENB_HUMAN and MCMI_YEAST, and are not necessarily long sequences.
• Even long proteins (e.g., >2115 amino acids) tend to have unique or small numbers of reconstructions, indicating that length alone does not predict reconstruction multiplicity.
• At K=11, all proteins with R > 1 have R = 1, indicating that reconstruction becomes fully unique for K ≥ 11 in all tested cases.
• The study identifies a small group of proteins with complex repeated structures that are highly sensitive to K-string decomposition, suggesting they may be candidates for structural or evolutionary analysis.