A Position-Weighted Asymmetric Embedding for DNA Sequence Representation: Mathematical Characterization and a BRCA1 Case Study

Source abstract

Compact numerical representations of DNA sequences can reduce the dimensionality of genomic data, but aggregation may result in the loss of positional information. This study investigates a position-weighted representation in which each nucleotide at position k contributes to one of four nucleotide channels according to a strictly increasing positional weighting function. The resulting sequence representation is a fixed four-dimensional vector, independently of sequence length. For a single nucleotide substitution at position j relative to a fixed reference, one nucleotide channel decreases and another increases by the same position-dependent magnitude, producing a zero-sum and antisymmetric change vector whose magnitude identifies the substitution position in exact arithmetic. A finite-precision analysis was additionally developed to derive a conservative sufficient precision condition for discrete position recovery. The approach was evaluated using the complete 126,033-nucleotide BRCA1 reference sequence. A deterministic single-nucleotide substitution was introduced independently at every position, and all 126,033 tested substitutions were correctly recovered using double-precision arithmetic under the evaluated pairwise-summation procedure, despite the conservative theoretical bound requiring 59 significand bits. The maximum continuous position error was 0.2227 positions. Computational benchmarking against one-hot encoding (OHE) and k-mer count representations showed that the proposed representation reduced a 5000-nucleotide sequence to four floating-point values (32 bytes), although its direct implementation required greater encoding time. In a complementary controlled Gradient Boosting experiment, the sequence-length-normalized weighted representation, padded OHE, and normalized 1-mer frequencies all produced classification performance close to chance after controlling for genomic window and sequence length, while the four-dimensional weighted representation substantially reduced memory requirements and downstream training time relative to padded OHE. These results characterize the proposed method as a compact, position-dependent DNA representation with analytically interpretable behavior for single nucleotide substitutions relative to a fixed reference. The findings distinguish positional identifiability and computational compactness from pathogenicity prediction and do not imply global sequence injectivity or diagnostic capability.