Descriptors for DNA Sequences Based on Joint Diagonalization of Their Feature Matrices from Dinucleotide Physicochemical Properties

For all the multiple sequences, we can use Approximate Joint Diagonalization (AJD) of their corresponding transformed matrices $\mathit{ℳ}=\{{\mathit{M}}^{(1)},{\mathit{M}}^{(2)},\mathrm{\cdots },{\mathit{M}}^{(N)}\}$ , which has been successfully applied in the dataset with the first exon from sequences of 11 beta globin genes^{[ 24 ]}.

In brief, AJD corresponds to the problem of seeking a matrix V, which will lead ${\mathit{V}}^{\text{H}}{\mathit{M}}^{(n)}\mathit{V}$ to be the diagonal as possible for all $n$ , where V is a unitary matrix. This is based on the premise that a set of matrices $\{{\mathit{M}}^{(1)},{\mathit{M}}^{(2)},\mathrm{\cdots },{\mathit{M}}^{(N)}\}$ consists of common statistical information of the observations which are the estimates of matrices in the form ${\mathit{V}}^{\text{H}}{\mathit{M}}^{(n)}\mathit{V}$ .

In general, for any $n\times n$ matrix V, the AJD criterion may be defined as the following non-negative function of V:

(3) $$\mathit{J}(\mathit{V},{𝚲}^{(1)},{𝚲}^{(2)},\mathrm{\cdots },{𝚲}^{(N)})\stackrel{\text{def}}{=}\underset{i=1}{\overset{N}{\sum }}{||{𝚲}^{(i)}-{\mathit{V}}^{\text{H}}{\mathit{M}}^{(i)}\mathit{V}||}^2$$

Usually, AJD does not require that the involved matrix set $\mathit{ℳ}$ be exactly simultaneously diagonalized by a common unitary matrix. Mostly, the criterion for AJD, as indicated by Eq. ( 3 ), cannot be zeroed, and the matrices can only be approximately jointly diagonalized. Thus, AJD deals with a kind of an "average eigen-structure", which is particularly convenient for statistically inferring the structural information extracted from sample statistics.

Considering two transformations:

$${\tau }_1:{\text{Sequence}}^{(i)}\mapsto {\mathit{M}}^{(i)}.$$

${\text{Sequence}}^{(i)}$ denotes the $i$ -th sequence, where the length of the sequence is $L$ , and $i=1,2,\mathrm{\cdots },N$ , while ${\mathit{M}}^{(i)}\in {ℝ}^{12\times 12}$ stands for the corresponding matrices mapped from each primary DNA sequences, and ${\mathit{M}}^{(i)}$ is a symmetric matrix, which can be determined by scanning along the sequence $S^{(i)}$ via the decision criterion listed in Table 2 .

$${\tau }_2:{\mathit{M}}^{(i)}\mapsto ({\lambda }_1^{(i)},{\lambda }_2^{(i)},\mathrm{\cdots },{\lambda }_{12}^{(i)}).$$

The feature vector ${\mathit{F}}_{12}^{(i)}=({\lambda }_1^{(i)},{\lambda }_2^{(i)},\mathrm{\cdots },{\lambda }_{12}^{(i)})$ is a 12-tuple vector consisting of all the eigenvalues extracted by AJD upon ${\mathit{M}}^{(i)}$ . Thus, compound transformation may be obtained as follows:

(4) $${\tau }_2\circ {\tau }_1:{\text{Sequence}}^{(i)}\mapsto ({\lambda }_1^{(i)},{\lambda }_2^{(i)},\mathrm{\cdots },{\lambda }_{12}^{(i)})$$

From Formula ( 4 ), we can freely extract the features of the DNA sequence. From the viewpoint of algebra space, the transformation may also be presented as

(5) $$\text{Ker}f:\mathit{S}\stackrel{\tau }{⟶}{\mathit{F}}^{1\times 12}$$

where S denotes the original sequence space comprising of primary DNA sequence having the length $L$ , while ${\text{𝑭}}^{1\times 12}$ indicates the objective feature space that is transformed from the original space. Further, the diagonal elements of $𝚲$ are simply the eigenvalues of the dinucleotide PhysicoChemical Matrix (PCM) via AJD.

According to the results from previous work^{[ 24 ]}, it was determined that the AJD-PCM algorithm has the property of distance-preserving. Thus, we can calculate all the Eigenvalue Vectors (EVs) for each obtained dinucleotide PCM, such as ${\mathit{F}}_{12}^{(i)}=({\lambda }_1^{(i)},{\lambda }_2^{(i)},\mathrm{\cdots },{\lambda }_{12}^{(i)})$ , $i=1,2,\mathrm{\cdots },N$ . Further, the $N$ corresponding 12-tuple vectors may be obtained that can be regarded as features extracted from the original DNA sequence. The AJD-PCM algorithm is shown in Algorithm 1.