Embeddability of centrosymmetric matrices capturing the double-helix structure in natural and synthetic DNA

TL;DR

This paper investigates the conditions under which centrosymmetric matrices, representing DNA substitution models, can be embedded into continuous-time models, with implications for understanding natural and synthetic DNA structures.

Contribution

It characterizes the embeddability of centrosymmetric matrices related to DNA models, extending the theory to higher order matrices relevant in synthetic biology.

Findings

Criteria for embeddability of centrosymmetric matrices.

Application to DNA substitution models like Kimura and Jukes-Cantor.

Extension to matrices of various sizes in synthetic biology.

Abstract

In this paper, we discuss the embedding problem for centrosymmetric matrices, which are higher order generalizations of the matrices occurring in Strand Symmetric Models. These models capture the substitution symmetries arising from the double helix structure of the DNA. Deciding whether a transition matrix is embeddable or not enables us to know if the observed substitution probabilities are consistent with a homogeneous continuous time substitution model, such as the Kimura models, the Jukes-Cantor model or the general time-reversible model. On the other hand, the generalization to higher order matrices is motivated by the setting of synthetic biology, which works with different sizes of genetic alphabets.