[Paper Review] Category theory for genetics I: mutations and sequence alignments

Rémy Tuyéras|arXiv (Cornell University)|May 18, 2018

RNA and protein synthesis mechanismsBiochemistry, Genetics and Molecular Biology16 references4 citations

TL;DR

This paper introduces a categorical framework using limit-sketches and pedigrads to formalize genetic mutation mechanisms—such as insertions, deletions, substitutions, duplications, and inversions—within a unified algebraic structure. By modeling sequence alignments as functors and leveraging right Kan extensions, the approach enables the detection of evolutionary mechanisms directly during alignment, with uncertainty in data revealing underlying mutation events.

ABSTRACT

The present article is the first of a series whose goal is to define a logical formalism in which it is possible to reason about genetics. In this paper, we introduce the main concepts of our language whose domain of discourse consists of a class of limit-sketches and their associated models. While our program will aim to show that different phenomena of genetics can be modeled by changing the category in which the models take their values, in this paper, we study models in the category of sets to capture mutation mechanisms such as insertions, deletions, substitutions, duplications and inversions. We show how the proposed formalism can be used for constructing multiple sequence alignments with an emphasis on mutation mechanisms.

Motivation & Objective

• To develop a formal, abstract language rooted in category theory to reason about genetic phenomena, particularly focusing on mutation mechanisms.
• To address the limitation of current alignment methods that treat genetic variation as random by instead identifying mutation mechanisms during the alignment process.
• To formalize multiple sequence alignments using categorical constructions such as right Kan extensions and limits, enabling a mechanistic understanding of sequence evolution.
• To distinguish between data consistency and uncertainty in alignments, linking surjective or isomorphic mappings to biological consistency and uncertainty to mutation events.
• To provide a foundation for a broader categorical formalism in genetics that can be extended to model genotypes, phenotypes, and recombination events in future work.

Proposed method

• Define 'chromologies' as a class of limit-sketches whose models (called 'pedigrads') formalize genetic operations in a category-theoretic framework.
• Model sequence alignments as functors $ T: B \to \mathbf{Set} $, where $ B $ is a category of sequence segments, capturing alignment data as sets of aligned sequences.
• Use right Kan extensions $ \mathsf{Ran}_{\iota}T $ to integrate multiple sequence alignments from partial data, reconstructing global alignments from local alignments.
• Apply the concept of 'slices' (Definition 4.7) to resolve uncertainty in alignments by identifying lifts that correspond to specific mutation mechanisms.
• Utilize limits and cone structures to assess data consistency: isomorphisms indicate perfect consistency, while surjections signal uncertainty linked to biological mechanisms.
• Construct a functor $ \pi_{\mathtt{c}}^*E_b^\varepsilon \circ \iota $ to model environmental constraints and detect mutation events via pullback constructions.

Experimental results

Research questions

• RQ1 How can category theory be used to formalize genetic mutation mechanisms such as insertions, deletions, substitutions, duplications, and inversions?
• RQ2 Can multiple sequence alignments be constructed in a way that identifies mutation mechanisms during the alignment process rather than post hoc?
• RQ3 How can uncertainty in sequence alignment data be formally linked to the presence of specific biological mutation mechanisms?
• RQ4 In what way do categorical constructions like right Kan extensions and limits provide a mechanistic understanding of sequence evolution?
• RQ5 How can the structure of a functor’s image and its morphisms (e.g., surjections vs. isomorphisms) reflect biological consistency or uncertainty in genetic data?

Key findings

• The right Kan extension of a sequence alignment functor $ T $ captures both local and global alignment information, enabling the construction of multiple sequence alignments from partial data.
• Surjective functions in the Kan extension structure indicate uncertainty in the data, which is shown to correlate with the presence of mutation mechanisms such as insertions or deletions.
• Isomorphic mappings in the Kan extension correspond to perfectly consistent data, indicating no uncertainty and high confidence in alignment integrity.
• The use of Craig’s slice construction allows for the detection of specific mutation mechanisms—such as inversions—by identifying lifts that match known sequence rearrangements.
• A triple of aligned sequences (e.g., $ \mathtt{x_1x_2ABCx_3x_4x_5x_6} $, $ \mathtt{x_1x_2CBAx_3x_4x_5x_6} $) in the image of the Kan extension reveals an inversion event, confirming mechanistic consistency.
• The framework successfully identifies that a sequence like $ \mathtt{x_1x_2EFGx_3x_4x_5x_6} $ does not produce such a triple, indicating it is unlikely to be related by inversion to the original sequence, thus validating mechanism detection.