Sheaf-theoretic representation of the proteolipid code

TL;DR

This paper introduces a topological and category-theoretic framework using sheaves to model membrane particles and zones, capturing multiscale interactions and dynamics in biological membranes.

Contribution

It develops a novel mathematical formalism combining sheaf theory and Hamiltonian mechanics for membrane structure and multiscale coupling.

Findings

Provides a formalism for representing membrane particle interactions.

Enables dynamical modeling of membrane processes.

Integrates topological and category-theoretic approaches.

Abstract

Membrane particles such as proteins and lipids organize into zones that perform unique functions. Here, I introduce a topological and category-theoretic framework to represent particle and zone intra-scale interactions and inter-scale coupling. This involves carefully demarcating between different presheaf- or sheaf-assigned data levels to preserve functorial structure and account for particle and zone generalized poses. The framework can accommodate Hamiltonian mechanics, enabling dynamical modeling. This amounts to a versatile mathematical formalism for membrane structure and multiscale coupling.