# Single-cell 3D genome reconstruction in the haploid setting using rigidity theory

## TL;DR

This paper introduces a method for reconstructing 3D genome structures in haploid cells using mathematical rigidity theory and applies it to real and synthetic data.

## Contribution

A novel 3D genome reconstruction method for haploid cells using rigidity theory and semidefinite programming.

## Key findings

- New results on realisability and uniqueness of 3D genome reconstructions in haploid organisms.
- A semidefinite programming-based method successfully applied to synthetic and real data.
- Multiple graph models derived from Hi-C and microscopy data improve reconstruction accuracy.

## Abstract

This article considers the problem of 3-dimensional genome reconstruction for single-cell data, and the uniqueness of such reconstructions in the setting of haploid organisms. We consider multiple graph models as representations of this problem, and use techniques from graph rigidity theory to determine identifiability. Biologically, our models come from Hi-C data, microscopy data, and combinations thereof. Mathematically, we use unit ball and sphere packing models, as well as models consisting of distance and inequality constraints. In each setting, we describe and/or derive new results on realisability and uniqueness. We then propose a 3D reconstruction method based on semidefinite programming and apply it to synthetic and real data sets using our models.