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

TL;DR

This paper bridges single-cell 3D genome reconstruction in haploid organisms with graph rigidity theory. It develops and analyzes multiple geometric models (unit-ball, inequality/equality, penny/marble, and interval radii penny/marble) that translate Hi-C and microscopy data into distance constraints, and it studies existence and identifiability within each model. A semidefinite programming framework is proposed to realize these constraints, complemented by reconstruction experiments on synthetic and real data, revealing that unit-ball models lack finite identifiability while more constrained models can achieve finite or unique reconstructions under suitable extra information. The work also introduces a practical SDP-based reconstruction algorithm and compares its performance to existing methods, highlighting trade-offs between accuracy and computational effort. Overall, the study provides a rigorous mathematical foundation, actionable reconstruction strategies, and guidance for integrating additional data to achieve reliable 3D genome reconstructions in the haploid single-cell setting.

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.

Figures (19)

• Figure 1: Graphs with different rigidity properties.
• Figure 2: A graph which would be infinitesimally rigid if realised as a generic framework, but the chosen framework has a non-trivial infinitesimal motion (as indicated).
• Figure 3: A schematic overview depicting the two main approaches to obtaining biological data, their mathematical counterparts and the advantages and disadvantages .
• Figure 4: A unit disk graph with a unit disk realisation (left) and another realisation which does not satisfy the unit disk condition (right).
• Figure 5: Some known minimal forbidden subgraphs in dimension two. The existence of such a subgraph prevents a graph from having a unit-disc realisation.

Theorems & Definitions (42)

• Lemma 2.1
• Lemma 2.2: Maxwell
• Lemma 2.3: AsimowRoth
• Definition 3.1: garamvolgyi2020global
• Lemma 3.2: Lekkeikerker1962
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5: garamvolgyi2020global