## Project: A Graph-Theoretic Approach for the Generation of Conformations of Macromolecules (Chromosomes) from Contact Matrices

**Description**: Consider two metric spaces. One is high dimensional and the other one
is the three dimensional euclidean space. The high-dimensional space is
derived from a contact matrix, i.e. a matrix where the row and column
sums are 1 which represents the normalize contacts in space that a
chromosome has made with itself (see for example the wicki on HiC
Chromosome Conformation Capture).

Hence in this contact matrix is encoded the topological properties of the ensemble of conformations that the chromosome has taken on.

One interpretation of this is a weighted graph, i.e. an adjacency matrix where the edges have weights according to the entires in the matrix connecting node i and j, i.e. aij (see for example)

This is augmented with a metric (let us call this the contact space). The mapping (for example multidimensional scaling (MDS) is now supposed to reconstruct in three-dimensional space possible conformations that are consistent with the given matrix and the corresponding metric. Since the matrix represents an ensemble average which comprises conflicting information, the task is to construct a method that goes beyond just to produce a single conformation representing the mean field conformation.

The central idea is to sample from the contact space. Essentially this means generating a sample graph representing the contacts and its strengths (probabilities) for its existence. This graph is then mapped into a conformation in three-dimensional dimensional euclidean space. This conformation may not obey the contraints of excluded volume and needs to be resampled in euclidean space to conform to the constraints. Furthermore there may be other constraints ( distance geometry), like known distances between specific sites that needs to be incorporated. This in turn may mean that the generated conformation is not entirely compatible with the graph sampled in the "contact space". Conditions for the acceptance of such a generated conformation need to be developed.

**Prerequisites**: Statistical Physics Course and an open mind

**Papers pertaining to the project**:

- Network concepts for analyzing 3D genome structure from chromosomal contact maps
- Resolving spatial inconsistencies in chromosome conformation measurements
- Topological properties of chromosome conformation graphs reflect spatial proximities within chromatin
- Graph rigidity reveals well-constrained regions of chromosome conformation embeddings
- Metagenomic chromosome conformation capture (meta3C) unveils the diversity of chromosome organization in microorganisms
- Detecting hierarchical 3-D genome domain reconfiguration with network modularity
- Interpretation of chromosome conformation capture data in terms of graphs
- Exploiting native forces to capture chromosome conformation in mammalian cell nuclei
- Three-Dimensional Folding and Functional Organization Principles of the Drosophila Genome