Algorithm for 3D Structural Reconstruction for Paternal and Maternal Chromosomes

Abstract

With the development of genetics, there is an increasing number of biological studies focusing on the chromosomes and acknowledge that spatial closeness between two segments within a chromosome contributes to the co-expression, namely, the expression of these closed genes tend to positively relate to each other. Therefore, this closeness could predicts the functional link between genes and also potentially reveals the underlying mechanism of gene regulation and expression. However, this closeness measurement is limited in 1D for a considerable period time so the spatial closeness in 3D is neglected which makes some actual functional linkages are behind the veil. Therefore, to discover the underlying mechanism of gene regulation, reconstructing the 3D chromosomal structure is of significance. With the help of Hi-C technology, there is possibility to reconstruct the structure with the interactions among genes which reflect the pairwise spatial distances. However, this technology neglects the distinction between alleles on paternal and maternal chromosomes. Generally, human chromosomes, which is our main research objects, are in pairs and from two sources, either father or mother, which have different structures. Neglecting the differences between paternal and maternal would, thus, causes inaccurate 3D reconstruction results. To solve this major problem, this project aims to design an algorithm to reconstruct the paternal and maternal chromosomal 3D structures separately, which is of significance for further accurate chromosomal studies. The algorithm models the problem with Generic Linear Model and assumes the interactions are subjected to Poisson distribution. Therefore, the reconstruction process is equivalent to maximize the total log-likelihood. Targeting on maximizing the likelihood, an iterative optimization algorithm is designed. It optimizes the target in a similar manner as Expectation Maximum, divides every iteration into two subproblems, and solve these subproblems with Multi-dimensional scaling (MDS). This report would provide mathematical proof on how the original maximizing likelihood could be reduced to this iterative Multi-dimensional scaling (MDS) problem. In conclusion, this report introduces an MDS assisted maximizing likelihood based algorithm for reconstructing 3D structures of paternal and maternal chromosomes assuming a Poisson distribution for the input Hi-C interaction matrix.