Orientation of Ordered Scaffolds

Abstract

Despite the recent progress in genome sequencing and assembly, many of the currently available assembled genomes come in a draft form. Such draft genomes consist of a large number of genomic fragments (scaffolds), whose order and/or orientation (i.e., strand) in the genome are unknown. There exist various scaffold assembly methods, which attempt to determine the order and orientation of scaffolds along the genome chromosomes. Some of these methods (e.g., based on FISH physical mapping, chromatin conformation capture, etc.) can infer the order of scaffolds, but not necessarily their orientation. This leads to a special case of the scaffold orientation problem (i.e., deducing the orientation of each scaffold) with a known order of the scaffolds.

We address the problem of orientation of ordered scaffolds as an optimization problem based on given weighted orientations of scaffolds and their pairs (e.g., coming from pair-end sequencing reads, long reads, or homologous relations). We formalize this problem within the earlier introduced framework for comparative analysis and merging of scaffold assemblies (CAMSA). We prove that this problem is $\mathsf {NP}$-hard, and further present a polynomial-time algorithm for solving its special case, where orientation of each scaffold is imposed relatively to at most two other scaffolds. This lays the foundation for a follow-up FPT algorithm for the general case. The proposed algorithms are implemented in the CAMSA software version 2.

Appendix. Pseudocodes

In the algorithms below we do not explicitly describe the function OrConsCount, which takes 4 arguments:

• 1. a subgraph c from ${{\mathrm{\mathsf {COG}}}}(\mathbb {O}_o)$ with 1 or 2 vertices;

• 2. a hash table so with scaffolds as keys and their orientations as values;

• 3. a set of orientation imposing assembly points $\mathbb {O}_o$;

• 4. an assembly $\mathbb {A}$

and counts the assembly points from $\mathbb {O}$ that have consistent orientation with $\mathbb {A}$ in the case where scaffold(s) corresponding to vertices from c were to have orientation from so in $\mathbb {A}$. With simple hash-table based preprocessing of $\mathbb {A}$ and $\mathbb {O}$ (can be done in $\mathcal {O}\left( k\log (k)\right)$ time, where $k=\max \{|\mathbb {O}|, \mathbb {S}(\mathbb {A})\}$) this function runs in $\mathcal {O}\left( n\right)$ time, where n is a number of assembly points in $\mathbb {O}$ involving scaffolds that correspond to vertices in c. So, total running time for all invocations of this function will be $\mathcal {O}\left( |\mathbb {O}|\right)$.