OMGS: Optical Map-Based Genome Scaffolding

Abstract

Due to the current limitations of sequencing technologies, de novo genome assembly is typically carried out in two stages, namely contig (sequence) assembly and scaffolding. While scaffolding is computationally easier than sequence assembly, the scaffolding problem can be challenging due to the high repetitive content of eukaryotic genomes, possible mis-joins in assembled contigs and inaccuracies in the linkage information. Genome scaffolding tools either use paired-end/mate-pair/linked/Hi-C reads or genome-wide maps (optical, physical or genetic) as linkage information. Optical maps (in particular Bionano Genomics maps) have been extensively used in many recent large-scale genome assembly projects (e.g., goat, apple, barley, maize, quinoa, sea bass, among others). However, the most commonly used scaffolding tools have a serious limitation: they can only deal with one optical map at a time, forcing users to alternate or iterate over multiple maps. In this paper, we introduce a novel scaffolding algorithm called OMGS that for the first time can take advantages of multiple optical maps. OMGS solves several optimization problems to generate scaffolds with optimal contiguity and correctness. Extensive experimental results demonstrate that our tool outperforms existing methods when multiple optical maps are available, and produces comparable scaffolds using a single optical map. OMGS can be obtained from https://github.com/ucrbioinfo/OMGS.

Appendix

1.1 A DAG Unique Ordering

1.2 B Statistical Test for Repetitive Regions

Here we provide additional details for the estimation of $\sigma ^2$ during the analysis of repetitive regions. Recall that we collect all estimated repetitive lists in set $R=\{D_{p} \text { is estimated repetitive}|p=1,\dots ,P \}$ and the estimated mean $\hat{\mu }_p$ for each distance list $D_p$ in the set R, where P is the total number of estimated repetitive lists. For each $D_p$, the distances $d_i$’s are distributed as a Gaussian with mean $\hat{\mu }_p$ and variance $\sigma ^2$. According to the density function of Gaussian distribution, the log likelihood of one $D_p$ is

$$ -\frac{|D_p|}{2}\log (2\pi )-\frac{|D_p|}{2}\log \sigma ^2-\frac{1}{2\sigma ^2}\sum _{d_{i} \in D_p} (d_i-\hat{\mu }_p)^2. $$

The total log likelihood is the sum of the log likelihoods across all $D_p$’s in R, which is

$$ \log L(\sigma ^2) = -\frac{\sum _{p=1}^{P}|D_p|}{2}\log \sigma ^2-\frac{1}{2\sigma ^2} \sum _{p=1}^{P} \sum _{d_{i} \in D_p} (d_i-\hat{\mu }_p)^2, $$

after ignoring all terms not related to $\sigma ^2$. To maximize $\log L(\sigma ^2)$, we require that the derivative of total log likelihood

$$ \frac{\partial \log L(\sigma ^2)}{\partial \sigma ^2}= 0, $$

that is,

$$ -\frac{\sum _{p=1}^{P}|D_p|}{2\sigma ^2}+\frac{1}{2(\sigma ^2)^2} \sum _{p=1}^{P} \sum _{d_{i} \in D_p} (d_i-\hat{\mu }_p)^2 = 0. $$

After some simplification, the estimator for variance becomes

$$ \hat{\sigma }^2=\frac{\sum _{p=1}^{P} \sum _{d_{i} \in D_p} (d_i-\hat{\mu }_p)^2}{\sum _{p=1}^{P}|D_p|}. $$

1.3 C Density Function of $d_{max}-d_{min}$

Here we provide additional details for calculating the density function of $d_{max}-d_{min}$. It is well-known that the joint density function of order statistics is

$$\begin{aligned} f_{X(i),X(j)}(u,v)=\frac{n!}{(i-1)!(j-1-i)!(n-j)!}f_x(u)f_x(v)[F_x(u)]^{i-1}[F_x(v)-F_x(u)]^{j-1-i}[1-F_x(v)]^{n-j} \end{aligned}$$

(1)

for $-\infty<u<v<+\infty$, where X(i) and X(j) are the i-th and j-th order statistics in $X_1,\dots ,X_n$ and $F_x$ and $f_x$ are the distribution function and density function of each $X_i$, respectively. Using (1), the joint density function of ($d_{max},d_{min}$) can be expressed as

$$ f_{d_{max}, d_{min}}(u,v)=n(n-1)f_{d_i}(u)f_{d_i}(v)[F_{d_i}(v)-F_{d_i}(u)]^{n-2} $$

for $-\infty<u<v<+\infty$, where $F_{d_i}$ and $f_{d_i}$ are the distribution function and density function of $d_i\sim N(\hat{\mu }_{j,q}, \hat{\sigma }^2)$, respectively.

Now, let $X= d_{max}-d_{min}$ and $Y=d_{min}$. Then $d_{max}=X+Y$ and $d_{min}=Y$, and the corresponding Jacobian determinant is

$$ J=\begin{vmatrix} \partial d_{\max }/\partial X&\partial d_{\max }/\partial Y\\ \partial d_{\min }/\partial X&\partial d_{\min }/\partial Y \end{vmatrix} = \begin{vmatrix} 1&1 \\ 0&1 \end{vmatrix} = 1. $$

Thus, the joint density function of (X, Y) is given by

$$ f_{X,Y}(x,y)=f_{d_{max},d_{min}}(x+y, y)|J|=n(n-1)f_{d_i}(y)f_{d_i}(x+y)[F_{d_i}(x+y)-F_{d_i}(y)]^{n-2}, $$

where $x\ge 0$ and $-\infty< y<+\infty$. By integrating over Y, the density function of $X=d_{max}-d_{min}$ becomes

$$ f_{d_{max}-d_{min}}(x)=\int _{-\infty }^{+\infty }n(n-1)f_{d_i}(y)f_{d_i}(x+y)[F_{d_i}(x+y)-F_{d_i}(y)]^{n-2}dy, x\ge 0. $$

1.4 D Gap Estimation

Here we provide additional details for calculating the log likelihood function when estimating gaps. Recall that $l_p,\dots ,l_{q-1}$ are independent chi-square random variables, and $\sum _{i=p}^{q-1}l_i$ is chi-square distributed with degree of freedom $\sum _{i=p}^{q-1}\alpha _i$. Since the density function of a chi-square random variable X with degree of freedom k is

$$ f_X(x)=\frac{1}{2^{k/2}\Gamma (k/2)}x^{k/2-1}e^{-x/2} $$

where $\Gamma$ is the gamma function, the likelihood of $\sum _{i=p}^{q-1}l_i$ with observation

$$ \gamma =d_j-\sum _{c=c_{p+1}}^{c_{q-1}}|c| $$

is

$$ \frac{1}{2^{\beta }\Gamma (\beta )}\gamma ^{\beta -1}e^{-\gamma /2}, $$

where $\beta =\sum _{i=p}^{q-1}\frac{\alpha _i}{2}$. Therefore, the log likelihood function for one sample is

$$ \log l=(\beta -1) \log \gamma -\frac{\gamma }{2}-\beta \log 2-\log \Gamma (\beta ). $$

The total log likelihood is the sum of the log likelihoods across all samples.