Statistical inference for homologous gene pairs between two circular genomes: a new circular–circular regression model

Abstract

In this paper, we investigate the problem of determining the relationship, represented by similarity of the homologous gene configuration, between paired circular genomes using a regression analysis. We propose a new regression model for studying two circular genomes, where the Möbius transformation naturally arises and is taken as the link function, and propose the least circular distance estimation method, as an appropriate method for analyzing circular variables. The main utility of the new regression model is in identification of a new angular location of one of a homologous gene pair between two circular genomes, for various types of possible gene mutations, given that of the other gene. Furthermore, we demonstrate the utility of our new regression model for grouping of various genomes based on closeness of their relationship. Using angular locations of homologous genes from the five pairs of circular genomes (Horimoto et al. in Bioinformatics 14:789–802, 1998), the new model is compared with the existing models.

Appendix

1.1 Proof of Theorem 1

The limiting distribution of $\zeta =\{a,b\}$ is obtained using an exact first-order Taylor series expansion of the first order condition, for some $\zeta ^+$ between $\hat{\zeta }$ and $\zeta _0$,

$$\begin{aligned} \frac{\partial Q_n(\zeta )}{\partial \zeta }= & {} \frac{1}{n}\sum _{j=1}^n \frac{\partial m_j(\zeta )}{\partial \zeta }\sin \{\theta _j-m_j(\zeta )\}\nonumber \\= & {} \frac{1}{n}\sum _{j=1}^n \frac{\partial m_j(\zeta )}{\partial \zeta }\sin \{\theta _j-m_j(\zeta )\}|_{\zeta _0}\nonumber \\&-\frac{1}{n}\sum _{j=1}^n\left[ \frac{\partial m_j(\zeta )}{\partial \zeta }\frac{\partial m_j(\zeta )}{\partial \zeta '}\cos \{\theta _j-m_j(\zeta )\}\right. \nonumber \\&\left. -\frac{\partial ^2m_j(\zeta )}{\partial \zeta \partial \zeta '}\sin \{\theta _j-m_j(\zeta )\}\right] \Bigg |_{\zeta ^+}(\hat{\zeta }-\zeta _0)=0.\\ \sqrt{n}(\hat{\zeta }-\zeta _0)= & {} \left\{ \frac{1}{n}\sum _{j=1}^n\left[ \frac{\partial m_j(\zeta )}{\partial \zeta }\frac{\partial m_j(\zeta )}{\partial \zeta '}\cos \{\theta _j-m_j(\zeta )\}\right. \right. \nonumber \\&\quad \left. \left. -\frac{\partial ^2m_j(\zeta )}{\partial \zeta \partial \zeta '}\sin \{\theta _j-m_j(\zeta )\}\right] \Bigg |_{\zeta ^+}\right\} ^{-1}\nonumber \\&\quad \cdot \frac{1}{\sqrt{n}}\sum _{j=1}^n \frac{\partial m_j}{\partial \zeta }\sin (\theta _j-m_j)\Bigg |_{\zeta _{0}}, \end{aligned}$$

where $Q_n(\zeta )=\frac{1}{n}\sum _{j=1}^n\left[ 1-\cos \{\theta _j-\mu -2\arctan (a+bx_j)\}\right]$. We apply the multivariate CLT for independent random vectors in the following, to obtain an asymptotic multivariate normality of $\frac{1}{\sqrt{n}}\sum _{j=1}^n \frac{\partial m_j}{\partial \zeta } \sin (\theta _j-m_j)|_{\zeta _{0}}$. Then,

$$\begin{aligned}&\frac{1}{\sqrt{n}}\sum _{j=1}^n \frac{\partial m_j}{\partial \zeta }\sin (\theta _j-m_j)\Bigg |_{\zeta _{0}}\mathop {\longrightarrow }\limits ^{d}N(0,B_0),\quad \text {where}\\&\quad B_0= \text {var} \left\{ \frac{\partial m_j}{\partial \zeta '}\sin (\theta _j-m_j)\right\} \Bigg |_{\zeta _{0}}= \text {E}\left\{ \frac{\partial m_j}{\partial \zeta }\frac{\partial m_j}{\partial \zeta '}\sin ^2(\theta _j-m_j)\right\} \Bigg |_{\zeta _{0}},\\&\quad \text {and}\quad \text {plim}\frac{1}{n}\sum _{j=1}^n\left[ \frac{\partial m_j(\zeta )}{\partial \zeta }\frac{\partial m_j(\zeta )}{\partial \zeta '}\cos (\theta _j-m_j)-\frac{\partial ^2m_j(\zeta )}{\partial \zeta \partial \zeta '}\sin (\theta _j-m_j)\right] \Bigg |_{\zeta ^+}\nonumber \\&\quad = \lim \frac{1}{n}\sum _{j=1}^n\left[ E\left\{ \frac{\partial m_j(\zeta )}{\partial \zeta }\frac{\partial m_j(\zeta )}{\partial \zeta '}\cos (\theta _j-m_j)\right\} -E\left\{ \frac{\partial ^2m_j(\zeta )}{\partial \zeta \partial \zeta '}\sin (\theta _j-m_j)\right\} \right] \Bigg |_{\zeta ^+}\nonumber \\&\quad =\left[ E\left\{ \frac{\partial m_j(\zeta )}{\partial \zeta }\frac{\partial m_j(\zeta )}{\partial \zeta '}\cos (\theta _j-m_j)\right\} -E\left\{ \frac{\partial ^2m_j(\zeta )}{\partial \zeta \partial \zeta '}\sin (\theta _j-m_j)\right\} \right] \Bigg |_{\zeta ^+}\nonumber \\&\quad =A_0 . \end{aligned}$$

Now, using Slutsky’s theorem, (or Product Limit Normal Rule) we get

$$\begin{aligned} \sqrt{n}(\hat{\zeta }-\zeta _0)\mathop {\longrightarrow }\limits ^{d}N(0,A_0^{-1}B_0A_0^{-1}). \end{aligned}$$

Then, the asymptotic distributions are given by

$$\begin{aligned} \hat{\zeta }\mathop {\sim }\limits ^{a} N(\zeta _0,n^{-1}A_0^{-1}B_0A_0^{-1}). \end{aligned}$$