Table 1 The definitions of six metrics used for motif comparison.

Average log-likelihood ratio (ALLR)

$ALLR\left(X,Y\right)=\frac{{\displaystyle \sum _b}{N}_{X_b}\left(\frac{Y_b}{q_b}\right)+{\displaystyle \sum _b}{N}_{Y_b}\left(\frac{X_b}{q_b}\right)}{{\displaystyle \sum _b}\left(N_{X_b}+N_{Y_b}\right)}$

Wang and Stormo [18]

Average Kullback-Leibler ( AKL)

$AKL\left(X,Y\right)=10-\frac{{\displaystyle \sum _b}{X}_{b}log\frac{X_b}{Y_b}+{\displaystyle \sum _b}{Y}_{b}log\frac{Y_b}{X_b}}{2}$

Kullback and Leibler [19]

Sum of squared distances (SSD)

$SSD\left(X,Y\right)=2-\left({\displaystyle \sum _b}{\left(X_b-Y_b\right)}^2\right)$

Schones et al. [17]

1-p-value of Chi-square (pCS)

${\chi }^2\left(X,Y\right)={\displaystyle \sum _b}\frac{{\left(N_{X_b}-N_{X_b}^e\right)}^2}{N_{X_b}^e}+{\displaystyle \sum _b}\frac{{\left(N_{Y_b}-N_{Y_b}^e\right)}^2}{N_{Y_b}^e}$

Schones et al. [17]

Pearson correlation coefficient ( PCC)

$PCC\left(X,Y\right)=\frac{{\displaystyle \sum _b}\left(X_b-\overline{X}\right)\left(Y_b-\overline{Y}\right)}{\sqrt{{\displaystyle \sum _b}{\left(X_b-\overline{X}\right)}^2{\displaystyle \sum _b}{\left(Y_b-\overline{Y}\right)}^2}}$

Pietrokovski [20]

Asymptotic Covariance (AC)

$AC\left(A,B\right)={\displaystyle \underset{m\to \infty }{lim}}{m}^{-1}\mathsf{\text{cov}}\left(N_A\left(m\right)+N_{A^{\prime }}\left(m\right),N_B\left(m\right)+N_{B^{\prime }}\left(m\right)\right)$

Pape et al. [24]