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# skbio.sequence.distance.tn93

skbio.sequence.distance.tn93(seq1, seq2, freqs=None, gamma=None)

  Calculate the TN93 distance between two aligned nucleotide sequences.

  Added in version 0.7.2.

  The Tamura and Nei 1993 (TN93) model assumes differential rates of the two types of transitions: between purines (R) (i.e., A <-> G) and between pyrimidines (Y) (i.e., C <-> T/U), and transversions (i.e., between a purine and a pyrimidine). It also allows varying base frequencies (\(\pi\)). The distance is calculated as:

  \[D = -2A\, ln\left(1-\frac{P_1}{2A}-\frac{Q}{2\pi_R}\right) -2B\, ln\left(1-\frac{P_2}{2B}-\frac{Q}{2\pi_Y}\right) -2C\, ln\left(1-\frac{Q}{2\pi_R\pi_Y}\right)\]

  Where \(P_1\) and \(P_2\) are the proportions of purine and pyrimidine transitions, respectively. \(Q\) is the proportion of transversions. And:

  \[\begin{split}\begin{aligned} &A = \frac{\pi_A\pi_G}{\pi_R} \\ &B = \frac{\pi_C\pi_T}{\pi_Y} \\ &C = \pi_R\pi_Y - A\pi_Y - B\pi_R \end{aligned}\end{split}\]

  The TN93 model can be corrected for site-rate heterogeneity by assuming that evolutionary rates follow a gamma distribution:

  \[\begin{split}\begin{aligned} D = 2\alpha\Bigg[ &A\left(1-\frac{P_1}{2A}-\frac{Q}{2\pi_R}\right)^{-\frac{1}{\alpha}} +B\left(1-\frac{P_2}{2B}-\frac{Q}{2\pi_Y}\right)^{-\frac{1}{\alpha}} \\ &\quad +\; C\left(1-\frac{Q}{2\pi_R\pi_Y}\right)^{-\frac{1}{\alpha}} - E\Bigg] \end{aligned}\end{split}\]

  Where \(\alpha > 0\) is the shape parameter of the gamma distribution. And:

  \[E = \pi_A\pi_G + \pi_T\pi_C + \pi_R\pi_Y\]
  Parameters:
    seq1, seq2{DNA, RNA}

      Sequences to compute the TN93 distance between.

    freqs array_like of float of shape (4,), optional

      Relative frequencies of nucleobases A, C, G, and T/U, respectively. Should sum to 1. If not provided, the observed frequencies from the two input sequences combined will be used.

    gamma float, optional

      Shape parameter (\(\alpha\)) of the gamma distribution for among-site rate heterogeneity. Must be a positive number. If not provided, no gamma correction will be applied.

      Added in version 0.7.3.

  Returns:
    float

      TN93 distance between the two sequences.

  See also

  `k2p`
  `f84`

  Notes

  The Tamura and Nei 1993 (TN93) model was originally described in [1] alongside with its gamma correction formula.

  This function returns NaN if any of the three logarithm arguments is zero or negative, which implicates over-saturation of substitutions.

  References

  [1]

  Tamura, K., & Nei, M. (1993). Estimation of the number of nucleotide substitutions in the control region of mitochondrial DNA in humans and chimpanzees. Molecular Biology and Evolution, 10(3), 512-526.