## GeneDoc Percent Identity Plots

The percent identity plots in GeneDoc are an alternative to tables showing the percentage of sequence residues that are identical in every pair of sequences in the alignment. Such tables have often been published as a way to describe the degree of divergence among the sequences in the alignment. This has been a useful guide to the probable amount of information available in the alignment and robustness of the alignment. More divergent sequences make the alignments more informative but also less robust and reliable.

The data in these plots can be either strict identities or the count conservative or favorable substitutions as if they were also identical residues. These are the same values that GeneDoc computes for the "Stats" or statistics view where they are shown in the upper right triangular part of the table.

This type of plot is called a cumulative distribution function (c.d.f.) or sometimes an empirical cumulative distribution function. In this context empirical refers to the fact that the plot is experimental, or empirical, data rather than derived from a mathematical equation describing a theoretical distribution such as the gaussian distribution. The plot is created by first sorting the data to be plotted into ascending order. Then for each data point you compute the fraction of data points that have the same or a smaller value. The data is then compressed to eliminate multiple points that have the same data value. The point with the highest value for the fraction of data items with the same or lower value is retained during compression. These compressed data points are the plotted as a step function.

The step function has as its horizontal axis the data values being plotted. The vertical axis is the fraction of data points with as small or smaller a data value. Thus it always ranges from 0.0 to 1.0. Plotting the data as a step function requires drawing two line segments for all of the data point except the first. For the first data point we draw a single vertical line from the horizontal axis up to the first data point. For all subsequent points we draw two line segments. The first line segment is a horizontal segment from the data value of the previous point to the data value of the current point. This line segment is drawn at the fractional value for the previous data point. This indicates the function retains this value until the next observed data value is reached. The second line segment is a vertical segment from the fractional value of the previous data point to the fractional value of the current data point. This gives the plot its characteristic appearance of a set of irregular steps.

The plot is related to the more commonly used histogram plot in a straight forward manner. As the name, c.d.f., implies it is the cumulative plot of the same data plotted in the histogram. The c.d.f. can be thought of as the integral of the distribution plotted in the histogram. This leads many users to ask "Well, why can�t I have the histogram, it is easier to look at?" Our response is that c.d.f.s are also easy to look at, you just have not looked at them enough to be comfortable looking at them yet. More importantly, because drawing a histogram of experimental data, an empirical histogram, always requires putting the data into bins. This process of putting the data into bins always results in the loss of some information that is not lost in the c.d.f.. One piece of information that is immediately obvious from the c.d.f. of any distribution is its median. With a histogram the approximate location of the median is easily seen only for smooth, symmetrical, unimodal, and unskewed distributions, such as the normal distribution.

Another important reason for using c.d.f.s rather than histograms is the Kolmogorov-Smirnov (K-S) test for the difference between distributions. The K-S test determines whether two distributions are different from each other and is performed on their c.d.f.s. Thus it can easily be presented in terms of the c.d.f.s, but not in terms of the histograms. GeneDoc uses this test in several places. It is used in conjunction with alignment scores to test whether the groups you have defined are statistically distinct. It does this by comparing the distribution of scores between pairs of sequences within groups with the distribution of scores between pairs of sequence in different groups. If the groups are distinct the scores for pairs of sequences in different groups are larger than the scores for pairs of sequences within the same group.

GeneDoc also uses the K-S test to tell you if the sequences in one group are more diverged from each other than are the sequences in a second group. The K-S test allows GeneDoc to compute a significance probability to tell you how likely the observed difference in the c.d.f.s is to have occurred by chance in groups of the same size. The K-S test is a useful general purpose test for differences between two groups because it measures all aspects of the differences between groups. That is, the K-S test will pick up differences in the shape of the c.d.f.s of the two groups as well as differences in the average of two groups.

More commonly used test such as Students t test and the F test detect differences only in the average or the variance of the groups respectively.

The percent identity plot is also useful for selecting a scoring matrix to use with the alignment. Particularly for protein sequence alignments the scoring matrix is important both for computing alignment scores and for determining which amino acids are included in the equivalence groups defined and used in some GeneDoc functions. The PAM matrices for both proteins and nucleic acids and the Blosum matrices for proteins are tuned for use with groups of sequences that have diverged to a different degree. The right hand column in the table below shows the degree of divergence appropriate for different PAM matrices shown in the middle column. The left hand column identifies the Blosum matrix that supplies the same amount of entropy per position in an alignment. Matrices that show the same amount of entropy per position in an alignment are suitable for sequences of the same degree of divergence.

Comparable Blosum and PAM Scoring Matrices based on Equivalent Entropy

 Blosum Table Degree of Identity Entropy of the Blosum Table PAM Value of the Table Entropy of the PAM Table Percent Sequence Identity of the PAM Table Blosum 90 1.18 PAM 100 1.18 43 Blosum 80 0.99 PAM 120 0.98 38 Blosum 60 0.66 PAM 160 0.70 30 Blosum 52 0.52 PAM 200 0.51 25 Blosum 45 0.38 PAM 250 0.36 20