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Local secondary structure

A separate $\chi^2$ analysis was performed to examine the distribution of amino acid patterns in different secondary structural contexts, as defined by the program DSSP[Kabsch & Sander, 1983] from atomic coordinates. This statistic will be referred to as $\chi_d^2$ (`d' for DSSP). For each pattern occurrence in the domain sequence, corresponding single character DSSP secondary structure definitions (character column 17 of the output) were retrieved. For an (i,i+3) pattern, two consecutive DSSP characters were extracted at positions i+1 and i+2. The frequency of occurrence of these two-character DSSP strings were then calculated for each amino acid pattern. Four character strings (i to i+3) were not used because too many sparsely populated combinations exist. In this study, the $\chi_d^2$ analysis is performed only for the patterns which have previously been identified using the $\chi _t^2$ analysis. Since the identity of these patterns is known a priori, the $P'$ adjustment (Equation 3.2) is not necessary. The number of degrees of freedom are large for these analyses (around 55 different two-character DSSP strings are observed). For many degrees of freedom (say, $f>30$) the value of $d$:

$$d = \sqrt{2\chi^2} - \sqrt{2f-1}$$ (4)

is approximately normally distributed around a mean of zero with unit standard deviation[Bailey, 1981]. The higher the value of $d$, the more certain one can be that the difference between observed and expected values has not occurred by chance. Values of $d>3$ are significant at the level of $P<0.001$ (one-tailed test).