# Use of the Codon Table to Quantify the Evolutionary Role of Random Mutations
## 4 Discussion
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{6}{c}{**All AA Mutations Included**} & \multicolumn{3}{c}{**AA Mutations Requiring Two/Three RNA Mutations**} \\ \hline  & \multicolumn{3}{c}{**Single Mutations**} & \multicolumn{3}{c}{**Double Mutations**} & \multicolumn{3}{c}{**Single Mutations**} & \multicolumn{3}{c}{**Double Mutations**} \\ \hline
**Code 1** & **Corr\_n 2** & **Corr\_w 2** & **Corr\_n** & **Corr\_w** & **Corr\_n** & **Corr\_w** & **Corr\_n** & **Corr\_w** \\ \hline ALTS910101 3 & 0.508 & 0.540 & 0.080 & \(-\)0.091 & 0.389 & 0.364 & 0.360 & 0.172 \\ \hline BENS940101 4 & 0.488 & 0.626 & 0.073 & \(-\)0.075 & 0.306 & 0.507 & 0.378 & 0.250 \\ \hline BENS940102 5 & 0.469 & 0.577 & 0.051 & \(-\)0.090 & 0.333 & 0.486 & 0.346 & 0.230 \\ \hline BENS940103 6 & 0.447 & 0.518 & \(-\)0.007 & \(-\)0.114 & 0.363 & 0.462 & 0.270 & 0.221 \\ \hline DAYM780301 7 & 0.449 & 0.480 & 0.055 & 0.085 & 0.338 & 0.326 & 0.307 & 0.167 \\ \hline JOND920103 8 & 0.476 & 0.596 & 0.057 & \(-\)0.096 & 0.318 & 0.486 & 0.376 & 0.237 \\ \hline JOND940101 9 & 0.404 & 0.591 & \(-\)0.043 & \(-\)0.111 & 0.169 & 0.444 & 0.216 & 0.243 \\ \hline DAYM780302 10 & 0.562 & 0.565 & 0.071 & \(-\)0.104 & 0.481 & 0.350 & 0.339 & 0.247 \\ \hline HENS920101 11 & 0.487 & 0.491 & \(-\)0.093 & \(-\)0.224 & 0.420 & 0.407 & 0.222 & 0.125 \\ \hline HENS920102 12 & 0.506 & 0.518 & \(-\)0.072 & \(-\)0.206 & 0.440 & 0.450 & 0.235 & 0.139 \\ \hline HENS920103 13 & 0.521 & 0.532 & \(-\)0.061 & \(-\)0.205 & 0.441 & 0.447 & 0.252 & 0.139 \\ \hline HENS920104 14 & 0.483 & 0.498 & \(-\)0.103 & \(-\)0.214 & 0.413 & 0.401 & 0.114 & 0.187 \\ \hline \hline \end{tabular}  
* 1: Code: the AAindex database identifier; 2 Corr\_n, Corr\_w: number of mutation paths calculated with \(W=1\) and \(W=\) cfreq(\(i\),\(j\),\(k\)) \(\times\) cfreq(\(i^{\prime}j^{\prime}j^{\prime}k\)), respectively; 3 PAM-120 [11]; 4 PAM-log-odds\_6.4–8.7 [12]; 5 PAM-log-odds\_22–29 [12]; 6 PAM-log-odds\_74–100 [12]; 7 PAM-250-log-odds\_100 [10]; 8 PAM-250\_PET91 [13]; 9 PAM-250\_TM [13]; 10 PAM-40-log-odds [10]; 11 BLOSUM45 [14]; 12 BLOSUM62 [14]; 13 BLOSUM62 [14]; 13 BLOSUM60 [14]; 14 BLOSUM50 [14].  
\end{table} 
Table 4: Correlation between PAM or BLOSUM matrices and the number of mutation paths. ments (like minimizing the number of RNA mutations, which would result in a drastic AA switch). In the study examining the correlations between the number of RNA mutations per AA mutations and the substitution matrices, the results were different for AA mutations that required single and those that required double RNA mutations. For single RNA mutations, the results showed significant correlations, both when using all matrix elements and when excluding the matrix elements that correspond to AA mutations that cannot be achieved with a single RNA mutation, although the correlations were weaker for the latter. Furthermore, it is gratifying that the correlations increased when the experimental codon usage probabilities were taken into account. The increased correlations provide additional confirmation for the role of random mutations. For the double RNA mutations, however, the results were different. When using all matrix elements and without using frequency weighting, the correlations were small, but at least positive, but they become negative when the usage frequencies were factored in. Omitting the \"forbidden\" AA mutations, contrary to the single RNA mutation case, the correlations increased, but remained small. This suggests that when an alignment indicates AA mutations requiring more than one RNA mutation, it is unlikely to be the result of just a pair of mutations. One explanation for the low or even negative correlations, using the assumption of random double mutations, could be that such AA changes are the result of insertions and/or frame shifts; this could also explain the occurrence of AA mutations requiring three RNA mutations (which is even more unlikely).