Some mathematical refinements concerning error minimization in the genetic code
The genetic code is known to have a high level of error robustness and has been shown to be very error robust compared to randomly selected codes, but to be significantly less error robust than a certain code found by a heuristic algorithm. We formulate this optimisation problem as a Quadratic Assignment Problem and use this to formally verify that the code found by the heuristic algorithm is the global optimum. We also argue that it is strongly misleading to compare the genetic code only with codes sampled from the fixed block model, because the real code space is orders of magnitude larger. This has consequences for the kind of mechanisms we envisage as having shaped the genetic code. We thus enlarge the space from which random codes can be sampled from approximately 2.433 x 10^18 codes to approximately 5.908 x 10^45 codes. We do this by leaving the fixed block model, and using the wobble rules to formulate the characteristics acceptable for a genetic code. By relaxing more constraints three larger spaces are also constructed. Using a modified error function, the genetic code is found to be more error robust compared to a background of randomly generated codes with increasing space size.