Some mathematical refinements concerning error minimization in the genetic code

TL;DR

This paper formulates the error minimization in the genetic code as a Quadratic Assignment Problem, verifies the global optimality of a heuristic code, and demonstrates that the genetic code's error robustness is significant even in larger, more realistic code spaces.

Contribution

It introduces a new larger code space based on wobble rules, formulates the problem as a Quadratic Assignment Problem, and compares error robustness across these expanded spaces.

Findings

The heuristic code is the global optimum in the formulated problem.

The genetic code is more error robust than random codes in larger, realistic spaces.

Expanding the code space affects the perceived optimality and robustness of the genetic code.

Abstract

The genetic code 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 thus verify that the code found by the heuristic 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. 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. We point out that these results do not necessarily imply that the code was optimized during evolution for error minimization, but that other mechanisms could explain this error robustness.