The emergence of genetic coding in physical systems

A simple model of molecular biological translation, based on the classification of polymers as either information carriers or functional catalysts, is used to analyse formal constraints on physical systems which utilise genetic coding. We investigate (i) how the structure-function relationship for coding assignment catalysts constrains the selection of genetic information which can sustain functional self-organisation and (ii) what general prerequisites must be satisfied for selection to give rise to an increase in functional complexity. This is done by considering two separate alphabets and defining the complete set of assignments from letters of one alphabet onto letters from the other. A code is defined as a set of assignments which maps each letter from the first alphabet onto a letter from the second alphabet. We enumerate all the embeddings of the assignment functions in the minimal sequence space of strings of letters from the second alphabet and demonstrate how the embeddings can be classified according to whether they allow different codes to be represented unambiguously in the minimal sequence space of strings of letters from the first alphabet. Non-minimal embeddings are also discussed. Finally, we consider how the mutual specification of letters of the two alphabets and assignment functions can be decomposed into more highly differentiated classes. Only a certain class of embeddings allows coding to be preserved under decomposition. We conclude that the evolution of increasing coding complexity can take place only when special conditions are satisfied regarding the structure-function relationship for the coding assignment catalysts.