The real game of life

I wanted to test some programming of Markov models and the creation of a biochemical nanomachine system from some fairly rudimentary elements that would be self replicating. The premise of the game or program or actual system is to devise and identify an amino acid sequence or a Markov chain that made a specific set of proteins in conjunction with a ligase which also could be a several step chain.
The process operates as this: A catalyst acts in a specific chemical environment that may be basic or acidic. It connects amino acids in a continuous sequence at random based on the probability distribution of the amino acids. This results in a very long chain of amino acids that continues to grow at some very slow rate. Due to ionizing radiation or mechanical stress it starts to split and produce fragments. One of those fragments is capable of splitting amino acid bonds. As a result it starts shredding all the amino threads until it creates sequences which catalyze chains and also break chains.

The game is to devise all the parametric equations, chains and Michaelis–Menten kinetics of the whole process.

It would seem that since a digestive enzyme breaks bonds that it would actually consume itself as well as the generative proteins. As a result the process of consumption would be self limiting as it would form a closed loop which had negative feedback.
The generative protein would appear again and again and eventually be reduced. It would also be limited by the concentration of amino acids and diffusion. The increase in proteins would serve as a sieve and also as a protection against consumption.

The end result is a system that would be a seed that served as a terra-forming agent. If water is present and amino acids it would proceed to generation of a soup that was a mixture of sequences that were self catalytic.
When consumption and creation are combined with random process the end result spreads to its maximum extent.

I wanted to have a program that identified the catalyst sequence , the first random sequence that could be a digestive enzyme in an acidic environment and thus determine what environments would naturally make this first step.
I believe that this experiment was tried with lightning ( electricity ) and got nowhere. I would say that the key element to all further evolution is the presence of the first catalyst. I presume that some sequence of amino acids will catalyze amino acids in different environments.

It would seem that by generating a DNA sequence that can be incorporated in a cell by electroporation, that it would continuously produce the core protein and allow testing of the hypothesis based upon the model.

It could be extended to identify other Markov chains in the concentration of different starting chemicals. I suspect that many different chemicals can serve as assembler-destructor pairs.