Markov modeling of DNA replication
This attached function seeks to demonstrate the evolutionary advantage of left-right symmetry breaking in the kinetic parameters of DNA replica construction on top of a template strand. The competition between the requirement of low kinetic barrier for monomer induction and the high kinetic barrier for monomer retention leads to the symmetry breaking.
There are two time-scales in the problem: The rate of H-bonding/dissociation between the free monomers and the template strand, and the rate of covalent bond formation between monomers on the replica strand. When the monomer supply is abundant and consequently, the H-bonding rates are high, there is no evolutionary pressure. But when the supply is scarce, the H-bonding rates are low and the ability to retain attached monomers becomes the deciding factor for successful replication. The DNA replication is modeled as a Markov chain below. '0' represents the absence of a monomer and '1', its presence. The script models the growth of a 5-nt long template. Covalent bond formation is not explicitly included in the chain. It is factored in indirectly as the retention-advantage factor below, calculated through the probability for the chain to stay in the '11111' state.
The DNA replica strand construction is assumed to take place cooperatively, with the neighboring monomers hydrogen-bonded to the template influencing the rate of monomer attachment. This rate is not symmetric w.r.t the left and right neighbors, and can be different. This difference is encoded in the variable 'a' below, which can be varied from zero (infinitely high kinetic barrier) to infinity (instantaneous formation of H-bond). The output produced by the script would show that it is evolutionarily advantageous for the DNA heteropolymer to have its left-right symmetry broken, with low barriers to the right (left) for rapid induction of monomers and high barriers to the left (right) for retention of the monomers attached to the template. The variable names are nearly self-explanatory.
For more information on the model, please refer to "H. Subramanian, R. A. Gatenby, 'Evolutionary advantage of directional symmetry breaking in self-replicating polymers', Journal of Theoretical Biology, vol. 446, pp. 128–136 (2018)".