Molecular Synthesis for Probability Theory and Stochastic Process

It is common that probability theory and stochastic process, especially Markov chains, have long been used to study and explain the behaviors of chemical reaction networks (CRNs). Nonetheless, this paper sees things from a reverse angle, devoting itself in synthesizing common probability theory and stochastic process with CRNs. The main motivation is to imitate and explore the evolution of large-scale and complex practical systems based on CRNs, by making use of the inherent parallelism and randomness. In our conference paper, a preliminary but concise approach has been put forward to synthesize the stand-alone examples such as law of total probability, Bayes’ theorem, and n-step transition of Markov chains. To make this methodology systematic and theoretically sound, we enrich and offer more solid foundation for the previous version. Rigorous stability analysis based on ordinary differential equations (ODEs) are provided. This paper further deeply discusses and distinguishes building stochastic models for CRNs and utilizing CRNs to solve stochastic problems. A joint distribution of Markov chains are implemented using molecular reactions as a showcase. In order to enhance the clearness of the results, all the simulations are done according to deterministic mass action. It is worth noting that an already mathematically proven conclusion, which states that nearly an arbitrary set of bimolecular or unimolecular reactions can be implemented by DNA strand displacement reactions, ensures the meaningfulness of our work. It is also believed that, though in its infancy, the proposed approach is also valid for other molecular synthesis than DNA, as long as the kinetics constraints are met.