Molecular Computing for Markov Chains

TL;DR

This paper introduces a biochemical methodology using chemical reaction networks to implement both discrete and continuous-time Markov chains, enabling their simulation and potential physical realization through DNA reactions.

Contribution

It presents a novel approach to implement Markov chains with biochemical systems, including a compilation scheme for DNA strand displacement for future physical applications.

Findings

Chemical reaction networks can accurately simulate Markov chain dynamics.

The methodology supports both discrete and continuous-time Markov chains.

Simulations validate the correctness and feasibility of the proposed biochemical implementations.

Abstract

In this paper, it is presented a methodology for implementing arbitrarily constructed time-homogenous Markov chains with biochemical systems. Not only discrete but also continuous-time Markov chains are allowed to be computed. By employing chemical reaction networks (CRNs) as a programmable language, molecular concentrations serve to denote both input and output values. One reaction network is elaborately designed for each chain. The evolution of species' concentrations over time well matches the transient solutions of the target continuous-time Markov chain, while equilibrium concentrations can indicate the steady state probabilities. Additionally, second-order Markov chains are considered for implementation, with bimolecular reactions rather that unary ones. An original scheme is put forward to compile unimolecular systems to DNA strand displacement reactions for the sake of future physical implementations. Deterministic, stochastic and DNA simulations are provided to enhance correctness, validity and feasibility.