Optimal information transmission in a non-Markovian auto-regulatory gene expression system

Gene expression is a multistep and inherently noisy process. Such multistep reactions introduces memory between molecular events, giving rise to non-Markovian dynamics. Simultaneously, noise in the gene regulatory network limits the fidelity with which input signals can be transmitted to gene expression outputs. These considerations raise a key unresolved question: how does molecular memory influence the maximum information transmission capacity of regulatory networks under varying physical conditions? To address this, we develop a theoretical framework for a non-Markovian auto-regulatory gene expression system with arbitrary feedback form. By introducing effective transition rates, we transform the original non-Markovian model into an equivalent Markovian formulation, enabling analytical treatment. We find that non-Markovianity confers a significantly stronger regulatory capacity than feedback alone. In monostable regimes, feedback strength shapes the pattern of optimal information transmission, giving rise to monotonic or non-monotonic dependencies on memory. In bistable regimes, increasing non-Markovianity leads to a consistent decrease in the upper bound of information transmission. Beyond these results, our approach establishes a generalizable strategy for analyzing information flow in complex gene regulatory systems involving memory, feedback, and noise.