  Synthetic biology aims at designing modular genetic circuits that can be
assembled according to the desired function. When embedded in a cell, a circuit
module becomes a small subnetwork within a larger environmental network, and
its dynamics is therefore affected by potentially unknown interactions with the
environment. It is well-known that the presence of the environment not only
causes extrinsic noise but also memory effects, which means that the dynamics
of the subnetwork is affected by its past states via a memory function that is
characteristic of the environment. We study several generic scenarios for the
coupling between a small module and a larger environment, with the environment
consisting of a chain of mono-molecular reactions. By mapping the dynamics of
this coupled system onto random walks, we are able to give exact analytical
expressions for the arising memory functions. Hence, our results give insights
into the possible types of memory functions and thereby help to better predict
subnetwork dynamics.
