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Synthetic biology aims at designing modular genetic circuits that can beassembled according to the desired function. When embedded in a cell, a circuitmodule becomes a small subnetwork within a larger environmental network, andits dynamics is therefore affected by potentially unknown interactions with theenvironment. It is well-known that the presence of the environment not onlycauses extrinsic noise but also memory effects, which means that the dynamicsof the subnetwork is affected by its past states via a memory function that ischaracteristic of the environment. We study several generic scenarios for thecoupling between a small module and a larger environment, with the environmentconsisting of a chain of mono-molecular reactions. By mapping the dynamics ofthis coupled system onto random walks, we are able to give exact analyticalexpressions for the arising memory functions. Hence, our results give insightsinto the possible types of memory functions and thereby help to better predictsubnetwork dynamics.

Context in Synthetic Biology: Memory Effects of Environments with Mono-molecular Reactions