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We show that in the generic situation where a biological network, e.g. aprotein interaction network, is in fact a subnetwork embedded in a larger"bulk" network, the presence of the bulk causes not just extrinsic noise butalso memory effects. This means that the dynamics of the subnetwork will dependnot only on its present state, but also its past. We use projection techniquesto get explicit expressions for the memory functions that encode such memoryeffects, for generic protein interaction networks involving binary and unaryreactions such as complex formation and phosphorylation, respectively.Remarkably, in the limit of low intrinsic copy-number noise such expressionscan be obtained even for nonlinear dependences on the past. We illustrate themethod with examples from a protein interaction network around epidermal growthfactor receptor (EGFR), which is relevant to cancer signalling. These examplesdemonstrate that inclusion of memory terms is not only important conceptuallybut also leads to substantially higher quantitative accuracy in the predictedsubnetwork dynamics.

Memory effects in biochemical networks as the natural counterpart of extrinsic noise