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