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Subnetworks. In this thesis we use the projection method to obtain a set of equations for a subnetwork embedded in a larger protein interaction network. Below we review other methods of model reduction that attempt to simplify an initial large model down to a subnetwork description. Sensitivity analysis tries to determine which molecular species are insigni cant to the dynamic system of interest [31]. A parameter is classi ed as insigni cant if it has a low sensitivity, in that its precise value does not have a large e ect on the concentrations of the rest of the species in the network. If we consider a reaction system ∂ x = Sfx = g(x) (2.3.1) ∂t then we can determine the importance of species i by considering Bi = (2.3.2) Σ ∂ xx xx 0 j ∂ ln x i where j indexes the components of x. Then any species i with sensitivity Bi < B, where B is a parameter re ecting the desired accuracy of the model, can be considered to have a low sensitivity. Low sensitivity parameters are then eliminated or replaced by a smaller number of e ective species. However, sometimes it is necessary to keep a low sensitivity parameter to ensure the results are biologically valid. This approach will not work with a predetermined subnetwork as that subnetwork may contain a mixture of low and high sensitivity parameters and therefore the subnetwork found using sensitivity analysis is e ectively chosen by the method. Timescale separation techniques are used to focus on the species that contribute most to the long-time dynamics of a system, by removing molecular species whose dynamics takes place on much shorter timescales. This is reasonable because biochemical processes occur on a range of timescales; changes in gene expression levels, for example, may take place over hours whereas protein signalling takes seconds. Timescale separation approaches have been used by x.x. Xxxxxxxx [32] and Xxxxxx et al. [33], with the subnetwork then containing all the slow molecular species and the bulk the fast ones. Thus, while these authors used projection techniques as we do, memory e ects did not arise: they become negligible if the bulk is fast enough to respond e ectively instantaneously on the timescale of the subnetwork dynamics to the state of the subnetwork. Here we consider signalling networks where the timescales of the dynamics of the subnetwork and the bulk are comparable, so that timescale separation methods are not directly applicable. Another way to reduce the system is to split it into modules where e...
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