  Double phosphorylation of protein kinases is a common feature of signalling
cascades. This motif may reduce cross-talk between signalling pathways, as the
second phosphorylation site allows for proofreading, especially when
phosphorylation is distributive rather than processive. Recent studies suggest
that phosphorylation can be `pseudo-processive' in the crowded cellular
environment, as rebinding after the first phosphorylation is enhanced by slow
diffusion. Here, we use a simple model with unsaturated reactants to show that
specificity for one substrate over another drops as rebinding increases and
pseudo-processive behavior becomes possible. However, this loss of specificity
with increased rebinding is typically also observed if two distinct enzyme
species are required for phosphorylation, i.e. when the system is necessarily
distributive. Thus the loss of specificity is due to an intrinsic reduction in
selectivity with increased rebinding, which benefits inefficient reactions,
rather than pseudo-processivity itself. We also show that proofreading can
remain effective when the intended signalling pathway exhibits high levels of
rebinding-induced pseudo-processivity, unlike other proposed advantages of the
dual phosphorylation motif.
