Box 1. Kinetic analysis of RFI action.

Box 1 Dynamics of gene expression: The expression of a gene into its protein is determined by four processes: transcription, translation, mRNA degradation, and protein degradation (schematically depicted above). As illustrated in panel A, the dynamics of gene expression may be described by two differential equations incorporating these four reactions. The steady-state protein concentration, P_SS, of a gene product is given by:

If a gene is regulated at the transcriptional level (i.e. if the transcription rate is changed to k₁ at t=0) the time course of protein expression P(t) is given by:

Thus, the response time, defined as the time required to reach the new steady state, is solely determined by the decay rates. The response time depends on both, d₁ and d₂, if the protein and the mRNA half-lives are of similar magnitude, while it is mainly set by the slowest decay in case mRNA and protein stability differ significantly from each other. This implies that for transcriptional regulation, both the mRNA and the protein have to be unstable to attain a new steady state rapidly.
Unstable proteins and mRNAs need higher translation or transcription rates, respectively, to reach the same steady-state protein concentration (equation (1)). Therefore, their production consumes more free energy, as the energy expenditure is proportional to the transcription and translation rates (k₁ and k₂[ mRNA] ). Thus, a trade-off exists between making the protein network flexible (by increasing d₁ and d₂, and simultaneously increasing k₁ and k₂ to maintain the expression level), and making it energy efficient (by decreasing k₁ or k₂ and, to compensate, simultaneously decreasing d₁ or d₂).
Transcriptional regulation and the dynamics of signal transduction: The activation of signalling networks can be modulated by transcriptional regulation of the concentrations of their components. The time required to attain a new signalling steady state defined by transcriptional regulation of a signal inhibitor is determined by the stability of the signal inhibitor mRNA and protein (see above). The behaviour is slightly more complex if feedback is involved: a negative feedback system subjected to activation reaches a steady state faster than expected from the decay rates of the feedback regulator, while no such acceleration is observed upon deactivation (Alon, 2007). Rapid transcriptional feedback regulation of the signalling network requires that both the mRNA and the protein of the transcriptional feedback regulator need to be unstable, since otherwise: (i) feedback induction upon stimulus addition implies continuously increasing feedback strength over many hours and (ii) long latency will be observed upon stimulus removal.
Transcriptional negative regulation of the signalling network can, in principle, occur by upregulation of signal inhibitors or by downregulation of signal transducers. In the following, we compare the dynamic behaviour of a generic protein kinase cascade for three different transcriptional regulatory designs to get insights into kinetic implications of RFI action: (i) repression of a kinase acting as a signal transducer (panel B, left); (ii) induction of a phosphatase acting as a catalytic RFI (panel B, middle); and (iii) induction of a stoichiometric inhibitor acting as a non-catalytic RFI (panel B, right).
In a weakly activated phosphorylation/dephosphorylation cycle (modelled with linear kinetics), the amount of active phosphorylated protein at steady state is proportional to the ratio of kinase to phosphatase concentrations (Heinrich et al, 2002). Thus, the signal can be reduced to 10% of its original value, either by reducing kinase expression to 10% or by a 10-fold phosphatase upregulation. The figure above shows how the signal cascade activation level (i.e. the ratio of kinase and phosphatase activities) follows a slow change in kinase or phosphatase expression, if modelled according to equation (2) (with d₁=2/h and d₂=1/h). A 10-fold phosphatase upregulation allows to switch off the signal much more quickly (middle graph, solid line) when compared to 10-fold kinase downregulation (panel B, left graph, solid line). We also analysed the recovery time after removal of the external activation if the kinase and phosphatase expression are regulated in the opposite direction. In this case, kinase upregulation (left graph, dashed line) allows for faster disappearance of the signalling than phosphatase downregulation (middle graph, dashed line). Thus, the signalling activity immediately follows transcriptional regulation of kinase expression (due to direct proportionality), while phosphatases regulate signalling pathways asymmetrically, with a long latency for recovery (this is due to the inverse proportionality). Similar conclusions also hold for strongly activated kinase cascades, although the difference between phosphatase and kinase regulation becomes less pronounced (not shown).
Several RFIs act as stoichiometric inhibitors, that is, they inhibit signal transduction non-catalytically by binding reversibly to their targets (as depicted schematically in the figure, right). We analysed a limiting case of stoichiometric inhibition, where the inhibitor binds to a kinase with very high affinity. Then, all available inhibitor I will be bound, unless the inhibitor is in present in excess over its target. Thus, the free, active concentration K of the targeted kinase with the total concentration K_T is given by:

The cascade activity was assumed to be proportional to the free kinase concentration K (see above), and was analysed for slow inhibitor up- and downregulation according to equation (2) (panel B, right; d₁=2/h and d₂=1/h). The change in the signal level (again 10-fold ultimately) immediately follows alterations in inhibitor protein expression. This statement holds true for as long as the inhibitor is not induced too strongly. Otherwise, the concentration I exceeds K_T, so that the system shows some latency before it recovers. In any case, the signalling dynamics in response to inhibitor regulation do not differ from those observed upon kinase regulation (compare left and right graphs on panel B).