Amplification

As we did with the Hill coefficient, we analyse the amplification for each step in the operating range. The whole amplification is then the product of the amplification at each step of the cascade:

$${\mathrm{d} \mathit{MAPK}\over \mathrm{d}\mathit{Input}} = ... ...}\, {\partial \mathit{MAPKKK} \over\partial \mathit{Input} }\,$$ (7)

So the whole amplification does not only depend on the maximal amplification in each step, but also on the operating range of the kinase. In case of a sigmoidal Hill-like response of each step, the amplification is maximal, when the substrate starts to get saturated. If the response is rather Michaelis-Menten-like (Hill coefficient of 1) than maximal amplification is provided in case of very low saturated kinase. Maximal amplification of the entire cascade can be reached by adjusting the operating range of the first step be in very low saturation and for the second step in middle activation. In Fig. X we see the amplification of all three steps and the amplification of the entire system for both models.

The main difference of both systems is that in the model according to Huang and Ferrell the maximal amplification is far much higher ($10^6$) than in the model according to Bhalla and Iyengar ($4$). The amplification of all steps are much higher in the model of Huang and Ferrell, since the velocity of dephosphorylation is lower, caused by lower concentrations of phosphatasis. Additionally the first step of the model according to Bhalla and Iyengar saturates in the working range.

Thus, the different amplification is due to saturation of the first step and small concatenations of phosphatasis.

Figure 4: The upper row shows amplification in the 3 steps and of the total system in the model according to Huang/Ferrell. The second row shows the same in the system according to Bhalla/Iyengar. Stimuli which result in $10\%$ and $90\%$ MAPK activation are marked with a vertical grid line.