The MAPK-cascade as a switch

The steady-state curves of both models show sigmoidal input response curves in the log-lin-plot (see Fig.2). The model according to Huang and Ferrell shows increasing switch-like behavior as one goes downstream in the cascade, and the stimulus needed to have a response of $50\%$ of activated kinase ($C50$) decreases. In the model according to Bhalla and Iyengar the sigmoidality increases only slightly from the first to the next step and the $C50$-value even increases.

To quantify how good a switch is realized, we used the Hill coefficient, which was originally introduced to describe cooperativity in enzyme kinetics. For a Hill function

$$A^*=A_{max}{E^h \over K_{0.5}^h + E^h}$$ (3)

where $A^{*}$ is the amount of activated substrate, $A_{max}$ is the total amount of substrate, the Hill coefficient can be obtained by

$$h={\log 81 \over \log C90/C10}$$ (4)

$C10$ and $C90$ are the stimulus concentrations where $10\%$ and $90\%$ of substrate are activated. Even though the sigmoidal curves in Fig. 2 are not true Hill functions we estimate the Hill coefficient from the $C10$ and $C90$ values using Eq. (4).

The steady-state of activated MAPK cascade can be treated as a three-step chain of functions:

$$\mathrm{MAPK^{**}}\left(\mathrm{Input}\right)=\mathrm{MAPK^{... ...\mathrm{MAPKKK^{*}} \left(\mathrm{Input}\right)\right)\right).$$ (5)

Ferrells simulations show, that the two-step phosphorylation of MAPKK and MAPK makes the signal-response-curves of the 2nd and 3rd layer of MAPK-cascade sigmoidal and Hill coefficients of 4 can be reached for each step [7]. However, for the mono phosphorylated MAPKKK the Hill-coefficient is approximately 1. It was shown that the maximum Hill coefficient of a cascade cannot exceed the product of the Hill coefficient of each step [5,7]:

$$h_{total}\le h_\mathrm{MAPKKK}\; h_\mathrm{MAPKK}\; h_\mathrm{MAPK}$$ (6)

So for the whole cascade only the 2nd and 3rd layer can contribute to the switch-like behavior of the entire system. We discuss in the following the operating-ranges of the upstream kinases to produce up to $90\%$ of activated MAPK. If all three functions are described by Hill curves, concatenation is only well approximated by a Hill curve, if both $\mathrm{MAPKK}^{**}$ and $\mathrm{MAPKKK}^{*}$ do not saturate in their operating-ranges. Saturation of activation is in this context not primary reached by saturation of the enzyme, but by the lack of unactivated substrate [8]. If saturation is avoided, the first layer can be approximated by a linear curve and the second layer by a power. In this case the Hill coefficient is maximal [6].

To understand how the switch-like behavior of the whole cascade is generated, we plot the response curve of all three layers in such a way, that the axis showing the output of the first layer (activated MAPKKK) is identical to the axis showing the input of the second layer (MAPKKK acts as input for the second layer) etc. This allows us to read off the Hill coefficient and operation ranges by drawing two lines in this graph corresponding to $10\%$ and $90\%$ of activated MAPK.

Figure 3: The realization of switch-like output in the models according to Bhalla/Iyengar (left graph) and Huang and Ferrell (right graph). The Hill coefficients of the first step (MAPKKK) are 1 for both models. The Hill coefficients of the second(third) step are $1.5$ ($3.4$) in the left and $2.7$ ($2.7$) in the right graph.

As one can see in Fig. 3, in the model published by Bhalla and Iyengar [4] the activation of MAPKK starts to saturate, when $10\%$ of MAPK is activated. Also the first layer (MAPKKK) saturates. So the Hill coefficient of the whole system is much lower than the product of Hill coefficients of the three layers ( $2.2\le 1 * 1.5 * 2.7$).

In the model according to Huang and Ferrell [9] the situation is different. The activation of MAPKKK is almost a linear function of input in the operating-range, and MAPKK activation is nearly unsaturated. So the Hill coefficient of the entire system ($5.8$) is close to the maximum of the product of the 2nd and 3rd step ( $2.7 * 2.7 = 7.3$).

Consequently, the pronounced differences of the signal-response curves in Fig. 2 can be traced back to different operating-ranges of the layers.