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Cellular signal transduction usually involves activation cascades, thesequential activation of a series of proteins following the reception of aninput signal. Here we study the classic model of weakly activated cascades andobtain analytical solutions for a variety of inputs. We show that in thespecial but important case of optimal-gain cascades (i.e., when thedeactivation rates are identical) the downstream output of the cascade can berepresented exactly as a lumped nonlinear module containing an incomplete gammafunction with real parameters that depend on the rates and length of thecascade, as well as parameters of the input signal. The expressions obtainedcan be applied to the non-identical case when the deactivation rates are randomto capture the variability in the cascade outputs. We also show that cascadescan be rearranged so that blocks with similar rates can be lumped andrepresented through our nonlinear modules. Our results can be used both torepresent cascades in computational models of differential equations and to fitdata efficiently, by reducing the number of equations and parameters involved.In particular, the length of the cascade appears as a real-valued parameter andcan thus be fitted in the same manner as Hill coefficients. Finally, we showhow the obtained nonlinear modules can be used instead of delay differentialequations to model delays in signal transduction.

Linear models of activation cascades: analytical solutions and coarse-graining of delayed signal transduction