Activation cascades are prevalent in cell signalling mechanisms . We study the classic model of linear activation cascades and find that in special but important cases the output of an entire cascade can be represented analytically as a function of the input and a lower incomplete gamma function. We also show that if the inactivation rate of a single component is altered, the change induced at the output is independent of the position in the cascade of the modified component. We use our analytical results to show how one can reduce the number of equations and parameters in ODE models of cell signalling cascades, and how delay differential equation models can sometimes be approximated through the use of simple expressions involving the incomplete gamma function.
Activation cascades are a prevalent feature in cellular mechanisms for signal transduction. Here we study the classic model of linear activation cascades and obtain analytical solutions in terms of lower incomplete gamma functions. We show that in the special but important case of optimal gain cascades (i.e., when all the deactivation rates are identical) the downstream output of an entire cascade can be represented exactly as a single nonlinear module containing an incomplete gamma function with parameters dependent on the input signal as well as the rates and length of the cascade. Our results can be used to represent optimal cascades efficiently by reducing the number of equations and parameters in computational ODE models under a variety of inputs. If the requirement for strict optimality is relaxed (under random deactivation rates), we show that the reduced representation can also reproduce the observed variability of downstream responses. In addition, we show that cascades can be rearranged so that homogeneous blocks can be lumped and represented by incomplete gamma functions. We also illustrate how the reduced representation can be used to fit data; in particular, the length of the cascade appears as a real-valued parameter and can thus be fitted in the same manner as Hill coefficients. Finally, we use our results to show how the output of delay differential equation models can be approximated with the use of simple expressions involving the incomplete gamma function.
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