 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.
<clarity>  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 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 , as well as parameters of the input signal. The expressions obtained can be applied to the non-identical case when the deactivation rates are random to capture the variability in the cascade outputs. We also show that cascades can be rearranged so that homogeneous blocks can be lumped and represented by incomplete gamma functions.