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

TL;DR

This paper provides analytical solutions for activation cascades in cellular signal transduction, enabling simplified modeling, efficient data fitting, and alternative delay modeling through nonlinear modules involving incomplete gamma functions.

Contribution

It introduces exact nonlinear modules for cascades with optimal gain, allowing coarse-graining and flexible modeling of delays and variability in signal transduction.

Findings

Exact solutions for cascade outputs with identical deactivation rates

Nonlinear modules can replace delay differential equations for delays

Cascade length can be treated as a real-valued parameter for fitting

Abstract

Cellular signal transduction usually involves activation cascades, the sequential activation of a series of proteins following the reception of an input signal. Here we study the classic model of weakly activated cascades and obtain analytical solutions for a variety of inputs. We show that in the special but important case of optimal-gain cascades (i.e., when the deactivation rates are identical) the downstream output of the cascade can be represented exactly as a lumped nonlinear module containing an incomplete gamma function with real parameters that depend on 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 blocks with similar rates can be lumped and represented through our nonlinear modules. Our results can be used both to represent cascades in computational models of differential equations and to fit data efficiently, by reducing the number of equations and parameters involved. 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 show how the obtained nonlinear modules can be used instead of delay differential equations to model delays in signal transduction.