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.
Highlights
IntroductionIn its simplest form, an activation cascade comprises a set of components (typically proteins) that become sequentially activated in response to an external stimulus (figure 1)
Activation cascades are pervasive in cellular signal transduction systems [1,2]
We focus on the case when the gain of the cascade is optimal, and find that a lower incomplete gamma function with only three real-valued parameters represents the output of the entire cascade
Summary
In its simplest form, an activation cascade comprises a set of components (typically proteins) that become sequentially activated in response to an external stimulus (figure 1). These systems have been the subject of numerous studies, experimental and theoretical [1,3,4,5,6,7,8,9]. When only one deactivation rate is different, the equations can be reordered, so that a lumped gamma function representation can be used for the block of identical proteins without altering the final output of the cascade. We show how the gamma function representation of a cascade can be used as a computationally efficient replacement of delay differential equations (DDEs)
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