Activation cascadesare a prevalent feature in cellular mechanisms for signaltransduction . 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 .
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 .
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Activation cascadesare a prevalent feature in cellular mechanisms for signaltransduction
Cellular signal transduction usually involves activation cascades, the sequential activation of a series of proteins following the reception of an input signal
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linear activation
weakly activated
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in terms of lower incomplete gamma functions
for a variety of inputs
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optimal gain
optimal-gain
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all
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an entire
the
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single
lumped
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parameters dependent on the input signal as well as the
real parameters that depend on the
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. 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
, 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
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homogeneous blocks
blocks with similar rates
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by incomplete gamma functions. We also illustrate how the reduced representation
through our nonlinear modules. Our results
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to fit data; in
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
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use our results to
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output of delay differential equation models can be approximated with the use of simple expressions involving the incomplete gamma function
obtained nonlinear modules can be used instead of delay differential equations to model delays in signal transduction