Long signaling cascades tend to attenuate retroactivity

Abstract

Signaling pathways consisting of phosphorylation/dephosphorylation cycles with no explicit feedback allow signals to propagate not only from upstream to downstream but also from downstream to upstream due to retroactivity at the interconnection between phosphorylation/dephosphorylation cycles. However, the extent to which a downstream perturbation can propagate upstream in a signaling cascade and the parameters that affect this propagation are presently unknown. Here, we determine the downstream-to-upstream steady-state gain at each stage of the signaling cascade as a function of the cascade parameters. This gain can be made smaller than 1 (attenuation) by sufficiently fast kinase rates compared to the phosphatase rates and/or by sufficiently large Michaelis-Menten constants and sufficiently low amounts of total stage protein. Numerical studies performed on sets of biologically relevant parameters indicated that ∼50% of these parameters could give rise to amplification of the downstream perturbation at some stage in a three-stage cascade. In an n-stage cascade, the percentage of parameters that lead to an overall attenuation from the last stage to the first stage monotonically increases with the cascade length n and reaches 100% for cascades of length at least 6.

Figure 1

A signaling cascade with n stages of PD cycles. The phosphorylated protein W^∗_i–1 of stage i–1 functions as a kinase for protein W_i of the next stage downstream. Dephosphorylation is brought about by the phosphatase E_i. A downstream perturbation in the concentration of D, in which D can be a substrate shared with other signaling pathways or an inhibitor of the active enzyme W^∗_n, results in a perturbation of protein concentration in all upstream stages.

Figure 2

A block diagram representation of the steady-state response of stage i to a small downstream perturbation in D_T. The downstream perturbation propagates upstream through perturbations x_i in the complexes of active proteins with their downstream substrates.

Figure 3

Attenuation and sign-reversal in a three-stage cascade. The x axis shows the value of the perturbation d_T and the y axis shows the steady-state value of the resulting perturbations w^∗₁, w^∗₂, and w^∗₃. Simulation is performed on the full nonlinear ODE model given by Eq. 2. The parameters of each stage i are taken from Huang and Ferrell (28) and are given by k_i = 150 (min)⁻¹, ${\overline{k}}_i=150{\left(\min \right)}^{-1}$, α_i = 2.5 (nM min)⁻¹, ${\overline{a}}_i=600{\left(\min \right)}^{-1}$, b_i = 2.5 (nM min)⁻¹, ${\overline{b}}_i=600{\left(\min \right)}^{-1}$, E_3T = 120 nM, E_2T = 0.3 nM, E_1T = 0.3 nM, W_3T = 1200 nM, W_2T = 1200 nM, W_1T = 3 nM, ${\overline{W}}_0^{\ast }=0.3\text{nM}$, and ${\overline{D}}_T=0\text{nM}$. As a result, K_i = 300 nM and ${\overline{K}}_i=300\text{nM}$.

Figure 4

Amplification in a three-stage cascade. Numerical simulation of system in Eq. 2: value of |w^∗_i| for i ∈{1,…,n} in response to a unit perturbation d_T = 1. This plot shows that violation of the necessary condition leads to amplification of the downstream perturbation as it transfers upstream in the cascade. Parameters of stage i are given by: k_i = 150 (min)⁻¹, ${\overline{k}}_i=150{\left(\min \right)}^{-1}$, a_i = 2500 (nM min)⁻¹, ${\overline{a}}_i=600{\left(\min \right)}^{-1}$, b_i = 2500 (nM min)⁻¹, ${\overline{b}}_i=600{\left(\min \right)}^{-1}$, E_3T = 120 nM, E_2T = 30 nM, E_1T = 0.3 nM, W_3T = 3 nM, W_2T = 30 nM, W_1T = 1200 nM, ${\overline{W}}_0^{\ast }=0.3\text{nM}$, and ${\overline{D}}_T=0.9\text{nM}$.

Figure 5

Percentage of simulations with overall attenuation (Ψ_tot < 1) as a function of the number of stages in a cascade with parameters randomly selected from the intervals of Table 1.