A hidden feedback in signaling cascades is revealed

Abstract

Cycles involving covalent modification of proteins are key components of the intracellular signaling machinery. Each cycle is comprised of two interconvertable forms of a particular protein. A classic signaling pathway is structured by a chain or cascade of basic cycle units in such a way that the activated protein in one cycle promotes the activation of the next protein in the chain, and so on. Starting from a mechanistic kinetic description and using a careful perturbation analysis, we have derived, to our knowledge for the first time, a consistent approximation of the chain with one variable per cycle. The model we derive is distinct from the one that has been in use in the literature for several years, which is a phenomenological extension of the Goldbeter-Koshland biochemical switch. Even though much has been done regarding the mathematical modeling of these systems, our contribution fills a gap between existing models and, in doing so, we have unveiled critical new properties of this type of signaling cascades. A key feature of our new model is that a negative feedback emerges naturally, exerted between each cycle and its predecessor. Due to this negative feedback, the system displays damped temporal oscillations under constant stimulation and, most important, propagates perturbations both forwards and backwards. This last attribute challenges the widespread notion of unidirectionality in signaling cascades. Concrete examples of applications to MAPK cascades are discussed. All these properties are shared by the complete mechanistic description and our simplified model, but not by previously derived phenomenological models of signaling cascades.

Figure 1. Schematic representation of a cascade of covalent modification cycles.

The i^th cycle is composed of two states of the same protein: the inactive and the active states, labeled Y _i and Y_i^*, respectively. In each step, the activation is catalyzed by the activated product of the previous step. The deactivation is performed by another enzyme, E'_i.

Figure 2. Performance of the new model compared to the mechanistic one.

Temporal evolution of the first unit in a chain of 10 units. (A) η = ε = 0.01, 0.1, 0.5 and µ = 1. (B) µ = ε = 0.01, 0.1, 0.5 and η = 1. Other parameters are K = 0.01, K' = K/µ, and S = 1. Dashed lines: output of the new model; filled lines: output of the complete mechanistic description.

Figure 3. Characterization of the new model's temporal dynamics.

(A) Parameter space, η on the horizontal axis, V = µη/ε on the vertical axis (notice that the axes are interrupted). The curves η = ε and µ = ε are indicated, and three pairs of values (η,V) over each of them were selected to show the temporal behavior of the chain. When η = ε, parameter V = µ was chosen as 1.2 (A₁), 1.0 (A₂), and 0.5 (A₃), respectively. In the same way, when µ = ε, parameter V = η was chosen as 1.2 (B₁), 1.0 (B₂), and 0.5 (B₃), respectively. (B) Temporal dynamics for the selected pairs depicted in (A). ε = 0.01, K = 0.01, K' = K/µ, and S = 1 for all the panels. The number of units in the chain, n, is 10, except for cases A₃ and B₃, where both n = 10 and n = 3 results are shown. In every case, time is plotted in arbitrary units along the horizontal axis and the temporal evolution y_i^* for three of the units in the chain are shown: y ₁ ^* (black), y ₄ ^* (blue), and y ₇ ^* (red). For the case n = 3, the same color pattern is used for y ₁ ^*, y ₂ ^* and y ₃ ^*, respectively. (C) Steady-state achieved by each unit in a chain with n = 10, plotted versus the unit number, for the cases A₁, A₃, B₁, and B₃. For cases A₁, A₃, and B₁, only the results in variable x_i are displayed. Both x_i and y_i^*are plotted for parameters B₃ (dashed-dotted lines with and without stars, respectively.)

Figure 4. Lateral input is propagated forwards and backwards in the new model.

y_i^* is plotted as a function of the index of the unit in the chain, for a chain of 15 units. The status of the chain at t = −1 (in arbitrary units) is indicated with the symbol +, and it corresponds to the steady-state situation. At t = 0, the indicated unit (see asterisk on the horizontal axis) receives a perturbation Δx, which is then propagated to other units. Times 1 to 10 are plotted in dotted lines. The parameters are (A) η = ε = 0.01 and µ = 0.5, (B) µ = ε = 0.01 and η = 0.5. The remaining parameters are K = 0.01, K' = K/µ, and S = 1 in both (A) and (B).

Figure 5. Stimulus-response curves.

Stimulus-response curves corresponding to parameters A₁ in Figure 3, for a chain with three units. The stimulus strength is the value of S and the response is y_i^*. Variables associated with units 1, 2, and 3 are plotted in filled black, blue, and red lines, respectively. The stimulus-response curve y ₃ ^* for the GK-like model is superimposed in red dotted lines.

Figure 6. Stimulus-response curves for a 3-unit chain.

Stimulus-response curves for a 3-unit chain involving only single phosphorylation (A) or with double phosphorylation in units 2 and 3, representing the MAPK cascade (B). The parameters are those indicated in Table 1. The responses were obtained by both the mechanistic and the reduced mechanistic descriptions, which are in perfect agreement. The input stimulus is given by E _1T , the total amount of kinase for the first unit. y ₁ ^*, y ₂ ^*, and y ₃ ^* are plotted with black, blue, and red filled lines, respectively. GK-like model predictions are also included (dotted lines). The Hill coefficients characterizing each curve are listed in the legend.

Figure 7. “Reverse” stimulus-response curves.

“Reverse” stimulus-response curves for a 3-unit chain involving only single phosphorylation (A) or with double phosphorylation in units 2 and 3, representing the MAPK cascade (B). The parameters are those indicated in Table 1. The responses were obtained by both the mechanistic and the reduced mechanistic descriptions, which are in perfect agreement. The input stimulus is given by E' _3T, which is the total amount of phosphatase for the last unit. y ₁ ^*, y ₂ ^*, and y ₃ ^* are plotted with black, blue, and red filled lines, respectively. GK-like model predictions are also included (dotted lines). Insets show details of the figures.

Figure 8. Modular response analysis (MRA).

Modular response analysis (MRA) applied to the new model for signaling cascades. MRA was applied to a 3-unit cascade involving only single phosphorylation and characterized by the parameters in Table 1. (A) Interaction map and reconstructed network topology regarding variables x_i. (B) Local response coefficients (regarding x_i) versus parameter E _1T. Black, blue, red, and green for r ₁₂, r ₂₁, r ₂₃, and r ₃₂, respectively. The asterisks over each curve indicate the values of the matrix in (A), corresponding to E _1T = 3×10^{− 4} µM. (C) Interaction map and reconstructed network topology regarding variables y_i^*.