Positional information generated by spatially distributed signaling cascades

Abstract

The temporal and stationary behavior of protein modification cascades has been extensively studied, yet little is known about the spatial aspects of signal propagation. We have previously shown that the spatial separation of opposing enzymes, such as a kinase and a phosphatase, creates signaling activity gradients. Here we show under what conditions signals stall in the space or robustly propagate through spatially distributed signaling cascades. Robust signal propagation results in activity gradients with long plateaus, which abruptly decay at successive spatial locations. We derive an approximate analytical solution that relates the maximal amplitude and propagation length of each activation profile with the cascade level, protein diffusivity, and the ratio of the opposing enzyme activities. The control of the spatial signal propagation appears to be very different from the control of transient temporal responses for spatially homogenous cascades. For spatially distributed cascades where activating and deactivating enzymes operate far from saturation, the ratio of the opposing enzyme activities is shown to be a key parameter controlling signal propagation. The signaling gradients characteristic for robust signal propagation exemplify a pattern formation mechanism that generates precise spatial guidance for multiple cellular processes and conveys information about the cell size to the nucleus.

Figure 1. Cascade signaling scheme.

Figure 2. Active form concentration profiles, , obtained by numerical integration of Eq. (8) with γ = 0.1 at different times: (A) t = 1; (B) t = 10; (C) t = 50; (D) .

Figure 3. Stationary concentration profiles, , for different values of γ: (A) γ = 0.05; (B) γ = 0.25; (C) γ = 4; (D) γ = 50.

The dotted lines in Fig. (A) and (B) are given by Eq. (10). Dashed lines in Figs. (C) and (D) are the exact solutions to Eq. (9) given in the supporting information by Eq. (A2).

Figure 4. Concentration profiles as in Fig. 3A shifted to the left by a distance given by Eq. (11) with .

Figure 5. Stationary concentration profiles, , obtained by numerical integration of Eqs. (13) with γ = 0.1 for different values of and : (A) , ; (B) , .

Figure 6. Stationary concentration profiles, , obtained by numerical integration of Eqs. (13) with γ = 10 for different values of and : (A) , ; (B) , .

Figure 7. Parameter regions corresponding to propagating (red crosses) and decaying signals (blue circles) as a function of γ and or for: (A) , and (B) .

Dashed lines in Fig A and B are given by the Eqs. (16) and (18), respectively.

Figure 8. Parameter regions corresponding to propagating (red crosses) and decaying signals (blue circles) as a function of γ and small values of .