Response dynamics of phosphorelays suggest their potential utility in cell signalling

Abstract

Phosphorelays are extended two-component signalling systems found in diverse bacteria, lower eukaryotes and plants. Only few of these systems are characterized, and we still lack a full understanding of their signalling abilities. Here, we aim to achieve a global understanding of phosphorelay signalling and its dynamical properties. We develop a generic model, allowing us to systematically analyse response dynamics under different assumptions. Using this model, we find that the steady-state concentration of phosphorylated protein at the final layer of a phosphorelay is a linearly increasing, but eventually saturating function of the input. In contrast, the intermediate layers can display ultrasensitivity. We find that such ultrasensitivity is a direct result of the phosphorelay biochemistry; shuttling of a single phosphate group from the first to the last layer. The response dynamics of the phosphorelay results in tolerance of cross-talk, especially when it occurs as cross-deactivation. Further, it leads to a high signal-to-noise ratio for the final layer. We find that a relay length of four, which is most commonly observed, acts as a saturating point for these dynamic properties. These findings suggest that phosphorelays could act as a mechanism to reduce noise and effects of cross-talk on the final layer of the relay and enforce its input-response relation to be linear. In addition, our analysis suggests that middle layers of phosphorelays could embed thresholds. We discuss the consequence of these findings in relation to why cells might use phosphorelays along with enzymatic kinase cascades.

Figure 1.

Dynamics of linear phosphorelays. (a) Cartoon representation of the linear phosphorelay considered. The phosphotransfer reactions among each layer are shown for a four-layer relay. Note that other configurations of specific proteins (RR, HK, etc.) are possible that would lead to the same dynamical effects. The generic relay structure is indicated by referring to different layers as L1, L2, etc. (b) Steady-state input–response curves are shown for each layer in relays with increasing relay length (number of relays indicated on top of each panel). Dark blue lines, L1p; brown lines, L2p; green lines, L3p; purple lines, L4p; light blue lines, L5p. (c) Time course showing system response (phosphorylated form of each layer), obtained from deterministic (middle) and stochastic (bottom) models of a four-layer system. Changes in input during the time course of simulation are shown in the upper panel. For the stochastic model, the input values are multiplied by 100 (§4).

Figure 2.

Effects of a bifunctional HK. (a) Cartoon representation of a four-layer relay, with a bifunctional HK sitting at the top and capable of dephosphorylating L4p (§4 and electronic supplementary material, equation (S1)). (b) Effect of the Michaelis constants (K_m) of the bifunctional HK on input–response curves. Dashed arrows show the trend how the responses in each layer (noted on each panel) are changing with increasing dephosphorylation efficiency of the bifunctional HK. K_m=0.1 (dark blue lines), 1 (orange lines), 10 (light blue lines), 100 (brown lines), NO (black lines). (c) Sensitivity in L4p as K_m of the bifunctional enzyme is varied. Sensitivities were calculated between input levels 0–2 (left panel) and 2–10 (right panel) from the changes in the response and input (i.e. ΔL4p/Δinput) in the investigated input regimes. In other words, sensitivity corresponds to the slope of the input–response curve for L4p (as shown in (b)). (d) Heat map of input–response relation of a four-layer relay, with varying values of the Michaelis constant (K_m) of the bifunctional HK in layer 1 (§4 and electronic supplementary material, equation (S1)). Bottom plot with the NO label shows the heat map for a system without the phosphatase activity of the HK.

Figure 3.

Possible cross-talks in a phosphorelay. (a) Four-layer phosphorelay with cross-activation—‘talk in’ (green)—and cross-inhibition—‘talk out’ (red). Note that these cross-talk reactions are modelled as self-phosphorylation (k₂₃, k₃₃, k₄₃) and self-dephosphorylation (k₂₂, k₃₂, k₄₂) in the model (equation (4.1)). (b) Heat map showing response level given the input at the top of the relay and the secondary signal coming from cross-talk. The location where cross-talk occurs and its nature (i.e. activating or deactivating) are indicated on each panel (see also electronic supplementary material, figure S5).

Figure 4.

Noise tolerance of phosphorelays. (a) Average (dark colour curves) and standard deviation (lighter shading) of steady-state input–response curves for each layer in a four-layer system with noisy input. (b) SNR (§4) at each layer of a four-layer relay. The black line shows the SNR expected from a random Poisson process. See also electronic supplementary material, figure S10, which shows changes in the SNR from the last layer with changes in relay length. (c) Dependence of standard deviation on input strength at each layer of the four-layer system. All results were calculated from approximately 400 000 stochastic simulation data points at each of the 200 input levels. Blue lines, L1p; brown lines, L2p; green lines, L3p; purple lines, L4p, black lines, input.