A combination of multisite phosphorylation and substrate sequestration produces switchlike responses

Abstract

The phosphorylation of a protein on multiple sites has been proposed to promote the switchlike regulation of protein activity. Recent theoretical work, however, indicates that multisite phosphorylation, by itself, is less effective at creating switchlike responses than had been previously thought. The phosphorylation of a protein often alters its spatial localization, or its association with other proteins, and this sequestration can alter the accessibility of the substrate to the relevant kinases and phosphatases. Sequestration thus has the potential to interact with multisite phosphorylation to modulate ultrasensitivity and threshold. Here, using simple ordinary differential equations to represent phosphorylation, dephosphorylation, and binding/sequestration, we demonstrate that the combination of multisite phosphorylation and regulated substrate sequestration can produce a response that is both a good threshold and a good switch. Several strategies are explored, including both stronger and weaker sequestration with successive phosphorylations, as well as combinations that are more elaborate. In some strategies, such as when phosphorylation and dephosphorylation are segregated, a near-optimal switch is possible, where the effective Hill number equals the number of phosphorylation sites.

Figure 1

(a) A diagram depicting the phosphorylation and sequestration of a substrate B. B_i (i = 0, 1, 2,…, n) represents free substrate B that has been phosphorylated on a total of i sites, S represents the sequestering entity (e.g., membrane surface, nucleus, or a binding protein), and B_iS represents the sequestered phosphoforms of B. The values k_i and d_i are rates for the phosphorylation and dephosphorylation reactions, respectively. Rate coefficients $k_{\mathrm{i}}^{\mathrm{a}}$ and $k_{\mathrm{i}}^{\mathrm{d}}$ characterize binding to and dissociation from S. (b) The corresponding reaction diagram. A represents the kinase; the phosphatase is not explicitly shown.

Figure 2

Strategy 2, sequestration of the more phosphorylated phosphoforms. (a) A scheme of stronger binding with each phosphorylation. (b) The fraction of B that is fully phosphorylated and sequestered varies with the concentration of kinase A. For each value of c = λ_i−1/λ_i, the corresponding effective Hill number n_H is also shown. Each curve is normalized by setting its EC50 equal to 1. (c) Hill coefficients as a function of the number of phosphorylation sites and fold change of binding ratios with each phosphorylation. (d) Same as panel b, but not normalized to EC50, so as to emphasize threshold (t_H), as measured by the EC10 metric. (e) Threshold as a function of the number of phosphorylation sites and fold change of scaffold binding ratios with each phosphorylation.

Figure 3

Strategy 3, sequestration of the unphosphorylated and fully phosphorylated phosphoforms. (a) Reaction scheme, envisioned by showing the unphosphorylated phosphoform preferentially binding to the plasma membrane, and the fully phosphorylated phosphoform preferentially translocating to the nucleus. (b) Typical cases showing the fraction of fully phosphorylated and sequestered B, as a function of the concentration of kinase and the total number of phosphosites n. (c) Hill coefficients as a function of the number of phosphosites and fold increase in the dissociation constant for sequestration experienced by the intermediate phosphostates. (d) Same as in panel b, but with the inset indicating threshold (t_H), as measured by the EC10/EC50 metric. (e) Threshold as a function of the number of phosphorylation sites and fold increase in the dissociation constant for sequestration experienced by the intermediate phosphostates.

Figure 4

Strategy 4, regulation by release from sequestration. In this scenario, the sequestered form is the active (or inactive) entity, so the appropriate functional output is total substrate bound. (a) A scheme showing weaker binding with each phosphorylation. (b) Plot, assuming eight phosphosites (n = 8), showing how the fraction of B that is sequestered (total substrate bound) varies with the concentration of kinase A. For each value of c = λ_i−1/λ_i, the corresponding effective Hill number n_H is also shown. Each curve is normalized by setting its EC50 equal to 1. (c) Hill coefficients as a function of the number of phosphorylation sites and fold change of binding ratios with each phosphorylation. (d) Hill coefficients as a function of the number of phosphorylation sites and total S, assuming the concentration of S is limiting (e.g., S is a protein and not a compartment).

Figure 5

(a) A model with kinase colocalized on a scaffold protein S. Phosphorylation takes place only when the substrate B is bound to the scaffold, whereas dephosphorylation only takes place when B is unbound. (b) The corresponding reaction diagram.

Figure 6

Typical cases for the scaffold protein model shown in Fig. 5. (a) Plot, assuming eight phosphosites (n = 8), showing how the fraction of free B that is fully phosphorylated varies with the concentration of kinase A. The value α is the fold-change in the scaffold binding rates (i.e., $\alpha =k_{\mathrm{i}}^{\mathrm{a}}/k_{\mathrm{i}-1}^{\mathrm{a}}$). For each value of α, the corresponding effective Hill number n_H is also shown. Each curves is normalized by setting its EC50 to 1. (b) Hill coefficients as a function of number of phosphosites and fold change of scaffold binding rates with each phosphorylation. (c) Hill coefficients as a function of number of phosphosites and fold change of scaffold dissociation rates with each phosphorylation. (d) Hill coefficients as a function of number of phosphosites and simultaneous fold change of scaffold binding and dissociate rates at each phosphorylation, with λ_i = $k_{\mathrm{i}}^{\mathrm{a}}/k_{\mathrm{i}}^{\mathrm{d}}$ = 1. (e) The same as panel c except k_i/k_i−1 = 2 and d_i/d_i–1 = 1/2 for all i. (f) Hill coefficients as a function of the number of phosphorylation sites and the total amount scaffold. In this case, the parameters are α = β = 4, λ_i = $k_{\mathrm{i}}^{\mathrm{a}}/k_{\mathrm{i}}^{\mathrm{d}}$ = 1, k_i/k_i−1 = 2, and d_i/d_i–1 = 1/2. In panels a–f, S_t = 15, B_t = 10, with all other parameters equal to 1 except as specified otherwise in the figure.