Polyelectrostatic interactions of disordered ligands suggest a physical basis for ultrasensitivity

Abstract

Regulation of biological processes often involves phosphorylation of intrinsically disordered protein regions, thereby modulating protein interactions. Initiation of DNA replication in yeast requires elimination of the cyclin-dependent kinase inhibitor Sic1 via the SCF(Cdc4) ubiquitin ligase. Intriguingly, the substrate adapter subunit Cdc4 binds to Sic1 only after phosphorylation of a minimum of any six of the nine cyclin-dependent kinase sites on Sic1. To investigate the physical basis of this ultrasensitive interaction, we consider a mean-field statistical mechanical model for the electrostatic interactions between a single receptor site and a conformationally disordered polyvalent ligand. The formulation treats phosphorylation sites as negative contributions to the total charge of the ligand and addresses its interplay with the strength of the favorable ligand-receptor contact. Our model predicts a threshold number of phosphorylation sites for receptor-ligand binding, suggesting that ultrasensitivity in the Sic1-Cdc4 system may be driven at least in part by cumulative electrostatic interactions. This hypothesis is supported by experimental affinities of Cdc4 for Sic1 fragments with different total charges. Thus, polyelectrostatic interactions may provide a simple yet powerful framework for understanding the modulation of protein interactions by multiple phosphorylation sites in disordered protein regions.

Fig. 1.

Schematic of a conformationally disordered ligand (yellow) with both positive (blue) and negative (red) charges (for a total charge, q_l) that is bound to a receptor (green) with a binding site having charge q_r. The distance 〈r〉 is between the binding site of the receptor and the center of mass of the disordered ligand.

Fig. 2.

Fraction of bound ligand θ as a function of the net charge (q_l) and the number of phosphorylations (n) on the ligand. (A) Variation of θ for different ligand concentrations in the mean-field model with E_b = −10 k_BT, q_r = 3, q_l⁰ = 11 and 〈r〉 = 12 Å. Ligand concentrations, ρ_l, equal [in units of (δV)⁻¹] 10⁻⁶ (short-dashed line), 10⁻⁷ (long-dashed line), and 10⁻⁸ (solid line). (B) Variation of θ for different numbers of binding motifs, n, for an alternate model in which K_d = (nδV)⁻¹ exp(E_b/k_BT), with E_b values −12 (solid line), −15 (long-dashed line), and −20 (short-dashed line) k_BT, and ρ_l = 10⁻⁷ (δV)⁻¹. The parameters n and q_l are shown as continuous variables; however, only integer numbers of n and the corresponding values of q_l are of physical interest. Numerical results presented in this and subsequent plots are for γ = 1, as discussed in the text, and T = 298 K.

Fig. 3.

Binding curves and binding constants as a function of the net charge (q_l) and the number of phosphorylations (n) on the ligand for mean-field models, with q_l⁰ = 11, ε_d = 20 as in Fig. 2, and ρ_l = 10⁻⁶ (δV)⁻¹. (A) θ vs. n for different contact energies. E_b = −35 (dotted line), −20 (short-dashed line), −15 (long-dashed line), −10 (solid line) k_BT. q_r = 3, and 〈r〉 = 12 Å. (B) θ vs. q_l for different effective charges on the receptor. q_r = 5 (short-dashed line), 3 (long-dashed line), and 1(solid line). E_b = −5 k_BT, and 〈r〉 = 12 Å. (C) K_d vs. n for different charges on the receptor. q_r = 2 (solid line), and q_r = 3 (dashed line). E_b = −10 k_BT, and 〈r〉 = 12 Å. (D) K_d vs. n for different effective distances for the polyelectrostatic interaction between the receptor and the ligand. 〈r〉 = 12 Å (solid line), 〈r〉 = 14 Å (dashed line), q_r = 3, and E_b = −10 k_BT.

Fig. 4.

Sic1 peptide binding to Cdc4. (A) Schematic of a short peptide fragment bound to a receptor protein. (B) Binding constants vs. ligand net charge (q_l) in the mean-field model, with q_r = 3.0, E_b = −15 k_BT, 〈r〉 = 12 Å, and ε_d = 44. K_d values are shown for n = 1 (solid line), 2 (dashed line), and 3 (dotted line). (C) Peptide fragment sequences. Blue (red) background corresponds to positively (negatively) charged residues. pS and pT denote phosphorylated serine and threonine, respectively. (D) Experimental dissociation constants vs. ligand net charge for peptide fragments shown in C. The line is a least-square fit of log(K_d) vs. q_l. Error bars are one standard deviation of the results from three independent titrations. The number labels for the fragment sequences correspond to those in C.