Utilization of extracellular information before ligand-receptor binding reaches equilibrium expands and shifts the input dynamic range

Abstract

Cell signaling systems sense and respond to ligands that bind cell surface receptors. These systems often respond to changes in the concentration of extracellular ligand more rapidly than the ligand equilibrates with its receptor. We demonstrate, by modeling and experiment, a general "systems level" mechanism cells use to take advantage of the information present in the early signal, before receptor binding reaches a new steady state. This mechanism, pre-equilibrium sensing and signaling (PRESS), operates in signaling systems in which the kinetics of ligand-receptor binding are slower than the downstream signaling steps, and it typically involves transient activation of a downstream step. In the systems where it operates, PRESS expands and shifts the input dynamic range, allowing cells to make different responses to ligand concentrations so high as to be otherwise indistinguishable. Specifically, we show that PRESS applies to the yeast directional polarization in response to pheromone gradients. Consideration of preexisting kinetic data for ligand-receptor interactions suggests that PRESS operates in many cell signaling systems throughout biology. The same mechanism may also operate at other levels in signaling systems in which a slow activation step couples to a faster downstream step.

Keywords: binding kinetics; cellular signaling; dose–response.

Fig. 1.

Time-dependent shift of the binding dose–response curve. (A) Time course of receptor/ligand complex formation for different ligand concentrations (relative to the affinity dissociation constant, K_d), in the range 0.1–100 K_d. We computed values for the case of one-step binding using the reaction $L+R\begin{array}{c}\stackrel{k_{+}}{\to }\\ \underset{k_{-}}{\leftarrow }\end{array}C$, with the following αF binding reaction rates: k₊ = 1.9 10⁵ M⁻¹⋅s⁻¹ and k₋ = 0.001 s⁻¹ (15, 16). We assumed that the concentration of free L over time is constant. Norm., normalized. (B) Dose–response curves for the receptor/ligand reaction computed at different times, as indicated over the curves (and with a red arrow). Two concentrations, L₁ and L₂, result in well-separated levels of occupied receptor C₁ and C₂ at 10 s (0.4211 and 0.5483, respectively), but not at the equilibrium values C_1-eq and C_2-eq (0.9821 and 0.9877, respectively). (C) Yeast cells of strain YAB3725 [∆bar1, P_STE2 STE2(T305)-CFP] were grown and stimulated with the indicated amounts of a fluorescent αF derivative (37), and the binding was determined by fluorescence microscopy as explained in Materials and Methods. The mean and 95% confidence interval for the mean are shown for each concentration and time. A simple binding model was globally fitted to the data, resulting in K_d = 23 ± 3 nM and k₋ = 1.0 ± 0.2 10⁻⁴⋅s⁻¹ (solid lines).

Fig. 2.

PRESS. Coupling a slow binding reaction to a fast and transient response can expand the input dynamic range beyond equilibrium saturation. (A) Inset shows a toy model with a downstream response activated by the ligand-receptor complex computed in Fig. 1A. Occupied receptor activates effector X; then, X* converts into X^, which slowly converts back to X, closing the cycle (details are provided in SI Appendix, section 2). The plot shows X* vs. time for all of the concentrations of L included in Fig. 1A. (B) Shift and expansion of the input dynamic range are due to PRESS. L-receptor complex at equilibrium (C_eq, solid black line), as well as peak X* resulting from using slow (red dotted line) or fast (blue dashed line) binding/unbinding rates of L to R, vs. input L, are shown. The resulting input dynamic ranges (the fold change required in input to elicit a change from 10 to 90% of maximum output) are indicated. The two doses indicated, L1 = 55 K_d and L2 = 80 K_d, result in an 8.5% difference in peak X* for slow-fast coupling, and only a 0.75% difference for fast-fast coupling. (C) Graphical description of the expansion of the input dynamic range in the toy model. The plot shows L-receptor complex at equilibrium (C_eq, solid black line), or at the indicated times before equilibrium (C_(t), solid gray line), as well as peak X* (solid red line), all as a function of input L. Note, as shown in A, that peak X* occurs at different times for different concentrations of input L. Therefore, the values of C at the time when X* peaks (C_tmax, solid green line) correspond to different C_(t) gray curves for each dose of L (green ○). The C_tmax curve is itself shifted to higher doses than C_eq, and it has a larger input dynamic range (less steep) than either C_eq or any C_(t) curves [EC₅₀ = 17.7 and sensitivity (n_H) = 0.9]. In B and C, the x axis corresponds to L normalized by the K_d of the ligand-receptor reaction. All data correspond to simulations, done using the following parameters for the X cycle: r₁ = 0.1, r₂ = 0.08, r₃ = 0.001, and X_tot = 10 (X_tot being the total amount of X). Binding/unbinding rates were k₋ = 0.001 1/s, k₊ = 0.00019 (nM · s)⁻¹ for slow binding (A, red line in B and C) and 100-fold those values for fast binding (blue line in B) (SI Appendix, Fig. S1).

Fig. 3.

Mathematical framework to study the detection of a stationary spatial gradient. (A, Left) Cell modeled as an impermeable sphere. a, radius; b, back (θ = π); d, distance to the point source of the ligand (αF); f, front (θ = 0); θ, angle away from the line connecting the center of the sphere to the point source. (A, Right) αF profile as a function of θ for two cells located at the same d from a strong (cell 1, black) or weak (cell 2, red) source, modeled with the following parameters: q/(4πDd) = 10 K_d and 1 K_d, respectively, and a/d = 0.3 (for details and definitions for q and D, see SI Appendix, section 5.1). The maximum and minimum concentrations of αF on the cell surface are 16.7 K_d and 6.6 K_d for cell 1, and 1.7 K_d and 0.7 K_d for cell 2. (B) Normalized equilibrium bound receptor (C) vs. αF concentration, in units of K_d, for cell 1 (○; f₁, b₁) and cell 2 (□; f₂, b₂). Dotted lines indicate the difference in normalized bound receptor between the front and back for both cells (Delta_eq-1 and Delta_eq-2) at equilibrium binding. (C) Normalized bound receptor vs. time at the front (solid lines) and back (dashed lines) for cell 1 (black, 10 K_d) and cell 2 (red, 1 K_d). Maximum Delta (Delta_max) is indicated by dotted lines. (D) Delta vs. time for cell 1 (black) and cell 2 (red). The amplitude and duration of the overshoot for cell 1 are indicated with dotted lines. (E) Time derivative (the rate of receptor occupation) vs. time of the data presented in C, for cell 1 (black) and cell 2 (red) at the front (solid line) and back (dashed line). Close to t = 0, the rate is directly proportional to the αF concentration (k_on * αF, ○). Delta_max (the peak of the overshoot) occurs at t_max (x), when the curves corresponding to the rates at the front and back cross (become equal). (Inset) Zoomed-in view of the same plot showing t_max for cell 2. (F) Fast polarization of the αF pathway machinery in living yeast. We stimulated yeast expressing the MAPK scaffold protein Ste5 fused to YFPx3 with 1 μM αF (isotropic stimulation) at time 0 and then followed cells by time-lapse confocal fluorescence microscopy at the time points indicated. Images show polarization (formation of an Ste5 patch) in one cell, which is evident starting at 1 min. Numbers correspond to time after αF addition (Movie S1 and SI Appendix, Fig. S4). (G) Fast relocalization of the polarization site in response to an external cue. We stimulated yeast expressing Bem1 fused to three mNeonGreen fluorescent proteins (mNG) with a 0–50 nM linear gradient (high on the right side), such that cells experience a 1 nM difference in pheromone concentration from “front” to “back” (SI Appendix, Fig. S5). Then, we followed cells by time-lapse confocal fluorescence microscopy at the time points indicated. Images show the location of the Bem1 patch. Light blue and yellow arrows mark the initial and new Bem1 patch, respectively. The white arrow marks the daughter cell bud-neck Bem1 patch. Red bar marks spans the repositioning time. (Scale bars: F and G, 2 μm.)

Fig. 4.

Slow binding receptors efficiently convey gradient information to a cell polarization model. (A) Schema of the early events in the PRS (αF) related to gradient detection. αF stimulates recruitment and activation of Cdc42 to sites of receptor activation. αF binds to the receptor (Ste2), causing dissociation of the G protein into Gα (Gpa1) and Gβγ (Ste4–Ste18) heterodimers. Gβγ recruits the MAPK scaffold (Ste5) to the membrane, leading to the activation of the MAPK Fus3. Gβγ also recruits Far1. Far1 recruits Cdc24, the activator of the small G protein Cdc42 (55, 56, 83). Cdc42 stimulates its own activation by binding to Bem1. Bem1 binds Cdc24, which further activates Cdc42 (7, 49). αF stimulates this positive feedback loop further via recruited Ste5, which also binds Bem1 (green dotted line) (84). Active Cdc42 directs the assembly of actin filaments in a later phase of the gradient sensing process. Solid arrows correspond to activation, dotted arrows correspond to molecule movement (e.g., membrane recruitment), dotted lines correspond to protein–protein interactions, and the double arrow indicates dissociation. (B) Schema of the Altschuler model for spontaneous emergence of cell polarity (54). It has four reactions: binding/unbinding of Cdc42 to/from the plasma membrane, recruitment of cytoplasmic Cdc42 to sites where Cdc42 has already been recruited (positive feedback), and lateral diffusion of Cdc42 through the membrane. The associated parameters spontaneous association rate (k_on), random dissociation rate (k_off), recruitment rate (k_fb), lateral diffusion (D), and the total number of signaling molecules (N) were estimated by Altschuler et al. (54) from experimental data. (C, Left) Input to the gradient sensing model: Steady-state αF spatial profile, as in Fig. 3A, corresponds to a cell located at a point in a gradient with an average αF concentration of 10 K_d. (C, Center) Sensing [C(θ,t)], representing normalized bound receptor. Plots correspond using a color scale on the right, at different angles θ (as in Fig. 3A) vs. time, for slow [WT (wt), Left] or fast binding (fast, Right) receptors. For slow dynamics, we used published binding rates of αF to Ste2, k₊ = 1.9 × 10⁵ M⁻¹⋅s⁻¹ and k₋ = 0.001 s⁻¹ (15). For fast dynamics, we used k₊ and k₋ that are 100-fold greater, maintaining the same K_d. Thus, the parameters are $\frac{q}{4\pi Dd}$ = 10, a/d = 0.3, and k₋ = 0.001 (slow) or 0.1 (fast) (SI Appendix, section 5.1). (C, Right) Example of simulation output: Plot corresponds to particle density at the membrane using a color scale, indicated by a bar on the right, at different angles θ (as in Fig. 3A) vs. time. (D, Left) Percentage of polarizations in the front quadrant as a function of binding dynamics (rate k₋ at constant K_d) for a cell located as in C after 5 min (black ●) or 15 min (red ■) of simulation. Of the tested k₋ rates, only those rates lower than or equal to 0.001 1/s resulted in polarizations in the front quadrant that were significantly different from random (25%, black line), P < 0.05. Arrows indicate the results shown in the histograms. At the slowest rate tested, there were no polarizations in the first 5 min. (D, Right) Histograms showing the number of stochastic simulations with polarization at the indicated angles $\mathit{\theta }$, using slow (Left) or fast (Right) ligand-receptor binding dynamics (2,000 simulation runs each). The size of the bins is π/6.5. The polarization state was measured at t = 5 min. Coupling between occupied receptor and this model was done through parameters k_on and k_fb as follows: $k_{on}\left(\theta ,t\right)=A_{on}*C\left(\theta ,t\right)$ and $k_{fb}\left(\theta ,t\right)=A_{fb}*C\left(\theta ,t\right),$ where A_on and A_fb are the coupling parameters. A_on = 0.0012, A_fb = 23, k_off = 9 min⁻¹, D = 1.2 μm², and N = 10³ (SI Appendix, Fig. S6).