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 Kd and 1 Kd, 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 Kd and 6.6 Kd for cell 1, and 1.7 Kd and 0.7 Kd for cell 2. (B) Normalized equilibrium bound receptor (C) vs. αF concentration, in units of Kd, 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 (Deltaeq-1 and Deltaeq-2) at equilibrium binding. (C) Normalized bound receptor vs. time at the front (solid lines) and back (dashed lines) for cell 1 (black, 10 Kd) and cell 2 (red, 1 Kd). Maximum Delta (Deltamax) 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 (kon * αF, ○). Deltamax (the peak of the overshoot) occurs at tmax (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 tmax 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.)