Modelling cell polarization driven by synthetic spatially graded Rac activation

Abstract

The small GTPase Rac is known to be an important regulator of cell polarization, cytoskeletal reorganization, and motility of mammalian cells. In recent microfluidic experiments, HeLa cells endowed with appropriate constructs were subjected to gradients of the small molecule rapamycin leading to synthetic membrane recruitment of a Rac activator and direct graded activation of membrane-associated Rac. Rac activation could thus be triggered independent of upstream signaling mechanisms otherwise responsible for transducing activating gradient signals. The response of the cells to such stimulation depended on exceeding a threshold of activated Rac. Here we develop a minimal reaction-diffusion model for the GTPase network alone and for GTPase-phosphoinositide crosstalk that is consistent with experimental observations for the polarization of the cells. The modeling suggests that mutual inhibition is a more likely mode of cell polarization than positive feedback of Rac onto its own activation. We use a new analytical tool, Local Perturbation Analysis, to approximate the partial differential equations by ordinary differential equations for local and global variables. This method helps to analyze the parameter space and behaviour of the proposed models. The models and experiments suggest that (1) spatially uniform stimulation serves to sensitize a cell to applied gradients. (2) Feedback between phosphoinositides and Rho GTPases sensitizes a cell. (3) Cell lengthening/flattening accompanying polarization can increase the sensitivity of a cell and stabilize an otherwise unstable polarization.

Figure 1. Schematics of a sequence of models explored in this paper.

a) A basic single GTPase (“wave pinning”) module with crosstalk to the phosphoinositides (PIs). The GTPase module can only polarize on its own , but not when connected to PIs in this way. b) As before but with an additional passive Rho module: still no polarization possible with PI crosstalk. c) Mutual inhibitory Rac-Rho module: Polarization observed both with and without the PI layer. d) A more complete Cdc42-Rac-Rho module that exhibits polarization both with and without PIs. Model equations are shown in (1), (8), (16) and represents the strength of PI feedback to Rac. Arrows represent upregulation and bars represent inhibition. In all cases, proposed interactions between GTPases and PIs are taken from the literature , –.

Figure 2. a) Membrane-cytosolic exchange for a single small GTPase.

Activation and inactivation of membrane bound forms occur via GEF phosphorylation and GAP dephosphorylation respectively. The inactive form can cycle on and off the membrane aided by GDI's. b) Approximation of cell geometry with a box of dimensions . The width is constrained by the microfluidic channels in experiments .

Figure 3. Basic “default” Local Perturbation Analysis (LPA) bifurcation diagram obtained using the LPA approximation of the PDEs (1), (8) using the reduction (18) (described in the Methods).

Shown is steady state active (local) Cdc42 () with , the basal Rac GEF activity level, as bifurcation parameter. Here (no PI feedback), and all other parameters as in Table 1. The monotone increasing (blue) curve represents the HSS of the original system and is stable to homogeneous perturbations. Elliptical (red) arcs represent additional equilibria found in the LPA-system. Stability to small heterogeneous perturbations is indicated by solid lines vs instability shown by dotted lines. Region I is insensitive to perturbations, II is polarizable with sufficiently large perturbations, III is hypersensitive (Turing unstable), IV is insensitive but overstimulated. Similar results are seen when plotting or on the vertical axis.

Figure 4. Kymographs (-plots) of active Cdc42 concentration for the full PDE system with no PI feedback (), and parameters as in Figure 3.

Left panel: (Region II in Figure 3). Patterning is induced by a large local perturbation applied to active Rac at . Identical behaviour is seen when this perturbation is applied to active Cdc42. Right panel: (Region III in Figure 3). Patterning is induced by random noise of size in the initial conditions. Similar (complementary) kymographs of Rac (Rho) are obtained (not shown).

Figure 5. Effect of feedback from Cdc42 to Rac.

LPA bifurcation diagram of (1) as in Figure 3, showing the effect of increasing values. For larger values, the model is more sensitive to heterogeneous stimuli.

Figure 6. Effect of PI feedback to Rac.

LPA bifurcation diagrams for (1) as in Figure 3, with and multiple values of . Left panel: PI variables treated as fast (global) LPA variables. Right panel: PI variables treated as slow (local) LPA variables. Note the simple linear leftwards shift as increases in both panels.

Figure 7. Two-parameter bifurcation plot for feedback from PIs to Rac () versus stimulus strength obtained via batch simulation of the full PDE system.

, and other parameters as in Table 1. The grey region is bistable and the white is Turing unstable. The linearity of this bifurcation curve is both qualitatively and quantitatively consistent with the linear shift of the bifurcation diagrams seen in Figure 6.

Figure 8. Effect of cell length: LPA bifurcation diagram with , , showing two values of .

As is increased, the stable Region I of Figure 3 at low values vanishes, eliminating the hysteresis associated with the stable to bistable transition.

Figure 9. Schematic of the applied local perturbation in the LPA method.

represents the slow diffusing active form which has a local component near the applied pertubation at and a global behaviour away from it. Since diffusion is slow, they do not directly influence each other on a short time scale. is fast diffusing and takes on only a global behaviour away . Solid curves qualitativly represent this pulse in the idealized diffusion limit and dashed curves represent the same situation with finite rates of diffusion.