A bistable model of cell polarity

Abstract

Ultrasensitivity, as described by Goldbeter and Koshland, has been considered for a long time as a way to realize bistable switches in biological systems. It is not as well recognized that when ultrasensitivity and reinforcing feedback loops are present in a spatially distributed system such as the cell plasmamembrane, they may induce bistability and spatial separation of the system into distinct signaling phases. Here we suggest that bistability of ultrasensitive signaling pathways in a diffusive environment provides a basic mechanism to realize cell membrane polarity. Cell membrane polarization is a fundamental process implicated in several basic biological phenomena, such as differentiation, proliferation, migration and morphogenesis of unicellular and multicellular organisms. We describe a simple, solvable model of cell membrane polarization based on the coupling of membrane diffusion with bistable enzymatic dynamics. The model can reproduce a broad range of symmetry-breaking events, such as those observed in eukaryotic directional sensing, the apico-basal polarization of epithelium cells, the polarization of budding and mating yeast, and the formation of Ras nanoclusters in several cell types.

Figure 1. Prototypical model of cell polarization.

A system of receptors transduces an external distribution of chemotactic cues into an internal distribution of activated enzymes , which catalyze the switch of a signaling molecule from an unactivated state to an activated state . A counteracting enzyme transforms the state back into . The network contains a couple of amplifying feedback loops: the signaling molecule activates and acvivates . The signaling molecules , are permanently bound to the cell surface and perform diffusive motions on it, while the , enzymes are free to shuttle between the cytosolic reservoir and the membrane. The result of the polarization process is the formation of separate domains with -rich patches and, respectively, -rich patches.

Figure 2. Stable chemical phases.

Left: relative concentrations of signaling molecules in the stable chemical phase and unstable chemical phase , as a function of the renormalized activation signal (19) (black) and for different values of the saturation constant . Right: Behavior of the potential , as a function of the phase , see (13). The potential has two minima: the left-hand one corresponding to a stable -rich and the right-hand one corresponding to a stable -rich phase. The two phases are separated by an effective energy barrier. The existence of the two distinct stable chemical phases is called bistability.

Figure 3. Bistability region, yellow region (II)-(III), as a function of the level of external renormalized stimulation

for . The purple line corresponds to phase coexistence (polarization) and is an attractor for the polarization dynamics. The two stable domains, blue (IV) and red (I), correspond to the two and stable phases.

Figure 4. In the equilibrium state the circular patches occupied by the

and phases have areas, respectively, and . Here we show the ratio at different values of the stimulation . Curves are plotted from top to bottom with increasing ratio of the initial enzymes quantities . Each curve shows two plateaux that are approximatively independent of the signal . For small the system is dominated by the mutual interaction between and , i.e., by the feedback loop, whilst for large the system is dominated by the interaction with receptors, i.e., by the external signal.

Figure 5. Physical analogy: membrane polarization and precipitation from a supersaturated solution.

At initial time, the concentration of some molecule is higher than the critical value , so that a small fluctuation, or an impurity, can easily give rise to the formation of small germs of precipitate. Germs larger than a critical size grow steadily, while germs smaller than are dissolved by diffusion. As the size of the germs grows, the molecule is extracted from the hydrated phase and transferred to the solid phase, moving the concentration closer to the critical value , increasing the value of , and correspondingly slowing down the process of germ growth. Grains that were initially larger than are dissolved, so that larger grains grow at the expense of the smaller grains. Eventually, an equilibrium is reached when and a single large grain of precipitate survives.

Figure 6. Model of chemotactic polarization.

With respect to the abstract scheme in Fig. 1 we have the identification  = PIP3,  = PIP2, REC. The PIP3-rich domain corresponds to the presence of a high concentration of chemoattractant factor.

Figure 7. We simulated a spatially homogeneous version of the Model.

We mimick the experimental conditions by switching on receptor activation at initial time. The phospholipid concentration field is driven by the slow variation in time of the effective potential , that follows the slow variation of the enzyme ratio . Receptor activation at (blue line) induces a uniform increase of PI3K, PIP3 on the whole plasmamembrane, which corresponds to the appearance of a single potential well centered in the PIP3-rich region. The enzyme ratio decreases, corresponding to PI3K recruitment to the plasmamembrane and PTEN relocation to the cytosol (blue line). When the enzyme ratio crosses the boundary of the bistable region (light blue area) the effective potential develops a secondary potential well centered in the PIP2-rich region.

Figure 8. Kimograph for a simulation of the full, spatially distributed, chemotaxis system.

In the simulation, before starting to stimulate cells with a uniform concentration of cAMP, the system is left to relax with zero signal until the levels of the relevant factors become stationary. Then, the stimulation is switched on at time , when we also impose a gaussian noise on the uniform concentration background in order to mimick random inhomogeneities. We compare the experimental results reported in Reference with the simulations of model (1–8). The kimograph shows the time evolution of simulated PIP3 levels along the major cell perimeter. Time in the simulation is to be compared with time s in the experiment.

Figure 9. The dynamics of the simulated 3D spatially distributed model for different times.

The colorbar is the same as in Fig. 9, the major cell perimeter is the one considered in Fig. 9.

Figure 10. Model of epithelial polarization, with respect to the scheme in Fig. 1 we identify

, , , , and C/M. To bind PI3K, cadherins must be activated by engagement with cadherins of a neighboring cell. The PIP2, PIP3 localization is central in the establishment of epithelial apico-basal orientation.

Figure 11. Growth of the PIP2-rich phase (blue lower patch).

The color scale shows the gradation of PIP2 content: the color is the relative concentration difference between PIP3 and PIP2 at a given site. The system at initial time is in a uniform PIP3-rich phase (red), apart from an initial PIP2-rich seed germ of size larger than the threshold radius. Then, a PIP2-rich patch becomes apparent and its radius saturates to an equilibrium value.

Figure 12. Model of cell polarization for budding yeast.

With respect to the scheme in Fig. 1, we identify , , , .

Figure 13. Intermittent and persistent polarization obtained by simulation of model (1–8).

In the graphs we plot concentrations of membrane-bound molecules along a 1 thick cross section of the plasmamembrane vs. time, normalized with the average membrane concentration. Upper three rows: small number of A-molecules (PTEN in Ref. , or Cdc24 in Ref. [45]). Intermittent polarization as shown here was already described in . The graphs of our realistic surface model are similar to those obtained in Ref. in the monodimensional case. Patches of signaling molecules randomly form and disappear. Observe that patches are the macroscopic counterpart of clans of signaling molecules, as defined in . Parameter values were taken as follows: diffusivity of membrane-bound molecules is , , [A] = 1, 10, 50 nM, the decay rate of is adjusted in order to get 10% of A molecules bound to the plasmamembrane, all other parameters are as in , .

Figure 14. The 3D behavior of intermittent Cdc42

patches. The graphs of our realistic surface model are similar to those obtained in the one-dimensional model of Ref. . It is worth observing here that intermittent, as opposed to stable, patch formation is here a consequence of the particular, small-concentration limit considered in .

Figure 15. Ras activation pathway.

With respect to the scheme in Fig. 1., we identify , , , -GAP.