A Bistable Model of Cell Polarity

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.


Stable phases
We start by looking for solutions of equation (12) in the main text, corresponding to a uniform chemical phase. They are obtained from the condition which, once solved, gives two stable equilibrium values ϕ + = c, ϕ − and one unstable equilibrium ϕ unst . The ϕ + , ϕ − values correspond to distinct, stable, uniform chemical phases, enriched respectively in the signaling molecules Φ + and Φ − . We refer to the existence of two distinct stable chemical phases as bistability. The explicit concentration values are Eqs.
(2-4) show that the concentration values (2) are completely controlled by the enzyme ratio ρ, which measures the relative strength of the counteracting X and Y enzimes, and by the renormalized activation signal σ. A graph of the concentration values in the two stable phases is given in Figure 2 of the main text. An important consequence of the existence of two distinct, locally stable phases is that different regions of the cell membrane can be occupied by different phases, giving rise to patterning into distinct signaling domains.
The ϕ ± , ϕ u values have actual meaning only if they correspond to non-negative concentrations, i.e. if they satisfy the inequality: which is equivalent to the following condition on the enzyme ratio ρ and saturation constant κ: These formula define the bistability region shown in Figure 3 of the main text.
In the limit of negligible feedback k c k a /k d → 0 the conditions (6) reduce to the result in [3], see main references.

Coexistence line
The cell membrane is polarized when it divides into two complementary regions, stably occupied by one of two distinct chemical phases, and separated by a thin diffusive interface. Stable polarized equilibria are reached when the effective energy F is minimal, i.e. when both terms in (15) take on their minimal values. If e.g. V (ϕ + ) < V (ϕ − ), no polarized configuration can be stable because the energy can still decrease by extending the area covered by the ϕ + phase, which has lower energy than the ϕ − phase. The same is true if V (ϕ + ) > V (ϕ − ). Therefore, stability of polarized equilibria (or phase coexistence) is possible only if the following mathematical condition is satisfied: that can be set in the form of an implicit integral equation for the value at equilibrium of the enzyme ratio ρ eq : This equation can be numerically solved to determine the phase coexistence line (Figure 3) where stable polarized configurations are possible. For κ = 1 an approximate expression for the coexistence line, valid for both small and large σ, is

Patch area
In the equilibrium state characterized by the equilibrium value ρ eq of the enzyme ratio (19), the A + and A − areas of the circular caps occupied respectively by the ϕ + and ϕ − phases are determined by the integral constraints. The A + and A − areas can be explicitly computed if the area of the interfacial region separating the two caps is negligible with respect to the cell membrane area A, so that e.g. φ + ϕ + A + /A: