Phase transitions of the coupled membrane-cytoskeleton modify cellular shape

Abstract

Formation of protrusions and protein segregation on the membrane is of a great importance for the functioning of the living cell. This is most evident in recent experiments that show the effects of the mechanical properties of the surrounding substrate on cell morphology. We propose a mechanism for the formation of membrane protrusions and protein phase separation, which may lay behind this effect. In our model, the fluid cell membrane has a mobile but constant population of proteins with a convex spontaneous curvature. Our basic assumption is that these membrane proteins represent small adhesion complexes, and also include proteins that activate actin polymerization. Such a continuum model couples the membrane and protein dynamics, including cell-substrate adhesion and protrusive actin force. Linear stability analysis shows that sufficiently strong adhesion energy and actin polymerization force can bring about phase separation of the membrane protein and the appearance of protrusions. Specifically, this occurs when the spontaneous curvature and aggregation potential alone (passive system) do not cause phase separation. Finite-size patterns may appear in the regime where the spontaneous curvature energy is a strong factor. Different instability characteristics are calculated for the various regimes, and are compared to various types of observed protrusions and phase separations, both in living cells and in artificial model systems. A number of testable predictions are proposed.

FIGURE 1

(a) A schematic picture of our model: an overall flat membrane (solid line) that is free to deform locally (h) and contains a population of membranal clusters (φ) that have a convex spontaneous curvature and are free to diffuse in the plane of the membrane with diffusion coefficient D. These membranal clusters (MPs) contain proteins that promote actin polymerization and protrusive force f, and adhesion to the surrounding ECM (inducing effective negative surface tension α). (b) The cell membrane-cortical cytoskeleton can be in the mixed state (i), which is featureless and uniform, or can become unstable (ii); phase-separated, with large aggregation of the MPs and protrusions/adhesions structures.

FIGURE 2

Instability patterns of the system. (a) Type-I instability: the instability band starts at q = 0, until q_p, and has a maximum at q*. This instability leads to global phase-separation. (b) Type-II instability: the instability band first appears at and then grows into an unstable band. This instability gives rise to patterns with typical length scale.

FIGURE 3

(a and b) Stability phase diagrams in the f-versus-T plane. The actin force leads to an increase in the critical temperature. The type-II instability is given by the shaded region. (a) We use σ = 10 and α = 10. In all the graphs in this work we use ɛ = 0.05. In the inset of panel a, we plot the same diagram but now for ɛ = 0.2, and we find that the temperature range drastically increases. (b) The solid lines represent the phase-separation transition (dashed lines give the type II-type I transition), using σ = 50 and blue α = 0, green α = 70, and red α = 90. The passive system is represented by the line f ≡ 0. (c and d) Stability phase diagrams in the f-versus-α plane. Here we find that increase in both f and α have a similar effect in driving the system into the unstable regime. (c) T = 0.997T₀ and σ = 10. (d) The solid lines represent the phase-separation transition (dashed lines give the type II-type I transition), using σ = 10 and blue T = 1.005T₀, green T = 0.9985T₀, and red T = 0.997T₀. For Σ > Σ_cr, only type-I instability is possible.

FIGURE 4

Graphs of versus T, for σ = 10 and various values of f and α. (a) f = 0, α grows counterclockwise (α = 0, 10, 20, 40, 60); the scale is Log () versus linear (T). (b) Same graph as panel a, but on linear-linear scale. The solid line give q_p while the dashed lines give q_n. Both are in units of 1/R, where R = 100 nm. (c) α = 0, f grows counterclockwise (f = 0, 10, 30, 45, 60); the scale is Log () versus linear (T); the cutoff temperature is T_cut = (J(σ – α(φ₀ + 1/2)) + ɛ²κ(f + α (φ₀ + 1/2) – σ))/(4(σ – α(φ₀ + 1/2))(1 + 4 )). (d) Same graph as panel c, but on linear-linear scale; again, has the same linear asymptotic dependence on T for T ≪ T₀.

FIGURE 5

Stability phase diagram in the adhesion(α)-temperature plane. We find that the adhesion can drive the instability, increasing the transition temperature. (a) General features of the phase transition line (solid line), for f = 4. The shaded region shows the type-II instability (bounded by solid and dashed lines). The passive system is represented by the x axis where α = 0. At α₀ (horizontal dashed line) the transition temperature diverges. (b) Transitions lines (solid line) for various values of the actin force f: from top to bottom, f = 0, 4, 10.

FIGURE 6

Stability phase diagram in the surface-tension(σ)-temperature plane. We find that the increase in the surface tension decreases the instability temperature below the thermodynamic phase transition temperature. Increasing f or α increases again the instability temperature. (a) Phase-separation lines (solid lines) for various values of f (and zero adhesion α = 0); from top to bottom, f = 20, 4, 0.1. The dashed lines give the transition between type-II and type-I instability. (b) Phase-separation lines (solid lines) for various values of α (and negligible actin force f = 0.1); from top to bottom, α = 20, 0. The dashed lines give the region of type-II instability. The horizontal black line indicates the value σ = αφ₀, below which the system is unstable at all temperatures. The vertical black line gives the lower critical temperature T₁₀, below which the system is unstable (type-I) for all surface tensions.

FIGURE 7

(a) Contours of the wavevector of the most unstable mode q^* in the (α, f) stability phase diagram (T = 0.997T₀). The heavy black dashed line shows the mixed-type-II phase transition and the heavy solid black line shows the type-II→type-I phase transition. In panels b and c, we plot the wavevectors of the unstable region (q^*, blue line; q_p, red line; and q_n, green line) for f = 10 and f = 0, respectively (the wavevectors are in units of 1/R, where R = 100 nm). Below the contour plot, we show the expected shapes of cells along the trajectory indicated by the heavy arrow.