Mechanochemical Rules for Shape-Shifting Filaments that Remodel Membranes

The sequential exchange of filament composition to increase filament curvature was proposed as a mechanism for how some biological polymers deform and cut membranes. The relationship between the filament composition and its mechanical effect is lacking. We develop a kinetic model for the assembly of composite filaments that includes protein–membrane adhesion, filament mechanics and membrane mechanics. We identify the physical conditions for such a membrane remodeling and show this mechanism of sequential polymer assembly lowers the energetic barrier for membrane deformation.

Figure 1

Shape-shifting by sequential monomer exchange. (a) In vitro experiments, tracking the presence of different monomer types (termed Snf7, Vps2, and Did2) in membrane-bound yeast ESCRT-III filaments, showed that the filament composition changes according to a precise sequence [11], which is accompanied by the membrane deformation. (b) The geometry of a membrane-bound filament has been proposed to change together with its composition, driving membrane remodeling for cargo transport.

Figure 2

Model setup. (a) Pure filaments made up of type-$i$ monomers ($i=0$, 1, 2) have in-plane curvature $c_i$ and are associated with an effective pitch $p_i$, causing the preferred membrane deformation depth $p_i$. (b) Three examples of the lattice composition over time with the associate overall filament shape, which gives an average pitch $P$, an average radius $1/C$, and a corresponding membrane shape. For a shallow pitch and large radius, the geometry of the membrane is that of a spherical cap; whereas with increasing pitch or decreased radius, a tube with a spherical cap end forms.

Figure 3

Geometry and adhesion control sequentiality. The effect of the pitches, curvatures, and adhesion energies for the three types of monomers on the filament composition (percent of lattice coverage). When monomers are all equal, (a) the filament stabilizes as a composite mixture of the three monomers in equal parts. When the monomer types have different pitches, (b) the binding of high-pitch monomers is delayed. Differences in curvature and adhesion energy together (c) result in the unbinding of weak-binding monomers, due to bending frustration. (d) Differences in pitch, curvature, and adhesion energy are all three needed to achieve sequential assembly and disassembly. Parameters are as follows: equal pitches [(a) and (c)], $p_0=p_1=p_2=0\mathrm{nm}$; otherwise, [(b) and (d)], $p_0=0\mathrm{nm}$, $p_1=30\mathrm{nm}$, and $p_2=45\mathrm{nm}$. Equal curvatures [(a) and (b)], $c_0=c_1=c_2=(15\mathrm{nm}{)}^{-1}$ and $E_0^{\mathrm{adh}}=E_1^{\mathrm{adh}}=E_2^{\mathrm{adh}}=6.5k_{\mathrm{B}}T$; otherwise, [(c) and (d)], $c_0=(30\mathrm{nm}{)}^{-1}$, $c_1/c_0=2.5$, $c_2/c_0=8$, and $E_0^{\mathrm{adh}}=6.5k_{\mathrm{B}}T$, $E_1^{\mathrm{adh}}/E_0^{\mathrm{adh}}=3$, $E_2^{\mathrm{adh}}/E_0^{\mathrm{adh}}=12$. Time is in units of ${10}^3$ simulation steps.

Figure 4

Sequentiality and membrane deformation. (a) The energy of the system as a function of type-2 monomers in the filament: when the full $0\to 1\to 2$ sequence is present (turquoise), and when monomer 2 is the only one present (gray). The inset shows an energy barrier of $39k_{\mathrm{B}}T$ for polymerization of monomer 2 on the membrane when no other monomers are involved. (b) Dependence of membrane deformation on monomer unbinding. If the earlier monomer types do not unbind ($c_0=c_1=c_2$), the membrane indentation is decreased. In all the other cases ($p_0=p_1=p_2$, $E_0^{\mathrm{adh}}=E_1^{\mathrm{adh}}=E_2^{\mathrm{adh}}$), type-2 monomers do not dominate in the final filament. The largest membrane deformation is achieved with full sequential binding and unbinding. (c) Absence of activity in our model (in turquoise) leads to sequentiality and optimal membrane indentation only for very large filament rigidities. Releasing a sufficiently large energy, $3E_{\mathrm{ATP}}$, to elastically frustrated monomers recovers large indentation at physiologically relevant rigidities, which are estimated from experiments and models [13, 28] (orange band).