Chiral twist drives raft formation and organization in membranes composed of rod-like particles

Phase separation into a raft phase containing only fd virus (orange) and a background phase containing both fd and M13 (purple) viruses. (A) For a completely phase-separated membrane (Left), the area fraction of the raft phase is α2=αt2; equivalently, if the raft phase formed a single circular domain as depicted, it would have radius αtRt. As fd viruses enter into the M13-rich phase (Center), the area fraction of the raft phase decreases to α2<αt2. For a completely mixed membrane (Right), α2=0. (B) Competition between the entropy of mixing and depletant entropy determines α. At low depletant concentration (Top), the mixed state is entropically preferred. Phase separation reduces the excluded volume and is preferred at high depletant concentration (Bottom). Green circles represent depletants and blue regions represent the excluded volume. (C) Introducing a shorter virus into a sea of longer ones (Top) increases the excluded volume less than introducing a longer virus into a sea of shorter ones (Bottom). (D) α for various αt and depletant concentrations c (Eq. 5). Values for other parameters are provided in Table 1. Schematics are not drawn to scale.

An Ising-like model for depletion-induced phase separation in the limit of small virus half-length difference d. (A) A membrane composed of longer L viruses (Left) contains more excluded volume per particle (blue) than a membrane composed of shorter S viruses (Center), which leads to a linear term in the Ising-like Hamiltonian Eq. S21. A virus pair consisting of one L and one S particle (Right) occupies more excluded volume than the average of a pair of two L particles and a pair of two S particles. This extra excluded volume (green) has approximate cross-sectional area ad2/2ξ and leads to quadratic terms in Eq. S21. (B) Assuming a hexagonal lattice of viruses, a nearest-neighbor triplet consisting of one L virus and two S viruses (red dotted triangle, for example) produces slightly more excluded volume than a triplet consisting of two L viruses and one S virus (blue dotted triangle, for example). These small additions (dark red regions) and reductions (dark blue regions) in excluded volume contribute cubic terms in Eq. S21. Schematics are not drawn to scale.

Raft size and chiral structure. (A) Schematics of two membranes with the same degree of phase separation and thus the same raft area fraction α2 containing either several smaller rafts (Left) or one larger raft (Right). (B) A single circular domain with a single circular raft is repeated to approximately tile the membrane. (C) Structure of the domain along the light blue plane in B. Along the radial coordinate r, the fd viruses (orange) twist from θ(0)=0 to θ(αR)=θ0 at the raft–background interface with one handedness, and the background viruses, containing mostly M13 virus (purple), twist from θ(αR)=θ0 to θ(R)=0 at the domain edge with the other handedness. (D and E) The effect of depletants (green circles) on raft structure and organization. (D) Between two membranes of equal volume, the one with more interface between raft and background (Right) has greater excluded volume (blue), leading to an interfacial line tension proportional to d. (E) Between two membranes of equal volume, the one whose viruses are tilted at angle θ (Right) has greater excluded volume, leading to a free energy term proportional to θ2 to leading order. (F) Tilt angle θ(r) (Eq. 11) for domains whose common twist penetration depth λ≡λ1≈λ2 is much less or much greater than their radius R. (G) Maximum twist angle θ0 (Eq. 12) as a function of λ and the twist wavenumber difference Δq≡q1−q2. Darker cyan indicates larger θ0. (H) Raft radius αR as a function of λ and Δq, calculated numerically. Darker red indicates smaller αR. We assume the large membrane limit Rt→∞. The maximum raft radius αRt corresponds to a membrane having only a single raft, a regime separated by a gray dashed line from membranes with multiple smaller rafts (Eq. 15). This line is reproduced in G. For G and H, α=0.3 and values for other parameters are provided in Table 1. Schematics are not drawn to scale.

Retardance values D for rafts of various radii αR. The points indicate experimental data and the lines indicate theoretical results calculated with αt=0.5 and the parameter values in Table 1, corresponding to twist penetration depth λ∼0.8μm and chiral wavenumber difference Δq=0.5μm−1. α is given by Eq. 5 and R is adjusted to produce rafts of different radii. Experimental data and methods are reported in ref. 18.

Raft–raft repulsion. (A) The approach of two rafts is modeled as raft shifts b0 with respect to their circular tiling domains. (B) Shifted polar coordinate system of the background membrane (Eq. 17). Dashed lines indicate curves of constant r from r=αR (red) to r=R (blue), which are circles of radius r whose centers (dots) lie at x=b(r) and y=0. (C) Raft–raft repulsion energy ΔF divided by temperature T for rafts of various radii αR. The points indicate experimental data and the lines indicate theoretical results calculated with αt=0.5 and the parameter values in Table 1, corresponding to twist penetration depth λ∼0.8μm and chiral wavenumber difference Δq=0.5μm−1. α is given by Eq. 5 and R is adjusted to produce rafts of different radii. Experimental data and methods are reported in ref. 18. Schematics are not drawn to scale.