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Research Article

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

View ORCID ProfileLouis Kang and Tom C. Lubensky
PNAS January 3, 2017 114 (1) E19-E27; first published December 20, 2016; https://doi.org/10.1073/pnas.1613732114
Louis Kang
aDepartment of Physics & Astronomy, University of Pennsylvania, Philadelphia, PA 19104
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  • For correspondence: lkang@mail.med.upenn.edu
Tom C. Lubensky
aDepartment of Physics & Astronomy, University of Pennsylvania, Philadelphia, PA 19104
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  1. Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved November 21, 2016 (received for review August 25, 2016)

This article has a Correction. Please see:

  • Correction for Kang and Lubensky, Chiral twist drives raft formation and organization in membranes composed of rod-like particles - August 06, 2018
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  • Fig. 1.
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    Fig. 1.

    Overview of two-species colloidal membrane experiments. (A) Virus particles and dextran molecules act as rod-shaped colloids and spherical depletants, respectively. fd viruses are shorter and prefer right-handed twist. M13 viruses are longer and prefer left-handed twist. (B–D) Differential interference contrast image (Top Left), fluorescence image with fd labeled (Top Right), and schematic (Bottom) of colloidal membranes. (B) At a low dextran concentration of 41,000 μm−3, the two virus species completely mix. (C) At an intermediate dextran concentration of 46,000 μm−3, several smaller rafts of fd virus form in a partially phase-separated background. (D) At a high dextran concentration of 62,000 μm−3, the two virus species completely phase separate. (E) Rafts exchange rods with the background membrane to attain a thermodynamically preferred size. Fluorescence images with fd labeled are taken 6.7 h apart. Green and purple circles track two rafts that start, respectively, smaller and larger than the preferred raft size. (F) Viruses adopt a twisted chiral structure. Shown is an LC-PolScope birefringence map with pixel brightness representing retardance, which indicates virus tilt toward the membrane plane. (G) Rafts repel one another. Fluorescence images with fd labeled are taken 5 s apart. Two optical plows consisting of multiple light beams (red circles) bring two rafts together and are then switched off. (All scale bars, 5 μm.) Experimental data and methods are reported in ref. 18. Schematics are not drawn to scale. Microscopy images were reprinted by permission from Macmillan Publishers Ltd. (18).

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    Fig. 2.

    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.

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    Fig. S1.

    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.

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    Fig. 3.

    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.

  • Fig. 4.
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    Fig. 4.

    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.

  • Fig. 5.
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    Fig. 5.

    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.

  • Fig. S2.
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    Fig. S2.

    Phase separation using a total free energy incorporating raft structure and organization. α2 is the area fraction of the raft phase, and it is obtained by numerically minimizing the sum of Eqs. 4 and 13 of the main text over a range of depletant concentrations c. We use the same parameter values given in Table 1 of the main text, except that the chiral wavenumbers q𝑓𝑑 and qM13 are multiplied by a factor of 1 (solid lines), 2 (dashed lines), or 2.5 (dotted lines). The solid lines are indistinguishable from Fig. 2D of the main text, which plots Eq. 5. As the chiral wavenumber difference increases in magnitude past a critical value, α increases beyond its value in Eq. 5 and phase separation begins at lower values of c.

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    Table 1.

    Membrane parameters and their values

    ParameterVariableExperimental estimateSourceModel value
    fd-Y21M half-lengthl𝑓𝑑430 nm(18)*Same
    M13KO7 half-lengthlM13560 nm(18)*Same
    Virus half-length differenced130 nmlM13−l𝑓𝑑Same
    Virus diameter7 nm(18)
    Virus nearest-neighbor distanceξ12 nm(19)Same
    Virus 2D concentrationcv∼9,000 μm−21/π(ξ/2)28,500 μm−2
    fd-Y21M Frank constantK𝑓𝑑∼2 pN(1)†,‡4 pN
    M13KO7 Frank constantKM13∼4 pN(6)‡10 pN
    fd-Y21M twist wavenumberq𝑓𝑑∼0.1 μm−1(7)‡0.11 μm−1
    M13KO7 twist wavenumberqM13∼ − 0.5 μm−1(6)‡− 0.55 μm−1
    fd-Y21M birefringenceΔn𝑓𝑑∼0.008(8)†,§0.011
    M13KO7 birefringenceΔnM13∼0.008(8)†,§0.011
    Dextran concentrationc48,000 μm−3(18)Same
    Dextran radiusa∼25 nm(23–25)¶Same
    TemperatureT22 °C(18)Same
    • ↵* Half the end-to-end length estimated from contour lengths and persistence lengths.

    • ↵† Measured for fd-wt virus.

    • ↵‡ Imprecise estimates extrapolated to membrane virus concentration ∼200 mg ⋅ mL− 1 (corresponding to cv∼ 9,000 μm−2) based on concentration-dependent behavior of fd-wt suspensions (1).

    • ↵§ Assuming membrane nematic order parameter of 1 and virus concentration ∼200 mg⋅mL− 1 (corresponding to cv∼9,000 μm−2).

    • ↵¶ Hydrodynamic radii for dilute solutions of 500 kDa dextran, whereas our experiments are in the semidilute regime.

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Liquid crystal chirality stabilizes membrane rafts
Louis Kang, Tom C. Lubensky
Proceedings of the National Academy of Sciences Jan 2017, 114 (1) E19-E27; DOI: 10.1073/pnas.1613732114

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Liquid crystal chirality stabilizes membrane rafts
Louis Kang, Tom C. Lubensky
Proceedings of the National Academy of Sciences Jan 2017, 114 (1) E19-E27; DOI: 10.1073/pnas.1613732114
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  • Article
    • Abstract
    • Phase Separation Between Virus Species
    • Landau Coefficients for Phase Separation
    • Raft Organization and Structure
    • Derivation of the Single-Domain Free Energy
    • Calculation of the Virus Tilt Angle
    • Raft–Raft Repulsion
    • Calculation of the Raft Shift Free Energy
    • Discussion
    • General Membrane Rafts Formed from Chiral Rod-Like Particles
    • Chiral Contribution to Phase Separation
    • Overview of the Linear System
    • Acknowledgments
    • Footnotes
    • References
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