Chiral twist drives raft formation and organization in membranes composed of rod-like particles
CChiral twist drives raft formation and organization in membranes composed ofrod-like particles
Louis Kang ∗ and T. C. Lubensky Department of Physics & Astronomy, University of Pennsylvania,209 South 33rd Street, Philadelphia, Pennsylvania 19104, USA (Dated: September 29, 2018)Lipid rafts are hypothesized to facilitate protein interaction, tension regulation, and traffickingin biological membranes, but the mechanisms responsible for their formation and maintenance arenot clear. Insights into many other condensed matter phenomena have come from colloidal systems,whose micron-scale particles mimic basic properties of atoms and molecules but permit dynamic vi-sualization with single-particle resolution. Recently, experiments showed that bidisperse mixtures offilamentous viruses can self-assemble into colloidal monolayers with thermodynamically stable raftsexhibiting chiral structure and repulsive interactions. We quantitatively explain these observationsby modeling the membrane particles as chiral liquid crystals. Chiral twist promotes the formation offinite-sized rafts and mediates a repulsion that distributes them evenly throughout the membrane.Although this system is composed of filamentous viruses whose aggregation is entropically drivenby dextran depletants instead of phospholipids and cholesterol with prominent electrostatic interac-tions, colloidal and biological membranes share many of the same physical symmetries. Chiral twistcan contribute to the behavior of both systems and may account for certain stereospecific effectsobserved in molecular membranes.
I. INTRODUCTION
Filamentous viruses have proven to be a fruitful col-loidal system [1–19]. They serve as monodisperse, rigid,and chiral rods that are approximately one micron inlength and interact effectively through hard-core re-pulsion [2, 7]. When suspended in an aqueous solu-tion at increasing concentrations, they transition froma disordered isotropic phase to a cholesteric (chiral ne-matic) phase characterized by alignment along a directorfield that twists with a preferred handedness and wave-length [1, 6]. The addition of a non-adsorbing polymersuch as dextran induces lateral virus-virus attraction viathe depletion interaction [10, 12, 20, 21]. The virusesself-assemble into monolayers that exhibit fluid-like dy-namics internally [10] and sediment to the bottom ofglass containers, which are coated with a polyacrylamidebrush to suppress depletion-induced virus-wall attrac-tions [22]. The rich physics and phenomenology of mem-branes formed from single virus species have been thor-oughly studied [8–17, 19]. However, two-species mem-branes demonstrate a novel set of behaviors which arenot adequately understood [18]. We will review these be-haviors now before describing a theory that can explainthem. fd -Y21M and M13KO7, which we will shorten to fd and M13 for convenience, are two species of filamen-tous virus that have slightly different lengths and formcholesteric phases of opposite handednesses (Table Iand Fig. 1a). Membranes composed of both fd andM13 viruses are circular with interior particles alignedlargely perpendicularly to the membrane plane and edge ∗ [email protected] particles tilted azimuthally, as in single-species mem-branes [13]. At low dextran concentrations, the twospecies are fully mixed, and at high dextran concentra-tions, the two are fully phase-separated with M13 virusessurrounding a single fd domain (Fig. 1b,d). At intermedi-ate concentrations, membranes exhibit partial phase sep-aration with several smaller circular rafts of fd virusesdistributed within a mixed background of both species(Fig. 1c).Particle tracking experiments show that fd viruses dif-fuse in and out of these rafts [18], allowing for equili-bration to a thermodynamically preferred raft size over ∼
24 h (Fig. 1e). Polarized light microscopy suggests thatthe raft system has a chiral structure, with particles tilt-ing around the interfaces between rafts and backgroundmembrane and around the membrane edge (Fig. 1f). Fi-nally, the rafts are distributed homogeneously through-out the membrane and never coalesce, indicating a long-ranged repulsion between rafts (Fig. 1g). This interac-tion can be measured quantitatively by bringing two raftsclose together with optical traps and tracking their tra-jectories upon release of the traps [18].The simplicity of this colloidal membrane system al-lows us to study it theoretically with a model builtfrom established physical principles and experimentallymeaningful parameters. Its components have well-characterized interactions: dextran molecules act as de-pletants that interact with viruses through hard-bodyinteractions [10, 12, 20, 21, 26], and the hard-body in-teractions between viruses can be coarse-grained as theFrank free energy for chiral liquid crystals [1, 6]. We pre-viously used such a model to investigate single-speciesmembranes and succeeded in reproducing a variety ofstructural, dynamical, and phase phenomena with a sin-gle set of realistic parameter values [19]. Extending themodel to the two-species system will demonstrate how a r X i v : . [ c ond - m a t . s o f t ] O c t b xy Complete mixing d Partial phase separationwith numerous rafts Dextran + a + M13KO7 virusDextranconcentration: Low Intermediate HighThermodynamically preferred size e xy Raft-raft repulsion g xy Raft properties fd -Y21M virus xy c f Chiral structureComplete phase separationwith a single raft
FIG. 1. Overview of two-species colloidal membrane experiments. a , Virus particles and dextran molecules act as rod-shapedcolloids and spherical depletants, respectively. fd viruses are shorter and prefer right-handed twist. M13 viruses are longerand prefer left-handed twist. b – d , Differential interference contrast image (top left), fluorescence image with fd labeled (topright), and schematic (bottom) of colloidal membranes. b , At a low dextran concentration of 41 000 µ m − , the two virus speciescompletely mix. c , At an intermediate dextran concentration of 46 000 µ m − , several smaller rafts of fd virus form in a partiallyphase-separated background. d , At a high dextran concentration of 62 000 µ m − , the two virus species completely phaseseparate. e , Rafts exchange rods with the background membrane to attain a thermodynamically preferred size. Fluorescenceimages with fd labeled taken 6 . f , Viruses adopt a twisted chiral structure. LC-PolScope birefringence map with pixel brightnessrepresenting retardance, which indicates virus tilt toward the membrane plane. g , Rafts repel one another. Fluorescence imageswith fd labeled taken 5 s apart. Two optical plows consisting of multiple light beams (red dots) bring two rafts together andare then switched off. All scale bars, 5 µ m. Experimental data and methods are reported in Ref. [18]. Schematics not drawnto scale. Microscopy images reproduced with permission from Nature Publishing Group. the intruiging behaviors depicted in Fig. 1 emerge fromFrank free energy, depletant entropy, and mixing entropy.The fundamental principles we encounter on the col-loidal scale may apply to similar but less tractable molec-ular systems whose particles and interactions share thesame physical symmetries. Colloidal systems have per-mitted the investigation of many quintessential con-densed matter phenomena with single-particle resolu-tion and exquisite control. For example, spherical col- loids exhibit crystal nucleation [27, 28] and glassy dy-namics [29, 30]; the addition of an isotropic attractionwith depletants allows them to demonstrate liquid-gasphase separation [31], thermal capillary waves [32], andwetting [33]. And in addition to the aforementionedwork in which filamentous viruses form nematic andcholesteric liquid crystal phases, plate-like and rod-likecolloids have shed insight on columnar and smectic liquidcrystal phases, respectively [34, 35]. Phospholipid fluid TABLE I. Membrane parameters and their values.Parameter Variable Experimental estimate Reference(s) Model value fd -Y21M half-length l fd
430 nm [18] a sameM13KO7 half-length l M13
560 nm [18] a sameVirus half-length difference d
130 nm l M13 − l fd sameVirus diameter 7 nm [18]Virus nearest-neighbor distance ξ
12 nm [19] sameVirus 2D concentration c v ∼ µ m − /π ( ξ/ µ m − fd -Y21M Frank constant K fd ∼ bc K M13 ∼ c
10 pN fd -Y21M twist wavenumber q fd ∼ . µ m − [7] c . µ m − M13KO7 twist wavenumber q M13 ∼− . µ m − [6] c − . µ m − fd -Y21M birefringence ∆ n fd ∼ .
008 [8] bd . n M13 ∼ .
008 [8] bd . c
48 000 µ m − [18] sameDextran radius a ∼
25 nm [23–25] e sameTemperature T ◦ C [18] same a Half the end-to-end length estimated from contour lengths and persistence lengths. b Measured for fd -wt virus. c Imprecise estimates extrapolated to membrane virus concentration ∼
200 mg mL − (corresponding to c v ∼ µ m − ) based onconcentration-dependent behavior of fd -wt suspensions [1]. d Assuming membrane nematic order parameter of 1 and virus concentration ∼
200 mg mL − (corresponding to c v ∼ µ m − ). e Hydrodynamic radii for dilute solutions of 500 kDa dextran, whereas our experiments are in the semidilute regime. membranes are another important soft-matter system;yet, due to our inability to directly visualize real-time dy-namics of lipid bilayers at the nanometer scale, many pro-cesses remain poorly understood. Following the analogybetween colloids and molecular substances, our theoreti-cal investigation of two-component colloidal membranesmay provide new, universal understanding about mem-brane rafts, which have been observed in experimentalphospholipid membranes [36, 37] but remain controver-sial in the case of biological membranes [38].As shown in previous work [39, 40] based on phe-nomenological models, the difference in chirality betweentwo coexisting phases, which favors different twist ratesof viruses relative to membrane normals, is the primarydriver of raft formation in viral membranes. When twoachiral phases coexist, the interface separating them hasa positive line tension (or surface tension in three di-mensions) that favors the smallest possible interfaciallength (or area). Chirality difference introduces an ef-fective negative contribution to the line tension, whichfor large enough difference becomes negative and favorsas much interfacial length as possible. Finite-size raftsare a result of the competition between negative line ten-sion and either repulsive interaction between segments ofinterface or interfacial curvature energy. The repulsiveenergy between rafts as they approach each other arisesfrom compression of the twist in the membranes’ back-ground phase. The formation of rafts and their mutualinteraction follows this fundamental physics in our calcu-lations that are based on the particular depletion physicsof viral rafts.The next few sections describe, respectively, the pro-cess of phase separation that generates raft and back- ground phases, the organization of the raft phase into do-mains with a preferred size and chiral structure, and therepulsion between rafts mediated by the chiral structureof the background phase. Each section includes theoret-ical development, results, and comparison to experimen-tal data. In the last section, we discuss the assumptionsmade by our theory, its contribution to the literature onheterogeneous membranes, and implications for phospho-lipid membranes.
II. PHASE SEPARATION BETWEEN VIRUSSPECIES
We start by investigating the separation of membraneparticles into two phases, one which we call the “back-ground” phase containing mostly M13 viruses completelysurrounding the other which we call the “raft” phasecontaining fd viruses, in accordance with experiment(Fig. 1). The structure of the phases, including the num-ber and size of rafts present, does not yet concern us.We assume a large circular membrane of radius R t → ∞ and henceforth ignore effects of the outer boundary. Thedegree of phase separation is parametrized by α , thearea fraction of the raft phase (Fig. 2a). It ranges be-tween α = 0, which corresponds to complete mixing,and α = α , which corresponds to complete phase sep-aration. α is determined experimentally by the fractionof fd virus provided in the initial suspension. For in-termediate values of α , some fd particles leave the raftand enter the background, producing a partially mixedbackground phase containing both viruses.Competition between two factors determines the de-
35 40 45 50 55 60 65 70 7500.20.40.6 c @ m m - D a a t = α R tt R t α R t a d R t R t z x d a c z x b ξ d a 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 phaseis α = α ; equivalently, if the raft phase formed a single cir-cular domain as depicted, it would have radius α t R t . As fd viruses enter into the M13-rich phase (middle), the area frac-tion of the raft phase decreases to α < α . For a completelymixed membrane (right), α = 0. b , Competition betweenthe entropy of mixing and depletant entropy determines α .At low depletant concentration (top), the mixed state is en-tropically preferred. Phase separation reduces the excludedvolume and is preferred at high depletant concentration (bot-tom). Green circles represent depletants and blue regionsrepresent the excluded volume. c , Introducing a shorter virusinto a sea of longer ones (top) increases the excluded volumeless than introducing a longer virus into a sea of shorter ones(bottom). d , α for various α t and depletant concentrations c (Eq. 5). Values for other parameters are provided in Table I.Schematics not drawn to scale. gree of phase separation. Thermal forces encourage thedepletants to explore as much physical space as possible.To do so, they must minimize the volume excluded totheir centers of mass by the membrane, which can be ac-complished by separating viruses of different lengths intodifferent phases. A shorter fd particle produces moreexcluded volume when surrounded by longer M13 parti-cles (Fig. 2b). For depletant particles small compared tothe dimensions of the membrane, the excluded volumeis approximately V + aA , where V is the volume of themembrane, A is the surface area of the membrane, and a is the depletant radius [41]. Their free energy is cal-culated via the ideal gas partition function V N a /N !Λ N applied to N depletant molecules, where Λ is their ther-mal de Broglie wavelength. The volume available to thedepletants can be written as V a = V t − V − aA , where V t (cid:29) V is the total volume of the virus-and-depletantsuspension [42]. Ignoring constant terms, the depletantfree energy is generically F dep = − N T log V t − V − aAV t ≈ cT ( V + aA ) (1)where c is the depletant concentration and T is the tem-perature. We use units in which the Boltzmann constantis unity.However, thermal forces also encourage binary fluidsto adopt disordered phases in which the two species aremixed. This tendency is described quantitatively by theentropy of mixing [43]. As depicted in Fig. 2a, the mixedbackground phase of total area (1 − α ) πR is formedfrom an area ( α − α ) πR of fd viruses and an areaof (1 − α ) πR of M13 viruses, yielding respective areafractions φ fd = α − α − α and φ M13 = 1 − α − α = 1 − φ fd . (2)The entropy of mixing per particle of the backgroundphase is s mix = φ fd log φ fd + φ M13 log φ M13 . (3)We only consider mixing in the background phase be-cause introducing the longer M13 viruses into the raftphase is disfavored by the depletants. Their surface pro-trusions would be surrounded by extra excluded volumeof order da per M13 particle, unlike the smaller amountof excluded volume of order d ( ξ/ per fd particle re-quired to introduce the shorter fd viruses into the back-ground phase (Fig. 2c). d ≡ l M13 − l fd is the virus half-length difference, a is the depletant radius, and ξ is thenearest-neighbor virus separation (Table I). We thus ig-nore mixing in the raft phase due to these asymmetriceffects of surface convexity and concavity on the deple-tion free energy.Combining the mixing entropy Eq. 3 and the depletionfree energy Eq. 1, which respectively disfavor and favorphase separation, gives the free energy F sep πR T = c v (cid:20) (1 − α ) log 1 − α − α + ( α − α ) log α − α − α (cid:21) + 2 cd ( α − α ) , (4)where c is the 3D depletant concentration and c v is the2D virus concentration in the membrane. Minimizing F sep with respect to α produces the result α = (cid:115) α − e − cd/c v − e − cd/c v cd/c v ≥ log 1 /α t cd/c v ≤ log 1 /α t . (5)where c is the 3D depletant concentration, c v is the 2Dvirus concentration in the membrane, and d is the half-length difference between the two species. In Fig. 2d, α ( c ) is plotted for various α t ’s using values in Table I.For each α t , there is complete mixing ( α = 0) below acritical depletant concentration ( c v /d ) log 1 /α t . Abovethis critical c , the system partially phase-separates andapproaches complete phase separation for c → ∞ . Thisbehavior qualitatively agrees with experimental resultsin Fig. 1b–d over the experimental range of depletantconcentrations c . III. RAFT ORGANIZATION AND STRUCTURE
Assuming we are in the regime cd/c v > log 1 /α t inwhich rafts exist, we now analyze their structure. Equa-tion 5 determines the total amount of fd virus sequesteredinto the raft phase by setting the value of α , but doesthis phase form a single large raft or several smaller rafts(Fig. 3a)? And how are the virus particles aligned? Wewill see that these questions are related via the naturaltendency of chiral rods to adopt twisted configurations.To answer them, we need to derive the structural freeenergy of the membrane.We take the membrane of radius R t → ∞ to be approx-imately tiled by circularly symmetric domains of radius R (Fig. 3b), as in the muffin-tin approximation of solidstate physics [44]. There are R /R domains and thetotal membrane free energy is F struct = R R F domain , (6)where F domain is the free energy of a single domain, whichcontains one raft of radius αR . The fd particles point ver-tically at the center of the raft and twist azimuthally withone handedness to their interface with the background,where they attain twist angle θ . The background parti-cles, which are mostly M13 with a smaller amount of fd ,twist with opposite handedness from θ at the interface to0 at the domain edge, where the next domain would begin(Fig. 3c). Once the membrane separates into its thermo-dynamically preferred raft and background phases, weassume zero net particle current between the phases and between each phase and the aqueous environment. Wealso assume that the 2D particle concentration c v in themembrane is constant. Thus, the volume of each phaseis conserved, so any effects of depletion must only act onthe surface area of the membrane (Eq. 1). For mathe-matical tractability, we assume the particles do not twistvery much, so their tilt angle satisfies θ (cid:28)
1, and thetwo virus species have similar half-lengths l fd and l M13 ,so their half-length difference satisfies d (cid:28) l fd ≈ l M13 . Ascalculated in Ref. [19], virus position fluctuations perpen-dicular to the membrane are strongly suppressed in the θ (cid:28) l fd cos θ and 2 l M13 cos θ , respectively. F domain consists of three components. First, interfacesbetween raft and background have a half-height differ-ence of approximately d . These vertical offsets, which ap-pear as “corners” in Fig. 3d, contribute additional mem-brane surface area and, through the depletion free energyEq. 1, produce an effective interfacial line tension propor-tional to d . Second, virus tilt away from the membranenormal also increases the membrane surface area and,also through depletion, produces an effective alignmentenergy proportional to θ (Fig. 3e). Third, the virusparticles behave as chiral nematic liquid crystals [1, 6].That is, each species prefers to be aligned in a twistedconfiguration with wavenumber q , where the sign of q indicates the chirality of twist (positive corresponds toright-handed) and 2 π/ | q | is the wavelength. The ener-getic cost of deviations from this preferred configurationis given by the Frank free energy [45]: F Frank = K (cid:90) d x l cos θ × (cid:2) ( ∇ · n ) + ( ∇ × n ) − q n · ∇ × n (cid:3) . (7) n is the nematic director, K is the 3D Frank elastic con-stant in the one-constant approximation, q is the pre-ferred twist wavenumber associated with intrinsic chi-rality of the constituent particles, l is the particle half-length, and θ is the particle tilt angle. For raft domainsdepicted in Fig. 3b–c, the nematic director is circularlysymmetric and tilts away from the membrane normal inthe negative azimuthal direction: n ( r ) = − sin θ ( r ) ˆ φ + cos θ ( r ) ˆz . (8)The complete derivation of F domain is given in Supporting Information, and it leads to the structural free energy F struct πcaT = R R (cid:40) dαR − (cid:2) λ q − λ q (cid:3) αRθ + 12 (cid:2) λ + λ (cid:3) θ + (cid:90) αR d r (cid:20) rθ + λ (cid:18) r ( ∂ r θ ) + θ r (cid:19)(cid:21) + (cid:90) RαR d r (cid:20) rθ + λ (cid:18) r ( ∂ r θ ) + θ r (cid:19)(cid:21) (cid:41) . (9)The subscripts 1 and 2 refer to raft and background phases respectively. An important lengthscale λ j ≡ (cid:112) K j l j /caT arises from comparing the Frank twist and depletion contributions to the free energy, where j ∈ { , } . The latterpenalizes nonzero θ ( r ) and the former penalizes gradients in θ ( r ), so λ j acts like a twist penetration lengthscale. Since q = l @ m m D D q @ m m - D l ` R l p R a R R r q q H r L a f R α R zr R α R θ b c g d a z x z x L a θ L ∕ cos θ d e a R @ m m D = a R = a R t l @ m m D D q @ m m - D h FIG. 3. Raft size and chiral structure. a , Schematics of two membranes with the same degree of phase separation and thus thesame raft area fraction α containing either several smaller rafts (left) or one larger raft (right). b , A single circular domainwith a single circular raft is repeated to approximately tile the membrane. c , Structure of the domain along the light blue planein b . Along the radial coordinate r , the fd viruses (orange) twist from θ (0) = 0 to θ ( αR ) = θ at the raft-background interfacewith one handedness, and the background viruses, containing mostly M13 virus (purple), twist from θ ( αR ) = θ to θ ( R ) = 0at the domain edge with the other handedness. d – 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 greaterexcluded 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 proportionalto θ to leading order. f , Tilt angle θ ( r ) (Eq. 11) for domains whose common twist penetration depth λ ≡ λ ≈ λ is muchless or much greater than their radius R . g , Maximum twist angle θ (Eq. 12) as a function of λ and the twist wavenumberdifference ∆ q ≡ q − q . Darker cyan indicates larger θ . h , Raft radius αR as a function of λ and ∆ q , calculated numerically.Darker red indicates smaller αR . We assume the large membrane limit R t → ∞ . The maximum raft radius αR t correspondsto 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 – h , α = 0 . only fd viruses compose rafts, q = q fd and λ = (cid:112) K fd l fd /caT . The corresponding expressions for the backgroundmust account for a mixture of virus species. Experiments demonstrate that cholesteric mixtures of fd -wt and fd viruses have intermediate twist wavenumbers that linearly interpolate between their pure values as a function ofrelative concentration [7]. We assume that the same behavior applies here to Frank constants and twist wavenumbersfor fd and M13 viruses: q = 1 − α − α q M13 + α − α − α q fd and λ = (cid:114) K l M13 caT , where K = 1 − α − α K M13 + α − α − α K fd . (10)Experimental estimates for K and virus half-length l are of the same order of magnitude for the two species (Table I).For better mathematical insight and clearer presentation of results, we will sometimes imagine that they are equal,so the two phases share the same λ ≡ λ ≈ λ . Another important parameter is ∆ q ≡ q − q , the difference betweenthe chiral wavenumbers of the raft and background.Minimization of F struct over the tilt angle θ ( r ) and the domain radius R yields the thermodynamically preferredmembrane structure. We first minimize over θ ( r ) with the boundary conditions θ (0) = 0, θ ( αR ) = θ , and θ ( R ) = 0: θ ( r ) = θ I ( s ) I ( αS ) 0 ≤ r ≤ αRθ K ( s ) /K ( S ) − I ( s ) /I ( S ) K ( αS ) /K ( S ) − I ( αS ) /I ( S ) αR ≤ r ≤ R, (11)where I ν and K ν are modified Bessel functions of the first and second kind, respectively, of order ν (the latter shouldnot be confused for Frank constants). Distances are rescaled by the twist penetration depths as s j = r/λ j and S j = R/λ j , for j ∈ { , } . Solving the Euler-Lagrange equation is described in Supporting Information. Equation 11is plotted in Fig. 3f. If the common twist penetration depth λ is much less than R , then the twist is exponentiallylocalized to the interface between raft and background, but if it is much greater than R , then the twist ∂ r θ extendsuniformly throughout the membrane.We then substitute Eq. 11 into Eq. 9, perform the integrals over r , and minimize over θ , the tilt angle at theinterface: θ = λ q − λ q λ I ( αS ) I ( αS ) + λ K ( αS ) /K ( S )+ I ( αS ) /I ( S ) K ( αS ) /K ( S ) − I ( αS ) /I ( S ) . (12)This equation is plotted in Fig. 3g. The magnitude of θ increases with λ and ∆ q , and its sign is determined by thesign of ∆ q .Substituting Eq. 12 back into F struct yields F struct πR caT = αR d − (cid:0) λ q − λ q (cid:1) λ I ( αS ) I ( αS ) + λ K ( αS ) /K ( S )+ I ( αS ) /I ( S ) K ( αS ) /K ( S ) − I ( αS ) /I ( S ) , (13)which only depends on the free parameter R through S and S . By minimizing over R , we numerically calculatethe preferred raft radius αR , remembering that α was determined in the previous section. Figure 3h shows that atlow λ and ∆ q , R adopts its maximum value, R t , so the membrane contains one large raft. As ∆ q increases pasta critical value, R prefers a finite value and the raft phase separates into several smaller rafts of radius αR . Forconstant ∆ q , increasing λ —or equivalently decreasing c —leads to more numerous, smaller rafts, which qualitativelyagrees with experimental observations in Fig. 1c–d. Note that the chirality inversion q → − q and q → − q yieldsthe mirror-image configuration θ ( r ) → − θ ( r ) via Eqs. 11 and 12 with same free energy Eq. 13.A large chiral twist wavenumber difference ∆ q indi-cates the proclivity of fd and M13 viruses to twist backand forth with opposite handednesses; however, deple-tants favor particle alignment perpendicular to the mem-brane. A large number of small rafts can satisfy bothtendencies, since the particles can twist back and forthover short distances while largely maintaining perpendic-ular alignment. In opposition is the positive interfacialline tension also generated by depletion, which prefers asmall number of large rafts in order to reduce the totalinterfacial length between raft and background phases.The competition between these factors sets the raft size,which we can see explicitly by expanding the free en-ergy to leading orders in R − , corresponding to the phasetransition between single- and multiple-raft membranes.With the simplification λ ≡ λ ≈ λ , Eq. 13 becomes F struct πR caT ∼ αR (cid:26) d − λ ∆ q + 332 λ ∆ q α R + 14 λ ∆ q e − − α ) R/λ (cid:27) . (14)Thus, virus chirality appends a correction term to the bare interfacial tension to produce the effective interfa-cial line tension 2 caT ( d − λ ∆ q / | ∆ q | = 2 d / λ − / . (15)In the multiple-raft regime where | ∆ q | exceeds this crit-ical value, the preferred raft size is αR ∼ (cid:115) λ ∆ q λ ∆ q − d , (16)indicating a second-order phase transition. Noticethat Eq. 14 is analogous to the free energy of the2D Frenkel-Kontorova model around the commensurate-incommensurate transition, with the first two terms cor-responding to an effective interfacial line tension betweenrafts and background, the third corresponding to whatcan be interpreted as an effective interfacial bending en- ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊʇ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú ÚÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏ r @ m m D D @ n m D a R @ m m D = Ê ‡ Ú Ï ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊʇ‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏ
FIG. 4. Retardance values D for rafts of various radii αR .The points indicate experimental data and the lines indicatetheoretical results calculated with α t = 0 . λ ∼ . µ m and chiral wavenumber difference ∆ q = 0 . µ m − . α is given by Eq. 5 and R is adjusted to produce rafts ofdifferent radii. Experimental data and methods are reportedin Ref. [18]. ergy, and the fourth corresponding to raft-raft repul-sion [43, 46]. The higher-order terms prevent a negativeeffective interfacial tension from decreasing the raft sizeto 0 and set the preferred size Eq. 16.To assess the validity of our model, we can comparemeasurements of optical retardance (Fig. 1f) to valuescalculated by our model. When polarized light passesthrough a birefringent material, the “ordinary” and “ex-traordinary” components propagate at different speeds,leading to a phase difference called retardance that wemeasure in wavelengths. For our membranes, it is ap-proximately given by D = 2∆ nl sin θ and is thus anindirect measure of the tilt angle θ [47]. The raw calcu-lated retardance profiles are convolved with a Gaussianof width 0 . µ m representing the microscope’s resolutionfunction, exactly as previously reported [8]. Figure 4shows good agreement between theoretical and experi-mental retardance profiles using the physically reason-able birefringence values reported in Table I. IV. RAFT-RAFT REPULSION
To model the interaction between two neighboring raftsas they approach each other, we shift each circular raftwithin its circular tiling domain off-center by a distance b towards each other (Fig. 5a). To accomplish this, thebackground membrane must be deformed; for simplicity,we assume that the rafts themselves are unchanged bythis shift. We parametrize the deformation by a shiftprofile b ( r ) such that Cartesian coordinates are given interms of shifted polar coordinates by x = r cos φ + b ( r ) and y = r sin φ. (17)In other words, the curves of constant r are nested non-concentric circles of radius r centered at x = b ( r ) and y = 0 (Fig. 5b). Our shift ansatz Eq. 17 breaks circularsymmetry into dipolar symmetry, implying that θ can ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊʇ‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚ interface - to - interface separation @ m m D D F ê T a R @ m m D = Ê ‡ Ú ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊʇ‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚ a c b b α R rb ( r ) b y x FIG. 5. Raft-raft repulsion. a , The approach of two rafts ismodeled as raft shifts b with respect to their circular tilingdomains. b , Shifted polar coordinate system of the back-ground membrane (Eq. 17). Dashed lines indicate curves ofconstant r from r = αR (red) to r = R (blue), which are cir-cles of radius r whose centers (dots) lie at x = b ( r ) and y = 0. c , Raft-raft repulsion energy ∆ F divided by temperature T for rafts of various radii αR . The points indicate experimentaldata and the lines indicate theoretical results calculated with α t = 0 . λ ∼ . µ m and chiral wavenumberdifference ∆ q = 0 . µ m − . α is given by Eq. 5 and R is ad-justed to produce rafts of different radii. Experimental dataand methods are reported in Ref. [18]. Schematics not drawnto scale. vary with azimuthal angle φ and that particles can tiltin the ˆr direction. To dipolar order, θ ( r, φ ) = θ ( r ) + ϑ ( r ) cos φ , where ϑ ( r ) is the dipolar tilt component. Wemust carefully recalculate terms in the single-domain freeenergy F domain that would be changed by this coordinatetransformation: F shift caT = (cid:90) RαR d r (cid:90) π d φ h r h φ × (cid:26) θ + λ (cid:2) ( ∇ · n ) + ( ∇ × n ) (cid:3)(cid:27) . (18) h r and h φ are scale factors of the coordinate transforma-tion. The evaluation of Eq. 18 is provided in SupportingInformation, where we see that the ˆr component of n canbe ignored to leading order in tilt angles. The shift profile b ( r ) appears from the scale factors and the spatial deriva-tives. Since we assume the rafts are unchanged by thedeformation, θ ( αR ) = θ and ϑ ( αR ) = 0, correspondingto the unshifted interfacial tilt angle as given by Eq. 12.The deformation vanishes at the edge of the tiling do-main, so b ( R ) = 0, θ ( R ) = 0, and ϑ ( R ) = 0. To calculatethe interaction energy between two rafts as a functionof separation distance, we impose various shift distances b = b ( αR ); numerically minimize the free energy over b ( r ), θ ( r ), and ϑ ( r ); subtract the energy of the unshiftedmembrane with b ( r ) = 0; and double the result.Meanwhile, the repulsive free energy of this two-raftsystem has been measured experimentally via opticaltrapping by moving rafts toward each other, releasingthem, and tracking their subsequent trajectories (Fig. 1gand [18]). Using parameter values given in Table I, ourmodel agrees well with these measurements for variousraft radii αR (Fig. 5c). Thus, despite our relatively sim-ple ansatz, our results quantitatively demonstrate thatdeformation of the background membrane as two raftsapproach each other can explain the observed repulsionbetween rafts. V. DISCUSSION
Our model is designed to emphasize physical relevanceand minimize phenomenological contributions. To do so,we ignore many effects that may ultimately produce amore precise description of these colloidal membranes,but in the process add more fit parameters that ob-scure the underlying generalizable physical principles.For example, the viruses are idealized to be hard rodsthat form geometrically precise and homogeneous mem-branes. During phase separation, we disregard the in-creased translational entropy of the shorter fd viruseswhen they are embedded within the longer M13 viruses.Furthermore, for mathematical tractability, we expandthe membrane free energy to quadratic order in d/l j and θ , even though the values in Table I imply d/l fd = 0 . θ ≈ . ∼
24 h to reach their equilibrium size, but the membranereaches its equilibrium degree of phase separation muchmore quickly (the background fluorescence stays constantthroughout the three panels). Both processes undergoenergetic relaxation through diffusion of the same parti-cles, so their decay timescales scale as τ ∼ η/ε , where η is the viscosity and ε is an energy density scale. A larger τ for the process of raft organization corresponds to asmaller ε compared to that of membrane phase separa-tion, which our model explains.Despite these sweeping simplifications, our model canmatch measurements with quantitative accuracy whileusing physically reasonable parameter values. It is con- sistent with our single-component membrane model thatdescribed an independent set of experimental observa-tions [19]. Moreover, it provides meaningful insight intothe fundamental mechanisms that drive membrane raftformation and organization. Competition between mix-ing entropy and depletion entropy determines the degreeof phase separation of two virus species with differentlengths. This competition is independent of virus chi-rality can be easily and precisely tuned by adjusting thedepletant concentration. A difference in the natural ten-dency for chiral particles to twist with a preferred hand-edness and pitch endows the rafts with a chiral structure.This structure stabilizes small rafts against an interfacialline tension that would otherwise promote coarsening to asingle raft domain and establishes a preferred depletant-concentration-dependent raft size. The twisted structureof the background membrane transmits torques and me-diates an elastic repulsion between rafts.Previous theoretical reports have demonstrated thatchiral structure can establish a membrane lengthscale,but they differ from our theory in several crucial ways.Some describe single-component smectic- C membranesthat contain hexagonal cells with only one handednessof twist and arrays of defects at the corners of thecells [48, 49]. Selinger and colleagues investigate mem-branes formed from racemic mixtures that can form do-mains of alternating chirality upon spontaneous symme-try breaking [50]. They find a square lattice of domainsthat also contain defects at their corners. Simultane-ously with our work, their theory has been expanded tohexagonal domains without defects and applied to fila-mentous virus membranes [40]. These aforementionedtheories are based on phenomenological Landau expan-sions in the concentration difference between the two chi-ral components (we show how our model can provide val-ues for Landau coefficients in Supporting Information).Complementarily, Xie and colleages investigate raft-raftrepulsion by directly minimizing the free energy of bothraft and background [39]. They highlight the role ofbackground chiral twist and use values for Frank con-stants (5 pN) and twist wavenumbers ( ∼ µ m − ) that arewithin an order of magnitude of those we use (Table I).However, they assume well-defined rafts of a particularsize, ignoring the processes of phase separation and raftsize establishment, and use a phenomenological virus tiltmodulus without exploring its physical basis in depletionentropy. In contrast, our theory, which provides a moreunified microscopic approach that facilitates comparisonwith experiments, produces analytical expressions for thechiral raft structure, and provides mathematical intuitionfor raft-raft repulsion via a shift ansatz.Colloidal membranes composed of viruses share im-portant physical symmetries with their molecular coun-terparts, even though their characteristic lengthscalesand microscopic origins of interactions differ. In fact,a leading-order free energy for rafts in a flat molecularmembrane would look very much like Eq. 9. The inter-facial line tension between rafts and background would0replace the term proportional to d [51, 52]. Phases thatprefer alignment perpendicular to the membrane plane,such as the biologically-relevant L α phase, would requirea θ term [53, 54]. Molecular twist would be encapsu-lated by Frank free energy terms. A generalization of ourmodel which can be applied to other membrane systemsis provided in Supporting Information. Furthermore,experimentally-prepared and biological membranes haverafts enriched in cholesterol as compared to the back-ground [55, 56]. Cholesterol demonstrates a strong pref-erence for chiral twist—in fact, the chiral nematic, orcholesteric, phase was the first liquid crystalline phaseobserved in 1888 by Friedrich Reinitzer while investigat-ing cholesteryl esters [57]. Hence, we expect a significantdifference in chiral wavenumbers ∆ q which could stabilizesmaller rafts.Our theory contributes to a biologically-relevant andpoorly-understood niche in the rich literature on molecu-lar membranes. It may explain why Langmuir monolay-ers composed of multiple chiral molecules demonstrate alimit to domain coarsening [58] and biological lipid raftsare believed to have a finite size [59], in contradictionto continous coarsening predicted by the Cahn-Hilliardmodel of phase separation [43]. Our description of raft-raft repulsion is analogous to the twist-mediated inter-action of chiral islands in smectic- C films [60–62]. Itoffers an explanation for the mutual repulsion observedbetween transmembrane protein pores formed by certainantimicrobials, if one imagines that these chiral poresimpose phospholipid tilt at their interface with the back-ground membrane [63, 64]. Ultimately, the validity ofour theory in a particular membrane system hinges on the direct observation of twist, which can be achievedwith polarized optical microscopy if the twist penetra-tion depth is at least the wavelength of light [13].Moreover, phospholipid rafts demonstrate chiral phasebehavior that must be explained by a theory attunedto chirality. By either replacing naturally chiral sphin-gomyelins with a racemic mixture [65] or replacing choles-terol with its enantiomer [66, 67] (although these lat-ter studies disagree with subsequent work [68, 69]), thecritical point for phase separation changes. Althoughfor our model parameters, phase separation occurs inde-pendently from raft organization, other parameter valuescause the raft area fraction α to depend on the differ-ence in chiral twist wavenumbers ∆ q (Supporting Infor-mation). Furthermore, different enantiomers of the sameanesthetic molecule have been shown to have different po-tencies [70–72]. Our theory presents a paradigm throughwhich chirality affects physical membrane properties, inaccordance with the classic hypothesis that anestheticmolecules disrupt membrane phase behavior [73, 74]. ACKNOWLEDGMENTS
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