Conformational dynamics and phase behavior of lipid vesicles in a precisely controlled extensional flow
CConformational dynamics and phase behavior of lipidvesicles in a precisely controlled extensional flow
Dinesh Kumar,
1, 2
Channing M. Richter, and Charles M. Schroeder
1, 2, 3, ∗ Department of Chemical and Biomolecular EngineeringUniversity of Illinois at Urbana-Champaign, Urbana, IL, 61801 Beckman Institute for Advanced Science and TechnologyUniversity of Illinois at Urbana-Champaign, Urbana, IL, 61801 Department of Materials Science and EngineeringUniversity of Illinois at Urbana-Champaign, Urbana, IL, 61801 (Dated: October 18, 2019)Lipid vesicles play a key role in fundamental biological processes. Despite recent progress, we lacka complete understanding of the non-equilibrium dynamics of vesicles due to challenges associatedwith long-time observation of shape fluctuations in strong flows. In this work, we present a flow-phase diagram for vesicle shape and conformational transitions in planar extensional flow using aStokes trap, which enables control over the center-of-mass position of single or multiple vesiclesin precisely defined flows [Shenoy, Rao, Schroeder,
PNAS , 113(15):3976-3981, 2016]. In this way,we directly observe the non-equilibrium conformations of lipid vesicles as a function of reducedvolume ν , capillary number Ca , and viscosity contrast λ . Our results show that vesicle dynamicsin extensional flow are characterized by the emergence of three distinct shape transitions, includinga tubular to symmetric dumbbell transition, a spheroid to asymmetric dumbbell transition, andquasi-spherical to ellipsoid transition. The experimental phase diagram is in good agreement withrecent predictions from simulations [Narsimhan, Spann, Shaqfeh, J. Fluid Mech. , 2014, , 144].We further show that the phase boundary of vesicle shape transitions is independent of the viscositycontrast. Taken together, our results demonstrate the utility of the Stokes trap for the precisequantification of vesicle stretching dynamics in precisely defined flows.
I. INTRODUCTION
Vesicles are fluid-filled soft containers enclosed by amolecularly thin (3-4 nm) lipid bilayer membrane sus-pended in a liquid medium. In recent years, the me-chanics of giant unilamellar vesicles (GUVs) has beenextensively studied to provide insight into the mechan-ical properties of biological systems such as red bloodcells [1, 2]. To this end, vesicles have been used to un-derstand the equilibrium and non-equilibrium dynamicsof simplified cells that do not contain a cytoskeleton ora polymerized, protein-laden membrane commonly foundin living cells [3, 4]. Artificial vesicles have also been usedfor the triggered release of cargo in biomedical applica-tions such as drug delivery and micro/nanoscale reactors[5–7].Achieving a full understanding of the non-equilibriumdynamics of single-component lipid vesicles in preciselydefined flows is crucial for understanding cell mechan-ics. From this view, such studies can inform how thefluid dynamics and membrane properties inside and out-side the fluid-filled compartment contribute to cell shapechanges. From this view, a significant amount of priorwork has been focused on investigating the shape dy-namics of vesicles under different flow conditions, such asPoiseuille flow [8–10], shear flow [11–25], and extensionalflow [26–32]. Experiments and simulations on vesicles ∗ To whom correspondence must be addressed: [email protected] in shear flow have uncovered intriguing dynamic behav-ior including: (i) tumbling, where a vesicle undergoesa periodic flipping motion, (ii) trembling, where vesicleshape fluctuates and the orientation oscillates in time,and (iii) tank-treading, where an ellipsoid vesicle’s majoraxis maintains a fixed orientation with respect to the flowdirection while the membrane rotates about the vorticityaxis [11, 18, 21, 23]. The transitions between these dy-namical motions depend on shear rate ˙ γ , viscosity ratio λ between the inner µ in and outer µ out fluid viscosities,and reduced volume ν , which is a measure of vesicle’sasphericity [33]. Prior work has also focused on the in-duced hydrodynamic lift of a single vesicle near a wall inshear flow [34], pair interactions between vesicles in flow[35, 36], and measurement of the effective viscosity of adilute vesicle suspension [14].Despite recent progress in understanding vesicle dy-namics in shear flow, the behavior of vesicles in exten-sional flow is less well understood. Extensional flow isconsidered to be a strong flow that can induce high levelsof membrane deformation. Unlike simple shear, exten-sional flows consist of purely extensional-compressionalcharacter without elements of fluid rotation [37]. Innatural blood flows, red blood cells repeatedly undergoreversible deformations by a combination of shear andextension when passing through capillaries in the body[38, 39]. From this perspective, there is a clear needto understand the shape dynamics of cells when transit-ing through narrow capillaries. Interestingly, prior workhas shown that the extensional components of the veloc- a r X i v : . [ phy s i c s . b i o - ph ] O c t ity gradient tensor are crucial for predicting rupture ofred blood cells undergoing tank-treading motion in shearflow [39, 40]. From kinematic analysis, any general lin-ear flow can be decomposed into elements of rotation andextension/compression, thereby identifying the flow fieldcomponents associated with rupture. From this view, un-derstanding the dynamics of vesicles in extensional flowis of fundamental interest to elucidate dynamics in morecomplex mixed flows containing arbitrary amounts of ro-tation and extension/compression [17, 41].The physical properties of vesicles govern their dy-namic behavior in flow. Reduced volume ν is definedas the ratio of a vesicle’s volume V to the volume ofa sphere with an equivalent surface area A , such that ν = 3 V / πR where R = (cid:112) A/ π is the vesicle’s equiv-alent radius based on the total surface area. For a vesi-cle with a perfectly spherical shape, the reduced volume ν = 1, which means that there is no excess area to deformwhen subjected to hydrodynamic stress. In the weak-flowlimit, a vesicle is described by a constant total surfacearea [42] and substantially deforms only if the reducedvolume ν < ν < .
56 in extensionalflow [26]. At low reduced volumes, vesicles essentiallyadopt tubular shapes at equilibrium and are highly de-formable due to the large surface area to volume ratio.Under these conditions, the dynamics of highly deflatedvesicles in extensional flow was observed to be similarto the coil-stretch transition for flexible polymers in ex-tensional flow [43–45]. Above a critical strain rate ˙ (cid:15) c ,deflated vesicles were found to undergo a shape tran-sition from a tubular to a symmetric dumbbell confor-mation. Steinberg and coworkers reported a flow phase-stability diagram for such shape transitions [26], however,they did not directly characterize the bending modulus κ b for the deflated vesicles, and rather used an order-of-magnitude estimate of κ b from literature. Nevertheless,these experimental observations are in good agreementwith numerical simulations by Shaqfeh and coworkers[30, 31], which further confirm the tubular-to-symmetricdumbbell shape transition for deflated vesicles. More-over, these simulations examined vesicle dynamics undera wide range of reduced volumes, predicting that moder-ately deflated vesicles (0 . < ν < .
75) would undergoa spheroid to asymmetric dumbbell shape transition dueto destabilizing curvature changes in the membrane as aresult of modified Rayleigh-Plateau mechanism [30, 31].Recently, Muller and coworkers [32] studied the dy-namics of lipid vesicles in planar extensional flow usinga cross-slot microfluidic device [46]. The results fromthis study generally confirmed the spheroid to asymmet-ric dumbbell shape transition for moderately deflated(0 . < ν < .
75) vesicles in extensional flow. Interest-ingly, Muller and coworkers reported a stability bound-ary for shape transitions in reduced volume-capillarynumber ( ν, Ca ) phase-space, where the capillary num-ber Ca = µ out ˙ (cid:15)R /κ b is the ratio of the bending time scale to the flow time scale, ˙ (cid:15) is the fluid strain rate, and µ out is the viscosity of suspending medium. These ex-periments generally involved manual trapping of singlevesicles near the stagnation point of planar extensionalflow, which makes it challenging to observe dynamics overlong times while maintaining a stable center-of-mass po-sition of vesicles in flow. From this perspective, study-ing the non-equilibrium shape dynamics of lipid vesiclesin precisely defined extensional flows is critically neededto understand vesicle shape transitions and stability instrong flows.In this paper, we present a detailed flow-phase dia-gram of non-equilibrium vesicle shape transitions in ex-tensional flow using a Stokes trap [47–49], which enablesprecise control over the center-of-mass position of singleor multiple particles in flow. In this way, we directly ob-serve vesicle shape transitions over long observation timesacross a wide range of parameters including reduced vol-ume ν and capillary number Ca . We first discuss theimplementation of the Stokes trap technique in a PDMS-based microfluidic device. We then present a method toestimate the bending modulus of a vesicle at equilibriumusing thermal fluctuation analysis. Using this approach,we systematically determine the flow-phase diagram forvesicles in ( ν, Ca ) space, and we investigate the effectof viscosity contrast λ on the phase boundary. Finally,we discuss how the Stokes trap technique can be usedto investigate the transient stretching and relaxation dy-namics of vesicles under highly non-equilibrium flow con-ditions. II. EXPERIMENTAL METHODSA. GUV preparation
Giant unilamellar vesicles (GUVs) are prepared froma mixture of 1,2-dioleoyl-sn-glycero-3-phosphocholine(DOPC, Avanti Polar Lipids) and 0.12 mol % of 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-(lissaminerhodamine B sulfonyl) (DOPE-Rh, Avanti Polar Lipids)using the classical electroformation method describedin Angelova et al. [50]. The fluorescently labeledlipid DOPE-Rh contains a rhodamine dye (absorp-tion/emission maxima 560 nm/583 nm) on the lipidhead group, rather than the tail group, because it isknown that lipids with labeled hydrocarbon tails canresult in altered membrane properties if the charged dyemolecule flips into the hydrophilic head group space,which may affect the bending modulus of the membrane[51].For electroformation of GUVs, a stock lipid solutionis prepared with 25 mg/mL DOPC and 0.04 mg/mLDOPE-Rh. Next, 10 µ L of the lipid solution in chlo-roform is spread on a conductive indium tin oxide (ITO)coated glass slide (resistance Ω, 25 × × ≈ ◦ C). Underthese conditions, DOPC lipid remains in the fluid phase[26]. Most of the vesicles prepared by this method areunilamellar with few defects in the size range of 5-25 µm in radius. The viscosity of the 100 mM sucrose solution( µ = 1 . ◦ C. Following electroformation, mostvesicles are only weakly deflated and quasi-spherical innature. To generate moderately deflated (low reducedvolume) vesicles, 100 µ L of a 200 mM sucrose solutionis added to 2.0 mL of the electroformed vesicle suspen-sion, which increases the total sucrose concentration to105 mM. In this way, osmotic pressure differences tendto drive water out of the vesicle interior until the sucroseconcentrations are nearly equal on both sides of the mem-brane [32, 52]. The osmotic deflation method generatedreduced volume vesicles in the range 0 . < ν < . . < ν < .
50) in the suspension was relativelyrare. For experiments involving high solvent viscosities(viscosity ratio λ = 0.1), the viscosity of the suspendingmedium was increased to µ out = 10.4 mPa-s by addingglycerol to the 100 mM sucrose solution. B. Stokes trap
We use a Stokes trap [47] to generate controlled strainrate schedules while simultaneously achieving long-termconfinement of single vesicles near the stagnation pointof a planar extensional flow. A four-channel cross-slotmicrofluidic device is used for studying vesicle dynam-ics (Fig. 1a). In brief, single-layer polydimethylsiloxane(PDMS)-based microfluidic device (width = 400 µ m, anddepth = 100 µ m) is fabricated using standard techniquesin soft lithography [53]. The channel dimensions aremuch larger compared to the typical vesicle equilibriumsize R = 5-25 µ m, such that the effect of confinementis negligible. During device operation, fluid is injectedinto two opposing inlet channels and withdrawn throughthe two remaining outlet channels, thereby forming mu-tually perpendicular inlets and outlets. In this way, thesymmetry of the flow-field under low Reynolds numberconditions results in the formation of a fluid stagnationpoint (zero-velocity point) near the center of the cross-slot device, thereby generating a planar extensional flowin the vicinity of stagnation point as shown in Fig. 1b.The Stokes trap was used to enable the direct obser-vation of vesicle dynamics in extensional flow with a pre-cisely defined strain rate ˙ (cid:15) for long observation times[47, 54]. Briefly, the center-of-mass position of a tar-get vesicle is trapped in real-time using fluorescence mi-croscopy and model predictive control (MPC) algorithm. a Trapped vesicle PDMSPressure regulator
Fluid reservoir
Objective lensCover slip ba FIG. 1. Stokes trap for studying vesicle dynamics in flow. (a)Schematic of the experimental setup used to generate planarextensional flow. Inlet/outlet channels in the microfluidic de-vice are connected to fluidic reservoirs containing the vesiclesuspension and pressurized by regulators controlled by a cus-tom LabVIEW program, thereby generating pressure-drivenflow in the cross-slot. (b) Schematic of the cross-slot microflu-idic device showing a deformed vesicle trapped in extensionalflow near the stagnation point for illustrative purposes (notdrawn to scale). The width of channels is 400 µ m and theradii of the two spherical ends of the deformed vesicle are 12 µ m and 4 µ m, respectively. The MPC feedback controller determines the necessaryflow rates required to achieve trapping at a specific pointwhile maintaining a nearly constant strain rate in ex-tensional flow and is achieved using computer-controlledpressure regulators. In this way, the Stokes trap can beused to confine vesicles under zero-flow conditions (withno external or net flow) or under non-zero net flow con-ditions [47, 48], and the latter method was used to studynon-equilibrium flow dynamics in the work.
C. Flow-field characterization
Particle tracking velocimetry (PTV) is used to deter-mine the fluid strain rates ˙ (cid:15) as a function of the in- S t r a i n r a t e ( s - )
20 40 60 80Height( m)0246810 S t r a i n r a t e ( s - ) ab FIG. 2. Flow-field characterization of cross-slot microfluidicdevices. (a) Strain-rate determination at the center-plane ofa cross-slot microfluidic device as a function of inlet pressure.Bead tracking experiments are performed in 100 mM sucrosebuffer. (b) Strain-rate determination as a function of distancefrom the horizontal mid-plane in the device. put pressure from the pressure regulators (Elveflow OB1-MkIII). Experimental characterization of the fluid strainrate is performed to ensure that the flow field is uniformin the vicinity of the stagnation point and enables de-termination of the capillary number Ca = µ out ˙ (cid:15)R /κ b .A trace amount of fluorescent microbeads (2.2 µ m di-ameter, Spherotech, 0.01% v/v) was added to 105 mMsucrose buffer solutions ( µ = 1.1 mPa-s, matched to thesolution used for vesicle dynamics experiments) to enableparticle tracking. Microfluidic devices are mounted onthe stage of an inverted fluorescence microscope (Olym-pus IX71), which allows for real-time imaging of fluores-cent beads using a high numerical aperture (1.45 NA,63x) oil-immersion objective lens and a 100-W mercuryarc lamp (USH102D, UShio). The sucrose buffer is in- troduced into microfluidic devices using the fluidic reser-voirs, and images of bead positions are acquired using aCCD camera (GS3-U3-120S6M-C) as a function of theapplied inlet pressure. For data shown in Fig. 2a, thestrain rate was determined at the center plane of thechannel in the z -direction (direction orthogonal to the2D flow plane). As shown in Fig. 2a, the strain rateat the central plane of cross-slot device increases linearlywith pressure over the characteristic range of strain ratesused in this work. For data shown in Fig. 2b, fluid strainrate was determined as a function of z -position by focus-ing through the depth of the device as a function of inletpressure driving flow. A particle tracking and analysisprogram [55] is used to determine bead velocities for alltrajectories, thereby enabling determination of the fluidstrain rate ˙ (cid:15) using a non-linear least square algorithm: (cid:20) v x v y (cid:21) = (cid:20) ˙ (cid:15) − ˙ (cid:15) (cid:21) (cid:20) x − x y − y (cid:21) (1)where v x , v y , x , y are velocities and positions in the x and y directions, respectively, and ( x , y ) is the locationof the stagnation point in the 2D flow plane. D. Bending modulus determination
1. Vesicle imaging in observation chamber
For determination of bending modulus, vesicles are im-aged in a secure-seal imaging spacer (Grace Bio-Labs, 7mm diameter, 0.12 mm depth) using an inverted opti-cal microscope (Olympus IX71) in epifluorescence modeequipped with a 63x oil immersion objective lens (NA1.4, Zeiss Plan-Apochromat) and an electron multiplyingcharge coupled device (EMCCD) camera (Andor iXon-ultra, DU-897U-CSO, 512x512 pixel output). A 100-Wmercury arc lamp (USH102D, UShio) was used as theexcitation light source in conjunction with a neutral den-sity filter (Olympus), a 530 ±
11 nm band-pass excitationfilter (FF01-530/11-25, Semrock), and a 562-nm single-edge dichroic mirror (Di03-FF562-25 ×
36, Semrock) inthe illumination path.The vesicle suspension is first introduced into thespacer, and the top of spacer is then sealed with a cover-slip to minimize evaporation and convection within theobservation chamber. The temperature inside the cham-ber is measured using a thermocouple and found to be22 ◦ C for all experiments. The effect of gravity influenc-ing the vesicle shape is negligible because of the nearlyequivalent concentration of sucrose in the interior and ex-terior of the vesicle, yielding symmetry across the bilayermembrane. Imaging is performed at the central planeof the spacer, and the center-of-mass of vesicles remainsnearly constant during an observation time of 30-60 s.Images are acquired over at least 30 s (acquisition framerate of 30 Hz), which is much larger than the relaxationtime of the slowest decaying mode of the membrane [56].The approximate order-of-magnitude relaxation time fora typical lipid membrane vesicle of size R = 10 µ m is ≈
200 ms [56], yielding a bending modulus of 10 − J ina suspending medium with viscosity of 1 mPa-s. In thisway, long observation times ensure that the available con-figurational modes of vesicles are given sufficient time torelax. For these experiments, unilamellar and defect-freevesicles are selected, and the fluctuating vesicles in thespacer are spatially isolated from their neighbors.
2. Contour detection and determination of κ b We use the method proposed by P´ecr´eaux et al. [57]to determine vesicle bending modulus. In this way, wefollow a rigorous selection criteria outlined in prior work[32, 58] that provides an unbiased procedure for reject-ing unsuitable vesicles that do not fluctuate according toan analytical fluctuation spectrum given by the Helfrichmodel [59]. Vesicle contours are first detected in eachimage with high precision using a custom MATLAB pro-gram that relies on intensity gradient maxima values tolocate the edges ( ESI † , Fig. S1). The detected coor-dinate positions of the vesicle membrane ( x i , y i ) in eachmovie frame are transformed to polar coordinates ( r i , θ i )and projected into Fourier modes as follows: r ( θ ) = R (cid:32) ∞ (cid:88) n =1 a n cos ( nθ ) + b n sin ( nθ ) (cid:33) (2)where R is the radius of contour in each frame definedas: R = 12 π N (cid:88) i =1 (cid:18) r i + r i +1 (cid:19) ( θ i +1 − θ i ) (3)The magnitude of Fourier amplitudes is calculated as c n = (cid:112) a n + b n , and the mean square amplitude of fluc-tuation modes around a base spherical shape is given by: (cid:68) | u ( q x ) | (cid:69) = π (cid:104) R (cid:105) (cid:0) (cid:104) c n (cid:105) − (cid:104) c n (cid:105) (cid:1) (4)where q x = n/ (cid:104) R (cid:105) is the wavenumber and (cid:104) R (cid:105) is themean radius of contours determined over all images ina fluctuation experiment on a single vesicle. For deter-mining the bending modulus κ b of vesicles, the followingsteps are performed:(1) For each vesicle contour, the mean square ampli-tude of fluctuations is calculated using Eq. 4. For thisanalysis, the behavior over modes n = 6 −
25 is examined(see ESI † for details).(2) A one-sample Kolmogorov-Smirnoff test is used tocheck the exponential distribution of modes. In brief,vesicles maintain a constant volume and surface area overthe the timescale of these experiments, so the Fouriermodes in Eq. 4 are expected to be exponentially dis-tributed. For the modes that pass this test, the experi- mental mean square amplitude (cid:68) | u ( q x ) | (cid:69) is calculated.In this way, the objective function F is optimized: F ≡ n =25 (cid:88) n =6 (cid:68) | u ( q x ) | (cid:69) − (cid:68) | u H ( q x ) | (cid:69) σ (cid:104) | u ( q x ) | (cid:105) (5)where σ (cid:104) | u ( q x ) | (cid:105) is the measured standard deviation ofthe experimental amplitudes (cid:68) | u ( q x ) | (cid:69) according to aprocedure discussed in ESI † , and (cid:68) | u H ( q x ) | (cid:69) is themodified form of Helfrich’s spectrum after incorporatingthe effect of the finite camera integration time [59].(3) The quantity (cid:68) | u ( q x ) | (cid:69) versus q x is plotted andanalyzed for each vesicle to generate the experimentalfluctuation spectrum. In this way, a two-parameter fit isperformed using the modified form of Helfrich’s spectrumthat accounts for the effect of finite integration time ofcamera: (cid:68) | u H ( q x ) | (cid:69) = 1 π (cid:90) ∞−∞ kT µ out q ⊥ τ m τ m τ (cid:20) ττ m + exp (cid:18) − ττ m (cid:19) − (cid:21) dq y (6)where τ − m = µ out q ⊥ (cid:0) σq ⊥ + κ b q ⊥ (cid:1) and q ⊥ = (cid:113) q x + q y .In this way, we determine the bending modulus κ b andmembrane tension σ for each vesicle [57]. In this fittingprocedure, the smallest value of the quantity (cid:68) | u ( q x ) | (cid:69) is taken to be 10 − m , limited by the spatial resolutionof camera (1 pixel ≈
200 nm), determined in a separateexperiment by measuring the fluctuation amplitudes of astationary rigid fluorescently labeled polystyrene bead inthe focal plane of the microscope.We further estimate the correlation coefficient of thetwo parameters of fit ( κ b and σ ) (see ESI † for details). Ifthe correlation coefficient corr ( σ, κ b ) < − .
85, the vesicleis rejected for analysis because the membrane is generallytaken to be too tense to provide an accurate estimate ofthe bending modulus [58]. In determining bending mod-ulus κ b , we only consider the images in which the averagecontour length does not change by more than 5% to en-sure the constant surface area and volume. Finally, weonly consider quasi-spherical vesicles in the fluctuationanalysis for estimation of bending modulus. The vesi-cles used in non-equilibrium flow experiments are highlydeflated (non-spherical), though we follow prior work inassuming that the ensemble-averaged bending modulusmeasured for quasi-spherical vesicles is representative ofall vesicles in the sample [32, 58]. E. Flow experiments in extensional flow
Following flow field characterization and determinationof equilibrium bending modulus κ b , we studied the non-equilibrium deformation of vesicles in extensional flow.Non-equilibrium flow experiments were conducted using time t o S t r a i n r a t e , ሶ 𝜖 ሶ𝜖 𝑐 FIG. 3. Flow deformation protocol and time-dependentstrain rate schedule for the phase diagram experiments. Fluidstrain rate is increased in a systematic step-wise fashion, andafter each step change, vesicle shape is directly observed for ≈ (cid:15) critical is defined as thestrain rate at which a vesicle undergoes a global shape tran-sition. fluorescence microscopy at 10x magnification using an in-verted optical microscope (Olympus IX71) with mercurylamp as the illumination source (100-W mercury arc lampUSH102D, UShio). Images were captured using a CCDcamera (Pointgrey GS3 23S6M USB3 CMOS) at a framerate of 30 Hz with an exposure time of 10 ms. A dilutevesicle suspension in sucrose buffer was introduced intothe PDMS microfluidic device via sample tubing (PEEKtubing 1/16” OD x 0.02” ID) connected to fluidic reser-voirs (Fig. 1). The four fluidic reservoirs are pressur-ized using pressure transducers to drive the fluid intothe microfluidic chip. The fluid inside and outside thevesicle (105 mM sucrose buffer) are density matched, sothere is no significant drift of vesicles in the orthogonaldirection ( z -direction) during the timescale of the exper-iment. Vesicles were introduced into the cross-slot deviceby flowing through inlet channels at extremely low veloc-ities such that vesicles are negligibly deformed prior toflow experiments.In this work, we only consider vesicles that are unil-amellar (via visual inspection of contour brightness andsmoothness), defect-free, and completely isolated fromneighboring vesicles. Multilamellar vesicles are observedin the sample, typically showing defects such as a daugh-ter vesicle inside a parent vesicle, or lipid tubes pro-truding from the membrane, but these vesicles are notincluded in our analysis. Prior to performing a non-equilibrium flow experiment, individual vesicles are firsttrapped under zero-flow conditions for ≈ R and reduced volume ν for each vesicle aremeasured under zero-flow conditions. Reduced volume ν is defined as the ratio of a vesicle’s volume V to the vol-ume of an equivalent sphere with surface area A = 4 πR , such that: ν = 3 V √ πA / (7)A reduced volume of ν = 1 corresponds to a perfectlyspherical vesicle, whereas ν < et al. [32]. Inbrief, the surface area and volume of a vesicle are esti-mated by revolution of the observed 2D membrane con-tour along the vesicle’s short axis (ESI † , Fig. S2 andFig. S3). The equilibrium shape of a vesicle is not al-ways symmetric, so the volume and surface area are cal-culated from the top and bottom halves of the vesicleseparately by numerical integration [60], and the totalsurface area and volume are taken as the average valuewith uncertainty corresponding to one-half of the differ-ence between the top and bottom halves. In this way, theequivalent vesicle radius R and the reduced volume ν aredetermined from the mean of 100 images at equilibrium(ESI † , Fig. S4).Following determination of R and ν for a single vesi-cle, the non-equilibrium flow experiment is performed bydirectly observing shape dynamics for the same individ-ual vesicle in planar extensional flow. Vesicle dynamics q x (m -1 ) -22 -21 -20 < | u q x | > ( m ) Fluctuation spectrumFitted model
FIG. 4. Analysis of membrane fluctuations for deter-mining bending modulus κ b . The amplitude of fluctuations (cid:10) | u ( q x ) | (cid:11) is plotted as a function of wave vector q x for a rep-resentative DOPC vesicle. The solid red line corresponds tothe analytical model using Eq. 6, yielding κ b = 8.9 × − J and membrane tension 3.9 × − N/m (Inset): Detectedcontour of a fluctuating GUV at equilibrium using image pro-cessing methods. in flow are governed by three dimensionless parameters:reduced volume ν , capillary number Ca , and viscositycontrast λ . The capillary number Ca is the ratio of theviscous forces to bending forces on the interface, suchthat: Ca = µ out ˙ (cid:15)R κ b (8)where µ out is exterior fluid viscosity, and the viscositycontrast λ is the ratio of the fluid viscosities between theinterior ( µ in ) and exterior ( µ out ) regions of a vesicle: λ = µ in µ out (9)Using the Stokes trap, the fluid strain rate is increasedin a systematic step-wise fashion (Fig. 3) by changingthe pressure difference δP between the inlet and out-let channels in the microfluidic device (Fig. 1b). Aftereach step increase in the flow rate, a trapped vesicle isobserved for ≈ ν, Ca ) space with high resolution between experimentaldata points along the Ca -axis in the flow-phase diagram. III. RESULTS AND DISCUSSIONA. Bending modulus estimation
We began by determining the average bending mod-ulus for an ensemble of DOPC lipid vesicles using theprocedure described in the Experimental Methods (Sec-tion 2.4). In brief, this method relies on analyzing mem-brane fluctuations for weakly deflated vesicles at equi-librium (no flow conditions), followed determination ofbending modulus κ b and membrane tension σ using atwo-parameter fit to the Helfrich model given by Eq.6. The amplitude of membrane thermal fluctuations (cid:68) | u ( q x ) | (cid:69) as a function of wave vector q x is shownfor a characteristic lipid vesicle in Fig.4. Using thisapproach, we determined an average bending modulusof κ b = (9 . ± . × − J ( N = 21). The av-erage value of membrane tension was found to be σ =(1 . ± . × − N/m ( N = 21), which is consistentwith prior work reported in literature [32].The experimentally determined value of the bendingmodulus for DOPC vesicles ( κ b = 9 . × − J, DOPCwith 0.12 mol% DOPE-Rh, 100 mM sucrose, T = 24 ◦ C)is in reasonable agreement with the bending modulusmeasured for pure DOPC vesicles ( κ b = 9 . × − J, 300mM sucrose/307 mM glucose, T = 25 ◦ C) by Zhou et al. [60], which suggests that the bending modulus for DOPC vesicles does not significantly depend on sugar concentra-tion over the relatively narrow the range of 100-300 mM.Indeed, low angle X-ray scattering measurements by Na-gle et al. [61, 62] have recently shown that the bendingmodulus of DOPC vesicles does not depend on sucroseconcentration in the range between 100-450 mM. Gracia et al. measured the bending modulus of pure DOPC vesi-cles (10 mM glucose, T = 25 ◦ C) to be κ b = 10.8 × − J, which is consistent with the value of κ b measured inthis work at a higher sucrose concentration of 100 mM.Prior work [63, 64] has shown that increasing the sugarconcentration from 10 mM to 100 mM decreases the valueof bending modulus by a factor of two, though our resultstend to show less deviation in κ b over this range of sucroseconcentration. Indeed, such variabilities in experimentalmeasurements of bending moduli for DOPC vesicles havebeen reported in prior work [65].The DOPC vesicles in this work contain an exceedinglysmall amount of fluorescently labeled lipid (0.12 mol%DOPE-Rh), which suggests that such a low concentrationof labeled lipid does not substantially alter the bendingmodulus of the membrane compared to pure DOPC vesi-cles [66]. Our method for determining bending modulusrelies on a fairly strict set of statistical rejection criteriafor excluding vesicles that do not conform to an analyt-ical model (Eq. 6), which yields a relatively narrow dis-tribution in bending moduli values across the ensemble.Nevertheless, variability in bending modulus between in-dividual vesicles can be attributed to light-induced per-oxide formation in GUVs and/or precision of membraneedge detection in vesicle images [66]. Broadly speaking,the experimentally measured values of κ b in this workare consistent with prior work reported for DOPC vesi-cles [52, 61, 67]. B. Non-equilibrium flow-phase diagrams
Following determination of bending modulus κ b , wefurther studied the non-equilibrium dynamics and con-formation phase transitions of vesicles in extensional flowover a wide range of reduced volume ν and capillary num-ber Ca for a uniform viscosity contrast λ = 1 (Fig. 5).Using the Stokes trap, we confined single vesicles near thestagnation point of planar extensional flow and observedthe non-equilibrium shape dynamics while systematicallyincreasing the strain rate ˙ (cid:15) in a scheduled fashion (Fig.3). In this way, vesicles were observed to adopt a widevariety of shapes in flow, including a symmetric dumbbellshape (Fig. 5a, Movie S1 ESI † ), an asymmetric dumbbellshape (Fig. 5c, Movie S2 ESI † ), and a stable ellipsoidalshape (Fig. 5b, Movie S3 ESI † ) depending on the flowstrength Ca and amount of membrane floppiness ν .Fig. 5a shows a characteristic time series of images fora highly deflated ( ν = 0.53) vesicle initially in a tubularshape under zero flow conditions. In the presence of ex-tensional flow, the vesicle stretches along the extensionalaxis and eventually transitions to a symmetric dumbbell Reduced volume C ap ill a r y nu m be r C a Narsimhan BI simulation 2015Zhao-Shaqfeh 2013Scaling theoryQuasi-spherical vesicle, low deflationAsymmetric dumbbell conformationSymmetric dumbbell conformationSpheroid vesicle, moderate deflationTubular vesicle, high deflation quasi-spherical ba d Highlydeflatedregime c Moderatelydeflatedregime
FIG. 5. Flow-phase diagram for vesicle dynamics in extensional flow. Time series of images showing vesicle shape changes inextensional flow for: (a) a vesicle in the highly deflated regime ν = 0 .
53, having a tubular shape at equilibrium and undergoing asymmetric dumbbell shape transition at Ca = 2 .
3, (b) a vesicle in the weakly deflated regime ν = 0 .
95, having a quasi-sphericalshape at equilibrium and maintaining a stable ellipsoid shape upon extension up to Ca of ≈ ν = 0 .
73, having a spheroid shape at equilibrium and undergoing an asymmetric dumbbell shapetransition at Ca = 98 .
7. (d) Flow-phase diagram of vesicles in planar extensional flow as a function of reduced volume ν andcapillary number Ca at a viscosity ratio λ = 1. Open green squares represent vesicles in the highly deflated regime ν < . Ca < Ca C at which a vesicle does not undergo shape instability. Filled green squaresrepresent the Ca c phase boundary at which a tubular to symmetric dumbbell transition occurs. Open red squares representvesicles in the moderately deflated regime 0 . < ν < .
75 (spheroid shape at equilibrium) for
Ca < Ca C at which a vesicledoes not undergo a shape instability. Filled red squares represent the Ca c phase boundary at which a spheroid to asymmetricdumbbell transition occurs. Open magenta squares represent vesicles in the weakly deflated regime ν > .
75 where vesicles havequasi-spherical shape at equilibrium and transition to a stable ellipsoid shape. The grey curve represents the phase boundaryfrom boundary integral simulations [31]. shape at Ca = 2.3. Once the shape change occurs, thevesicle is observed to reach a steady-state conformationin flow. Similarly, Fig. 5c shows a characteristic time se-ries of images for a moderately deflated ( ν = 0.73) vesicleinitially in a spheroidal shape, eventually transiting to anasymmetric dumbbell shape at Ca = 98.7. Finally, Fig.5b shows a time series of images for a quasi-sphericalvesicle that largely retains an ellipsoidal shape as Ca in-creases and does not undergo a transition into a dumbbellshape.The experimental flow-phase diagram for vesicleshapes in extensional flow is shown in Fig. 5d. Ourresults reveal three distinct dynamical regimes in the ( ν , Ca ) plane attained by lipid vesicles. In general, highlydeflated ( ν < .
60) and moderately deflated (0 . < ν < .
75) vesicles are observed to transition into symmetricor asymmetric dumbbell shapes, respectively, at a criti-cal strain rate ˙ (cid:15) c (Movie S1,S3 ESI † ). The critical capil-lary number Ca c for the vesicle shape transition dependson the reduced volume ν . As shown in Fig. 5d, the filled green symbols (red symbols) represent the symmet-ric (asymmetric) dumbbell shape transition for vesicleswith reduced volume ν < .
60 (0 . < ν < . Ca values below the critical value for ashape transition. At higher reduced volumes ( ν > . Ca and do not undergo a symmetric/asymmetric dumbbellshape change over the entire range of Ca . The grey curveshows the predicted stability boundary from boundary-integral simulations [29–31], which is in good agreementwith our experimental data.In general, the flow-phase diagram reveals three dis-tinct regimes in vesicle shape dynamics defined by re-duced volume ν : (i) ν < .
60, (ii) 0 . < ν < . ν > .
75 corresponding to transitions to a sym-metric dumbbell, asymmetric dumbbell, or stable ellip-soid shape, respectively. Interestingly, the critical capil-lary number Ca c required to trigger a shape transition Reduced volume C ap ill a r y nu m be r C a Narsimhan BI simulation 2015Zhao-Shaqfeh 2013Scaling theoryAsymmetric dumbbell conformationSymmetric dumbbell conformationQuasi-spherical vesicleSpheroid vesicleTubular vesicle
Reduced volume C ap ill a r y nu m be r C a Phase boundary at = 0.1Phase boundary at = 1.0 ba FIG. 6. Flow-phase diagram for vesicle dynamics in extensional flow at viscosity ratio λ = 0 .
1. (a) Flow-phase diagram ofvesicle conformations as a function of reduced volume ν and capillary number Ca at viscosity ratio λ = 0 .
1. Green, red andmagenta markers have the same meaning as Fig. 5. (b) Comparison between phase boundaries for viscosity ratios λ = 1 and λ = 0 . decreases with higher levels of deflation (decreasing ν ).These observations are consistent with prior experimen-tal work [26, 32] and numerical simulations on vesicledynamics in extensional flow [30, 31]. Moreover, the pre-dicted phase boundary from a scaling analysis in priorwork [30] is also shown in Fig.5d, which appears to be inqualitative agreement with experiments.Our experimental results also reveal some degree ofvariability in the behavior of vesicle shape transitionsnear the critical stability boundary. For example, vesiclesmarked as ‘ ’ and ‘ ’ in Fig. 5d have approximately thesame reduced volume ν ≈ .
74, but they transition to anasymmetric dumbbell shape at different values ( Ca =133and Ca =17.8, respectively). Similarly, vesicles markedas ‘ ’ and ‘ ’ undergo an asymmetric shape transition at Ca numbers slightly above and below the curve predictedfrom simulations. In general, such variability in vesicledynamics near the phase boundary can arise due to sev-eral reasons. First, Ca is defined based on an ensembleaveraged value of bending modulus κ b determined fromthermal fluctuation analysis of quasi-spherical vesicles atequilibrium. In the non-equilibrium flow experiments,vesicles are osmotically deflated and have non-sphericalshapes, which may result in differences in bending modu-lus on an individual vesicle basis. Moreover, our methodof estimating reduced volume ν by assuming a 2D con-tour for vesicles as a body of revolution generally ignoresthermal wrinkles in the vertical direction ( z -direction),which may introduce minor variability in determining ν [12, 60]. Finally, numerical simulations of vesicle shape dynamics do not include thermal fluctuations of the vesi-cle membrane, which may lead to differences betweenexperimental results and numerical predictions. Indeed,our results show that the role of thermal fluctuations maybe important in describing the nature of vesicle shapetransitions in flow (Movies S1, S3, see ESI † for details).To investigate the influence of viscosity ratio λ on thestability boundary, we performed an additional set of ex-periments by increasing the viscosity of the suspendingmedium by adding glycerol, such that the viscosity ra-tio λ = 0.1. Fig. 6a shows the flow-phase diagram forDOPC vesicles in extensional flow as a function of Ca and ν at λ = 0.1. Overall, the dynamic behavior of vesi-cles at λ = 0.1 was similar to that observed at λ = 1.0.To quantitatively compare the dynamic behavior of vesi-cles at different viscosity ratios, we plotted the stabilityboundary for λ = 1.0 and 0.1 in Fig. 6b. The differencebetween these curves is not statistically significant as de-termined by a Mann-Whitney test ( p > . λ [68]. IV. CONCLUSIONS
In this work, we experimentally determine the flow-phase diagrams for vesicles in extensional flow with highresolution in (
Ca, ν ) space using a Stokes trap. Our re-sults show that vesicles undergo symmetric and asym-metric dumbbell shape transitions depending on Ca and ν over a wide range of conditions. Quantitative char-acterization of the phase diagram reveals three distinctdynamical regimes for vesicles in extensional flow namely,a tubular to symmetric dumbbell transition, a spheroidto asymmetric dumbbell transition, and quasi-sphericalto stable ellipsoid depending on the value of reduced vol-ume. We further demonstrate that the phase boundaryfor shape transitions in flow is insensitive to viscositycontrast between vesicle interior and exterior. Due tothe presence of the incompressible molecularly thin lipidbilayer membrane, vesicles exhibit very different dynam-ics compared to liquid drops in flow.Importantly, the trapping method used in this workallows vesicles to reach a steady-state conformation inextensional flow after experiencing a global change in shape. We emphasize that such experimental precisionwas enabled by using the Stokes trap, which allows forthe long-time observation of single or multiple particlesin an externally imposed flow. An intriguing questionrelates to vesicle dynamics at flow rates exceeding thecritical capillary number Ca c . Upon increasing the flowrates above Ca c , we anticipate that vesicles will continueto stretch and will likely undergo large deformations toextremely high large aspect ratios (ratio of a vesicle’sstretched length along the extensional axis to the equi-librium length). In future work, it will be interestingto investigate if additional membrane-bound soft materi-als such as polymersomes (polymer vesicles), capsules, orcells undergo similar shape changes under flow. Overall,our work establishes the utility of Stokes trap as a toolfor investigating vesicle dynamics and opens new avenuesfor investigating the non-equilibrium dynamics of soft de-formable particles in strong flows. ACKNOWLEDGEMENTS
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