When giant vesicles mimic red blood cell dynamics: Swinging of two-phase vesicles in shear flow

Red blood cells under shear flow present a specific swinging motion superimposed to a fluidlike tank-treading motion. Swinging is hypothesized to originate from periodic storage of shear energy in the cell membrane. Here we designed giant unilamellar vesicles with two lipid phases separated by a contact line, which swing and tank-tread like red cells. We propose a model that quantitatively fits our data, finds the value of the contact-line tension, and shows that swinging is due to the storage of elastic energy associated with the periodic modulation of the contact-line length during tank-treading.

Figure 1

(a)–(c) Schematics of RBCs successively undergoing tumbling, rolling, and tank-treading at increasing shear rate. The rolling regime is specific to RBCs and does not occur with vesicles. (d) During swinging the elastic energy stored by membrane elements moving along the capsule contour varies periodically.

Figure 2

(a) Vesicle at rest on the flow chamber surface. (b) Parameters measured to characterize vesicle motion: vesicle size and shape $(a_1,a_2,a_3)$; orientation to flow (θ); phase angle (α); contact-line length $\left(l\right)$. (c) Typical vesicle of high reduced volume (mean radius $\langle r\rangle =15.6\mu \mathrm{m}$, mean flattening $\langle f\rangle =0.3$, reduced volume $\upsilon \approx 0.95)$ exhibiting tank-treading and swinging motion under a shear rate of $3.9{\mathrm{s}}^{-1}$, with a mean inclination angle $\langle \theta \rangle v$ ≈ 30° (GUV 1 in Table 1). (d) Same vesicle as in (c) undergoing complex motion with out-of-plane orientational changes of the membrane under a higher shear rate of $11.5{\mathrm{s}}^{-1}$. The lipid phases change hemispheres during tank-treading. (e) A deflated vesicle with an intermediate reduced volume $\left(\upsilon \approx 0.8\right)$ strongly deforms during tank-treading under a shear rate of $3.9{\mathrm{s}}^{-1}$ (GUV 9 in Table 1). When the contact line is in the meridian zone the vesicle is no longer ellipsoidal but resembles two vesicles adhered together, each consisting of a lipid phase. When the contact line is in the equatorial zone the vesicle has an ellipsoidal shape. (f) A highly deflated vesicle with a low reduced volume $\left(\upsilon \approx 0.7\right)$ tumbling under a shear rate of $3.9{\mathrm{s}}^{-1}$ (GUV 10 in Table 1). Scale bars: $20\mu \mathrm{m}$.

Figure 3

(a) Temporal evolution of θ, α, $f$, and $l$ of a typical tank-treading and swinging vesicle (mean radius: $\langle r\rangle =15.8\mu \mathrm{m},\langle f\rangle \approx 0.2)$ under a shear rate of $3{\mathrm{s}}^{-1}$ (GUV 6 in Table 1). (b), (c) Phase (α) and inclination (θ) angles as function of the adimensioned time for the vesicle shown in Fig. 2, at three shear values (GUV 1 in Table 1). The solid lines are fits from Eqs. (6) and (7) using the vesicle shape parameters ( $a_1=19.2\mu \mathrm{m},a_2=a_3=12\mu \mathrm{m})$ and a line tension of 15 pN.

Figure 4

(a) $l$ dependence of the flattening $f$ and the inclination angle θ for the vesicle shown in Fig. 3 (GUV 6 in Table 1). Insets show the vesicle configurations at extremal $l$ values. (b) Shear-rate dependence of the tank-treading period $T$ and of the swinging amplitude Δθ in log-log scale, for all the analyzed vesicles. Diamond symbols correspond to the four vesicles studied under several shear rates (one color per vesicle) whereas all other symbols correspond to vesicles observed under a single shear rate (see Table 1 for details). The thick solid lines are specific fits for two vesicles (GUV 1, purple diamonds and GUV 7, blue diamonds) calculated from the model using a line tension of 15 pN and the vesicle respective shape parameters. Slope of −1 expected in the low-line-tension limit is indicated. The shaded region on the Δθ plot is a guide to the eye: It contains the scattered data of all vesicles including the two fits at different shear rates exhibiting the same slope.

Figure 5

Theoretical phase (top) and inclination (bottom) angles versus time for three shear rates calculated from the model using a line tension of 150 pN for a vesicle of flattening $f=0.5$ and reduced volume $\upsilon \approx 0.9$. Left: tumbling motion, α oscillates and θ rotates; center: swinging-tank-treading motion near the transition, α rotates and θ oscillates in an asymmetric manner and with a large amplitude; right: swinging–tank-treading motion far from the transition, α rotates and θ oscillates with a small amplitude. Note the difference in the form of the α$\left(t\right)$ curve near and close the tumbling–tank-treading transition.

Figure 6

Theoretical phase trajectories of θ versus α. Left: limit cycle for the tumbling motion (shear rate of $0.8{\mathrm{s}}^{-1})$, α oscillates and θ rotates; center: limit cycle of an intermittent regime at the tumbling–tank-treading transition (shear rate of $1.2{\mathrm{s}}^{-1})$; right: limit cycle for the swinging-tank-treading motion (shear rate of $6{\mathrm{s}}^{-1}$ ), α rotates and θ oscillates. The corresponding animations of the temporal evolutions of θ versus α are available in Movies S6–S8 in the Supplemental Material [21].