Vesicle dynamics: experiments

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Victor Steinberg (2009), Scholarpedia, 4(3):5475. doi:10.4249/scholarpedia.5475 revision #91910 [link to/cite this article]
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Curator: Victor Steinberg

Rheology of biofluids such as blood remains a great challenge and the progress towards its understanding requires detailed studies of dynamics of a single cell. A vesicle is a model system to study the dynamical behavior of biological cells and, in particular, red blood cells. A giant unilamellar vesicle is a droplet of a viscous fluid encapsulated by a lipid bilayer membrane and suspended in a fluid of either the same or different viscosity as the inner one. In typical biological systems as well as in the experiments described below the vesicle membrane is in fluid state, so the individual lipids can freely move around. Both volume and surface area of the vesicle are conserved. Constant vesicle volume means that the membrane is impermeable, at least on the time scale of the experiment. Constant vesicle area means that membrane stretching or compression are negligible as it is a 2D incompressible fluid. Despite of the two constraints, a vesicle shape is not uniquely defined and shape deformations are permitted under external perturbations. Fluidity of the membrane allows the vesicle to adopt any shape that satisfies the volume and surface area constraints. The constraints have far-reaching consequences on vesicle dynamics because the membrane surface tension, which is determined by external stresses exerted on a vesicle can adopt any value: positive as well as negative. The membrane surface tension plays the same role as the pressure in incompressible liquids: it is adjusted to external stresses to ensure the local membrane incompressibility.

Contents

Vesicle dynamics in shear flow

Figure 1: Dynamical states of a vesicle with R= 6.19 μm and Δ=0.82 during (a) tank-treading: ω/s=1.7, s=0.22 s-1; (b) tumbling: ω/s=5.3, s=0.07 s-1; (c) trembling: ω/s=3.1, s=0.12 s-1.

For a long time two regimes of vesicle dynamics in a shear flow, tank-treading (TT) and tumbling (TU), and transition between them were explained by a simple model based on a fixed shape ellipsoid with a membrane rotating around it [1]. In the TT regime both the inclination angle θ between the vesicle long axis and the shear flow direction and its shape remain stationary apart from thermal fluctuations, whereas the membrane undergoes TT motion (see Figure 1). In TU the vesicle long axis rotates with respect to the shear flow direction and the membrane rotates around a vesicle (see Figure 1). This model introduces two dimensionless control parameters, geometrical, the excess area Δ=A/R2-4π, and physical, the viscosity contrast λ=ηinout, which deternmine the region of existence of TT and TU and TT-TU transition. Here Δ defines the degree of asphericity, R is defined from a vesicle volume V=4πR3/3 and in the experiments has about 10 μm, A is the surface area, and ηin and ηout are the viscosity of inner and outer fluids, respectively. Both TT and TU are described by one dynamical equation for θ, and according to it their dynamics are independent of the shear rate \(\dot{\gamma}\) (at leading order in the excess area). The theory predicts that TT exists at λ<λc(Δ) and TU at λ>λc(Δ), where λc(Δ) is the transition line between TT and TU in the λ-Δ plane [1,2]. Effects of thermal noise in the framework of a variable shape model within the second order spherical harmonics and Δ<<1 approximation has been then considered for λ=1[3] and later for λ>1 [4]. The predictions of the models [1-5] and 2D numerical simulations for more realistic approximation [6] have been tested quantitatively by the experiment [7-9] over a wide range of λ between 1 and 10 and various Δ. Surprisingly, the both analytical models and numerical simulations describe rather accurately the data on θ decrease with increasing Δ, the dependence of θ on λ at a given Δ in the TT regime, and the transition line λc(Δ), in spite of the fact that vesicles with Δ from 0.1 up to 1.8 are used in the experiment [7,8]. According to [9], the Keller-Skalak model [1,2] also describes TU semi-quantitatively. However, the more surprising discovery is a new type of an unsteady motion, characterized by oscillations of |θ|<π/2 around the shear flow direction accompanied by strong vesicle shape deformations [8]. It is coined trembling (TR) (see Figure 1). A crucial aspect of TR is the dependence of its existence region, separated from the TU region, on \(\dot{\gamma}\ .\) Thus, both strong shape deformations and dependence on \(\dot{\gamma}\) are distinct signatures of TR. Precisely these features change our idea about vesicle dynamics as smooth and shape-preserved motion and call for an adequate theoretical description. First step in this direction has been taken by considering the equation for shape deformations in addition to the θ-equation [10,11]. However, an oscillating state observed independently and simultaneously with the experiment and dubbed a vacillating-breathing mode, does not exhibit the distinctive features of TR motion [11]. Further progress is reached in three theoretical models by including the next order terms. All of them are supposed to completely describe the dynamics of all three states in shear flow, their existence regions and transitions [12-14]. Recent experiments [15], designed to test the predictions and to verify which of the theories properly describes the experimental data, were conducted in a plane Couette flow with significantly reduced errors in determining geometrical and physical characteristics of each vesicle. The main qualitative result of the experiment is construction of a 2D phase diagram of the vesicle dynamical states in a space of two self-similar dimensionless parameters suggested in Ref. [13], namely \[ S=\frac{7\pi}{3\sqrt{3}}\frac{\dot{\gamma}\eta_{out}R^3}{\kappa\Delta}\] and \[\Lambda=\frac{4}{\sqrt{30\pi}(1+\frac{23}{32}\lambda)\sqrt{\Delta}} ,\] where κ is the bending modulus. The corresponding experimental control parameters in the experiment are \(\dot{\gamma}\) and λ. So each vesicle with a given λ is either observed only in TT for λ<λc(Δ) or can be driven from TU to TR by increasing \(\dot{\gamma}\) for λ>λc(Δ). The theory is developed in the approximation of a 2D shear flow, Δ<<1, second order spherical harmonics shape deformations and neglecting thermal noise. Besides the 2D plane shear flow, none of the approximations are fulfilled in the experiment. In spite of a qualitative nature of the main result and semi-quantitative agreement in a distinct separation of the existence regions of TT, TU and TR states, the statement about the self-similarity of the experimental data is rather strong. It means that independent of Δ all vesicles with different geometrical characteristics can be presented on the 2D phase diagram, an important result that was overlooked in Refs. [12,14].

A noticeable quantitative discrepancy between theory and experiment was found in the width of the TR region. It is attributed to the approximations used in the theory and mentioned above as well as the strong deviations of a vesicle from the elliptical shape during the TR cycle (see Figure 1). Note that only TR requires for its existence two dynamical variables, θ and the shape deformation, while TU exists also in the single θ description [1,2,9]. Thus a feedback reaction of the vesicle shape deformations on the θ variations near a zero angle results in the TR state. In contrast to TT, in both TU and TR regimes the vesicle is subjected to periodic stretching and compression, which cause shape deformations similar to that in a time-dependent elongation flow described in the next Section. Due to the constraints, strong shape deformations under compression are mostly realized via generation of concavities, which are indeed visible during a TR cycle (see Figure 1) and are associated with a negative surface tension. The latter excites higher order modes that are observed during the TR cycle (see Figure 1) and could be the main reason for the discrepancy with the theory (which accounts only for ellipsoidal modes) regarding the width of the TR region.

Vesicle dynamics in elongation flow

Figure 2: Wrinkling instability. Snapshots of vesicle dynamics in time-dependent elongation flow at λ=1, Δ≈1, and: (a) χ=8.1, (b) χ=81, (c) χ=323.5. The scale bar is 20 μm.

An elongation flow is a strong flow, in which at least an order of magnitude higher straining stress than in a shear flow can be achieved. Two new questions arise: First, what is the vesicle relaxation dynamics towards its steady state? And second, what is the stretching dynamics of a tubular vesicle and its conformational transformations in an elongation flow?

Vesicle relaxation dynamics towards a new stationary state in a time-dependent plane elongation flow is studied experimentally in a cross-slot geometry micro-channel with two inlets and two outlets, which can be interchanged very fast compared to the vesicle time scale \(\tau\propto\eta_{out}R^3/\kappa\ .\) After switching the flow direction a vesicle undergoes a relaxation from one stretched state to another. The sudden reversal of the velocity gradient causes switching from vesicle stretching to compression that results in generation of higher-order modes in shape deformations, called wrinkles (see Figure 2) [16].

Quantitative characterization of the harmonic content of the vesicle shape deformations is performed by measuring the amplitude of deviations of the vesicle contour from its elliptical fit as a function of a polar angle φ and time. The instantaneous Fourier transform of the amplitude in respect to φ provides information about the shape spectrum dynamics. Such spectrum analysis shows that above some threshold value of the normalized elongation rate \(\chi_c=\dot{\epsilon_c}\tau\) the higher order modes are excited. The dependence of the average in time power spectrum of the relaxation modes Pk as a function of the wave number k shows thermal noise spectrum Pk ~ k-4 at χ<χc, whereas at χ>χc the spectrum becomes flat at k>2 and the flat region increases with χ. The transition threshold is found from the dependence of the averaged over spectrum wave number k* as a function of χ, where a well-defined transition from the elliptical vesicle shape to the shape containing all modes in the spectrum with \(k^*\geq3\) is observed. It is rather obvious that generation of higher-order modes in the vesicle shape would be penalized by the bending energy increase that make them improbable to appear in a case of positive surface tension. On the other hand, under compression a vesicle is deformed by first generating concavities, which are associated with negative surface tension tuned itself to the alternating stress. Such shape deformations are a direct consequence of the constraints and very similar to undulations that appear in a rod subjected to a unidirectional compression (Euler buckling instability). Thus, as a simultaneously developed theory shows, a negative surface tension can lead to the wrinkling instability, which results in excitation of higher-order modes in the membrane, if χ>χc. The initial instability is followed by the process of wrinkle coarsening, when the wrinkle wavelength as well as their amplitude increases whereas the absolute value of surface tension decreases algebraically with time [17]. The theoretical instability threshold and the power law dependence of k* versus χ are found in a fair agreement with the experiment [16,17].

A steady elongation flow is used to study dynamics of stretching and conformal transformations of tubular shape vesicles [18]. Biological membranes often develop tubular structures and form dynamical tubular networks, which play a crucial role in many biological processes, including inner- and inter-cell transport. Conformational fluctuations of a vesicle membrane define its configuration entropy and therewith entropy-driven tension. The vesicle membrane in this respect behaves analogously to a linear flexible polymer molecule, whose elasticity is entropy-driven too. In elongation flow, a flexible polymer undergoes a conformational coil-stretch transition at a critical value of the velocity gradient (strain rate). Similarly, the transition from a floppy tubular shape vesicle with an initial length-to-diameter ratio L0/D0>4.2 to a dumbbell shape at a critical strain rate \(\dot{\epsilon_c}\) and further into a transient pearling state is found [17]. Measurements of the vesicle dynamics are conducted in the same cross-slot channel at \(\dot{\epsilon_c}\)=0.01-3 1/s. At \(\dot{\epsilon}\leq\dot{\epsilon_c}\) only a slight stretching of the tubular vesicle is observed. At \(\dot{\epsilon_c}\) tubular vesicles evolve into dumbbells. At\(\dot{\epsilon}\geq\dot{\epsilon_c}\ ,\) further significant thinning and extension of the tether in the dumbbell occur. At even higher\(\dot{\epsilon}\ ,\) the tether becomes unstable, and sequence of transitions to transient pearling configurations with different number of beads is observed. The number of beads decreases during the transient stretching, until a stationary stretched dumbbell configuration is established. Different conformation transitions with different order of unstable modes are found to depend on the value of \(\dot{\epsilon}\ .\) Stretching sequences showing the pearling are presented in Figure 3.

Figure 3: Stretching of a tubular vesicle. Parameters of the tube are L0=66 μm, D0=1.4 μm, L0/D0=47, η=5.4x10-3 Pas. (a) =0.09 1/s, (b) =0.14 1/s (dumbbell mode instability), (c) =0.19 1/s (second mode instability-pearling), (d) =0.92 1/s. The scale bar is 10 μm.

In regards to thresholds of the tubule-to-dumbbell conformation transition and further transition to pearling, a strong dependence on L0/D0 is found. So to collapse all data on the critical elongation rate versus L0/D0, they are scaled by the characteristic retraction time of the vesicle to its equilibrium state after flow cessation τr. The data are described rather close by \(\dot{\epsilon_c}\tau_r\propto(D_0/L_0)\ln(L_0/D_0)\) with \(\tau_r=D_0^2L_0\eta/4\kappa\) obtained from simple considerations [18]. And finally, the tubule-to-dumbbell transition is a continuous one and accompanied by the slowing down of the dynamics of the vesicle extension near \(\dot{\epsilon_c}\) due to presence of a large number of possible configurations available close to the transition point, where the entropic force (elasticity) is balanced by the hydrodynamic drag (stretching force). This critical effect, which is also observed in a case of the polymer coil-stretch transition, is a characteristic feature and unique signature of the stretching transition and shows how a simple elongation flow has a nontrivial effect on the dynamics of a single vesicle. Thus it can be suggested that the critical effect is a universal dynamical phenomenon, which occurs in any micro- and mesoscopic objects, where thermal noise leads to large variety of accessible configurations close to the conformation transition.

Vesicle dynamics in general flow

Figure 4: Transition of a vesicle from TT to TU and further to TR due to variations of ω/s and s. (a) θ versus time, (b) D=(L-B)/(L+B) versus time. L and B are the large and small semi-axis of the elliptical approximation of vesicle. R=14.4 μm, Δ=0.64.

A new approach to study not only vesicle dynamics but a wide variety of micro-objects including biologically-relevant ones in a fluid flow is based on a combination of a new tool and a novel control parameter, which allows scanning the entire phase diagram and provides a new way for quantitative investigation of dynamics and interaction of micro-objects [19]. An improved micro-fluidic four-roll mill device [20] is used as a dynamical trap for a vesicle. The flow can be continuously varied from purely rotational to purely elongational (note that the simple shear is a combination of rotation and elongation) by tuning just one parameter, the pressure difference between input and output channels ΔP. The latter is directly related to the ratio of the vorticity to the strain rate ω/s, which together with s are used as the physical control parameters that allow scanning the phase diagram of vesicle dynamical states in a general flow, according to the theory [13]. Such approach avoids the technical problems of varying the control parameters in a shear flow in order to scan the phase diagram discussed above. First, it is particularly relevant to λ, which is difficult to control and varies only in discrete manner. Second, all dynamical states can be observed on the same vesicle and even for λ=1 in contrast to the previous view based only on shear flow dynamics [1-12,14]. The corresponding self-similar parameters for the phase diagram of vesicle dynamical states in general flow are \[ S=\frac{14\pi}{3\sqrt{3}}\frac{s\eta_{out}R^3}{\kappa\Delta}\] and \[\Lambda=\frac{4}{\sqrt{30\pi}(1+\frac{23}{32}\lambda)}\frac{\omega}{s}\sqrt{\Delta} ,\] where sik is the symmetric strain tensor and ωj is the vorticity vector defined from the velocity gradient as \(\partial_iV_k=s_{ik}+\epsilon_{ikj}\omega_j\ .\) Further improvement of experimental techniques reduces the errors in S and Λ and a feedback on the flow velocity allows trapping a vesicle sufficient time to observe many cycles in each time-dependent state and to study transitions from TT to TU and back to TR motions ( Figure 4). To explore the whole space of parameters (S, Λ), vesicles with various values of R and Δ are loaded and individually observed, and ω/s is varied during the experiment by changing ΔP. In this way the phase diagram in a space of (S, Λ) is scanned along a particular path and populated with experimental points, whereas the different dynamical regimes are identified from the dynamics of the main vesicle axis.

Figure 5: Phase diagram of the vesicle dynamical states in general flow: green symbols-TU, red-TR, blue-TT. Grey bands are guides for the eye. Dotted, dashed and solid black lines are the theoretical boundaries between TT, TU and TR, respectively. ■- Δ Î [0-0.8], □- Î [0.8-1.3], ●- Î [1.3-1.8], ○- Î [1.8-2.6]. Two black arrows indicate the experimental path through the dynamical states followed by the vesicle presented in Figure 4.

In Figure 5 the data are plotted in the form of a phase diagram, identifying the different dynamical regimes attained by different vesicles for parameters spanning the (S,Λ)-plane. The grey bands separate regions with different dynamics. The lines separating the three regimes TT, TR, and TU predicted theoretically, taken from Ref. 13, are also shown. Distinct dynamical regimes are indeed achieved experimentally in the defined regions of S and Λ in the parameter plane, while no correlation with the value of Δ of the vesicles is seen. This agrees with the self-similar parameterization of the vesicle dynamics suggested in Ref. 13. As in the case of a shear flow, the experimental TR region is wider than the theoretical. Moreover, the TR region is divided into subregions in regards to higher order modes observed over a TR cycle: a significant third order harmonic is observed in TR at 2<S<15 (see Figure 4), and fourth and fifth orders at 15<S<50. The discrepancy between theory and experiment regarding the width of the TR region brings us back to the discussion on the TR motion. A vesicle deforms mostly via concavities as a response to compression during a part of the TR cycle. Hence, there is a connection between TR and the vesicle wrinkling in a time-dependent elongation flow discussed above. The main difference between these two cases is that in the latter the specific properties of the external flow are responsible to non-smooth dynamics, whereas in the TR state the appearance of the higher order harmonics on each cycle is the intrinsic feature of the TR motion. Quantitative analysis of a higher order mode content of the vesicle shape deformations during the TR cycle is currently in progress. Despite intense experimental work, number of puzzles remain such as why is the self-similar solution valid even at large excess areas \(\Delta\propto O(1)\ ?\) Furthermore, a search for a vesicle spinning motion predicted theoretically [13] and the dynamics of a vesicle in a random flow are on the way.

References

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Internal references

  • Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
  • Giovanni Gallavotti (2008) Fluctuations. Scholarpedia, 3(6):5893.
  • Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
  • Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

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