Amplification of Thermal Noise by Vesicle Dynamics - APS Link Manager

PRL 109, 268103 (2012)

week ending 28 DECEMBER 2012

PHYSICAL REVIEW LETTERS

Amplification of Thermal Noise by Vesicle Dynamics Michael Levant and Victor Steinberg Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, 76100, Israel (Received 21 March 2012; published 27 December 2012) A novel noise amplification mechanism resulting from the interaction of thermal fluctuations and nonlinear vesicle dynamics is reported. It is observed in a time-dependent vesicle state called trembling (TR). High spatial resolution and very long time series of TR compared to the vesicle period allow us to quantitatively analyze the generation and amplification of spatial and temporal modes of the vesicle shape perturbations. During a compression part of each TR cycle, a vesicle finds itself on the edge of the wrinkling instability, where thermally excited spatial modes are amplified. DOI: 10.1103/PhysRevLett.109.268103

PACS numbers: 87.16.D, 83.50.v, 43.50.+y

As was shown both theoretically [1–3] and experimentally [4–7], thermal noise is greatly amplified by time-periodic dynamical systems with only temporal modes due to the interaction between stochasticity and nonlinearity in the prebifurcation region of spontaneous symmetry breaking instabilities such as saddle point, pitchfork, period doubling, and Hopf. Then, the noise amplification is observed only in the frequency domain. Nonlinear response on the noise became a central issue in studies of various dynamical systems including climactic [8], geophysical [9], populational [10], and optical models [11]. In this Letter, we report a novel mechanism of noise amplification of both temporal and spatial modes due to the interaction of thermal noise and nonlinearity of vesicle dynamics. A vesicle is a droplet of a fluid encapsulated in a lipid bilayer membrane and suspended in either the same or a different fluid. It is considered to be a simple model to simulate red blood cell dynamics as the first step toward understanding blood rheology. Impermeability and inextensibility of a lipid membrane dictate conservation of both volume V and surface area A of a vesicle. Despite the two constraints on vesicle dynamics, a vesicle shape is not uniquely defined and shape deformations are permitted. A phase diagram of vesicle dynamical states contains at least three dynamical regimes: tank treading (TT) [12,13], tumbling (TU) [14,15], and intermediate trembling (TR) (also vacillating breathing and swinging) [14,16–19]. In TT, both the inclination angle between the vesicle long axis L and the shear flow direction and its shape remain stationary apart from thermal fluctuations, whereas the membrane undergoes TT motion. In TU, L rotates with respect to the shear direction and the membrane rotates around the vesicle. The TR motion, where L oscillates around the shear flow direction, is the key state to understanding vesicle dynamics due to the striking manifestations of nonlinearity, nonlocality, and coupling to noise as the result of the two constraints. In contrast to TT, the TR dynamics is not a smooth and shape-preserving motion. Due to stretching at 0031-9007=12=109(26)=268103(5)

positive and compression at negative of a vesicle during its oscillation cycle in a shear flow, strong shape deformations are observed. Then, to characterize the vesicle dynamics, at least two appropriate dynamical variables are required [14,16–25]. The compression at < 0 leads to a negative surface tension that results in an instability and concave shape perturbations resembling wrinkles that arise from the instability in a time-periodic elongation flow [26,27]. Similarly, during each TR cycle, the negative surface tension can lead to the instability, which results in particular in the enhancement of the third and higher modes in the vesicle shape perturbations. The details of the TR dynamics are shown in the snapshots and movies in Refs. [14,23–25]. Theory [16–22] and 3D simulations without thermal noise [28–30] of the vesicle dynamics and shape deformations in the TR regime are found to be in contrast with those observed experimentally. These theoretical models, which take into account only the lowest even modes dictated by the r ! r symmetry based on the flow symmetry, show the interplay solely between the even modes without breaking this symmetry [16,17,19–22]. Moreover, additive noise being included in each of the even mode equation also would not break the symmetry. The relevant numerical snapshots in Ref. [20] and movies in Refs. [21,30] present pure oscillations of the main axis of the vesicle elliptical shape with only even modes imitating ‘‘breathing,’’ never observed experimentally even at small excess areas A=R2 4 0:2, where R is the effective vesicle radius defined as V ¼ 4R3 =3. The theory of the vesicle dynamics is based on the following assumptions, which are not consistent with the experimental conditions [16,17,19–22]: (i) 1; (ii) thermal noise is neglected; (iii) only the second [16,17,19–21] and later on the fourth order [22] harmonics in shape perturbations are taken into account. In 3D simulations, Oð1Þ are also considered but the dynamics is similar [29,30]. On the contrary, 2D simulations with thermal noise show the vesicle shape deformations in TR to be similar to the experimentally observed ones [18,31].

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Ó 2012 American Physical Society

PRL 109, 268103 (2012)


Susceptibility of vesicles to thermal noise in equilibrium [32] and its role in TT dynamics [12,13] were investigated in detail during the years. There, the noise plays a minor role due to a small ratio kB T= 0:05, where is the membrane bending elasticity [33]. In spite of this, in the TR dynamics, the role of noise is probably decisive. This paradox raises the following questions: Is it possible to clarify the role of thermal noise and to single out characteristic features (such as, e.g., concavities), which can enhance it, in the vesicle TR dynamics? How can one analyze and quantify noise amplification in TR compared with the TT motion? In this Letter, we report new vesicle data and analysis of much longer time series and higher spatial resolution compared with our previous results [25] that allows us to verify the role of thermal noise and to conduct its quantification in TT versus TR in the vesicle dynamics. We observed 17 vesicles with R 2 ½10:45; 15:3 m, 2 ½0:43; 1:4 located at different coordinates on the phase diagram in the region S 2 pffiffiffi½2:5; 97:7, 2 ½0:82; 21:5, is the normalized strain where S ¼ 14out R3 s=3 pffiffiffiffi3pffiffiffiffiffiffiffiffiffi rate, ¼ ð23 þ 32Þ! =8s 30 is the normalized viscosity contrast, s is the strain rate, ! is the vorticity, ¼ in =out is the ratio of the inner and outer fluid viscosities, and ¼ 1 in the experiment [19]. Observations in TR were supplemented by observations of the TT dynamics for comparison. Observations and measurements of the vesicle dynamics were conducted inside a microfluidic four-roll mill device, implemented in silicone elastomer (Sylgard 184, Dow Corning) via soft lithography [24,34]. The key component of the device is a dynamical trap that allows long time observation of vesicles in a planar linear flow (see Fig. 1). Observation times were up to 100 times longer than the TR

FIG. 1 (color online). Experimental setup. PC, personal computer.


oscillation period. The physical control parameter !=s inside the trap is varied by the single experimental parameter, the pressure drop across the trap P. For 0 s < !, vesicles move along elliptical trajectories inside the trap, and for typical parameters R Oð10 mÞ, Oð1Þ all three dynamical states can be observed in this range. We used single particle tracking [35] to map the flow inside the dynamical trap as a function of P with suspension of fluorescent particles, R ¼ 0:5 m (Duke Scientific), 1:1000 v/v. Flow calibrations were performed prior to the experiment. Vesicles were prepared in water ( ¼ 1) via electroformation [36], using lipid solution, consisting of 85% dioleoyl-phosphatidylcholine lipids (Sigma) and 15% fluorescent-labeled phosphatidylcholine lipids (Molecular Probes), dissolved in 9:1 v/v chloroform-methanol solvent. The vesicle dynamics inside a 140 180 m observation window (spatial resolution 1px ¼ 0:27 m) was monitored using an inverted fluorescent microscope (IMT-2, Olympus). The images were collected with a Prosilica EC1380 CCD camera, aligned with the shear axis, and synchronized with a mechanical chopper on the path of the excitation beam to reduce exposure time (Fig. 1). It allows us to extend the total observation time up to 550 sec at 15 fps that provides sufficient data sets for statistic analysis. The vesicle suspension flow was driven by hydrostatic pressure P. Vesicles were introduced into the trap by rapid variations of P to change the flow inside the trap from rotational to extensional, i.e., from closed to open streamlines and vice versa. The trapped vesicles were manipulated by subtle variations of P to bring the vesicle as close as possible to the trap center and to expel other vesicles out of the trap. The required vesicle dynamical state was achieved by P variation. The radial position of the membrane at each polar angle , rð; tÞ was determined in the frame of reference of the vesicle using intensity variations along the radial directions [23,26]. For each image, we obtained up to 500 discrete positions sampled along the contour. To quantitatively analyze the TR dynamics, the dimensionless shape of the vesicle, uð; tÞ ¼ rð; tÞ=RðtÞ 1, 0 2, RðtÞ ¼ hrð; tÞi , was Fourier decomposed via fast Fourier transform; i.e., uð; tÞ was expressed as uð; tÞ ¼ Pn iq . We calculated the autocorrelation funcq¼1 uq ðtÞe tion of the time sequence of the angle of the second mode, which corresponds to . As pointed out in Ref. [25], the TR motion looks roughly periodic with an appearance of dips in the second harmonics amplitude during a compression part of each period. A detailed observation of the vesicle dynamics, presented together with the instantaneous shape spectra in Fig. 2, reveals a further fundamental discrepancy with the theoretical predictions discussed in the introduction. In the compression part of the first cycle [Fig. 2(a)], the triangular shape becomes predominant, so that the third order mode

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FIG. 2 (color online). Snapshots of two subsequent cycles in TR with R ¼ 15 m, ¼ 0:66 at S, ¼ 60:7 and 1.8, and the instantaneous shape spectrum at given times on a log-linear scale (right).

exceeds by 2 orders of magnitude the second one and so that the second order mode varies up to 3 orders of magnitude during the cycle [see the power spectra in Fig. 2(a)]. In the subsequent cycle [Fig. 2(b)], the compression at negative results in predominantly concave perturbations of short wavelengths due to negative surface tension that further leads to the amplification of higher order modes up to 10th order. The visual resemblance of vesicle shapes during the second cycle [Fig. 2(b)] to wrinkles and concavities shown in Ref. [26] qualitatively supports our argument that the strong shape fluctuations during TR cycles emerge as a result of the negative surface tension. Attempting to further quantitatively compare the TR dynamics and the wrinkling instability, we define similarly to Ref. [26] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P20 2 2 P20 2 k ¼ k¼3 k juk j = k¼3 juk j . During the compression in a significant part of the TR cycles, k reaches maximum values fluctuating between about 5 and 8, which are just above the onset of the wrinkling instability (see Fig. 3 in Ref. [26]). A comparison of probability distribution functions (PDFs) of ju2 j and ju3 j, where juq j ¼ juq j hjuq jit

and ¼ hi (Fig. 3) for several (S, ), indicates a continuous increase of the PDF variances with . 2 of ju2 j grows about twice close to the TT-TR transition, compared to the lowest in the TT region, where the PDF is Gaussian and 2 is equal to the thermal noise value [13]. Further growth of causes 2 to increase 1 order of magnitude in the TR region. A similar increase of 3 is found for ju3 j and of for (Fig. 3). In contrast, no visible changes in TR were observed due to S variations, which is consistent with our previous observations [25]. Another way to quantify the TR dynamics is to analyze the trajectories in the (D, ) parameter plane and the normalized autocorrelation functions Cð Þ ¼ hðt þ ÞðtÞi= 2 , where D ¼ ðL BÞ=ðL þ BÞ and L and B are the major and minor axes of the vesicle (Fig. 4). TT is represented by the fixed point in the (D, ) plane, slightly smeared due to thermal noise. The corresponding autocorrelation function is a -like peak with a short correlation time. Closer to the TT-TR transition, the fixed point is smeared more. Inside the TR region, a periodic component in the autocorrelation

FIG. 3. PDFs of ju2 j, ju3 j, at two extreme values of (S, ) out of four presented in the insets: (70.3,0.84) and (60.7,1.8). The values correspond to TT at ¼ 0:84, the vicinity of the TT-TR transition at ¼ 1:18, the inside of the TR region at ¼ 1:56, and a region close to the TR-TU transition at ¼ 1:8. Insets: PDF variance q of juq j and of versus .

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FIG. 4. Trajectories in the (D, ) plane and the normalized autocorrelation functions of ðtÞ for (top to bottom) ðS; Þ ¼ ð70:3; 0:84Þ; ð51:2; 1:18Þ; ð53:6; 1:56Þ; ð60:7; 1:8Þ.

function is evident; however, the correlation time is still rather short. In the (D, ) plane, the vesicle’s trajectory fills an area bounded by the limit cycle, in contrast to an ideal situation without any noise, as shown in Ref. [21]. Near the TR-TU transition, the area filled by the trajectory in the (D, ) plane grows as well as the correlation time, and occasional flips occur [open trajectories in the (D, ) plane]. The (D, ) diagrams presented in Fig. 4 as well as the vesicle shape shown in Fig. 2 resemble the images presented in the 2D simulations with thermal noise in Ref. [31]. The time averaged spectra of the vesicle shape show the difference of the mode contributions in the TT and TR regimes (Fig. 5). Thanks to higher spatial resolution and longer time series than in Ref. [25], a drastic suppression of odd modes in the TT spectrum is clearly visible up to 15th order. They are still observable, probably due to noise perturbations. On the contrary, in the TR motion the lower even and odd modes (excluding the second) are comparable (from the 3rd until the 6th). Moreover, the ‘‘plateau’’ at the highest even and odd modes (from about the 12th until the 30th) is considerably higher than the S=70.3, Λ=0.84 S=53.6, Λ=1.56

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FIG. 5 (color online). Time averaged spectra of the vesicle shape in TT and TR.

corresponding one in the TT regime. It should be compared with the observation in Fig. 2(b), where the higher order modes were amplified due to the concave perturbations. Both spectra differ from the equilibrium thermal noise spectrum exhibiting q4 decay [32]. The results of the current analysis of the TR dynamics and statistics lead us to the following conclusions. (i) The thermal noise is strongly amplified with in the vesicle shape perturbations as well as during TR. Upon crossing the TT-TR transition into the TR region, the dynamics become more complex and noise is drastically amplified while coupled with the time-dependent motion. The amplified noise is evident in the TR dynamics presentation via trajectories in the (D, ) plane, where instead of the expected limit cycle the TR trajectory fills an area bounded by the limit cycle. Moreover, upon approaching the TR-TU transition from below, isolated TU events occur that are not predicted by the noiseless models. This new noise amplification mechanism differs from the known one, associated with a bifurcation onset [1,2], and leads to the amplification of temporal as well as spatial modes. Our conjecture is that the noise amplification may be caused by the vesicle shape instability resulting in appearance of concavities due to compression at < 0 on each TR cycle, similar to the wrinkling instability of a vesicle in a time-dependent elongation flow [26]. (ii) Strong shape perturbations are quantified by the instantaneous power spectra of the vesicle shape, where the amplification of the odd modes is observed during only the compression in each TR cycle. The lowest odd mode (q ¼ 3) frequently becomes the dominant one, up to 2 orders of magnitude higher than the second mode, being accompanied by the enhancement of the higher order modes. This symmetry breaking of the even modes reveals the most fundamental discrepancy

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with the theoretical models. (iii) The TR dynamics can be substantially different in details even between two subsequent cycles in the same time series of the vesicle due to noise. The only reproducible behavior we found is the attenuation of the second mode and the strong amplification of the odd modes by up to 2 orders of magnitude during the compression in each TR cycle. Enhanced spatial resolution achieved in these experiments allowed us to visualize and to quantify the concave perturbations resulting from the wrinkling instability. We emphasize that the noise amplification mechanism is strikingly different from the known one in Refs. [1–7]. Indeed, in the presented mechanism, both the spatial and temporal modes are amplified in contrast to only the temporal modes in the latter one. The amplification of the spatial modes occurs in the vicinity of the wrinkling instability during the compression at < 0, although the instability does not have time to be further developed, being cut by the elongation at > 0 in the TR cycle. We thank G. Falkovich and S. Vergeles for remarks on symmetry breaking. This work is partially supported by grants from the Israel Science Foundation, the Minerva Foundation, and the German-Israel Foundation.

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