Absolute/Convective Instabilities and Front Propagation in Lipid Membrane Tubes
AAbsolute/Convective Instabilities and Front Propagation in Lipid Membrane Tubes
Joël Tchoufag, ∗ Amaresh Sahu, ‡ and Kranthi K. Mandadapu
1, 2, † Department of Chemical & Biomolecular Engineering, University of California, Berkeley, CA 94720 Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 (Dated: September 1, 2020)We analyze the stability of biological membrane tubes, with and without a base flow of lipids.Membrane dynamics are completely specified by two dimensionless numbers: the well-known Föppl–von Kármán number Γ and the recently introduced Scriven–Love number S L , respectively quantify-ing the base tension and base flow speed. For unstable tubes, the growth rate of a local perturbationdepends only on Γ , whereas S L governs the absolute or convective nature of the instability. Fur-thermore, nonlinear simulations of unstable tubes reveal an initially localized disturbance results inpropagating fronts, which leave a thin atrophied tube in their wake. Depending on the value of Γ , thethin tube is connected to the unperturbed regions via oscillatory or monotonic shape transitions—reminiscent of recent experimental observations on the retraction and atrophy of axons. We elucidateour findings through a weakly nonlinear analysis, which shows membrane dynamics may be approx-imated by a model of the class of extended Fisher–Kolmogorov equations. This model possessesLifshitz points, where the front dynamics undergo a steady-to-oscillatory bifurcation, thus sheddinglight on the pattern selection mechanism in neurons. Keywords: lipid membrane tube; absolute/convective instability; front propagation
Lipid membranes are ubiquitous in biology, as theyconstitute the cell boundary and the boundary of thecell’s internal organelles. In both in vivo and in vitro systems, lipid membranes are often found in cylindricalshapes [1–4]. When membrane tubes are under high ten-sion, they can undergo shape instabilities leading to newmorphologies. Such dynamical shape changes are trig-gered through various disturbances, which can be clas-sified as either global or local perturbations. In the for-mer case, tubes are disturbed along their entire length,for example via spontaneous thermal fluctuations [5], anapplied extensional flow [6, 7] or osmotic shocks [8–10]—with pearled configurations always ensuing. When a tubeis perturbed locally, on the other hand, different mor-phologies result from the initial disturbance invading theunstable tube via propagating fronts. For example, whena laser is aimed at a point on a membrane tether, pearlsform in the wake of the moving fronts [5, 11–13]. Incontrast, focused laser beam ablation applied to in vitro cultures of axons results in a thin atrophied tube, whichis connected to the unperturbed region via either a mod-ulated or monotonic transition [14]—the latter being sim-ilar to the nonlinear simulation results reported in Fig.1(b). Additionally, drug treatments disrupting actin fil-aments and microtubules in axons revealed that frontsonly propagate from the neuron’s growth cone to its cellbody [14]. However, the physical mechanisms govern-ing these behaviors, namely (i) the pattern selection inthe membrane region connecting thin atrophied tubes tomoving fronts and (ii) the direction of instability propa-gation, remain poorly understood.In this Letter, we shed light on the aforementionedmechanisms by incorporating two often overlooked traitsinto our description of membrane tubes. First, whilemost studies of cylindrical membrane instabilities assume AUCU ∼ Γ / (cid:0) c (cid:0) : (cid:24) ((cid:0) (cid:0) (cid:0) c ) = ∗‡ Γ S L (a) t z ∗ absolutely unstable ‡ convectively unstable V (b) FIG. 1. (a) Stability diagram of lipid membrane tubes in thespace of Γ and S L , showing stable (gray, Γ < Γ c = / ) , ab-solutely unstable (AU, red), and convectively unstable (CU,blue) regimes. The critical velocity of the AU/CU transitionscales as Γ / at large Γ , but as (Γ − Γ c ) / for Γ → Γ +c (seeinset). Dashed vertical lines mark the interval (Γ , Γ ) wherefronts invading the undeformed tube are oscillatory. At thepoints marked ‘ ∗ ’ and ‘ ‡ ’, snapshots from nonlinear simula-tions (b) reveal propagating fronts. The initial perturbation isat the vertical dashed line, flow is in the positive z -direction,the color bar indicates the surface tension, and for clarity thesnapshots are scaled by a factor of 40 in the z -direction. a r X i v : . [ c ond - m a t . s o f t ] A ug the bulk fluid viscosity is the main source of dissipation[5, 13, 15–20], here we consider the intramembrane vis-cosity to be the primary dissipative cause. Second, dueto the flow of lipids observed in both axons (where speedsreach up to ∼ µ m/min) [21, 22] and membrane teth-ers bridging spherical vesicles [23], we incorporate a baseflow of lipids in our description. In what follows, we be-gin by presenting the dimensionless equation governingcylindrical membrane dynamics, from which we obtainthe dispersion relation. We then elucidate the effects ofa nonzero base flow of lipids on membrane spatiotempo-ral stability. In particular, we demonstrate the relevanceof the hydrodynamic concepts of absolute and convec-tive instabilities [24, 25] to understanding the long-timebehavior of lipid membrane tubes subjected to initiallylocalized perturbations. We next investigate the dynam-ics of how the fronts ensuing from a local perturbationinvade the unperturbed membrane tube. We employ theso-called marginal stability criterion (MSC) [26, 27], andcorroborate our analytical results with direct simulationsand a weakly nonlinear analysis of the shape evolution ofan unstable tube. Importantly, we find both atrophiedand pearled morphologies can result from a local per-turbation, and our front speed calculations qualitativelyagree with previous experimental measurements. Membrane dynamics and dispersion relation. —We consider an unperturbed lipid membrane tube ofradius r , aligned along the z -axis. The base flow oflipids is captured by the velocity field v (0) = V e z forconstant speed V , and a force balance in the normaldirection reveals the base surface tension is given by λ = p r + k b / (4 r ) . Here, k b is the bending modu-lus and p is the jump in the normal traction across themembrane surface, which is treated as a constant sincedynamics of the surrounding fluid are not considered.We now linearize the general membrane equationsabout the chosen base state, as detailed in the Supple-mental Material (SM) [28, Sec. I.4]. All quantities arenon-dimensionalized with r , k b , and the membrane vis-cosity ζ [29]—for which ζr /k b is the fundamental timescale, k b /r is the fundamental tension scale, and inertialterms are found to be negligible in biological systems ofinterest [30]. Note lipid membrane tubes under tension,for which λ ≥ , are stable to all non-axisymmetric per-turbations (see Ref. [20] and Sec. II.3 of the SM [28]).Consequently, only axisymmetric perturbations are con-sidered here. In this case, the perturbed equations ofmotion can be combined into a single equation governingthe perturbed tube radius ˜ r , given by [28, Sec. I.4(c)] ˜ r ,t + S L ˜ r ,z = (cid:18) Γ − / (cid:19) ˜ r + (cid:18) Γ − / (cid:19) ˜ r ,zz −
18 ˜ r ,zzzz . (1)In the absence of bulk dynamics, membrane dynamicsare governed by two dimensionless numbers: the Föppl–von Kármán number Γ and the Scriven–Love number S L , defined respectively as Γ = λ r k b and S L = ζ V r k b . (2)The Föppl–von Kármán number compares tension tobending forces [30, 31], and the recently introducedScriven–Love number [30] compares out-of-plane viscousforces to bending forces.At this point, we decompose the perturbed radius ˜ r into normal modes of the form ∼ exp[i( qz − ωt )] , where q is a dimensionless wavenumber and ω is a dimensionlessfrequency. Substituting this decomposition into Eq. (1)leads to the dispersion relation ω = S L q + i4 (cid:104) Γ (cid:0) − q (cid:1) + 14 (cid:0) − q − q (cid:1) (cid:105) . (3) Temporal stability analysis. —Our first task is todetermine how global sinusoidal perturbations to themembrane evolve in time. We assess the tube’s temporalstability with normal modes of real wavenumber, q ∈ R ,and complex frequency, ω = ω ( r ) + i ω ( i ) ∈ C , for which ω ( r ) = S L q and ω ( i ) = [Γ(1 − q ) + ( − q − q )] .Given our normal mode decomposition, the tube is unsta-ble when there exists a q for which ω ( i ) > —a criterionwhich only depends on Γ . As shown in the SM [28, Sec.II.3], and corroborated by previous studies [13, 15–20],we find membrane tubes are unstable to a finite rangeof long wavelength perturbations when Γ > Γ c := / , asevident from the growth rate being ω ( i ) = (Γ − Γ c ) inthe limit of q → [32]. Though the growth rate of mem-brane perturbations is determined only by the Föppl–vonKármán number, the real part of the dispersion relation(3) reveals that a nonzero base flow ( S L (cid:54) = 0 ) leads totemporal oscillations in the membrane’s response, withfrequency proportional to the base flow speed. Addition-ally, an axial base flow confers a directionality to themembrane tube, which can bias the spatial evolution ofan initially local perturbation—bringing us to the con-cept of absolute and convective instabilities. Absolute vs. convective instabilities. —Considera membrane perturbation that is initially localized inspace, as is the case when a tube is subjected to laserablation [11] or when drugs are locally administered toan axon [14]. For any unstable tube with Γ > Γ c , thereis a competition between the local perturbation being (i)amplified, as characterized by Γ , and (ii) convected awayfrom its location, captured by S L . Figure 1(a) shows howunstable systems are classified as being either absolutelyunstable (AU) when amplification dominates convection,or convectively unstable (CU) when the instability is con-vected faster than it grows. The qualitative differencebetween AU and CU systems is highlighted in Fig. 1(b),in which we inquire for the long-time response as seenby a stationary observer—for example, one stationed atthe vertical dashed line. In the AU case (Fig. 1(b), leftcolumn), the base flow is sufficiently small and the initialperturbation grows to invade the entire domain. Con-sequently, our observer sees a deformed configuration atlong times. In the CU case (Fig. 1(b), right column), onthe other hand, the base flow is large enough to carrythe perturbation downstream. As a result, any station-ary observer may see a transient growth of the instabil-ity, yet at long times will see an unperturbed system—despite the disturbance continuing to grow as it is sweptdownstream. In what follows, we describe how to deter-mine when the system transitions from being AU to CU,known as the absolute-to-convective transition .To determine the boundary between domains of AUand CU membrane tubes, we carry out a spatiotemporalstability analysis following studies of instabilities in openflow systems [24, 25, 33], as detailed in the SM [28, Sec.III]. To this end, we consider normal modes with com-plex wavenumbers and frequencies: q = q ( r ) + i q ( i ) ∈ C and ω = ω ( r ) + i ω ( i ) ∈ C , as justified in Ref. [34]. Bydefinition, the long-time membrane response seen by astationary observer is dominated by zero group velocityperturbations—namely, those with (d ω/ d q ) | q = 0 and ω = ω ( q ) . Here, we introduce the so-called absolutefrequency ω = ω ( r )0 + i ω ( i )0 and absolute wavenumber q = q ( r )0 + i q ( i )0 , where ( q , ω ) corresponds to a saddlepoint of the system. If ω ( i )0 > , a stationary observersees the initial perturbation grow at long times and thesystem is AU, as in the left column of Fig. 1(b). On theother hand, if ω ( i )0 < , a stationary observer sees theinitial perturbation decay at long times and the systemis CU, as in the right column of Fig. 1(b). Consequently, ω ( i )0 = 0 at the boundary between AU and CU cylindricalmembranes.The real and imaginary parts of the two saddle pointcriteria, as well as the condition ω ( i )0 = 0 , are five equa-tions relating six quantities: ω ( r )0 , ω ( i )0 , q ( r )0 , q ( i )0 , S L , and Γ . As detailed in Sec. III.1 of the SM [28], for any choice Γ > Γ c , we solve this system of equations for S L ac (Γ) : thevalue of the Scriven–Love number corresponding to theabsolute-to-convective transition. As it turns out, how-ever, ω ( i )0 = 0 is a necessary but not sufficient require-ment for such a transition [24, 35, 36]. Consequently,solving the aforementioned system of equations leads toboth physical and spurious values of S L ac . We select onlyphysically relevant values by applying the so-called pinch-ing point criterion [25], as detailed in the SM [28, Sec.III.2]. Our calculation of the physical values of S L ac (Γ) isshown as the boundary between AU and CU domains inFig. 1(a). While the full analytical expression of S L ac (Γ) is lengthy and provided in Table 1 of the SM [28], wenote two limiting cases: S L ac ≈ Γ / when Γ → ∞ and S L ac ≈ √ (Γ − Γ c ) / when Γ → Γ +c . In calculat-ing S L ac (Γ) , we also determine the absolute wavenumber q (Γ) and frequency ω (Γ) , which dominate the long-timedynamics as seen by a stationary observer. While S L ac − − (cid:0) c (cid:0) (cid:0) : Γ Γ q ( r )0 q ( i )0 − . . (cid:0) c (cid:0) Γ Γ ω ( r )0 FIG. 2. Real and imaginary parts of the absolute wavenumber q (left) and real part of the absolute frequency ω ( r )0 (right),with ω ( i )0 = 0 . The solid (resp. dashed) lines correspond tothe relevant (resp. spurious) saddle points in searching forthe AU/CU boundary. As the Föppl–von Kármán numberincreases, the bifurcation at Γ signals a transition from steadyto oscillatory spatiotemporal dynamics, and vice versa at Γ . is a smoothly varying function of Γ , the saddle point ( q , ω ) undergoes two bifurcations as Γ is varied: one at Γ := / −√ ≈ . and another at Γ := / + √ ≈ . ,as shown in Fig. 2 and derived in the SM [28, Sec. III.1].We discuss how the saddle point bifurcations affect thelong-time response of a perturbed lipid membrane tubein the following section on front propagation. Front propagation. —When a membrane tube is lo-cally perturbed, both linear simulations (Fig. 5 in theSM [28]) and nonlinear simulations [Fig. 1(b)] show theperturbed region invades the unstable, unperturbed re-gion via a leading ( + ) and a trailing ( − ) front. We nowcalculate the leading and trailing front velocities V ± f pre-dicted by the linear theory. To this end, we employ theMSC [26, 27], which states that an observer travelingat the front speed would at long times see a system inits marginal state, i.e. one in which the growth rate iszero. If this observer travels at the dimensionless speed S L ± f := ζ V ± f r /k b , then they see normal modes of theform ∼ exp[i( qz ± f − ω ± f t )] . Here, z ± f = z − S L ± f t and ω ± f = ω − S L ± f q are the Doppler-shifted axial position andfrequency of the observer, respectively. At long times, theobserver sees membrane dynamics dominated by modesof zero group velocity, for which (d ω ± f / d q ) | q ± f = 0 ; here q ± f are the wavenumbers associated with the leadingand trailing fronts. Additionally, ω ± ( i )f ( q ± f ) = 0 in themarginal state. The aforementioned conditions in themoving frame are nearly identical to those of the AU/CUtransition, and yield (see SM [28, Sec. III.5]) S L ± f = S L ± S L ac , q f ± = q , ω f ± = ω − S L ± f q f . (4)We now proceed to compare the linear predictionsto nonlinear simulations of an axisymmetric membranetube subject to a local perturbation. Our computa-tions are based on the numerical method of Ref. [37],with additional details provided in the SM [28, Sec.IV.2]. Figure 1(b) shows the dynamics of an AU sys-tem ( S L = 0 . S L ac , left) and a CU system ( S L = 2 S L ac ,right) at Γ = 6 . In both cases, the leading and trailingfront speeds are calculated numerically and compared tothe linear prediction (4) , as shown in Fig. 3. It is well-known that a front can be either ‘pulled’ at the leadingedge or ‘pushed’ by the growing nonlinearities behind thefront, however one cannot in general anticipate whichtype of front will emerge from a local perturbation to in-vade a given nonlinear system [27, 38]. Our numericalsimulations reveal the trailing edge is a pulled front, inwhich the front speed agrees with the MSC, while theleading edge is a pushed front traveling faster than theMSC predicts (Fig. 3). In the latter case, however, thefront speed agrees with the linear theory at early timeswhen perturbations are small (see insets in Fig. 3).Equation (4) confirms the connection [39, 40] betweenthe MSC and the AU/CU transition: the saddle pointbifurcations shown in Fig. 2 also represent transitions inthe spatiotemporal dynamics of the propagating fronts.In particular, when Γ ∈ [Γ c , Γ ] ∪ [Γ , ∞ ) , q ( r )f = 0 and ω ( r )f = 0 , such that there are no oscillations in the lead-ing edge of the front. Consequently, the front evolvesas a steadily traveling envelope, as confirmed by non-linear simulations [see Fig. 4(a),(c)]. In contrast, when Γ ∈ (Γ , Γ ) , q ( r )f (cid:54) = 0 and ω ( r )f (cid:54) = 0 —for which the fronthas an oscillatory structure in both time and space. Asa result, a pattern is selected in the wake of the front,and a pearled morphology connects the thin and unper-turbed cylindrical regions [Fig. 4(b)]. We thus find thevalue of the Föppl–von Kármán number governs whetheror not a pattern is selected as the front propagates, whichcould explain why axons subjected to local laser ablationbeaded in some experiments, while others atrophied andonly formed long, thin tubes [14]. Weakly nonlinear analysis. —To better understandhow nonlinearities affect front dynamics, we develop aweakly nonlinear model of the membrane shape evolu-tion. In particular, for a tube with dimensionless radius r ( z, t ) = 1 + ˜ r ( z, t ) , we assume (i) ∂ jz ˜ r · ∂ kz ˜ r is negligiblewhen j ≥ and k ≥ , (ii) ˜ r ,t · ∂ jz ˜ r is negligible when j ≥ , and (iii) ˜ r j · ∂ kz ˜ r (cid:28) ˜ r j when k ≥ . With theseassumptions, the evolution equation for the tube radiusis found to be (see SM [28, Sec. IV.1]) r , T + S L r , Z = r , ZZ −
12 (Γ − Γ c )(Γ − / ) r , ZZZZ + f ( r ) , (5)where the nonlinear forcing term f ( r ) is given by f ( r ) = ( r −
1) + Γ − / Γ − Γ c ( r − + / Γ − Γ c ( r − r (6)and hereafter we refer to the coefficient of the th -order term as γ (Γ) := (Γ − Γ c ) / (Γ − / ) . Equa-tion (5) is written in terms of the rescaled variables T = t [(Γ − Γ c ) / , Z = z [(Γ − Γ c ) / (Γ − / )] / , and S L = 4 S L / [(Γ − Γ c )(Γ − / )] / to highlight its struc-ture [41]. Indeed, it is clear that the evolution equation t ( ∗ ) S L = 0 . S L ac S L +f S L − f t (‡) S L = 2 S L ac S L +f S L − f FIG. 3. Plot of leading (blue) and trailing (red) front veloci-ties over time in the simulations shown in Fig. 1, where
Γ = 6 and S L / S L ac = ].Insets show front speeds at early times, confirming the trail-ing front moves to the left for S L < S L ac and to the right for S L > S L ac ; the leading front moves to the right in all cases. belongs to the family of extended Fisher–Kolmogorov(EFK) equations—well-studied in the front propagationliterature and known to possess several universal proper-ties [27, 42, 43]. For example, as Γ → Γ +c the AU/CUtransition speed of the EFK equation is given by S L ac = 2 [27] or equivalently S L ac ≈ √ (Γ − Γ c ) / , in agreementwith our earlier calculations. Moreover, the EFK equa-tion is known to undergo steady-to-oscillatory bifurca-tions in the front dynamics at the universal value of γ = / [42, 43]. We confirm that γ (Γ ) = γ (Γ ) = / ,thus justifying the transitions in the front dynamics ob-served in Fig. 4. The particular values of Γ and Γ ,for which γ = / , can be understood as Lifshitz pointswhere the q term in the dispersion relation vanishes atthe saddle point [44], and for which higher order gradi-ents are required for an appropriate description of thesystem (see SM [28, Sec. III.6] and Refs. [43, 45]).To understand how an oscillatory pattern emergeswhen Γ ∈ (Γ , Γ ) , we recognize that γ characterizes frontdynamics in the EFK equation: fronts are monotonic for γ < / and oscillatory for γ > / [42, 43]. Accord-ingly, the th -order bending term in Eq. (5) favors oscil-lations at the leading edge of the front. In the limit of atube with vanishing bending modulus ( k b → , Γ → ∞ ),we find γ → —implying a monotonic front will dom-inate the long-time dynamics. Now imagine starting at Γ → ∞ and steadily increasing k b , such that Γ decreases.In doing so, Γ marks where bending forces are first largeenough to favor an oscillating front. As the bendingmodulus is further increased and Γ → Γ +c , however, thewavelength of every unstable mode diverges (see Eq. (3),SM [28, Sec. III.3(b)]); the bending cost of such modescommensurately decreases. Thus, Γ indicates where the th -order bending term becomes negligible once again,and we find the competition between bending forces andlong-wavelength oscillations is captured by γ .Finally, we compare predictions from the EFK model(5) with direct simulations of the full nonlinear mem- : : t zr (a) : :
130 140 150
110 130 150 17010 : t zr (b) : : : : t zr (c) FIG. 4. Plots of the propagating fronts over time, with no base flow ( S L = 0 ) and (a) Γ = 0 . ∈ [Γ c , Γ ] , (b) Γ = 2 ∈ (Γ , Γ ) ,and (c) Γ = 6 ∈ [Γ , ∞ ) . Solid lines are results from nonlinear simulations and dotted lines are numerical predictions ofthe weakly nonlinear EFK equation (5). Snapshots are separated by (a) τ , (b) τ , and (c) τ , with a portion of thethree-dimensional tube at the final snapshot shown above each subplot to highlight how the atrophied region connects with theunperturbed tube. When Γ ∈ (Γ , Γ ) as in (b), the front leaves pearls in its wake (left inset) and oscillates at the leading edge(right inset), in agreement with the linear theory (see also Fig. 2). The EFK equation (5) best predicts the front speed andshape when Γ → Γ +c (a) and cannot predict shapes when Γ ∈ (Γ , Γ ) due to the presence of outward bulges (b). brane equations in Fig. 4, from which we make sev-eral observations. First, both sets of dynamics resultin a thin, atrophied tube behind the front. We cal-culate the atrophied, ‘homogeneous radius’ r h by rec-ognizing it varies in neither time nor space, such thatour EFK equation requires f ( r h ) = 0 —for which theonly stable, physically meaningful solution is given by r h = 1 + (3 −
8Γ + √ − / (8Γ − [28, Sec. IV.1].Thus, for all Γ > Γ c , the evolution equation predicts < r h < . Moreover, r h → in the limit Γ → ∞ , i.e.for a fluid film with no bending modulus: a result consis-tent with our previous findings [46]. We also find that for Γ / ∈ (Γ , Γ ) , the evolution equation serves as a good pre-dictor of the front speed and final radius as Γ → Γ +c [Fig.4(a)], while it only predicts r h at large Γ [Fig. 4(c)]. Onthe other hand, when Γ ∈ (Γ , Γ ) , the front oscillates atthe leading edge and the quadratic forcing terms in f ( r ) amplify outward perturbations ( r > ), such that theEFK model predicts an unphysical, diverging radius atfinite times (see SM [28, Sec. IV.2(b)]). Despite the quan-titative shortcomings of the evolution equation (5), it stillprovides (i) a connection to previous studies of nonlin-ear dynamics and pattern formation, and (ii) a qualita-tive understanding of front propagation. Importantly, asthe evolution equation (5) gives rise to pulled fronts andagrees with nonlinear simulations when Γ → Γ +c , we ex-pect the front velocity of an initially static tube to scaleas S L f ∼ (Γ − Γ c ) / near the instability threshold (4) .Such a prediction could elucidate the previously unex-plained, experimentally measured front velocities near Γ c [12, Fig. 11]. That is, while the front speed is linear withthe tension over most of the unstable range, it deviatesfrom linearity and follows ∼ (Γ − Γ c ) / near the insta-bility threshold (see SM [28, Sec. III.5]). Conclusions. —In this Letter, we analyzed the spa- tiotemporal stability of lipid membrane tubes, with andwithout a base flow. Our findings are summarized in thestability diagram in Fig. 1(a), with tubes being eitherstable, AU, or CU depending on the values of Γ and S L .Moreover, recalling the connection between the AU/CUtransition and the well-known MSC, we showed that un-stable membrane tubes possess critical Lifshitz points atwhich propagating fronts bifurcate from a steady to anoscillating behavior, and vice versa (Fig. 4). In the SM[28, Sec. IV], we also explore the nonlinear evolution ofvarious types of initial perturbations (local constrictions,local expansions, and global sinusoids) by means of directnumerical simulations and analytical techniques. Ourwork was inspired by experimental observations of bothpearled and atrophied morphologies resulting from localmembrane perturbations [5, 14]. Interestingly, a previousanalysis of front propagation in membrane tubes—whichneglected the intramembrane viscosity yet included thebulk viscosity—always predicted a pearled configurationin unstable tubes [13]. In contrast, the present study con-siders only the intramembrane viscosity and finds thatdifferent patterns can be selected depending on the rest-ing tension. Thus, a natural extension of this work isto perform a spatiotemporal stability analysis which in-cludes both intramembrane and bulk viscosities. Ad-vanced numerical methods, such as those presented inRef. [46], are required to fully describe such a system. Acknowledgements. —We thank Mr. Yannick Omarfor helpful discussions regarding the axisymmetric mem-brane simulations, and Mr. Dimitrios Fraggedakis for in-sightful questions. J.T. acknowledges the support of U.T.Southwestern and U.C. Berkeley. A.S. acknowledges thesupport of the Computational Science Graduate Fellow-ship from the U.S. Dept. of Energy, as well as U.C. Berke-ley. K.K.M. is supported by U.C. Berkeley. ∗ [email protected] ‡ [email protected] † [email protected][1] M. Terasaki, L. B. Chen, and K. Fujiwara, J. Cell Biol. , 1557 (1986).[2] J. Nixon-Abell, C. J. 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