Vesicle shape transformations driven by confined active filaments
VVesicle shape transformations driven by confined active filaments
Matthew S. E. Peterson, Aparna Baskaran, ∗ and Michael F. Hagan † Martin A. Fisher School of Physics, Brandeis University, Waltham, MA, 02453 (Dated: February 5, 2021)The field of active matter studies materials whose microscopic constituents consume energy atthe particle scale to produce motion. Many biological functions are driven by confinement of activecomponents within a cell, including cytoplasmic streaming, morphogenesis, and cell migrationpowered by the actin cytoskeleton. As a minimal representation of such a cell, we consider aparticle-based simulation of an elastic vesicle containing a collection of polar active filaments. Theinterplay between the internal active stresses and vesicle elasticity leads to a variety of filamentspatiotemporal organizations that have not been observed in bulk systems or under rigid confinement,including highly-aligned rings and caps. In turn, these filament assemblies drive dramatic and tunabletransformations of the vesicle shape and its dynamics. We present simple scaling models that revealthe mechanisms underlying these emergent behaviors.
Functionalities of a biological cell, such as motility [1, 2],division [3–5], and endo-/exocytosis [6, 7] require largescale deformations and other dramatic cellular shapetransformations. These capabilities are powered by thecellular cytoskeleton, which dynamically self-organizesand reconfigures in response to internal and environmen-tal cues. There are two complementary paradigms forquantifying the mechanisms that underlie these behav-iors. First, the molecular paradigm seeks to identify theproteins whose interactions and associated biochemicalregulatory mechanisms govern cytoskeletal organization.Second, the physical paradigm aims to understand theemergent phenomenology in terms of force transmissionand stress organization. Active matter has played a keyrole in developing the physical paradigm, through study-ing minimal biological systems and theoretical modelsinspired by them (e.g. [8–23]).From the physical perspective, cell shape transforma-tions are driven by internal active filaments, whose self-organization simultaneously drives and responds to de-formations of the cell boundary, allowing them to exertwork on and adapt to the external environment. Thus,a starting point for modeling such a system is a mini-mal model of active matter confined within boundaries.Experimental and theoretical studies have shown thatconfining an active material can dramatically alter itsbehaviors, since boundaries generate system-spanning ef-fects [24–72]. In turn, internal active components candrive non-equilibrium boundary fluctuations and shapetransformations [49–51, 73–75]. However, most studieshave focused on rigid boundaries [45–47, 76], active agentsthat lack internal degrees of freedom [49, 52, 73, 74, 77–79],or have been in 2D [80–82]. Further, recent experimentsshow that changing passive properties of the active mate-rial, such as filament length, can dramatically reorganizeactive stresses within the material [83, 84].In this work, we seek to leverage these insights fromactive matter to advance the physical paradigm of shapetransformations. We uncover how active stresses are or-ganized by the interplay between filament clustering and passive stresses from deformable boundaries, leading toemergent shape deformations in vesicles. By performingBrownian dynamics simulations of polar self-propelledsemiflexible filaments confined within 3D flexible vesicles,we identify a rich variety of steady-state behaviors thathave not been observed in bulk systems or under rigidconfinement. These include highly-aligned rings, andcaps that have tunable self-limited sizes, number, andsymmetry. Each filament organization drives a character-istic large-scale vesicle shape transformation that can beselected by varying parameters such as filament length,density, and activity level. We present simple scalingmodels that reveal how the feedback between vesicle ge-ometry and filament organization drives and stabilizesthese emergent behaviors.Understanding the fundamental mechanisms that gov-ern this coupling between self-organization of active stressand deformations of a flexible boundary will establish de-sign principles for soft robotics, artificial cells, or otheradvanced materials that mimic the capabilities of livingorganisms, and could elucidate the behaviors of naturalcells.
Model:
We simulate a system of N fil active filamentsconfined within an elastic vesicle, which has radius R ves in its undeformed state. We represent active filamentsusing a modified version of the model in Joshi, et al. [86],in which each filament is a nearly-inextensible, semiflexi-ble chain of M beads of diameter σ [12]. Bonded beadsinteract through an expanded FENE potential [87], whilenon-bonded beads interact through a purely repulsive ex-panded Weeks-Chandler-Andersen (eWCA) potential [88]with strength (cid:15) . The equilibrium bond length is set to b fil = σ/ κ through a harmonicangle potential applied to each set of three consecutivebeads along the chain. A polar active force of magnitude f a acts on each bead, in a direction tangent to the filamentand toward the filament head. The filament volume frac- a r X i v : . [ c ond - m a t . s o f t ] F e b FIG. 1. Snapshots illustrating steady-state configurations ofthe vesicle and enclosed active filaments as a function of fila-ment aspect ratio a and initial volume fraction φ . See SI Movie1 for animations of the corresponding simulations [85]. Themarked regions of parameter space indicate the typical vesicleconformation: (I) spherical, (II) oblate, (III) polar-prolate, (IV) apolar-prolate, and (V) polyhedral. The symbols asso-ciate the conformation with the internal filament organization:homogeneous throughout the bulk or on the surface, with novesicle deformation ( • ); transient clusters and/or bands, withoblate vesicle shapes ( ? ); stable polar rings ( ◦ ); and caps ( ◦ ,with a number of intersecting lines equal to the median numberof caps). The dashed line shows the transition to aligned statespredicted from the competition between the characteristic col-lision and reorientation timescales ( φ = ( π/ /a ) describedin the text. Other parameters are filament bending modulus κ fil = 10 and vesicle radius R ves = 25. tion in the undeformed state is given by φ = N fil V fil /V ves ,where V fil = πσ / M − πbσ / V ves = 4 πR / σ ves > σ . Bonded vesiclebeads interact through the expanded FENE potential [87]with bond length b ves and spring constant k stretch whichcontrols the vesicle stretching modulus. Vesicle beadsinteract with filament beads through the eWCA poten-tial. The vesicle bending energy is calculated from theangle difference between normal vectors of neighboringtriangles, with bending modulus κ ves .We simulate the coupled Langevin equations for thefilament and vesicle bead dynamics using LAMMPS [98],modified to include the active force. We neglect long-ranged hydrodynamic interactions for this system of highfilament density; we will investigate their effect in a futurestudy. We have set units such that the mass of all beads is FIG. 2. a) The onset of ring and cap formation is deter-mined by a competition of timescales: the timescale associatedwith rotations parallel to the vesicle (top) and the timescaleassociated with collisions that tend to orient the filamentsperpendicular to the vesicle (bottom). b) Schematic of thetheory for the number of caps (Eqs. (1) and (2)). We as-sume an activity-induced effective attractive interaction thatis quadratic in the rod-rod contact length ∆ l . The cap isassumed circular, with size parameterized by the angle θ be-tween the cap center and edge. Vesicle curvature leads to ashearing of rods within the cap. m = 1, and energies, lengths, and time are respectively inunits of k B T , σ , and τ = p mσ /(cid:15) . The friction constantis set to γ = 1 /τ . For additional model details, see theSupplemental Materials [85]. Results:
To discover the steady-state conformationsthat arise due to coupling between active propulsionand elasticity, we have performed simulations over awide range of control parameters (Fig. 1, Fig. 3, and SIFig. S2): the volume fraction of filaments in the vesicle φ ∈ [0 . , . a = 1 + ( M − b fil /σ ) ∈ [3 , . f a ∈ [0 , κ fil = 10 k B T , but results forvarying κ fil are shown in SI Fig. S2 [85].Fig. 1 shows the steady-states as a function of filamentvolume fraction and aspect ratio for moderate activity f a = 8. At this activity and vesicle size, for aspectratios a & (8 πR ves /f a ) / ≈ . strong confinement limit : because the persistence length l COMp ∝ f a a of the filament center-of-mass motion islarger than the vesicle size l COMp > R ves , most filamentsare found on the vesicle surface at all times [55] (see SISec. A [85]).Under these conditions we can classify the steady-statevesicle conformations into several categories: (I) spherical, (II) oblate, (III) polar-prolate, (IV) apolar-prolate, and (V) polyhedral. These vesicle configurations are tightlycoupled to the spatiotemporal organization of the fila-ments within, as follows. (I) : Spherical vesicle shapes arise at low filament vol-ume fractions and aspect ratios. Under these conditions,filament-filament collisions are rare and inter-filamentaligning forces are weak [99–105]. Thus, filament posi-tions and orientations are homogeneous (throughout thevesicle interior below strong confinement, or on the vesiclesurface above strong confinement), leading to little defor-mation of the vesicle. (II) : For low volume fraction buthigh aspect ratios, such that the filament length L = aσ iscomparable to the unperturbed vesicle radius, L ∼ R ves ,the vesicle deforms into oblate spheroid conformations.This transition is driven by the filaments organizing intoa stable polar band, which deforms the vesicle along ageodesic. This filament arrangement closely resembles thepolar bands observed on the surface of rigid spheres [52],which arises due to topological requirements for a surface-constrained polarization field [47]. However, in our systemthe band is further stabilized by a positive feedback be-tween finite vesicle flexibility and filament activity—theactive forces deform the vesicle into oblate configurations;in turn, the high local vesicle curvature at its equatorreinforces alignment of the filament band. (III-IV) : Forintermediate volume fractions and aspect ratios, the vesi-cle deforms into a prolate spheroid. These prolate vesicleconformations can be further classified by their motion,either polar (III) or apolar (IV) . Further increasing thevolume fraction or decreasing the aspect ratio leads topolyhedral conformations, (V) .States (III-V) all result from filaments assembling intocrystalline caps in which the rods are highly aligned andperpendicular to the vesicle surface. Interestingly, thecaps are ‘self-limited’ in that their typical size decreaseswith decreasing aspect ratio, but is roughly independent ofthe total number of filaments N fil in the vesicle. Increasing N fil at fixed aspect ratio increases the number of caps; weobserve up to 12 caps for the finite vesicle size that weconsider (Fig. 4). Further, caps drive local curvature of thevesicle, leading to elasticity-mediated cap-cap repulsionswhich favor symmetric arrangements of caps. Thus, thevesicle morphology can be sensitively tuned by controllingfilament aspect ratio and density to achieve a specificnumber of caps. The polar-prolate (III) , apolar-prolate (IV) , and multifaceted states (V) respectively have 1, 2,and ≥ (I) , to the highlydeformed oblate, prolate, and polyhedral vesicle shapesof states (II-V) . Our simulations demonstrate that suchsignificant vesicle shape deformations occur when filament-filament interactions mediate the organization of orderedstructures either in the plane of the vesicle or orthogonal FIG. 3. Steady-state configurations as a function of φ andactive force f a . The marked regions are defined as in Fig. 1.The dashed line shows the transition to aligned states predictedby the timescale competition, which is independent of f a .Other parameters are a = 10 . κ fil = 10 . See SI Movie 2for corresponding animations. to it.The onset of this transition can be understood by consid-ering a competition between two characteristic timescalesthat respectively govern collision-induced filament-vesiclealignment and filament-filament alignment (see Fig. 2).Filament-vesicle collisions, which tend to reorient fila-ments parallel to the surface [58, 107], have a charac-teristic timescale τ rot ∼ L/v . We can estimate thetimescale for filament interactions by considering filament-filament pairwise collisions whose timescale is given by τ coll ∼ σ/v φ , with v = f a /γ the filament self-propulsionvelocity (see SI [85]). Thus, deformed vesicle states willarise when τ coll < τ rot or equivalently aφ > c , where c ∼ = ( π/ is independent of activity and filament length(see [85]). This defines a boundary separating highly de-formed states of the vesicle from the undeformed sphericalstates (the dashed line in Fig. 1).Notably, the active force drops out of this argumentbecause both collision and reorientation times are ∝ f a .Thus, the theory predicts that the emergence of deformedvesicle states is independent of activity of the enclosedfilaments (above a threshold activity, discussed next).As a test of this prediction, Fig. 3 shows the steady-states as a function of φ and f a for fixed aspect ratio a = 10 .
5. Indeed, formation of large deformations does notdepend on activity, with non-spherical shapes forming for φ ≥ c/a ≈ .
06 (as predicted by the timescale argument)for all f a > f a . FIG. 4. The number of caps measured in simulations (symbols)compared to the theory (Eqs. (1) and (2), line), with A =4 R ves / √ πσ and a ∗ ≈
130 (chosen by eye). Diamonds indicatedynamic cap states. Note that the number of caps in thesimulation results is likely undercounted for the dynamic statesdue to the caps’ motility. The simulation data is the sameas in Fig. 1. Active filaments are colored by which cap theybelong to for visual clarity. fluctuations. Also, cap formation (and thus vesicle shapetransitions) do not occur when the filaments are belowthe strong confinement limit discussed above ( a . ≈ . U shear ( θ ) = n cap πR G [cos θ + sec θ − θ theangle subtended by the cap on the vesicle surface, n cap the number of caps, and a ‘shear modulus’ G ∼ f a (butindependent of L rod ). In the strongly deformed regionthe caps are roughly circular, so the interfacial energy isgiven by U int ( θ ) = n cap πR ves γ sin θ , with with the ‘inter-facial tension’ γ ∼ L rod f a accounting for the diminishedinteractions at the cap boundary. This results in a freeenergy as a function of cap size [85]: f ( θ ) = 11 − cos θ (cid:20)
12 (cos θ + sec θ −
2) + Γ R ves sin θ (cid:21) (1)where Γ = G/γ ∼ L is given by the balance between theeffective interfacial tension and shear modulus, and shouldbe linear in filament length but roughly independent of f a since both of these effects are driven by activity.Minimizing this per-filament free energy yields an op-timal θ [108, 109] corresponding to the self-limited capsize. Assuming that we are well above the onset of capformation so that essentially all filaments are in caps, n cap ∝ φa − [1 + ( a/a ∗ ) − / ] (2)where a ∗ ∝ R ves /σ is an adjustable parameter that maydepend on activity. This expression holds provided a (cid:28) a ∗ . For the data in Fig. 1, we obtain a ∗ ≈ n cap &
7) motile caps,there is close agreement between the observed and pre-dicted n cap . Above this threshold our cap-counting al-gorithm likely undercounts n cap , since different caps areoften adjacent and interacting. Further, the predictionof Eqs. (1) and (2) that the self-limited cap size is in-dependent of activity is consistent with observations atdifferent f a (see Fig. 3). The motile cap states appear toarise when the curved vesicle geometry forces interactionsbetween the inward-facing ends of adjacent caps. Suchinteractions occur above a threshold number and aspectratio of filaments, given by N fil & C (1 − aσ/R ves ) , where C is a constant (see SI Sec. D [85]).Finally, we note that the emergent vesicle shapes alsodepend on the filament flexibility, with the ‘phase’ bound-aries shifting to higher aspect ratios for more flexiblefilaments (SI Fig. S2 [85]). Moreover, for finite filamentflexibility, there is an upper-threshold activity level, abovewhich aligned rings and caps are suppressed. These re-sults can be understood from the fact that increasingflexibility reduces propulsion-induced alignment forces,and that activity renormalizes filament flexibility to lowervalues [85, 86].In summary, our simulations demonstrate that confin-ing active filaments within a deformable vesicle leads tomultiple transformations of the vesicle shape and mo-tions, which can be precisely tuned by control parametersincluding filament length, density, and flexibility. Sim-ple scaling arguments indicate that these behaviors arisegenerically due to feedback between vesicle elasticity andactive filament organization, independent of the specificmodel. Thus, these results suggest strategies for engineer-ing active vesicles with designable shapes and dynamics,with capabilities resembling those of living cells.We acknowledge support from NSF DMR-1855914and the Brandeis Center for Bioinspired Soft Materi-als, an NSF MRSEC (DMR-2011846). We also acknowl-edge computational support from NSF XSEDE comput-ing resources allocation TG-MCB090163 (Stampede andComet) and the Brandeis HPCC which is partially sup-ported by DMR-MRSEC 2011486. ∗ [email protected] † [email protected][1] P. Sens, Proc Natl Acad Sci USA , 202011785 (2020).[2] R. Ananthakrishnan and A. Ehrlicher, Int. J. Biol. Sci. ,303 (2007).[3] M. Leptin and B. Grunewald, Development , 73(1990).[4] A. L. Miller, Current Biology , R976 (2011).[5] O. Polyakov, B. He, M. Swan, J. W. Shaevitz,M. Kaschube, and E. Wieschaus, Biophysical Journal , 998 (2014).[6] J. M. Besterman and R. B. Low, Biochemical Journal , 1 (1983).[7] G. J. Doherty and H. T. 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CONTENTS
Supplementary Figures 2Supplementary Text 4A. Filament persistence time 4B. Competition between rotation and collision timescales 4C. Calculation of number of caps 5D. Onset of dynamic caps 7E. Effects of filament semiflexibility 7F. Equations of motion and simulation details 8G. State detection algorithm 9H. Movie Descriptions 11References 12 a r X i v : . [ c ond - m a t . s o f t ] F e b SUPPLEMENTARY FIGURES
FIG. S1. The filament orientation persistence time scales with aspect ratio as τ corr ∝ a in the rigid rod limit ( κ fil = 10 ). Theblue dots and orange dots are data from simulations with the active force turned off and on, respectively, showing that activitydoes not affect the persistence time. The dashed line follows τ corr = a / πγ .FIG. S2. a) Vesicle conformations and rod organizations as a function of filament rigidity κ fil and active force strength f a ,for volume fraction φ = 0 . a = 10 .
5. For a given filament stiffness, increasing activity reduces thenumber of caps until an upper-threshold activity value f SCa , beyond which the system transitions into an undeformed state. Asdescribed in SI section E, this transition occurs because activity renormalizes the filament bending modulus to smaller values,thus reducing filament alignment interactions and causing the system to leave the strong confinement limit. The dashed lineshows the prediction for f SCa given by Eq. (S35). Note that there is no adjustable parameter. In the rigid rod limit κ fil & , allnon-zero active force values that we simulated led to cap formation. b) Selected snapshots of states shown in (a). Animations ofthese states can be found in SI Movie 3.
FIG. S3. Caps that form at lower active force strengths (left) tend to be more ragged and drive less vesicle deformation thanthose that form at larger active force strengths (right). For both snapshots, φ = 0 . a = 10 .
5, and κ fil = 10 .FIG. S4. The vesicle is constructed from a triangulated mesh of N ves beads, and contains N fil polar active filaments. Eachfilament consists of M bonded monomers of diameter σ . A propulsion force with magnitude f a is applied to each monomer inthe direction of the local filament tangent, and the filament bending modulus is κ fil . SUPPLEMENTARY TEXTA. Filament persistence time
In this section we compute the persistence length of the center of mass motion of the active filaments, l COMp , todetermine the regions in parameter space that correspond to the strong confinement limit. Under strong confinement,almost all rods are on the vesicle surface at all times, while below this limit rods are found throughout the vesicleinterior. Strong confinement occurs when the filament motion persistence length is larger than the vesicle size, i.e., l COMp & R ves [1, 2]. This limit can be understood by noting that when a self-propelled filament collides with thevesicle surface it is reoriented into the surface tension plane. When l COMp & R ves , rotation of the filament away fromthe surface occurs more slowly than realignment with the surface due to subsequent collisions.The autocorrelation timescale for rotation of a rigid rod scales as τ corr ∝ a [3]. Consistent with this result,measurements on simulations of isolated active filaments give an autocorrelation time of τ corr ≈ a / πγ (dashed line inFig. S1). The persistence length of motion l COMp is therefore l COMp = τ corr v ≈ f a a πmγ , (S1)where we have used v = f a /mγ as the active propulsion velocity. Recalling that the system enters the strongconfinement limit when this persistent length is on the order of the vesicle diameter, 2 R ves , the threshold aspect ratiofor strong confinement is a & (cid:18) πmγ R ves f a (cid:19) / . (S2)For the activity of Fig. 1, f a = 8 k B T /σ , the threshold aspect ratio is a SC ≈ . B. Competition between rotation and collision timescales
Rotation timescale:
Consider a rigid, self-propelled rod, consisting of N + 1 monomers of mass m and bond length b , that is moving toward a flat boundary. The rod is propelled by an active force with magnitude f a along the rod’stangent ˆ p . At the point of collision with the boundary, the rod will begin to rotate parallel to the wall. The rotationwill happen over a timescale equal to the that over which the rod would have moved its own length, τ rot ∼ mγL/f a ,where L = N b is the rod’s length. More explicitly, the (overdamped) equation of motion for the n th monomer of therod is mγ ∂ r n ( t ) ∂t = f a ˆ p ( t ) + N ( s, t ) , (S3)where r n ( t ) = R ( t ) + nb ˆ p ( t ) is the position of the n th monomer. The last term is the normal force due to the boundary.Note that we have assumed the activity-dominated regime, so thermal forces are negligible.Without loss of generality, we take the boundary to be oriented such that its normal vector is ˆ y , and the rod to liein the x - y plane. The normal force can then be written as N ( s, t ) = f a (ˆ p · ˆ y ) (cid:20)(cid:18) ˆ p · ˆ y ˆ p · ˆ x (cid:19) ˆ x − ˆ y (cid:21) δ n,N . (S4)Subtracting Eq. (S3) for n = 0 from that of n = N yields mγL ∂ ˆ p ∂t = f a (ˆ p · ˆ y ) (cid:20)(cid:18) ˆ p · ˆ y ˆ p · ˆ x (cid:19) ˆ x − ˆ y (cid:21) . (S5)The only timescale in this equation is τ rot = mγLf a . (S6) Collision timescale:
To determine the collision timescale, we first must compute the collisional cross section, σ coll ,of two self-propelled rods. Consider two rigid rods of length L and diameter σ . One rod is located at r with orientationˆ p , and the other is located at the origin with orientation ˆ q . Again neglecting thermal motion, we assume the rodsmove in the direction of their tangents with an active velocity v = f a /mγ . We choose our reference frame such thatthe rod at the origin is stationary, and the other moves at the relative velocity v = v ( p − q ). This defines an axisˆ w = v / | v | . Projecting the rods onto the plane perpendicular to ˆ w , we find that both rods have the projected length L = (cid:18) p · ˆ q (cid:19) L. (S7)We can additionally project the rods’ orientations on to this plane:ˆ p = ˆ p − (ˆ p · ˆ w ) ˆ w | ˆ p − (ˆ p · ˆ w ) ˆ w | = ˆ p + ˆ q | ˆ p + ˆ q | (S8)ˆ q = ˆ q − (ˆ q · ˆ w ) ˆ w | ˆ q − (ˆ q · ˆ w ) ˆ w | = ˆ p . (S9)Notably the rods are always parallel when projected into this plane. Treating the projected shapes of the rods assimple rectangles of length L and width σ , the collision region is also a rectangle, with side lengths 2 L and 2 σ , givinga collision area of A (ˆ p , ˆ q ) = 4 L σ = 2 Lσ (1 + ˆ p · ˆ q ) . (S10)This approximation of the projected rod shape is valid in the long rod limit, L (cid:29) σ , since accounting for the exactshape of the ends of the rods will only contribute a term of order O ( σ ). Averaging over all orientations ˆ p and ˆ q leadsto the total collisional cross section: σ coll = Z A (ˆ p , ˆ q ) dˆ p π dˆ q π = 2 π Lσ. (S11)If the number density of rods is ρ , then the collision timescale is defined by the relation ρσ coll ( L + v τ coll ) = 1 . (S12)That is, a rod will explore a volume V ( t ) = σ coll ( L + v t ) in time t , and therefore will collide with ρV ( t ) rods onaverage in that time. The collision timescale is the time τ coll such that it will have collided with one rod on average.Thus, τ coll = 1 ρσ coll v − Lv (S13) Competition of timescales:
Recall the rotation timescale τ rot = L/v . The collision timescale can be written interms of τ rot as τ coll = (cid:18) ρσ coll L − (cid:19) τ rot , (S14)We expect that highly-aligned configurations, such as polar rings and caps, will form when τ coll . τ rot . The numberdensity is related to the volume fraction as ρ = 4 φ/πLσ (treating the rods as cylinders). Thus, we expect a transitionprovided 2 ρσL &
1, or φa & ( π/ , (S15)where a = L/σ is the rod aspect ratio. Finally, we note that performing the same calculation in 2D results in the samescaling.
C. Calculation of number of caps
Derivation of the effective ‘free energy’:
We assume that activity induces an effective attractive interaction betweenactive rods, leading rods to preferentially form smectic layers. These smectic layers are sheared by the curvature of theconfining vesicle. We assume that there is a force that is linear in displacement that resists this shear, as well as aninterfacial tension at the edges of the cap. Using the parameterization shown in Fig. 2b, we align the cap to the z -axisand write the height field of the cap as a function of the polar angle θ : h ( θ ) = R ves cos θ. (S16)Note that we have assumed the cap radius is the same as the undeformed vesicle radius. Active forces may deform thevesicle at the caps, leading to caps having a radius that is different from the vesicle’s unperturbed radius. Thus, thiscalculation will be most accurate when the vesicle is not deformed too much from its initial state (high vesicle stiffnessor low activity).The gradient in the height field gives the local shear energy density, which can be integrated over the cap surface tofind the total shear energy, with shear modulus G ∼ f a : u shear ( θ ) = 12 G Z d S (cid:12)(cid:12)(cid:12)(cid:12) ∂h∂ ( R ves sin θ ) (cid:12)(cid:12)(cid:12)(cid:12) = πR G Z θ d θ sin θ tan θ = πR G (1 − cos θ ) cos θ . (S17)The cap has radius R ves sin θ , and therefore the interfacial energy can be written as u int ( θ ) = 2 πR ves γ sin θ, (S18)where γ ∼ f a L is the interfacial tension.If the total number of rods in the system is N fil , and we assume that all caps are the same size, then the totalnumber of caps is given by n cap = N fil πR ρ (1 − cos θ ) , (S19)where ρ = 2 / √ σ is the hexagonal close packing density, and σ is the rod diameter. Therefore, the total energy(relative to a reference state corresponding to a perfectly aligned smectic layer of rods) is U ( θ ) = n cap ( θ )[ u shear ( θ ) + u int ( θ )]= GNρ (1 − cos θ ) (cid:20) (1 − cos θ ) θ + γGR ves sin θ (cid:21) . (S20)From this, we define the (normalized) free energy per filament f ( θ ) as f ( θ ) = ρU ( θ ) GN = 11 − cos θ (cid:20) (1 − cos θ ) θ + ζ sin θ (cid:21) , (S21)where ζ = γ/GR ves = cL/R ves , with the proportionality constant c the only free parameter. Computing the number of caps:
Using Eq. (S21), we first compute the optimal θ as θ ∗ = arg min θ f ( θ ) (S22)and then evaluate n cap ( θ ∗ ). This yields the number of caps as a function of the parameter ζ , which we can fit to oursimulation data.Provided ζ is small, we can find an asymptotic solution in the following way. Let z = cos θ/ (1 − cos θ ) for convenience.Since z ( θ ) is a monotonic function of θ (for 0 ≤ θ < π ), minimizing Eq. (S21) with respect to z is equivalent tominimizing with respect to θ . The free energy can then be written as f ( z ) = 12 z + ζ √ z. (S23)Taking the derivative and setting it to zero, we obtain the following equation for z : ζ √ z − z = 0 . (S24)Squaring both sides, we can rearrange this into the form z z = 14 ζ . (S25)If ζ (cid:28)
1, then the right hand side is very large and so we can assume that z (cid:29) z ≈ z ,this reduces to z ≈ (2 ζ ) − / (S26)or, equivalently, cos θ ≈
11 + (2 ζ ) / . (S27)Recall that we expect ζ ∝ L/R ves . We can instead write ζ as a ratio of the rod aspect ratio a to a critical aspect ratio, ζ = a/a ∗ , with a ∗ ∼ = R ves /σ (cid:29)
1. Inserting Eq. (S27) into Eq. (S19), and absorbing constants into a ∗ , we obtain theresult shown in the main text n cap = Aφa − [1 + ( a/a ∗ ) − / ] , (S28)where we have used N = φV ves /V fil , with V ves = 4 πR / V fil = πLσ /
4, and A = 8 R ves / πρσ . Dependence of a ∗ on activity. We anticipate that the critical aspect ratio may depend on activity for several reasons.First, the critical aspect ratio will depend on the local vesicle curvature in the vicinity of the cap, a ∗ ∼ = R localves /σ , with R localves ≤ R ves because the active force from the rods will locally deform the vesicle from its unperturbed curvature.The extent of deformation will increase with activity. Second, at very low activity f a . k B T /σ (or f a . D. Onset of dynamic caps
Assuming all N fil rods are contained within a cap, the total area occupied by rods is A tail = N fil /ρ, (S29)where, as before, ρ = 2 / √ σ is the hexagonal close packing density. Now, let us consider a state with all rodsorganized into caps, and assume that the vesicle is approximately spherical with radius R ves . Then, the tails of therods will be located approximately on the surface of an inner spherical region, with radius R = R ves − L rod . If thetotal area of rod tails A tail occludes a large fraction of the surface area of this inner region, then the tails of the rodsare likely to interact. Such interactions can disrupt the stability of the caps, potentially leading to motions and/orbreakup of caps. This occurs when N fil /ρ & π ( R ves − L rod ) → N fil & πρR (1 − L rod /R ves ) . (S30)Based on this analysis, caps will become motile above threshold values of the length and number of filaments given byEq. (S30).We can recast these results in terms of the volume fraction φ and aspect ratio a of the filament. Using N fil ∼ φ/a ,we write φ & Ca (1 − aσ/R ves ) , (S31)where C ∝ L rod /R ves . Consistent with this result, our simulation results show that motile caps arise in the lower-rightregion of the parameter space of Fig. 1. E. Effects of filament semiflexibility
To focus on the interplay between activity and vesicle deformability, in the main text we have focused on simulationsperformed in the rigid rod limit κ fil = 10 . We now briefly discuss the effect of allowing for finite filament flexibility.Fig. S2 shows the vesicle conformation and filament organization states as a function of filament bending modulus andactivity, for fixed filament volume fraction φ = 0 .
2. We see that for finite filament flexibility, the transition to alignedring and cap states is suppressed above a threshold activity, which decreases with decreasing κ fil .This result can be understood as follows. On generic grounds, decreasing the filament rigidity will reduce thetendency for filaments to align and thus impede the formation of aligned rings and caps. For filament stiffness wellbelow the rigid rod limit, the process by which caps and rings form is more complicated than considered in thetimescale and cap-counting arguments discussed in the main text and below. The upper-threshold activity for filamentorganization can be, at least in part, explained by the observation of Joshi et al. [4] that activity renormalizes filamentrigidity to smaller values according to κ efffil ∼ = κ fil / (cid:0) f (cid:1) . That is, the active energy preferentially dissipates intobend modes, effectively increasing filament flexibility and therefore suppressing filament alignment.In particular, the upper-threshold activity corresponds to the point when the activity-renormalized flexibility offilaments causes the system to leave the strong confinement limit. (As noted in the main text, the system must bein the strong confinement limit to observe cap and ring formation.) To show this correspondence, we perform thefollowing calculation to estimate the effect of activity on the strong confinement limit.As we show in Sec. A, the filament orientation correlation timescale is τ corr = a πγ . (S32)We assume that for semiflexible rods we can make the substitution a → l p /σ , where l p is the filament persistencelength. Further, simulations show [4] l p = 2 κb/k B T f a σ/k B T ) , (S33)where b = σ/ κ is strength of the harmonic angle potential thatcontrols the semiflexibility of the filament. Assuming that f a is large, we find l p /σ ≈ k B Tσ κf . (S34)The system enters the strong confinement limit when τ corr & R ves /v , where v = f a /mγ is the filament propulsionvelocity. Using Eq. (S34), this can be rearranged to find f SCa σ/kT & C ( κ/k B T ) / , (S35)where C = (cid:16) k B T πmγ σR ves (cid:17) / . This is shown as a dashed line in Fig. S2.Note that there are no fitting parameters used in this analysis. F. Equations of motion and simulation details
We simulate a vesicle with N ves = 2432 monomers and a nominal radius of R ves ≈ σ , measured as the distancefrom the center of mass of the vesicle to the center of any given monomer within the vesicle in its undeformed state.For this radius, we set the vesicle monomer diameter to σ ves = aσ with a ≈ .
934 to ensure that there were no holes inthe vesicle that the active filaments could escape from. A filament is a linear chain of M monomers, each with diameter σ . The vesicle is filled with N fil such filaments, with N fil varied to control the volume fraction φ = N fil V fil /V ves . SeeFig. S4.Bonded monomers in both the vesicle and the filaments interact through an expanded FENE potential: U FENE ( r ) = − k stretch ∆ ln " − (cid:18) r − b ∆ (cid:19) , (S36)where r is the distance between the two bonded monomers, k stretch is the bond strength, b is the preferred bondlength, and ∆ is the maximum deviation; that is, | r − b | < ∆. For the vesicle, we set k stretch = 1000 k B T /σ , b ves ≈ . σ (for hexamer bonds) or b pent ≈ . σ (for pentamer bonds), and ∆ = b/
2. For the active filaments, weset K = 2000 k B T /σ , b fil = 0 . σ , and ∆ = 0 . σ . These stiff bond potentials act to nearly constrain the length andarea of the filaments and vesicle, respectively.To penalize curvature, neighboring triangles on the vesicle (those that share an edge) also interact through aharmonic dihedral potential. Each triangle i defines a unique normal vector ˆ n i . For neighboring triangles i and j , theinteraction potential is given by U dih ( φ ) = κ dih (1 − cos φ ) , (S37)where cos φ = ˆ n i · ˆ n j . We set κ dih = 5000 k B T so that the vesicle has a relatively large bending modulus κ ves = √ κ dih [5].Filament semiflexibility is incorporated through a harmonic angle potential of the form: U angle ( θ ) = κ fil θ , (S38)where θ is the angle between two neighboring bonds i and j ; that is, cos θ = ˆ b i · ˆ b j , where ˆ b { i,j } are neighboring bondvectors.Non-bonded monomers interact sterically through an expanded WCA potential given by U WCA ( r ) = 4 (cid:15) "(cid:18) σr − ∆ (cid:19) − (cid:18) σr − ∆ (cid:19) r < r c , (S39)where r is the distance between two interacting monomers, (cid:15) is the strength of the interaction, ∆ shifts the potential,and r cut = 2 / σ + ∆ is a cutoff distance. For interactions between two filament monomers, we set ∆ = 0, and forinteractions between a filament monomer and a vesicle monomer we set ∆ = σ ves /
2. This ensures that active filamentscannot escape the vesicle. In all cases, we set (cid:15) = k B T . We do not consider excluded volume interactions between twovesicle monomers as the stiff bond and dihedral potentials act to keep vesicle monomers far apart.Given these interactions, the total energy is the sum U tot = X pairs U WCA ( r ) + X bonds U FENE ( r )+ X angles U angle ( θ ) + X dih U dih ( φ ) . (S40)The equation of motion for any given atom i within molecule α is then given by m ∂ r αi ∂t = − ∂U total ∂ r αi − mγ ∂ r αi ∂t + (1 − δ α, ) f a r αi +1 − r αi − | r αi +1 − r αi − | + ξ αi ( t ) . (S41)Here, α = 0 represents the vesicle, while α > γ = 1 /τ ), the third term applies an active force to the atoms of each filament along thetangent, and the last term adds random thermal forces. The thermal forces are modeled as Gaussian white noise withmoments h ξ αi ( t ) i = 0 , and D ξ αi ( t ) · ξ βj ( t ) E = 6 γk B T δ ij δ αβ δ ( t − t ) . (S42)Simulations are run with a timestep of δt = 10 − τ for a total time of 10 τ . For the two state diagrams shown inthe main text (Fig. 1 and Fig. 3), we ran 9 trials for each set of parameters. For the state diagram shown in this SI(Fig. S2) we only ran one trial. G. State detection algorithm
We classify states by a combination of the vesicle conformation and the organization of the rods within the vesicle.The classifications are as follows• undeformed/spherical : the vesicle is in a nearly spherical state (symbolized by • )0• deformed/other : the vesicle is in a deformed state, but the rod organization is not consistent or difficult to detect;typically, this occurs for states in which the filaments form unstable polar rings that form and break apart overtime (symbolized by ? )• oblate/ring : the vesicle is an oblate spheroid, and the filaments are organized into a stable polar ring (symbolizedby ◦ )• capped : the filaments are organized into stable caps, with varying amounts of vesicle deformation (symbolized by ◦ with the number of intersecting lines in the symbol equal to the median number of detected caps)• dynamic capped : the filaments are organized into caps that are dynamic and/or motile (symbolized by (cid:5) )Our classification algorithm is constructed as follows. We first classify the vesicle conformation. We define theasphericity α of the vesicle to be α = 16 √ π S / V , (S43)where S is the surface area of the vesicle and V is the volume. Note that α = 1 for a sphere, and α > G as G = 1 N ves N ves X i =1 ~r i ⊗ ~r i , (S44)where N ves is the number of monomers making up the vesicle, and ~r i is the position of the i th monomer. Let a , b , and c be the eigenvalues of G such that a ≤ b ≤ c . Then define x = ( b − a ) / ( b + a ) y = ( c − b ) / ( c + b ) . Using α , x , and y , we classify the vesicle conformation into four categories:• spherical : α ≤ .
01 and x ≤ . y ≤ . oblate : x > . y ≤ . prolate : x ≤ . y > . non-spherical : otherwise.Next, we attempt to classify the filament organization.First, we determine if there are any caps in the system. To do this, we compute two fields on the vesicle surface: apolarization field p i and an order parameter q i that measures how perpendicular the filaments are to the vesicle, bothof which are evaluated at each vesicle monomer i . Let ˆ n i be the unit normal vector of the vesicle at the i th vesiclemonomer, and let M i be the set of filament monomers that are within 4 σ of the i th vesicle monomer. Then p i = P j ∈ M i ˆ t j (cid:12)(cid:12)(cid:12)P j ∈ M i ˆ t j (cid:12)(cid:12)(cid:12) , (S45)where ˆ t j is the unit tangent vector of the filament evaluated at filament monomer j . Similarly, q i = 1 | M i | X j ∈ M i ˆ t j · ˆ n i (S46)To find potential caps, we first find the vesicle monomer i ∗ such that q i ∗ is maximized. We then use the proceduredescribed in Algorithm 1 to find a cap based on this seed monomer. To find additional caps, we choose a new seed pointby finding another monomer j ∗ such that q j ∗ is the maximal value among all monomers that are not yet a member ofa cap. We continue this process until all vesicle monomers have been visited, or until we do not find any additionalcaps. Once all candidate caps are found, we discard any caps that comprise fewer than d√ N ves e = 50 monomers.Finally, as done for the vesicle itself, we compute the smallest and largest eigenvalues, λ min and λ max respectively,of the gyration tensor computed using only the filament monomers. Let z = 1 − λ min /λ max . Then we classify thefilament organization as one of the following:1 input : A seed vesicle monomer i ∗ output : A list of vesicle monomers belonging to a cap initialize cap to empty list; initialize visited to empty set; initialize todo to stack containing only i ∗ ; while todo is not empty do i ← value popped from todo ; insert i into visited ; if ˆ p i · ˆ p i ∗ ≥ cos( π/ then append i to cap ; N i ← set of monomers neighboring monomer i ; for j ∈ N i do if j not in visited and ˆ p j · ˆ p i ∗ ≥ cos( π/ then push j on to todo ; end end end end Algorithm 1:
Algorithm to determine vesicle monomers belonging to a cap• capped : if there are a non-zero number of caps,• polar ring : if there are no caps, and z > . other : otherwise.The class other is very broad, and can include a range of filament organizations and behaviors such as polar flockson the vesicle surface, or transient polar rings that continuously form and break apart. Usually these states do notresult in any large scale deformation of the vesicle, but in some cases the transient ring states can lead to measurabledeformation.For each simulation, we take 11 samples from the last 10% of the trajectory and measure the vesicle conformationand filament organization at each sample. If we also have multiple trials, then we aggregate these identifications.Finally, we assign an conformation and organization based on the most common measurement.Having information on both the vesicle conformation and filament organization, we can revisit our earlier classificationscheme:• undeformed/spherical : the filament organization is other , and the vesicle conformation is spherical ,• deformed/other : the filament organization is other , and the vesicle conformation is not spherical ,• oblate/ring : the filament organization is polar ring , and the vesicle conformation is oblate ,• capped : the filament organization is capped ,• dynamic capped : the filament organization is capped , but the number of caps is inconsistent.For the dynamic capping states, we classify the number of caps as inconsistent if the majority of states are capped, butno particular number of caps forms a majority of these states. H. Movie Descriptions • movie-1.mp4 : Animations corresponding to the data shown in Fig. 1. Each element of the figure corresponds toan independent simulation performed at the indicated parameter values, with the volume fraction φ increasingalong rows, and the aspect ratio a increasing along the columns. Other parameters are f a = 8 and κ fil = 10 .• movie-2.mp4 : Animations corresponding to the data shown in Fig. 3, with the volume fraction φ increasingalong rows, and the active force strength f a increasing along the columns. Other parameters are a = 10 . κ fil = 10 .2• movie-3.mp4 : Animations corresponding to the data shown in Fig. S2, with the filament stiffness κ fil increasingalong rows, and the active force strength f a increasing along the columns. Other parameters are φ = 0 .
20 and a = 10 . movie-4.mp4 : Animations of simulations performed at f a ∈ { , , } for φ = 0 . a = 10 .
5, and κ fil = 10 . Thesevideos show that decreasing the activity leads to smaller vesicle deformations, more ragged caps, and morefrequent rod dissociation from caps. ∗ [email protected] † [email protected][1] Y. Fily, A. Baskaran, and M. F. Hagan, Soft Matter , 5609 (2014).[2] Y. Fily, A. Baskaran, and M. F. Hagan, https://arxiv.org/abs/1601.00324 (2016).[3] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, reprinted ed., International Series of Monographs on PhysicsNo. 73 (Clarendon Press, Oxford, 2007).[4] A. Joshi, E. Putzig, A. Baskaran, and M. F. Hagan, Soft Matter , 94 (2019).[5] G. Gompper and D. M. Kroll, J. Phys. I France6