Self-organized motility of vesicles with internal active filaments

Self-organized motility of vesicles with internal active filaments

Clara Abaurrea-Velasco Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany c.abaurrea@fz-juelich.de; t.auth@fz-juelich.de; g.gompper@fz-juelich.de    Thorsten Auth Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany    Gerhard Gompper Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany
July 20, 2019
Abstract

Self-propulsion and navigation due to the sensing of environmental conditions — such as durotaxis and chemotaxis — are remarkable properties of biological cells that cannot be reproduced by single-component self-propelled particles. We introduce and study “flexocytes”, deformable vesicles with enclosed attached self-propelled pushing and pulling filaments that align due to steric and membrane-mediated interactions. Using computer simulations in two dimensions, we show that the membrane deforms under the propulsion forces and forms shapes mimicking motile biological cells, such as keratocytes and neutrophils. When interacting with walls or with interfaces between different substrates, the internal structure of a flexocyte adapts, resulting in a preferred angle of reflection or deflection, respectively. We predict a correlation between motility patterns, shapes, characteristics of the internal forces, and the response to micropatterned substrates and external stimuli. We propose that engineered flexocytes with desired mechanosensitive capabilities enable the construction of soft-matter robots.

In active matter, collective behavior of simple mesoscopic agents can lead to complex structure and dynamics. Examples are self-organization of active granular matter dauchot_membrane_2017; giomi_bots_2013, clustering of self-propelled colloidal particles palacci_living_2013, turbulent motion in bacterial colonies peruani_bacteria_2012, vorticity in motility assays of microtubules sumino_microtubules_2012, and the dance of topological defects in active nematics dogic_drops_2012. In living systems, a sensing capability of single agents, such as chemotaxis for neutrophils nuzzi_neutrophil_2007 and mechanosensitivity for epithelial cells wickert_hierarchy_2016, is often essential for biological function.

The self-organization of force-generating units is underlying the motility of biological cells svitikina_myosin_1997 and of cellular aggregates brochard_aggregates_2018. Here, motility is controlled mainly by actin polymerization and actomyosin contractility mogilner_shape_2009; mogilner_rotating_2017. Interestingly, self-organized machinery constructed from various active and passive components inherently includes sensing capability. The components react differently to stimuli and the machinery can therefore process external information. This construction principle is not restricted to biological macromolecules and cells. The interplay between active and spatial self-organization can also been found in engineered colloidal systems and in biological or artificial systems on significantly larger length scales. For example, diffusiophoretic Janus colloids form spinning superstructures that can autonomously regulate themselves aubret1811, groups of ants collectively carry a large cargo to their nest gelblum_ant_2015, and active agents and microbots in deformable mobile confinement deblais_bots_2018; dileonardo_vesicle_2016 demonstrate complex responses to environmental cues.

Because of the large length-scale gap between single filaments and entire cells, two kinds of theoretical approaches have been employed. One the one hand, continuum models that do not take into account the internal filamentous structure of the cytoskeleton have been developed to predict cell shape and motility mogilner_shape_2009; ziebert_cell_2012; marenduzzo_cell_2015; kruse_lamellipodium_2006; rappel_cell_2010. On the other hand, filament-based models are often used to study some aspects of the internal cytoskeletal structure and dynamics weichsel_actin_2010; schmeiser_cell_2015 and specific biological processes, such as filopodia formation fletcher_bundling_2008 and lamellipodial waves shlomovitz_membrane_2007.

Here, we introduce self-propelled “flexocytes”, a minimal model system for the motility of mechanosensitive active vesicles. The model is based on active filaments in vesicles pulling or pushing on the membrane, and is suitable for single-cell and multi-cell simulations. We perform Brownian dynamics simulations for this system in the overdamped regime to model substrate friction and internal noise. We find that the filaments in the flexocytes self-organize to reproduce cellular shapes and motility patterns, giving rise to dynamical phase transitions. Furthermore, we show that explicit pulling forces are sufficient to recover behavior of keratocytes: the flexocytes are reflected at walls and deflected at friction interfaces. Here, additional pushing forces lead to less persistent motion, induce trapping at walls, and reduce the deflection at interfaces. Therefore, we propose “cell scattering” at walls and interfaces as a new approach to probe the internal dynamics and feedback loop of complex self-propelled particles.

I Results

i.1 Flexocyte Model

Our study is based on a two-dimensional (2D) model system consisting of semiflexible membrane rings, which represent the contact line between the vesicle and the substrate, with attached pushing and pulling filaments, see Fig. 1. Flexocytes of puller type have only pulling filaments, and flexocytes of mixed type have an equal number of pushing and of pulling filaments. The membrane ring has an equilibrium radius and is subject to a bending elasticity with rigidity , a target area , an area compressibility with modulus , and a perimeter-length compressibility with modulus . The filaments are modeled as stiff rods of length , which are propelled along their long axis with a force . Passive filaments experience the frictions parallel and perpendicular to their long axis, and are subject to rotational diffusion with single-rod autocorrelation time . Filament length and mutual penetrability determine filament alignment. We use a high penetrability for pulling filaments and a low penetrability for pushing filaments. For details, see Methods section.

In applications of such a flexocyte model to crawling biological cells on a substrate, the membrane ring represents the contact line where the membrane detaches from the substrate, the area the contact interface with the substrate, and the pushing and pulling filaments actin polymerization and actomyosin contraction, respectively. The target area of the contact interface in cells is determined by adhesion, actin polymerization at the lamellipodium, and the elastic energy of membrane and cortical cytoskeleton. Filament attachment to the membrane can be achieved by proteins that bind actin filaments to the cell membrane, such as myosin 1 bassereau_myosin1_2014, formin bershadsky_formin_2016, and ezrin bosk_2011_ezrin. The membrane-substrate friction represents the rates of breaking and forming receptor-ligand bonds at rear and front, respectively, when the cell moves.

In the following, we present our results in terms of the reduced membrane-substrate friction with the filament longitudinal friction coefficient , the reduced area compression modulus , where is the thermal energy with an effective temperature , and the filament propulsion force characterized by the Péclet number .

i.2 Shapes and Motility

Figure 1: Shape and motility. a) Flexocytes of mixed type: a -flexocyte with , , , , a -flexocyte with , , , , a -flexocyte with , , , , a -flexocyte with , , , . b) Flexocytes with the same parameters as in subfigure (a), but of puller type. c) Scatter plots for the product of flexocyte velocity and membrane-substrate friction , and the signed asphericity . The lines are guides to the eye. Flexocytes of mixed type with to 128, to 100, and 100, and to 25. d) Same plot as in subfigure (c) but with flexocytes of puller type. Fluctuating flexocytes (blue circles), keratocytes (orange squares), neutrophils with none or a single cluster of pushing filaments (upward green triangles), and neutrophils with two clusters of pushing filaments (downward purple triangles) occupy different parts of the phase space. The open black symbols correspond to the flexocytes shown in subfigures (a) and (b). See movies in the SI M1 to M7.

Flexocytes that are initialized as circular membrane rings with randomly positioned and oriented pushing and pulling filaments assume various stationary shapes, see Fig. 1(a) and (b). For filament propulsion forces that are weak compared with the membrane elastic and friction forces, filaments that pull on the membrane rings point toward the flexocyte centers, while filaments that push circle along the boundaries abaurrea_rigid-ring_2017, both leading to nearly stationary quasi-circular, fluctuating “” shapes. For strong propulsion forces, cluster formation of filaments deforms and propels the flexocytes. For small membrane-substrate frictions motile flexocytes have keratocyte-like “” shapes with round apical (front) and flat dorsal (rear) ends barnhart_adhesion-dependent_2011. The flat dorsal ends are stabilised by the membrane-mediated alignment of the pulling filaments. For intermediate and large membrane-substrate frictions the dorsal ends assume pointed neutrophil-like “” shapes nuzzi_neutrophil_2007. The pointed dorsal ends occur for strong membrane-substrate friction where an instability of the distribution of the filament pulling forces induces a strong deformation force on the membrane ring that overcomes its bending stiffness. For mixed systems, we can further distinguish between neutrophile “” flexocytes with only one cluster (“mono-modal”) of pushing filaments at the apical end, and neutrophile “” flexocytes with two pushing clusters (“bi-modal”), a large (small) cluster at the apical (dorsal) end, which together elongate the shape and slow down the motion. -flexocyte-like cells with two lamellipodia have been reported in populations of slow-moving keratocytes jurado_retrograde_2005.

We quantify flexocyte motility by the reduced (dimensionless) center-of-mass velocity

(1)

The velocity is the ‘instantaneous’ center-of-mass velocity, calculated over a time interval . It is scaled by that quantifies the maximum propulsion of a flexocyte when all filaments point in the same direction. The shapes are characterized by the signed asphericity

(2)

where and are the eigenvalues of the gyration tensor that correspond to the lengths of the long and short axes, respectively. The asphericity is positive (negative) for flexocytes where the long axis is oriented parallel (perpendicular) to the direction of motion.

The stationary shapes and motility of flexocytes are dictated by the interplay of the active forces due to the internal degrees of freedom, the conservative forces due to membrane elasticity and area compressibility, and substrate friction. Importantly, we find strong correlations between emergent shape and motility of flexocytes, see Fig. 1(c) and (d). The various types of flexocytes occupy distinct, well-defined regions in a state diagram in which instantaneous total propulsion force and asphericity are employed as main parameters 111It is important to note that the universal relation between shape and motility is only obtained when the flexocyte velocity multiplied by the substrate friction is employed as relevant observable, not for the flexocyte velocity. The relevant observable is therefore the instantaneous total propulsion force, .. The sign of the asphericity distinguishes between -flexocytes, and all other shapes. Flexocytes that are elongated parallel to the direction of motion strongly vary in their motility. Not surprisingly, -flexocytes are the slowest, because they have no polarity, and therefore mainly fluctuate without directed translational motion. -flexocytes are polar, and indeed show a pronounced velocity. For mixed systems, -flexocytes are slower than -flexocytes, but are found for larger values of because they are stable at larger substrate frictions. -flexocytes are found at smaller values of because their motility is hindered by the cluster of pushing filaments at their dorsal end. For -flexocytes of puller type, not all filaments point in the direction of motion, which leads to reduced velocities compared with -flexocytes.

Figure 2: Trajectories and mean-squared displacements MSDs. a) Center-of-mass trajectories of flexocytes; the starting point is marked by a blue dot. The upper row shows flexocytes of mixed type: -flexocyte with , , , , a -flexocyte () with , , , , a -flexocyte () with , , , , a -flexocyte with , , , , and -flexocyte with , , , . b) Center-of-mass trajectories for the same parameters as the systems in subfigure (a), but for flexocytes of puller type. c) Center-of-mass MSDs of flexocytes shown in subfigures (a) and (b). Solid lines represent flexocytes of mixed type, and dashed lines represent flexocytes of puller type. The left column of the legend refers to flexocytes of mixed type, and the right column refers to flexocytes of puller type. The black lines are guides to the eye showing ballistic and random walk-like relations. d) Reduced velocities , obtained from the ballistic regime, for different substrate frictions for flexocytes of mixed type and of puller type , and 100, and . The flexocyte shapes are indicated: -flexocytes (circles), -flexocytes (squares), and -flexocytes (upward triangles), and (downward triangles).

To study the motility of flexocytes in more detail, we analyze center-of-mass trajectories of flexocytes of mixed type and of puller type for the same parameter values, see Fig. 2(a) and (b), respectively. We characterize the motility by the reduced mean squared displacement

(3)

As for other self-propelled agents that experience noise, such as active Brownian particles, we expect to find passive diffusive motion at short times, active ballistic motion at intermediate times, and active diffusive motion at long times golestanian_propulsion_2005; abaurrea_rigid-ring_2017. For typical -flexocytes of puller type, shown in Fig. 2(b), upon addition of pushing filaments the trajectories become less persistent or circling, and the shape can even change to -flexocytes.

Figure 3: Flexocytes at walls. a) Center-of-mass trajectories for various angles of incidence ( is measured with respect to the y-axis). Flexocyte shapes and orientations for are indicated, with symbols refering to the position along the trajectory. Flexocytes of mixed type with , , , . b) Center-of-mass trajectories for various angles of incidence . Flexocytes of puller type , , , and ; flexocyte shapes and orientations for are indicated. c) Angles of reflection versus angle of incidence for flexocytes of puller type , and various values of and . Solid lines indicate the angle of reflection just after the flexocytes have left the wall, dashed lines indicate the angle of reflection after the flexocytes have relaxed to their new steady-state shape. d) Time dependence of the signed asphericity of the flexocytes of puller type , and various values of and . coincides with the time when the vesicle first comes in contact with the wall. The points indicate the time when the dorsal end leaves the wall. See movies in the SI M8 to M11.

In general, -flexocytes have the lowest MSD in the considered range of delay times, followed by -flexocytes, and -flexocytes, see Fig. 2(c), consistent with the shape-motility diagrams in Figs. 1(c) and (d). For the same parameters, selected flexocytes of mixed type and of puller type show similar propulsion velocities, but different persistence of their motion. The stable -flexocyte of puller type shows very persistent ballistic motion. Additional pushing filaments at the apical end act as a fluctuating steering wheel that destabilises the direction of motion. The -flexocyte of mixed type even shows circling motion, with periodic oscillations as characteristic signature in the MSD; the superimposed linear increase indicates an overall drift. Here, a dynamic instability of the pushing filaments breaks the left-right symmetry; the high number of pushing filaments and the motility-induced alignment stabilize the asymmetric shape. The -flexocyte, which in parameter space is close to -flexocytes, has an unstable dorsal end. Because the pulling filaments may temporarily point in different directions, the persistence of motion is strongly reduced. -flexocytes show less persistent motion than stable -flexocytes, because of decreased filament alignment and flexocyte motility. However, in -flexocytes of mixed type, pushing-filament clusters at the apical and the dorsal end stretch the flexocytes and increase their persistence of motion. Thus, the -flexocyte has a longer ballistic regime than the -flexocyte.

Figure 2(d) shows the effective propulsion force extracted from the ballistic regime of the MSD, , as function of . For a simple self-propelled particle, is independent of the friction. The dependence on membrane-substrate friction thus reflects the different internal organisation of the filaments for the various shapes. Here, we briefly discuss the two cases with a pronounced dependence on the friction coefficient, both for . For flexocytes of puller type, -flexocytes for small are fast movers, because almost all filaments are aligned with the direction of motion. They become unstable and transform into slower -flexocytes with increasing membrane-substrate friction. For flexocytes of mixed type, the emergent immediate total propulsion force of the flexocyte interestingly increases with increasing friction . This is related to the cluster formation from the random initial state, where a slower deformation of the membrane results in a larger apical/dorsal asymmetry in the pushing-filament distribution.

i.3 Flexocytes at Walls

Figure 4: Flexocytes at friction interfaces. Center-of-mass trajectories, deflection, and shape changes of flexocytes for various membrane frictions and angles of incidence . All flexocytes initially move in positive direction. Selected simulation snapshots indicate instantaneous shapes, with symbols showing corresponding locations on the trajectories. a) Flexocytes of mixed type, , and . Systems with angle of incidence , friction for and various frictions for . b) Flexocytes with the same parameters as in subfigure (a). Systems with for , for , and various . c) Data for the same parameters as in subfigure (a), but for flexocytes of puller type. d) Data for the same parameters as in subfigure (b), but for flexocytes of puller type. e) Deflection of flexocytes versus substrate frictions and for normal angle of incidence . Flexocytes of mixed type , , , and corresponding data for flexocytes of puller type. For the systems where , varies and vice-versa. No data exists for the flexocytes of mixed type for and . In this case the flexocytes show no broken symmetries which leads to persistent motion and as such they do not cross the friction interface. f) Deflection of flexocytes for various angles of incidence . Systems where and , or and . For positive the trajectories become more perpendicular to the interface, for negative more parallel.

The exposure of flexocytes to external forces and the measurement of their response allows us to probe both their internal architecture and their dynamic behavior. We first consider the interaction of flexocytes with hard walls and obstacles. The behavior of the two types of flexocytes is fundamentally different in such an encounter. Whereas -flexocytes of mixed type get stuck at the walls because of the accumulation of pushing filaments at the apical end, see Fig. 3(a), those of puller type are reflected, see Fig. 3(b). Wall adhesion and reflection are accompanied by major shape changes of the -flexocytes. Flexocyte deformation thus links the angles of reflection with the flexocyte elastic and active properties. Directly after contact with a wall, -flexocytes of puller and mixed type elongate along their long axis (parallel to the wall). The symmetric state is unstable, because small fluctuations lead to a symmetry breaking, which is amplified by the propulsion forces. For flexocytes of puller type, the shapes subsequently become round, both during reorientation and shortly after the flexocytes detach from the wall. Finally, the flexocytes recover their stationary shapes. To characterize this evolution, we determine the reflection angle just after detachment from the wall and the reflection angle after complete shape relaxation, see Fig. 3(c), and the time evolution of the asphericity , see Fig. 3(d). All trajectories are more tangential to the wall directly after wall detachment compared with the angle of incidence, i.e., . Furthermore, for , we find independent of the angle of incidence. The reflection angles after shape relaxation are either comparable to or larger than . For almost tangential impact with , the flexocytes do not deform much, as shown by a nearly constant asphericity, and are almost specularly reflected.

Wall absoption and reflection depend on the flexocyte elastic properties, see Fig. 3(c) and (d). Whereas the qualitative behavior remains unchanged for varying the compression modulus , both deflection angle and asphericity change quantitatively. The far-field angle decreases with increasing , while the near-field angle is hardly affected and nearly independent of the angle of incidence for . This implies that because the shapes become more circular, there is less internal reorganization after detachment from the wall. Correspondingly, the shape deformation decreases with increasing , as signalled by the reduced variation of the asphericity. Although the initial flattening for normal impact mimics the elastic deformation of a bouncing ball, the microscopic mechanism is very different because the flexocytes lack inertia; they deform due to the persistence of the pulling-filament motion and their deformability.

i.4 Flexocytes at Interfaces

Another interesting approach to study the response to external perturbations is the motility on patterned substrates. We focus here on friction interfaces, such that the flexocytes move from an area of low friction to one of high friction, or vice versa. Figure 4 shows center-of-mass trajectories of flexocytes at such friction interfaces. The flexocytes are initialized in their stationary shapes with their directions of motion oriented toward the interface. Trajectories for -flexocytes of mixed type are only slightly deflected, see Figs. 4(a) and (b). Here, the pushing filaments at the apical end prevent major shape deformations and therefore stabilize the direction of motion. Trajectories for -flexocytes of puller type can be strongly deflected, see Figs. 4(c) and (d), because the delicate balance of pulling and friction forces is disturbed at the interface, which leads to strong deformations of the apical ends. This is demonstrated also in Fig. 4(e), which quantifies deflection as a function of friction jump at the interface. For almost normal angles of incidence, , flexocytes of mixed type, stabilizing pushing filaments can reduce the deflection angle by a factor compared to flexocytes of puller type 222Positive values of indicate a deflection towards the -axis and negative values indicate a deflection towards the -axis..

It is important to note that shape changes and trajectories at friction interfaces break time-reversal symmetry, i.e., the behavior from high to low friction is very different than in the opposite direction. This is demonstrated in Fig. 4(e) for initial -flexocytes; for normal incidence and increasing friction the trajectories can be strongly deflected, while for decreasing friction they remain nearly perpendicular to the interface. However, trajectories can also be strongly deflected for decreasing friction, in particular for small angles of incidence, see Fig. 4(f). For , after impact the flexocyte attains a state of motion along the interface with parts of its membrane in both the large- and small-friction regions. Because the membrane in the small-friction region moves faster, the flexocyte settles in a persistent tank-treading motion, which stabilizes the motion along the interfaces and traps it there, see Fig. 4(d). Interestingly, for flexocytes of puller type friction interfaces in both directions of impact lead to a deflection toward motion parallel to the interface. The dynamics of transient and persistent shape changes is discussed in the Supplementary Information.

Ii Discussion

The dynamical phases of our active vesicles are dictated by filament propulsion, membrane deformability, and membrane-substrate friction. Such ’responsive’ active particles show complex feedback between movement, shape, and their mechanical environment. This is reminiscent of motile cells, where actin polymerization, contractile actomyosin structures, and membrane elasticity determine both cell shape and motility tojkander_actin_2012; hotulainen_fiber_2006. Furthermore, we show that flexocytes with pulling filaments at the dorsal end are sufficient to mimic shapes and motility of biological cells. This agrees well with the importance of pulling forces \colorblack at the back of keratocytes that break symmetry spontaneously with myosin-driven actin flow preceding rear retraction barnhart_balance_2015. In C. elegans sperm cells the alignment of the pushing filaments in the lamellipodium is dictated by the membrane tension. The persistent motion of the cell increases with increased filament alignment batchelder_tension_2011. Although it might seem counterintuitive, for flexocytes of mixed type aligned pushing filaments at the apical end lead to less ballistic motility compared with flexocytes of puller type. The additional pushing filaments at the apical end can even cause circling motion, in agreement with predictions for keratocytes mogilner_rotating_2017.

Micropatterned structures can be used to further characterize shape relaxation, motility, and internal feedback. Such studies can thus contribute to elucidate the common generic mechanisms and design principles of responsive biological and engineered active systems. Cells have indeed been studied experimentally on micropatterned and functionalized substrates miyoshi_characteristics_2012; miyoshi_control_2010; barnhart_adhesion-dependent_2011.

Our model predicts flexocytes of puller type to be reflected by walls with a preferred reflection angle. This finding is very different from steady-state accumulation of active Brownian particles at walls elgeti_surface_2013; brady_pressure_2014. It reflects the internal reorganisation of the filaments and reproduces the behavior of keratocytes at interfaces between adhesive and passivated, microgrooved interfaces miyoshi_characteristics_2012; miyoshi_control_2010. Flexocytes at friction interfaces experience transient and stationary shape changes depending on their internal architecture. The deformation and relaxation processes to the new shapes lead to deflection of their trajectories. Our observations for flexocytes of puller type qualitatively reproduce the changes in shape and motility of keratocytes at interfaces between substrates with various adhesion strengths barnhart_adhesion-dependent_2011. The simulation results therefore predict an internal force distribution based on observations of cells at interfaces between substrates.

Our minimal model predicts how sensing capabilities can be realized in engineered active-matter composite agents and in soft robots. Furthermore, it facilitates studies of the active properties of cells, such as an activity-induced tension and active fluctuations of cells lieber_tension_2013; betz_fluctuations_2016; monzel_fluctuations_2016. Future work may include many-flexocyte simulations to study collective behavior ranging from the circling observed in small keratinocyte colonies nanba_rotation_2015 to wound healing wickert_hierarchy_2016.

Iii Methods

We study self-propelled rod-like filaments attached to semiflexible membrane rings two-dimensional systems using Brownian dynamics simulations. The attachment is either in the direction of the propulsion for pushing filaments or in the opposite direction for pulling filaments abaurrea_rigid-ring_2017.

iii.1 Filaments

The system consists of rod-like filaments with length , where each filament consists of beads. The filaments are characterized by their center-of-mass positions , their orientation angles with respect to the axis, their center-of-mass velocities , and their angular velocities abkenar_collective_2013. Filament positions and orientations are initialized randomly along the membrane, with the only constraint that filaments and membrane cannot overlap.

Filament-filament and filament-membrane interactions are modeled by the repulsive separation-shifted Lennard-Jones (SSLJ) potential for the bead-bead interaction abkenar_collective_2013

(4)

where is the distance between two beads, characterizes the capping of the potential, and shifts the potential to avoid a discontinuity at . The length is calculated by requiring the potential minimum to be at . Hence is the potential energy barrier. Once has been set, we obtain . Neighbouring beads overlap by , such that the friction for filament-filament sliding is small and no interlocking occurs abaurrea_rigid-ring_2017.

iii.2 Membrane

The membrane ring is characterized by its equilibrium radius , bending rigidity , target area , compression modulus , and perimeter-length compression modulus . It is discretized into beads that are separated from each other by a distance , such that the membrane is smooth and the friction between the filaments and the membrane is minimal abaurrea_rigid-ring_2017.

The membrane beads are connected by harmonic bonds with spring constant and rest length . The total stretching energy

(5)

thus controls both bond length and total contour length. Here, is the bond vector from monomer to monomer . In our simulations, is chosen sufficiently large to prevent changes of the membrane perimeter that are larger than .

The bending energy kierfeld_stretching_2004

(6)

controls membrane shape and deformations. Here, is bending rigidity.

The area-compression energy

(7)

controls the area enclosed by the membrane. Here, is the bulk modulus and is the membrane target area. The area of triangle is

(8)

calculated using the position vectors for the membrane beads and , .

iii.3 Filament-Membrane Interaction

Filaments and membrane interact sterically via the SSLJ potential with a large energy barrier , such that the filaments cannot exit the membrane ring for all considered propulsion forces. The filaments are attached to the membrane by their first bead. The attachment is modeled via a harmonic-spring potential

(9)

where is the spring constant and the rest length of the spring. This attachment controls the radial distance between the filaments and the membrane, while still allowing the filaments to slide along the membrane abaurrea_rigid-ring_2017.

The direction of the propulsion force with respect to the attachment defines whether an attached filament pushes or pulls. For attached-pushing filaments, the propulsion force points towards the membrane, while for attached-pulling filaments, the propulsion force points away from the membrane. Filaments of different type do not interact with each other. However, pulling filaments interact with a small energy barrier , and pushing filaments with a large energy barrier .

iii.4 Simulation Technique

Brownian dynamics simulations are employed for all our systems. The filament velocity is decomposed into parallel and perpendicular components for the center-of-mass velocity, , and the angular velocity . The filament velocities are given by

(10)
(11)
(12)

Here, and are unit vectors parallel and perpendicular to the filament orientation, respectively, and is a vector oriented normal to the plane of filament motion. is the propulsion force for each filament. and are the steric force and the torque from filament to filament , and and are the steric force and the torque from of the membrane on filament , respectively. and are the attachment force and torque from the membrane on filament .

The velocity of a membrane bead is then

(13)
(14)

where is the steric force of filament on the membrane bead , and represents the stretching, bending, and compression forces of the membrane bead.

The filament friction coefficients in three dimensions are , and , and , , and are the corresponding noise terms, respectively; is the membrane-substrate friction, and is the corresponding noise. All noises are drawn from Gaussian distributions with variances , such that the fluctuation-dissipation theorem is fulfilled, at equilibrium abaurrea_rigid-ring_2017; loewen_spherocylinders_1994.

For systems where the friction becomes inhomogeneous the Ito-Stratonovich dilemma has to be taken into account farago_friction_2014; durang_friction_2015. Here, we modify the Brownian dynamics equations by adding an extra term containing the gradient of the diffusion coefficient , with . The value of distinguishes the Ito , the Stratonovich , and the anti-Ito approach. We employ the Ito approach, which has been applied earlier for active Brownian particles in an environment with an anisotropic friction bechinger_friction_2011, where it has also been shown that the numerical results coincide with the particle velocities measured in experiments. If the substrate friction is independent of the position, , the last term of Eqs. (13) and (14) disappears because .

iii.5 Friction Interfaces

For flexocytes at friction interfaces, the friction interface is taken to be a smoothed-out step function

(15)

where is the center of the interface, is the membrane-substrate friction for , is the membrane-substrate friction for and characterizes the interface width. In our simulations, we use , and , which corresponds to an interface width ; thus the interface is essentially discontinuous on the scale of the flexocyte.

iii.6 Simulation Parameters

The flexocytes enclose and pushing and pulling filaments with Péclet numbers and . Each rod-like filament consists of beads. The membrane ring with radius consists of beads. The energy barrier for filament-membrane interaction is , which prevents the filaments from escaping from the vesicle. The energy barrier between pulling filaments is , and between pushing filaments . Pushing and pulling filaments do not interact with each other.

For the membrane, we employ reduced membrane-substrate frictions and , reduced area compression moduli and , a reduced bending rigidity , and a reduced spring constant . The harmonic springs that attach an end bead of a filament to the nearest bead of the membrane has rest length and spring constant .

Acknowledgements.
C.A.V. acknowledges support by the International Helmholtz Research School of Biophysics and Soft Matter (IHRS BioSoft). CPU time allowance from the Jülich Supercomputing Centre (JSC) is gratefully acknowledged.

References

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