Interacting filaments drive vesicle morphogenesis

Domain: physics
Published: 2025-12-06
Editorial outcome: major_revision

Abstract
The interplay between vesicles and their enclosed filaments is fundamental to the morphogenesis, motility, and mechanical response of biological cells, artificial cells, and biomimetic robotic systems. By engineering responsiveness or interaction capabilities—such as long-range filament interactions—these filaments can function as active elements that regulate system behavior. Here, we combine theoretical modeling and molecular dynamics simulations to demonstrate how interacting filament loops within vesicles induce diverse, system-wide morphological transformations. These transformations are driven by inter- and intrafilament interactions, as well as the competing deformations of both the vesicle and its encapsulated filaments, with interfilament interactions playing a dominant role. We observe phenomena including filament buckling and reorientation, vesicle stretching, and convex-to-concave shape transitions. Morphological phase diagrams are constructed for both vesicles under zero osmotic pressure and those with a fixed relative volume, and we further explore the packing of inhomogeneous filament loops. These results offer quantitative design principles for artificial cellular systems in which filament interactions act as levers to control and stabilize emergent morphologies, laying the groundwork for the development of adaptive soft robotics.

Introduction
The interplay between deformable membranes and enclosed filaments governs key aspects of morphogenesis, motility, and mechanical adaptability in biological cells as well as in vesicle-based artificial cellular systems1,2. From a mechanical perspective, cells and vesicles encapsulate the fundamental physics of membrane-mediated confinement3–5, while their enclosed filamentous components—whether cytoskeletal polymers, nucleated microtubules, or engineered materials—act as structural and functional scaffolds that regulate system-wide behavior through mechanical and biophysical feedback.

In living cells, this physical dialogue manifests in diverse processes such as microtubule-driven platelet activation6, actin-mediated pseudopod formation during cell locomotion1, and membrane tubulation in cell uptake7. Synthetic analogs such as vesicles encapsulating actin filaments and microtubules8–15, or self-propelled rigid rods and cargoes16–20, recapitulates similar behaviors on similar length scales yet with reduced biological complexity. These systems offer valuable platforms for uncovering design principles of shape-programmable structures with potential applications in artificial cells. Recent theoretical advances have established a full spectrum of packing morphologies for single flexible filaments—both open and closed—within vesicles, delineating the conditions for transitions between configurations21–23.

While these investigations have provided valuable insights, a key unresolved question is how competing energy hierarchies—spanning filament flexibility, tunable interfilament interactions, and membrane deformation—collaborate to orchestrate and stabilize morphological transitions in filaments–vesicle systems. Understanding these interactions is essential for bridging the gap between the inherent complexity of biological systems and the rational design of synthetic cellular systems. Notably, long-range interfilament interactions (e.g., electrostatic forces), which are prevalent in biological processes like polyelectrolyte condensation24,25 and viral genome packaging26,27, remain underexplored in the context of vesicle morphogenesis.

In this work, we employ theoretical modeling and molecular dynamics (MD) simulations to elucidate the mechanisms underlying vesicle morphogenesis driven by enclosed flexible filament loops, with long-range interactions capturing both inter- and intrafilament forces. We systematically explore how variations in filament stiffness, length, and interaction strengths influence system morphologies in vesicles under zero osmotic pressure or in those with a fixed volume. By identifying the morphological phase diagrams and the underlying energy landscapes, we reveal how filament interactions act as levers to control and stabilize emergent morphologies, including filament buckling, reorientation, and convex-to-concave shape transitions. Membrane tension analysis further reveal the mechanical pathways driving these transitions. We also discuss possible implications of our results on programmable cellular systems.

Results
Model and parametrization
Consider two inextensible, uniformly charged filament loops, f1 and f2, confined within a lipid vesicle of fixed area A(=4πR2), where R is the effective vesicle radius. The vesicle can either freely adjust its volume at zero osmotic pressure difference (Δp=0) or maintain a fixed volume V. Filaments f1 and f2 have lengths L1 and L2, respectively, and are uniformly charged with like charges (e.g., negative in Fig. 1) at densities λ1 and λ2. Here, electrostatic interactions are employed to model the tunable filament interaction. The nonlocal Coulomb interaction causes the filaments to repel each other, while the vesicle confinement imposes elastic and geometric constraints to the filaments. The competition between the repulsive Coulomb interaction and the bending resistance of the filaments and vesicle membrane leads to a complex morphological evolution of the filaments–vesicle system.

It is important to note that the electrostatic interaction here does not correspond to a specific physical mechanism, but rather serves as a generic, tunable model for long-range filament-filament interactions. This abstraction enables us to systematically probe how nonlocal repulsion modulates vesicle morphology. Experimentally, such interactions can be realized using stimulus-responsive microscale filaments, for example, through magnetic self-assembly of lipid-coated nanoparticles or electrospinning of magnetically or optically active polymers. Related experimental strategies and their relevance to our findings are further discussed in the “Discussion”.

Throughout the main text, we focus on the long-range limit, corresponding to a salt-free environment where electrostatic interactions are effectively unscreened. This regime provides a simplified setting allowing us to isolate the conceptual role of unscreened electrostatics in filaments-vesicle organization. While not intended to replicate specific biological conditions, this limit serves as a reference point for understanding the qualitative role of interaction range in shaping vesicle morphology.

To assess the robustness of our findings with respect to this modeling assumption, we also perform simulations incorporating electrostatic screening, modeled via a screened Coulomb potential with an adjustable Debye length. The results on screening-dependent modulation are analyzed in detail in the “Discussion” and Supplementary Information. Briefly, we find that under weak screening, the filament configurations and vesicle morphologies remain qualitatively similar to those observed in the unscreened case, indicating that the shape transitions are robust against mild attenuation. In contrast, strong screening substantially reduces interfilament repulsion, requiring higher interaction strength to achieve comparable morphological changes.

The total system energy is Etot = Em + Ef + Eintra + Einter, consisting of the bending energy of the vesicle membrane Em, bending energy of two filaments Ef, intrafilament interaction energy Eintra, and interfilament interaction energy Einter. Adopting the Helfrich theory for the membrane28, the wormlike chain model for the filaments29, and the Coulomb’s inverse-square law for interaction potentials, these four energy components are

Em = 2κ ∫ H2dA,
Ef = ∫0L1 Y1I1C12 ds1 + ∫0L2 Y2I2C22 ds2,
Eintra = ∑i=1,2 8π λi2 / ε0εr ∫0Li ∫0Li |rf i(si) - rf i(si')|-1 dsidsi',
Einter = λ1λ2 / 4πε0εr ∫0L1 ∫0L2 |rf1(s1) - rf2(s2)|-1 ds1ds2,

where κ, H, and dA represent the bending rigidity, mean curvature, and surface element of the membrane, respectively; Ci is the local curvature of the filament fi (i=1,2) modeled as a space curve with its centerline position vector rfi(si), and Yi, Ii, si, and dsi denote the Young’s modulus, moment of inertia, arclength, and curve element of the filament fi, respectively; λi is the uniformly distributed charge density (per unit length), ε0 is the vacuum permittivity, εr is the relative permittivity of the solvent (e.g., εr ≈ 80 for water at room temperature). Hereinafter, the subscripts 1 and 2 refer to quantities pertaining to filaments f1 and f2, respectively. In Eintra, a mollifier M is introduced to regularize the singular self-interaction electrostatic potential. A specific form is taken as M=ηi4/(ηi+e-7ηi)4 with ηi=sin(π|si-si'|/Li)30,31. For two circular charged filament loops infinitely far away from each other as a reference state, the corresponding intrafilament interaction energy is E0intra = 0.1286(λ12L1 + λ22L2)/(ε0εr).

For simplicity in the following analysis, we consider two identical filaments (L1=L2=L, Y1I1=Y2I2=YI, and λ1=λ2=λ) unless otherwise specified. At zero osmotic pressure (Δp=0) or a given reduced vesicle volume v= V/(4πR3/3), the total system energy is Etot = Etot(YI/(κR), L/(πR), χ), a function of the relative stiffness YI/(κR) and length L/(πR) of the filaments, as well as the dimensionless Coulomb interaction strength χ=λ2R/(4πε0εrκ).

Here, we employ both numerical optimization and MD simulations to identify the minimum energy state for a given set of parameters (YI/(κR), L/(πR), χ), with methodological details in the Methods section and Supplementary Figs. S1–S4.

At Δp=0, the spherical filament-free vesicle offers isotropic confinement. In contrast, when the vesicle’s reduced volume v is specified, its shape depends on v: at v=1, it is spherical, but as v decreases, it transforms from prolate to oblate and then to a stomatocyte shape32. These nonspherical forms provide anisotropic confinement. The behavior of the interacting filaments–vesicle system under both the isotropic and anisotropic confinements is elucidated in the following analysis.

Interacting filaments–vesicle system at zero osmotic pressure difference
We first investigate system morphologies for highly flexible filaments (YI/(κR)=0.1) at Δp=0 (Fig. 2), focusing on how filament-vesicle interactions are regulated by the relative filament length L/(πR) and Coulomb interaction strength χ=λ2R/(4πε0εrκ). Based on point group symmetry, the system morphologies at different (L/(πR), χ) values fall into four symmetry groups: D∞h, D2, C2h, and Ci. For small values of (L/(πR), χ) (e.g., morphology at (1, 0.4) in Fig. 2a), weak repulsion between the charged filaments results in minimal vesicle deformation, with only slight stretching due to weak pushing forces on the inner membrane. As χ increases, the vesicle transitions from an axisymmetric convex to a concave morphology (e.g., from (1, 3) to (1, 30) in Fig. 2a). Increasing L/(πR) leads to lateral inflation and vertical shortening (comparing morphologies at (1, 0.4) and (5/3, 0.4)). MD simulations (Fig. 2b) yield morphologies consistent with numerical results.

With a further increase in L/(πR), the compressive forces exerted by the vesicle membrane on the filaments intensify, destabilizing their position and morphology and triggering a system-wide transition. At small χ, axisymmetric morphologies with inversion symmetry D∞h transform into those with D2 symmetry (e.g., morphologies at (2, 2) and (7/3, 2) in Fig. 2a). At high χ, these D∞h morphologies change into C2h (e.g., morphology at (4/3, 15)). The D2 symmetry lacks mirror planes but features three twofold rotation axes, and C2h has a twofold axis of rotation and a reflection plane. Regarding the filament loops as hand contours, the long, flexible filaments in D2 system resemble a palm-to-palm hand holding configuration with curled fingers. Increasing χ, the inter- and intrafilament interactions strengthen, repelling and straightening the saddle-shaped filaments. This drives the transition from D2 to C2h symmetry (e.g., morphology at (2, 10)), where the vesicle adopts a rodlike shape with inversion as its sole symmetry element besides the identity. For the D2 symmetry system at large L/(πR) and small χ, the highly flexible filaments with YI/(κR)=0.1 minimally deform the vesicle, and their configurations approximate perturbed spherical curves (Supplementary Figs. S5 and S6)33,34. For YI/(κR)=1 and 10, representative system morphologies are presented in Supplementary Figs. S7 and S8, respectively.

With knowledge of all possible system morphologies at Δp=0, we construct the corresponding morphological phase diagrams in Fig. 2c–e as functions of the relative filament length L/(πR) and Coulomb interaction strength χ. The morphological transitions at YI/(κR)=0.1 and 1 exhibit qualitative similarity but differ quantitatively. One difference is that in the D∞h and D2 phases, with relatively large L/(πR) and small χ, stiffer filaments deform the vesicle into an oblate shape (Supplementary Fig. S7a), whereas highly flexible filaments induce only minor deviations from a spherical morphology (Fig. 2a). Additionally, increasing YI/(κR) from 0.1 to 1 expands the D∞h region, shifts the D2 region rightward, and moves the C2h region upward at large L/(πR), consistent with the increased resistance of stiffer loops to buckling at longer critical lengths. At YI/(κR)=10, these trends intensify: the D∞h region further expands, the D2, C2h, and Ci regions shift rightward and upward, respectively, and the D2 phase disappears entirely within the studied L/(πR)−χ region (Fig. 2e).

To elucidate the mechanisms underlying the morphological phase transitions, the total energy change and its components are analyzed in Fig. 3. In the Δp=0 case, the ground energy state consists of a spherical vesicle with bending energy 8πκ and two infinitely separated circular charged filaments, each of length L and charge density λ. The total energy change is ΔEtot=ΔEm+ΔEf+ΔEintra+Einter, where ΔEm = Em − 8πκ, ΔEf = Ef − 4π2YI/L, and ΔEintra = Eintra − 1.029πκχL/R. Energy profiles at χ=20 (Fig. 3) show that the interfilament interaction energy Einter dominates ΔEtot throughout the vesicle evolution. As L/(πR) increases, ΔEtot exhibits a kink, marking a discontinuous morphological transition. In the transition from D∞h (orange) to the C2h phase (cyan), the filaments reorient and undergo significant in-plane bending, stretching the vesicle and sharply increasing both ΔEm and ΔEf. The transition from the C2h to the Ci phase (purple) is continuous. With increasing filament bending stiffness, the energy contribution from filament bending grows. In the Ci and C2h phases, ΔEf plays a minor role at YI/(κR)=0.1 and 1 (Fig. 3a, b) but becomes comparable to ΔEm and ΔEintra at YI/(κR)=10 (Fig. 3c).

The morphological transition can also be understood from the evolution of the membrane tension. As χ increases, the growing interfilament repulsion stretches the confining vesicle, raising membrane tension σ (Fig. 4). This increased tension σ enhances radial contraction of the elongated vesicle. When the resulting distributed compressive force on the filaments becomes sufficiently large, they destabilize, undergoing rotation and bending, which drives the system’s morphological transitions.

While this general mechanism applies across a range of parameters, the specific evolution of the membrane tension during the transition depends on the filament stiffness. For example, although both cases with YI/(κR)=1 and 10 undergo a similar morphological transition from an oblate axisymmetric shape (D∞h) to a centrosymmetric elongated shape (C2h) with a central neck, the associated membrane tension exhibit opposite sharp changes: it drops for flexible filaments but jumps for stiff ones. This contrast arises from how filament stiffness modulates the geometric and mechanical constraints on the vesicle.

In systems with flexible filaments, the C2h configuration allows the filaments to deform and adapt more easily, thereby exerting weaker geometric and mechanical constraints on the vesicle. This flexibility leads to a lower membrane tension compared to the case with stiff filaments, which resist deformation and act as stronger internal constraints. As a result, the vesicle must accommodate these stiff structures by increasing its membrane tension, even when the overall morphology appears similar, as confirmed in Supplementary Fig. S9. Supplementary Fig. S9 further indicates that at low filament stiffness, the vesicle membrane tension is lower than that of the corresponding oblate axisymmetric state. As YI/(κR) increases, the tension surpasses that of the axisymmetric configuration and eventually saturates at a plateau. This behavior explains the sharp drop observed at YI/(κR)=1 and the jump at YI/(κR)=10 in the σ–χ profiles (Fig. 4).

Interacting filaments–vesicle system at a given vesicle volume
In preceding cases, the vesicles at Δp=0 provide an isotropic confinement for the filaments. Here, we investigate how the mechanical interplay between the charged filaments and the vesicle is influenced by a nonspherical, anisotropic confinement, achieved by specifying a reduced vesicle volume at v=0.9 for example.

Figure 5a, b shows representative system morphologies for highly flexible filaments (YI/(κR)=0.1) at v=0.9, and the corresponding morphological phase diagram is established in Fig. 5c. The system morphologies at different (L/(πR), χ) values are categorized into symmetry groups: D∞h, C2h, D2h, and D2, where D2h symmetry has three mutually perpendicular reflection planes and an inversion center. At small L/(πR), the filaments arranged in circular configurations with axisymmetric prolate vesicles (D∞h symmetry), which transition from convex at smaller L/(πR) (e.g., morphologies at (1, 1) and (1, 10) in Fig. 5a) to concave shapes at larger L/(πR) (e.g., morphology at (11/8, 1)). Increasing Coulomb interaction strength χ leads to progressive elongation of the axisymmetric vesicles.

As in the Δp=0 case, increasing L/(πR) further destabilizes filament loops, driving system transitions from D∞h to C2h and D2 morphologies. However, a key difference at v=0.9 is that C2h morphologies appear at small χ, while D2 morphologies emerge at large χ (Fig. 5c). This difference arises from the anisotropic confinement provided by the prolate vesicle, which naturally offers the filament loops suitable inclined positions to form the C2h morphologies (e.g., morphology at (1.6, 1) in Fig. 5a), eliminating the need for high χ, whereas the round vesicles at Δp=0 require elongation (induced by stronger interfilament repulsion at high χ) to accommodate the inclined loops. The D∞h-to-C2h transition is discontinuous, evidenced by abrupt filament sliding and reorientation (e.g., morphologies at (11/8, 1) and (1.6, 1)). Morphologies from MD simulations closely match those from numerical optimization (Fig. 5b).

The D2 system morphologies, where the filaments resemble a palm-to-palm hand holding configuration, are energetically preferred by round vesicles. For prolate vesicles at v=0.9, achieving this state requires longer filament loops and higher χ to induce a more spherical geometry. This translates to expanded lateral dimensions and a reduced vertical dimension for the prolate vesicle (e.g., morphologies at (5/3, 10) and (2.033, 40) in Fig. 5a). Consequently, the D∞h-to-D2 transition occurs at larger values of (L/(πR), χ), compared to the D∞h-to-C2h transition (Fig. 5c).

For the C2h morphologies, a further increase in L/(πR) leads to a transition to D2 at high χ (e.g., morphology at (2.267, 10) in Fig. 5a), with longer loops exhibiting greater bending. In D2h morphologies, the filaments elongate and reorient parallel to the vesicle’s long axis. Unlike other discontinuous transitions (solid phase boundaries in Fig. 5c), the C2h-to-D2h transition is continuous (dashed phase boundary).

Comparing phase diagrams at YI/(κR)=1 and 10 in Fig. 5d, e with that at YI/(κR)=0.1 in Fig. 5c, structural changes in morphology emerge as YI/(κR) increases. In D2h and D2 morphologies, the filament loops experience significant in-plane or out-of-plane bending (Supplementary Figs. S10 and S11). As YI/(κR) increases, this bending instability necessitates larger L/(πR) and χ, shifting the D2h and D2 regions toward the upper right. The gaps between C2h and D2h, as well as between C2h and D2, are occupied by the D∞h phase (Fig. 5d, e), where the filament loops remain circular. Further morphological results indicate that, for YI/(κR)=1 and 10 (Fig. 5d, e), the left D∞h-to-C2h transition is discontinuous, marked by abrupt filament sliding and reorientation, whereas the right C2h-to-D∞h transition is continuous, exhibiting smooth filament rearrangement (Supplementary Figs. S10 and S11).

The energy analysis at v=0.9 is given in Supplementary Fig. S12. Here, the ground state comprises a prolate vesicle with bending energy 29.90κ32 and two circular, charged filaments (length L, charge density λ), positioned at infinite separation. Consequently, the changes in energy are ΔEm = Em − 29.90κ, ΔEf = Ef − 4π2YI/L, and ΔEintra = Eintra − 1.029πκχL/R. The energy profiles at χ=1 in Supplementary Fig. S12 indicate that the interfilament interaction energy Einter dominates the total energy evolution, similar to the Δp=0 case. For small values of (L/(πR), χ), the filaments in circular configurations are far apart, resulting in negligible ΔEintra. With increasing L/(πR), the reorienting filaments undergo pronounced in-plane and out-of-plane bending, which raises ΔEf. In the D2 phase (green), the two approaching filaments bend more severely, causing substantial increases in both ΔEf and ΔEintra. For filaments with YI/(κR)=1 and 10 (Supplementary Fig. S12b, c), ΔEf remains minor relative to ΔEm in the C2h region (cyan), reflecting only slight filament deformation.

For a filament-free vesicle of v=0.9, the membrane tension is σR2/κ ≈ −5.11. The presence of repelling filaments increases σ as the vesicle is stretched. Fig. 6 shows the evolution of the membrane tension σ with the relative filament length L/(πR) for different filament stiffnesses at χ=1 and 3. Positive (negative) σ means in-plane membrane tension (compression). Several sharp jumps and drops in σ mark discontinuous phase transitions.

Packing of inhomogeneous filament loops
Filaments within cells can experience localized mechanical property changes due to the influence of specific proteins35. For instance, cofilin-decorated filaments are more flexible than native filaments, and in some cases, cofilin molecules can induce filament breakage36, whereas tropomyosin-troponin complexes increase the bending stiffness of actin filaments by stabilizing the filament structure37. To explore how filament inhomogeneity affects system morphologies, we consider two identical inhomogeneous filaments confined within a vesicle, each composed of flexible (s1) and stiffer (s2) segments (Fig. 7).

Figure 7 shows the system morphologies at Δp=0 and v=0.9 for a given filament interaction strength. At Δp=0, the filaments have YIs1s1/(κR)=0.1, YIs2s2/(κR)=1, and segment lengths Ls1=Ls2=0.95πR at χ=1. As shown in Fig. 7a, they resemble a palm-to-palm arrangement with an approximate 90° relative orientation. At v=0.9, filaments with YIs1s1/(κR)=1, YIs2s2/(κR)=10, and Ls1=Ls2=7πR/6 at χ=1 exhibit nearly identical orientations (Fig. 7b), with stiffer segments aligned with each other and softer segments similarly positioned. In both cases, the filament deformation localizes at the ends of the relatively flexible segment (light brown or blue) to minimize the energy cost associated with bending the stiffer segment.

Methods
Numerical optimization
In the numerical optimization, the vesicle morphology is depicted by the membrane position vector rm using a spherical harmonic parameterization21,51,52. In a Cartesian coordinate system Oxyz (Fig. 1), the membrane position is taken as rm = (xm, ym, zm) = ∑lm (Cx,y,z lm) Ylm(θ, φ), where Cx,y,z lm are the coefficients for xm, ym, and zm, respectively. Here, Ylm(θ, φ) denotes the real spherical harmonic basis function in terms of the polar angle θ ∈ [0, π] and azimuthal angle φ ∈ [0, 2π] of the spherical coordinate system (see Supplementary Note 1). The charged filaments, which repel each other, bend against the vesicle membrane and are assumed to be constrained to the membrane surface in our theoretical model, consistent with qualitative suggestions from MD simulations. The filaments could freely slip along the vesicle surface. Each filament fi (i=1,2) is characterized by its position vector rfi(φ)=rm(θi, φ), with a specified polar angle θi. The vesicle surface and filament geometries can then be derived from rm(θ, φ) and rfi(φ) using classical differential geometry (Supplementary Note 1). Consequently, the total energy Etot is expressed as a function of undetermined coefficients Cx,y,z lm and minimized numerically via sequential quadratic programming. The fixed vesicle area A and filament length L i serve as equality constraints, with vesicle volume V included if a reduced volume is specified. Once values of Cx,y,z lm are obtained, the system morphology and corresponding energy components are determined.

MD simulations
In the coarse-grained MD simulations, the lipid vesicle membrane is represented by a one-particle-thick layer53–55, where each lipid molecule is modeled as a particle with translational and rotational degrees of freedom. The interaction between these lipid particles depends on their relative distances and orientations. For vesicles at Δp=0, solvent is not introduced; for vesicles with a reduced volume v, solvent is included to control volume56. Filaments are modeled as chains of unit-charged particles connected by harmonic bonds and angles. The electrostatic interaction is computed using the Particle-Particle Particle-Mesh method57 with the interaction strength proportional to 1/εr. The discretization of charges, finite simulation box size, and periodic boundary conditions complicate the mapping of the interaction strength from MD simulations to the theoretical model58. Lennard-Jones potentials govern lipid, solvent, and filament interactions54. For elongated vesicle morphologies at Δp=0, the simulation box size bs is adjusted to adequately accommodate the morphologies. All MD simulations are conducted with the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)59 and visualized using OVITO60. For clarity, simulation boxes and solvent particles are not shown in the main text. More detailed descriptions can be found in Supplementary Note 2 and Figs. S1–S4.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors use computer simulations (both energy minimization and molecular dynamics) to study the organization of interacting filaments within a fluid vesicle. They specifically focus on the case of two like-charged, loop-shaped filaments. The authors investigate the morphologies that emerge from this model over a wide range of parameters, including filament length, stiffness, and charge density, as well as for either zero osmotic pressure or fixed volume vesicles. They construct phase diagrams and present many representative configurations to showcase the different possible morphologies, and quantify how elastic energy costs vary over these different structures. The novel feature of this model compared to those in the existing literature is the inclusion of electrostatic interactions between filaments.\n\nAssessment:\nI recommend major revisions. The simulations appear technically sound – I appreciate the detailed and careful simulation protocols explained in the supporting information – and the authors have thoroughly studied their model over a wide range of parameters. The morphologies they observe are interesting and the classification of different morphologies by their point group symmetries is neat. However, I think that the model needs to be better motivated and connected to potential experimental realizations. The authors should also address the extent to which such long-ranged electrostatic forces are realistic, since biological filaments are typically immersed in salty solutions where electrostatics are screened.\n\nMajor points:\n● The long-ranged repulsive forces between the like-charged filaments seem crucial for obtaining the observed morphologies, in which filaments tend to push each other away and hence deform the vesicle. Yet in typical biological settings (and many if not most experiments) the solution contains many salt ions, which screen long ranged electrostatic interactions beyond distances of more than a few nanometers, at most. This would seem to undermine electrostatic repulsion as a mechanism for generating interfilament repulsion over the hundreds of nanometers to microns typical of lipid vesicles. Do the authors’ findings hold up if they use screened electrostatics instead?\n● I am not aware of cytoskeletal filaments that form loops individually. (Of course, cytoskeletal filaments can collectively form loops, e.g. FtsZ cell division rings.) Can the authors comment on how their loops might be realized experimentally? Alternatively, do they observe similar results if they use non-closed-loop filaments?\n● In Fig. 4: why does tension jump up for Y = 10 but down for Y = 1, accompanying what looks like the same morphological transition?\n\nMinor points:\n● In Fig. 3, please label where different morphological changes take place.\n● In Fig. 3c, what is going on with the kink at 5/3? This should be explained.\n● Please consider citing more recent work on reconstituting filaments within vesicles, which might be useful for thinking about experimental realizations of the simulated system:\n○ Y. Mulla. et al., Phys. Biol. 2018\n○ L. Baldauf et al., ACS Synth. Biol. 2022\n○ A. Sciortino et al., Nat. Phys. 2025\nAlso, other recent work on how vesicle morphologies are impacted by interacting enclosed cargo, e.g.:\n○ P. Iyer et al., Soft Matter 2022\n● Related to Major Point 1, there needs to be more references to back up the statement on p. 3 about long-range interfilament interactions being biologically important. This is certainly true e.g. for viral genome packaging, but the reference cited (23) hardly discusses the role of electrostatic interactions in viral genome packaging. References like V.A. Belyi & M. Muthukumar, PNAS 2006 would be more appropriate.", "reviewer_2": "In the present manuscript the authors present a theoretical framework to study the interaction of bending energies of lipid membranes and the bending and interaction energies of filaments. A Phase diagram of vesicle and filament configurations is computed. The manuscript adresses a timely issue, with intereting predictions.\nI actually cannot judge the theoretical calculations and simulations - From an experimental point of view the paper adresses interesting predictions, which are worthwhile to be pursued experimentally. albeit a bit different and truely actively transported, there is just a recent publication in nature physics where it appears that similar configurations were observed - with microtubules in vesicles.", "reviewer_3": "The study explores how interacting filaments drive vesicle morphogenesis, using theoretical modeling and molecular dynamics (MD) simulations to investigate morphological transformations of vesicles induced by filament loops. The authors demonstrate that filament stiffness, length, and inter- and intra-filament electrostatic interactions dictate the equilibrium shapes of vesicle-filament systems, with increasing filament stiffness inducing stronger vesicle deformations. Combining continuum mechanics, filament elasticity, and electrostatics, the work provides a systematic framework for understanding cytoskeleton-driven membrane remodeling, and the results are well-supported by agreement between theoretical predictions and simulations. However, the study lacks direct experimental validation, leaving it unclear how well the predicted morphologies apply to real biological systems. Additionally, key biological factors such as filament polymerization, active forces, and membrane-protein interactions are not considered, potentially limiting the broader applicability of the findings. While the work is well-organized and solid, it does not offer sufficiently novel physical insight to meet the highest impact criteria, making it more suitable for publication in a specialized journal. In addition, the following points needed to be clarified:\n1. In this work, the filament interaction is described by electrostatic Coulomb interaction that is long-ranged. However, there are different kinds of ions in experimental environment, which can significantly screen the Coulomb interaction. So, the electrostatic interaction becomes the screened Coulomb interaction. The short-ranged interaction is expected to weaken the effects of the filaments on the vesicle morphologies. Have the authors considered charge screening effects in a physiological environment? How might ion concentrations influence the findings? The authors should discuss this point carefully.\n2. In general, vesicle morphologies from MD simulations closely match those from numerical optimization. However, it looks that the MD simulations cannot reproduce the numerical optimization results with parameter (5/3,10) in Fig.5a. Please explain this.\n3. The discussion section lacks a deeper connection to biological systems. For instance, the authors could explore how the observed morphological transitions relate to cellular processes like endocytosis or cytokinesis, and provide more discussions about possible biological analogs.\n4. This study validates theoretical predictions using numerical optimization and molecular dynamics (MD) simulations. Have they compared their results with existing experimental data on filament-induced vesicle deformations? How to realize the numerical results experimentally? How does the bending rigidity obtained from their simulations compare to experimentally measured values for real lipid bilayers? I suggest that the authors propose potential experimental systems or techniques that could validate the theoretical predictions.\n5. This study focuses on equilibrium morphologies, but many biological vesicle-filament interactions involve dynamic remodeling. How would the findings change if active processes such as filament polymerization or motor-protein forces were included?\n6. C2h-to-D2h and D∞h-to-C2h transitions are claimed as continuous and discontinuous transitions, respectively. The evidence is not sufficient. According to thermodynamics, the criterions for discontinuous or continuous transition include latent heat, phase coexistence, and the divergence of correlation length etc. However, no solid evidence is shown in this study."}, "author_response": "Response to Comments of the First Reviewer\n“Reviewer #1:\nSummary:\nThe authors use computer simulations (both energy minimization and molecular dynamics) to study\nthe organization of interacting filaments within a fluid vesicle. They specifically focus on the case\nof two like-charged, loop-shaped filaments. The authors investigate the morphologies that emerge\nfrom this model over a wide range of parameters, including filament length, stiffness, and charge\ndensity, as well as for either zero osmotic pressure or fixed volume vesicles. They construct\nphase diagrams and present many representative configurations to showcase the different possible\nmorphologies, and quantify how elastic energy costs vary over these different structures. The novel\nfeature of this model compared to those in the existing literature is the inclusion of electrostatic\ninteractions between filaments.”\nOur response: We sincerely thank the reviewer for the careful evaluation of our manuscript\nand for the constructive comments, which have helped us improve the work. We are especially\ngrateful for the reviewer’s recognition of the novelty in incorporating long-range filament\ninteractions, which we believe are essential for understanding filament–membrane coupling\nunder confinement. Below, we provide detailed responses to each of the reviewer’s comments.\n“Assessment:\nI recommend major revisions. The simulations appear technically sound – I appreciate the detailed\nand careful simulation protocols explained in the supporting information – and the authors have\nthoroughly studied their model over a wide range of parameters. The morphologies they observe\nare interesting and the classification of different morphologies by their point group symmetries is\nneat. However, I think that the model needs to be better motivated and connected to potential\nexperimental realizations. The authors should also address the extent to which such long-ranged\nelectrostatic forces are realistic, since biological filaments are typically immersed in salty solutions\nwhere electrostatics are screened.”\nOur response: We sincerely thank the reviewer for the constructive and thoughtful\nassessment. A central issue raised concerns the physical relevance of the long-range\nelectrostatic interactions that drive the observed morphologies. As this issue is closely related\nto Major Point #1, we provide a detailed and unified response in our reply to that point. There,\nwe clarify the modeling assumptions, explain the motivation for using unscreened Coulomb\ninteractions, and present new simulation results based on screened electrostatics to assess the\nexperimental feasibility of the proposed mechanism.\n“Major points:\nThe long-ranged repulsive forces between the like-charged filaments seem crucial for obtaining the\nobserved morphologies, in which filaments tend to push each other away and hence deform the\nvesicle. Yet in typical biological settings (and many if not most experiments) the solution contains\nmany salt ions, which screen long ranged electrostatic interactions beyond distances of more than\na few nanometers, at most. This would seem to undermine electrostatic repulsion as a mechanism\nfor generating interfilament repulsion over the hundreds of nanometers to microns typical of lipid\nvesicles. Do the authors’ findings hold up if they use screened electrostatics instead?”\nOur response: We thank the reviewer for raising this important concern about the physical\nrelevance of long-ranged electrostatic interactions in biological environments. We fully agree\nthat, under physiological or experimental conditions such as saline buffers, electrostatic\ninteractions are screened by mobile ions in the medium, resulting in a finite interaction range.\nIn our original model, we employed the classical Coulomb potential to represent long-range\nrepulsive interactions. This choice was motivated by the desire to construct a minimal\nconceptual framework that systematically probes how nonlocal interactions drive filament\norganization and vesicle shape transitions. To clarify this modeling assumption, we have now\nexplicitly stated in the second paragraph of page 4 of the main text that the baseline\nsimulations correspond to an environment where screening is negligible and electrostatics\nremain effectively long-ranged.\nTo directly address the reviewer’s concern, we have extended our analysis in the Note 4 of\nthe Supplementary Information by incorporating screened electrostatic interactions modeled\nusing a screened Coulomb potential of the form as (refs. 42 and 43)\n2 L L M(|s −s |)e−|r i f(s i )−r i f(s i )|/ D\nE =  i  i i i i dsds ,\nintra 8 0 0 |rf(s )−rf(s )| i i\ni=1,2 0 r i i i i\n L L e−|r 1 f(s 1 )−r 2 f(s 2 )|/ D\nE = 1 2  2 1 dsds ,\ninter 4 0 0 |rf(s )−rf(s )| 1 2\n0 r 1 1 2 2\nwhere  is the Debye length, quantifying the effective range of electrostatic interactions. It\nD\nreads  = [4πƩ(cq2)]-1/2, where  = e2/(4πε ε k T) is the Bjerrum length with e as the\nD B s s B 0 r B\nelementary charge, and c and q are the concentration and valence of small ions, respectively.\ns s\nAt 25 °C and pH 7, pure water’s autoionization yields an ionic concentration of approximately\n10⁻7 M, giving  ≈ 1 μm and thus minimal electrostatic screening.\nD\nUsing this screened interaction model with an exponential decay factor, we performed\nadditional numerical simulations to examine how varying screening conditions influence\nfilament conformations and vesicle morphologies. The results, summarized in Fig. S13, reveal\nthe following trends:\n(1) Under weak screening (e.g., /R = 10), the resulting filament configurations and vesicle\nD\nshapes are qualitatively similar to those in the unscreened case, demonstrating that the shape\ntransitions remain robust when electrostatic interactions are weakly screened.\n(2) Under strong screening (/R = 0.1), the effective interfilament repulsion is significantly\nD\nreduced, which results in a downward shift of the phase boundary and necessitates stronger\ninteraction strength to achieve comparable morphological transitions.\nThese results provide quantitative insight into how electrostatic screening modulates shape\ntransitions. They also support the physical intuition that long-range interactions are\nparticularly effective in driving system morphological changes. While our main model\nemploys the unscreened Coulomb interaction to isolate the role of nonlocal repulsion, the\nadditional simulations with screened potentials demonstrate that the underlying\nmorphological mechanisms are robust across a wide range of interaction ranges and decay\nlengths.\nThe corresponding discussion on the screening modulation has been added in on page 16 of\nthe revised manuscript, and in Note 4 of the Supplementary Information.\nFig. S13. Representative equilibrium system morphologies for highly flexible filaments (YI/(κR)\n= 0.1) obtained from numerical optimization at different screening lengths /R = 10, 1, and\nD\n0.1. The corresponding values of (L/(πR), χ) are listed below each morphology.\n“I am not aware of cytoskeletal filaments that form loops individually. (Of course, cytoskeletal\nfilaments can collectively form loops, e.g. FtsZ cell division rings.) Can the authors comment on\nhow their loops might be realized experimentally? Alternatively, do they observe similar results if\nthey use non-closed-loop filaments?”\nOur response: We thank the reviewer for this insightful question. While our theoretical\nmodel considers closed-loop filaments for simplicity, similar ring-like structures can also\nemerge collectively from multiple filaments in realistic systems. For example, Miyazaki et al.\n(Nat. Cell Biol. 17, 480 (2015); ref. 8) demonstrated that polymerizing actin filaments confined\nwithin cell-sized spherical droplets spontaneously self-organize into a single equatorial ring,\nsuggesting that topological closure at the single-filament level is not a prerequisite for ring\nformation. Similarly, Limozin and Sackmann (Phys. Rev. Lett. 89, 168103 (2002); ref. 9)\nexperimentally showed that actin filaments cross-linked by proteins such as α-actinin and\nmuscle filamin can assemble into ring-like bundles within giant unilamellar vesicles under\nappropriate conditions. Another relevant example is the marginal band found in blood\nplatelets, a peripheral ring of microtubules that provides internal mechanical support and\ndrives cell shaping (Diagouraga et al. J. Cell Biol. 204, 177 (2014); ref. 6). Upon biochemical\nactivation, the platelet undergoes a morphological transition from a flat, discoid shape to a\nspherical shape, accompanying the marginal band buckling from a planar circle- or oval-\nshape to a spatial curved shape. In our model, such collective or bundled filament assemblies\nare represented as effective closed-loop filaments, capturing essential features of curvature-\ninduced organization under confinement.\nIn addition, recent advances in soft matter systems provide promising routes toward\nexperimental realization of synthetic, responsive filament loops. For example, Bharti et al.\n(Nat. Mater. 14, 1104 (2015); ref. 44) reported the magnetic assembly of ultraflexible\nmicrofilaments from lipid-coated magnetic nanoparticles via nanocapillary forces. These\nfilaments exhibit reconfigurability, extreme flexibility, and the ability to self-close into loops\nor figure-eight structures under external fields. The fluidity of the capillary bridges between\nnanoparticles enables both reversibility and curvature adaptability, highlighting a materials\nplatform that mimics key features of our modeled system at the microscale.\nRegarding open filaments, we agree that it is important to assess whether similar membrane\ndeformation can be achieved without ring topology, and we are actively exploring their effects\nas a direction for future work. We note, however, that closed loops and interconnected\nfilamentous networks are subject to topological constraints that enforce global coupling along\nthe filament contour. Unlike open filaments, which can locally relax through end displacement\nor reorientation, closed loops resist such relaxation, thereby promoting cooperative\ndeformation and enhancing confinement-induced geometric frustration. These topological\nconstraints could lead to in-plane and out-of-plane buckling, saddle-like deformation, and\nsymmetry breaking.\nWe have added the corresponding discussion on experimental realization on pages 16 and\n17 of the revised manuscript.\n“In Fig. 4: why does tension jump up for Y = 10 but down for Y = 1, accompanying what looks like\nthe same morphological transition?”\nOur response: We thank the reviewer for raising this question. The opposite trends in\nmembrane tension arise from the different ways filament stiffness modulates the mechanical\nconstraints imposed on the vesicle. For flexible filaments with low bending stiffness, the\ncentrosymmetric vesicle configuration allows the filaments to deform and adapt more easily,\nthereby exerting weaker geometric constraints on the vesicle. This flexibility leads to a lower\nmembrane tension compared to the case with stiff filaments, which resist deformation and act\nas stronger internal constraints. As a result, the vesicle must accommodate these stiff\nstructures by increasing its membrane tension, even when the overall morphology appears\nsimilar.\nFurther numerical calculations (new Fig. S9, also shown below) indicate that within the\ncentrosymmetric branch, membrane tension increases monotonically with filament stiffness.\nAt low filament stiffness, the membrane tension is lower than that of the corresponding oblate\nshape, but as stiffness increases, the tension surpasses that of the axisymmetric configuration\nand eventually approaches a plateau.\nWe have added the corresponding discussion on pages 10 and 11 of the revised manuscript.\nFig. S9. Evolution of the vesicle membrane tension σ with varying filament bending stiffness\nYI/(κR) at Δp = 0, χ = 10, and L/(πR) = 1.867. The system morphologies at YI/(κR) = 1, 10, and\n100 are shown as insets and indicated by arrows.\n“Minor points:\nIn Fig. 3, please label where different morphological changes take place.”\nOur response: We thank the reviewer for the helpful suggestion. In the original versions of\nFigs. 3 and S11, we did not include explicit morphological labels because the color coding was\ndesigned to correspond to the morphologies shown in Figs. 2 and 5, respectively. Following the\nreviewer’s suggestion, we have revised Figs. 3 and S12 to label the regions corresponding to\ndifferent morphological regimes. This clarification makes it easier for readers to directly\ncorrelate energy profiles with the associated system morphologies.\n“In Fig. 3c, what is going on with the kink at 5/3? This should be explained.”\nOur response: We thank the reviewer for raising this point. Although the energy curve in\nFig. 3c appears to exhibit a kink near L/(πR) = 5/3, the energy remains a smooth and\ncontinuous function of filament length. This feature corresponds to the case with YI/(κR) = 10,\nwhere the filament undergoes a gradual geometric transition from a nearly circular planar\nshape to a more elliptical configuration. The point group symmetry remains unchanged\nthroughout this process, and there is no discontinuity or non-differentiability in the energy.\n“Please consider citing more recent work on reconstituting filaments within vesicles, which might\nbe useful for thinking about experimental realizations of the simulated system:\n○ Y. Mulla. et al., Phys. Biol. 2018\n○ L. Baldauf et al., ACS Synth. Biol. 2022\n○ A. Sciortino et al., Nat. Phys. 2025\nAlso, other recent work on how vesicle morphologies are impacted by interacting enclosed cargo, e.g.:\n○ P. Iyer et al., Soft Matter 2022”\nOur response: Following the reviewer’s suggestion, we have now cited the recommended\nstudies on reconstituted filaments in vesicles and enclosed cargoes as refs. 10, 13, 15, and 19 in\nthe revised manuscript.\n“Related to Major Point 1, there needs to be more references to back up the statement on p. 3 about\nlong-range interfilament interactions being biologically important. This is certainly true e.g. for\nviral genome packaging, but the reference cited (23) hardly discusses the role of electrostatic\ninteractions in viral genome packaging. References like V.A. Belyi & M. Muthukumar, PNAS 2006\nwould be more appropriate.”\nOur response: We fully agree with the reviewer that long-range interfilament interactions\nplay a critical role in genome organization and packaging, particularly in viral systems.\nFollowing the reviewer’s suggestion, we have now cited the PNAS work by Belyi and\nMuthukumar (PNAS, 2006; now ref. 26), which provides a quantitative demonstration of how\nlong-range electrostatics govern the packaging of ssRNA genomes in viruses. This study\ndirectly demonstrates that long-range interactions can serve as a major organizing principle\nin biological filamentous systems. The analogy between genome–capsid electrostatics in ref. 26\nand filament–filament repulsion in our model illustrates the broader relevance of long-range\ninteractions in driving confined structural organization across diverse biological contexts.\nResponse to Comments of the Second Reviewer\n“Reviewer #2:\nIn the present manuscript the authors present a theoretical framework to study the interaction of\nbending energies of lipid membranes and the bending and interaction energies of filaments. A Phase\ndiagram of vesicle and filament configurations is computed. The manuscript addresses a timely issue,\nwith interesting predictions.”\nOur response: We sincerely appreciate the reviewer’s positive evaluation of our work and\nthe recognition of its timeliness and significance. We are also grateful for the constructive\ncomments and thoughtful review, which have helped us to improve the manuscript.\n“I actually cannot judge the theoretical calculations and simulations - From an experimental point\nof view the paper addresses interesting predictions, which are worthwhile to be pursued\nexperimentally. albeit a bit different and truly actively transported, there is just a recent publication\nin nature physics where it appears that similar configurations were observed - with microtubules in\nvesicles.”\nOur response: We thank the reviewer for drawing attention to the recent publication in\nNature Physics (A. Sciortino et al., Nat. Phys. 21, 799 (2025); now ref. 15), which presents\nexperimental and DPD-based simulation results on membrane deformations driven by\nmicrotubule–motor assemblies. That study explores a dynamic, non-equilibrium system in\nwhich kinesin motors slide antiparallel microtubules, generating internal stresses that lead to\ntime-dependent vesicle shape fluctuations. As the authors demonstrate through quantitative\nanalysis, active forces set the temporal scale of fluctuations, producing fluctuation spectra that\ndiffer markedly from those observed in thermal equilibrium—both in spatial and temporal\ndecay.\nIn contrast, our study focuses on a different physical regime and employs a distinct\ntheoretical framework. We investigate equilibrium shape transitions in vesicles decorated with\ninteracting elastic filaments, where filament–filament interactions are direct, tunable, and\nfurther coupled to the membrane via mechanical feedback. This is in contrast to the Nat. Phys.\nsystem, where filament interactions are mediated indirectly through membrane deformation\nand motor-driven activity. By incorporating both direct inter-filament repulsion and\nmembrane elasticity into a unified framework, we reveal how these coupled effects dictate\nvesicle morphology and membrane tension under equilibrium conditions.\nMethodologically, our approach combines continuum elastic theory with molecular\ndynamics simulations to construct detailed morphological phase diagrams and quantify\nfilament-induced membrane tension. Taken together, our work and the Nat. Phys. study\nprovide complementary insights into filament–membrane composite systems: from active,\nfluctuation-driven remodeling to programmable shape control via tunable filament\ninteractions.\nIn response to the reviewer’s comment, we have added corresponding discussion in the\nrevised manuscript (Discussion section, page 19), highlighting the conceptual distinction and\ncomplementarity between our work and that of the Nat. Phys. study.\nResponse to Comments of the Third Reviewer\n“Reviewer #3:\nThe study explores how interacting filaments drive vesicle morphogenesis, using theoretical\nmodeling and molecular dynamics (MD) simulations to investigate morphological transformations\nof vesicles induced by filament loops. The authors demonstrate that filament stiffness, length, and\ninter- and intra-filament electrostatic interactions dictate the equilibrium shapes of vesicle-filament\nsystems, with increasing filament stiffness inducing stronger vesicle deformations. Combining\ncontinuum mechanics, filament elasticity, and electrostatics, the work provides a systematic\nframework for understanding cytoskeleton-driven membrane remodeling, and the results are well-\nsupported by agreement between theoretical predictions and simulations. However, the study lacks\ndirect experimental validation, leaving it unclear how well the predicted morphologies apply to real\nbiological systems. Additionally, key biological factors such as filament polymerization, active\nforces, and membrane-protein interactions are not considered, potentially limiting the broader\napplicability of the findings. While the work is well-organized and solid, it does not offer sufficiently\nnovel physical insight to meet the highest impact criteria, making it more suitable for publication in\na specialized journal. In addition, the following points needed to be clarified:”\nOur response: We thank the reviewer for the summary and acknowledge the points raised,\nand we sincerely appreciate the reviewer’s constructive comments and thoughtful review,\nwhich have helped us to improve the manuscript. However, we respectfully disagree with the\nassessment that the work lacks sufficient novelty or physical insight. To the best of our\nknowledge, this is the first systematic study to demonstrate how filament–filament long-range\ninteractions, when confined within deformable vesicles, can drive large-scale membrane\nmorphogenesis in a tunable way. This is a nontrivial and previously unexplored physical\nmechanism, which we investigate using a combination of continuum modeling and molecular\ndynamics simulations. The agreement between the two methods supports the robustness of the\nresults, and the findings open new conceptual directions for understanding confinement-\ninduced membrane remodeling.\nWhile we agree that additional biological features such as polymerization, active stresses,\nand membrane–protein interactions play important roles in cells, our aim here is to isolate the\nphysical effects of long-range filament repulsion and elastic coupling with the membrane,\nunder well-controlled and tractable conditions. We believe such abstraction is essential for\nuncovering fundamental principles that may otherwise be masked in fully complex biological\nsystems.\nTo further enhance the biological relevance, we have now included the effect of screened\nfilament interactions, and clarified that the model corresponds to some conditions relevant for\ncertain in vitro experiments. In response to the reviewer’s suggestion below, we also outline\npotential experimental systems and techniques that could be used to validate our theoretical\npredictions. We believe this work provides a useful theoretical basis upon which more\nbiologically detailed and experimentally guided models can be developed in future work.\n“1. In this work, the filament interaction is described by electrostatic Coulomb interaction that is\nlong-ranged. However, there are different kinds of ions in experimental environment, which can\nsignificantly screen the Coulomb interaction. So, the electrostatic interaction becomes the screened\nCoulomb interaction. The short-ranged interaction is expected to weaken the effects of the filaments\non the vesicle morphologies. Have the authors considered charge screening effects in a\nphysiological environment? How might ion concentrations influence the findings? The authors\nshould discuss this point carefully.”\nOur response: We thank the reviewer for raising this important point regarding the impact\nof ionic screening on filament–filament interactions in physiological environments. We fully\nagree that in experimental or biological conditions, such as saline buffers, mobile ions in the\nsolvent screen electrostatic interactions, reducing their effective range.\nIn our baseline model, we employ unscreened Coulomb interactions to represent long-range\nfilament interaction. This choice is not intended to model a specific physical mechanism, but\nrather to construct a minimal conceptual framework that isolates the role of nonlocal\ninteractions in driving vesicle morphogenesis. To clarify this modeling assumption, we have\nnow explicitly stated in the second paragraph of page 4 of the main text that the baseline\nsimulations correspond to an environment where screening is negligible and electrostatics\nremain effectively long-ranged.\nTo directly address the reviewer’s concern, we have extended our analysis in the Note 4 of\nthe Supplementary Information by incorporating screened electrostatic interactions modeled\nusing a screened Coulomb potential of the form as (refs. 42 and 43)\nE\nE\nin\nin\ntra\nte r\ni\n4\n1 ,2\n1\n8\n0\n2\nr\n2i\n0 r\nL\n2\ni\n0 |\nL\ni\n0\ne\nf1\nM\nf | 1\n( s\n(\n(s1\n)\n1\n|\n)\ns\ni|\nf2\ns |)\ni\nf ( s )\ni i\n( s )|/ 2 D\nf ( s )\n2 2\ne\n|\nd\nf f | (s ) ( i i i\nf ( s ) |\ni i\ns d s ,\n1 2\ns )|/ i D\nd s di s\ni\n,\n \n\n \n\n=\n=\n=\n\n\n\n\nr\n− r −\n−\nr\nr\nr\n−\n−\n− r\nr\n− r\nwhere  is the Debye length, quantifying the effective range of electrostatic interactions. It\nD\nreads  = [4πƩ(cq2)]-1/2, where  = e2/(4πε ε k T) is the Bjerrum length with e as the\nD B s s B 0 r B\nelementary charge, and c and q are the concentration and valence of small ions, respectively.\ns s\nAt 25 °C and pH 7, pure water’s autoionization yields an ionic concentration of approximately\n10⁻7 M, giving  ≈ 1 μm and thus minimal electrostatic screening.\nD\nUsing this screened interaction model with an exponential decay factor, we performed\nadditional numerical simulations to examine how varying screening conditions influence\nfilament conformations and vesicle morphologies. The results, summarized in Fig. S13, reveal\nthe following trends:\n(1) Under weak screening (e.g., /R = 10), the resulting filament configurations and vesicle\nD\nshapes are qualitatively similar to those in the unscreened case, demonstrating that the shape\ntransitions remain robust when electrostatic interactions are weakly screened.\n(2) Under strong screening (/R = 0.1), the effective interfilament repulsion is significantly\nD\nreduced, which results in a downward shift of the phase boundary and necessitates stronger\ninteraction strength to achieve comparable morphological transitions.\nThese results provide quantitative insight into how electrostatic screening modulates shape\ntransitions. They also support the physical intuition that long-range interactions are\nparticularly effective in driving system morphological changes. While our main model\nemploys the unscreened Coulomb interaction to isolate the role of nonlocal repulsion, the\nadditional simulations with screened potentials demonstrate that the underlying\nmorphological mechanisms are robust across a wide range of interaction ranges and decay\nlengths.\nThe corresponding discussion on the screening modulation has been added in on page 16 of\nthe revised manuscript, and in Note 4 of the Supplementary Information.\nFig. S13. Representative equilibrium system morphologies for highly flexible filaments\n(YI/(κR) = 0.1) obtained from numerical optimization at different screening lengths\n/R = 10, 1, and 0.1. The corresponding values of (L/(πR), χ) are listed below each\nD\nmorphology.\n“2. In general, vesicle morphologies from MD simulations closely match those from numerical\noptimization. However, it looks that the MD simulations cannot reproduce the numerical\noptimization results with parameter (5/3,10) in Fig.5a. Please explain this.”\nOur response: We thank the reviewer for pointing out this case. In the original version of\nthe manuscript, the MD simulation result for the parameter set (5/3, 10) was not included. We\nhave now added the corresponding MD simulation result in the updated Fig. 5b (page 12). The\nvesicle morphology obtained from the MD simulations agrees well with that from the\nnumerical optimization.\n“3. The discussion section lacks a deeper connection to biological systems. For instance, the authors\ncould explore how the observed morphological transitions relate to cellular processes like\nendocytosis or cytokinesis, and provide more discussions about possible biological analogs.”\nOur response: We thank the reviewer for the constructive suggestion. Following the\nreviewer’s suggestion, we have expanded the Discussion section (pages 17 and 18) to highlight\nthe biological relevance of the observed morphological transitions.\nOne biologically relevant analogy is the remodeling of the nuclear envelope during mitosis.\nIn this process, mitotic spindles generate internal forces that elongate and constrict the nucleus\ninto configurations that resemble the filament-induced vesicle morphologies observed in our\nsimulations (refs. 46 and 47). Similarly, giant unilamellar vesicles subjected to extensional flow\nundergo elongation and symmetry breaking due to flow-induced membrane tension (refs. 48\nand 49), which is conceptually similar to the repulsive interactions produced by confined\nfilament loops in our model. In both cases, membrane deformation arises from internally\ngenerated or externally imposed forces that mirror the mechanical principles encoded in our\nframework.\nWe also note that endocytosis involves shape transformations, such as membrane elongation,\nnecking, and symmetry breaking, that share qualitative similarities with those observed in our\nstudy. In particular, membrane-associated filaments such as dynamin and ESCRT-III play\ncentral roles in these processes. For example, dynamin helices constrict membrane necks\nduring scission events (ref. 50), while ESCRT-III filaments undergo supercoiling and\nreorganization to drive membrane deformation and fission (ref. 51). These systems illustrate\nhow filament–membrane coupling can induce complex shape transitions analogous to those\ncaptured in our mechanical model.\n“4. This study validates theoretical predictions using numerical optimization and molecular\ndynamics (MD) simulations. Have they compared their results with existing experimental data on\nfilament-induced vesicle deformations? How to realize the numerical results experimentally? How\ndoes the bending rigidity obtained from their simulations compare to experimentally measured\nvalues for real lipid bilayers? I suggest that the authors propose potential experimental systems or\ntechniques that could validate the theoretical predictions.”\nOur response: We thank the reviewer for these valuable and forward-looking questions\nregarding the experimental relevance and validation of our results. Below we address each\naspect in turn:\n(1) Comparison with existing experimental data\nTo the best of our knowledge, the particular filament–vesicle system studied here, involving\nclosed-loop filaments that interact via long-range repulsion under confinement, represents a\nnew theoretical construct that has not yet been directly implemented in experiments. We hope\nour findings will motivate future experimental exploration and provide a tractable and\nphysically grounded basis that could guide future synthetic realizations of shape-\nprogrammable vesicles.\nNonetheless, as described in our response to Comment 3, our results share key\nmorphological features—such as elongation, necking, and symmetry breaking—with those\nobserved in mitosis (e.g., mitotic spindle-induced nuclear deformation; refs. 46 and 47) and\nvesicles under external flow (refs. 48 and 49). These mechanisms are analogous to the repulsive\nforces generated by filament loops in our model. Such analogies suggest that the mechanical\nmechanisms underlying our model are relevant to a broad range of biological and synthetic\nsystems, even though a one-to-one experimental counterpart for our specific setup has not yet\nbeen established. These biological and biomimetic analogies are discussed in the revised\nmanuscript as part of a broader expansion motivated by Comments 3 and 4 (Discussion section,\npages 17 and 18).\n(2) Bending rigidity of the membrane\nIn our continuum modeling, the membrane bending rigidity is set to 20 k T. In the molecular\nB\ndynamics simulations, the membrane bending rigidity is governed by the underlying inter-\nparticle interaction parameters. Specifically, the solvent-free model yields a membrane\nbending rigidity of 26 k T, while the explicit solvent model gives a value of 45 k T (top of page\nB B\nS4 in Supplementary Information). These values lie within the experimentally reported range\nfor lipid bilayers (ref. 1). Since all numerical results are presented in a dimensionless form,\nthey can be readily mapped to physical systems by rescaling with the appropriate bending\nrigidity.\n(3) Experimental realization and proposed techniques\nTo experimentally realize the predicted shape transformations, we suggest designing\nstimulus-responsive microscale filaments that can exert tunable long-range interactions under\nconfinement. Recent advances in soft matter and nanomanufacturing provide a promising\nfoundation. For example, Bharti et al. (Nat. Mater. 2015; ref. 44) demonstrated the magnetic\nassembly of ultraflexible microfilaments composed of lipid-coated magnetic nanoparticles\nheld together by nanocapillary forces. These filaments exhibit extreme flexibility, structural\nreversibility, and the ability to form closed loops or figure-eight shapes under external\nmagnetic fields. Beyond self-assembly, complementary fabrication approaches include\nembedding superparamagnetic nanoparticles into electrospun or wet-spun polymer\nnanofibers, and incorporating photoresponsive moieties, gold nanorods, or liquid crystal\ndroplets into filament backbones using coaxial electrospinning or direct ink writing. These\nmethods allow for programmable stiffness, shape adaptability, and external field sensitivity,\nmaking them well-suited to recapitulate the confinement-driven morphologies predicted by\nour model.\nBuilding on these advances, we propose a magnetically responsive filament system as a\nconcrete route to emulate the nonlocal repulsion in our simulations. In particular, we consider\na pair of confined filaments composed of one paramagnetic and one diamagnetic component.\nWhen placed in an external magnetic field, the paramagnetic filament acquires a magnetic\nmoment aligned with the field, while the diamagnetic filament develops an anti-aligned\nmoment. This antiparallel dipole configuration gives rise to a net repulsive dipole–dipole\ninteraction when the dipoles are aligned along the field direction (Fig. S14). Such filaments\ncan be fabricated using established colloidal techniques—for example, by linking\nsuperparamagnetic beads into flexible chains for the paramagnetic component, and using\nweakly diamagnetic materials (such as carbon-based or polymer-coated particles) for the\nopposing filament.\nTo demonstrate this idea, we have extended our modeling to include magnetically induced\ninteractions between confined filaments. Specifically, we considered a system composed of one\nparamagnetic and one diamagnetic filament, whose opposing magnetic moments in an\nexternal field give rise to a net repulsive dipole–dipole interaction (Supplementary Note 5 and\nFig. S15). The resulting simulations show that this magnetically induced repulsion\nqualitatively reproduces a similar spectrum of vesicle deformation—such as elongation,\nflattening, and neck formation—as those observed in the electrostatic case. These findings\nsuggest that magnetic contrast between filaments can serve as a physically realizable surrogate\nfor the nonlocal repulsion used in our theoretical model.\nWe believe this hybrid magnetic system offers a feasible and tunable experimental platform\nfor validating our theoretical predictions, and we are actively exploring potential collaborative\nefforts to test these concepts in future in vitro experiments. The corresponding discussion on\nexperimental realization has been added on pages 15 and 16 of the revised manuscript.\n“5. This study focuses on equilibrium morphologies, but many biological vesicle-filament\ninteractions involve dynamic remodeling. How would the findings change if active processes such\nas filament polymerization or motor-protein forces were included?”\nOur response: We thank the reviewer for this insightful question regarding the role of active\nprocesses in vesicle–filament systems. We fully agree that many biological systems,\nparticularly those involving cytoskeletal filaments, exhibit dynamic remodeling driven by\npolymerization kinetics, motor-protein forces, and ATP-dependent activity. These active\nmechanisms are essential for functions such as cell shape adaptation and motility.\nOur study focuses on equilibrium morphologies and aims to establish a mechanically\ngrounded baseline that isolates the effect of long-range filament–filament interactions under\nconfinement. Although our model does not include active driving, we find that internal\nmechanical stresses—arising purely from filament interaction and elasticity—can induce\nrobust shape transformations, symmetry breaking, and filament reorganization. These\nfeatures qualitatively resemble those observed in active systems and highlight the central role\nof mechanical interactions in shaping membrane-bound structures.\nFrom a biophysical perspective, we view this model as a minimal mechanical prototype,\nwhich captures how internal forces—whether generated actively (e.g., by motors or\npolymerization) or passively (e.g., through confinement and repulsion)—can couple to\nmembrane elasticity and geometry to produce emergent morphologies. The framework thus\nprovides a tractable starting point to dissect physical principles that may also operate in more\ncomplex, nonequilibrium biological contexts.\nWe fully acknowledge that including explicit active components, such as time-dependent\nfilament growth, force-generating motor activity, or chemically driven filament turnover,\nwould significantly enrich the model and alter the morphology landscape. In future extensions,\nthese effects could be incorporated via active stress terms in the energy functional, filament-\nlength evolution governed by polymerization kinetics, or effective force dipoles representing\nmotor-filament interactions. Such additions would enable exploration of nonequilibrium\nshape fluctuations, remodeling pathways, and steady-state cycles, thereby bringing the model\ncloser to living systems.\nIn response to the reviewer’s comment, we have added new discussion of the role of active\nprocesses in the revised Discussion section (Toward dynamic and nonequilibrium remodeling,\npages 18 and 19).\n“6. C2h-to-D2h and D∞h-to-C2h transitions are claimed as continuous and discontinuous\ntransitions, respectively. The evidence is not sufficient. According to thermodynamics, the criterions\nfor discontinuous or continuous transition include latent heat, phase coexistence, and the divergence\nof correlation length etc. However, no solid evidence is shown in this study.”\nOur response: We thank the reviewer for this insightful comment and for emphasizing the\nthermodynamic criteria such as latent heat, phase coexistence, and divergence of correlation\nlength for distinguishing between continuous and discontinuous phase transitions. These\ncriteria are indeed central in statistical thermodynamic systems.\nHowever, we would like to clarify that our study is based on a deterministic energy\nminimization framework, which does not incorporate temperature, thermal noise, or entropy\ncontributions. Consequently, classical thermodynamic signatures such as latent heat or\ndiverging correlation length are not defined within our model.\nIn this context, our classification of the transitions as “continuous” or “discontinuous” is\nbased on geometric and energetic features observed during the minimization process.\nSpecifically, we examine whether the system exhibits smooth or abrupt changes in symmetry\nand morphology as control parameters are varied, and whether the energy landscape shows\nsmooth continuity or kinks. This mechanics-based approach is commonly used in nonlinear\nstability analyses of soft materials, where morphological bifurcations and non-smooth energy\nresponses serve as reliable indicators of mechanically induced transitions. For example, the\nD -to-C transition involves an abrupt morphological change, manifested as a sudden\n∞h 2h\nreorientation of the filament relative to the vesicle’s long axis, accompanied by a non-smooth\nenergy response (Figs. 5, S11, and S12). In contrast, the C -to-D transition proceeds through\n2h 2h\na gradual evolution of the filament configuration and vesicle shape, with no abrupt symmetry\nbreaking or energy discontinuity, indicative of a continuous transition (Figs. 5 and S12a)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have thoroughly addressed my concerns, showing how screening affects their results and providing a detailed discussion of potential experimental realizations. I recommend acceptance of the revised manuscript.", "reviewer_2": "I thank the authors for addressing all concerns and recommend acceptance of the manuscript", "reviewer_3": "The authors have responded adequately to my concerns. I would like to suggest the manuscript to be accecpted."}, "author_response": "Response to Comments of the First Reviewer\n“Reviewer #1:\nThe authors have thoroughly addressed my concerns, showing how screening affects their results\nand providing a detailed discussion of potential experimental realizations. I recommend acceptance\nof the revised manuscript.”\nOur response: We thank the reviewer for the positive evaluation and recommendation to\naccept the manuscript. We appreciate the reviewer’s time and valuable input on our work.\nResponse to Comments of the Second Reviewer\n“Reviewer #2:\nI thank the authors for addressing all concerns and recommend acceptance of the manuscript.”\nOur response: We thank the reviewer for the positive evaluation and recommendation to\naccept the manuscript. We appreciate the reviewer’s time and valuable input on our work.\nResponse to Comments of the Third Reviewer\n“Reviewer #3:\nThe authors have responded adequately to my concerns. I would like to suggest the manuscript to\nbe accepted.”\nOur response: We thank the reviewer for the positive evaluation and recommendation to\naccept the manuscript. We appreciate the reviewer’s time and valuable input on our work."}]

Decision letter