Design principles for transporting vesicles with enclosed active particles^(a)

Our goal is to characterize the motility of the vesicle. We begin by focusing on its center of mass, $\bar {r}(t)$. We find that the dynamics of $\bar {r}(t)$ is well described by a persistent random walk with $\langle |\bar {r}(t)-\bar {r}(t+\Delta t)|^{2}\rangle =D[\Delta t-\lambda +\lambda e^{-\Delta t/\lambda }]$, where λ is the persistence time and D is the diffusion coefficient describing the vesicle motions at times $t\gg \lambda$ (fig. 1(a)). Further, we measure ${\bar {F}_{\text {net}}}$, the net active force on the vesicle, exerted by the enclosed rods that are in contact with it and ${\bar {v}_{\text {com}}}$, the center of mass velocity. We find that both quantities are distributed normally (fig. 1(b)) and exponentially correlated in time (fig. 1(c)), while being proportional to each other instantaneously (fig. 1(d)). While it is suggestive to map the emergent persistent random walk of our vesicle to the well-studied Active Brownian Particle model [12,48], we find that the dynamics is more complex due to timescales arising from reorganization of the enclosed rods. Therefore, as we vary the physical properties of the vesicle building blocks, we characterize its motility by the diffusion coefficient D, and the correlation times of the velocity and the force, ${\lambda _{\text {v}}}$ and ${\lambda _{\text {f}}}$, respectively (further information is given in the Supplementary Material, Supplementarymaterial.pdf (SM), sect. IIF).

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To highlight the key findings of this study, let us consider the physics we know from prior studies on self-propelled particles at curved walls [32,35–37,37,38,49,50]. Self-propelled particles cluster at walls [26,27,29,30,51], and tend to accumulate in regions of high curvature [32,49,52–54]. This can lead to spontaneous motion of rigid containers [55–57]. Since our vesicle is deformable in the present work, we expect curvature to increase in regions of accumulation and thereby lead to recruitment of more particles into the cap [34–38,40,41,54]. This feedback mechanism should be greater for vesicles with lower bending rigidity, since the curvature amplification will be larger for a given active force magnitude and local particle density. Thus, we might naively expect softer vesicles to have higher motility compared to stiff ones, since they provide a pathway to the formation of a few larger caps rather than a more uniform distribution of particles along the boundary.

Our first observation is that this simple argument is not the full story. Figure 2 characterizes the vesicle motility for fixed ${N_{\text {rods}}}=30$ rods and activity ${f_{\text {a}}}=3{k_{\text {B}}T}/\sigma$ as a function of the rod aspect ratio $\ell /\sigma$ and vesicle stiffness ${\kappa _{\text {B}}}$. We identify three distinct behaviors: i) for $\ell /\sigma \leq3$, the motility is independent of vesicle bending stiffness, ii) for $3<\ell /\sigma \leq5$, the motility increases as the vesicle softens, and iii) for longer rods a strong non-monotonic dependence on stiffness emerges.

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We can understand these changes in phenomenology as a function of aspect ratio by considering the limiting case of $\ell /\sigma =1$, i.e., spheres, in an infinitely stiff, i.e., rigid, circular vesicle. Supposing that the system was initialized with isotropically distributed self-propulsion forces, as all the simulations in this study are, we expect a uniform distribution of particle orientations on the vesicle boundary. Therefore, the net force on the center of mass is small and its fluctuations are correlated on the same timescale as the bare correlations of isolated active particles. Hence, the enclosed active particles will not enhance the vesicle motility to any significant degree. When we move away from the rigid limit, deformations of the vesicle do induce larger caps and appreciable density variation along the boundary. But, as $\ell /\sigma \to1$, neither the boundary nor the inter-particle interactions can exert significant torques to realign the active forces. Thus, the net force on the vesicle remains small and enhancement of motility negligible. Given this picture, we can understand the low aspect ratio results in fig. 2(a) as the generalization of this asymptotic case of spherical particles.

Now let us switch focus to the stiffness dependence of motility for longer rods. Again, it is useful to first build a heuristic based on known phenomenology [58]. Let us consider long rods accumulated at a rigid circular boundary. Suppose the rods do not interact with each other, i.e., in the dilute limit, they would slide along the boundary until they become tangential to it [11,26,30,32,49]. But, when the rods collide at the boundary, the inter-particle interactions will lead the rods to align such that they form polar caps that are normal to the boundary [11,26,33]. These polar caps push on the boundary, thus inducing and amplifying local curvature fluctuations when the boundary is no longer rigid. In turn, the rods' forward motion couples to the vesicle curvature to result in an effective attraction between neighboring rods. For softer vesicles, the induced curvature and corresponding effective rod-rod attraction increases, thus resulting in larger and longer-lived polar caps. For moderate aspect-ratio rods, this effect leads to an increase in vesicle motility with decreasing vesicle rigidity (most easily seen in fig. 2(a) for $\ell /\sigma =4.5$). However, for longer rods, this phenomenology is overwhelmed by the emergence of an optimal stiffness, around which there is a huge enhancement of motility above the basic trend (see fig. 2(b), (d)).

To elucidate the mechanism underlying the dramatic motility enhancement for optimal vesicle stiffness, we now examine the organization of active rods within the vesicle (fig. 3). We find that the enhanced motility states are characterized by the presence of a single (fig. 3(a)) long-lived (SM, fig. 5) polar cluster of rods. Hence, essentially all encapsulated rods point in the same direction, leading to a large net active force on the vesicle ${\bar {F}_{\text {net}}} \thicksim {f_{\text {a}}} {N_{\text {rods}}}$ and correspondingly a large persistence length for the vesicle's motion. For vesicles that are softer than the optimal stiffness, the rods organize into two or more polar caps. Through the combination of the capss polar motion and the consequent local curvature of the membrane, there is an effective repulsion between caps driving them into a steady-state configuration in which their active forces tend to cancel (see fig. 3(c), fig. 5(a), and SM, sect. I). Consequently, the vesicle exhibits a much smaller persistence length than for the 1-cap state. For stiffer-than-optimal vesicles, the rods can only weakly deform the vesicle boundary. Since local curvature of the vesicle is essential to stabilize the polar cap (see below), only transient caps form. Therefore, the direction of ${\bar {F}_{\text {net}}}$ fluctuates rapidly and the vesicle motion is characterized by a small persistence length. These observations suggest that both the aligning interactions among the rods due to their self-propulsion and the vesicle deformability are crucial to the emergence of the enhanced motility states. As evidence of the importance of direct rod-rod alignment interactions, we observe large net active forces on the vesicle only for large aspect-ratio rods (fig. 3(b)).

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Now, we seek to quantify the emergence of a single cap in terms of the physical parameters of the building blocks of our motile vesicle. As we have shown in previous work [33], long-lived caps can be fruitfully described as self-limited structures whose size can be accurately captured using the concepts of self-assembly. We lay out below a physical description of the model that predicts the number and sizes of caps as a function of parameters, and we use these estimates to predict the optimal stiffness at which the huge enhancement in motility occurs. The mathematical details are given in the SM, sect. I.

We know from previous work that self-propulsion together with the presence of the wall drives rods to align with each other, perpendicular to the wall, and to maximize the overlap along their length with their neighbors. That is, the rods tend to form smectic layers at the wall [58]. We can capture this phenomenology through an effective "energy" of the form $U_{\text {active}}+U_{\text {interfacial}}$, where $U_{\text {active}}=-C\sigma {f_{\text {a}}} n_{\text {r}}$, an attractive interaction that scales with the activity and number of rods in the cap $n_{\text {r}}$, and $U_{\text {interfacial}}=-2\,\gamma$ is a surface tension that accounts for the absence of neighbors at either edge of a cap. The surface tension encodes the tendency for caps to grow. While this captures the phenomenology at a flat wall, the curvature of the vesicle "opposes" the growth of the cap by forcing rods within a cap to shear relative to their neighbors, thus preventing the perfect overlap that flat walls would allow. We can capture this effect through an energy $U_{\text {shear}}=-2k_{\text {shear}}\log (\cos \theta )$, where θ is the angular extent of the cap (see fig. 4(b)) and the constant $k_{\text {shear}}$ also scales with activity ${f_{\text {a}}}$, consistent with the attractive interactions that lead to the clustering.

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For a given cap size, the angular extent of a cap depends on the local radius of curvature of the vesicle, which is determined by a balance of active force of the rods and the elastic response of the vesicle. We describe this phenomenology through an energy $U_{\text {ves}}=U_{\text {bend}}+U_{\text {f}}$, where $U_{\text {bend}}$ is the standard Helfrich free energy [59–61] and $U_{\text {f}}=-{f_{\text {a}}} n_{\text {r}}l$ is the work done by the active force (see SM, sect. I). Putting all this together, the number and size of caps, and correspondingly the steady-state geometry of the vesicle, can be described by finding the minimum of a free energy of the form $U({\kappa _{\text {B}}},{f_{\text {a}}},{N_{\text {rods}}})=U_{\text {rods}}+U_{\text {ves}}$, where $U_{\text {rods}}=U_{\text {active}}+U_{\text {interfacial}}+U_{\text {shear}}$, described above.

In particular, the analysis shows that, starting from the floppy vesicle limit ${\kappa _{\text {B}}}\to0$, the number of stable caps decreases with increasing bending modulus. By calculating when the system transitions from a two-cap state to a single cap state, corresponding to the highly motile states observed in the simulations, we can estimate that the optimal stiffness scales as $\kappa _{\text {B}}^*/{f_{\text {a}}}\sim N_{\text {rods}}^{5}$ (see eq. (20) in sect. ID of the SM). The predicted scaling with both activity and number of enclosed rods shows good agreement with our simulation results (fig. 4(a)), although the accessible range of ${N_{\text {rods}}}$ within the vesicle size that we focus on $({R_{\text {ves}}}=80\,\sigma )$ is insufficient to rigorously test the scaling exponent.

The analysis predicts that this trend continues until a threshold value of the bending modulus $\kappa _{\text {B}}^{\text {max}}\sim {R_{\text {ves}}^{2}f_{\text {a}}}$, above which the active force is insufficient to deform the membrane. Because membrane deformation is essential to stabilize long-lived polar caps, we expect only transient caps for stiffer vesicles. Furthermore, equating this limit with the transition value for the single cap state, i.e., $\kappa _{\text {B}}^*={\kappa _{\text {B}}^{\text {max}}}$, identifies a maximum number of enclosed rods ${N_{\text {rods}}^{\text {max}}}\sim \left ({R_{\text {ves}}} \ell \right )^{2/5}$ above which the system will transition directly from multiple-cap states to transient caps with increasing bending modulus. Thus, for ${N_{\text {rods}}} > {N_{\text {rods}}^{\text {max}}}$ the system will not exhibit the single-cap high-motility state for any parameter values. For our vesicle size ${R_{\text {ves}}}=80\,\sigma$ and the range of aspect-ratios that lead to polar caps $(\ell /\sigma \ge6)$ this estimates ${N_{\text {rods}}^{\text {max}}}$ is order 10 to within a scaling constant, which is consistent with the simulation results that we do not observe the single-cap state for ${N_{\text {rods}}}\gtrsim60$.

Since the number of caps is closely linked to the steady-state geometry of the vesicle configuration (see fig. 5), further analysis of this theoretical model can be leveraged to design shapes of 2D active vesicles (similar to [33,54]). We defer this to future work, in order to keep the focus of this article on motility of the vesicle.

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The scaling relationship above provides a quantitative design principle for building highly motile vesicles. But one important limitation it identifies is that motility enhancement through the formation of the single cap state only applies in the dilute limit. This corresponds to ${N_{\text {rods}}}<{N_{\text {rods}}^{\text {max}}} \approx60$ for the vesicle size that we focus on ${R_{\text {ves}}}=80\,\sigma$. Above this limit, there are too many rods to form a state in which there is a single polar-aligned cap. Consequently, the simple picture of three distinct dynamical regimes for large aspect-ratio rods $(\ell >5\,\sigma )$ breaks down (fig. 5).

Figure 5(a) is a visual representation of the structures formed by the enclosed rods as the number of rods increases. In the intermediate regime ($60\,\lesssim {N_{\text {rods}}} \lesssim100$ for ${R_{\text {ves}}}=80\,\sigma$), the dominant steady-state configurations gradually shift from 2-cap to 3-cap states with increasing ${N_{\text {rods}}}$ in floppy vesicles, 1-cap to 2-cap states for intermediate stiffness, and one or more transient caps at high vesicle stiffness. However, even though self-limited caps continue to form in softer vesicles, the caps lack perfect polar alignment and tend to become multilayered. We speculate that this phenomenology reflects the onset of motility induced phase separation (MIPS) related behaviors at these densities [58,62–64], which unevenly and transiently enhance clustering. At higher densities (${N_{\text {rods}}}>100$ for ${R_{\text {ves}}}=80\,\sigma$ and ${f_{\text {a}}}=3{k_{\text {B}}T}/\sigma$), the MIPS dominates the organization of the self-propelled rods at the boundary. In this regime, the rods tend to form aggregates comprising multiple sub-clusters with different orientations. For example, in floppy vesicles, the dominant rod organization corresponds to three aggregates sitting at three vertices of a triangular configuration. As noted above, the combination of rod-propulsion and membrane-curvature-mediated interactions tends to drive caps away from each other, stabilizing the triangular arrangement. However, because the rods within a given aggregate are not all aligned, aggregates are transient, tending to break apart and merge with other aggregates. With increasing ${\kappa _{\text {B}}}$ the mean number of caps decreases, but the caps remain transient and thus do not lead to the highly motile vesicle dynamics characterized by a long persistence length. Furthermore, even in the 1-cap states at these higher rod densities, there are additional rods forming transient caps elsewhere on the boundary, which have orientations for which the propulsion forces partly cancel those of the rods in the cap.

Figure 5(b) characterizes the motility of the vesicle for different numbers of enclosed rods. For the intermediate densities, the diffusion coefficient as a function of vesicle stiffness shows similar trends as seen in fig. 2(b), with motility increasing with ${\kappa _{\text {B}}}$ for floppy vesicles and the value of optimal stiffness $\kappa _{\text {B}}^*$ shifting toward larger values with increasing number of rods. This is consistent with the fact that the number of caps decreases with stiffness, thus increasing the average magnitude of the net active force. However, the motility enhancement is significantly smaller than for ${N_{\text {rods}}}<{N_{\text {rods}}^{\text {max}}}$, due to the imperfect alignment of the caps. At higher densities $({N_{\text {rods}}}>100)$, the motility becomes largely independent of stiffness. This reflects the fact that, when the MIPS phenomenology dominates, the caps lack polar order and are transient. Thus, the net active force lacks the large magnitude and long-lived orientation that drive directed vesicle motion.