Designing Compliant Substrates to Regulate the Motion of Vesicles

By integrating mesoscale models for hydrodynamics and micromechanics, we examine fluid-driven motion of vesicles on compliant surfaces. The vesicles, modeled as fluid-filled elastic shells, represent biological cells or polymeric microcapsules. Focusing on nonspecific interactions between these vesicles and synthetic substrates, we isolate mechanically and topographically patterned surfaces that transmit stop and go instructions, causing the vesicles to halt at specific locations, and with an increase in the flow velocity, to resume moving. For surfaces containing arrays of compliant posts, the substrates also affect the vesicles’ gait, causing them to “crawl,” “walk,” or “jump.” The latter behavior could promote the intermixing of reactants that are encapsulated within the microcapsules. Such control over vesicle dynamics can facilitate various biological assays and fabrication of arrays of mobile microreactors.

Figure 1

Vesicle rolling on a flat, uniform substrate for $\gamma =2\times {10}^{-4}$ and varying stiffness and interaction strength: (a) $E_{\mathrm{s}}/E_{\mathrm{v}}=1$ and $ϵ={10}^{-2}$; (b) $E_{\mathrm{s}}/E_{\mathrm{v}}=1.2\times {10}^{-2}$ and $ϵ={10}^{-2}$; (c) $E_{\mathrm{s}}/E_{\mathrm{v}}=3\times {10}^{-2}$ and $ϵ={10}^{-3}$. The color bar indicates the local strain: $\delta =0.05$ for the vesicle’s shell, while for the substrate $\delta =0.02$, 0.5, and 0.05, for (a), (b), and (c), respectively. (d) Rolling velocity vs $E_{\mathrm{s}}/E_{\mathrm{v}}$ for: $ϵ={10}^{-3}$ (●), $ϵ=2\times {10}^{-3}$ (◆), $ϵ=5\times {10}^{-3}$ (▲), and $ϵ={10}^{-2}$ (■). Positive strain indicates compression.

Figure 2

(a) Snapshot of a vesicle that is stopped along a flat substrate with periodic stiffness variations ($\gamma =6.67\times {10}^{-5}$ and $P/R=1.44$). (b) Similarly, for $P/R=4.32$. (c) Close-up of (a) showing the detailed flow field and the interaction force (the $x$ component is magnified threefold for clarity). The colors are as in Fig. 1 with $\delta =0.05$ and 0.1 for the vesicle’s shell and the substrate, respectively. (d) Summary of the vesicle’s behavior for varying $\gamma$ and $P/R$; crosses are for situations where the vesicles stops, while circles indicate parameters for which it keeps rolling.

Figure 3

Snapshots of vesicle rolling on a substrate with topological patterning for varying pattern period $P$ and shear rate $\gamma$: (a) $P/R=0.576$ and $\gamma ={10}^{-4}$ (crawling regime); (b) $P/R=1.728$ and $\gamma ={10}^{-4}$ (walking regime); (c) $P/R=1.728$ and $\gamma =2\times {10}^{-4}$ (jumping regime). The colors are as in Fig. 1 with $\delta =0.02$ for both the vesicle’s shell and the substrate. Time $t$ is measured from the moment that the vesicle’s center of mass passed the leading edge of a post. (d) Rolling velocity vs position for the system in (a) (solid line), (b) (dashed line), and (c) (dash-dotted line). The length of the domain shown in (a)–(c) is 210 lattice units; for clarity, we show only a limited range in (d).

Figure 4

Average rolling velocity of a vesicle moving along a topographically patterned substrate vs the pattern period $P$, for varying shear rates $\gamma$.