Forced transport of deformable containers through narrow constrictions

We study, numerically and analytically, the forced transport of deformable containers through a narrow constriction. Our central aim is to quantify the competition between the constriction geometry and the active forcing, regulating whether and at which speed a container may pass through the constriction and under what conditions it gets stuck. We focus, in particular, on the interrelation between the force that propels the container and the radius of the channel, as these are the external variables that may be directly controlled in both artificial and physiological settings. We present lattice Boltzmann simulations that elucidate in detail the various phases of translocation and present simplified analytical models that treat two limiting types of these membrane containers: deformational energy dominated by the bending or stretching contribution. In either case we find excellent agreement with the full simulations, and our results reveal that not only the radius but also the length of the constriction determines whether or not the container will pass.

Figure 1

(a) Model system for the transport of containers through a narrow constriction of length $L$ and radius $d$. (b) A typical time sequence of the transport of a deformable container through a relatively short and narrow constriction. Position of the container's center of mass $x$ as function of time $t$ for (c) varying the relative radius of the constriction $d/R_0$ (solid line, Pass, $d/R_0=0.92–;0.68$ and dashed line; Stuck, $d/R_0=0.67$) and a fixed applied body force $F_d=1.5\times {10}^{-4}$ and (d) where we vary the applied force $F_m$ (dashed line: Stuck, $F_m=2\times {10}^{-5}$ and $6.5\times {10}^{-5}$; and solid line: Pass, $F_d=7\times {10}^{-5}-2\times {10}^{-4}$) and fix $d/R_0=0.7$. In (e) we fix the applied body force $F_d=1.5\times {10}^{-4}$, and in (f) we fix $d/R_0=0.7$ and we calculate the minimal velocity of the particle during transportation. For the containers that remained stuck $v_{\min }=0$. In (c), (d), (e), and (f), the length of the constriction is $L=100$, and the system size is $b=208$ and $a=52$ lattice units [see panel (a)].

Figure 2

(a) Phase diagram indicating whether the container passes the constriction as function of the relative size of the constriction $d/R_0$ and the applied body force $F_m$. (b) Phase diagram as function of the neck length $L$ and the relative size of the constriction $d/R_0$, where $F_m=1.5\times {10}^{-4}$.

Figure 3

(a) Translocation sequence for the passage of a container through a narrow constriction. Stage I: free capsule in solution (the reference configuration), stage II: partial entry of the capsule into the constriction, stage ${\mathrm{III}}_a$: intermediate stage for short channels or large containers, stage ${\mathrm{III}}_b$: intermediate stage for long channels or small containers, stage IV: partial exit out of the constriction. Stretching and bending energy $E_A$ (d) and $E_B$ (e), respectively, as function of $x$ for various lengths of the constriction. Decreasing the length of the neck strongly diminishes the height of the energy barrier the container has to overcome. For (b) and (c) we assumed $R_0=1\mu \mathrm{m}$, for (b) $d=0.2\mu \mathrm{m}$ and ${\kappa }_A=1\mathrm{J}{\mathrm{m}}^{-2}$, and for (c) ${\kappa }_B=2\times {10}^{-19}$  J and $d=0.3\mu \mathrm{m}$. In (d) and (e) we show the height of the energy barrier as function of the radius of the constriction $d$ relative to the initial radius of the vesicle $R_0$, where for (d) ${\kappa }_A=1\mathrm{J}{\mathrm{m}}^{-2}$ and for (e) ${\kappa }_B=2\times {10}^{-19}\mathrm{J}$. If we decrease the length of the constriction we find that the height of the barrier strongly diminishes. For an infinite neck length, the height of the barrier strongly increases for decreasing $d/R_0$. For finite neck lengths, however, there is a decrease at low $d/R_0$ in the stretching energy (d).

Figure 4

The minimal velocity during the entrance of the constriction $v_{\min }$ as function of (a) the relative radius of the constriction $d/R_0$ (${\kappa }_A={10}^5$ $\mathrm{J}{\mathrm{m}}^{-2}$, $F_m=2$ $\mathrm{pN}$) and (b) as function of the applied force $F_m$ (${\kappa }_A={10}^5\mathrm{J}{\mathrm{m}}^{-2}$, $d/R_0=0.4$). Phase diagram indicating if the stretch-dominated container Passes or gets Stuck inside the constriction as function of (c) $F_m$ and $d/R_0$ (${\kappa }_A={10}^5\mathrm{J}{\mathrm{m}}^{-2}$) and (d) ${\kappa }_A$ and $d/R_0$ ($F_m=4$ pN). $v_{\min }$ as function of (e) $d/R_0$ (${\kappa }_B=4\times {10}^{-19}\mathrm{J}$ and $F_m=10$ pN) and (f) $F_m$ (${\kappa }_B=8\times {10}^{-19}\mathrm{J}$ and $d/R_0=0.4$). Phase diagram indicating if a bend-dominated container Passes or gets Stuck as function of (g) $F_m$ and $d/R_0$ (${\kappa }_B=4\times {10}^{-19}\mathrm{J}$) and (h) ${\kappa }_B$ and $d/R_0$ ($F_m=10$ pN). These figures show a very generic phase behavior for both the stretch- and bend-dominated containers.