Hydrodynamics of a semipermeable inextensible membrane under flow and confinement

Lipid bilayer membranes have a native (albeit small) permeability for water molecules. Under an external load, provided that the bilayer structure stays intact and does not suffer from poration or rupture, a lipid membrane deforms and its water influx/efflux is often assumed negligible in the absence of osmolarity. In this work we use boundary integral simulations to investigate the effects of water permeability on the hydrodynamics of an inextensible membrane under a mechanical load, such as the viscous stress from an external flow deforming an inextensible membrane in free space or pushing it through a confinement. Incorporating the membrane permeability into the framework of Helfrich free energy for an inextensible, elastic membrane (a vesicle), we illustrate that, in the absence of an osmotic stress gradient, the semipermeable vesicle is affected by water influx/efflux over a sufficiently long time or under a strong confinement. Our simulations quantify the conditions for water permeation to be negligible in terms of the timescales, flow strength, and confinement. These results shed light on how microfluidic confinement can be utilized to estimate membrane permeability.

Figure 1

Schematics showing the effects of semipermeability on vesicle hydrodynamics in various configurations. The lipid bilayer membrane (modeled as an interface $\gamma$ of zero thickness and an outward normal $\mathbf{n}$) is permeable to water molecules, depending on the mechanical normal stress balance on the membrane. The slipper in (a) is the steady-state shape of a semipermeable vesicle in a Poiseuille flow.

Figure 2

Phase diagram of an impermeable vesicle with various reduced areas submerged in a Poiseuille flow with various flow rates. The shapes agree with Kaoui et al. [46] (see Fig. 2).

Figure 3

(a) Snapshots of four semipermeable inextensible membranes with $\beta ={10}^{-8}$ submerged in a fluid with no background flow. Time goes from left to right. The red regions correspond to influx and the blue regions correspond to efflux. In all cases, the vesicle reaches a steady-state circular shape. (b) The reduced area of each of the simulations. The particular snapshots in (a) occur at the marks along the curve. Three dimensioned times are denoted by the dashed lines.

Figure 4

The distribution of membrane tension corresponding to the relaxation dynamics in Fig. 3.

Figure 5

(a) The reduced area of a semipermeable inextensible membrane with $\beta ={10}^{-3}$ initialized with four different reduced areas in a shear flow with two different flow rates. (b) The equilibrium membrane shape with varying flow rates and membrane permeability. The red regions correspond to influx and the blue regions correspond to efflux. The boxed membranes correspond to the simulation results in part (a).

Figure 6

The tension distribution along a semipermeable inextensible membrane in a linear shear flow. The semipermeability constant is $\beta ={10}^{-4}$ for all these cases, and the elastic capillary numbers are (a) ${10}^{-1}$, (b) ${10}^0$, (c), ${10}^1$, and (d) ${10}^2$ as in Fig. 5.

Figure 7

(a) A semipermeable membrane suspended in a Poiseuille flow with maximum velocity $800\mu \mathrm{m}/\mathrm{s}$. The channel width is 12.5 times larger than the vesicle radius. (b) The reduced area of the vesicle as a function of time. The red marks correspond to the vesicle shapes in (a). (c) The equilibrium shape of a semipermeable inextensible membrane submerged in a Poiseuille flow with varying initial reduced areas and flow rates. The red regions correspond to influx and the blue regions correspond to efflux. (d) The steady-state vertical displacement at four different flow velocities.

Figure 8

The equilibrium tension distribution of a semipermeable inextensible membrane with $\beta ={10}^{-3}$ in a parabolic flow (Fig. 7). The maximum flow velocities are (a) $200\mu \mathrm{m}/\mathrm{s}$, (b) $400\mu \mathrm{m}/\mathrm{s}$, (c) $600\mu \mathrm{m}/\mathrm{s}$, and (d) 800 $\mu \mathrm{m}/\mathrm{s}$.

Figure 9

The rate of change of the reduced area decomposed into contributions from membrane tension (blue curves) and membrane bending (red curves). The dashed line is the sum of these two contributions. (a) The quiescent example in Fig. 3 with $\alpha \left(0\right)=0.66$. (b) The quiescent example in Fig. 3 with $\alpha \left(0\right)=0.38$. (c) The planar shear example in Fig. 5 with $\alpha \left(0\right)=0.45$ and ${\mathrm{Ca}}_E={10}^0$. (d) The Poiseuille example in Fig. 7 with $\alpha \left(0\right)=0.60$ and maximum velocity $U=600\mu \mathrm{m}/\mathrm{s}$.

Figure 10

(a) A semipermeable inextensible membrane passing through a closely fit channel similar to the experimental device in Abkarian et al. [52]. (b) A semipermeable inextensible membrane passing through a (three-times) longer closely fit channel than (a). In (a) and (b), red regions correspond to influx and blue regions correspond to efflux. [(c) and (d)] The tension in N/m of the vesicle shapes in (a) and (b). (e) The excess pressure of an impermeable and semipermeable inextensible membrane in the short geometry in (a). (f) The reduced area in the geometries in (a) and (b) as a function of the membranes' center-of-mass. In (e) and (f), the $x$ axis is scaled by $1/L$ where $L$ is half the length of the constriction. (g) The reduced area in the geometries in (a) and (b) as a function of time. The time axis is scaled by $1/t_R$ where $t_R$ is the residency time. In (e) and (f), the dashed lines correspond to the locations of the inlet and outlet. In (g), the dashed lines correspond to the times that the vesicle enters and exits the constriction.

Figure 11

The rate of change of the reduced area due to the membrane tension (blue curves) and membrane bending (red curves). The dashed line is the sum of these two contributions. (a) The short stenosis geometry in Fig. 10. (b) The long stenosis geometry in Fig. 10.

Figure 12

A contracting channel similar to the experimental configuration in Wu et al. [55]. The channel width varies from 18 to $1.8\mu \mathrm{m}$, and it repeats every $45\mu \mathrm{m}$. The red regions correspond to influx and blue regions correspond to efflux [(a)–(c)] The ability for a semipermeable vesicle to pass through the contraction depends on the permeability $\beta$. [(d)–(f)] The ability for a semipermeable inextensible membrane to pass through the contraction depends on the flow rate. [(g)–(i)] The tension distribution and maximum tension in N/m of the vesicle shapes in (a) and (b). [(j)–(l)] The tension distribution and maximum tension in N/m of the vesicle shapes in (d) and (e).

Figure 13

The rate of change of the reduced area due to membrane tension (blue curves) and membrane bending (red curves). The dashed black line is the sum of these two contributions. Only simulations where the vesicle passes through the contraction are decomposed into the contributions due to tension and bending. (a) The low semipermeability coefficient in Fig. 12. The dash-dotted green curve corresponds to the simulation in Fig. 12. (b) The high semipermeability coefficient in Fig. 12. The dash-dotted green curve corresponds to the simulation in Fig. 12.

Figure 14

The reduced area and velocity of a semipermeable inextensible membrane vs. time. The permeability coefficient is $\beta ={10}^{-3}$ and the maximum velocity is $18\mu \mathrm{m}/\mathrm{s}$. (a) A semipermeable inextensible membrane passing through six periods of the contracting geometry. (b) The reduced area with the marks representing shapes in (a). (c) The center-of-mass velocity with the marks representing shapes in (a). In (b) and (c), the lower $x$ axis is the $x$ coordinate of the center-of-mass, and the upper $x$ axis is time. (d) The membrane tension in N/m. (e) The maximum membrane tension.