Dynamics of multicomponent vesicles in a viscous fluid

Abstract

We develop and investigate numerically a thermodynamically consistent model of two-dimensional multicomponent vesicles in an incompressible viscous fluid. The model is derived using an energy variation approach that accounts for different lipid surface phases, the excess energy (line energy) associated with surface phase domain boundaries, bending energy, spontaneous curvature, local inextensibility and fluid flow via the Stokes equations. The equations are high-order (fourth order) nonlinear and nonlocal due to incompressibil-ity of the fluid and the local inextensibility of the vesicle membrane. To solve the equations numerically, we develop a nonstiff, pseudo-spectral boundary integral method that relies on an analysis of the equations at small scales. The algorithm is closely related to that developed very recently by Veerapaneni et al. [81] for homogeneous vesicles although we use a different and more efficient time stepping algorithm and a reformulation of the inextensibility equation. We present simulations of multicomponent vesicles in an initially quiescent fluid and investigate the effect of varying the average surface concentration of an initially unstable mixture of lipid phases. The phases then redistribute and alter the morphology of the vesicle and its dynamics. When an applied shear is introduced, an initially elliptical vesicle tank-treads and attains a steady shape and surface phase distribution. A sufficiently elongated vesicle tumbles and the presence of different surface phases with different bending stiffnesses and spontaneous curvatures yields a complex evolution of the vesicle morphology as the vesicle bends in regions where the bending stiffness and spontaneous curvature are small.

Fig. 1

Comparison of the Fourier spectra |r̂(k, T) for an explicit Adams–Bashforth method (a and c) and our semi-implicit non-stiff algorithm (b and d) for N grid points, time step Δt and final time T as indicated.

Fig. 2

Multicomponent vesicle evolution with a 50–50 mixture of lipid phases: ū = 0.5. (a) The evolution of the total energy E^M. Insets: vesicle morphologies and surface phase concentration u at indicated times (color online: blue and red correspond to the u = 0 and u = 1 phases, respectively). (b) The energy components: the line energy E^T and the bending energy E^b. (c) The normal velocity.

Fig. 3

Multicomponent vesicle evolution with a 50–50 mixture of lipid phases: ū = 0.5 with fixed ∊ = 0.1 and varying Pe = ∊ (dash), 1 (dash dot) and 1/∊ (solid). (a) The surface phase concentration u at indicated times. (b) The energy components: the line energy E^T and the bending energy E^b.

Fig. 4

Multicomponent vesicle evolution with a 70–30 mixture of lipid phases: ū = 0.7 with ∊ = 0.1 and Pe = 1. (a) The evolution of the total energy E^M. Insets: vesicle morphologies and surface phase concentration u at indicated times (color online: blue and red correspond to the u = 0 and u = 1 phases, respectively). (b) The energy components: the line energy E^T and the bending energy E^b. (c) The normal velocity. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5

Multicomponent vesicle evolution with a 80–20 mixture of lipid phases: ū = 0.8 with ∊ = 0.1 and Pe = 1. (a) The evolution of the total energy E^M. Insets: vesicle morphologies and surface phase concentration u at indicated times (color online: blue and red correspond to the u = 0 and u = 1 phases, respectively). (b) The energy components: the line energy E^T and the bending energy E^b. (c) The normal velocity. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6

Multicomponent vesicle evolution with a 85–15 mixture of lipid phases: ū = 0.85 with ∊ = 0.1 and Pe = 1. (a) The evolution of the total energy E^M. Insets: vesicle morphologies and surface phase concentration u at indicated times (color online: blue and red correspond to the u = 0 and u = 1 phases, respectively). (b) The energy components: the line energy E^T and the bending energy E^b. (c) The normal velocity. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 7

Multicomponent vesicle evolution with a 30–70 mixture of lipid phases: ū = 0.3 with ∊ = 0.1 and Pe = 1. (a) The evolution of the total energy E^M. Insets: vesicle morphologies and surface phase concentration u at indicated times (color online: blue and red correspond to the u = 0 and u = 1 phases, respectively). (b) The energy components: the line energy E^T and the bending energy E^b. (c) The normal velocity. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 8

Multicomponent vesicle evolution with a 30–70 mixture of lipid phases: ū = 0.3 with ∈ = 0.05 and Pe = 1. (a) The evolution of the total energy E^M. Insets: vesicle morphologies and surface phase concentration u at indicated times. (color online: blue and red correspond to the u = 0 and u = 1 phases, respectively). (b) The energy components: the line energy E^T and the bending energy E^b. (c) The normal velocity. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 9

Multicomponent vesicle tank-treading under an applied shear with a 30–70 mixture of lipid phases: ū = 0.3 with Δ = 1.7244, ∈ = 0.1 and Pe = 1. (a) The evolution of the vesicle at the indicated times (color online: blue and red correspond to the u = 0 and u = 1 phases, respectively). (b) The concentration u at the corresponding times. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 10

The normal (a) and tangential (b) velocities for the tank-treading vesicle shown in Fig. 9.

Fig. 11

A multicomponent vesicle tumbling under an applied shear with a 30–70 mixture of lipid phases: ū = 0.3 with Δ = 6.5682, ∊ = 0.1 and Pe = 1. (a) The evolution of the vesicle at the indicated times (color online: blue and red correspond to the u = 0 and u = 1 phases, respectively). (b) The concentration u at the corresponding times. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 12

The minimum distance of neck region from the vesicle shown in Fig. 11 is shown in (a) with different spatial resolutions and time-step sizes. The solid line indicates N = 512 with Δt = 1 × 10⁻⁵, the dashed line indicates N = 512 with Δt = 0.5 × 10⁻⁵ and dash-dotted line represents N = 1024 with Δt = 1 × 10⁻⁵.

Fig. 13

The normal velocity of the vesicle shown in Fig. 11 is shown in (a). In (b), the evolution of the maximal distance from the vesicle membrane to the origin. In (c), the temporal spacing between the consecutive valleys in the maximal distance is shown.

Fig. 14

A homogeneous membrane corresponding to u = 0 in an applied shear flow. Compare to Fig. 11 showing the evolution of a multicomponent membrane under the same conditions.