Shape dynamics of nearly spherical, multicomponent vesicles under shear flow

TL;DR

The paper addresses how lipid composition and line tension affect the shape dynamics of nearly spherical vesicles under linear shear. It develops a semi-analytical, small-deformation framework that couples Canham-Helfrich membrane mechanics with a Cahn-Hilliard phase-field on a spherical surface, solved via vector spherical harmonics and Stokes flow. The study delivers reduced-order ODEs for shape modes $f_{lm}$ and concentration modes $q_{lm}$, analyzes regimes with Peclet numbers $Pe \gg 1$ and $Pe \sim O(1)$, and presents phase diagrams predicting swinging, tumbling, tank-treading, phase treading, and breathing motions, with validation against numerical simulations and connections to 2D results. This framework provides experimentalists with quantitative guidance on how membrane composition and line tension influence vesicle deformation in shear, enabling better interpretation of lipid raft dynamics and membrane mechanics in biological contexts.

Abstract

In biology, cells undergo deformations under the action of flow caused by the fluid surrounding them. These flows lead to shape changes and instabilities that have been explored in detail for single component vesicles. However, cell membranes are often multi-component in nature, made up of multiple phospholipids and cholesterol mixtures that give rise to interesting thermodynamics and fluid mechanics. Our work analyses linear flows around a multi-component vesicle using a small-deformation theory based on vector and scalar spherical harmonics. We set up the problem by laying out the governing momentum equations and the traction {balance } arising from the phase separation and bending. These equations are solved along with a Cahn-Hilliard equation that governs the coarsening dynamics of the phospholipid-cholesterol mixture. We provide a detailed analysis of the vesicle dynamics (e.g., tumbling, breathing, tank-treading, swinging, and phase treading) in two regimes -- when flow is faster than coarsening dynamics (Peclet number $Pe \gg 1$) and when the two time scales are comparable ($Pe \sim O(1)$) -- and provide a discussion on {when these behaviours occur}. The analysis aims to provide an experimentalist with important insights pertaining to the phase separation dynamics and their effect on the deformation dynamics of a vesicle.

Figures (13)

• Figure 1: Schematic for a Newtonian fluid enclosed by a spherical multicomponent lipid bilayer separating the fluid from a surrounding Newtonian fluid. The inset figure shows a zoomed-in version of the lipid bilayer and its properties.
• Figure 2: Error convergence plots for $Cn=0.5$. The parameters for the simulations are $\alpha=1,\beta=0.1,\Delta=0.01,\chi=0.5$ The $y-$axis represents the ratio of the difference in the order parameter over the vesicle surface divided by the average magnitude of the order parameter of the case with most modes $l_{max} = 11$
• Figure 3: Comparison of vesicle shape profiles against single component results (VG(2007)-Vlahovska2007). The parameters are $Cn=0,\tilde{a}=0,\tilde{b}=0,\alpha=0,\beta=0,\lambda=20,\chi=0.6,\Delta=0.2$. The initial condition is $f'_{22}(0)=-\sqrt{\frac{2\Delta\pi}{15}},f"_{22}(0)=-\sqrt{\frac{2\Delta\pi}{15}},f_{20}(0)=-\sqrt{\frac{2\Delta\pi}{15}}$
• Figure 4: Swinging visualization. The parameters are $Pe=10000,a=-1,b=1,Cn=0.5,\alpha=1,\beta=0.5,\Delta=0.1,\chi=0.5$. The blue phase represents the softer phase and the yellow phase represents the stiffer phase. See supplementary video for visualization.
• Figure 5: Phase diagram for $Pe \to \infty$ limit at $\Delta=0.0001$ and $\lambda = 2$.