Ohta-Kawasaki Model Reveals Patterns on Multicomponent Vesicles

TL;DR

The paper addresses how membrane curvature and protein-driven microphase separation jointly shape multicomponent vesicles. It introduces a unified mechanochemical model that couples a phase-field description of membrane geometry with a membrane-bound Ohta–Kawasaki energy for surface protein patterning, solved via a force-balance equation for the membrane and an advection–diffusion–reaction dynamics for the protein density on the moving interface. Key contributions include the membrane localization of the OK energy, a two-field formulation with $\phi$ and $u$, and efficient spectral/numerical methods that reproduce experimentally observed patterns in both 2D and 3D settings, including the influence of the long-range repulsion parameter $\gamma$ and biochemical activity $\alpha$. The framework provides a predictive platform for studying curvature-driven pattern formation and morphogenesis in multicomponent membranes and offers avenues to incorporate adhesion and hydrodynamics for broader biological relevance.

Abstract

We present a new mechanochemical modeling framework to explore the shape deformation and pattern formation in multicomponent vesicle membranes. In this framework, the shape of the membrane is described by an elastic bending model, while phase separation of membrane-bound activator proteins is determined by an Ohta-Kawasaki (OK) model. The coupled dynamics consist of an overdamped force-balanced equation for the membrane geometry and an OK-type advection-reaction-diffusion equation on the deformable membrane. We implement efficient spectral methods to simulate these dynamics in both two- and three-dimensions. Numerical experiments show that the model successfully reproduces a wide range of experimentally observed membrane morphologies \cite{baumgart2003imaging}. Taken together, the framework unifies curvature mechanics, microphase separation, and active forcing, providing new insight into membrane-bounded multicomponent vesicle dynamics and a practical platform for studying multicomponent biomembrane morphology.

Figures (9)

• Figure 4.1: Example illustrating phase separation and membrane deformation with three protein-rich subdomains. First three columns (from left): the evolution of membrane-bound protein domains starting from a random initial distribution under the OK model on a fixed membrane. Next two columns: the subsequent membrane deformation governed by the coupled system. Final column: the corresponding biological experiment in Figure 1(e) of baumgart2003imaging. Parameters are given as: $\lambda_{\mathrm{surf}} = 2$, $\lambda_{\mathrm{line}} = 6$, $\kappa=1$, $\alpha = 1.5$, $u_0 = 0.2$, $\bar{u} = 0.8$, $\gamma = 80$, and $\epsilon_u = 20h_x$.
• Figure 4.2: Effect of long-range repulsive strength $\gamma$ for the phase separation on membrane surface, $\gamma = 60, 90, 125, 155, 190, 210$ for three, four, five, six, seven, and eight protein-rich regions, respectively. Other parameters are given as: $\lambda_{\mathrm{surf}} = 3$, $\lambda_{\mathrm{line}} = 3$, $\kappa=1$, $\alpha = 2$, $u_0 = 0$, $\bar{u} = 0.75$, and $\epsilon_u = 20h_x$.
• Figure 4.3: Four examples with a single protein-rich subdomain on membrane at the dynamical steady state. Top left three: Volumes of the protein-rich and protein-poor domains are equal. Parameter values are taken as: $\lambda_{\mathrm{surf}} = 1$, $\lambda_{\mathrm{line}} = 25$, $\kappa=1$, $\alpha = 0.1$, $u_0 = 9$, $\bar{u} = 0.5$, $\gamma = 5$, and $\epsilon_u = 20h_x$; Top right three: volume of the red region is slightly lower than that of the white region. Parameter values are: $\lambda_{\mathrm{surf}} = 2$, $\lambda_{\mathrm{line}} = 35$, $\kappa=1$, $\alpha = 0.15$, $u_0 = 9$, $\bar{u} = 0.4$, $\gamma = 5$, and $\epsilon_u = 20h_x$; Bottom left three: membrane with a small protein-rich domain. Parameter values are: $\lambda_{\mathrm{surf}} = 1$, $\lambda_{\mathrm{line}} = 15$, $\kappa=1$, $\alpha = 0.4$, $u_0 = 9$, $\bar{u} = 0.2$, $\gamma = 5$, and $\epsilon_u = 20h_x$; Bottom right three: A membrane with a small protein-poor domain. Parameter values are: $\lambda_{\mathrm{surf}} = 1$, $\lambda_{\mathrm{line}} = 15$, $\kappa=1$, $\alpha = 0.1$, $u_0 = 7$, $\bar{u} = 0.8$, $\gamma = 5$, and $\epsilon_u = 20h_x$.
• Figure 4.4: Cell velocity versus biochemical strength $\alpha$ for a cell with a small protein-rich domain (red region). Left insert subfigure is for $\alpha=0.5$, and right insert subfigure is for $\alpha=0.7$. Other parameter values are: $\lambda_{\mathrm{surf}} = 1$, $\lambda_{\mathrm{line}} = 15$, $\kappa=1$, $u_0 = 9$, $\bar{u} = 0.2$, $\gamma = 5$, and $\epsilon_u = 20h_x$.
• Figure 4.5: Three examples with multiple protein-rich subdomains. Top row: membrane with two protein-rich subdomains. Parameter values are given as: $\lambda_{\mathrm{surf}} = 1$, $\lambda_{\mathrm{line}} = 17$, $\kappa=1$, $\alpha = 0.5$, $u_0 = 0.2$, $\bar{u} = 0.5$, $\gamma = 30$, and $\epsilon_u = 20h_x$. Middle row: eight-bump membrane. Parameter values are $\lambda_{\mathrm{surf}} = 3$, $\lambda_{\mathrm{line}} = 3$, $\kappa=1$, $\alpha = 3$, $u_0 = 0.2$, $\bar{u} = 0.75$, $\gamma = 200$, and $\epsilon_u = 20h_x$. Bottom row: ten-finger membrane. Parameter values are: $\lambda_{\mathrm{surf}} = 0.7$, $\lambda_{\mathrm{line}} = 3$, $\kappa=1$, $\alpha = 10$, $u_0 = 0$, $\bar{u} = 0.6$, $\gamma = 400$, and $\epsilon_u = 20h_x$.

Theorems & Definitions (1)

• Remark 2.1