Interaction between cell membranes and protein inclusions in the large-deformation regime

TL;DR

This study addresses how biological membranes interact with protein inclusions in the large-deformation regime, where perturbative small-deformation theories fail. It combines finite-element simulations with an approximate analytical LD framework to predict membrane shapes, forces, and inter-protein interactions, including the effects of flows. Key findings include a non-monotonic normal force on a single inclusion, a sub-power-law decay of membrane-mediated interactions between two inclusions with orientation-dependent repulsion/attraction, and a flow-driven threshold velocity $v_*$ separating bending- and viscous-dominated deformations, governed by the characteristic length $\ell = \sqrt{\kappa/\sigma}$. These results provide quantitative predictions for membrane–protein systems in biologically relevant LD scenarios and point to future work on multi-protein assemblies, clustering mechanisms, and stability under flow.

Abstract

Biological membranes are dynamic surfaces whose shape and function are critically influenced by protein inclusions (PIs). While membrane deformations induced by PIs have been extensively studied in the small-deformation regime, a variety of processes involves strong membrane deformations. We investigate the interaction between lipid membranes and PIs in the large deformation (LD) regime, with the finite-element method. We develop an approximate analytical solution that captures key features of the LD regime. We show that the force exerted by the membrane on a PI displays a non-monotonic behavior with respect to the PI vertical displacement. The qualitative features of this force appear to be independent of the protein geometry. For two interacting PIs, the membrane-mediated potential exhibits sub-power-law decay with inter-protein distance, reflecting the complex nature of the elastic medium. The interaction potential shows that conical PIs with identical and opposite orientations repel and attract, respectively, confirming the analogy between PI orientation and electric charge, in the LD regime. In the presence of membrane flows, we identify a characteristic velocity that separates two regimes in which bending rigidity and viscous effects dominate, respectively, implying the onset of flow-induced deformations above such velocity threshold. Overall, our results provide quantitative predictions for membrane-protein systems in biologically relevant scenarios involving LDs, with implications for protein sorting, clustering, and membrane trafficking.

Figures (11)

• Figure 1: Sketch of the domain $\Omega$ and its boundaries for the radially symmetric steady state of a membrane and prin, in the absence of flows. The center of ${\partial \Omega}_{ {\mathord{ }}}$ (red dashed circle) is located at $x^1=x^2=0$, and its radius is denoted by $r_0$. The center of ${\partial \Omega}_{ {\mathord{ }}}$ (blue solid circle) is located in the same position, and its radius is denoted by $R$.
• Figure 2: Comparison between analytical and numerical membrane shapes obtained with irene worthmullerIRENEFluIdLayeR2025. A) Comparison for fixed contact angle and different values of the protein vertical displacement $h$, with bc \ref{['linearized_BC_1', 'linearized_BC_2', 'linearized_BC_3', 'linearized_BC_4']} . B) Same as A, with fixed protein displacement and different values of the contact angle $\alpha$.
• Figure 3: Membrane shapes in the large-deformation regime, obtained with the irene worthmullerIRENEFluIdLayeR2025. A) Membrane deformation induced by a positive vertical displacement of the protein and positive contact angle, $h_0 = 4.2 \, r_0$ and $\tan \alpha = 0.5$, cf. \ref{['fig:BCs-scheme_radial_symmetry']}. B) Membrane deformation induced by a negative vertical displacement of the protein and positive contact angle angle, $h_0 = -6.5\, r_0$, $\tan \alpha = 0.5$, cf. \ref{['fig:BCs-scheme_radial_symmetry']}. The corresponding membrane profiles are shown in panels C) and D), respectively. A convergence analysis of such type of radially symmetric solution with respect to the finite-element mesh resolution is reported in worthmullerIRENEFluIdLayeR2025.
• Figure 4: Comparison between the solution with zero mean curvature \ref{['eq_zero_curvature_solution']} and the numerically exact, fe solution, in the large-deformation regime. The fe solution (thin black curves) has been obtained with the irene worthmullerIRENEFluIdLayeR2025, and the zero-mean-curvature solution is shown as thick colored curves. Both solutions are shown for different values of the membrane radius $R$ and contact angle $\alpha$.
• Figure 5: Force exerted by the membrane on a prin along the $z$-axis, in the large-deformation regime. The force has been computed numerically by using the irene worthmullerIRENEFluIdLayeR2025. Different panels show the force as a function of the membrane vertical displacement $h$, for different values of the size $R$ of the membrane domain. A) $R = 5 \, r_0$, B) $R = 10\, r_0$, C) $R =30\, r_0$, D) $R = 100\, r_0$.