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

Technical Summary: Interaction between Cell Membranes and Protein Inclusions in the Large-Deformation Regime

Problem Statement
Biological membranes are dynamic surfaces whose morphology and function are critically governed by protein inclusions (PIs). While the interaction between PIs and lipid membranes has been extensively modeled in the small-deformation (SD) regime using perturbative expansions around flat configurations, many biologically relevant processes involve strong membrane deformations. These include scenarios where PIs move at high velocities, inducing significant invaginations, or where multiple PIs are positioned in close proximity, creating steep gradients that cannot be described by linearized theories. This work addresses the gap in understanding membrane-protein interactions in the large-deformation (LD) regime, specifically focusing on the breakdown of SD approximations and the emergence of complex elastic behaviors.

Methodology
The authors employ a hybrid approach combining finite-element (FE) numerical simulations with approximate analytical derivations.

• Numerical Framework: The study utilizes the finite-element software IRENE to solve the full non-linear shape equation derived from the Helfrich free energy. The membrane is modeled as a two-dimensional manifold embedded in three dimensions using the Monge gauge ($z(x)$). The governing equation includes terms for mean curvature ($H$), Gaussian curvature ($K$), bending rigidity ($\kappa$), and surface tension ($\sigma$).
• Boundary Conditions: Simulations enforce specific constraints, including pinned membrane boundaries at a radius $R$, fixed contact angles ($\alpha$) at the protein interface (modeled as infinitely rigid), and no-slip conditions for membrane flows.
• Analytical Approach: To gain physical insight, the authors derive an approximate analytical solution for the LD regime by assuming a zero mean curvature ($H=0$). This leads to a catenoid profile, allowing for explicit expressions of membrane height and force as functions of contact angle and displacement.
• Flow Dynamics: In the presence of membrane flows, the study solves coupled partial differential equations (PDEs) describing mass conservation, momentum balance (including convective terms and viscous stresses), and the shape equation, while neglecting the external fluid to avoid the Stokes paradox within the computational domain.

Key Contributions and Results

1. Limitations of Linearized Theory:
   By comparing FE solutions with the linearized equation (valid for small gradients), the authors demonstrate that the linear approximation fails when the contact angle exceeds $\tan \alpha \approx 0.4$ or when vertical displacements are large. In these LD regimes, the full non-linear shape equation must be solved to capture the correct membrane geometry.

2. Approximate Analytical Solution (Zero Mean Curvature):
   The authors propose an analytical solution based on the assumption of zero mean curvature, which yields a catenoid profile. While this is an approximation for the general case, it successfully captures the qualitative features of the LD regime. It predicts that the spontaneous vertical displacement ($h$) of a protein with a fixed contact angle scales logarithmically with the domain size ($h \sim \ln R$) for large $R$.

3. Force-Displacement Relationship:
   The study calculates the force exerted by the membrane on a PI. A key finding is that this force exhibits non-monotonic behavior with respect to the protein's vertical displacement. The force increases to a maximum and then decreases, a feature that appears independent of the specific protein geometry. This non-monotonicity suggests the potential onset of dynamic instabilities relevant to tubule formation and retraction. The magnitude of these forces is estimated to be in the range of 1–10 pN, consistent with forces exerted by molecular motors.

4. Membrane-Mediated Interaction Between Two PIs:
   For a system of two interacting PIs, the membrane-mediated potential energy does not follow a simple power-law decay as seen in SD regimes. Instead, the potential exhibits a sub-power-law decay, decreasing more slowly than a power law with inter-protein distance. This reflects the complex nature of the elastic medium. Furthermore, the interaction potential is an odd function of the protein orientation:

   • PIs with identical orientations (same "charge") repel.
   • PIs with opposite orientations (opposite "charges") attract.
     This confirms the analogy between PI orientation and electric charge in the LD regime. The system energetically favors an antisymmetric configuration, potentially leading to antiferromagnetic-like patterns of alternating proteins.

5. Flow-Induced Deformations and Characteristic Velocity:
   In the presence of membrane flows, the authors identify a characteristic velocity scale, $v^* = \kappa / (L\eta)$, which corresponds to a Scriven-Love number of unity. This velocity separates two distinct regimes:

   • $v < v^*$: Flow has negligible effect on membrane shape; bending rigidity dominates.
   • $v > v^*$: Flow-induced deformations emerge, with a characteristic wavelength $\lambda \sim \kappa/(v\eta)$.
     The study suggests that above this threshold, the competition between viscous and bending forces leads to significant shape changes.

Significance and Claims
The paper claims to provide quantitative predictions for membrane-protein systems in biologically relevant scenarios involving large deformations, moving beyond the limitations of perturbative SD theories. The results have direct implications for understanding:

• Protein Sorting and Clustering: The non-monotonic forces and the specific attraction/repulsion rules based on orientation suggest mechanisms for how proteins might organize into clusters or patterns on the membrane.
• Membrane Trafficking: The identification of a critical velocity for flow-induced deformations implies that protein motion can actively drive membrane shape changes, potentially influencing vesicle formation and transport.
• Biological Examples: The authors illustrate the relevance of their findings using bacteriorhodopsin (BR) on giant unilamellar vesicles (GUVs). They estimate that the lateral diffusion velocity of BR molecules is sufficient to induce flow-driven deformations with wavelengths comparable to the inter-protein spacing, suggesting that flow-mediated interactions could significantly influence BR patterning.

Overall, the work establishes a framework for analyzing the complex, non-linear interplay between membrane elasticity, protein geometry, and hydrodynamic flows, offering a more realistic description of membrane mechanics in dynamic cellular environments.