[Paper Review] Solvated Membrane Nanodiscoids: A Probe For The Effects Of Gaussian Curvature

Richard G. Morris, Timothy R. Dafforn+1 more|arXiv (Cornell University)|Apr 1, 2019

Lipid Membrane Structure and BehaviorBiochemistry, Genetics and Molecular Biology21 references3 citations

TL;DR

This paper proposes solvated membrane nanodiscoids as a probe for studying the effects of Gaussian curvature on lipid membranes, leveraging their high boundary-to-area ratio to stabilize saddle-shaped (negative curvature) structures. Using cryo-EM analysis and theoretical modeling, the authors show that high surface tensions (>10⁻⁴ N/m) enable solvation of high-curvature discoids and enable fractionation of lipids by their Gaussian rigidity modulus, $ί{\kappa}$, with potential applications in probing curvature-dependent protein behavior.

ABSTRACT

Several methods now exist to solvate lipid bilayer discoids at the scale of tens of nanometres. Due to their size, such nanodiscoids have a comparatively large boundary-to-area ratio, making them unusually well-suited to probing the effects of Gaussian curvature. Arguing that fluctuations in discoid size and shape are quenched on formation, we quantify the stability, in terms of size and shape, of near-solvation discoid-like flaps that are subject to thermal fluctuations. Using cryo-Electron Microscopy images of Styrene Maleic Acid stabilised discoids, we deduce that stable, saddle-like discoids (with high Gaussian curvature) can likely be solvated from bulk lamellar ($L_α$) phase at moderate-to-high surface tensions ($>10^{-4}$ N/m). We then describe how such tension-controlled solvation can be used for both measuring, and fractionating membrane components according-to, the modulus of Gaussian rigidity $\barκ$. Opportunities for investigating the effects of Gaussian curvature on membrane-embedded proteins, which can be co-solvated during the formation process, are also discussed.

Motivation & Objective

• To investigate the role of Gaussian curvature in membrane stability and phase behavior using nanodiscoids with high boundary-to-area ratios.
• To develop a method for measuring and fractionating lipids based on their Gaussian rigidity modulus, $ί{\kappa}$.
• To explore the influence of Gaussian curvature on membrane-embedded proteins during nanodisc formation.
• To quantify the stability of discoid-like flaps subject to thermal fluctuations during solvation.
• To establish a link between surface tension during solvation and the formation of high-curvature, saddle-shaped nanodiscoids.

Proposed method

• Cryo-electron microscopy (cryo-EM) was used to image SMA-stabilized nanodiscoids, followed by image processing using ImageJ and OpenCV for noise reduction and contour detection.
• Elliptical fitting of discoid contours via least-squares optimization enabled quantification of size and shape distributions.
• A theoretical model based on membrane Hamiltonian formalism incorporated bending energy, line tension, and Gaussian rigidity contributions, including a Flory-like parameter $\chi$ for compositional inhomogeneity.
• The model derived energy expressions for pringle-shaped fluctuations, with curvature-dependent terms involving $\bar{\kappa}$, $\delta\bar{\kappa}$, and $\psi$, the area fraction of a component with different rigidity.
• Stability conditions were derived by minimizing the total energy $\mathcal{H} = \mathcal{H}_m + \mathcal{H}_b + \mathcal{H}_\psi$, leading to expressions for principal curvatures and critical $\bar{\kappa}$ values.
• The analysis identified a critical threshold for stable, non-spherical shapes based on $\bar{\kappa} > \mathrm{max}(\bar{\kappa}^*_{p}, \bar{\kappa}^\dagger_{\chi})$, with $\bar{\kappa}^\dagger_{\chi}$ accounting for compositional effects.

Experimental results

Research questions

• RQ1 Can solvated nanodiscoids with high Gaussian curvature be stabilized under experimentally accessible surface tensions?
• RQ2 How does the modulus of Gaussian rigidity $\bar{\kappa}$ influence the size and shape of nanodiscoids during formation?
• RQ3 Can differences in $\bar{\kappa}$ between lipid components lead to selective partitioning into nanodiscoids during solvation?
• RQ4 What is the role of thermal fluctuations in determining the final size and shape of nanodiscoids?
• RQ5 Can nanodiscoids be used as a tool to fractionate membrane components based on their $\bar{\kappa}$ values?

Key findings

• Cryo-EM analysis of SMA-stabilized nanodiscoids revealed an average semi-major axis of 3.48 nm with a variance of 1.68 nm², indicating a polydisperse distribution.
• Stable, saddle-shaped (high Gaussian curvature) nanodiscoids can be solvated from the bulk $L_{\alpha}$ phase at surface tensions exceeding $10^{-4}$ N/m.
• Theoretical modeling predicts that stable, non-spherical shapes emerge when $\bar{\kappa} > \mathrm{max}(\bar{\kappa}^*_{p}, \bar{\kappa}^\dagger_{\chi})$, with $\bar{\kappa}^\dagger_{\chi}$ corrected for compositional inhomogeneity.
• The area fraction $\psi$ of a component with higher $\bar{\kappa}$ in a pringle-shaped fluctuation scales as $\psi^2 \propto \alpha^4 / R_0^4$, indicating curvature-dependent enrichment.
• The energy penalty for compositional inhomogeneity in curved regions is $\mathcal{H}_\psi \approx \frac{\pi (\delta\bar{\kappa})^2}{2\chi R_0^2}$, which stabilizes curvature-specific phase separation.
• The framework enables potential fractionation of lipids by $\bar{\kappa}$, as components with different rigidity moduli partition differently into high-curvature nanodiscoids.