An official website of the United States government
Curvature mediated elastic interactions between inclusions in lipid membranes have been analyzed using both theoretical and computational methods. Entropic corrections to these interactions have also been studied. Here we show that elastic and entropic forces between inclusions in membranes can compete under certain conditions to a yield a maximum in the free energy at a critical separation. If the distance between the inclusions is less than this critical separation then entropic interactions dominate and there is an attractive force between them, while if the distance is more than the critical separation then elastic interactions dominate and there is a repulsive force between them. We assume the inclusions to be rigid and use a previously developed semi-analytic method based on Gaussian integrals to compute the free energy of a membrane with inclusions.Weshow that the critical separation between inclusions decreases with increasing bending modulus and with increasing tension. We also compute the projected area of a membrane with rigid inclusions under tension and find that the trend of the effective bending modulus as a function of area fraction occupied by inclusions is in agreement with earlier results. Our technique can be extended to account for entropic effects in other methods which rely on quadratic energies to study the interactions of inclusions in membranes.
more »
« less
Axisymmetric membranes with edges under external force: buckling, minimal surfaces, and tethers
We use theory and numerical computation to determine the shape of an axisymmetric fluid membrane with a resistance to bending and constant area. The membrane connects two rings in the classic geometry that produces a catenoidal shape in a soap film. In our problem, we find infinitely many branches of solutions for the shape and external force as functions of the separation of the rings, analogous to the infinite family of eigenmodes for the Euler buckling of a slender rod. Special attention is paid to the catenoid, which emerges as the shape of maximal allowable separation when the area is less than a critical area equal to the planar area enclosed by the two rings. A perturbation theory argument directly relates the tension of catenoidal membranes to the stability of catenoidal soap films in this regime. When the membrane area is larger than the critical area, we find additional cylindrical tether solutions to the shape equations at large ring separation, and that arbitrarily large ring separations are possible. These results apply for the case of vanishing Gaussian curvature modulus; when the Gaussian curvature modulus is nonzero and the area is below the critical area, the force and the membrane tension diverge as the ring separation approaches its maximum value. We also examine the stability of our shapes and analytically show that catenoidal membranes have markedly different stability properties than their soap film counterparts.
more »
« less
- Award ID(s):
- 2020098
- PAR ID:
- 10287703
- Date Published:
- Journal Name:
- Soft Matter
- Volume:
- 17
- Issue:
- 31
- ISSN:
- 1744-683X
- Page Range / eLocation ID:
- 7268 to 7286
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Curvature mediated elastic interactions between inclusions in lipid membranes have been analyzed using both theoretical and computational methods. Entropic corrections to these interactions have also been studied. Here we show that elastic and entropic forces between inclusions in membranes can compete under certain conditions to a yield a maximum in the free energy at a critical separation. If the distance between the inclusions is less than this critical separation then entropic interactions dominate and there is an attractive force between them, while if the distance is more than the critical separation then elastic interactions dominate and there is a repulsive force between them. We assume the inclusions to be rigid and use a previously developed semi-analytic method based on Gaussian integrals to compute the free energy of a membrane with inclusions.Weshow that the critical separation between inclusions decreases with increasing bending modulus and with increasing tension. We also compute the projected area of a membrane with rigid inclusions under tension and find that the trend of the effective bending modulus as a function of area fraction occupied by inclusions is in agreement with earlier results. Our technique can be extended to account for entropic effects in other methods which rely on quadratic energies to study the interactions of inclusions in membranes.more » « less
-
We develop a theoretical framework to quantify how active forces renormalize the effective bending rigidity, Gaussian modulus, and surface tension of thermally fluctuating membranes. Building on classical statistical mechanics, we extend the analysis to include nonequilibrium active forces, both direct forces and those coupled to membrane curvature, within a nonlinear continuum formulation. Our model also incorporates hydrodynamic interactions mediated by the surrounding viscous fluid, which significantly alter the fluctuation spectrum. We find that direct active forces enhance long-wavelength undulations, leading to a substantial reduction in both the effective bending rigidity and surface tension, with the extent of softening strongly modulated by fluid viscosity. In contrast, curvature-coupled active forces primarily influence intermediate and short-wavelength fluctuations and show minimal sensitivity to viscosity. Together, these findings provide key insights into the nonequilibrium mechanics of active membranes and yield testable predictions for interpreting fluctuation spectra in both biological contexts and engineered membrane systems.more » « less
-
We show in the companion paper that the free membrane shape of lipid bilayer vesicles containing the mechanosensitive ion channel Piezo can be predicted, with no free parameters, from membrane elasticity theory together with measurements of the protein geometry and vesicle size [C. A. Haselwandter, Y. R. Guo, Z. Fu, R. MacKinnon, Proc. Natl. Acad. Sci. U.S.A. , 10.1073/pnas.2208027119 (2022)]. Here we use these results to determine the force that the Piezo dome exerts on the free membrane and hence, that the free membrane exerts on the Piezo dome, for a range of vesicle sizes. From vesicle shape measurements alone, we thus obtain a force–distortion relationship for the Piezo dome, from which we deduce the Piezo dome’s intrinsic radius of curvature, 42 ± 12 nm, and bending stiffness, 18 ± 2.1 k B T , in freestanding lipid bilayer membranes mimicking cell membranes. Applying these estimates to a spherical cap model of Piezo embedded in a lipid bilayer, we suggest that Piezo’s intrinsic curvature, surrounding membrane footprint, small stiffness, and large area are the key properties of Piezo that give rise to low-threshold, high-sensitivity mechanical gating.more » « less
-
null (Ed.)The Marangoni propulsion of spheres and elliptical disks floating on the air–water interface were studied to understand the effect of particle shape on its motion and its stability at moderate Reynolds numbers. Self-propulsion of the Marangoni surfer was achieved by coating half of the spheres and the elliptical disks with either a solution of soap or isopropyl alcohol (IPA). The presence of the soap or IPA resulted in a surface tension gradient across the particles which propelled the particles in the direction of increasing surface tension. Beyond a critical velocity, a transition was observed from a straight-line motion to a rotational motion. These vortices were observed to shed above a critical Reynolds number resulting in an unbalanced torque that caused the particles to rotate. Increasing the aspect ratio between the major and minor axes of the elliptical disks was found to decrease their stability and greatly enhance their rate of rotation. This was especially true for elliptical disks traveling in a direction parallel to their major axis. The interactions between the particles and the wall of a Petri dish were also studied. Repulsive, concave curvature was found to decrease stability and enhance rotational motion, while attractive, convex curvature was shown to stabilize the straight-line motion of the spheres. For the neutrally buoyant elliptical disks, the presence of the bounding wall was found to greatly stabilize the straight-line motion of the elliptical disks when they were traveling in a direction parallel to their minor axis.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1039/D1SM00827G
