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1. A method to compute elastic and entropic interactions of membrane inclusions

   Liang, Xiaojun; Purohit, Prashant K. (April 2018, Extreme mechanics letters)

   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. 

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2. Axisymmetric membranes with edges under external force: buckling, minimal surfaces, and tethers

   https://doi.org/10.1039/D1SM00827G  

   Jia, Leroy L.; Pei, Steven; Pelcovits, Robert A.; Powers, Thomas R. (August 2021, Soft Matter)

   null (Ed.)

   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. 

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3. Entropic Pressure Between Fluctuating Membranes With Surface Tension

   https://doi.org/10.1115/1.4068468  

   Hassan, Rubayet; Farokhirad, Samaneh; Ahmadpoor, Fatemeh (September 2025, Journal of Applied Mechanics)

   Entropic pressure, a longstanding topic of interest in biophysics and biomechanics, has been studied for over four decades. Similar to an ideal gas, fluctuating surfaces can generate entropic pressure through thermally driven motions. These thermal fluctuations impact a wide range of biological activities, including but not limited to vesicle fusion, cell adhesion, exocytocis, and endocytocis among many others. It has been proposed (and validated) by many researchers that the entropic pressure near a fluctuating confined fluid membrane without surface tension scales as p∝1/d3, where d is the confining distance, and this power law is size independent. In this article, we show that entropic pressure near a fluctuating fluid membrane could be strongly affected by the membrane’s size and surface tension. We show that while for membranes of size L=1μm and larger, the pressure is size independent, for smaller membranes, the pressure does indeed depend on the membrane’s size. Our findings also shows that the surface tension changes this scaling law and at larger distance makes the pressure decay exponentially. Our work provides insights into how surface tension enhances biological vesicles fusion by suppressing membrane fluctuations, and consequently, the repulsive entropic force, and impacts biomembranes interactions with external objects at the early stage of approaching. 

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4. Determining the Bending Rigidity of Free-Standing Planar Phospholipid Bilayers

   https://doi.org/10.3390/membranes13020129  

   Zabala-Ferrera, Oscar; Liu, Paige; Beltramo, Peter J. (February 2023, Membranes)

   We describe a method to determine membrane bending rigidity from capacitance measurements on large area, free-standing, planar, biomembranes. The bending rigidity of lipid membranes is an important biological mechanical property that is commonly optically measured in vesicles, but difficult to quantify in a planar, unsupported system. To accomplish this, we simultaneously image and apply an electric potential to free-standing, millimeter area, planar lipid bilayers composed of DOPC and DOPG phospholipids to measure the membrane Young’s (elasticity) modulus. The bilayer is then modeled as two adjacent thin elastic films to calculate bending rigidity from the electromechanical response of the membrane to the applied field. Using DOPC, we show that bending rigidities determined by this approach are in good agreement with the existing work using neutron spin echo on vesicles, atomic force spectroscopy on supported lipid bilayers, and micropipette aspiration of giant unilamellar vesicles. We study the effect of asymmetric calcium concentration on symmetric DOPC and DOPG membranes and quantify the resulting changes in bending rigidity. This platform offers the ability to create planar bilayers of controlled lipid composition and aqueous ionic environment, with the ability to asymmetrically alter both. We aim to leverage this high degree of compositional and environmental control, along with the capacity to measure physical properties, in the study of various biological processes in the future. 

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5. Kinetics of self-assembly of inclusions due to lipid membrane thickness interactions

   https://doi.org/10.1039/d0sm01752c  

   Liao, Xinyu; Purohit, Prashant K. (March 2021, Soft Matter)

   null (Ed.)

   Self-assembly of proteins on lipid membranes underlies many important processes in cell biology, such as, exo- and endo-cytosis, assembly of viruses, etc. An attractive force that can cause self-assembly is mediated by membrane thickness interactions between proteins. The free energy profile associated with this attractive force is a result of the overlap of thickness deformation fields around the proteins which can be calculated from the solution of a boundary value problem. Yet, the time scales over which two inclusions coalesce has not been explored, even though the evolution of particle concentrations on membranes has been modeled using phase-field approaches. In this paper we compute this time scale as a function of the initial distance between two inclusions by viewing their coalescence as a first passage time problem. The mean first passage time is computed using Langevin dynamics and a partial differential equation, and both methods are found to be in excellent agreement. Inclusions of three different shapes are studied and it is found that for two inclusions separated by about hundred nanometers the time to coalescence is hundreds of milliseconds irrespective of shape. An efficient computation of the interaction energy of inclusions is central to our work. We compute it using a finite difference technique and show that our results are in excellent agreement with those from a previously proposed semi-analytical method based on Fourier–Bessel series. The computational strategies described in this paper could potentially lead to efficient methods to explore the kinetics of self-assembly of proteins on lipid membranes. 

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