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Small equatorial deformation of homogeneous spherical fluid vesicles

Andrés Solís-Cuevas, Pablo Vázquez-Montejo

TL;DR

This work addresses the initiation of equatorial deformation in a spherical vesicle subjected to a circular rigid ring, within the linear regime. It derives a rotationally symmetric, spontaneous-curvature membrane model and uses the first integral of the Euler-Lagrange equation under fixed area and volume to obtain the linearized governing equations. By performing a perturbative expansion about the sphere, it obtains explicit first-order corrections for the generating curve coordinates in three regimes determined by $C_{(0)}$, including a Legendre-form solution for the general case; it also identifies the critical external force corresponding to $C_{(0)}=6$. The results show that the total bending energy and curvature do not change to first order, provide the initial forces and deformations needed to start the process, and agree with nonlinear numerical findings, offering a solid baseline for nonlinear analyses and potential applications to cellular cytokinesis.

Abstract

We examine the reaction of a homogeneous spherical fluid vesicle to the force exerted by a rigid circular ring located at its equator in the linear regime. We solve analytically the linearized first integral of the Euler-Lagrange equation subject to the global constraints of fixed area and volume, as well as to the local constraint imposed by the ring. We determine the first-order perturbations to the generating curve of the spherical membrane, which are characterized by the difference of the radii of the membrane and the ring, and by a parameter depending on the physical quantities of the membrane. We determine the total force that is required to begin the deformation of the membrane, which gives rise to a discontinuity in the curvature of the membrane across the ring.

Small equatorial deformation of homogeneous spherical fluid vesicles

TL;DR

This work addresses the initiation of equatorial deformation in a spherical vesicle subjected to a circular rigid ring, within the linear regime. It derives a rotationally symmetric, spontaneous-curvature membrane model and uses the first integral of the Euler-Lagrange equation under fixed area and volume to obtain the linearized governing equations. By performing a perturbative expansion about the sphere, it obtains explicit first-order corrections for the generating curve coordinates in three regimes determined by , including a Legendre-form solution for the general case; it also identifies the critical external force corresponding to . The results show that the total bending energy and curvature do not change to first order, provide the initial forces and deformations needed to start the process, and agree with nonlinear numerical findings, offering a solid baseline for nonlinear analyses and potential applications to cellular cytokinesis.

Abstract

We examine the reaction of a homogeneous spherical fluid vesicle to the force exerted by a rigid circular ring located at its equator in the linear regime. We solve analytically the linearized first integral of the Euler-Lagrange equation subject to the global constraints of fixed area and volume, as well as to the local constraint imposed by the ring. We determine the first-order perturbations to the generating curve of the spherical membrane, which are characterized by the difference of the radii of the membrane and the ring, and by a parameter depending on the physical quantities of the membrane. We determine the total force that is required to begin the deformation of the membrane, which gives rise to a discontinuity in the curvature of the membrane across the ring.
Paper Structure (11 sections, 94 equations, 5 figures, 3 tables)

This paper contains 11 sections, 94 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Geometric quantities of axisymmetric membranes
  3. Bending energy and first integral of the Euler-Lagrange equation
  4. Perturbative expansion about the sphere
  5. Weak deformation of the sphere
  6. Solution for $C_{(0)}=0$
  7. Solution for $C_{(0)} = 2$
  8. Solution for $C_{(0)} \neq 0,2$
  9. Discussion and conclusions
  10. List of useful integrals
  11. Derivation of the trigonometric expasion of $P_\nu(\sin \ell)$

Figures (5)

  • Figure 1: The profile curve of the axisymmetric surface is defined by the scaled radial and height coordinates $r$ and $z$, which are parametrized by the scaled arc length $\ell$. The tangent vector $\mathbf{T}$ forms an angle $\Theta$ with the radial direction $\hat{\bm{\rho}}$.
  • Figure 2: (a) Dilation and (b) constriction of a spherical vesicle with parameter $C_{(0)}=1$. The initial spherical configuration is shown with a black dashed line. The magnitude of the deformations has been augmented for illustration purposes. The bending energy density is color coded.
  • Figure 3: Values of $\dot{\Theta}_{1}$ at the equator as a function of $C_{(0)}$. The red (blue) line corresponds to dilation (constriction) of the membrane, with $r_{(1)0}>0$ ($r_{(1)0}<0$).
  • Figure 4: Total force as a function of $C_{(0)}$. The red (blue) line corresponds to dilation (constriction) of the membrane, with $r_{(1)0}>0$ ($r_{(1)0}<0$).
  • Figure 5: Magnitude of the first-order correction to the radial coordinate at the pole as a function of the parameter $C_{(0)}$.