Small equatorial deformation of homogeneous spherical fluid vesicles

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.

💡 Research Summary

In this work the authors investigate the initial response of a homogeneous spherical fluid vesicle when it is constrained at its equator by a rigid circular ring. The problem is treated analytically in the linear regime, i.e., for infinitesimal differences between the vesicle radius and the ring radius. The membrane is modeled by the spontaneous curvature (Helfrich) energy, which depends quadratically on the deviation κ of the mean curvature from a prescribed spontaneous curvature c_s. Global constraints of fixed surface area and enclosed volume are enforced with Lagrange multipliers µ (surface tension) and p (osmotic pressure), while the ring imposes a local geometric constraint r(0)=r_0 (the nondimensional equatorial radius) through an additional multiplier φ_0 that represents the linear force density exerted by the ring.

The authors first introduce an axisymmetric parametrisation (radial coordinate r(ℓ), height z(ℓ), arclength ℓ, and tangent angle Θ(ℓ)) and nondimensionalise all quantities with the initial sphere radius R_S. The bending energy, together with the constraints, yields a fourth‑order Euler–Lagrange (EL) equation. Because of axial symmetry the EL equation admits a first integral, Eq. (11), which is a first‑order relation among κ, Θ, and r.

To study the onset of deformation the authors expand all fields about the exact spherical solution (r_0=cosℓ, z_0=sinℓ, Θ_0=ℓ+π/2, κ_0=2−c_s) and introduce a small parameter ε = R_0−R_S (the difference between the ring radius and the sphere radius). The expansions r = r_0 + r_1 + …, etc., lead to a linear system of ordinary differential equations for the first‑order corrections r_1(ℓ), z_1(ℓ), Θ_1(ℓ) and κ_1(ℓ). The authors derive four coupled linear equations (23a–d) and, after eliminating r_1 and z_1, obtain a second‑order ODE for κ_1(ℓ). The coefficients of this ODE contain the combination C(0)=c_s−p_0^2, which is fixed by the zeroth‑order balance of bending, tension and pressure (Eq. 18). Depending on the value of C(0) three distinct analytical families of solutions arise:

1. C(0)=0 (a special balance of pressure and spontaneous curvature);
2. C(0)=2 (another special balance);
3. General case C(0)≠0,2.

For each case the authors solve the ODE analytically, obtaining expressions for κ_1(ℓ) in terms of elementary or hyperbolic functions, and then reconstruct r_1, z_1 and Θ_1 using the linearized geometric relations (23a–c). The boundary conditions at the equator (r(0)=r_0, Θ(0)=π/2) and at the pole (regularity of curvature) lead to two possible sets of conditions: either both r_1 and Θ_1 vanish at the pole, or they cancel each other (r_1(π/2)+Θ_1(π/2)=0). These conditions fix the integration constants and also determine the first‑order correction to the Lagrange multiplier φ_0.

The total force exerted by the ring is related to the second derivative of the tangent angle at the equator, φ_0 = –2 Θ̈_0, and the total axial force is f_0 = –4π r_0 Θ̈_0. In the undeformed sphere Θ̈_0 = 0, but the linear analysis shows that a non‑zero Θ̈_0 appears as soon as ε ≠ 0, signalling a discontinuity in the curvature across the ring. The critical force required to initiate deformation, f_c, is obtained by evaluating φ_0 with the first‑order solutions; its sign indicates whether the ring is pulling (dilative, ε>0) or compressing (constrictive, ε<0) the vesicle.

Overall, the paper provides a complete analytical treatment of the small‑amplitude equatorial deformation of a fluid vesicle, linking the mechanical response (critical force, curvature jump) to the physical parameters (bending modulus, spontaneous curvature, surface tension, osmotic pressure) and to the geometric mismatch between vesicle and ring. The results give insight into the early stage of processes such as cytokinetic ring constriction in cells, where a contractile actomyosin ring exerts a localized equatorial force on the plasma membrane. The methodology also offers a benchmark for numerical simulations of vesicle deformation in the nonlinear regime.