Fluctuation tension and shape transition of vesicles: renormalisation calculations and Monte Carlo simulations

Abstract

It has been known for long that the fluctuation surface tension of membranes r, computed from the height fluctuation spectrum, is not equal to the bare surface tension σ, which is introduced in the theory either as a Lagrange multiplier to conserve the total membrane area or as an external constraint. In this work we relate these two surface tensions both analytically and numerically. They are also compared to the Laplace tension γ, and the mechanical frame tension τ. Using the Helfrich model and one-loop renormalisation calculations, we obtain, in addition to the effective bending modulus κ, a new expression for the effective surface tension σ = σ - εkT/(2a) where kT is the thermal energy, a the projected cut-off area, and ε = 3 or 1 according to the allowed configurations that keep either the projected area or the total area constant. Moreover we show that the crumpling transition for an infinite planar membrane occurs for σ = 0, and also that it coincides with vanishing Laplace and frame tensions. Using extensive Monte Carlo (MC) simulations, triangulated membranes of vesicles made of N = 100-2500 vertices are simulated within the Helfrich theory. As compared to alternative numerical models, no local constraint is applied and the shape is only controlled by the constant volume, the spontaneous curvature and σ. It is shown that the numerical fluctuation surface tension r is equal to σ both with radial MC moves (ε = 3) and with corrected MC moves locally normal to the fluctuating membrane (ε = 1). For finite vesicles of typical size R, two different regimes are defined: a tension regime for [small sigma, Greek, circumflex] = σR/κ > 0 and a bending one for -1 < [small sigma, Greek, circumflex] < 0. A shape transition from a quasi-spherical shape imposed by the large surface energy, to more deformed shapes only controlled by the bending energy, is observed numerically at [small sigma, Greek, circumflex] ≃ 0. We propose that the buckling transition, observed for planar supported membranes in the literature, occurs for [small sigma, Greek, circumflex] ≃ -1, the associated negative frame tension playing the role of a compressive force. Hence, a precise control of the value of σ in simulations cannot but enhance our understanding of shape transitions of vesicles and cells.

A Rationale for Mesoscopic Domain Formation in Biomembranes

“…Here σ appears as a Lagrange multiplier controlling the membrane area. There are several alternative definitions of the surface tension that coincide in the high tension limit [65]. It is imposed by external constraints, and cannot exceed the so-called “lysis” tension, on the order of 10−2 N/m for usual lipids such as 1,2-Dioleoyl-sn-glycero-3-phosphocholine (DOPC).…”

“…On the contrary, Gueguen et al [85] construct a phase diagram at fixed temperature in the (m0,C1/C1∗) plane. In principle many parameters indeed depend on T in an ill-defined way, notably the masses but also the surface tension [65] and anticipating how the phase diagram behaves with T is quite challenging. Since the same mixture of lipids is assumed to be present on each leaflet, the critical temperature Tc is the same.…”

A rationale for mesoscopic domain formation in biomembranes

“…This pressure jump can be an osmotic one due to solutes to which the membrane is impermeable, or from a theoretical perspective, a Lagrange multiplier to enforce the volume constraint. This pressure jump fixes therefore the projected area A s = 4πR 2 of the sphere having the same volume, related to R by V = 4πR 3 /3, and plays exactly the same role as the frame tension for planar membranes [65]. At the Gaussian order and for large surface tensions, σ κ/R 2 , both are connected through the Laplace law…”

“…Here σ appears as a Lagrange multiplier controlling the membrane area. There are several alternative definitions of the surface tension that coincide in the high tension limit [65]. It is imposed by external constraints, and cannot exceed the so-called "lysis" tension, on the order of 10 −2 N/m for usual lipids such as 1,2-Dioleoyl-sn-glycero-3-phosphocholine (DOPC).…”

“…In contrast, for low surface tensions, σ κ/R 2 , the vesicle is no more quasi-spherical and a local equilibrium shape condition has been derived by Zhong-can and Helfrich [99] ↵(T T c )/(C 2 1 ) which generalizes the Laplace law. Note however, that in the limit σ κ/R 2 , high order terms must be taken into account in the Hamiltonian, which renormalize the surface tension [65].…”