This article aims to study the effect of non-uniform distribution of spontaneous curvature on shape transformation of two-phase vesicles via an evolutionary method. Their dynamic evolution is developed based on conventional Helfrich theory, considering bending of the membrane and friction in the surrounding fluid in each phase with variable spontaneous curvature. The variation of spontaneous curvature is assumed to be a function of arc length in each domain considering the effects of inducing factors (surrounding solution concentration and the membrane-protein interactions such as scaffolding and insertion). Membrane pearling from a large vesicle is simulated by the model and compared with the result of constant curvature and also with empirical observations. It can be shown that accurate simulation of some membrane deformation mechanisms depends on careful consideration of key factors such as the SC variations. In addition, the importance of different uniform and non-uniform distributions of spontaneous curvature is discussed with reference to specific cases.
Keywords: Two-phase vesicle; Evolutionary model; Variable spontaneous curvature; Protein distribution; Membrane pearling
Lipid bilayer membranes consist of polar molecules which are positioned across the membrane and any deflection in their normal arrangement requires a consumption of energy. These membranes are naturally symmetric and have no orientation. However, some factors like proteins can locally change the bilayer structure arrangement, resulting in curvature induction to the membrane . The effects of spontaneous curvature (SC) on studying behavior of lipid membranes have been considered in many theoretical and experimental researches [2-4].
Although SC is considered as an invariable parameter in common known continuum models of vesicles [5,6], non-uniform distribution of curvature is a usual phenomenon [7,8]. Reference  presents an equilibrium model of a flat membrane with variable SC from the perspective of elastic shell theory. The effect of variable SC on local tension was also studied for a two-dimensional flat membrane . An  equilibrium model is developed regarding variable curvature in which SC is considered as a function of polymer concentration gradient. Also in uniform vesicles, variable SC has been surveyed using local changes in molecular density . The present study introduces an axisymmetric evolutionary model of two-phase vesicles with variable SC. Two-phase vesicles as examples of heterogeneous vesicles have a better efficiency to simulate closed membranes.
Several factors may cause SC in lipid membranes (for example, different molecular properties of the constituent membrane lipids, mechanisms by which proteins can generate membrane curvature such as scaffolding and insertion and local concentrations of adsorbing particles). Using the model, the effect of these mechanisms can be considered for more accurate simulation.
In the previous paper, the dynamic evolution of two-component vesicle has been developed using the equilibrium between the membrane bending potential and local fluid friction in each phase. Also, stationary and evolutionary methods were compared and stated that evolutionary method is a generalized form and evaluation criteria for related stationary/equilibrium model . In this study, firstly equations dominating each phase of the membrane are presented considering variable SC. Then regarding the matching conditions of two domains, updated expressions of the membrane section reactions are introduced. Finally the model is described for some specific cases. By examining some examples of curvature distribution, the importance of this factor on shape transformation of vesicle membranes is highlighted. Next, a giant vesicle is simulated under the influence of protein insertion, with protein insertion into a small area of the membrane assumed to be a hyperbolic function with significant changing values and slope. In our results, budding and pearling processes are observed similar to experimental observations. We also show that it is not easy to simulate the pearling process with constant SC.
The current study considers a lipid bilayer axisymmetric closed membrane consisting of two separate phase. r and θ are polar coordinates in the x-y plane and s represents the measured arc length from the south pole of the vesicle, z=0 (Figure 1). Assuming constant bending rigidity of the membrane, k the elastic free energy of the membrane, EB, is proportional to an integral of the square of the membrane curvature,
Due to various mechanisms of inducing SC in membranes, nearing real-life conditions for this parameter, for example nonuniform distribution, can make for a better understanding of the subject. In the current study, a model of a two-phase vesicle with varying distribution of SC is developed which relies on conventional continuum theory of lipid membranes. Non-uniform SC is considered as a function of arc length in each phase. In this way, the effect of different origins of SC like type and distribution of proteins and changes in solution density can be introduced in the model vesicle. Regarding each SC mechanism, a function can be introduced to represent the curvature distribution that considers the properties of the mechanism by determination of magnitude, gradient and affected area. As previously stated, the curvature distribution of a mechanism (protein, concentration, etc.), is the required input which can be calculated using laboratory test or molecular dynamics calculation. However, the challenge of this method is defining the curvature distribution which needs to be characterized either by molecular calculation or experiment. The shape transformation of a giant vesicle introduced by SC in two phases, and the simulation of bud, tube and pearl-shaped sphere are simulated by this model. It has also been shown that constant values and variable distributions of SC affect the final shape of a vesicle.
The authors declare that they have no conflict of interest.