1 3 Eur Biophys J DOI 10.1007/s00249-016-1177-3 ORIGINAL ARTICLE Two-phase vesicles: a study on evolutionary and stationary models MohammadMahdi Sahebifard 1 · Alireza Shahidi 1 · Saeed Ziaei-Rad 1 Received: 17 May 2016 / Revised: 27 August 2016 / Accepted: 9 September 2016 © European Biophysical Societies’ Association 2016 Introduction The identification of vesicle behaviour (e.g. shape trans- formation, budding, fission, fusion) can play a major role in advancing our understanding of cell functions, owing to their ready availability and simple structure. Furthermore, vesicles with lipid bilayer membranes are of great impor- tance in biological systems. The concurrent existence of different phases in their membranes is a common and widely studied phenomenon. The likelihood of the effect of various domains existing in the membrane on the basic processes of biological systems has further motivated stud- ies in which membrane signalling, lipid sorting, trafficking, and drug delivery can be revealed (Ikonen 2001; Simons and Ikonen 1997; Simons and Vaz 2004). Non-homogeneous vesicles have been largely regarded in experimental research (Hess et al. 2007; Knorr et al. 2012; Vequi-Suplicy et al. 2010) as examples of multi- phase membranes. Accordingly, extensive research has been directed toward developing different mathemati- cal models and numerical methods for two-phase vesi- cles in both stationary (equilibrium) and dynamic forms (Campelo and Hernandez-Machado 2006; Jülicher and Lipowsky 1996). Phase field models (Campelo and Hernández-Machado 2007; Wang and Du 2008) and con- tinuum models (Rahimi and Arroyo 2012) have also been employed for simulating multicomponent vesicles, and have been solved using relatively expensive numerical methods such as the finite element method (FEM). The equilibrium and stability of multicomponent membranes was recently investigated (Givli et al. 2012), and a ther- modynamic model was presented for multi-component two-dimensional membranes using Stokes flow and solved by the boundary integral method (Bonito et al. 2010; Sohn et al. 2010). Abstract In the current article, the dynamic evolution of two-phase vesicles is presented as an extension to a previ- ous stationary model and based on an equilibrium of local forces. In the simplified model, ignoring the effects of mem- brane inertia, a dynamic equilibrium between the membrane bending potential and local fluid friction is considered in each phase. The equilibrium equations at the domain bor- ders are completed by extended introduction of membrane section reactions. We show that in some cases, the results of stationary and evolutionary models are in agreement with each other and also with experimental observations, while in others the two models differ markedly. The value of our approach is that we can account for unresponsive points of uncertainty using our equations with the local velocity of the lipid membranes and calculating the intermediate states (shapes) in the consequent evolutionary, or response, path. Keywords Two-phase vesicle · Dynamic evolution · Membrane elastic force · Friction force · Membrane section reactions Abbreviations V Volume v Reduced volume v Membrane local velocity vector v Membrane normal velocity (scalar) Electronic supplementary material The online version of this article (doi:10.1007/s00249-016-1177-3) contains supplementary material, which is available to authorized users. * MohammadMahdi Sahebifard mm.sahebifard@me.iut.ac.ir 1 Isfahan University of Technology, Isfahan, Isfahan, Islamic Republic of Iran