  Lipid vesicles composed of a mixture of two types of lipids are studied by
intensive Monte-Carlo numerical simulations. The coupling between the local
composition and the membrane shape is induced by two different spontaneous
curvatures of the components. We explore the various morphologies of these
biphasic vesicles coupled to the observed patterns such as nano-domains or
labyrinthine mesophases. The effect of the difference in curvatures, the
surface tension and the interaction parameter between components are thoroughly
explored. Our numerical results quantitatively agree with previous analytical
results obtained by [Gueguen et al., Eur. Phys. J. E, 2014, vol. 37, p. 76] in
the disordered (high temperature) phase. Numerical simulations allow us to
explore the full parameter space, especially close to and below the critical
temperature, where analytical results are not accessible. Phase diagrams are
constructed and domain morphologies are quantitatively studied by computing the
structure factor and the domain size distribution. This mechanism likely
explains the existence of nano-domains in cell membranes as observed by
super-resolution fluorescence microscopy.
