  We describe a complete methodology to bridge the scales between nanoscale
Molecular Dynamics and (micrometer) mesoscale Monte Carlo simulations in lipid
membranes and vesicles undergoing phase separation, in which curving molecular
species are furthermore embedded. To go from the molecular to the mesoscale, we
notably appeal to physical renormalization arguments enabling us to rigorously
infer the mesoscale interaction parameters from its molecular counterpart. We
also explain how to deal with the physical timescales at stake at the
mesoscale. Simulating the so-obtained mesoscale system enables us to
equilibrate the long wavelengths of the vesicles of interest, up to the vesicle
size. Conversely, we then backmap from the meso- to the nano- scale, which
enables us to equilibrate in turn the short wavelengths down to the molecular
length-scales. By applying our approach to the specific situation of the
patterning of a vesicle membrane, we show that macroscopic membranes can thus
be equilibrated at all length-scales in achievable computational time offering
an original strategy to address the fundamental challenge of time scale in
simulations of large bio-membrane systems.
