  We consider a generic proto-cell model consisting of any conservative
chemical reaction network embedded within a membrane. The membrane results from
the self-assembly of one of the chemical species (membrane precursor) and is
semi-permeable to some other chemical species (nutrients) diffusing from an
outside growth medium into the proto-cell. Inside the proto-cell, nutrients are
metabolized into all other chemical species including the membrane precursor,
and the membrane grows in area and the proto-cell in volume. Investigating the
conditions under which such a proto-cell may reach stationary growth, we prove
that a simple necessary condition is that each moiety be fed with some nutrient
flux; and that a sufficient condition for the existence of a stationary growth
regime is that every siphon containing any species participating in the
membrane precursor incorporation kinetics also contains the support of a moiety
that is fed with some nutrient flux. These necessary and sufficient conditions
hold regardless of chemical reaction kinetics, membrane parameters or nutrient
flux diffusion characteristics.
