  Biological cells with all of their surface structure and complex interior
stripped away are essentially vesicles - membranes composed of lipid bilayers
which form closed sacs. Vesicles are thought to be relevant as models of
primitive protocells, and they could have provided the ideal environment for
pre-biotic reactions to occur. In this paper, we investigate the stochastic
dynamics of a set of autocatalytic reactions, within a spatially bounded
domain, so as to mimic a primordial cell. The discreteness of the constituents
of the autocatalytic reactions gives rise to large sustained oscillations, even
when the number of constituents is quite large. These oscillations are
spatio-temporal in nature, unlike those found in previous studies, which
consisted only of temporal oscillations. We speculate that these oscillations
may have a role in seeding membrane instabilities which lead to vesicle
division. In this way synchronization could be achieved between protocell
growth and the reproduction rate of the constituents (the protogenetic
material) in simple protocells.
