  The apparent difficulty of designing simple autocatalysts that grow
exponentially in the absence of enzymes, external drives or ingenious internal
mechanisms severely constrains scenarios for the emergence of evolution by
natural selection in chemical and physical systems. Here, we systematically
analyze these difficulties in the context of one of the simplest and most
generic autocatalysts: a dimeric molecule that duplicates by templated
ligation. We show that despite its simplicity, such an autocatalyst can achieve
exponential growth autonomously. This only requires that the rate of the
spontaneous dimerization, the interactions between molecules, and the
concentrations of substrates and products are in appropriate ranges. We also
show, however, that it is possible to design as simple sub-exponential
autocatalysts that have an advantage over exponential autocatalysts when
competing for a common resource. We reach these conclusions by developing a
general theoretical framework based on kinetic barrier diagrams. Besides
challenging commonly accepted assumptions in the field of the origin of life,
our results provide a blueprint for the experimental realization of elementary
autocatalysts exhibiting a form of natural selection, whether on a molecular or
colloidal scale.
