  An important transition after the origin of life was the first emergence of a
Darwinian population, self-reproducing entities exhibiting differential
reproduction, phenotypic variation, and inheritance of phenotypic traits. The
simplest system we can imagine to have these properties would consist of a
compartmentalized autocatalytic reaction system that exhibits two growth states
with different chemical compositions. Identifying the chemical composition as
the phenotype, this accounts for two of the properties. However, it is not
clear what are the necessary conditions for such a chemical system to exhibit
inheritance of the compositional states upon growth and division of the
compartment. We show that for a general class of autocatalytic chemical systems
subject to serial dilution, the inheritance of compositional information only
occurs when the time interval between dilutions is below a critical threshold
that depends on the efficiency of the catalytic reactions. Further, we show
that these thresholds provide rigorous bounds on the properties required for
the inheritance of the chemical compositional state for general growth and
division cycles. Our result suggests that a serial dilution experiment, which
is much easier to set up in a laboratory, can be used to test whether a given
autocatalytic chemical system can exhibit heredity. Lastly, we apply our
results to a realistic autocatalytic system based on the Azoarcus ribozyme and
suggest a protocol to experimentally test whether this system can exhibit
heredity.
