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An important transition after the origin of life was the first emergence of aDarwinian population, self-reproducing entities exhibiting differentialreproduction, phenotypic variation, and inheritance of phenotypic traits. Thesimplest system we can imagine to have these properties would consist of acompartmentalized autocatalytic reaction system that exhibits two growth stateswith different chemical compositions. Identifying the chemical composition asthe phenotype, this accounts for two of the properties. However, it is notclear what are the necessary conditions for such a chemical system to exhibitinheritance of the compositional states upon growth and division of thecompartment. We show that for a general class of autocatalytic chemical systemssubject to serial dilution, the inheritance of compositional information onlyoccurs when the time interval between dilutions is below a critical thresholdthat depends on the efficiency of the catalytic reactions. Further, we showthat these thresholds provide rigorous bounds on the properties required forthe inheritance of the chemical compositional state for general growth anddivision cycles. Our result suggests that a serial dilution experiment, whichis much easier to set up in a laboratory, can be used to test whether a givenautocatalytic chemical system can exhibit heredity. Lastly, we apply ourresults to a realistic autocatalytic system based on the Azoarcus ribozyme andsuggest a protocol to experimentally test whether this system can exhibitheredity.

Robustness of compositional heredity to the growth and division dynamics of prebiotic compartments