  Explanation of exponential growth in self-reproduction is an important step
toward elucidation of the origins of life because optimization of the growth
potential across rounds of selection is necessary for Darwinian evolution. To
produce another copy with approximately the same composition, the exponential
growth rates for all components have to be equal. How such balanced growth is
achieved, however, is not a trivial question, because this kind of growth
requires orchestrated replication of the components in stochastic and nonlinear
catalytic reactions. By considering a mutually catalyzing reaction in two- and
three-dimensional lattices, as represented by a cellular automaton model, we
show that self-reproduction with exponential growth is possible only when the
replication and degradation of one molecular species is much slower than those
of the others, i.e., when there is a minority molecule. Here, the synergetic
effect of molecular discreteness and crowding is necessary to produce the
exponential growth. Otherwise, the growth curves show superexponential growth
because of nonlinearity of the catalytic reactions or subexponential growth due
to replication inhibition by overcrowding of molecules. Our study emphasizes
that the minority molecular species in a catalytic reaction network is
necessary to acquire evolvability at the primitive stage of life.
