  In this thesis we present a mathematical model describing the population
dynamics of molecules in an artificial chemistry where large molecules can be
produced by successive ligation of pairs of smaller molecules. The chemistry
contains a large number of spontaneous reactions of which a small subset could
be catalyzed by molecules produced in the chemistry with varying catalytic
strengths. We show ACSs, if present in the catalytic network, can focus the
resources of the system into a sparse set of molecules. ACSs can produce a
bistability in the population dynamics and, in particular, steady states
wherein the ACS molecules dominate the population, i.e., have higher
concentrations compared to the rest of molecules in the chemistry (background).
In this thesis we attempt to address two main questions: First, under what
circumstances do molecules belonging to the ACSs dominate over the background,
and second, starting from an initial condition that does not contain good
catalysts, can a sparse set of large molecules (containing several tens or a
few hundred monomers) that are good catalysts arise and be maintained in the
system at concentrations significantly above the background? We show that if an
ACS catalyzed by large molecules contains within it (or partially overlaps
with) a smaller ACS catalyzed by smaller molecules (referred to as a `nested
ACS' structure), the catalytic strength required for the large ACS to dominate
comes down significantly. We show that when the network contains a cascade of
nested ACSs with the catalytic strengths of molecules increasing gradually with
their size, a sparse subset of molecules including some very large molecules
can come to dominate the system.
