  We prove several limits on the behavior of a model of self-assembling
particles introduced by Dabby and Chen (SODA 2013), called insertion systems,
where monomers insert themselves into the middle of a growing linear polymer.
First, we prove that the expressive power of these systems is equal to
context-free grammars, answering a question posed by Dabby and Chen.
  Second, we give tight bounds on the maximum length and minimum expected time
of constructed polymers in systems of three increasingly restricted classes. We
prove that systems of $k$ monomer types can deterministically construct
polymers of length $n = 2^{\Theta(k^{3/2})}$ in $O(\log^{5/3}(n))$ expected
time. We also prove that if non-deterministic construction of a finite number
of polymers is permitted, then the expected construction time can be reduced to
$O(\log^{3/2}(n))$ at the trade-off of decreasing the length to
$2^{\Theta(k)}$. If the system is allowed to construct an infinite number of
polymers, then constructing polymers of unbounded length in $O(\log{n})$
expected time is possible. We follow these positive results with a set of lower
bounds proving that these are the best possible polymer lengths and expected
construction times.
