and addition, accounts for the fact that a monomer can be added in $i-1$ places on the backbone a polymer of size $i$, and that any one of $i-1$ monomers can be removed from a polymer of size $i+1$ because removal from the ends is forbidden for this pathway.

The reactions are assumed to be reaction limited rather that diffusion limited, implying that the reaction rates are constant in time. The indices $i$ and $j$ for reaction iv) obey $i,j \ge n_c$, where $n_c \ge 2$. We consider the case $n_c \ge 2$ in order to be able to close the sums in the master equation and obtain the dynamical equations for the first two principal moments of the polymer length distribution. In the case $n_c \ge 2$, a monomer is not counted as a polymer, whereas for $n_c=1$, a monomer can be an active monomer, i.e., a polymer of size one, or an inactive monomer. Hence, the master equations for $n_c \ge 2$ are fundamentally different from that for $n_c=1$ polymerisation.

One additional assumption that we employ in order to close the sums in our master equations is that we assume $k_n^-=k_e^-$. Later we will see that this approximation 