an energy scale for monomer removal or addition along the backbone of a polymer would be needed to characterise the *body evaporation and addition* in addition to an energy scale associated with the monomer addition to or removal from the ends of a polymer.

Notice that Eqs. \[eq6\] and \[eq7\] are highly nonlinear offering little hope of exact analytical solution. This implies that we have not improved the state of affairs significantly in the context of the various simplifications of the equations that we have already implemented. Of course, one may linearise Eqs. \[eq6\] and \[eq7\] and obtain linear solutions, but this is of limited help as a linear solution can never result in sigmoidal kinetics. Hence, with the aim to reduce the nonlinearity of the equations but preserve the most important and generic behaviour of the kinetics of self-assembly, we employ several additional approximations. Our first approximation is to restrict our analysis to the limit $k_n^+ \rightarrow 0$. This way we reduce the degree of the polynomial, which is essential to obtain an analytical solution. In addition, we also neglect the polymer recombination $\kpf P(t)^2$ term in Eq. \[eq6\]. By doing so, we break the reversibility condition and hence 