the *body evaporation and addition*. The last four terms result from *scission and recombination*. Here, $\delta_{i,n_c}$ denotes Kronecker delta that acquires value of 1 when $i=n_c$ and is zero otherwise. Notice that Eq. (5) is missing the term for nucleus disintegration. This is due to our approximation of $k_n^- = k_e^-$ that allows nucleus to disintegrate via monomer removal from an end hence the nucleus disintegration term gets absorbed in the end evaporation terms.

The factor of $(i- 2 n_c +1)$ in the tenth term on the right-hand side of Eq. \[eq2\] accounts for the number of bonds allowed to break such that the fragmenting filaments are larger than the nucleus size $n_c$. It should be mentioned that the same term, in principle, should include a factor of $\theta(i-2 n_c)$. However, this factor would prevent us from closing the summations and obtaining the dynamical equations for the first two moments of the full polymer length distribution. Hence, we choose not to include it in our master equation. As a consequence, an inconsistency arises for the dimers and trimers, at least if we focus on the case of $n_c=2$. Indeed, this choice would in principle allow a dimer and a trimer 