  Magombedze and Mulder (2013) studied the gene regulatory system of
\textit{Mycobacterium Tuberculosis} (\textit{Mtb}) by partitioning this into
three subsystems based on putative gene function and role in dormancy/latency
development. Each subsystem, in the form of $S$-system, is represented by an
embedded chemical reaction network (CRN), defined by a species subset and a
reaction subset induced by the set of digraph vertices of the subsystem. Based
on the network decomposition theory initiated by Feinberg in 1987, we have
introduced the concept of incidence-independent and developed the theory of
$\mathscr{C}$- and $\mathscr{C}^*$-decompositions including their structure
theorems in terms of linkage classes. With the $S$-system CRN $\mathscr{N}$ of
Magombedze and Mulder's \textit{Mtb} model, its reaction set partition induced
decomposition of subnetworks that are not CRNs of $S$-system but constitute
independent decomposition of $\mathscr{N}$. We have also constructed a new
$S$-system CRN $\mathscr{N}^*$ for which the embedded networks are
$\mathscr{C}^*$-decomposition. We have shown that subnetworks of $\mathscr{N}$
and the embedded networks (subnetworks of $\mathscr{N}^*$) are digraph
homomorphisms. Lastly, we attempted to explore modularity in the context of
CRN.
