Simplified answer:
These models tend to be object-oriented in the sense that a genetics "module" interacts with a protein signaling module, etc. In each module, you'd have the member data (say, a list of all proteins) and member functions (say, a model of the reaction network that discretizes the massive system of ODEs).
The objects then interact. You have well-defined interfaces between these modules to codify currently known (or hypothesized!) biology. For example, members of the proteins module activate certain genes in the genetics module to (eventually) drive synthesis of more proteins.
You write the rules based upon our current state-of-the-art in understanding cell biology, simulate, and see what happens. To the extent that it quantitatively matches experiments, we can assess the underlying hypotheses, refine them, or toss them out.
In this work, it looks like they pulled information from 900 papers on this species of bacterium to simulate 525 genes, God knows how many proteins (genes can encode multiple proteins), and 28 processes.
Notably, there is no spatial component (e.g., transport of proteins, RNAs, cell volume changes, cell mechanics, etc.), but it's an incredible set of work. And to be able to predict phenotype solely based upon the emergent behavior of this network is pretty incredible.
