## Appendix A  
```
0: Graph G(V,E), node (compartment) \(i\) receives input
0: Modes[\(j\)] mode-count for all compartments, for all \(j\) e \(V\) // Determine compartments reachable from the input while\(v\in V\) if v is reachable from i then // Using DFS or BFS: \(\textit{TrueMode}[v]\gets 1\) endif endwhile // Count modes based on connectivity: while\(v\in V\)do while\(v^{\prime}\in V\)do if\(v\) is reachable from \(v^{\prime}\)then // Using DFS or BFS: if\(\textit{TrueMode}[v]=1\)then \(\textit{Modes}[v]\leftarrow\text{Modes}[v]+1\) endif endif endwhile
```  
This algorithm actually finds the maximum number of modes in each compartment because it is completely symbolic--with no regard for numerical values of model parameters. _Hidden_ modes can be missed, e.g., those due to model parameters coinciding like the \(k_{22}=k_{33}\) in Example 4. Appendix B Full Michaelis-Menten (M-M) Dynamics Reduced to 2 nd Order--A Very Common Paradigm in Biochemistry  
The following is adapted from [7]. The stoichiometric equation:  
\[S+E\underset{k_{-1}}{\overset{k_{1}}{\rightleftharpoons}}ES\overset{k_{2}}{ \rightarrow}P+E\] (A1)  
describes the Michaelis-Menten (M-M) reaction, in which enzyme \(E\) and substrate \(S\) initially react to reversibly form a _complex ES_. The complex then breaks down to form the _product P_ plus _free_ (not complexed) enzyme \(E\). Let \(c_{S}\), \(c_{E}\), \(c_{ES}\) and \(c_{P}\) represent concentrations of the four molecular species (state variables) in (A1). Then, flux balance and/or the law of mass action applied to the three elementary reactions in (A1) leads directly to the following four coupled nonlinear ODEs:  
\[\frac{dc_{S}}{dt}=-k_{1}c_{E}c_{S}+k_{-1}c_{ES}\] (A2)  
\[\frac{dc_{E}}{dt}=-k_{1}c_{E}c_{S}+k_{-1}c_{ES}+k_{2}c_{ES}\] (A3)  
\[\frac{dc_{ES}}{dt}=k_{1}c_{E}c_{S}-(k_{-1}+k_{2})c_{ES}\] (A4)  
\[\frac{dc_{P}}{dt}=v_{2}=k_{2}c_{ES}\] (A5)The first three equations have a \"hidden\" redundancy and can be reduced to two ODEs, using an algebraic _conservation equation_ derived simply by realizing that enzymes are present in the mix in either free \(E\) or complexed form \(ES\) and must always have a sum equal to the initial \(c_{E}(0)\):  
\[c_{E}(t)+c_{ES}(t)=c_{E}(0) \tag{10}\]  
for \(t>0\). Equation (10) is then used to eliminate \(c_{E}\equiv c_{E}(t)\) from the RHS of Equations (11) and (12). This gives:  
\[\frac{dc_{S}}{dt}=-k_{1}c_{E}(0)c_{S}+(k_{1}c_{S}+k_{-1})c_{ES} \tag{11}\]  
\[\frac{dc_{ES}}{dt}=k_{1}c_{E}(0)c_{S}-(k_{1}c_{S}+k_{-1}+k_{2})c_{ES} \tag{12}\]  
The _quasi-steady state approximation (QSSA)_ of M-M theory is based on the fact that equilibrium is very rapidly established among \(E\), \(S\) and \(ES\) in accordance with the first (the reversible) reaction, virtually independent of the second reaction. Kinetically (and mathematically), this means that the change in complex \(ES\) concentration is _approximately_ zero, relatively speaking, i.e., \(\frac{dc_{ES}(t)}{dt}\simeq 0\) for all \(t>0\). This then means that the LHS of Equation (12) is approximately equal to zero, making it an _algebraic constraint equation_ instead of a third ODE in the M-M model. After some algebraic rearrangements of the equations, the reduced M-M ODE model has only _two_ ODEs, combined in (11), representing the (approximate) coupled nonlinear relationship between product \(P\) and substrate \(S\) concentrations in the reaction mix, parameterized by two constants, \(v_{max}\) and \(K_{m}\), in (12). \[\frac{dc_{p}}{dt}=-\frac{dc_{S}}{dt}\cong\frac{v_{\max}\;c_{S}(t)}{K_{m}+c_{S} (t)}\quad\text{ for all }t>0\quad\text{conc/time units} \tag{13}\]  
Parameters \(v_{max}\), the **maximum value** of \(v\), and \(K_{m}\), the **Michaelis constant**, depend on the rate constants and \(e(0)\), i.e.,  
\[v_{\max}=k_{2}c_{E}(0)\quad\text{and}\quad K_{m}=\frac{k_{-1}+k_{2}}{k_{1}} \tag{14}\]  
The _necessary condition for validity of the QSSA_ is \(K_{m}+c_{S}(0)>>c_{E}(0)\)[26].