Chemical reaction networks and linear systems

Let's say we have a set of reactions with the structure:

$$\begin{align*} \ce{2A <=>[R_1] B}& \\ \ce{B <=>[R_2] C}& \\ \end{align*}$$

A common way to model reaction networks is with power law kinetics (sometimes called mass action kinetics). This means that the rate of one direction of a reaction is proportional to the concentration of the reactants raised to some power, usually chosen to be the stoichiometry of the reactant in the reaction.

For example, the rates of the forward and reverse reactions of reaction $R_1$ would be:

$$\begin{align*} v_f &= k_{f} A^2 \qquad v_r = k_{r} B \\ \end{align*}$$ Where $k_f$ and $k_r$ are parameters that depend on the specific energetics of the reaction, and $A$ and $B$ are the concentrations of the molecules. One way to model the change in concentration of $A$ would be with a simple derivative: $$ \frac{dA}{dt} = 2(-k_{f} A^2 + k_{r} B) $$ This is unfortunately not a linear system, i.e. there is no way to express the derivative as a product between a coefficient matrix and a vector of states: $$ \dot{x} = Ax $$

Linear systems are great for many reasons, among others because they're easy to control with external input. I was recently reading a paper by Steven Diamond that presents an interesting idea: what if we can express the dynamics of a chemical reaction network as linear system in log space?

$$ \frac{dA}{dt} = 2(-k_{f} A^2 + k_{r} B) $$