Fig. 5: A model reduction recipe for Intein-based controllers.

Under the conditions of Theorem 2, all controllers comprised of M species (where M can be large) that respect the flexible structure depicted in Fig. 2, reduce to the simple motif shown here. The reduced model is shown schematically as a motif comprised of only three effective species (Z⁺, Z⁻, Z⁰), and mathematically as a set of Differential Algebraic Equations (DAEs) comprised of only three differential equations in (z⁺, z⁻, z⁰) and M − 3 algebraic equations in \(\tilde{z}\). Note that (S_B, λ_B) and (S_C, λ_C) denote the stoichiometry matrices and total propensity functions (forward minus backward) of the reversible binding and conversion reactions, respectively. Furthermore \({\mathbb{1}}(.),\circ,{I}_{M}\) and (. )^T denote the indicator function, the Hadamard (element-wise) product, the identity matrix of size M and the transpose of a matrix, respectively. In certain scenarios (see Fig. 6), the algebraic equations can be solved explicitly and thus further simplifying the model to only three Ordinary Differential Equations (ODEs). Observe that the schematic of the simple motif is fully determined once the three vectors q⁺, q⁻, q⁰ and the function ψ(z^tot) are calculated. The vectors q⁺, q⁻ and q⁰ are easily calculated by counting active split inteins (see Theorem 2); whereas, ψ(z^tot) can be calculated by solving the algebraic equations for \(\tilde{z}\ge 0\) as a function of z^tot.