Fig. 3. Intein-based implementation of RPA-achieving integral controllers using ZF, TetR and Gal4 as DBDs.

In all four circuits, the protein Z₁ is constitutively expressed at a rate μ₁ from Plasmid 1 to encode for the desired setpoint. One of the two main tasks of Z₁, which contains one Int^C within a TF, is to either directly actuate the regulated network by producing the input species X₁, or to dimerize first and then actuate. The second task of Z₁ is to undergo an intein-splicing reaction with the second split intein Int^N, denoted by Z₂, whose production is driven by the regulated output X_L at a rate θ₂x_L to encode for the “sensing” reaction. Different positions of the Int^C segment and different TF structures yield different control topologies. Controller species containing DDs undergo reversible homo- or hetero-dimerization reactions with association and dissociation rates of a_i and d_i; whereas, controller species containing active Int^C and Int^N segments undergo irreversible intein-splicing reactions with rate η multiplied by an integer that depends on the number of participating inteins. Note that inactive splicing products are omitted here for simplicity. The control action u is mathematically expressed as a (Hill-type) function of the repressors and activators depicted in the dashed bubbles. Every reaction is labeled from 1 to 6 according to the permitted reaction rules stated in Fig. 2. Furthermore, every monomer, independent of its dimerization status, is labeled in the yellow boxes with one of the following “charges”: + , − , 0, according to Theorems 1 and 2. This is also repeated in the charge vectors q⁺, q⁻ and q⁰ that encode, for each controller species, the number of active Int^C, Int^N, and monomers with no active inteins. Furthermore, in the blue boxes, all controller species are grouped into $\mathcal{C},\mathcal{N}$ and $\mathcal{S}$ classes according to the split inteins they contain (see Species Rules in Fig. 2). Since all the Species and Reaction Rules of Fig. 2 are respected, then by Theorem 1, we conclude that all four controllers ensure RPA with a setpoint of μ₁/θ₂.