Fig. 7: Inactive-intein controller: theoretical and experimental analysis.

a A schematic of the inactive-intein controller. This controller consists of two genes, realized on separate plasmids. The gene in Plasmid 1 encodes for a protein (Z₁) comprised of Int^C-AD; whereas, the gene in Plasmid 1' encodes for a protein (Z₄) comprised of TetR-inactive Int^N. Both genes are driven by a strong constitutive promoter (EF-1α), and their expression rates are denoted by μ₁ and μ₄, respectively. Z₁ and Z₄ can reversibly bind to form a heterodimeric transcription factor, which positively actuates the regulated network via the production of the input species X₁. The production of the second split intein Int^N, denoted by Z₂, is driven by the regulated output X_L at a rate θ₂x_L to encode for the “sensing" reaction. Controller species containing DDs undergo reversible homo- or hetero-dimerization reactions with association and dissociation rates of a_i and d_i. Here, only Z₁ and Z₂ can directly undergo the intein-splicing reaction (at a rate η), because Z₁ is the only species that contains an active Int^C segment not bound to the inactive Int^N segment. 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. The entire charge matrix can be viewed in the blue shaded box where, additionally, all controller species have been grouped into the three classes, \({{{{{{{\mathcal{C}}}}}}}}\)-class, \({{{{{{{\mathcal{N}}}}}}}}\)-class and \({{{{{{{\mathcal{S}}}}}}}}\)-class, according to the species rules of Fig. 2. Since all the Species and Reaction Rules of Fig. 2 are respected, then by Theorem 1, we conclude that this controller ensures RPA with a setpoint of μ₁/θ₂ and is thus interestingly independent of μ₄. b Experimental demonstration of RPA. The performance was tested using the same setup as in Fig. 4(a). The only difference here is that the Int^N segment (Gp41-1) is replaced by an orthogonal Int^N (NrdJ-1) for the open-loop setting, and thus no intein-splicing reaction can occur. The results are demonstrated in a fashion similar to that of Fig. 4. c Reduced model. Unlike the three-dimensional reduced models in Fig. 6 that are obtained by directly applying Theorem 2, the reduced model here is four dimensional because it was necessary to introduce the dynamics of an additional state variable z^⋆. The functions ψ and ϕ are given implicitly in Supplementary Information Section 3.E. Note that the cartoon describing the reduced model is non-physical because the mathematical equations do not satisfy the structure of a simple motif like the models that satisfy Theorem 2. d Simulation Results. A closed-loop system is simulated for four increasing setpoints, where a model of a gene expression network is controlled by the inactive-intein controller. The simulations results demonstrate that the reduced model indeed accurately captures the dynamics of the original full model. The numerical values are provided in Supplementary Information Section 3.E.