Equations

The equations below model the concentrations of bound transcription factors. That is, they serve to model the concentration of araC bound to pBAD and tetR bound to pTET given the concentrations of the ligands, arabinose and aTc.

The subsequent equations model the probability of active complex for each element in our circuit. P_BAD represents the probability that the pBAD promoter will be unbound by araC and thus active. P_TET represents the probability that the pTET promoter will be unbound by tetR and thus active. P_Riboswitch expresses the probability that the riboswitch is bound by theophylline, and thus active. For simplicity, it has been modeled here as an activator-controlled promoter. P_{Tale Binding Site}, which may be abbreviated to P_TBS expresses the probability that the TALe binding site is unbound by the TAL repressor, and thus active.

The third set of equations are ordinary differential equations modeling the change in concentration over time of the riboswitch-TALe transcript, TAL repressor, GFP mRNA, GFP protein intermediate, and GFP protein. In this model we have taken into account the maturation time of GFP.

Parameters

Included here are the parameters used in this model. Please refer to the References section of this page for the source of each parameter value.

MATLAB Simulation

TALe Binding Site K_D as a source of tunability

Each state variable in the system of ODEs was given an initial condition of 0. The dynamic response of the system was calculated and plotted over a time span of 10 hours. The results of the model support our data in that the RiboTALe with the larger dissociation constant (RiboTALe 1) is less effective at repressing GFP than RiboTALe 8 under the same induction conditions. The peak seen in the dynamic response of both simulations is a result of the kinematics of the system; there is lag between the initiation of GFP production and when the concentration of active TAL repressors is enough to tip the system.

RiboTALe Modulation Through Theophylline Induction Levels

This simulation was carried out under the same conditions defined above, but interrogated only one RiboTALe, RiboTALe 1 with a K_D of 240 nM. The concentration of theophylline, however, was varied over a range of 1 mM to 10 mM and the results plotted. This simulation also supports our data in that it is clear that the riboswitch is, in fact, responsive to theophylline and that final GFP counts are inversely proportional to the amount of theophylline added.

Amplifying System Response Through Transcript Induction

To investigate the effects of increasing GFP transcript while maintaining constant levels of arabinose and theophylline, the dynamic response of the system under RiboTALe 1 repression was simulated for aTc levels of 0, 25 ng/mL, and 100 ng/mL, where aTC is the inducer of the GFP transcript. The model results show the expected behavior: at higher concentrations of aTc GFP reaches greater peak concentration before repression by the RiboTALe becomes evident. Moreover, this event occurs later in the simulation under conditions of 100 ng/mL of aTc than it does for the other two simulated responses.




This model can be further developed to take into account riboswitch leakiness and system stochasticity, and the parameters fine-tuned. It is, however, a useful model in that it provides a mathematical basis that supports the functionality of our RiboTALe devices and shows the wide variety of system responses achievable through the modulation of the engineerable and tunable elements of our construct. We tested combinations of two TAL repressors and two theophylline riboswitches. With this model we will be able to predict the response of a library of RiboTALes, composed a much greater variety of riboswitches and TAL repressors, and perhaps identify with which combination and under what induction conditions a desired system response may be achieved.