Team:UC Davis/Modeling

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Revision as of 22:55, 28 October 2013

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. PBAD represents the probability that the pBAD promoter will be unbound by araC and thus active. PTET represents the probability that the pTET promoter will be unbound by tetR and thus active. PRiboswitch 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. PTale Binding Site, which may be abbreviated to PTBS 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 ^back to top

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 SimulationSee the Code!^back to top

TALe Binding Site KD 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.



It should be noted that "less effective" does not mean that RiboTALe 1 is an inferior part, merely that it generates a distinct system response and displays kinematic behavior that may be specifically needed in a future circuit design. We have demonstrated that through their engineerable tunability RiboTALes are capable of achieving a broad range of system responses, a conclusion that is supported by this model.


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 KD 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.



Future work with this model may involve the simulation of RiboTALe activity under different non-theophylline riboswitches and an investigation of the orthogonality achievable when using multiple riboswitches in a system.


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.