Team:UC Davis/Modeling
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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.
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Parameters Return 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 Simulation-- See the Code:(1) and (2)!Return to Top
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.
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 8 with a K_{D} of 1.3 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.
The expected effect of theophylline induction levels on the expression of the gene of interest can be calculated and plotted. The expression levels have been normalized to the expected expression of GFP under conditions of 1% arabinose and 10 mM theophylline for RiboTALe 8.
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 8 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.
Anderson Promoter Model Return to Top
To determine whether our synthetic transcription factors would effectively repress the constitutive family of Anderson promoters, we placed the a TALe target sequence downstream of a number of Anderson promoters. The model presented above for the GFP testing construct holds for the repressible Anderson promoter constructs. However, instead of the pTET term we now deal with simply 'P', the relative promoter strength of the promoter in question. Solving the model at steady state, the following expression for GFP concentration is derived.
Given arabinose concentration data, theophylline concentration data, and corresponding GFP fluorescence levels, a nonlinear regression may be performed in order to estimate the parameters of the model. This model is intended to estimate relative levels of GFP fluorescence given the relative promoter strength and the induction levels of arabinose and theophylline. To this end, the derived expression may be simplified as follows.
For simplicity, the activity of the pBAD promoter has been modeled as an activator-controlled promoter in the expression above. After performing a nonlinear regression using relative GFP values (fluorescence divided by maximum fluorescence under the case of no repression), the following parameter estimates for A, B, C, D, and E were generated.
Parameter | Value | |
---|---|---|
A | 1.0338 | |
B | 0.9669 | |
C | 0.0129 | |
D | 3.273*10^{3} | |
E | 5.8811 |
These parameter estimates were generated from data for the RiboTALe under control of Riboswitch 2 and expressing TALe 8. Plugging these parameter estimates back into the model, the following expected relative GFP fluorescence levels were calculated for five different relative promoter strengths across ranges of both theophylline and arabinose induction levels. The effects of theophylline induction levels on system behavior were investigated for 0.01% arabinose. The effects of arabinose induction levels on system behavior were investigated for 1 mM theophylline.
With the parameter estimates generated from a nonlinear regression, the model displays a decrease in GFP fluorescence as induction levels of arabinose and theophylline are increased. Furthermore, the effect of the relative promoter strength is accurately reflected in the graphs generated by the model. A small relative promoter strength results in a a lower baseline fluorescence under conditions of no repression, and lower fluorescence levels across all ranges of arabinose and theophylline. This model may be used to approximate the behavior of systems under control of Anderson promoters that have been engineered to be repressible by our synthetic transcription factors.
ReferencesReturn to Top
[1] "Key Numbers for Cell Biologists." Bionumbers: The Database of Useful Biological Numbers
[2] Tamsir et al. 'Robust multicellular computing using genetically encoded NOR gates and chemical ‘wires’: Supplementary Information'. Nature 469, 212–215 (13 January 2011)
[3] J. F. Meckler, M. S. Bhakta, M. S. Kim, R. Ovadia, C. H. Habrian, A. Zykovich, et al., "Quantitative analysis of TALE-DNA interactions suggests polarity effects," Nucleic Acids Res, vol. 41, pp. 4118-28, Apr 2013.
[4] Wachsmuth M., Findeiss S., Weissheimer N., Stadler P., Morl M., "De novo design of a synthetic riboswitch that regulates transcription termination," Nucleic Acids Res, 2012
[5] "Part:BBa_K750000" The Registry of Standard Biological Parts
[6] Cormack et al. 'FACS-optimized mutants of the green fluorescent protein (GFP).' Gene. 1996;173(1 Spec No):33-8.