Quantitative model and stability analysis

Model description and parameter determination

In order to be able to predict the quantitative behavior of in vivo application of the genetic switches, we set up a quantitative model of a Transcription-activator like effector (TAL) based bistable switches. We apply experimental measurements for individual parts and predict their behavior when applied into a single system. The essential characteristics of a switch are responsiveness (reaching a stable state with the introduction of the corresponding inducer), stability (not leaving the state it after inducer removal) and robustness (expressing equivalent qualitative behavior under a wide range of conditions). In the following subsections, modeling approaches for individual parts are described.

A variety of well-established models in the literature were examined, which resulted in choosing ordinary differential Hill equations model, presented in (Tigges et al., 2009). The former work presents a tunable synthetic mammalian oscillator, whose model was readjusted to present a switch structure. On top of that, the host choice of mammalian cells and pristinamycin-based inducible system corresponds to our system.

To get a better idea of the functioning of the model, an interactive web application was developed and is available here.

TAL:KRAB repressor constructs

For repressor constructs, the percentage of remaining transcription rate comparing to the unrepressed reporter plasmid was measured. Intuitively, the remaining rate was directly proportional to the repressor/reporter ratio, but not in a linear fashion. To approximate this relation for TAL:KRAB constructs, the following analytical function was fitted to the measurements:

where x is repressor/reporter plasmid ratio, and a, b, c are parameters subject to least square error fitting. This produced decreasing function for remaining transcription rate q, depicted in Figure 1a. The achieved percentage of remaining transcription rate q was directly employed in repressed promoter rate equation. The dynamics were modeled using Hill equations for repression and activation (Alon, 2007). For example, the effect of a repressor on transcription rate is modeled as a factor in a rate equation:

Intuitively, the unrepressed transcription rate r is inversely proportional to the concentration of TAL-A:KRAB protein with non-linearity coefficient n. The remaining rate percentage q was used to determine the s constant, which determines the concentration at which half of the unrepressed rate is reached, after the concentration threshold t is crossed.

TAL:VP16 activator constructs

Using a similar approach to that described above, we fitted the data achieved for activator constructs TAL-A:VP16 and TAL-B:VP16. As expected, the transcription rate increased proportionally with increasing activator/reporter ratio of plasmids transfected, again in a non-linear fashion. To approximate the behavior, the following increasing function was fitted:

where x is activator/reporter plasmid radio and c, b are parameters subject to least square error fitting. The plot of the function is shown in Figure 1b. The factor of activation a is again directly employed as a rate equation factor for activated promoter:

where x is activator/reporter plasmid radio and c, b are parameters subject to least square error fitting. The plot of the function is shown in Figure 1b. The factor of activation a is again directly employed as a rate equation factor for activated promoter: where the increasing concentration of TAL-B:VP16 protein reaches half the maximum activation rate a when crossing the concentration threshold t, with non-linearity exponent n. Figure 1: Functions fitted to experimental data for repressor and activator constructs. (a) Data measured for TAL-A:KRAB (blue) and TAL-B:KRAB (red) constructs and averaged afterwards (black). (b) Data measured for TAL-A:VP16 (blue) and TAL-B:VP16 (red) constructs and averaged afterwards (black).

References

Alon U. (2007), An introduction to systems biology: design principles of biological circuits. Chapman & Hall/CRC.

Chatterjee A., Kaznessis Y., and Hu W. (2006) Tweaking biological switches through a better understanding of bistability behavior. Current opinion in biotechnology, 19(5):475-81.

Gardner T., Cantor C., and Collins J. (2000) Construction of a genetic toggle switch in Escherichia coli. Nature, 403(6767):339-42.

Batard P., and Wurm F. (2001) Transfer of high copy number plasmid into mammalian cells by calcium phosphate transfection. Gene: An international Journal on Genes and Genomes, 270:61-68.

Malphettes L., and Fussenegger M. (2006) Impact of RNA interference on gene networks. Metabolic engineering, 8(6):672-83.

Tigges M., Marquez-Lago T., Stelling J., and Fussenegger M. (2009) A tunable synthetic mammalian oscillator. Nature, 457(7227):309-12.

Zakharova A., Kurths J., Vadivasova T., and Koseska A. (2011) Analysing dynamical behavior of cellular networks via stochastic bifurcations. PloS one, 6(5):e19696.