Team:Michigan/Modeling
Modeling
The action of thrombin switch 2.0 can be modeled as the system of chemical reactions below:
This allows us to treat transcription and translation as first order chemical reactions (assuming an excess of ribosomes, tRNA’s, and dNTP’s. The activation of the RNA switch by thrombin can be treated as a second order chemical reaction because it requires a bimolecular collision. Dissociation of thrombin and the RNA switch is omitted from this model for simplicity (Kd of aptamers produced by SELEX is typically in the nanomolar to picomolar range). This yields the following system of differential equations where [Thrombin] indicates the concentration of thrombin in micromolar, [DNA] indicates the concentration of DNA in micromolar, etc.:
In-vitro transcription of this type is expected to achieve a maximum rate of approximately 20ug of RNA per hour from 84 ng of template DNA. This gives a K1 of approximately .3776s-1. The amount of protein produced from active RNA per second (K3) varies depending on promoter strength, codon optimization, and other factors; however, a rough estimate for K3 might fall between 0.05s-1 and .5s-1 (.278s-1 was used in this model, but the exact value has not yet been determined, see future plans section). K2 must be determined experimentally. Solving the above system of differential equations gives:
Where T0 is the initial concentration of thrombin (micromolar), and t is the time in seconds. As expected, this gives a [GFP] curve which is concave upward initially and then linearizes after all thrombin is bound to an RNA molecule (at this point a fixed concentration of active RNA is available for translation). As t increases to infinity, [GFP] approaches the expression .278T0t. Therefore, because the fluorescence of GFP is proportional to its concentration, the fluorescence output (RFU) observed is expected to be proportional to T0t as the reaction time approaches infinity; however, limitations such as the finite supply of tRNA in the reaction mixture cause the fluorescence to plateau after a time.
As expected, our experimental results for thrombin switch 2.0 contain a linear portion (r squared > .98 for all curves) between 25 minutes and 85 minutes (marked as 20 and 80 minutes on graph in results section due to differences between reaction start time and measurement start time). Interestingly, the slope of the regression varied by up to a factor of 2.3, with higher slopes observed at lower concentrations of thrombin. This suggests that K3 may actually vary significantly depending on the initial thrombin concentration present, possibly a result of ribosome saturation at higher levels of thrombin.
In order to more fully model and characterize our system we could perform a number of future experiments. A standard curve of thrombin concentration (determined via Bradford or similar assay) and observed fluorescence could allow us to fit the observed data more quantitatively to our mathematical model. After this, K3 could be determined more accurately by using a known amount of pre-activated RNA directly in the in-vitro transcription/translation kit (without a DNA template) and measuring the observed fluorescence.