Experiments

Besides modelling the overall system to gain a theoretical understanding as to how the system should behave, we modelled individual experiments so that we could help the biologists and directly compare results by fitting data. We formulated our models using the literature available on the topic as a primary source. This coupled with the simple mass acion law allowed us to formulate models that theoretically should represent the individual experiments quite well. Upon obtaining the results for each experiment, the results would be analysed in the context of the original model's predicted outcome and compared. Due to experimental setbacks this was not possible for all of the experiment models. For some models analytical solutions were obtainable, however for others this was not possible, as a result, we used MATLAB to explore the dynamics of these systems under varying constants. These were the experiments that we modelled:

IRES (Internal Ribosome Entry Site)

We obtained two different IRES sequences whilst constructing our system, one for the NKRF IRES and one for the EMCV IRES. We knew that the EMCV had been shown to work in human cells (Huh-7 cells) [Lohmann 1999] but NKRF had been introduced into HeLa cells successfully [1] and be 92 times more efficient than EMCV in this cell line. We wanted to determine if the same held in Huh-7.5 cells which was our cell line of choice. We modelled the efficiency of the two IRESs and then fitted the data to our model to determine its accuracy. The equations are as follows: \[ \frac{dR_{+}}{dt} = \tau(t) - \mu_{R_{+}}R_{+} \] \[ \frac{dG}{dt} = \beta R_{+} - \mu_{G}G \] This set of equations can be solved analytically to give solutions: \[ R_{+}(t) = \int_{0}^{t} e^{\mu_{R_{+}}(s-t)} \tau(s) ds \] \[ G(t) = \frac{\beta}{\mu_{G} - \mu_{R_{+}}} \int_{0}^{t} ( e^{\mu_{R_{+}}(s-t)} - e^{\mu_{G}(s-t)} ) \tau(s) ds \] where $R_{+}$ is the positive strand of RNA, $G$ is the GFP, $\mu$ is a degradation rate with the subscript denoting the subject, $\beta$ is a translation constant and $\tau$ is the transfection function.

3’ UTR

We found sequences for several different 3’ promoters, and similar to the IRES we wanted to determine which was most efficient. Our goal with the modelling was to determine which would be the best for our system, so that we could then incorporate it into our overall replicon. Due to experimental constraints, the testing of the promoters was only undertaken in E.Coli. The equations governing this experiment are: \[ \frac{dR_{+}}{dt} = c - \mu_{R_{+}}R_{+} \] \[ \frac{dR_{-}}{dt} = \alpha_{-} R_{+} E - \mu_{R_{-}}R_{-} \] \[ \frac{dE}{dt} = 0 \] \[ \frac{dG}{dt} = \beta R_{-} - \mu_{G}G \] where $R_{+}$ is the positive strand of RNA, $c$ is a constant, $\mu$ is a degradation rate with the subscript denoting the subject, $R_{-}$ is the negative strand of RNA, $E$ is RdRp and we assume it's rate of change to be zero indicating that the system has reached equilibrium, $\beta$ is a translation constant and $G$ is the GFP.

Aptazyme

This functions as the kill switch for our system. We wanted to model the effectiveness of this switch, by using values obtained from the literature to obtain theoretical time scales as to how long it would take to stop replication of the system. We then conducted the experiment and fit the data to obtain a comparison as to how close the theoretical model and real values differ. \[ \frac{dR_{+}}{dt} = \tau(t) - \rho X R_{+} - \mu_{R_{+}}R_{+} \] \[ \frac{dX}{dt} = \beta R_{+} - \rho X R_{+} - \mu_{X}X \] where $R_{+}$ is the positive strand of RNA, $\tau$ is the transfection function, $\rho$ is a constant governing the probability of binding, $X$ is the theophylline and$\mu$ is a degradation rate with the subscript denoting the subject.

siRNA

This model was to illustrate the effects of siRNA as a method of gene silencing. We took to the literature and followed previously developed models from these papers, and then explored the dynamics of them using values closer to our systems to gauge a response as to how effective siRNA targeting is at silencing. In the testing this was specific to DPP-IV, however in practice we can take this and apply it to any siRNA targeting sequence. Therefore we made our model general to keep with our aim of reusability so that others in the future could take our model, and with a few tweaks be able to apply it to their systems. \[ \frac{dR_{i}}{dt} = \tau(t) - \rho R_{i} T_{R} - \mu_{R_{i}} R_{i} \] \[ \frac{dR_{ds}}{dt} = \rho R_{i} T_{R} - \lambda_{C} C - ( \lambda_{R_{ds}} + \mu_{R_{ds}} ) R_{ds} \] \[ \frac{dT_{R}}{dt} = - \rho R_{i} T_{R} - \mu_{T_{R}} T_{R} \] \[ \frac{dG}{dt} = \beta T_{R} - \mu_{G} G \] \[ \frac{dC}{dt} = \lambda_{R_{ds}} R_{ds} - \mu_{C} C \] where $R_{i}$ is the "interfering" strand of RNA, $\tau$ is the transfection function, $\rho$ is a constant governing the probability of binding, $T_{R}$ is the target RNA, $\mu$ is a degradation rate with the subscript denoting the subject, $R_{ds}$ is double stranded RNA, $\lambda_{C}$ is a constant, $C$ is the RISC complex, $\beta$ is a translation constant and $G$ is the GFP.