The Dynamics of the system

Differential Equations and parameters

The dynamic model of both of the pathways were analysed using Simbiology toolbox in Matlab and some of the modelling and parameters are based on the paper "Frequency domain analysis of small non-coding RNAs". The dynamic time domain analysis of the system was done using the combined model in Figure 1 with different initial conditions corresponding to negative and positive pathways separately. It is worth mentioning that data fitting was used to determine some of the parameters. A table of the parameters and data is available in the Appendix of the report. The differential equations describing the systems are as follow.

• $$\frac{d[\text{sRNA}]}{dt} = \Gamma_1 - \alpha_1[\text{sRNA}] - k_1[\text{IL10 mRNA}][\text{sRNA}], \;$$
• $$\frac{d[\text{IL10 mRNA}]}{dt} = \Gamma_2 - \alpha_2[\text{IL10 mRNA}] - k_1[\text{IL10 mRNA}][\text{sRNA}], \;$$
• $$ \frac{d[\text{IL10 - i}]}{dt} = k_2[\text{IL10 mRNA}] - \alpha_3[\text{IL10 - i}] - k_3[\text{IL10 - i}], \;$$
• $$\frac{d[\text{IL10 - s}]}{dt} = k_3[\text{IL10 - i}] - \alpha_4[\text{IL10 - s}], \;$$

where \(\Gamma_i\) are the Hill functions describing the rate of transcription with inputs of Adenosine and NO for \(\Gamma_1\) and \(\Gamma_2\) respectively as represented by \(u_i\) below. $$\Gamma_i = \frac{\beta [u_i]^n}{K^n+[u_i]^n}$$

In which \(\beta\) is the maximal transcription factor, \(K\) the dissociation coefficient and \(n\) the Hill coefficient. \(k_1\) is the rate of sRNA binding to IL10 mRNA which heavily depends on the length of the sRNA, making it easy to exploit this relation for a better control over the fate of the system. \(k_2\) is the translation rate of IL10 mRNA and \(k_3\) is the secretion and diffusion rate of IL10. \(\alpha_i\) also represent the dilution + degradation rate.

NO response dynamics

The negative pathway dynamics is fully described by the ODEs stated earlier and it should be noted that based on the second assumption, the concentration of NO would not change with time and hence \(\frac{d[NO]}{dt}= 0\). An initial amount of \(19.88 \mu M\) was used for elevated level of NO corresponding to IL10 deficiencies in patients with IBD, UC and CD as well as the nominal concentration of \(18 \mu M\) for Adenine. The evolution of the system to NO stimuli is illustrated below in Figure below.





As it could be seen from Figure~\ref{negativepathwaydynamics}, $[IL10]$ reaches a concentration of $13.54 \; \mu M$ after roughly 120 seconds. it is important to note that the initial amount of IL10 is set to 0 for simplicity in the model meaning that in reality the model can correct deficiencies of IL10 to a margin of $13.54 \mu M$. For bacteria to be able to follow and maintain a healthy level of IL10, a control mechanism and design is essential. To develop notions into how the system would respond and whether or not it would be able to stabilise the concentration of IL10, we need to construct a bigger picture of the model that has the body as well as the system embedded in a negative feedback loop~\cite{feedback}. The control modelling is discussed in more details later in the paper. Lastly, the model behaviour is as expected as the production rate is constant as opposed to the degradation rate and hence a bounded steady state value is expected.

Adenine response dynamics