Transfer Functions

As stated in the assumptions, model has been linearised due to non-linearity raised by the Hill functions and sRNA binding; Hence, Hill functions have been replaced by \(\gamma_i^*\) -described below. The block diagram and signal flow graph for the linearised system are shown in Figure 2 , where transcription of IL10 mRNA, sRNA and translation of IL10 could be seen clearly.

$$ \gamma_i^* = \beta_i \frac{n_iK_i^{n_i}[u_i]^{n_i-1}}{(K_i^{n_i}+u_i^{n_i})^2} $$

Transfer functions for both negative and positive pathway were found using Mason's gain relation (Bolton &Newnes, 1998) and were double checked against(Steel &Papachristodoulou, 2017) . Transfer functions of both positive and negative pathways could be found below.

$$ G_1(s) = \frac{\text{IL10(s)}}{\text{Adenine(s)}} = \frac{k_2\gamma_1^*(s+\alpha_1+k_1[\text{IL10 mRNA}]^*)}{(s+\alpha_3)(s+s_+)(s+s_-)}, $$ $$ G_2(s) = \frac{\text{IL10(s)}}{NO(s)} = \frac{-k_1[\text{IL10 mRNA}]^*k_2\gamma_2^*}{(s+\alpha_3)(s+s_+)(s+s_-)}, $$
$$s_\pm = \frac{1}{2}(\alpha_2 + k_1[\text{sRNA}]^* + \alpha_1 +k_1[\text{IL10 mRNA}]^*)\pm \frac{1}{2}\sqrt{d},$$ $$d = (\alpha_2 + k_1[\text{sRNA}]^*- \alpha_1-k_1[\text{IL10 mRNA}]^*)^2+4k_1^2[\text{IL10 mRNA}]^*[\text{sRNA}]^*$$

Body response transfer function

Laplace transformed could be taken from both sides of the Equation [label] and by ignoring \(c\), being very small and negligible, we would be able to derive the transfer function for the body as shown below.

$$ s[\text{NO,Adenine(s)}] = -a_i[\text{NO,Adenine(s)}]+b_i[\text{IL10(s)}] \implies F_i(s) =\frac{[\text{NO,Adenine(s)}]}{[\text{IL10(s)}]} = \frac{b_i}{s+a_i} $$

Stability

The closed loop transfer function for cascade cascade controller design given in Figure 1 would contain importantinformation regarding steady state error and relative stability. It should be mentioned that similar to most control problems where the controller is designed for specific specifications (Dorf & H.Bishop, 1998), our reaction pathway could also be changed by changing promoter strength and length of sRNA, making it almost possible to adopt desired structures for the controller. The closed loop transfer function forpositive and negative pathways is as follow.

$$ T_i(s) = \frac{F_i(s)G_i(s)}{1+F_i(s)G_i(s)} $$

Based on equation above, the stability of the system could be found by looking at the location of the poles which are the roots to the characteristic polynomial \(1+Fi(s)Gi(s)=0\). The characteristic equation was solved numerically and smallest poles and zeros for both reaction pathways are summarised in the Argand diagram below, Figure 3. All poles and Zeros are summarised after the figure. It should be noted that some poles and zeros were omitted as they were significantly larger than those shown in Figure 3 and would have thus hindered comparison with those illustrated. All poles and zeros are given below.

DC Gain and Steady State Error

The steady state error could be easily found using the cascade compensation model demonstrated earlier and the Final value Theorem(FVT). The steady state error would be \( \lim_{t\rightarrow \infty} e(t) \) which coupled to FVT could be written as \(\lim_{s \rightarrow 0} sE(s)\), where \(E(s)\) is the Laplace transform of the error. \(E(s)\) could be written as follow.

$$ E(s) = \frac{1}{1+G_i(s)F_i(s)}R(s) \; , \; R(s) = \frac{C}{s} \quad C \in \mathbb{R} $$ $$FVT \, :\quad \lim_{t\rightarrow \infty} e(t) = \lim_{s \rightarrow 0} sE(s)$$ $$\therefore \; \lim_{t \rightarrow \infty} e(t) = \lim_{s\rightarrow 0} \frac{1}{1+G_i(s)F_i(s)} $$

By substituting back Transfer functions for both E. Coli,\(G_i(s)\), and the body,\(F_i(s)\), the steady state error can be found as follows:

• For NO response \(\lim_{t\rightarrow\infty} e(t) =-2.9501\times10^{-4} \).
• For adenine response \(\lim_{t\rightarrow\infty} e(t) = -2.1912\times10^{-3} \).

It is evident that the combined model has a very small DC gain and hence, a very small margin of error is estimated for the steady concentration of IL-10.