Team:Oxford/Model

Full Width Pics - Start Bootstrap Template

Model

Modeling Description

Different parts of the project have been modelled for a better understanding of the validity of our proposed solution as well as predicting different aspects of the system. Firstly, a system of non-linear Ordinary Differential Equations was written for all reactions taking place; then, they were solved numerically over 250 seconds to explore and understand both transient and steady-state response of the system. The time-domain analysis of the system was proceeded by looking at steady-state curves and output/input behaviour of the system in the steady state as well as body response dynamics, which gave an overall prediction about the fate of the combined system-body model. Secondly, optimisation and modelling were used to determine the optimum number of base pairs needed for sRNA binding as well as promoters' strength used in transcriptions. Modelling was then taken to the frequency domain for transfer function derivation and cascade controller design.

The detailed reaction pathway is shown below in Figure 1.

Figure 1 - Detailed reaction pathways

Methodology

A list of general assumptions is summarised below. It should be noted that more specific assumptions are stated in our modelling report at the end of the page.

  • item Adenosine has been replaced by Adenine as the Hydrolase reaction is believed to be much faster than the body response.
  • item Concentration of Adenine and NO was kept constant for dynamic analysis as there is plenty of these chemicals available outside and inside the cell.
  • item The initial conditions used for time domain analysis correspond to the worst case scenarios.

Dynamics of the system

ODE based model

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 could be found at the end of the section and also in our detailed paper. 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 2.


    Figure 2 - NO dynamic response with the nominal level of Adenine

    Adenine response dynamics

    The complexity of the positive pathway could clearly be seen in Figure 1. The system was kept as detailed as possible for the dynamic analysis based on ODEs stated earlier. It should be noted that based on the second assumption, the concentration of Adenine would not change with time and hence \(\frac{d[Adenine]}{dt}= 0\). An initial amount of \(100 \mu M\) was used for elevated level of NO corresponding to IL10 abundance in patients with IBD, UC and CD as well as the nominal concentration of \(13.24 \mu M\) for NO. The evolution of the system to Adenine stimuli is illustrated below in Figure 3.


    Figure 3 - Adenine dynamic response with the nominal level of NO

Sensitivity analysis and Optimisation

Sensitivity analysis was done separately for each reaction pathway in order to find the most significant steps and parameters and if possible exploit their significance to make the system more responsive and sensitive to smaller perturbations in concentrations. The sensitivity of \([IL10]\) with respect to parameters used in simulations is shown in Figure 4.


Figure 4 - Sensitivity analysis for [IL10] with respect to parameters

The length of sRNA was optimised to 24 base pairs yielding \(\Delta G = -45.97 Kcal\cdot mole^{-1}\). The graph of \(\Delta G \) against number of base pairs is illustrated in Figure 5.

Figure 5 - \(\Delta G\) for sRNA and mRNA binding against the number of base pairs.

Section 3

Lorem ipsum dolor sit amet, consectetur adipisicing elit. Vero odio fugiat voluptatem dolor, provident officiis, id iusto! Obcaecati incidunt, qui nihil beatae magnam et repudiandae ipsa exercitationem, in, quo totam.