# Model Overview

## Modeling Description

Various features of the project have been modelled to verify the validity of our proposed solution and to enable us to make predictions based on different aspects of the system. Firstly, nonlinear ordinary differential equations (ODEs) were written for all reactions taking place. These were then solved numerically over 250 seconds to characterise both the 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 promoter strength and number of base pairs needed for sRNA binding. Modelling was then taken to the frequency domain for transfer function derivation and cascade controller design.

The detailed reaction pathway is shown in Figure 1.

## Methodology

A list of general assumptions is summarised below. It should be noted that more specific assumptions are stated where they have been used for simplification.

- Adenosine substituted with adenine as the hydrolase reaction is believed to be much faster than the bodily response.
- Adenine and NO concentrations kept constant for dynamic analysis due to the intracellular and extracellular abundance.
- Initial conditions used for time domain analysis correspond to maximal immunodeficiency.
- The dynamic model has been developed based on the assumption that all reactions are irreversible and they all assumed to behave as if they were taking place in a cell-free reaction vessel. It should be noted that later in our model we introduce correction factors to take into account concentration differences inside and outside of the cell.
- The stochastic response has been ignored due to the large number of
*E.coli*that is used.

The following biochemical reactions have been modelled for both time domain and frequency domain analysis.

**Parameters used in modelling.**

Parameters | Description | Value from Literature | Value used | Units | Optimised | Reference |
---|---|---|---|---|---|---|

\(\beta_1\) | Maximal Transcription rate | \(1\) | \(10\) | \(nMmin^{-1}\) | Yes | A |

\(K_1\) | Dissociation constant | \(300\) | \(300\) | \(nM\) | No | B |

\(n_1\) | Hill Coefficient | \(0.7-3.5\) | \(2\) | \(Dimensionless\) | No | A |

\(\beta_2\) | Maximal Transcription rate | \(1\) | \(1\) | \(nMmin^{-1}\) | Yes | A |

\(K_2\) | Dissociation constant | \(10\) | \(10\) | \(nM\) | No | A |

\(n_2\) | Hill Coefficient | \(0.7-3.5\) | \(2\) | \(Dimensionless\) | No | A |

\(\alpha_1\) | Degradation rate of sRNA | \(0.03\) | \(0.03\) | \(min^{-1}\) | No | A |

\(\alpha_2\) | Degradation rate of mRNA | \(0.14\) | \(0.14\) | \(min^{-1}\) | No | A |

\(\alpha_3\) | Degradation rate of intracellular IL10 | \(0.03\) | \(0.03\) | \(min^{-1}\) | No | A |

\(\alpha_4\) | Degradation rate of extracellular IL10 | \(0.03\) | \(0.03\) | \(min^{-1}\) | No | A |

\(k_1\) | sRNA-mRNA binding rate | \(100\) | \(100\) | \(nM^{-1}min^{-1}\) | Yes | C |

\(k_2\) | Translation rate of IL10 | \(0.3\) | \(0.3\) | \(min^{-1}\) | No | A |

\(k_3\) | secretion rate of IL10 | \(0.13\) | \(0.13\) | \(min^{-1}\) | No | D |

\([NO]\) | Nominal concentration of NO | \(13.24\) | \(13.24\) | \(\mu M\) | No | E |

\([NO]\) | Elevated concentration of NO | \(19.88\) | \(19.88\) | \(\mu M\) | No | E |

\([Adenine]\) | Nominal concentration of Adenine | \(18\) | \(18\) | \(\mu M\) | Yes | F |

\([Adenine]\) | Elevated concentration of Adenine | \(100\) | \(100\) | \(\mu M\) | Yes | F |

Reactions were modelled based on ODEs which were solved numerically using ODE15s with an absolute tollerance of \(10^{-30}\) and a relative tolerance of \(10^{-7}\). The differential equations and their response is summarised in the following section.

We have developed a Matlab script that would allow any two input system to be modelled and studied. The code is given below and it's free to use by any future igem teams.

## Detailed Models

## References

Index | References |
---|---|

A | Steel, H., & Papachristodoulou, A. (2017). "Frequency domain analysis of small non-coding rnas shows summing junction-like behaviour" |

B | Mandal, M., & Breake, R. R. (2004). "Adenine riboswitches and gene activation by disruption of a transcription terminator" |

C | generated by http://rna.tbi.univie.ac.at/ |

D | Michael H. H. Lenders, Tobias Beer, Sander H. J. Smits & Lutz Schmitt "In vivo quantification of the secretion rates of the hemolysin A Type I secretion system" |

E | Avdagic, N., & Zaciragic, A. (2013). "Nitric oxide as a potential biomarker in inflammatory bowel disease". |

F | Traut, T. W. (2006). "Physiological concentrations of purines and pyrimidines" |

G | Karshenas.A, Windo.J, IGEM 2018 (2018). "Frequency and Time Domain Analysis of sRNA-based Treatment for IBD" |