Team:Dundee/Model

Modelling

Mathematical Modelling

Introduction

Mathematical modelling is fundamental for synthetic biology, as it acts as a link between the interpretation of experiments and helps to improve our system in a systematic, rational way. Simulating the behaviour of the experimental systems makes possible to improve and refine experiments carried out in the lab. The advantage of using mathematical modelling is that we do not need to continuously measure data from experiments, as this would be a difficult process. Also using mathematics to model systems reduces the system to its core elements and therefore makes it easier to analyse. These are the reasons why we developed a mathematical model of our project. The treatment we developed would theoretically be consumed orally by livestock and would travel through the gastrointestinal (GI) tract - so modelling the amount of BactiFeed that is fed to each animal is essential in calculating whether the target bacteria will be completely eradicated from the animal.

Design: We decided that the most important piece of information that would be needed was if our BactiFeed product could completely kill enough target bacteria to cure the animal of its symptoms - therefore allowing it to be more healthy, and also trying to avoid the transmision of such diseases to humans. We took a chicken as an example, a schematic diagram can be found in Figure 1.1, the BactiFeed product would travel through the oesophagus, into the crop and then through the proventriculus (glandular stomach) and gizzard (muscular stomach) where the colicins would be produced due to the low levels of pH and then through the intestine where the cells lyse due to the detection of bile salts, which are then deposited into the liver, they then start to release the colicins with the corresponding warheads.

Figure 1.1: Schematic diagram of a chicken’s GI tract, blue circle represents where the target cells are most likely to be located, red circle represents where the colicins will be produced by the cell and the yellow region is where the cells will lyse because of the detection of bile salts - The colicins can then go on to kill any target bacteria that they come across in the gut.

Here is an animation about the behaviour of our system:

Parameter Fitting

Initially we created a simple model to illustrate how the data fitted from the growth culture results which was carried out by doing a plate reader experiment. Exponential growth was not considered as our cultures saturate. Therefore, logistic and Gompertz growth models were considered to fit the data. Both models consider two parameters, a growth rate r and a carrying capacity K (which is the highest value at which a culture can grow. The logistic growth is modelled as follows:

The equation for Gompertz growth is the following:

DH5α and MG1655 cells are two of the many strains of E.coli , which were being utilised in the lab, therefore it was decided to model these strains, using both models, to find the best fit. Firstly, analysis was carried out for the DH5α strain. Figures 2.1 and 2.2 show the fit for data using logistic or Gompertz growth, respectively.

Figure 2.1: Growth of DH5α cells, fitted with the Gompertz equation. The blue line represents the actual growth of the cell and the red line is the line of best fit.

Figure 2.2: Growth of DH5α cells, fitted with the Logistic growth equation. The blue line represents the actual growth of the cell and the red line is the line of best fit.

After analysing Figures 2.1 and 2.2, it was evident that the cells were more inclined to following a logistic growth compared to Gompertz growth. Therefore the parameter fitting was carried out based on the logistic growth equation which can be seen in Figures 2.3, 2.4 and 2.5. The best fit was obtained by minimisation using Least Squares Estimation.

Figure 2.3: pSB1C3 plasmid grown in MG1655 cells, clearly following a logistic growth. The blue dotted line represents the actual growth of the cell and the red line is the line of best fit.

Figure 2.4: pSB1C3-PgadA (pH promoter) plasmid grown in MG1655 cells. The blue line represents the actual growth of the cell and the red line is the line of best fit.

Figure 2.5: pSB1C3-Pasr (pH promoter) plasmid grown in MG1655 cells. The blue line represents the actual growth of the cell and the red line is the line of best fit.

Two Compartments Model

To model our system, we devised a two compartment model using a set of ordinary differential equations: the first compartment represents the chicken's stomach. Due to the low pH, our cells will produce colicins. In the second part, cells will go on to lyse as they detect the presence of bile salts and thus releasing the colicins, killing the target bacteria present in the gut. A scheme of this system is depicted in Figure 3.1. The equations are as follows:

P_s - producing cell in the stomach

k_in - BactiFeed fed to the chicken

d_s - death rate of any of the producing cell through natural causes

k_trans - transit of cells from stomach to the liver

P_g - producing cell in the gut

k_lyse - how fast the cells lyse

k_out - any cells/colicins that may leave the cell without actually doing anything

L - number of colicins (per cell)

d_out - decay and follow out

T - target bacteria

Figure 3.1: Simplistic diagram of a chicken’s GI tract, illustrating where the different parameters of the equations are located in a chicken.

These equations were modelled in MATLAB in order to determine the efficiency of BactiFeed. The relative cell density vs time was measured corresponding to the amount of BactiFeed consumed by a chicken. The steady state of the target bacteria (e.g. salmonella or E.coli ) is at its optimum, the aim was to get it below a critical level. It can be seen in Figure 3.2 that as the BactiFeed is increased the steady state load decreases managing to kill all the target bacteria and going below the critical level which is ideal. However, it would be no use if it took months for the target bacteria to die as the chicken would be dead by then, therefore it was essential to know how fast the steady state load could be decreased and how low it could go.

Figure 3.2: This graph represents how the target bacteria steady state load (blue line) decreases as the consumption of BactiFeed decreases. The vertical line represents when the consumption of BactiFeed begins and the horizontal line represents the critical level that the target bacteria must pass. The orange, yellow and purple lines represent the increase in BactiFeed consumption.

This brings us to analysing the key features of our model. All parameters can affect the steady state. However, only a handful of them can be modified in our system. Therefore, we would like to find which ones are better to modify to increase the efficiency of our system. These parameters are:

• BactiFeed consumption rate
• Killing rate
• Number of colicins produced per cell
• Lyse rate

The number of colicins produced per cell and the lysis rate were chosen to analyse further as they are parameters that can be easily modified experimentally. A numerical analysis of these features is shown in Figures 3.3 and 3.4.

Figure 3.3: Represents the decrease of the bacteria that is being targeted, when the number of colicins that our cells can produce is increased.

Figure 3.4: Represents the decrease of the target bacteria when the lysis rate is increased.

Looking at Figures 3.3 and 3.4, the analysis suggests that both features can increase the efficiency of our system in decreasing the levels of target bacteria (or even eliminating them). However, the relationship is linear when the number of colicins is modified and asymptotic when the lysis rate is increased. Thus, a greater improvement of our system will be obtained if the number of colicins are increased rather than the lysis rate. This relationship can be observed when the steady state for the bacteria is computed:

Therefore, if time permitted, we could have tried to use a different plasmid backbone, one of a higher copy so that more colicins could be produced per cell or alternatively used a stronger ribosome binding site to try to improve our system. Nonetheless, our modelling can help us to increase the efficiency of our system in a clever way.

Conclusion

Mathematical modelling is a helpful tool for cases when experiment are difficult to perform or expensive. Even for simple models, like ours, it is expected to be a good model of reality, giving helpful insight on the behaviour of our system. Furthermore, it can help us to improve our system by tweaking the correct parameters and saving time and money. From our analysis, the best method to increase the efficiency of Bactifeed would be changing the levels of colicins per cell. However, we were unable to perform such experiments, and future work will have to validate our model. Nonetheless, this model-driven approach has been proved useful and can lead us for further development of our system.