Team:Dundee/Model

Modelling

Mathematical Modelling

Introduction

Mathematical modelling is essential for synthetic biology, it acts as a link between the interpretation of experiments and helps to improve our system. Simulating the behaviour of the experimental systems, makes it 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 principal components therefore makes it easier to analyse. Our treatment that we are creating 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. 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.

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, logistic growth and the Gompertz function were used initially to test what function gave the best fit. 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. Firstly analysis was carried out for the DH5α strain, figures 2.1 and 2.2 illustrate this.

Figure 2.1: Growth of DH5α cells, fitted with the Gompertz equation. One time step was equal to 3 mins, the plate reader experiment was run for approximately 22 hours. 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. One time step was equal to 3 mins, the plate reader experiment was run for approximately 22 hours.

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. Also it was obvious that the cells did not grow exponentially therefore this was not modelled. 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 parameters were changed until we got the best fit and this helped shape the modelling. The blue line represents the actual growth of the cell and the red line is the line of best fit.

Figure 2.3: MG1655 cells grown in pSB1C3, clearly following a logistic growth. One time step was equal to 15 mins, the plate reader experiment was run for approximately 24 hours. The blue dotted line represents the actual growth of the cell and the red line is the line of best fit.

Figure 2.4: MG1655 cells grown in pSB1C3 with the promoter gadA. One time step was equal to 15 mins, the plate reader experiment was run for approximately 24 hours. The blue line represents the actual growth of the cell and the red line is the line of best fit.

Figure 2.5: MG1655 cells grown in pSB1C3 with the promoter asr. One time step was equal to 15 mins, the plate reader experiment was run for approximately 24 hours. The blue line represents the actual growth of the cell and the red line is the line of best fit.

Two Part Model

Sticking with the chicken as an example, it was decided to utilise the fact that the they have a low pH level in their stomachs therefore the cells would produce colicins here and then go on to lyse when bile salts were detected - this the technique that was decided when choosing suitable promoters for our system. ODEs were then used again but for a two part model of the stomach and the rest of the gut. 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.

A schematic diagram shown in figure 3.1 illustrates the equations in regards to a chickens gut. 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 it’s 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: Represents how the target bacteria steady state load 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.

This bring us to analysing the key features of our model. We determined what features could affect the steady state:

• 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 based on the analysis of the steady state, these are 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 these key features are crucial in eradicating a sufficient amount of target bacteria to go below the critical level and therefore removing any infection that may 9 be present. Also since increasing the number of colicins that are produced by cells, results in an almost linear decrease of the target bacteria - this method would be preferred over increasing the lysis rate. 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.

Conclusion

It can be concluded that after analysing the graphs, that there is a possibility of our product, BactiFeed, being able to kill the target bacteria to the extent that - if any bacterial infections are present they can be eradicated. This is shown by the analysis of the key features mentioned in section 3, Now that we know we know this information, in the future we could work on trying to find alternative ways to make our cells produce more colicins therefore improving the efficiency of BactiFeed.