# Output Amplification Model

## Why do we want to model the output amplification?

• To determine whether simple production is better than 1- or 2-step amplification.

## Model pre-A: Simple production of Dioxygenase

Simple production of Dioxygenase

This model involves the simple transcription and translation of Dioxygenase. Combining the trancription and translation step, we have one differential equation:

• [p'] = s - d*[p]
• [p]...concentration of dioxygenase
• s...production rate of dioxygenase

Using the built-in Matlab function ode45, which is a solver for differential equations, we obtain the following graph:

Simulation of Model pre-A

The concentration of Dioxygenase produced tends towards a value of 8*10^-6 mol/dm^3. This could indicate the maximum concentration of Dioxygenase that will be produced in a cell.

## Model A: 1-step amplification

1-step amplification

The 1-step amplification model involves the following enzymatic reaction:

• TEV + sD <--> TEV-sD --> TEV + D
• TEV...Tobacco Etch Virus
• sD...split Dioxygenase
• TEV-sD...intermediate complex
• D...Dioxygenase

Note that this enzymatic reaction cannot be simplified using Michaelis-Menten kinetics because of the assumptions of Michaelis-Menten. Michaelis-Menten assumptions that our system does not meet:

• Vmax is proportional to the overall concentration of the enzyme.

But we are producing enzyme, so Vmax will change! Therefore, the conservation E0 = E + ES does not hold for our system.

• Substrate >> Enzyme

Since we are producing both substrate and enzyme, we have roughly the same amount of substrate and enzyme.

Therefore, we have to solve from first principle by using the law of mass action. Assumptions of mass action:

• Chemical reaction rate is directly proportional to the concentration of the reacting substances

The reaction can then be rewritten in these four sub-equations:

1. [T'] = -k1[T][sD] + (k2+k3)[TsD] + sT - dT[T]
2. [sD']= -k1[T][sD] + k2[TsD] + ssD - dsD[sD]
3. [TsD'] = k1[T][sD] - (k2+k3)[TsD] - dTsD[TsD]
4. [D'] = k3[TsD] - dD[D]

Implementing these differenital equations in the ode45-solver does not give reasonable results (e.g. concentrations fall below zero). We found out that this is due to our set of differential equations being stiff, hence, we use a different Matlab solver (ode15s). The results from the Matlab simulation are highly dependent on the initial concentration of split Dioxygenase, as well as other constants that we have approximated.

Here is the Matlab result for an initial split dioxygenase concentration of 10^-5 mol/dm^3:

Commparison between Model pre-A and A

It can be seen that the 1-step amplification is more efficient if we wait for longer than 1000s. However, the most important information for this graph is the threshold concentration, for which Dioxygenase becomes visible. If this threshold concentration corresponds to a time greater than 1000s, then the 1-step amplification will be more efficient.

### Varying constants

We want to determine how our system reacts if different parameters are changed. This is to find out which parameters our system is very sensitive to.

Parameter Sensitivity
Initial concentration of split Dioxygenase Change of one order of magnitude in the initial concentration, c0, gives change of oen order of magnitude in the output concentration (range: 1>c0>10^-5). Loses sensitivity for extremely high or low values.
Km Change of one order of magnitude results in change of output concentration by one order of magnitude (0.01<Km<100). At values smaller than 0.01, the sensitivity is lost. For higher values than 100 the sensitivity is at least the same as the change of order of magnitude.
kcat kcat proportional to dioxygenase production (1-to-1 sensitivity for all values) for an initial concentratio of 0.01 mol/dm^3. For very high initial concentrations, the system is very sensitive to changes in kcat.
Production rate of TEV 1-1 sensitivity for most values. At some point the systemâ€™s response is limited by the initial concentration of sD, so for very high TEV production rates not much change is observed.
Production rate of split Dioxygenase Not much influence on 1-step amplification. However, the value seems to be crucial for simple production of Dioxygebase (1-1 order of magnitude sensitivity).
Degradation rates Sensitive within the relevant range. Not very sensitive for values smaller than 10^-6;. For high degradation rates (1>degradation rate>0.01): unexplainable behaviours.

Hence, the system is sensitive to most of the constants (given a particular range of values). The most crucial one, however, seems to be the initial concentration of split Dioxygenase.

## Model B: 2-step amplification

2-step amplification

The only difference between this model and the 1-step amplification is the added step of amplification. Similar to 1-step amplification, this system can be modelled using mass action.

Implementing the 2-step amplification model in Matlab (using ode15s) gives the following graph, comparing all three different models:

Comparison between Model pre-A, A and B

It can be seen that 2-step amplification is only marginally more efficient than 1-step amplification. Therefore, it would be more sensible to implement the 1-step amplification.

# Receptor and Surface Protein Model

## Why do we want to model the protein display?

The aim of this model is to determine the concentration of Schistosoma elastase or TEV protease that should be added to bacteria to trigger the response. It is also attempted to model how long it takes for the protease or elastase to cleave enough peptides.

## Our model

Similarly to the 1-step amplification model, cleavage of proteins is an enzymatic reaction, which can be written as:

Substrate + Enzyme <--> Substrate-Enzyme complex --> Product + Enzyme

In our case:

• Substrate = Protein
• Enzyme = TEV Protease (or Schistosoma Elastase)
• Product = Peptide

Modelling of this reaction is very similar to the 1-step amplification. However, all constants and initial concentrations need to be changed.

Threshold concentration of peptide The peptide concentration required to activate ComD is 10 ng/ml [1]. This is the threshold value for ComD. Our model will tell us how long it takes until this threshold is reached.

Protein production in Bacillus Subtilis This paper mentions that each cell of B.sub expresses 2.4*10^5 peptides, which equals 0.398671*10^-18 mol. Given the volume of a B.sub cell (V = 2.79*10^-15 dm^3), we can determine the concentration of peptide in a B.sub cell: c = 1.4289*10^-4 mol/L.

Diffusion Since the peptide is displayed on the cell wall, the diffusion of peptide into the extracellular space had to be considered. However, it is impossible to consider diffusion for a very long distance (since this will take a long time). We decided to focus on diffusion within a defined space around each bacterial cell, which we will call control volume (CV). To find this Control Volume, we used data from iGEM Imperial 2008.

CFU/ml vs. Optical Density at 600nm (OD600). Figure taken from Imperial iGEM 2008 Wiki page.

From this data, we deduced that the control volume for one bacterial cell is 2*10^(-12) dm^3/cell. For simplicity, we will assume that CV is a cube. Therefore, each side length of the CV is 1.26*10^(-5) m.

Choice of control volume allows simplifications

• Assuming that the bacterial cell is placed at the centre of each CV, we calculated that the maximum time allowed for diffusion is 0.74s.

After this time we can assume that the concentration around the bacterial cell will be uniform.

• Since the cells are not in isolation, but each CV is adjacent to other CVs, we can neglect diffusion out of the CV because fluxes across the border, in and out, will be of similar values (see figure below).
Two adjacent cells with their control volumes. Shows diffusive fluxes in and out of each cell for one dimension.

## Matlab Simulation

Graphs showing the simulation using [TEV]0 = 4*10^-4 mol/dm^3. The graph on the bottom right hand-side shows that the AIP threshold (red line) is reached after 22 s.

## Sensitivty of the Protein Display Model

• Changing initial concentration of TEV

Whether the threshold concentration of AIP is reached is highly dependent on the initial concentration of TEV. The smallest initial concentration of TEV, [TEV]0, for which the threshold is reached is 6.0*10^-6 mol/dm^3. On the graoh below, it can be seen that the optimal [TEV]0 is a concentration higher than 10^-4 mol/dm^3, which corresponds to a threshold being reached within 1.5 minutes.

Graph showing when threshold AIP concentration is reached (for different initial TEV concentrations). Notice log-log scale.
• Changing the production rate

1 order of magnitude change in production rate results in at least 50s (50 seconds is the smallest step for 1 order change - for others is way bigger) delay of the AIP concentration riching the threshold concentration.

• Changing production rate

It influences a lot the time duration of the AIP concentration being above the threshold level. The higher it is the shorter the receptor is activated (at extreme values, AIP concentration never get to the threshold). However, it has not much influence on how fast the threshold is being reached.