IGEM:Imperial/2010/Overview

=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


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
 * d...degradation rate of dioxygenase

Using the built-in Matlab function ode45, which is a solver for differential equations, we obtain the following graph: 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.
 * Here is the Matlab code for Model pre-A.

Model A: 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: But we are producing enzyme, so Vmax will change! Therefore, the conservation E0 = E + ES does not hold for our system. Since we are producing both substrate and enzyme, we have roughly the same amount of substrate and enzyme.
 * Vmax is proportional to the overall concentration of the enzyme.
 * Substrate >> 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: 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.


 * Here is the Matlab code for Model A.

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.

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


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: 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.


 * Here is the Matlab code for Model B.

=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. 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.

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



Matlab Simulation



 * Here is the Matlab code for the Matlab simulation.

Sensitivty of the Protein Display Model
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. 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. 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.
 * Changing initial concentration of TEV
 * Changing the production rate
 * Changing production rate