Team:Imperial College London/Modelling/Output/Detailed Description

The following page presents the details of the models that have been developed. Firstly, assumptions that have been exploited are explained. Then every model is presented separately as each of them has slightly different elements of the system and the interactions between them. However, there are only 3 fundamental biochemical processes that will be analysed:

1. Transcription
2. Translation
3. Enzymatic reactions

1. Law of Mass Action

During a meeting with our advisors, it was noted that our initial models (in which it was assumed that our system obeyed Michaelis-Menten kinetics) were wrong as the assumptions made by Michaelis-Menten approximation were not obeyed by the system.

A few of Michaelis-Menten assumptions were not met by our system:

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

Since we are continuously producing enzyme, Vmax will change. Therefore the conservation of enzymes E₀ = E + E_S does not hold for our system.

• Substrate >> Enzyme

We are producing both substrate and enzyme, so we have approximately the same amount of substrate and enzyme.

• Enzyme affinity to the substrate has to be high.

Click on the button below, in order to get to know more on Michaelis-Menten kinetics based models.

Elements of the system
Depending on which amplification it there is several species appearing. The full list is:

• GFP (on diagrams shown green) which was out original choice for output later on changed to dioxygenase acting on catechol. It was supposed to be split and attached to coiled coils. It would get activated by TEV protease.
• TEV protease (shown orange on diagrams)and split TEV (TEVs is shown yellow). Split TEV would be inactive form mounted on the coiled coils to be activated by HIV1 or another active TEV.
• HIV1 protease (shown blue on diagrams) which would be produced active upon receiving the activation signal in case of 2 amplification step outputs.

Simple models
Our first models only were accounting for the protein and mRNA production:

2-step amplifier with distinct protease at each level.
	Where: m - is the concentration of mRNA, p - is the protein concentration (subscript indicates which protein)
2-step amplifier with TEV protease implemented at both amplification levels.

Implementation of Michaelis-Menten kinetics
So, far the models were considering only the production and degradation of species. The example presented below includes enzymatic interaction between the species governed by Michaelis-Menten.

2-step amplifier with distinct protease at each level.

Implementation in Matlab
The Matlab code for the different stages of amplification and diagrams can be found [http://www.openwetware.org/wiki/Image:Modelling.docx here].

Kinetic constants

	GFP	split GFP	TEV	split TEV
Km and kcat	-	Km = 0.061; kcat = 0.16; [1]	40% of value for TEV	-
Half-life or degradation rate	Half-life in B.sub approximately 1.5 hours	-	-	Half-life shorter than GFP
Production rate in B.sub	-	-	-	-

Conclusion

We were not able to obtain all the necessary constants. Hence, we decided to make educated guesses about possible relative values between the constants as well as varying them and observing the change in output.

As the result, we concluded that the amplification happens at each amplification level proposed. The magnitude of amplification varies depending on the constants. There is not much difference between using TEV or HIV1.

Change of output

During our literature research, we came across a better output, so we abandoned the idea of using GFP as an output. Instead, we are using catechol. An enzyme, dioxygenase, will be acting on the catechol, which will then result in a coloured output. Catechol will be added to the bacteria manually (i.e. the bacteria will not produce catechol). Hence, in our models dioxygenase will be treated as an output as this enzyme is the only activator of catechol in our system. This means that the change of catechol into its colourful form is dependent on the dioxygenase concentration.

References

1. Kapust R. et al (2001) Tobacco etch virus protease: mechanism of autolysis and rational design of stable mutants with wild-type catalytic proficiency. Protein Engineering. [Online] 14(12), 993-1000. Available from: http://peds.oxfordjournals.org/cgi/reprint/14/12/993 [Accessed 28th July 2010]

As we could not use the Michaelis-Menten simplification to model enzymatic reactions in our system, we will had to solve the problem from first principle. It meant referring to more general set of assumptions called Law of Mass Action. This allowed us to model our enzymatic reactions without making assumptions about amounts of particular species as long as the amounts are bigger than single molecule level. This resulted in bigger number of partial differential equations as there was one per each species instead of 1 per reaction.

2. Model preA: Simple production of dioxygenase

This model was developed to illustrate 1-step amplification output.

Elements of the system

1. Dioxygenase (C230 chown blue) is the enzyme that starts getting produced upon activation signal (eg. detection of Schistosoma).
2. Catechol (cat. shown white with orange edge) is artificially added chemical during the experiment. It is acted on by dioxygenase to produce yellow muconic acid.

Transcription and translation(simple production) of dioxygenase and its enzymatic activation of yellow colour.

Interactions between elements

Initially, dioxygenase has to be transcribed and translated. This has been described using 1 ODE equation:

The enzymatic interactions can be described in the following way:

It is important to mention dioxygenase molecules tretramerize before becoming active enzymes. It has been very simplified in the model. Tetramerization is accounted for by simply dividing dioxygenase concentration by 4 before it acts on catechol.

Differential equations

The above reaction can be written in terms of ordinary differential equations:

Implementation in Matlab

These equations were implemented in Matlab, using a built-in function (ode15s) which solves ordinary differential equations. For the code please refer to the Download MatLab Files section.

3. Model A: Activation of Dioxygenase by TEV enzyme

This model was developed to illustrate 1-step amplification output.

Elements of the system

1. TEV protease (shown pink) is simply produced upon detection of schistosoma.
2. Dioxygenase (C230 chown blue) is pre-produced in inactive form by being attached to GFP. TEV releases dioxygenase and allows it to tetramerize and activate.
3. Catechol (cat. shown white with orange edge)

The species that are shown in front of vertical line which indicates beginning of experiment mean that they have been accumulated beforehand in the cell. TEV protease activates inactive dioxygenase which acts on catechol to produce colour.

Interactions between elements

This model includes 2 enzymatic reactions:

1. Reaction between dioxygenase and catechol, which is exactly the same as in model preA
   
   In the equation above : D is Dioxygenase, C is Catechol, C-D is catechol-dioxygenase transition complex, M_a is Muconic acid - the yellow ouptput.
2. Reaction between TEV and inactive dioxygenase
   
   In the equation above: TEV = Protease of the Tobacco Etch Virus, sD = split Dioxygenase, TsD = intermediate complex of the enzymatic reaction,
   D = Dioxygenase

Differential equations

This is a simple enzymatic reaction, where TEV is the enzyme, Dioxygenase the product and split Dioxygenase the substrate. Choosing k₁, k₂, k₃ as reaction constants, the reaction can be rewritten in these four sub-equations:

Implementation in Matlab

These equations were implemented in Matlab, using a built-in function (ode15s) which solves ordinary differential equations. For the code please refer to the Download MatLab Files section.

Implementation in TinkerCell

Another approach to model the amplification module would be to implement it in a program such as TinkerCell (or CellDesigner). This was used to check whether implementation in Matlab generates similar results. If happened otherwise, we would need to look for reasons for those differences in the programs. As the results, generated by Matlab were the same, only Matlab code has been developed further as it allows more flexibility, control and insight.

Model A implemented in TinkerCell.

4. Model B: Activation of Dioxygenase by TEV or activated split TEV enzyme

Elements of the system

1. split TEV protease (shown purple) is introduced as the 3rd amplification step. It exists in 2 versions (not shown on diagram): a and b as it does not separate into 2 identical halves.
2. TEV protease (shown pink) is simply produced upon detection of schistosoma.
3. Dioxygenase (C230 chown blue) is pre-produced in inactive form by being attached to GFP. TEV releases dioxygenase and allows it to tetramerize and activate.
4. Catechol (cat. shown white with orange edge)

This diagram introduces inactive split TEV protease attached to a coiled-coil as the third amplification step. Both inactive compounds have active site for TEV to activate tehm which results in multiple possibilities of action.

Interactions and assumptions

This version includes the following features:

• Split TEV is specified to have a and b parts
• TEVa is forbidden to interact with TEVa (though in reality there could be some affinity between the two). Same for the interaction between Tevb and Tevb
• Both TEV and TEVs are allowed to activate dioxygenase
• Dioxygenase is assumed to be active as a monomer
• Activate split TEV (TEVs) is not allowed to activate sTEVa or sTEVb (this kind of interaction is accounted for in the next model version)
• This model does not include any specific terms for time delays

Implementation in Matlab

For the code please refer to the Download MatLab Files section.

5. Model C: Further improvements

This model has not been implemented because of the conclusions that we reached from Models A and B. It would include the following features:

• activated split TEV (TEVs) is allowed to activate not only sD but sTEVa and sTEVb