IGEM:Imperial/2010/Modelling

Have a look at this link: Synthetic Biology (Spring2008): Computer Modelling Practicals

Have a look at Cell Designer to easily generate images of the system.

Example on how Valencia 2006 team used SimulLink to simulate their project: Valencia 2006 PowerPoint presentation

Objectives

Week 6

Day	Monday	Tuesday	Wednesday	Thursday	Friday	Weekend
Date	09	10	11	12	13	14-15
Objective				Find constants	Find protein production constants and TEV reaction rate constants
Completion				We didn't manage to complete the task	The orders of magnitude established - ready to run simulations

Week 7

Day	Monday	Tuesday	Wednesday	Thursday	Friday	Weekend
Date	16	17	18	19	20	21-22
Objective	Implementing the constant ranges in the output model. Comparing the results between the models.	Start modelling the protein display signalling to find the concentrations. Explain the oscillations that are occuring in the output amplification model	Research stiff differential equations Research on receptor (especially MAP kinase) Finalise results for amplification model Prepare presentation
Completion	The task is accomplished. However, unexplained oscillations are observed for some specific values	We got rid of the oscillations in our model (by using ode15s instead of ode45).

Output amplification model

Motivation: We have come up with a simple concept of amplification of output done by enzymes. Before the final constructs are assembled within the bacterial ogranism, it is beneficial for us to model the behaviour of our design.

The questions to be answered:

1. How beneficial use of amplification is? (compare speeds of response of transcription based output to amplified outputs)
2. How many 'amplification levels' are beneficial to have? (if too many amplification steps are involved, the associated time delay with expressing even amplfiied output may prove it not to be beneficial.
3. Does mixing of amplfication levels have a negative infleunce on output? Is it better to use TEV all the way or HIV1? Modelling should allows us to take decision which design is more efficient.

First attempt

At each stage of amplification a distinct protease is being used

At each stage of amplification a distinct protease is being used

---

TEV is used at both stages of amplification

TEV is used at both stages of amplification

Second attempt

Model improved to account for the enzymes (protease action)

Implementation in Matlab

The Matlab code for the different stages of amplification and diagrams can be found here.

Kinetic constants

Quality	GFP	TEV	split TEV	split GFP
Km and Kcat	Doesn't apply	TEV constants (Km and kcat)	40% of whole TEV	Doesn't apply
half-life or degradation rate	Half-life of GFP in Bacillus = 1.5 hours - ref. Chris	?	?	Half-life shorter than GFP
production rate in B.sub	?	?	?	?

Conclusions

We couldn't 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. It's magnitude varies depending on the constants. There doesn’t seem to be much difference in substitution of TEV with HIV1.

Modified version

Michaelis Menten kinetics does not apply

We cannot use Michaelis-Menten kinetics because of its preliminary assumptions, which our system does not fulfil. These assumptions are:

• 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.

• Enzyme affinity to substrate has to be high.

Therefore, the model above is not representative of the enzymatic reaction. As we cannot use the Michaelis-Menten model we will have to solve from first principle (which just means writing down all of the biochemical equations and solving for these in Matlab).

Changes in the system

GFP is not used any more as an output. It is dioxygenase acting on the catechol (activating it into colourful form). Catechol will be added to bacteria, it won't be produced by them. Hence, basically in our models dioxynase is going to be treated as an output as this enzyme is recognised as the only activator of catechol in our system. This means that change of catechol into colourful form is dependent on dioxygenase concentration.

Models:

Model preA: Production of Dioxygenase

This model includes transcription and translation of the dioxygenase. It does not involve any amplification steps. It is our control model against which we will be comparing the results of other models.

Model A: Activation of Dioxygenase by TEV enzyme

The reaction can be rewritten as: TEV + split Dioxygenase <-> TEV-split Dioxygenase -> TEV + Dioxygenase. 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:

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]

These four equations were implemented in Matlab, using a built-in function (ode45) which solves ordinary differential equations. The Matlab code for this module can be found here.

Results of the Matlab simulation, setting all constants to 1

Implementation in TinkerCell

Another approach to model the amplification module would be to implement it in a program such as TinkerCell (or CellDesigner). It would also be useful to check whether the Matlab model works.

LHS: Network implemented in TinkerCell, RHS: constants and results

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

This version includes the following features:

• 2 amplification steps (TEV and split TEV)
• Split TEV is specified to have a and b parts
• TEVa is forbidden to interact TEVa (though in reality there could be some affinity between the two). Same for interaction between Tevb and Tevb
• Both TEV and TEVs are allowed to activate dioxugenase molecule
• Dioxugenase 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)
• There is no specific terms for time delays included

The MatLab code can be found here. Note that no final conclusions should be drawn before realistic estimates for kinetic constants are included. It wasn’t done so far.

All chemical species appearing in the model	Network of the improved model
Resulting graphs part 1. Compare the production graphs of TEV (transcribed and translated from scratch and the Dioxugenase which is the final species in the whole cascade	Resulting graphs part 2.

Model C: Further improvement

This model is not implemented yet.

This version adds the following features:

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

Network of the further improved model

Network of the further improved model (continued)

Results

The major concern of the results that we got (in particular for small concentrations < 10^(-4) mol*dm^(-3)) was that the solver was oscillating about positive or zero values but marking concentration values below zero too. It was recognised as faulty and probably leading the solver to false solutions.

Trying to prevent the ode solver going crazy, the following precuations were implemented:

• function preventing solver from going to negative values (does it really work) - still some marginally negative values show
• Scaling - all the values were scaled up by a factor of 10^6 as working on small numbers could be problematic for matlab (especially at the beggining of the cascade). Once the result is generated by the solver the resulting matrix is scaled back down by 10^6

Model pre-A

This is the result for the simulation of simple production of Dioxygenase. It can be seen that the concentration will tend towards a final value of around 8.5*10^-6 mol dm^(-3). This final value is dependent on the production rate (which we have just estimated!).

Results of the Matlab simulation of Model preA

Model A

When we entered production and degradation rates into our model, it did not seem to work properly (e.g. we got negative concentrations as our output). We found out that this is due to our set of differential equations being stiff. Since ode45 cannot solve stiff differential equations, we had to switch to using ode15s.

• Initial Concentration

The initial concentration of split Dioxygenase, c0, determines whether the system is amplifying. The minimum concentration for any amplification to happen is 10^-2 mol dm^(-3). If the initial concentration of split Dioxygenase is higher, then the final concentration of Dioxygenase will be higher as well (see graphs below). Note that the obtained threshold value is above the maximum value that can be generated in the cell according to Model pre-A!!!

Initial concentration of split Dioxygenase: 0.01 mol dm^(-3)

• Changing Km:

Km is indirectly proportional to the "final concentration" (which is the concentration at the end of the simulation), i.e. the bigger the evalue of Km, the smaller the "final concentration" will be. Different Km values determine how quickly the amplification will take place.

(Also, it was found that the absolute value of k1 and k2 entered into Matlab does not change the outcome as long as the ratio between them (Km~k2/k1) is kept constant. This is important when simulating (in case entering very high values for k1 and k2 takes too long to simulate).

• Changing kcat

Model predicts the concentration values for dioxygenase to raise quicker and to higher values for increasing values of kcat=k3. That indicates that k3 is actually the slowest step in enzymatic reaction and allows us to appreciate how our system is dependent on kinetic properties of enzyme.

• Changing production rate

At the moment, our biggest source of error could be the production rate, which we weren't able to obtain from literature. So, we had to estimate (see below) the value of the production rate. We hope to be able to take rough measurements of that value in the lab as it has big effect on models' behaviour.

Model B

• Initial Concentration

The initial concentration of split Dioxygenase, c0, determines whether the system is amplifying. The minimum concentration for any amplification to happen is noted to be higher than in case of ModelA. It is between 10^(-3) and 10^(-2) mol dm^(-3).

The behaviour in varying the initial concentration obeys similar relationship as the one of Model A: If the initial concentration of split Dioxygenase is higher, then the final concentration of Dioxygenase will be higher as well (see graphs below).

• Model A vs. B

Having run both models with the same initial conditions (c0=0.01 mol dm^(-3)). It has been noted that Model B does not generate very siginificant amplfication over the Model A. It seems as well that it takes as much time for it to reach its pick values.

Model A

Model B

Constants for Modelling

Type of constant	Derivation of value
TEV Enzyme dynamics	Enzymatic Reaction: E + S <-> ES -> E + P Let k1 = rate constant for E + S -> ES k2 = rate constant for E + S <- ES kcat = rate constant for ES -> E + P We know that Km = (kcat + k2)/k1 Assuming that kcat << k2 << k1, we can rewrite Km ~ k2/k1 From this paper constants for TEV can be found: e.g. wildtype TEV Km = 0.062 +/- 0.010 mM kcat = 0.16 +/- 0.01 s^-1 These values correspond with our assumption that kcat ~ 0.1 s^-1 and Km ~ 0.01 mM. Hence, we can estimate the following orders of magnitude for the rate constants: k1 ~ 10^5 M^-1 s^-1 k2 ~ 10^3 s^-1 Using these values should be a good approximation for our model.
Degradation rate (common for all)	Assumption: To be approximated by cell division (dilution of media) as none of the proteins are involved in any active degradation pathways Growth rate (divisions/h): 0.53<G.r.<2.18 Hence on average, g.r.=1.5 divisions per hour => 1 division every 1/1.5=0.6667 of an hour (40mins) To deduce degradation rate use the following formula: τ_(1⁄2)=ln2/k Where τ_(1⁄2)=0.667 hour, k – is the degradation rate k=ln2/τ_(1⁄2) = 0.000289 s^(-1)
Production rate (TEV and dioxygenase)	We had real trouble finding the production rate values in the literature and we hope to be able to perform experiments to obtain those values for (TEV protease and catechol 2,3-dioxygenase). The experiments will not be possible to be carried out soon, so for the time being we suggest very simplistic approach for estimating production rates. We have found production rates for two arbitrary proteins in E.Coli. We want to get estimates of production rates by comparing the lengths of the proteins (number of amino-acids). As this approach is very vague, it is important to realise its limitations and inconsistencies: Found values are taken from E.Coli not B.sub. The two found rates are of the same value for quite different amino-acid number which indicates that protein folding is limiting the production rates (we use the chosen approach as the only way of getting the estimate of order of reaction) LacY production = 100 molecules/min (417 Amino Acids) LacZ production = 100 molecules/min (1024 AA) Average production ≈ 100molecules/min 720 AA That gives us: TEV production ≈ 24 molecules/min = 0.40 molecules/s (3054 AA) As production rate needs to be expressed in concentration units per unit volume, the above number is converted to mols/s and divided by the cell volume → 2.3808*10^(-10) mol*dm^(-3)*s^(-1) C23D production ≈ 252 molecules/min = 4.2 molecules/s (285 AA) → 2.4998*10^(-9) mol*dm^(-3)*s^(-1) We will treat these numbers as guiding us in terms of range of orders of magnitudes. We will try to run our models for variety of values and determine system’s limitations.
Kinetic parameters of dioxygenase	Initial velocity of the enzymatic reaction was investigated at pH 7.5 and 30 °C. Wild type (we use that one) Km = 10μM kcat = 52 s^(−1) Mutated type Km = 40μM kcat = 192 s^(−1) Consequently, the kcat/Km 4.8 of the mutant was slightly lower than kcat/Km 5.2 of the wild type, indicating that the mutation has little effect on catalytic efficiency. reference
Dimensions of Bacillus subtillis cell	Dimensions of Basillus subtilis (cylinder/rod shape) in reach media (ref. bionumbers): diameter: d= 0.87um length: l=4.7 um in rich media This gives: Volume= π∙(d/2)^2∙l=2.793999 μm^3≈ 2.79∙10^(-15) dm^3
Split TEV production rates	*Assume the both parts of split TEV are half of size of the whole TEV (3054/2=1527 AA) The length of the coil is 90 AA The whole construct is then: 1617 AA → split TEV production rate ≈ 1.2606*10^(-10) mol*dm^(-3)*s^(-1)