Team:Imperial College London/Modelling/Output/Results and Conclusion

Objectives; Detailed Description; Parameters & Constants; Results & Conclusion;Download MatLab Files;

Results & Conclusion

Initial Results

Trying to correct this problem with the ode-solver, the following precuations were implemented:

• NonNegative function in Matlab preventing solver from reaching negative values - still some marginally negative values show
• Scaling - all the values were scaled up by a factor of 10⁶ as working on small numbers could be problematic for Matlab. Once the result is generated by the solver the resulting matrix is scaled back down by 10⁶.

When we entered the real production and degradation rates into our model, we once again obtained nagetive values. This was due to our set of differential equations being stiff. Since ode45 cannot solve stiff differential equations, we had to switch to using ode15s - an ode-solver designed to handle stiff equations.

Model pre-A This is the result of the simulation of simple production of Dioxygenase. It can be seen that the concentration will tend towards a final value of approximately 8×10^-6 mol/dm³. This final value is dependent on the production rate (which has been estimated for all of the models).

Model A

• 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^-5 mol/dm³. 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 higher than the maximum value that can be generated in the cell according to Model pre-A.

• Changing K_m:

K_m is indirectly proportional to the "final concentration" (which is the concentration at the end of the simulation), i.e. the bigger the value of K_m, the smaller the "final concentration" will be. Different K_m 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 (K_m≈k₂/k₁) is kept constant. This is important when simulating (in case entering very high values for k₁ and k₂ slows down the simulation).

• Changing k_cat

If k_cat = k₃ is increased, the model predicts that the dioxygenase concentration will rise quicker and to higher values. This indicates that k₃ is the slowest step in the enzymatic reaction.

• Changing production rate

At the moment, our biggest source of error could be the production rate, which we could not obtain from literature. Hence, we had to estimate the value of the production rate (see variables). We hope to be able to take a measurement of this value in the lab as it has a big effect on model's behaviour.

Sensitivity of Model A

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

• Initial Concentration

The initial concentration of split Dioxygenase, c₀, determines whether the system is amplifying.

If the initial concentration is changed, the observed behaviour is similar to the one from Model A. If the initial concentration of split Dioxygenase is increased, then the final concentration of Dioxygenase will increase as well (see graphs below).

• Model A vs. B

Running both models with the same initial conditions (c₀=10^-5 mol/dm³), it has been noted that Model B does not generate a siginificant amplfication over Model A. Hence, it would be more sensible to integrate a one step amplification module into our system.

Colour response model Initially, dioxygenase was being treated as "output" because the last reaction (catechol and dioxygenase react to form a coloured output) is common to all models. We concluded that 2-step amplification (Model B) presented little improvement over 1-step amplification (Model A). However, we realized that it is very important to add this last reaction, since it is an enzymatic reaction, hence an amplification step by itself. Adding it to all models means that Model A becomes a 2-step amplifier and Model B becomes a 3-step amplifier. Having previously drawn the conclusion that 2-step amplification is not much better than 1-step amplification, we decided to model this last step to deduce whether our initial conclusion was valid. If it was true, this would mean that our construct is not innovative at all.

The important information about the last amplification step is that the coloured compound (i.e. the product of the last enzymatic reaction) is toxic to the cells. It is suggested that product of Catechol destroys the cell membrane by inhibiting lipid peroxidation. It causes significant changes in the structure and functioning of membrane components (e.g. disruption of membrane potential, removal of lipids and proteins, loss of magnesium and calcium ions). These effects cause the loss of membrane functions, leading to cell death.

Since the product of Catechol acts on the cell membrane, it might not affect our enzymatic reaction immediately. In our simulation, we will try to model immediate cell death as well as neglecting the effect that the coloured output has on the cell. Comparing these two models will show if there are significant differences in the results.

Important Articles:

• Toxicity of three phenolic compounds and their mixtures on the gram-positive bacteria Bacillus subtilis in the aquatic environment[http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6V78-4XXNV4Y-4-7&_cdi=5836&_user=217827&_pii=S0048969709011668&_orig=search&_coverDate=02%2F01%2F2010&_sk=995919994&view=c&wchp=dGLbVzz-zSkzk&_valck=1&md5=f8dca6227c29db659ddbeb588ad115e7&ie=/sdarticle.pdf]
• Toxic Effects of Catechol and 4-Chlorobenzoate Stresseson Bacterial Cells[http://www.msk.or.kr/jsp/downloadPDF1.jsp?fileName=393-15.pdf]

Initial conclusions Despite our model not working entirely correctly (some negative concentrations were obtained), it was possible to deduce several points.

• The images presented below show cathecol being added at 3 different points in time. Cross-section refers to a point in time at which concentration of dioxygenase in the amplified systems crosses the concentration of the non-amplified system. From these graphs it can be seen that the output amplification is only visible after the cross-section has been reached. Note that these simulations were run for 1M solutions of catechol (which is quite high). This allows to see the differences between various amplification models easily.

• We noticed that amplification is only benefical if the initial concentration of catechol is quite high (>0.01M). For smaller concentrations of catechol, the dioxygenase conetration in different systems do not seem to be crucial for the speed of response (no difference between all 3 models). Hence, for small catechol concentrations the amplified systems are redundant (dioxygenase is overproduced) as concnetration of dioxygenase from simple production seems to be high enough to convert catechol almost instantenously. Amplification models become useful when there is a lot of substrate present to act on (ie. high concentration of catechol). Therefore, we need to determine the threshold value of coloured output for visibility. It will be a crucial factor in deciding whether the amplifiers designed by our team obtain the response faster than by simple production.

Problems with the simulation Once we have implemented the colour change into the models, we noticed that there are some inconclusive results. After adding catechol, some concentrations were reaching negative values. We checked our equations and constants but could not find a mistake. Hence, we concluded that there must be something wrong with the way that MatLab evaluates the equation or deals with the numbers. The problem seems to originate from the very rapid concentration change of catechol which disrupts the whole system.

Prospective solutions

• Implementation in TinkerCell

We hoped that TinkerCell imposes non-negative conditions on its solutions. Hence, we implemented the whole amplification model (including coloured output) in TinkerCell. However, we realized that TinkerCell does not deal well with very high or low numbers (For example, values higher than 10⁵ are not acceptable - this is important since our rate constants (k₁) are usually bigger than 10⁵. Also, the low degradation rates (10^-9) result in a zero output line). However, TinkerCell can still be used for testing that our Matlab programs behave the way we anticipated (by using default values of 1), as well as producing nice diagrams of our system.

• Varying ODE solver options in MatLab

We had a close look at the ODE solver options in MatLab. However, we were already using the one that produced the most reasonable results (ode15s). We found that decreasing Relative and Aboslute tolerances (to values as small as 10^-15) significantly improved the simulation. However, this is not an ultimate solution as in the simulations negative numbers still appear (order of 10^-15). We decided that such small negative concentrations were acceptable. We also decided that the point of interest lies between the first 100 to 150 seconds after adding catechol, while concentrations hit the negatve values at much higher time values.

The images below show the influence of the relative and absolute tolerance values on the model. Note that it was important to allow the ode-olver to adjust the time step automatically, as big time steps (1 second) were generating wrong answers for the catechol model. Adjusting time steps manually to very small values was not efficient (the whole simulation does not require very high definition simluation).

• Using SimBiology

We hoped that SimBiology could be more suited for our modelling than using ode-solvers, so we implemented our models into SimBiology. This package offers an interactive user interface similar to Tinker Cell, but uses MatLab to simulate. Initially, we confirmed that our simple production model (Model PreA) and 1-step amplification model (Model A) implemented in SimBiology generated exactly the same results as our ODE equation based models. The interface allowed us to have clearer control over paramters. It also allowed modelling special events, for example, adding catechol at certain point in time. Previously we had to split simulation into to two parts.

Final Conclusion

• Changing time when catechol is added

If Catechol is added before t= 1000s, then the coloured output will reach its threshold value faster by simple production. If Catechol is added when t>1000s, then the coloured output will increase (marginally) faster through the amplification step in Model A. There does not seem to be a significant difference between the two models (Model preA and Model A). These observations are true for intial concentration of dioxygenase equal to 10^-5mol/dm³. However, we noticed that if the initial concentration is raised to 10^-4mol/dm³, then Model A can be more beneficial than Model preA after only 100 seconds.

Hence, the question arises whether the concentration of protein in the cell can be as high as 10^-4mol/dm³. Our simple production model predicted that the concentration of protein could not reach such a high value. However, we decided to research more on ribosomal concentrations in bacteria to determine whether it is possible to establish such a high concentration in the cell.

On the website E.coli Statistics [http://gchelpdesk.ualberta.ca/CCDB/cgi-bin/STAT_NEW.cgi] it is stated that number of ribosomal proteins per cell is 900,000. In a cellular volume of order of 1μm³ = 10^-15dm³=10^-15L, the above number of ribosomes converts to 1.5×10^-3mol/L. This means that a concentration of 10^-4mol/dm³ is not completely out of scale.

• Changing concentration of catechol added

There seem to be 3 regions of catechol concentration that influence the system in different ways. These regions are: c>1M, 1M>c>0.01M, 0.01M>c. The boundaries of these regions tend to vary depending on the choice of other initial conditions. The values given above apply to boundary conditions that are currently considered to be physiologically relevant. Varying the initial concentration of catechol within the highest region does not result in any change of colour output response (It is possible that all enzymes are occupied and the solution is over saturated with catechol). In the middle region the catechol concentration influences the amplfication. Amplification decreased when the concentration tends towards 0.01M. When this region is entered, there is no difference in output production by the two models.

• Cell death

The coloured product of catechol kills cells by destroying the cell membrane. However, we do not know how quickly the cells will die. Therefore, we examined two different cases: immediate cell death and negligible cell death (i.e. cells death is negligible because it takes too long)

Running the simluation in Matlab (not Simbiology!), our conclusions are:

1. Immediate cell death slows down production of coloured output. Depending on the threshold concentration this can delay the detectable response by a few minutes.
2. If Catechol is added before t=1000s, then cell death slows down the response considerably.
3. In case of cells being modelled as alive, the difference between the amplified and the simple production model is smaller than it is in case of cell death.

Since it appears that the time of cell death is important, we decided to discuss this issue with Wolf and Harriet. Referring to this paper [1] we decided that cell death induced by catechool is a very slow process (we estimate that it will take a few hours) in comparison to the time scale that we are interested in (several seconds to minutes).

References

1. Chen, H. et al (2009) Toxicity of three phenolic compounds and their mixtures on the gram-positive bacteria Bacillus subtilis in the aquatic environment. Science of the Total Environment. [Online] 408(2010), 1043-1049. Available from: http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6V78-4XXNV4Y-4-7&_cdi=5836&_user=217827&_pii=S0048969709011668&_orig=search&_coverDate=02%2F01%2F2010&_sk=995919994&view=c&wchp=dGLbVzz-zSkzk&_valck=1&md5=f8dca6227c29db659ddbeb588ad115e7&ie=/sdarticle.pdf [Accessed 6th September 2010]
2. Park, S., Ko, Y. & Kim, C. (2001) Toxic Effects of Catechol and 4-Chlorobenzoate Stresses on Bacterial Cells. The Journal of Microbiology. [Online] 39(3), 206-212. Available from: http://www.msk.or.kr/jsp/downloadPDF1.jsp?fileName=393-15.pdf [Accessed 6th September 2010]
3. Habibi Nazhad, B. (2008) E. coli Statistics. [Online] Available from: http://gchelpdesk.ualberta.ca/CCDB/cgi-bin/STAT_NEW.cgi [Accessed 6th September 2010]