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Team:Imperial College London/Modelling/Output/Results and Conclusion
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1. Initial conclusions based on levels of dioxygenaseThe conclusions below are based on models that were implemented up to the dioxygenase. Catechol was not added to the system, so conclusions are based on the concentrations of dioxygenase. Results of full models, including colour output, are included in section 2 below. Challenges Despite being quite thoughtful in translating our models into MatLab, unexpected outputs were observed. For example some species were generating negative concentrations. In particular, for concentrations smaller than 10-4 mol/dm3 the results were inconclusive since they were oscillating around zero. We realised that this could be due to the ode-solver that we were using initially (ode45 in Matlab). Trying to correct this problem with the ode-solver, the following precuations were implemented:
When we entered the real production and degradation rates into our model, we again obtained negative 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 and degradation rate(which has been estimated for all of the models).
Model A 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 and kcat. Model B Sensitivity of Model B This model has shown very similar sensitivity results as Model A, eg.:
The initial concentration of split Dioxygenase, c0, determines whether the system is amplifying. 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.
2. Final conclusions based on the colour outputThe important information about the coloured compound (i.e. the product of the last enzymatic reaction) is toxic to the cells ([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],[http://www.msk.or.kr/jsp/downloadPDF1.jsp?fileName=393-15.pdf]). It is suggested that product of by-product of catechol-dioxygenase reaction 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 by-product of catechol-dioxygenase acts on the cell membrane, which usually is quite a slow process that takes more than an hour. We were interested in seconds to several minutes after catechol addition, so we decided to neglect its toxic effect on our cell. Out of curiosity, model with immediate cell death was implemented as well to allow comparison of the two approaches. Challenges Once we have implemented the colour change into the models, we noticed that there are some odd 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. Tried solutions:
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 parameter values of 1), as well as producing illustrative diagrams of our system.
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 Absolute 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 negative 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-solver 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).
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 PDE equation based models. The interface allowed us to have clearer control over parameters. 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. Conclusions
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.
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.
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:
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).
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