Team:Imperial College London/Modelling/Output/Results and Conclusion
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Trying to correct this problem with the ode-solver, the following precuations were implemented: | 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 | *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 <math>10^6</math> 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 < | + | *Scaling - all the values were scaled up by a factor of <math>10^6</math> 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 <html>10<sup>6</sup></html>. |
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. | 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 <math>8*10^{-6} mol/dm^3</math>. This final value is dependent on the production rate (which has been estimated for all of the models). | 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 <math>8*10^{-6} mol/dm^3</math>. This final value is dependent on the production rate (which has been estimated for all of the models). | ||
[[Image:Model_prea.bmp|450px|thumb|center|alt=A|Results of the Matlab simulation of Model preA]] | [[Image:Model_prea.bmp|450px|thumb|center|alt=A|Results of the Matlab simulation of Model preA]] | ||
- | + | Model A | |
*'''Initial Concentration''' | *'''Initial Concentration''' | ||
- | The initial concentration of split Dioxygenase, < | + | The initial concentration of split Dioxygenase, <html>c<sub>0</sub></html>, determines whether the system is amplifying. The minimum concentration for any amplification to happen is <html>10<sup>-5</sup> mol/dm<sup>3</sup></html>. 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.''' |
- | [[Image:Comparison_a%2C_prea.bmp|450px|thumb|center|alt=A|Comparison between Model pre-A and Model A. Initial concentration of split Dioxygenase: < | + | [[Image:Comparison_a%2C_prea.bmp|450px|thumb|center|alt=A|Comparison between Model pre-A and Model A. Initial concentration of split Dioxygenase: <html>10<sup>-5</sup> mol/dm<sup>3</sup></html>]] |
- | *'''Changing < | + | *'''Changing <html>K<sub>m</sub></html>:''' |
- | < | + | <html>K<sub>m</sub></html> is indirectly proportional to the "final concentration" (which is the concentration at the end of the simulation), i.e. the bigger the value of <html>K<sub>m</sub></html>, the smaller the "final concentration" will be. |
- | Different < | + | Different <html>K<sub>m</sub></html> values determine how quickly the amplification will take place. |
(Also, it was found that the absolute value of <math>k_1</math> and <math>k_2</math> entered into Matlab does not change the outcome as long as the ratio between them (<math>K_m</math>≈<math>k_2/k_1</math>) is kept constant. This is important when simulating (in case entering very high values for <math>k_1</math> and <math>k_2</math> slows down the simulation). | (Also, it was found that the absolute value of <math>k_1</math> and <math>k_2</math> entered into Matlab does not change the outcome as long as the ratio between them (<math>K_m</math>≈<math>k_2/k_1</math>) is kept constant. This is important when simulating (in case entering very high values for <math>k_1</math> and <math>k_2</math> slows down the simulation). | ||
- | *'''Changing < | + | *'''Changing <html>k<sub>cat</sub></html>''' |
- | If < | + | If <html>k<sub>cat</sub> = k<sub>3</sub></html> is increased, the model predicts that the dioxygenase concentration will rise quicker and to higher values. This indicates that <html>k<sub>3</sub></html> is the slowest step in the enzymatic reaction. |
*'''Changing production rate''' | *'''Changing production rate''' | ||
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! Parameter !! Sensitivity | ! Parameter !! Sensitivity | ||
|- | |- | ||
- | | Initial concentration of split Dioxygenase || Change of one order of magnitude in the initial concentration, < | + | | Initial concentration of split Dioxygenase || Change of one order of magnitude in the initial concentration, <html>c<sub>0</sub></html>, gives change of one order of magnitude in the output concentration (range: <html>1>c<sub>0</sub>>10<sup>-5</sup></html>). Sensitivity is lost for extremely high or low values. |
|- | |- | ||
- | | < | + | | <html>K<sub>m</sub></html> ||Change of one order of magnitude results in change of output concentration by one order of magnitude (<html>0.01>K<sub>m</sub>>100</html>). At values smaller than 0.01, the sensitivity is affected. For higher values than 100 the sensitivity is slightly higher than the change of order of magnitude. |
|- | |- | ||
- | | < | + | | <html>k<sub>cat</sub></html> || <html>k<sub>cat</sub></html> is proportional to dioxygenase production (1-to-1 sensitivity for all values) for an initial concentration of <html>0.01 mol/dm<sup>3</sup></html>. For very high initial concentrations, the system is very sensitive to changes in <html>k<sub>cat</sub></html>. |
|- | |- | ||
| Production rate of TEV || 1-1 sensitivity for most values. At some point the system’s response is limited by the initial concentration of sD, so for very high TEV production rates not much change is observed. | | Production rate of TEV || 1-1 sensitivity for most values. At some point the system’s response is limited by the initial concentration of sD, so for very high TEV production rates not much change is observed. | ||
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| Production rate of split Dioxygenase || Not much influence on 1-step amplification. However, the value seems to be crucial for simple production of Dioxygenase (1-1 order of magnitude sensitivity). | | Production rate of split Dioxygenase || Not much influence on 1-step amplification. However, the value seems to be crucial for simple production of Dioxygenase (1-1 order of magnitude sensitivity). | ||
|- | |- | ||
- | | Degradation rates || Sensitive within the relevant range. Not very sensitive for values smaller than < | + | | Degradation rates || Sensitive within the relevant range. Not very sensitive for values smaller than <html>10<sup>-6</sup></math>. For high degradation rates (1>degradation rate>0.01): unexplainable behaviour. |
|} | |} | ||
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. | 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''' | *'''Initial Concentration''' | ||
- | The initial concentration of split Dioxygenase, < | + | The initial concentration of split Dioxygenase, <html>c<sub>0</sub></html>, 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). | 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''' | *'''Model A vs. B''' | ||
- | Running both models with the same initial conditions (< | + | Running both models with the same initial conditions (<html>c<sub>0</sub>=10<sup>-5</sup> mol/dm<sup>3</sup></html>), 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. |
[[Image:Comparison_prea%2C_a%2C_b.bmp|450px|thumb|center|alt=A|Comparison between Models pre-A, A and B]] | [[Image:Comparison_prea%2C_a%2C_b.bmp|450px|thumb|center|alt=A|Comparison between Models pre-A, A and B]] | ||
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*Toxic Effects of Catechol and 4-Chlorobenzoate Stresseson Bacterial Cells[http://www.msk.or.kr/jsp/downloadPDF1.jsp?fileName=393-15.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. | 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. | *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. | ||
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*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. | *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. | 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''' | '''Prospective solutions''' | ||
- | *'''Implementation in TinkerCell | + | *'''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 < | + | 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 <html>10<sup>5</sup></html> are not acceptable - this is important since our rate constants (<html>k<sub>1</sub></html>) are usually bigger than <html>10<sup>5</sup></html>. Also, the low degradation rates (<html>10<sup>-9</sup></html>) 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. | 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 | + | *'''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 < | + | 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 <html>10<sup>-15</sup></html>) significantly improved the simulation. However, this is not an ultimate solution as in the simulations negative numbers still appear (order of <html>10<sup>-15</sup></html>). 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). | 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). | ||
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|| [[Image:High_tolerance_inaccurate.png|420px|thumb|center|alt=A|Graphs representing conctrations of catechol, dioxygenase, catechol-dioxygenase complex and colur compound when adding catechol at t=2000s. Note the negative values that concentration of catechol hits.]] | || [[Image:High_tolerance_inaccurate.png|420px|thumb|center|alt=A|Graphs representing conctrations of catechol, dioxygenase, catechol-dioxygenase complex and colur compound when adding catechol at t=2000s. Note the negative values that concentration of catechol hits.]] | ||
- | || [[Image:Low_tolerance_accurate.png|600px|thumb|center|alt=A|Here the Relative and the Absolute Tolerances are chosen to be <math>10^{-18}</math> and <math>10^{-12}</math>, respectively. When lookin closely it is clear that catechol and the catechol-dioxygenase complex reach slightly negative values (eg. < | + | || [[Image:Low_tolerance_accurate.png|600px|thumb|center|alt=A|Here the Relative and the Absolute Tolerances are chosen to be <math>10^{-18}</math> and <math>10^{-12}</math>, respectively. When lookin closely it is clear that catechol and the catechol-dioxygenase complex reach slightly negative values (eg. <html>10<sup>-120</sup></html>]] |
|} | |} | ||
- | *'''Using SimBiology | + | *'''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. | 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. | ||
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|} | |} | ||
- | <html><h3>References</ | + | <html><h3>References</h3></html> |
#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] | #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] | ||
#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] | #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] | ||
#Habibi Nazhad, B. (2008) E. coli Statistics. [Online] Available from: http://gchelpdesk.ualberta.ca/CCDB/cgi-bin/STAT_NEW.cgi [Accessed 6th September 2010] | #Habibi Nazhad, B. (2008) E. coli Statistics. [Online] Available from: http://gchelpdesk.ualberta.ca/CCDB/cgi-bin/STAT_NEW.cgi [Accessed 6th September 2010] |
Revision as of 16:49, 17 October 2010
Temporary sub-menu: Objectives; Detailed Description; Parameters & Constants; Results & Conclusion;Download MatLab Files; |
Output Amplification Model | |||||||||||||||||||||
Results & ConclusionInitial Results The first results that were obtained seemed to be flawed since they indicated negative concentrations would be obtained from the amplification step. In particular, for concentrations smaller than <math>10^{-4} mol/dm^3</math> 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 (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 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 A
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.
Km is indirectly proportional to the "final concentration" (which is the concentration at the end of the simulation), i.e. the bigger the value 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 <math>k_1</math> and <math>k_2</math> entered into Matlab does not change the outcome as long as the ratio between them (<math>K_m</math>≈<math>k_2/k_1</math>) is kept constant. This is important when simulating (in case entering very high values for <math>k_1</math> and <math>k_2</math> slows down the simulation).
If kcat = 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.
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.
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.
|