Simulations
Now, based on our equations, we can run simulations to estimate αY, the parameter of interest of our model. For that, we decided to try to find the maximum αY that will, according to our model, fit the EC50 time and the maximum value of ρ.
For that, we must assume a new assumption: That all the proteins in the systems (synthases, receptors and reporters) are approximately equal, therefore we can say that their production and degradation rates are the same, if not explicitly given a value.At first, we defined our parameters based on literature values. (To know precisely the parameters used in the runs, see the table at the end of the page).
After establishing the parameters of interest, we created an algorithm that will receive the EC50 time (tEC50) and the maximum ρ (MAX) and return the values of αY. The algorithm is resumed at the flowchart below:
In this flow chart, we can see the process of the algorithm's workings. Where a parallelogram represents the initial data, round rectangles represent processes and diamonds represent decision processes. Af is used as αY and ε is a small increase. From that, we’ve been able to, comparing to our data and using some literature data as proxy, estimate the alphas!
We've been able to predict the values of α and, also, estimate a basal expression, called now γ, for each promoter, by adding a single term in the differential equation:
The first value represents the value of γ and the second of α.
- Lux:Lux - .39; .693
- Lux:Las - -.14; 1.18
- Lux:Rpa - -.301; 1.185
- Lux:Tra - 0.0; 0.0
- Rhl:Lux - .011; .11
We also, using the work of Scott and Hasty, we estimated some theoretical values of other systems
- Tra:Tra - -; 0.089
- Rpa:Lux - -; .347
- Rpa:Rpa - -; 1.035
- Las:Rpa - -; 1.43
- Las:Las - -; 2.772
Using those parameters, we constructed a graph to show each one of them. The first comparing the values from our data, the second is from the the literature data
However, to do a proper comparison, we’ve set up a relative measure, we decided to call λ. Is a very simple proposal:
$$\lambda N \mid M = \frac{ ( \frac{\alpha M}{\gamma M}) ^2 } { ( \frac{\alpha N}{\gamma N}) ^2}$$
Where N is the native system, the system of the HSL and Receptor, and M is the system of the promoter. Alfa is the alfa that represents the promoter interaction, and gamma is the basal of expression. With that in mind, we found the following values Lambda:
- Lux:Las - 4,74
- Lux:Rpa - 2,21
- Las:Rpa - 0,52
- Rpa:Lux - 0,34
The most interesting part of this comparison is see, that even with Lux:Las and Lux:Rpa where the expression is essential the same, as show by the simulations, Las has twice more of cross-talk then Rpa with Lux!
Parameters
| Name | Value | Description | Reference |
|---|---|---|---|
| $$\alpha_{S}$$ | .005μMmin-1 | Production/Expression of the variable, indicated in the index | Basu, S. et al. 2005 |
| $$\alpha_{H}$$ | 3.22e2 | Production/Expression of the variable, indicated in the index | Pai, A. et al. 2018 |
| $$\alpha_{R}$$ | .05μMmin-1 | Production/Expression of the variable, indicated in the index | Basu, S. et al. 2005/Estimated |
| $$\mu_{S}$$ | .0231min-1 | Degradation of the variable, indicated in the index | Basu, S. et al. 2005 |
| $$\mu_{H}$$ | 1e-2min-1 | Degradation of the variable, indicated in the index | Pai, A. et al. 2018 |
| $$\mu_{R}$$ | .0231min-1 | Degradation of the variable, indicated in the index | Basu, S. et al. 2005 |
| $$\mu_{C}$$ | .0231min-1 | Degradation of the variable, indicated in the index | Basu, S. et al. 2005 |
| $$\mu_{Yf}$$ | 2.14e-3min-1 | Degradation of the variable, indicated in the index | iGEM MIT Team 2010 |
| $$\beta$$ | 0.1 | Formation of the HSL/Receptor Complex | Weber, M & Buceta, J. 2013 |
| $$\theta$$ | 10 | Separation of the Complex HSL/Receptor | Weber, M & Buceta, J. 2013 |
Parameter References
- MIT's 2010 iGEM Team
- S.Basu , Y.Gerchman, C.H. Collins, F.H. Arnold, R.Weiss. “A synthetic multicellular system for programmed pattern formation” Nature volume 434, pages 1130–1134 (28 April 2005)
- M.Weber, J.Buceta. “Dynamics of the quorum sensing switch: stochastic and non-stationary effects” BMC Systems Biology 7:6 (2013) doi: 10.1186/1752-0509-7-6
- Pai, Anand, and Lingchong You. “Optimal Tuning of Bacterial Sensing Potential.” Molecular Systems Biology 5 (2009): 286. PMC. Web. 17 Oct. 2018.