Team:Imperial College London/Modelling/Protein Display/Parameters and Constants

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Department of Bioengineering

Division of Molecular Biosciences

Modelling	Overview \| Detection Model \| Signaling Model \| Fast Response Model \| Interactions
A major part of the project consisted of modelling each module. This enabled us to decide which ideas we should implement. Look at the Fast Response page for a great example of how modelling has made a major impact on our design!

Objectives | Description | Results | Constants | MATLAB Code

Parameters and Constants

Type of Constant	Derivation of Value
TEV Enzyme Dynamics	Enzymatic Reaction: E+S `↔` ES `→` E+P k₁ = rate constant for E + S `→` ES = 10⁸ M^-1s^-1 k₂ = rate constant for E + S `←` ES = 10³ s^-1 k_cat = rate constant for ES `→` E + P = 0.16 `±` 0.01 s^-1 We are assuming the same cleaving rates of TEV as on other substrates. However, we are planning to measure them to gain more confidence in the model.
Production Rate of Surface Proteins	It was found that each cell displays 2.4×10⁵ peptides [1]. Hence, we adjusted our simple production of display protein model to converge to that value. As production rate was the constant that we could not obtain, that value was manipulated. The result 4.13×10^-8mol/dm³/s seemed to be of reasonable order of magnitude. Ideally, we would like to get this value measured as it is resulting from a very vague estimate.
Degradation Rate of Surface Proteins (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. k_deg= 0.000289s^-1 For all proteins that are outside of cells or the timescale that is short enough to neglect cell division effect: k_deg=0
Diffusion Coefficient of Proteins	We have found two references which quote very similar values for very different media. For protein in agarose gel: D_average = 1.07×10^-10m²/s - for a protein in agarose gel for pH=5.6 [2] In the final model the following was used: For protein in water: D=10^-10m²/s [3]
Localised Concentration Coefficient	The localised concentration coefficient seems to be the weakest point of this model. We have tried to rationalise it as much as we could. However, errors seem to be unavoidable. It is important to realise that the value of localised concentration coefficient probably would require adjusting for different bacterial concentrations.
Cell Dimensions	Bacillus subtillis was treated as a rod unless specifically state otherwise. The following dimensions were used [4]: Diameter: 0.87 μm Length: 4.7 μm
Cell Wall thickness	The cell wall thickness was found to be around 35nm. [5]:
Diffusion constant	The diffusion coefficient of small protein in water is of order = 10^-6 cm² s^-1 [6]:

Click here to download the MATLAB code for this model...

References

Kobayashi, G. et al (2000) Accumulation of an artificial cell wall-binding lipase by Bacillus subtilis wprA and/or sigD mutants. FEMS Microbiology Letters. [Online] 188(2000), 165-169. Available from: http://onlinelibrary.wiley.com/doi/10.1111/j.1574-6968.2000.tb09188.x/pdf [Accessed 27th August 2010]
Gutenwik, J., Nilsson, B. & Axelsson, A. (2003) Determination of protein diffusion coefficients in agarose gel with a diffusion cell. Biochemical Engineering Journal. [Online] 19(2004), 1-7. Available from: http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6V5N-4B3MXDC-2-K&_cdi=5791&_user=217827&_pii=S1369703X03002377&_origin=search&_coverDate=07%2F01%2F2004&_sk=999809998&view=c&wchp=dGLzVtb-zSkzS&md5=c17d0e7320f03931006f9b1a10a438b9&ie=/sdarticle.pdf [Accessed August 20th 2010]
Crofts, A. (1996) Biophysics 345. [Online] Available from: http://www.life.illinois.edu/crofts/bioph354/diffusion1.html [Accessed 1st September 2010]
BioNumbers is the on-line database storing values of biologically related parameters with references; [Online] Available from: http://www.life.illinois.edu/crofts/bioph354/diffusion1.html [Accessed August 19th 2010]
Graham L. L. & Beverisge T. J. (1993) Structural Differentiation of the Bacillus subtilis 168 Cell Wall. Journal of Bacteriolofy. [Online] 5(1994), 1413-1420. Available from: http://jb.asm.org/cgi/reprint/176/5/1413?ijkey=27dafbac7e23dee50390d3fe67d9d1bab0c6f48c [Accessed October 26th 2010]
Value was presented at Home Page for the Crofts laboratory (A. R. Crofts), at the University of Illinois at Urbana-Champaign; [Online] Available from: http://www.life.illinois.edu/crofts/bioph354/diffusion1.html [Accessed September 15th 2010]

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