Our novel concept of amplifying our output by incorporating enzymes raised some important questions that needed to be answered:
How beneficial is it to incorporate the enzyme amplification step? We need to compare the speed of response between transcription and translation with 1- or 2-step ampification.
How many amplification steps are beneficial? Will the addition of further amplification steps introduce considerable time delays?
Which enzymes should be used? TEV or HIV1?
In order to answer these questions, we would have to model this system. The results of the computer models would enable us to decide which design would be the most efficient one. This design would then be put forward to be built in labs.
Detailed Description
Parameters & Constants
Type of Constant
Derivation of Value
TEV Enzyme Dynamics
Enzymatic Reaction: E+S ↔ ES → E+P
Let
k1 = rate constant for E+S → ES
k2 = rate constant for E+S ← ES
kcat = rate constant for ES → E+P
We know that Km = (kcat + k2)/k1
Assuming that kcat << k2 << k1, we can rewrite Km≈ k2/k1
From this paper [1] the constants for TEV can be found:
For example, for wildtype TEV: Km = 0.061±0.010mM and kcat = 0.16±0.01s-1
These values correspond with our assumption that kcat = 0.1 s-1 and Km = 0.01 mM.
Hence, we can estimate the following orders of magnitude for the rate constants:
k1 = 108M-1s-1
k2 = 103s-1
Using these values should be a good approximation for our model.
Degradation rate (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
Growth rate, gr (divisions/h): 0.53 ≤ gr ≤ 2.18 [2]
Hence on average, gr = 1.5 divisions per hour, which gives one division every 40mins
To deduce degradation rate we use the following formula:
τ1/2 = ln2/k, where τ1/2 = 0.667 hours and k = degradation rate
k = ln2/τ1/2 = 0.000289s-1
Production rate (TEV and Dioxygenase)
We had difficulties finding values of the production rate in the literature and we hope to be able to perform experiments to obtain those values (for TEV protease and catechol 2,3-dioxygenase). Before any values can be obtained from the Lab, we suggest very simplistic approach for estimating production rates.
We have found production rates for two arbitrary proteins in E.Coli. We want to get estimates of production rates by comparing the lengths of the proteins (number of amino-acids).
As this approach is very vague, it is important to realise its limitations and inconsistencies:
Values are taken from E.Coli not B.sub.
The two production rates are of the same value for quite different amino-acid number which indicates that protein folding is limiting the production rates.
LacY production = 100 molecules/min[3] (417 Amino Acids[4])
LacZ production = 100 molecules/min[5] (1024 AA[6])
Average production ≈ 100molecules/min 720 AA
This gives us:
TEV production ≈ 24 molecules/min = 0.40 molecules/s (3054 AA[7])
As production rate needs to be expressed in concentration units per unit volume, the above number is converted to mols/s and divided by the cell volume: 2.3808×10-10 mol/dm3/s
C23D production ≈ 252 molecules/min = 4.2 molecules/s (285 AA[8]) → 2.4998×10-9 mol/dm3/s
We will treat these numbers as guiding us in terms of range of orders of magnitudes. We will try to run our models for variety of values and determine system’s limitations.
Kinetic Parameters of Dioxygenase
Initial velocity of the enzymatic reaction was investigated at pH 7.5 and 30 °C.
Wild type (used for our simulations): Km = 10 μM; kcat = 52s-1
Mutated type: Km = 40 μM; kcat = 192s−1
Consequently, the ratio of Km/kcat of the mutant (Km/kcat = 4.8) is slightly lower than the ratio of the wild type (Km/kcat = 5.2), indicating that the mutation has little effect on the catalytic efficiency [9].
Dimensions of B.sub cell
Dimensions of B.sub (cylinder/rod shape) in rich media:
diameter: d = 0.87μm; length: l = 4.7μm
This gives: Volume= πd2l/4 = 2.793999μm3≈ 2.79×10-15 dm3
Production Rate of split TEV
Assuming that both parts of split TEV are half the size of the whole TEV (3054/2=1527 AA).
The length of the coil is 90 AA.
The whole construct is then: 1617 AA
Therefore, split TEV production rate ≈ 1.2606×10-10 mol/dm3/s
Relevant concentrations of Catechol
We have catechol in the lab in powder form so we are only limited by it's solubility.
For a concentration of 0.1 M with built up levels of dioxygenase the colour change happens within seconds.
We will run our models for 0.1M ± several orders of magnitude to determine the smallest catechol concentration that will give a significant difference between the simple production response and the amplified response.
References
Kapust, R. et al (2001) Tobacco etch virus protease: mechanism of autolysis and rational design of stable mutants with wild-type catalytic proficiency. Protein Engineering. [Online] 14(12), 993-1000. Available from: http://peds.oxfordjournals.org/content/14/12/993.full.pdf+html [Accessed 20th August 2010]
Sargent, M. (1975) Control of Cell Length in Bacillus subtilis. Journal of Bacteriology. [Online] 123(1), 7-19. Available from: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC235685/pdf/jbacter00326-0019.pdf [Accessed 20th August 2010]
Milo, R., Jorgensen, P. & Springer, M. (2007) BioNumbers. [Online] Available from: http://bionumbers.hms.harvard.edu/bionumber.aspx?s=y&id=100738&ver=0&hlid=29205 [Accesed 25th August 2010]
UniProt Consortium (2002-2010) UniProt. [Online] Available from: http://www.uniprot.org/uniprot/P02920 [Accessed 24th August 2010]
Milo, R., Jorgensen, P. & Springer, M. (2007) BioNumbers. [Online] Available from: http://bionumbers.hms.harvard.edu/bionumber.aspx?s=y&id=100737&ver=0&hlid=29206 [Accesed 25th August 2010]
UniProt Consortium (2002-2010) UniProt. [Online] Available from: http://www.uniprot.org/uniprot/P00722 [Accessed 24th August 2010]
UniProt Consortium (2002-2010) UniProt. [Online] Available from: http://www.uniprot.org/uniprot/P04517 [Accessed 24th August 2010]
UniProt Consortium (2002-2010) UniProt. [Online] Available from: http://www.uniprot.org/uniprot/P54721#section_x-ref [Accessed 24th August 2010]
Wei, J. et al (2009) Rational Design of Catechol-2, 3-dioxygenase for Improving the Enzyme Characteristics. Appl Biochem Biotechnol. [Online] 162, 116-126. Available from: http://www.springerlink.com/content/e3718758m5052214/fulltext.pdf [Accessed 25th August 2010]
Results & 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 [1] 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:
Immediate cell death slows down production of coloured output. Depending on the threshold concentration this can delay the detectable response by a few minutes.
If Catechol is added before t=1000s, then cell death slows down the response considerably.
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).
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