# IGEM:Imperial/2010/Variables1

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Constants for Modelling

+ + + + + +
Type of Constant + Derivation of Value +
TEV Enzyme Dynamics + Enzymatic Reaction: + $E+S\rightleftarrows ES \rightarrow E+P$ + +
Degradation rate (common for all) + +
Production rate (TEV and Dioxygenase) + +
Kinetic Parameters of Dioxygenase + +
Dimensions of B.sub cell + +
Production Rate of split TEV + +
Relevant concentrations of Catechol + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

# Constants for Modelling

Type of constant Derivation of value
TEV Enzyme dynamics Enzymatic Reaction:

$E+S\rightleftarrows ES \rightarrow E+P$

Let

• k1 = rate constant for $E+S\rightarrow ES$
• k2 = rate constant for $E+S\leftarrow ES$
• kcat = rate constant for $ES\rightarrow E+P$

We know that $K_m = \dfrac{k_{cat} + k_2}{k_1}$

Assuming that kcat < < k2 < < k1, we can rewrite Kmk2 / k1

From this paper [1] the constants for TEV can be found:

• e.g. wildtype TEV
• $K_m = 0.061\pm0.010mM$
• $k_{cat} = 0.16\pm0.01s^{-1}$

These values correspond with our assumption that kcat = 0.1s − 1 and Km = 0.01mM.

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.

(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 => 1 division every 40mins

To deduce degradation rate use the following formula:

$\tau_{\tfrac{1}{2}}=\frac{ln2}{k}$

Where $\tau_{\tfrac{1}{2}}=0.667 hour$

$k=\frac{ln2}{\tau_\tfrac{1}{2}}=0.000289s^{-1}$

Production rate

(TEV and dioxygenase)

We had real trouble finding the production rate values in the literature and we hope to be able to perform experiments to obtain those values for (TEV protease and catechol 2,3-dioxygenase). The experiments will not be possible to be carried out soon, so for the time being 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:

• Found values are taken from E.Coli not B.sub.
• The two found rates are of the same value for quite different amino-acid number which indicates that protein folding is limiting the production rates (we use the chosen approach as the only way of getting the estimate of order of reaction)

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

That 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 − 10mol / dm3 / s

• C23D production ≈ 252 molecules/min = 4.2 molecules/s (285 AA[8]) → 2.4998 * 10 − 9mol / 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 (we use that one)

Km = 10μM

kcat = 52s − 1

• Mutated type

Km = 40μM

kcat = 192s1

Consequently, the $\tfrac{k_{cat}}{K_m} = 4.8$ of the mutant was slightly lower than $\tfrac{k_{cat}}{K_m} = 5.2$ of the wild type, indicating that the mutation has little effect on catalytic efficiency[9].

Dimensions of

Bacillus subtillis cell

Dimensions of Basillus subtilis (cylinder/rod shape) in reach media (ref. bionumbers):

1. diameter: d = 0.87μm
2. length: l = 4.7μm in rich media

This gives: ${Volume}= \pi\tfrac{d^2}{4}l=2.793999\mu m^3\approx 2.79∙10^{-15} dm^3$

Split TEV

production rates

• Assume the both parts of split TEV are half of size of the whole TEV (3054/2=1527 AA)
• The length of the coil is 90 AA

The whole construct is then: 1617 AA

→ split TEV production rate ≈ 1.2606 * 10 − 10mol / dm3 / s

Relevant concentrations of catechol We have catechol in the lab in powder form so we're limited only by catechol's solubility.

For 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 appreciable difference between simple production response and the amplified response.

# Constants for Modelling

 Type of Constant Derivation of Value TEV Enzyme Dynamics Enzymatic Reaction: $E+S\rightleftarrows ES \rightarrow E+P$ Degradation rate (common for all) Production rate (TEV and Dioxygenase) Kinetic Parameters of Dioxygenase Dimensions of B.sub cell Production Rate of split TEV Relevant concentrations of Catechol

## References

1. 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]
2. 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]
3. 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]
4. UniProt Consortium (2002-2010) UniProt. [Online] Available from: http://www.uniprot.org/uniprot/P02920 [Accessed 24th August 2010]
5. 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]
6. UniProt Consortium (2002-2010) UniProt. [Online] Available from: http://www.uniprot.org/uniprot/P00722 [Accessed 24th August 2010]
7. UniProt Consortium (2002-2010) UniProt. [Online] Available from: http://www.uniprot.org/uniprot/P04517 [Accessed 24th August 2010]
8. UniProt Consortium (2002-2010) UniProt. [Online] Available from: http://www.uniprot.org/uniprot/P54721#section_x-ref [Accessed 24th August 2010]
9. 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]