Team:Imperial College London/Modelling/Output/Parameters and Constants

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|style="font-family: helvetica, arial, sans-serif;font-size:2em;color:#ea8828;"|Output Amplification Model
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|<html><h2>Parameters & Constants</h2></html><html>
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<table width="775px" border="0">
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<tr>
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    <td style="background-color:#FFFF66;height:50px;width:75;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;"><b>Type of Constant</b>
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    </td>
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    <td style="background-color:#FFFF99;height:50px;width:700;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;"><b>Derivation of Value</b>
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    </td>
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</tr>
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<tr>
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    <td style="background-color:#FFCC66;height:100px;text-align:center; font-family: helvetica, arial, sans-serif;color:#555555;"><b>TEV Enzyme Dynamics</b>
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    </td>
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    <td style="background-color:#e7e7e7;height:100px;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;">Enzymatic Reaction: E+S <var>&harr;</var> ES <var>&rarr;</var> E+P
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<br />
 +
Let
 +
<ul>
 +
<li>k<sub>1</sub> = rate constant for E+S <var>&rarr;</var> ES
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<li>k<sub>2</sub> = rate constant for E+S <var>&larr;</var> ES</math>
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<li>k<sub>cat</sub> = rate constant for ES <var>&rarr;</var> E+P
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</ul>
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We know that K<sub>m</sub> = (k<sub>cat</sub> + k<sub>2</sub>)/k<sub>1</sub>
 +
 +
Assuming that k<sub>cat</sub> << k<sub>2</sub> << k<sub>1</sub>, we can rewrite K<sub>m</sub> <var>&asymp;</var> k<sub>2</sub>/k<sub>1</sub>
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<br />
 +
From this paper <a href="http://peds.oxfordjournals.org/cgi/reprint/14/12/993">[1]</a> the constants for TEV can be found:
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<br />
 +
For example, for wildtype TEV: K<sub>m</sub> = 0.061<var>&plusmn;</var>0.010mM and k<sub>cat</sub> = 0.16<var>&plusmn;</var>0.01s<sup>-1</sup>
 +
<br />
 +
These values correspond with our assumption that k<sub>cat</sub> = 0.1 s<sup>-1</sup> and K<sub>m</sub> = 0.01 mM.
 +
<br />
 +
Hence, we can estimate the following orders of magnitude for the rate constants:
 +
<br />
 +
k<sub>1</sub> = 10<sup>8</sup>M<sup>-1</sup>s<sup>-1</sup>
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<br />
 +
k<sub>2</sub> = 10<sup>3</sup>s<sup>-1</sup>
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<br />
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Using these values should be a good approximation for our model.
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    </td>
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</tr>
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<tr>
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    <td style="background-color:#FFCC66;height:100px;text-align:center; font-family: helvetica, arial, sans-serif;color:#555555;"><b>Degradation rate (common for all)</b>
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    </td>
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    <td style="background-color:#e7e7e7;height:100px;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;">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 <var>&le;</var> gr <var>&le;</var> 2.18 <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC235685/pdf/jbacter00326-0019.pdf">[2]</a>
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<br />
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Hence on average,  gr = 1.5 divisions per hour, which gives one division every 40mins
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<br />
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To deduce degradation rate we use the following formula:
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<br />
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<var>&tau;</var><sub>1/2</sub> = ln2/k, where <var>&tau;</var><sub>1/2</sub> = 0.667 hours and k = degradation rate
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<br />
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k = ln2/<var>&tau;</var><sub>1/2</sub> = 0.000289s<sup>-1</sup>
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    </td>
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</tr>
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<tr>
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    <td style="background-color:#FFCC66;height:100px;text-align:center; font-family: helvetica, arial, sans-serif;color:#555555;"><b>Production rate (TEV and Dioxygenase)</b>
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    </td>
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    <td style="background-color:#e7e7e7;height:100px;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;">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.
 +
<br />
 +
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).
 +
<br />
 +
As this approach is very vague, it is important to realise its limitations and inconsistencies:
 +
<ul>
 +
<li>Values are taken from E.Coli not B.sub.</li>
 +
<li>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.</li>
 +
</ul>
 +
LacY production = 100 molecules/min<a href="http://bionumbers.hms.harvard.edu/bionumber.aspx?s=y&id=100738&ver=0&hlid=29205">[3]</a> (417 Amino Acids<a href="http://www.uniprot.org/uniprot/P02920">[4]</a>)
 +
<br />
 +
LacZ production = 100 molecules/min<a href="http://bionumbers.hms.harvard.edu/bionumber.aspx?s=y&id=100737&ver=0&hlid=29206">[5]</a> (1024 AA<a href="http://www.uniprot.org/uniprot/P00722">[6]</a>)
 +
<br />
 +
Average production ≈ 100molecules/min 720 AA
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<br />
 +
This gives us:
 +
TEV production ≈ 24 molecules/min = 0.40 molecules/s (3054 AA<a href="http://www.uniprot.org/uniprot/P04517">[7]</a>)
 +
<br />
 +
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&times;10<sup>-10</sup> mol/dm<sup>3</sup>/s
 +
<br />
 +
C23D production ≈ 252 molecules/min = 4.2 molecules/s (285 AA<a href="http://www.uniprot.org/uniprot/P54721#section_x-ref">[8]</a>) → 2.4998&times;10<sup>-9</sup> mol/dm<sup>3</sup>/s
 +
<br />
 +
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.
 +
    </td>
 +
</tr>
 +
 +
<tr>
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    <td style="background-color:#FFCC66;height:100px;text-align:center; font-family: helvetica, arial, sans-serif;color:#555555;"><b>Kinetic Parameters of Dioxygenase</b>
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    </td>
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    <td style="background-color:#e7e7e7;height:100px;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;">Initial velocity of the enzymatic reaction was investigated at pH 7.5 and 30 °C.
 +
<br />
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Wild type (used for our simulations): K<sub>m</sub> = 10 <var>&mu;</var>M; k<sub>cat</sub> = 52s<sup>-1</sup>
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<br />
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Mutated type: K<sub>m</sub> = 40 <var>&mu;</var>M; k<sub>cat</sub> = 192s<sup>−1</sup>
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<br />
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Consequently, the ratio of K<sub>m</sub>/k<sub>cat</sub> of the mutant (K<sub>m</sub>/k<sub>cat</sub> = 4.8) is slightly lower than the ratio of the wild type (K<sub>m</sub>/k<sub>cat</sub> = 5.2), indicating that the mutation has little effect on the  catalytic efficiency <a href="http://www.springerlink.com/content/e3718758m5052214/fulltext.pdf">[9]</a>.
 +
    </td>
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</tr>
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<tr>
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    <td style="background-color:#FFCC66;height:100px;text-align:center; font-family: helvetica, arial, sans-serif;color:#555555;"><b>Dimensions of B.sub cell</b>
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    </td>
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    <td style="background-color:#e7e7e7;height:100px;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;">Dimensions of B.sub (cylinder/rod shape) in rich media:
 +
<br />
 +
diameter: d = 0.87<var>&mu;</var>m; length: l = 4.7<var>&mu;</var>m
 +
<br />
 +
This gives:  Volume= <var>&pi;</var>d<sup>2</sup>l/4 = 2.793999<var>&mu;</var>m<sup>3</sup> <var>&asymp;</var> 2.79&times;10<sup>-15</sup> dm<sup>3</sup>
 +
    </td>
 +
</tr>
 +
 +
<tr>
 +
    <td style="background-color:#FFCC66;height:100px;text-align:center; font-family: helvetica, arial, sans-serif;color:#555555;"><b>Production Rate of split TEV</b>
 +
    </td>
 +
    <td style="background-color:#e7e7e7;height:100px;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;">Assuming that both parts of split TEV are half the size of the whole TEV (3054/2=1527 AA).
 +
<br />
 +
The length of the coil is 90 AA.
 +
<br />
 +
The whole construct is then: 1617 AA
 +
<br />
 +
Therefore, split TEV production rate ≈ 1.2606&times;10<sup>-10</sup> mol/dm<sup>3</sup>/s
 +
    </td>
 +
</tr>
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 +
<tr>
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    <td style="background-color:#FFCC66;height:100px;text-align:center; font-family: helvetica, arial, sans-serif;color:#555555;"><b>Relevant concentrations of Catechol</b>
 +
    </td>
 +
    <td style="background-color:#e7e7e7;height:100px;text-align:center;font-family: helvetica, arial, sans-serif;color:#555555;">We have catechol in the lab in powder form so we are only limited by it's solubility.
 +
<br />
 +
For a concentration of 0.1 M with built up levels of dioxygenase the colour change happens within seconds.
 +
<br />
 +
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.
 +
    </td>
 +
</tr>
 +
</table>
 +
</html>
 +
 +
<html>
 +
<b>References</b>
 +
<ol>
 +
<li>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]</li>
 +
<li>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]</li>
 +
<li>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]</li>
 +
<li>UniProt Consortium (2002-2010) UniProt. [Online] Available from: http://www.uniprot.org/uniprot/P02920 [Accessed 24th August 2010]</li>
 +
<li>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]</li>
 +
<li>UniProt Consortium (2002-2010) UniProt. [Online] Available from: http://www.uniprot.org/uniprot/P00722 [Accessed 24th August 2010]</li>
 +
<li>UniProt Consortium (2002-2010) UniProt. [Online] Available from: http://www.uniprot.org/uniprot/P04517 [Accessed 24th August 2010]</li>
 +
<li>UniProt Consortium (2002-2010) UniProt. [Online] Available from: http://www.uniprot.org/uniprot/P54721#section_x-ref [Accessed 24th August 2010]</li>
 +
<li>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]</li>
 +
</ol>
 +
</body>
 +
</div>

Revision as of 15:13, 17 October 2010

Temporary sub-menu:

Objectives; Detailed Description; Parameters & Constants; Results & Conclusion;Download MatLab Files;

Output Amplification Model

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

  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]
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