Team:UIUC Illinois/Modeling

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  <h2>Mathematical Modeling of Caffeine Degradation Pathway</h2>
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<center><p style="font-size: 400% ">Modeling</p></center>
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<center><p style="font-size: 150% "><i>"All science requires mathematics. The knowledge of mathematical things is almost innate in us.... This is the easiest of sciences, a fact which is obvious in that no one's brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon."</i></p></center>
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<br><center>-Roger Bacon</br></center>
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<p><b>Overview:</b>
  To predict the result of bioreactor, we used mathematica to solve differential using Michaelis-Menton equation. The strength of utilizing mathematica rather than matlab was that it allowed us to set up the value of several constants as varying rather than setting it as invariant. Kcat & Km values were obtained through research papers by Swati & Sathyanarayana (2006), and Ryan M.Summers (2010).  
  To predict the result of bioreactor, we used mathematica to solve differential using Michaelis-Menton equation. The strength of utilizing mathematica rather than matlab was that it allowed us to set up the value of several constants as varying rather than setting it as invariant. Kcat & Km values were obtained through research papers by Swati & Sathyanarayana (2006), and Ryan M.Summers (2010).  
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<h2>Modeling Dog's intestine </h2>
<h2>Modeling Dog's intestine </h2>
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<p><b>Description:</b></p>
<p><b>Description:</b></p>
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<p>In addition to the model of the degradation of caffeine through the two pathways as shown above, it is possible to model the transport of caffeine through two body compartments: blood, small intestine. By doing so, we could understand the optimal levels of bacteria that we would need in order to degrade the maximum concentration of caffeine. In the research book “Solving Ordinary Equations in R” by Soetaert (2012), they list two equations which could model any drug concentration in the intestine represented by y_1 and in the blood represented by y_2 </p>
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K<sub>T</sub> is the rate of transfer, C<sub>i</sub> is concentration of caffeine in intestine of dog,  
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C<sub>b</sub> is concentration of caffeine in blood which should be coming down from intestine.<br>
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A<sub>i</sub> is the surface area of the intestine, V<sub>i</sub> is the volume of the intestine, V<sub>b</sub> is the volume of blood.
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K<sub>cl</sub> is the clearance rate.
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<p> a is the absorption rate of drug from the intestine , b is the removal rate of drug from the blood, and u(t) represent a time dependent dosage of the drug into the intestine. In our case, since bacteria is involved in degrading caffeine, there would be an extra variable in the equationy_1': the degradation rate from bacteria.  With these equations, the model should have a similar behavior as below </p>
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<p>Where initial concentration of the drug in the blood is 0 but rises exponentially until the drug gets absorbed by the intestine, in which case, concentration of the drug in the blood starts to fall. This pattern of increasing to decreasing concentration of drug in the blood over a period of time is a result of the daily dosage of drug given. Similarly, the concentration of drug in the intestine increases at a rapid rate once the drug is absorbed into the intestine and decreases over time  as the drug is being removed from the system. However, in our case with the bacteria, the rate at which the concentration of the drug is decreasing will be faster. </p>
 
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         <br><h2><center>Fig3. Dog's Intestine & Blood Model</center></h2></br>
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         <h2><center>Fig3. Dog's Intestine & Blood Model</center></h2>
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<center>Initial Concentration of Caffeine decreases exponentially in the intestine as caffeine is delivered into the blood and degraded by bacteria. Concentration of caffeine increases in the blood then decreases due to the clearance factor K<sub>cl</sub> </center>
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Latest revision as of 03:45, 18 October 2014


Modeling

"All science requires mathematics. The knowledge of mathematical things is almost innate in us.... This is the easiest of sciences, a fact which is obvious in that no one's brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon."


-Roger Bacon

Overview: To predict the result of bioreactor, we used mathematica to solve differential using Michaelis-Menton equation. The strength of utilizing mathematica rather than matlab was that it allowed us to set up the value of several constants as varying rather than setting it as invariant. Kcat & Km values were obtained through research papers by Swati & Sathyanarayana (2006), and Ryan M.Summers (2010).

This is caffeine demehtlyation pathway by demethlyase. It goes from Caffeine to theobromine to 7-methylxanthine to xanthine. The other pathway is caffeine dehydrogenase. It goes from Caffeine to Trimethyl Uric acid.

Parameters

Name Description
Vm Maximum rate of system
Kcat Maximum number of substrate molecules converted into products
Km Substrate concentration where the reaction rate is half of maximum (depend on both enzyme and substrate)




Fig1. Caffeine Demethylation Pathway

Initial concentration of caffeine, 10 uM and decreases as it degrades to theobromine, 7-methylxanthine, and finally xanthine with final concentration of 10uM, only if our designed bacteria has efficiency of 100%.





Fig2. Caffeine Dehydrogenase Pathway

Denotes initial concentration of caffeine 10 uM and gradually degrades to Trimethyl Uric Acid with final concentration of 10uM

Modeling Dog's intestine

Description:

KT is the rate of transfer, Ci is concentration of caffeine in intestine of dog, Cb is concentration of caffeine in blood which should be coming down from intestine.
Ai is the surface area of the intestine, Vi is the volume of the intestine, Vb is the volume of blood.
Kcl is the clearance rate.




Fig3. Dog's Intestine & Blood Model

Initial Concentration of Caffeine decreases exponentially in the intestine as caffeine is delivered into the blood and degraded by bacteria. Concentration of caffeine increases in the blood then decreases due to the clearance factor Kcl


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