Difference between revisions of "Team:Jilin China/Model/degradation model"

Revision as of 12:14, 26 October 2017

Degradition Model

Overview

Our degradation model contained a set of biochemical reactions, as other metabolic models might be. We generated several ordinary differential equations from these reactions with unknown kinetic constants. These equation sets might not have analytic solutions and numeric methods always fail when dealing with unknown parameters in differential equations. One way to fix such problem is to make an approximation. For example, if a reaction X is not highly reversible, the kinetic constant k_X for that reaction would be regarded as zero. However, if any of the reaction in a chain process is not that irreversible, such approximations could also fail. This year, we involved a parameter optimizing method and combined the optimization with numeric solvers, which allowed us to get more accurate values for the kinetic constants from an initial guess. As soon as we get the accurate values for the kinetic constants, the degradation model could then predict the degradation rate of our pollutant. We’ve developed this method (we call it a numeric solving – parameter optimizing method) for our future work – since the enzyme could deal with many kinds of pollutant (theoretically), we couldn’t test them all and fit the model with experimental data.

Mathematical modelling

Once the population of the engineered E.coli reached a stable level (and the expression of enzymes would reach a stable level as well), the degradation rate of the pollutant could be measured. The degradation process contains the following progress:

$\mathrm{TfdB}+\Phi\rightleftharpoons\mathrm{TfdB}\Phi$      $k_{1+},k_{1-}$

$\mathrm{TfdB}\Phi\rightleftharpoons\mathrm{TfdB}\Phi'$      $k_{2+},k_{2-}$

$\mathrm{TfdB}\Phi'\rightleftharpoons\mathrm{TfdB}+\Phi'$      $k_{3+},k_{3-}$

$\mathrm{CphA_{-1}}+\Phi'\rightleftharpoons\mathrm{CphA_{-1}}\Phi'$      $k_+',k_-'$

$\mathrm{CphA_{-1}}\Phi'\rightarrow\mathrm{CphA_{-1}}+\Omega$      $k'$

Where Φ represents the pollutant, Φ' represents the intermediate product (the production from TfdB) and Ω represents the final production.

Parameters

Variable	Explanation
$[\mathrm{substance}]$	concentration of the substance
$t$	time
Constant	Explanation
$[\mathrm{TfdB}]$	concentration of TfdB
$[\mathrm{CphA_{-1}}]$	concentration of CphA-1
$k_{?}$	rate constant, see reaction equations

Equations

$\begin{cases} \dfrac{\mathrm{d}[\Phi]}{\mathrm{d}t}=k_{1-}[\mathrm{TfdB}\Phi]-k_{1+}[\mathrm{TfdB}][\Phi]\\\\ \dfrac{\mathrm{d}[\mathrm{TfdB}\Phi]}{\mathrm{d}t}=k_{1+}[\mathrm{TfdB}][\Phi]+k_{2-}[\mathrm{TfdB}\Phi']-(k_{2+}+k_{1-})[\mathrm{TfdB}\Phi]\\\\ \dfrac{\mathrm{d}[\mathrm{TfdB}\Phi']}{\mathrm{d}t}=k_{3-}[\mathrm{TfdB}][\Phi']+k_{2+}[\mathrm{TfdB}\Phi]-(k_{3+}+k_{2-})[\mathrm{TfdB}\Phi']\\\\ \dfrac{\mathrm{d}[\Phi']}{\mathrm{d}t}=k_{3+}[\mathrm{TfdB}\Phi']+k_-'[\mathrm{CphA_{-1}}\Phi']-k_+'[\mathrm{CphA_{-1}}][\Phi']-k_{3-}[\mathrm{TfdB}][\Phi']\\\\ \dfrac{\mathrm{d}[\mathrm{CphA_{-1}}\Phi']}{\mathrm{d}t}=k_+'[\mathrm{CphA_{-1}}][\Phi']-(k'+k_-')[\mathrm{CphA_{-1}}\Phi']\\\\ \dfrac{\mathrm{d}[\Omega]}{\mathrm{d}t}=k'[\mathrm{CphA_{-1}}\Phi'] \end{cases}$

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