Team:UANL Mty-Mexico/Modeling/transport and accumulation

From 2012.igem.org

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<br><p><b>Equation 6 and 7</b></p>
<br><p><b>Equation 6 and 7</b></p>
\begin{equation}
\begin{equation}
-
\large\frac{d[ArsR|As]}{dt} = (\frac{d[ArsR]}{1+ (\frac{[As_{FREEin}]}{k_{[ArsR|As]})^h_{1} - \delta_{[ArsR|As]}
+
\large\frac{d[ArsR|As]}{dt} = (\frac{d[ArsR]}{1+ (\frac{[As_{FREEin}]}{k_{[ArsR|As]}})^h_{1} - \delta_{[ArsR|As]}
\end{equation}
\end{equation}
<br><p>
<br><p>
\begin{equation}
\begin{equation}
-
\large\frac{d[MT|As]}{dt} = (\frac{d[MT]}{1+ (\frac{[As_{FREEin}]}{k_{[MT|As]})^h_{1} - \delta_{[MT|As]}
+
\large\frac{d[MT|As]}{dt} = (\frac{d[MT]}{1+ (\frac{[As_{FREEin}]}{k_{[MT|As]}})^h_{1} - \delta_{[MT|As]}
\end{equation}  
\end{equation}  
</br></p>
</br></p>

Revision as of 05:48, 17 September 2012

iGEM UANL 2012


Transport and accumulation


Before us, team iGEM Groningen 2009 made a model for an arsenic accumulator at the population level; that is, they set some ODEs that represent the change on the total intracellular arsenic (considering not a single cell, but the whole culture, or more exactly, the total cell volume) with respect to time. Nevertheless, as the precise value for some parameters were unavailable, specially for the ArsB effect, part of their model remains aparameterized and they perform a quasi-steady state analysis.

After considering the effect of their metallothioneins (As-binding proteins), GlpF, ArsB and ArsR, they ended with the following time derivative:


Equation 1

\begin{equation} \large\frac{\mathrm{d[As(III)in] } }{\mathrm{d} x} = -ArsR_{As}-MBPArsR_{As} -n_{f}\cdot fMT_{As} -k_{1} ArsB_{As} + \frac{k_{2}V_{s}GlpF_{As}}{V_{c}} \end{equation}


Where As(III)in is the total intracellular arsenic; ArsRAs, MBPArsRAs, fMTAs, ArsBAs and GlpFAs are the arsenic bound proteins; nf is the Hill coefficient for the interaction between As and fMT; k1 and k2 are the kinetic constants for the interaction between As and ArsB and GlpF, respectively; finally, Vs/Vc represents the proportion between the total solution volume (Vs) and the total cell volume (Vc).


Core model



We built upon their model and made the following modifications, which we'll call the "core modifications" from now on:

  1. We assume that ArsB is non functional, so that the only protein affecting As transport is GlpF.
  2. GlpF effect is masked by the population level kinetics.
  3. We assume that the intracellular As concentration and the GlpF effect at the population level (that is, considering total cell volume) are homogeneously distributed and should be the same as in a single cell.
  4. The protein MBPArsR is not present in our system, so the variable MBPArsRAs is not considered for our model.

The next equation shows the application of those modifications:


Equation 2

\begin{equation} \large\frac{\mathrm{d[As(III)in] } }{\mathrm{d} x} = -ArsR_{As} -n_{f}\cdot fMT_{As} + (\frac{V_s}{V_c} \cdot V_{max} \cdot \frac{As_{ex}}{K_{t}+As_{ex}} \end{equation}

Now, let us introduce a variable called AsTOTALin, which represents the total amount of arsenic inside a cell (recall core modification number 2), whether free or bound to whatever protein. Let's also call AsFREEin the amount of free intracellular arsenic and AsBOUNDin the protein-bound As. In this way, AsTOTALin can be represented as follows:

Equation 3

\begin{equation} \large[As_{TOTALin}] = [As_{FREEin}] + [As_{BOUNDin}] \end{equation}


In the iGEM Groningen 2009 model, Asin represents the free intracellular arsenic, as this variable has negative terms related to the binding of As to proteins; to avoid further confusions, we'll establish this equivalence as follows: AsFREEin, so that equation 2 changes to:


Equation 4

\begin{equation} \large[As_{FREEin}] = [As_{in(iGEM Groningen 2009)}] \end{equation}


We'll only use from now on the variable called AsFREEin , which we'll assume to be transitory, at least for the scope of our core model. Equation 3 with the transient AsFREEin assumption, turns to be:

If we further analyze the AsBOUNDin variable, considering that in our system only ArsR and a methalothionein (which we'll simply call MT) are being expressed, then equation 5 turns to be:


Equation 5

\begin{equation} \large\large[As_{TOTALin}] = [ArsR|As] + [MT|As] \end{equation}


Finally, taking into consideration the work of team Cambridge 2009 and assuming that equilibrium is reached quickly, we can describe the formation kArsR|As and kMT|As as follows:

Equation 6 and 7

\begin{equation} \large\frac{d[ArsR|As]}{dt} = (\frac{d[ArsR]}{1+ (\frac{[As_{FREEin}]}{k_{[ArsR|As]}})^h_{1} - \delta_{[ArsR|As]} \end{equation}

\begin{equation} \large\frac{d[MT|As]}{dt} = (\frac{d[MT]}{1+ (\frac{[As_{FREEin}]}{k_{[MT|As]}})^h_{1} - \delta_{[MT|As]} \end{equation}

In equations 5 to 7, kArsR|As and kMT|As are the kinetic constants for the interaction of arsenic with ArsR and MT, respectively, using a different nomenclature as in equations 1 and 2, where the binding of two molecules is represented as "moleculeA|moleculeB". The deltas are the degradation constants for the protein|As complexes. The unbound As that results from complex degradation then goes to the AsFREEin pool and is ready to bind again available ArsR or MT.


ODEs



The core modifications and equation 6 allow us to propose a set of ODEs that describe the change of the concentrations of intracellular As, ArsR|As, MT|As and the unbound protein species.



Core model ODEs

mRNAs

\begin{equation} \large \frac{d[mRNA_{ArsR}]}{dt} = \alpha _{mArsR}\cdot (pro_{ars})\cdot(\frac{k_{D1}^{h_{1}}}{k_{D1}^{h_{1}}+[ArsR]^{h_{1}}})- \delta _{mRNA_{ArsR}} \end{equation}

\begin{equation} \large \frac{d[mRNA_{MT}]}{dt} = \alpha _{mArsR}\cdot(pro_{cons})- \delta _{mRNA_{MT}} \end{equation}

Proteins

\begin{equation} \large \frac{d[ArsR]}{dt} = \alpha _{pArsR}\cdot[mRNA_{ArsR}]- \delta _{ArsR} - ArsR|As \end{equation}

\begin{equation} \large \frac{d[MT]}{dt} = \alpha _{pMT}\cdot[mRNA_{MT}]- \delta _{MT} - MT|As \end{equation}

Proteins with arsenic

See equations 7 and 8

Arsenic

\begin{equation} \large\frac{\mathrm{d[As_{FREEin}] } }{\mathrm{dt}} = (\frac{V_s}{V_c} \cdot V_{max} \cdot \frac{As_{e}}{K_{t}+As_{e}} + h_1 \delta _{ArsR|As} + h_2 \delta _{MT|As}) \end{equation}


Parameters



Simulations



Steady state analysis



Model considering lethal level of intracellular free As



Population level model


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