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Team:UANL Mty-Mexico/Modeling/transport and accumulation
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\begin{equation} | \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} | \end{equation} | ||
<p></br></p> | <p></br></p> | ||
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<p> | <p> | ||
\begin{equation} | \begin{equation} | ||
| - | $\ | + | $\large\frac{\mathrm{d[As(III)in] } }{\mathrm{d} x} = -ArsR_{As} -n_{f}\cdot fMT_{As} + \frac{k_{2}V_{s}GlpF_{As}}{V_{c}}$ |
\end{equation} | \end{equation} | ||
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\begin{equation} | \begin{equation} | ||
| - | $\ | + | $\large[As(III)_{TOTALin}] = [As(III)_{FREEin}] + [As(III)_{BOUNDin}]$ |
\end{equation} </br></p> | \end{equation} </br></p> | ||
Revision as of 22:51, 16 September 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:
- We assume that ArsB is non functional, so that the only protein affecting As transport is GlpF.
- We assume that the intracellular As concentration at the population level (that is, considering total cell volume) is homogeneously distributed and should be the same as in a single cell.
- 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}
$\large\frac{\mathrm{d[As(III)in] } }{\mathrm{d} x} = -ArsR_{As} -n_{f}\cdot fMT_{As} + \frac{k_{2}V_{s}GlpF_{As}}{V_{c}}$
\end{equation}
Now, let us introduce a variable called As(III)TOTALin, 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 As(III)FREEin the amount of free intracellular arsenic and BOUNDin the protein-bound As. In this way, As(III)TOTALin can be represented as follows:
Equation 3
\begin{equation} $\large[As(III)_{TOTALin}] = [As(III)_{FREEin}] + [As(III)_{BOUNDin}]$ \end{equation}In the iGEM Groningen 2009 model, As(III)in 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 call this variable As(III)FREEin, so that equation 2 changes to:
They also characterized the BioBrick corresponding to the GlpF transporter
ODEs
Parameters
Simulations
Steady state analysis
Model considering lethal level of intracellular free As
Population level model
