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

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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. 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.
  3. 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 As(III)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 establish this equivalence as follows: As(III)FREEin, so that equation 2 changes to:


Equation 4

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


We'll only use from now one the variable called As(III)FREEin , which we'll assume to be transitory and can be discarded, at least for the scope of our code model. The time derivative for equation 3 with the transient As(III)FREEin assumption, turns to be:


Equation 5

\begin{equation} \large\frac{d[As(III)_{TOTALin}]}{dt} = \frac{d[As(III)_{BOUNDin}]}{dt}] \end{equation}


If we further analyze the As(III)BOUNDin 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 6

\begin{equation} \large\frac{d[As(III)_{TOTALin}]}{dt} = (\frac{d[ArsR]}{k_ArsR::As+[ArsR]})+(\frac{d[MT]}{k_MT::As+[MT]}) \end{equation}


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


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