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

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; ArsR_As, MBPArsR_As, fMT_As, ArsB_As and GlpF_As are the arsenic bound proteins; n_f 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 MBPArsR_As 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 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 on the variable called As(III)_FREEin , which we'll assume to be transitory, at least for the scope of our core 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}

Where k(III)_ArsR|As and k(III)_MT|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".

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 6

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

\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}

\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}

Arsenic

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

Parameters

Simulations

Steady state analysis

Model considering lethal level of intracellular free As

Population level model