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:

\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:

\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_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_FREEin the amount of free intracellular arsenic and As_BOUNDin the protein-bound As. In this way, As_TOTALin can be represented as follows:
\begin{equation} \large[As_{TOTALin}] = [As_{FREEin}] + [As_{BOUNDin}] \end{equation}

In the iGEM Groningen 2009 model, As_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_FREEin, so that equation 2 changes to:

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

If we further analyze the As_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:

\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 k_ArsR|As and k_MT|As as follows:
\begin{equation} \large \frac{d[ArsR|As]}{dt} = (\frac{[ArsR]}{1+(\frac{[As_{FREEin}])}{k_{[ArsR|As]}})^{h_2}} - \delta_{ArsR|As}[ArsR|As] \end{equation}

\begin{equation} \large \frac{d[MT|As]}{dt} = (\frac{[MT]}{1+(\frac{[As_{FREEin}])}{k_{[MT|As]}})^{h_3}} - \delta_{MT|As}[MT|As] \end{equation}

In equations 5 to 7, k_ArsR|As and k_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; here, in equations 6 and 7, the binding of two molecules is represented as "moleculeA|moleculeB". The indexes h2 and h3 are the Hill coefficients for the interaction between arsenic and ArsR and MT, respectively. The deltas are the degradation constants for the protein|As complexes. The unbound As that results from complex degradation then goes to the As_FREEin pool and is ready to bind again available ArsR or MT.

ODEs

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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}}[mRNA_{ArsR}] \end{equation}

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

Proteins

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

\begin{equation} \large \frac{d[MT]}{dt} = \alpha _{pMT}\cdot[mRNA_{MT}]- \delta _{MT}[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_2 \delta _{ArsR|As}[ArsR|As] + h_3 \delta _{MT|As}[MT|As]) \end{equation}

Parameters

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Parameter	Description	Value	References
αmArsR	Maximal transcription rate of ArsR	Value	References
αmMT	Maximal transcription rate of MT	Value	References
αpArsR	Maximal translation rate of ArsR	Value	References
αpMT	Maximal translation rate of MT	Value	References
δmRNAArsR	Degradation rate of ArsR mRNA	Value	References
δmRNAMT	Degradation rate of MT mRNA	Value	References
δArsR	Degradation rate of ArsR	Value	References
δMT	Degradation rate of MT	Value	References
δArsR|As	Degradation rate of ArsR|As complex	Value	References
δMT|As	Degradation rate of MT|As complex	Value	References
proars	Concentration of ars promoter	Value	References
procons	Concentration of constitutive promoter	Value	References
KD1	Dissociation constant for the interaction of ArsR and proars	Value	References
kt	Kinetic constant for the transport of extracellular arsenic	Value	References
h1	Hill coefficient for the interaction between ArsR and proars	Value	References
h2	Hill coefficient for the interaction of ArsR and As	Value	References
h3	Hill coefficient for the interaction of MT and As	Value	References
VS/VC	Relation between total solution volume(VS) and total cell volume (C)	Value	References
Vmax	Vmax for the Michaelis-Menten equation that describes As transport	Value	References

Simulations

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Steady state analysis

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Model considering lethal level of intracellular free As

Population level model