Team:Virginia/Model

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Model Development


In order to develop our model, we decided on a system of nonlinear ordinary differential equations (ODEs) to describe the concentrations of arsenate (As(V)) and arsenite (As(III)). This system of nonlinear ODEs are based on the concept of mass balance. The model is divided into two compartments: intracellular and extracellular. The above schematic was used to develop the model.

The equation below describes the first part of the model whereby extracellular arsenate concentration is determined by diffusional flux into and out of the cell. This flux is determined by the concentration gradient as reflected in the equation below (the k12 terms).

http://people.virginia.edu/~bsf2u/Extracellular%20Arsenate%20Concentration_files/image002.gif

The next part of the model represents the intracellular arsenate concentration. This is a function of the influx of arsenate into the cell by diffusion as well as the eflux of arsenate from the cell (again, the k12 terms). Furthermore, there is a term (the kc term) which represents the conversion of arsenate into arsenite by ArsC reductase.

http://people.virginia.edu/~bsf2u/Extracellular%20Arsenate%20Concentration_files/image004.gif

Intracellular arsenite is generated by the ArsC reductase (again, the kc term) and is also subject to the usual diffusional flux into and out of the cell (represented by the k34 terms). Furthermore, since arsenite is so water soluble, it has a difficult time passing through the nonpolar cell membrane by simple diffusion and therefore requires the cell to actively pump the arsenite out of the cell. This is the action, indeed the purpose of the ArsB pump: to get the more soluble, and more toxic arsenite out of the cell. This action of the ArsB pump is represented by the kB term in the equation below.

http://people.virginia.edu/~bsf2u/Extracellular%20Arsenate%20Concentration_files/image006.gif

Finally, extracellular arsenite concentration is a function of the arsenite pumped out by ArsB (again represented by the kB term), as well as the much less important simple diffusion of arsenite.

http://people.virginia.edu/~bsf2u/Extracellular%20Arsenate%20Concentration_files/image008.gif

Model Results/Validation
http://people.virginia.edu/~bsf2u/Extracellular%20Arsenate%20Concentration_files/model_output_1.png

Using HPLC measurements of arsenate and arsenite concentrations, in intracellular or extracellular compartments, and at various time points, we used a novel parameter fit algorithm developed in Matlab in order to determine the parameters.

Our model predicts exactly the expected results as is shown above. In the upper left graph, extracellular arsenate decays as it diffuses into the cell. In the upper right graph, intracellular arsenate initially peaks as the arsenate diffuses into the cell and before the ArsC reductase kicks in. Eventually however, the ArsC kicks in and arsenate is converted to arsenite at which point intracellular arsenate also decays.

In the lower right graph, intracellular arsenite initially peaks as the ArsC kicks in and converts arsenate to arsenite. It eventually peaks however, as ArsB, the arsenite pump kicks in pumps arsenite out of the cell. As we can see in the lower left extracellular arsenite concentration graph, as the cellular system functions, extracellular arsenite levels rises as extracellular arsenate is converted by the cell.

--Bsf2u 00:59, 22 October 2009 (UTC)

ArsB Pump Knockout
http://people.virginia.edu/~jdb2jf/vgem/ArsB_KO.png

The model can be used to predict the concentrations of intracellular and extracellular arsenate and arsenite if the ArsB transporter pump gene is removed via knockout.

The results show that the arsenate transport is unaffected, but intracellular arsenite concentration will saturate and extracellular arsenite will rise at a constant rate but at a very low concentration. Intuitively, the results make sense because the pump will not affect arsenate, but pump knockout will remove the active transport mechanism for getting arsenite out of the cell. However, since the model accounts for very low simple diffusion, it is seen that some arsenite will escape the cell based on this route.

--JBozzay 02:54, 22 October 2009 (UTC)

Sensitivity Analysis
Sensitivity analysis was performed to better understand the model. The different methods can be used to better understand how particular variables of interest are affected by different parameter perturbations.

http://people.virginia.edu/~jdb2jf/vgem/sens_fig1.png

These figures show the sensitivity coefficients for the particular output based on different parameter values.

http://people.virginia.edu/~jdb2jf/vgem/sens_fig2.png

These figures show how the output changes based on a range of parameter values.

http://people.virginia.edu/~jdb2jf/vgem/sens_fig3.png

These figures show the output range based on a coefficient of variation of ±0.1.

http://people.virginia.edu/~jdb2jf/vgem/sens_fig4.png

These figures depict 2D sensitivity matrices based on normalized sensitivity coefficients, which are useful for visualizing how parameter perturbations will affect model output.

--JBozzay 03:22, 22 October 2009 (UTC)