Team:Edinburgh/Modelling/Biosensors/Bio-el-Sensor

Equations involved in using bioelectric interface as a biosensor

Our MATLAB modellers decided to take on the task of modelling the bioelectric interface as a biosensor similar to the one created by the Edinburgh iGEM team in 2006 in order to investigate whether it is currently . While the KAPPA modellers were looking at the electron transport going on inside the cell between molecules, the MATLAB modellers wanted to look at how the cell becomes an electric interface and if this can become a useful model.

As described in the video abstract, an inducer molecule starts the process by interacting with the mtrCAB gene which will then lead to the production of the mtrCAB proteins required to make an ordinary cell into a bioelectric interface. If we are to use the bioelectric interface as an arsenic biosensor, the inducer molecule will be the arsenic molecule being detected and the protein being expressed will be the mtrCAB proteins.


Figure 1 – diagram to represent mtrCAB operon

In the above diagram, the arsenic molecule (As(III)) binds to the arsD protein expressed by the operon, stopping repression of the mtrCAB promoter which then means mtrCAB can be expressed.

First we found equations to represent the following:
1) formation of [arsD – 2As(III)] complex: arsD + 2As(III) <-> arsD[arsD-2As(III)]
2) arsD binding to promoter: 2arsD + P <-> (2arsD-P)
3) arsD degradation: arsD -> degraded arsD
4) mtrCAB degradation: mtrCAB -> degraded mtrCAB
5) arsD expression: V_m*arsDgene/(K_m+arsDgene)
6) mtrCAB expression: V_m*mtrCABgene/(K_m+mtrCABgene)

Each of the six processes has a rate equation based on mass transfer on Michaelis Menten:

1) r_1,forward = k₁[arsD][As(III)]¬¬² + and r_1,backwards = k_-1[arsD-2As(III)]
2) r_2,forward = k₂[arsD]²[P] and r_2,backwards = k_-2 [2arsD-P]
3) r₃ = k₃[arsD]
4) r₄ = k₄[mtrCAB]
5) r₅ = V_m,5*[g_arsD]/(K_m,5 + [g_arsD])
6) r₆ = V_m,6*[g_mtr]/(K_m,6 + [g_mtr])
n.b square brackets refer the concentration of the component inside the brackets in mol/l, k, V_m and K_m are rate constants, g_i refers the the gene for protein i, P refers to the promoter and r is the rate in mol/l/s

This allows for a system of ordinary differential equations to be created based on the rate of change of each component present:

d[arsD]/dt = V_m,5*[g_arsD]/(K_m,5 + [g_arsD]) - k₁[arsD][As(III)]¬¬² + k_-1[arsD-2As(III)] - k₂[arsD]²[P] + k_-2 [2arsD-P] - k₃[arsD]
d[arsD-As(III)]/dt = k₁[arsD][As(III)]¬¬² - k_-1[arsD-2As(III)]
d[2arsD-P]/dt = k₂[arsD]²[P] - k_-2 [2arsD-P]
d[P]/dt = k_-2[2arsD-P] - k₂[arsD]²[P]
d[As(III)]/dt = k_-1[arsD-2As(III)] - k₁[arsD][As(III)]¬¬²
d[mtrCAB]/dt = V_m,6*[g_mtr]/(K_m,6 + [g_mtr]) - k₄[mtrCAB]

The main problem with using this method is that the constants which determine the rate cannot be found in literature for mtrCAB protein synthesis and degradation as it is a very new area of research, meaning these values must be estimated.