Team:Linkoping Sweden/Modeling

The purpose

To get an estimate of how the detection system would work under ideal conditions we created two mathematical models of the system. These two models represent two likely scenarios of how the biological process works. Using these models we can simulate the biological reactions that occur within the sample area when the

peanut protein Ara h 1 is introduced and thus estimate parameters and variables involved in the process. Furthermore, by estimating the detectors dimensional, optical and electrical restrictions we can estimate the final signal strength of the detector.

The models

By researching the many reactions involved in the biological process we finally created the following mass action models for the system.

Model 1

`(dEAC)/dt=k1f*Ab*E-k1b*EAC-k2f*EAC*Ar`

`(dA AC)/dt=k2f*EAC*Ar-k3f*A AC+k3b*Ar*Ab`

`(dAb)/dt=k3f*A AC-k3b*Ab*Ar-k1f*Ab*E+k1b*EAC`

`(dE)/dt=k2f*EAC*Ar-k1f*Ab*E+k1b*EAC`

`(dAr)/dt=k3f*A AC-k2f*Ar*EAC-k3b*Ar*Ab`

Model 2

`(dEAC)/dt=k1f*Ab*E-k1b*EAC`

`(dA AC)/dt=k3f*Ab*Ar-k3b*A AC`

`(dAb)/dt=k3b*A AC-k3f*Ab*Ar-k1f*Ab*E`

`(dE)/dt=k1b*EAC-k1f*Ab*E`

`(dAr)/dt=k3b*A AC-k3f*Ar*Ab`

EAC and AAC are the epitope- and Ara h 1-antibody complexes. Ab, E and Ar are the antibodies, epitopes and Ara h 1 proteins that create the complexes. The differential equations above describes the interactions between these molecules with the association (f) and dissociation (b) rate constants k1-3.1, 2, 3

For model 1 we assume that the change from epitope to Ara h 1 is done by changing one protein for the other, independently of the dissociation of proteins. In model 2 on the other hand, the change from epitope to Ara h 1 is instead dependant of the protein dissociation.

Using the modelling and simulation tool Wolfram SystemModeler the following models were then created:

Figure 1. Model 1. Mathematical model for the biological process. White circles indicate product and blue indicates substrate. Green circles represent the concentrations of the different constituents that build the process. The reaction rates in the process are denoted k1-2-3.

Figure 2. Model 2. Mathematical model for the biological process. White circles indicate product and blue indicates substrate. Green circles represent the concentrations of the different constituents that build the process. The reaction rates in the process are denoted k1 and k3.

These models were then used in combination with an estimate, `U_(conversion)`, of the detectors electrical, optical and biological effects to describe the full detection system.

`U_(RFPconversion)=I_(flux)*epsilon*l*Bp*D_(befo re)*E_(FRET)*Q_(RFP)*D_(after)*Lp_(RFP)*E_(RFP)*R/(6.241*10^18)`

`U_(FITCconversion)=I_(flux)*epsilon*l*Bp*D_(befo re)*Q_(FITC)*D_(after)*Lp_(FITC)*E_(FITC)*R/(6.241*10^18)`

The optical restrictions that the filters exert on the photon flux are described by BP and LP for the band pass and long pass filters respectively4, 5, 6. `D_(befo re)` and `D_(after)` are estimates of how the dimensions of the detector affect the photon flux. The fluorophores, fluorescein isothiocyanate (FITC) and red fluorescent proteins (RFP) quantum yield, Q, affects the flux as well as the förster resonance energy transfer (FRET) interaction, `E_(FRET)`, between the fluorophores7, 8, 9. The amount of absorbed photons by the FITC fluorophores is determined by the absorbance of the solution, calculated with the path length, l, the molar attenuation coefficient, `epsilon`, and the molar concentration used in later equations10. `I_(flux)` is an estimate of how many photons that are

emitted from the LED light after passing the first filter in the detection system4, 11. In order to estimate the electrical restrictions of the detector we need the sensors photon to electron conversion efficiency `E_(FITC)` and `E_(RFP)` for FITC and RFP emitted photons12. When used in combination with a resistance, R, and the amount of electron, `6.241*10^18`, required for 1 ampere current we can calculate the current created from a specific concentration of antibody complexes.

`U_(conversion)` is multiplied with the molar concentration change of the antibody complexes which yields the voltage change over time.

`(dU_(senso r1))/dt=(dU_(RFP))/dt=U_(RFPconversion)*(dEAC)/dt`

The Results

By using the model described in the section “The Model” the following simulations were created using Wolfram SystemModeler.

When comparing the two models we can see in Figure 3 that both models could in theory achieve high enough voltage, `10^-6`, for the detector to be able to read the signal. The main difference, for our purposes, seems to be that the process of changing from epitope to Ara h 1 takes up to 3 minutes to reach steady state in model 2, seen in Figure 4.

In model 1 on the other hand the biological process requires only 5-6 seconds to reach steady state after Ara h 1 has been introduced to the system, also seen in Figure 4. The detector should therefore in theory be able to give the user results within seconds after a sample has been taken.

Furthermore, a molar concentration for the epitope complex of `10^-6` seems to be optimal for the system, any lower and the detections systems hardware would not be able to detect the signal, any higher and the molecules in the detector would disrupt the photon flux too much resulting in a major signal disruption.13

The simulation of the detection systems sensor output as well as the calculated voltage generated by FITC is show in Figure 3. By using these two sensor systems it will be possible to determine if the signal change is created due to a complex change or simply because more molecules have been introduced to the sample area. When comparing the signal strength of the sensors in Figure 3 it is clear that the two sensor system is able to generate a larger signal change than either sensor on its own.

Figure 3. Simulation of the voltage change with model 1 and 2. X-axis is to the left in model 1 time 0-60 seconds, to the right in model 2 time 0-300 seconds. Y-axis is the voltage, V. The antigen Ara h 1 is introduced at 40 seconds.

Figure 4. Simulation of the Bioprocess with model 1 and 2. X-axis, to the left is model 1 time 0-60 seconds, to the right is model 2 time 0-300 seconds. Y-axis is the molar concentration, M. The antigen Ara h 1 is introduced at 40 seconds.

References