Team:BGIC-Union/Model
Introduction
In the modeling part, we decide to prove that using NASBA to amplify the circulating tumor DNA is necessary and to estimate the optimal concentration of the circulating tumor DNA for detection.The influence of cell free DNA on the effectiveness of the detection device:
In real physiological environment of the plasma, there are many factors influencing the effectiveness of dcas9 protein binding with the targeted DNA sequence (EML4-ALK), the most prominent one of which is cell free DNA. Cell free DNAs come from apoptosis and necrosis of normal cells and may include gene EML4 and gene ALK, which are potential ligands bind with dcas9 protein and lead to a decrease of the effective concentration of NT7-dcas9 and CT7-dcas9 protein. The effective concentration is referred as the concentration of NT7-dcas9 and CT7-dcas9 which bind to the target sequence (EML4-ALK fusion gene), leading to one DNA molecule bears both NT7-dcas9 and CT7-dcas9 to enable the transcription of report gene. In order to determine the magnitude of the influence from the cell free DNAs on the effectiveness of split-T7-dcas9 in binding with EML4-ALK, the following model is made. Assumption that NT7-dcas9 binds with sequence of EML4 and CT7-dcas9 binds with sequence of ALK is made in the formula.
As shown in the formula above, given the total concentration of NT7-dcas9 and CT7-dcas9, the effective concentration of NT7-dcas9 and CT7-dcas9 decreases as the concentration of cell free DNA increase. In real physiological environment of plasma, the concentration of cell free DNAs is seven orders of magnitude larger than the concentration of circulating tumor DNAs. Therefore, it can be assumed that d[EML4] and d[ALK] approach to infinity. Hence, in this scenario, the change in effective concentration of NT7-dcas9 and CT7-dcas9 is equal to negative infinity, revealed as no signal emitted even if EML4-ALK fusion gene exists.
Based on the modeling, in order to gain valid result of the presence and concentration of EML4-ALK fusion gene, amplifying the concentration of EML4-ALK fusion gene is necessary. By employing the Nucleic Acid Sequence-Based Amplification (NASBA) (Figure 1), the concentration of EML4-ALK fusion gene increases at an exponential rate (Figure 2) . Within eight rounds, the concentration of EML4-ALK approaches to a detectable value. Therefore, it is proved that NASBA is a necessary and effective signal amplification approach before detection.
Determining the upper limit of target DNA concentration:
We decide to determine whether there is an upper limit of the target DNA concentration for split-T7-dcas9 system, since it is hypothesized that the effective concentration of NT7-dcas9 and CT7-dcas9 will decrease after the concentration of the target DNA reaches an upper limit; since the increase in potential binding sites increases the probability of one DNA molecule bears only one protein (Figure 3). The upper limit is the optimal point for the highest activity of the detection device.
To determine the optimal point for detection, the concentration of NT7-dcas9, the concentration of CT7-dcas9, and their fraction of binding with DNA are considered. The formula for the optimal point is established as:
2 × [DNAoptimal] = [NT7 total] × θN + [CT7 total] × θC,
where [DNAoptimal] represents the optimal concentration of DNA for detection system.
In the situation that the concentration of DNA is lower than the optimal concentration, the concentration of NT7-dcas9 and CT7-dcas9 binding with the DNA increases linearly as the concentration of target DNA increases. The change in effective concentration of NT7-dcas9 and CT7-dcas9 are represented in the formula below:
When the concentration of target DNA is really low, the dissociation between the DNA and the proteins almost does not occur, therefore, the dissociation constant is negligible. However, after the concentration of DNA reaches a tipping point, the dissociation becomes prominent, thus, θ is considered and the value of the slope decreases by θ.
After the concentration reaches the optimal concentration for detection, the probability of one DNA molecule bearing both NT7-dcas9 and CT7-dcas9 and enabling transcription of report gene decreases as the concentration continually increases. The probability of one DNA molecule bearing both NT7-dcas9 and CT7dcas9 is represented as the formula below:
Employing the formula above and the data from Peking 2015, the best-fitted curve for the relationship between relative fluorescent unit and target DNA concentration before and after the optimal concentration is reached has been made in Figure 4 and Figure 5. Revealed in the curves, before the optimal concentration is reaches, in the interval of low concentration (0-0.6), the line has a greater slope than the slope after the concentration reaches a certain value (0.6-1.5), decreasing by multiplying the factor of θ. After the optimal concentration is reaches, the slope is negative and the value of the negativity increases as a factor of 
Therefore, the best-fitted curves prove that our hypothesis is correct.
