Team:QHFZ-China/Model
Model for describing the efficiency of the uric acid (UA) degradation device.
The schematic diagram of UA degradation device in HeLa cells is shown below (Fig. 1).
Figure 1. Gene circuits designed for UA degradation. (A) Schematic diagram showed the main gene parts. (B) Schematic diagram showed the operating principle of the gene circuits.
In this design, the enzyme for clearing UA is urate oxidase (smUOX). It can be secreted into tissue and convert UA to allantoin. To control the efficiency of smUOX, we set a controller, HucR-hucO8 system, upstream of smUOX. The express of HucR is constitutive and it works as a transcription inhibitor. In the absence of UA, HucR will bind to a special DNA sequence named hucO, to repress the expression of downstream gene. When the UA concentration is high enough, smUOX will be expressed. However, due to the structure of UA molecules, it is difficult for them to complete transmembrane transport, which leads to differences between in intracellular and extracellular UA concentrations. To balance the differentiation, we introduced a constitutively expressed human urate transporter 1 (URAT1) into engineered cells, which can enhance the transmembrane transport efficiency of urate. So that our cells can sense the concentration of UA in tissue, and express smUOX to clear UA if the concentration is high.
To characterize the degradation efficiency of extracellular UA by our engineered cells, and to evaluate the performance of the gene circuit, which could provide guidance for subsequent improvements, we established a mathematical model to describe the relationship between UA concentration inside or outside of our engineered cell and time.
First, in order to describe the state of UA concentration inside and outside the cell at a certain time, we proposed the following equation (1.1). Where [U1] is the extracellular UA concentration, [U2] is the intracellular UA concentration, k1 is the parameter describing the efficiency of URAT1, and t is the time. k2 and km are constants that describe the properties of smUOX, smUOX is a UA oxidase with exocrine function which we use to remove UA.
d[U1]/dt=-k1*([U1]-[U2]) -k2*y(4)*y(1)/(y(4)+km)..........(1.1)
d[U2]/dt= k1*([U1]-[U2])..........(1.2)
To describe the process about that UA molecules bind to the HucR and cause it to lose its inhibitory effect, we use a hill equation. It is an equation type used to describe ligand-receptor interactions. In our case, UA and hucO sequence compete for the binding to HucR protein. So, the hill equation is deformed into the following form. [HucR] is the concentration of HucR protein. r1 is the natural degradation rate of HucR protein. In order to simplify the equation, we proposed that all HucR proteins bind to the downstream DNA, and established the following equation. k3 is the parameter describing the ligands. And n1 is hill coefficient which describes the ligand synergy.
d[HucR]/dt=C2*k3^n1/(k3^n1+[U2]^n1)-r1*[HucR]..........(1.3)
According to the change of [HucR], we can obtain an equation to describe the expression of smUOX. We define smUOX is a UA oxidase with exocrine function, but secretion was not considered in this model because it was found to be unsatisfactory in the experiment. [smUOX] is the concentration of smUOX. Since the HucR protein binds to DNA, it can also be treated as a ligand-binding process, and the hill equation is used. [smUOX] is the concentration of smUOX protein. C4 is the expression level of smUOX in the absence of inhibition of HucR. k4 is the coefficient of HucR regulation of gene expression, and r2 is the rate of natural degradation of smUOX.
d[smUOX]/dt=C4*k4^n2/(k4^n2+[HucR]^n2)-r2*[smUOX]..........(1.4)
Finally, [U1] is used as the dependent variable, and t is the independent variable to draw an image, and the following image is obtained (Fig. 2).
Figure 2. Change of UA concentration under the action of UA degradation device. (A) Output of the model. (B) Concentration curve of UA drawn by experimental data.
The image on the left is result obtained from the model, and the image on the right is the experimental data image. We can find that the model correctly predicts the change in the rate of UA concentration, and by changing the parameters in the model, we can predict the strength or specificity of the promoter in the regulatory gene loop. After the rate of protein degradation, the rate of UA decomposition changes.
Model for describe the behavior of the UA detector in E.coli.
In our UA detector design, we used the same UA sensation mechanism as shown above. But as a detector, the engineered bacteria need to sense the concentration of UA in the treated blood sample or other samples, quickly and accurate. In our experiments, we found that if we need to use a blood sample for testing, we need to dilute the sample 1000 times to make the engineered bacteria work normally. However, the original design cannot have such a sensitivity. Therefore, we added a cascading amplification system, RinA_p80α - PRinA_p80α, which can rapidly and drastically increase sensitivity and output amplitude of cell-based biosensors. To shorten the waiting time, we selected sfGFP with short maturation time and high fluorescence intensity, which is convenient for reading by instruments such as a microplate reader.
To describe the behavior of the detector, we set several equations. The first equation is as same as equation (1.3) mentioned above. Considering there is no significant difference of intracellular and extracellular UA levels for downstream gene expression from the experimental data, we ignored the differences between intracellular and extracellular UA levels in this model. C1 is the constant expression rate of HucR protein, k1 is the parameter describing the ligands. And n1 is hill coefficient which describes the ligand synergy. [u] is UA concentration. r1 is the rate of natural degradation of HucR.
d[HucR]/dt=C1*k1^n1/(k1^n1+[U]^n1)-r1*[HucR]..........(2.1)
The second equation describes the regulation process of HucR on the expression of amplifier gene RinA_p80α. [RinA_p80α] is the concentration of RinA_p80α. The form is the same as equation (2.1) except the parameters. C2 is the constant expression rate of RinA_p80α protein, k2 is the parameter describing the ligand ligands. And n2 is hill coefficient which describes the ligand synergy. r2 is the rate of natural degradation of RinA_p80α. [RinA_p80α] is the concentration of RinA_p 80α protein.
d[RinA_p80α]/dt=C2*k2^n2/(k2^n2+[HucR]^n2)-r2*[RinA_p80α]..........(2.2)
The third equation describes the expression of sfGFP protein by RinA_p80α - PRinA_p80α, where the hill equation is still used, but it is modified. [C3] is the rate of sfGFP expression. k3 is the parameter describing the ligand ligands. And n3 is hill coefficient which describes the ligand synergy. [sfGFP] is the concentration of sfGFP protein.
d[sfGFP]/dt=C3*[RinA_p80α]^n3 / (k3^n3+[RinA_p80α]^n3)-r3*[sfGFP]..........(2.3)
Finally, we use [sfGFP] as the dependent variable, and t as the independent variable. In order to reflect the effects of the amplifier, we also built a model that does not contain the amplifier. Similar to the combination of equations (2.1) and (2.2). The parameters here are the same as they in the former model.
d[HucR]/dt=C1*k1^n1/(k1^n1+[U]^n1)-r1*[HucR]..........(3.1)
d[sfGFP]/dt=C3*k2^n2/(k2^n2+[hucR]^n2)-r3*[sfGFP]..........(3.2)
We draw the two curves simulated by the model (Fig. 3). As shown in the figure, we can find the amplifier gene increases the signal output significantly, and it help [sfGFP] enter the numerically stable interval earlier, which means that the available readings can be obtained earlier and the detection is faster. As for the reason that the amplifier gene can accelerate the detection speed, we speculated that the addition of the amplifier gene lowered the threshold of sfGFP gene initiation, and the response speed of the whole gene loop was accelerated.
Figure 3. The curve of fluorescence intensity with time simulated by the model.
Finally, UA concentration was taken as the independent variable and the readable output (stable state fluorescence value) of the bacterial UA sensor as the dependent variable, and the following image is obtained (Fig. 4).
Figure 4. Curve of detector output (numerical stability interval value) and UA concentration.
As can be seen, the addition of amplifier gene improved the sensitivity of bacterial sensor, which could output readable signal at lower UA concentration.
Attention: in two models, we selected all the parameter values in the real value interval. It can be considered that the effect of the model can be achieved by improving the design.
