Our model is mainly composed of two parts: the first part is the biological part, and the first part is the Michaelis-Menten equation model, which is used to describe the relationship between the starting speed of enzymatic reaction and the substrate concentration, and then the negative feedback model of huc-huco-uric Acid gene, which is used to describe the regulation of Uric Acid on the negative feedback model. The second part is a one - chamber model of the variation of uric acid oxidase concentration over time with single dose and multiple doses. According to the knowledge of pharmacology, the intensity of drug action is proportional to its concentration. When taking single and periodic drugs, we can characterize the action strength of uric acid oxidase by the concentration of uric acid oxidase in blood.
Team:DLUT China/Model
Model
First, the reaction rate and the concentration of substrate and enzyme can be obtained from the Michaelis-Menten equation.
Our chemical reaction equation can be written as:
Thus, we can infer that:
We assume that :
So:
The reaction rate after reaching equilibrium is:
Our chemical reaction equation can be written as:
Thus, we can infer that:
We assume that :
So:
The reaction rate after reaching equilibrium is:
Negative feedback model
HucR in the figure is a transcriptional regulatory factor, and hucR gene generates hucR protein after being expressed. Two HucR proteins can bind to form stable dimers. The dimer can interact with the promoter of flagellin and the common sequence of hucR, which is called hucO.
When uric acid did not exist, the gene expressed HucR protein, forming dimer.
![]()
The dimer can bind to genes,![]()
This prevents the promoter from binding to the RNA polymerase, further inhibiting the expression of related genes.
However, when uric acid is present, HucR dimer will bind to uric acid to form stable complexes, and will not bind to hucO, so that related genes can be transcribed and translated normally. Inactive genes can also bind to uric acid to restore activity.
![]()
From the Griffith model, we can know: In the negative feedback model of a single gene, gene activity can be expressed by a formula.
![]()
To represent.
In this formula, m is the number of uric acid acting on genes, which can be seen from equation (10), m=4. The activity of plasmid gene expression (proportion) was![]()
mRNA concentration (M) can change over time by the end of this rate M0, under a certain level of uric acid gene transcription rate of k_1 U ^ M/(K_eq + U ^ M) and mRNA degradation rate k2M decided that:
![]()
Where k1 and k2 are rate coefficients and both are positive values.
Changes in uric acid oxidase concentration (E1) and uric acid transport protein concentration (E2) can be given by the generation and degradation of mRNA:![]()
In equation (14)(15), c1,c2,d1 and d2 are rate constants, which are positive values.
The increase of uric acid concentration from extracellular to intracellular is mediated by uric acid transporter protein, while the decomposition of uric acid depends on the uric acid oxidase. As can be seen from the mistral equation (7) derived from the first part:
![]()
Proportional coefficient are positive.
HucR in the figure is a transcriptional regulatory factor, and hucR gene generates hucR protein after being expressed. Two HucR proteins can bind to form stable dimers. The dimer can interact with the promoter of flagellin and the common sequence of hucR, which is called hucO.
When uric acid did not exist, the gene expressed HucR protein, forming dimer.
The dimer can bind to genes,
This prevents the promoter from binding to the RNA polymerase, further inhibiting the expression of related genes.
However, when uric acid is present, HucR dimer will bind to uric acid to form stable complexes, and will not bind to hucO, so that related genes can be transcribed and translated normally. Inactive genes can also bind to uric acid to restore activity.
From the Griffith model, we can know: In the negative feedback model of a single gene, gene activity can be expressed by a formula.
To represent.
In this formula, m is the number of uric acid acting on genes, which can be seen from equation (10), m=4. The activity of plasmid gene expression (proportion) was
mRNA concentration (M) can change over time by the end of this rate M0, under a certain level of uric acid gene transcription rate of k_1 U ^ M/(K_eq + U ^ M) and mRNA degradation rate k2M decided that:
Where k1 and k2 are rate coefficients and both are positive values.
Changes in uric acid oxidase concentration (E1) and uric acid transport protein concentration (E2) can be given by the generation and degradation of mRNA:
In equation (14)(15), c1,c2,d1 and d2 are rate constants, which are positive values.
The increase of uric acid concentration from extracellular to intracellular is mediated by uric acid transporter protein, while the decomposition of uric acid depends on the uric acid oxidase. As can be seen from the mistral equation (7) derived from the first part:
Proportional coefficient are positive.
In pharmacology, the degree of action of the drug is directly proportional to the concentration of the drug in the blood, and the efficacy of the drug can be analyzed by measuring the concentration of uric acid oxidase in the blood.
We assume that the entire human body is treated as a central chamber, a part of the body's blood. Uric acid oxidase in the central chamber is uniformly distributed and ongoing and degradation, we can set the number of e. coli enters the body, the degradation rate constant, etc., according to the function of relationship between each variable, using the extremum and differential and so on many kinds of mathematical method and the MATLAB software to solve the model, calculated the medication and periodic medication for the first time in the blood uric acid oxidase a chamber model of change over time.
A The basic assumptions:
1. The central ventricular volume, or blood volume, remains unchanged.
2. The influence of individual differences on the model is not considered.
3. Before the first entry of uric acid oxidase into the central chamber, the blood drug concentration in the central chamber was zero.
4. Treat the whole body as a central chamber, considering only the variation of uric acid oxidase concentration in blood.
5. The degradation rate of uric acid oxidase is directly proportional to the concentration of urea oxidase in blood.
6. The rate of uric acid oxidase entering blood is directly proportional to the number of escherichia coli in the intestinal tract.
7. After taking a drink or medicine, the medicine is produced and immediately enters the blood and is distributed evenly.
8. Within a reasonable range of blood drug concentration, uric acid oxidase has no side effects on the body.
B Drug distribution model
We hypothesized that the number of escherichia coli taken was D, and the number of escherichia coli taken into the human intestinal tract was p(0), then there were![]()
The schematic diagram of blood drug concentration change in a compartment model is as follows:
Because the drug from the ventricular rate meet the first order reaction kinetics, which exclude rate with the moment center room are positively, total drug proportion coefficient is, the, room rate of drugs into the centre, by more than one room model assumption, the body change meet: of center indoor total drug, chamber volume as the center, so the center indoor blood drug concentration at any moment one room model
C. Variation of uric acid oxidase concentration and blood drug concentration curve under single dose administration
Based on the knowledge of pharmacokinetics, uric acid oxidase first reaches uniform distribution in the absorption area immediately, and then uric acid oxidase is absorbed into the central chamber, divided into two processes, which can be abstracted as:
1. Based on the knowledge of pharmacokinetics, the absorbed regional dosage X0(t) meets the initial value problem:
2. Changes of drug concentration in the center and curve of drug concentration
Based on the knowledge of pharmacokinetics, this model is satisfied, therefore, the 1-compartment model of blood drug concentration is determined as,
Its solution is:
Use MATLAB to draw the image as shown in figure 4 below.
D. multidose administration kinetics
Assuming that multidose function is:
In the time equation of blood drug concentration after single dose administration, each index is multiplied by the multi-dose function r, where the injection cycle is set to T2, and the time equation of blood drug concentration after repeated administration is obtained
When, the blood drug concentration reaches a steady state, and the relationship between the blood drug concentration and time is
If I take the derivative, I get
The derivative method is used to find the extremum of steady state.
Thus
So the range of D is the range of values of the fixed quantity to be determined.Combined with the expression of oral (or muscle injection) under the one-time administration mode, set C(T2)=C1 to obtain the time interval of oral (or muscle injection) under the fixed agent D.
We assume that the entire human body is treated as a central chamber, a part of the body's blood. Uric acid oxidase in the central chamber is uniformly distributed and ongoing and degradation, we can set the number of e. coli enters the body, the degradation rate constant, etc., according to the function of relationship between each variable, using the extremum and differential and so on many kinds of mathematical method and the MATLAB software to solve the model, calculated the medication and periodic medication for the first time in the blood uric acid oxidase a chamber model of change over time.
A The basic assumptions:
1. The central ventricular volume, or blood volume, remains unchanged.
2. The influence of individual differences on the model is not considered.
3. Before the first entry of uric acid oxidase into the central chamber, the blood drug concentration in the central chamber was zero.
4. Treat the whole body as a central chamber, considering only the variation of uric acid oxidase concentration in blood.
5. The degradation rate of uric acid oxidase is directly proportional to the concentration of urea oxidase in blood.
6. The rate of uric acid oxidase entering blood is directly proportional to the number of escherichia coli in the intestinal tract.
7. After taking a drink or medicine, the medicine is produced and immediately enters the blood and is distributed evenly.
8. Within a reasonable range of blood drug concentration, uric acid oxidase has no side effects on the body.
B Drug distribution model
We hypothesized that the number of escherichia coli taken was D, and the number of escherichia coli taken into the human intestinal tract was p(0), then there were
The schematic diagram of blood drug concentration change in a compartment model is as follows:
Because the drug from the ventricular rate meet the first order reaction kinetics, which exclude rate with the moment center room are positively, total drug proportion coefficient is, the, room rate of drugs into the centre, by more than one room model assumption, the body change meet: of center indoor total drug, chamber volume as the center, so the center indoor blood drug concentration at any moment one room model
C. Variation of uric acid oxidase concentration and blood drug concentration curve under single dose administration
Based on the knowledge of pharmacokinetics, uric acid oxidase first reaches uniform distribution in the absorption area immediately, and then uric acid oxidase is absorbed into the central chamber, divided into two processes, which can be abstracted as:
1. Based on the knowledge of pharmacokinetics, the absorbed regional dosage X0(t) meets the initial value problem:
2. Changes of drug concentration in the center and curve of drug concentration
Based on the knowledge of pharmacokinetics, this model is satisfied, therefore, the 1-compartment model of blood drug concentration is determined as,
Its solution is:
Use MATLAB to draw the image as shown in figure 4 below.
D. multidose administration kinetics
Assuming that multidose function is:
In the time equation of blood drug concentration after single dose administration, each index is multiplied by the multi-dose function r, where the injection cycle is set to T2, and the time equation of blood drug concentration after repeated administration is obtained
When, the blood drug concentration reaches a steady state, and the relationship between the blood drug concentration and time is
If I take the derivative, I get
The derivative method is used to find the extremum of steady state.
Thus
So the range of D is the range of values of the fixed quantity to be determined.Combined with the expression of oral (or muscle injection) under the one-time administration mode, set C(T2)=C1 to obtain the time interval of oral (or muscle injection) under the fixed agent D.