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# Model

*Overview*

*Overview*

With the aid of our mathematical model, we gain a better understanding of the enzymatic activity between uric acid(urate) and uricase(urate oxidase). Since the aim of our project is to eliminate the excessive uric acid, we want to first understand the rate of uric acid consumption. In order to optimize our product for removal of uric acid, we will further predict the optimum concentration of uricase and the time required for the reaction by using our mathematical model.

### Aim:

1. Study the uricase activity with different concentration of uric acid with time.

2. Study the uricase activity with different concentration of competent cell containing uricase with time.

3. Fit in our experiments data to Michaelis-Menten kinetics model and predict the relationship between rate of reaction and concentration of uric acid.

4. Study the order of reaction (uricase activity) with respect to the concentration of uricase.

### Assumptions:

1. The enzymatic reaction is irreversible and there is only one enzyme’s catalytic constant involved.

2. Neglect the decomposition of H2O2 within 60 minutes.

3. Since the absorbance of H2O2 is directly proportional to concentration of HIU, we use absorbance of H2O2 to represent the concentration of HIU.

4. Since the concentration of e-coli is directly proportional to the concentration of uricase, we use the concentration of e-coil to represent the concentration of uricase.

### Definitions of parameters and variables:

S |
Substrate ( Uric acid ) |

E |
Enzyme ( Uricase ) |

C |
enzyme-substrate Complex |

P |
Product ( HIU ) indicate by H2O2 |

k1 |
Reaction rate of reaction from substrate and enzyme to complex |

k−1 |
Reaction rate of reaction from complex to substrate and enzyme |

k2 |
Reaction rate of reaction from complex to product and enzyme |

*Concentration over time*

*Concentration over time*

According to mass action law, we know that the rate of reaction is directly proportional to the concentration of H2O2. We have the equation:

Moreover, we try to derive the equation to find the relationship between concentration of H2O2 and time:

Let C be the constant of this reaction, where C is equal to . We finally design this model and it is an exponential equation . After lab work is done, the constants C and k1 are calculated by using the data from experiments and applying the equation. For this experiment, the concentration of uric acid remains unchanged during all the process.

**Figure 1**

From figure 1, the relationship between absorbance of uric acid and time is shown. We used different concentration of uricase to build this model. The concentration of uric acid remains unchanged at 2500 μM.

There is a significant difference between concentration of uricase 1422 cell/mL and 948 cell/mL. In order to find the most suitable concentration of uricase required, 1422 cell/mL of concentration should be used to obtain a higher rate of reaction and efficiency. Our final model for this reaction is where C is 0.5566 , k1 is 0.01292/min.

**Figure 2**

From figure 2, the relationship between absorbance of uric acid and time is shown. For this one, we used different concentration of uric acid to build this model. The concentration of uricase remains unchanged at 50 U / mL. Since the normal blood uric acid level is around 200 – 500 μM, we try to test the concentration of uric acid from 50 to 1000 μM. From the graph, concentration with 50 / 100 / 200 uM uric acid is gradually increasing the absorbance but for other concentrations of uric acid, there is no significant difference.

*Michaelis-Menten kinetics*

*Michaelis-Menten kinetics*

Michaelis-Menten kinetics is a model used to examine enzyme kinetic – activity of uricase. Uricase may catalyze the conversion from uric acid to HIU.

At the beginning, we assume the reaction is an irreversible enzyme-catalysed reactions and leads to a chemical event:

After applying the law of mass action, we have the following differential equation model:

We reduce the model to describe this enzyme-substrate activity as a single reaction. The reaction rate is called Michaelis-Menten rate law. Define Vmax as the maximum rate of reaction and Km as the half-saturating constant:

We use this model to find the rate of reaction ( rate of Uric acid -> HIU ). Constant Km also represents the affinity of enzyme, which is the strength of uricase to bind to the substrate. An equation with a low Km value indicates a large binding affinity, as the reaction will approach Vmax more rapidly. Vmax is the maximal rate of the reaction. Vmax/Km, or more usually kcat/Km, is a measurement of "catalytic efficiency."

*Rate of reaction against concentration*

*Rate of reaction against concentration*

After we input the data from experiment, the result are as follow:

This model represents the activity of uricase and now we may predict the rate of reaction with different concentration of uric acid. The fixed concentration of uricase is 50 U / μM.

*Conclusion*

*Conclusion*

The ultimate goal of our modeling is to find the optimum concentration of uricase that the gout patient need, in order to solve their hyperuricemia problem.

Vmax is directly proportional to the initial concentration of uricase [E0], where kcat is a catalyst constant, therefore we can use the graph of rate of reaction at 50 U/μM of uricase to predict the rate of reaction with different concentration of uricase.

Using the result of part 4, we know that the 50 U/μM of uricase has 92.64 μM of KM. This Mathematically proves that the result of part 2, why there is gradual increase of HIU when we use 50 / 100 / 200 μM of uric acid. As the uricase should work best with 92.64 μM of uric acid. It is in range of 50 – 200 μM of uric acid.

By using the other equation :

Combining two equations and we can get:

Therefore the concentration of uric acid is directly proportional to that of uricase, we may follow the ratio of uric acid to uricase in order to find the optimum concentration that the patient should take in.

Here is an simple example, if we have 500 μM of blood uric acid level, we will need 50*2500/500 U / μM of uricase to solve the hyperuricemia.