### Introduction

In choosing the enzyme HpaBC to catalyze tyrosine into levodopa, we use experimental data to build up a model to simulate the catalysis process, testing its reaction rate in different substrate concentration. This model can direct our way for our experiment, and predict the rough trend of the yielding of levodopa.

In our experiment part we have explained the total catalyze process, which is presented below,

Our team uses Michaelis-Menten equation to simulate the catalysis of the key enzyme, HpaBC. The Michaelis-Menten equation is the equation that models the catalysis of reactions that involve enzyme as catalyst. The basic form is presented below:

Where E is the enzyme, S is the substrate, ES is the enzyme-substrate complex, and P is the product.

The rate constant of the forward reaction is , whereas that of the reverse reaction is

. The rate constant of the reaction is , also denoted as . Therefore, we have the overall reaction rate,

* *

Assume the total concentration of enzyme is and the steady rate constant to be, we get

When , v reaches its maximum. Therefore we have

### Assumptions

1. and stay constant. The two enzymes HpaB and HpaC will incessantly converting the two substrate circularly.

2. At cellular level, and stay constant by our assumption. Therefore the overall reaction can be modified as the following,

3. When plotting in Matlab, we assume that the formation rate of levodopa is the same as the consumption rate of tyrosine in a long time scale. That is,

### Notations

--- The total concentration of enzyme HpaB

--- The concentration of levodopa

--- The concentration of tyrosine

--- The rate constant that enzyme-substrate complex turns into product

--- The catalytic efficiency, Michaelis–Menten constant

--- The reaction rate

--- The maximum reaction rate

### Models

Replacing the notations in the model with the experimental factors, we derive the following equations,

To know the trend of how varying substrate concentration can affect reaction rate, the constants in this equation should be known. Therefore, we develop models to determine the constants.

*Determination of parameters*

To determine the rate constants, the experimental data that the yielding of levodopa change with respect to time with different initial concentration of tyrosine is needed. In this case, we apply the double reciprocal equations to help us determine those factors. Taking the reciprocal of equation (3) on both sides, we get the following equations,

Fitting the data points of verses into a linear line, we can find the absolute value of by the intercept of the X-axis. The value of is the intercept of the Y-axis. Using unary linear regression, we can successfully estimate the parameters.

(Figure.1 Double reciprocal graph of HpaBC-WT)

### Results

The determined parameters are listed below as a table.

Fitting the parameters in to equation (2), we plot the following graphs for HpaBC-WT and HpaBC-SMS.

(Figure.2 Reaction rate of HpaBC-WT)

(Figure.3 Reaction rate of HpaBC-SMS)

Then, we use built-in functions of excel to derive the fit line of reaction rate. To have a better visualization, we uses Matlab to solve the ordinary differential equations based on the graphs above. The classic ODE45 is applied in our code. The results are listed below.

(Figure.5 Left, HpaBC-WT. Right, HpaBC-SMS)

### Discussions

*Explanation of graphs*

Interestingly, as we can see on the graph, the reaction rate of HpaBC-SMS is actually lower than that of HpaBC-WT. However, the most intriguing thing is not presented on the graphs. When the concentration of tyrosine (substrate) is relatively high, higher than 5mM, the catalysis seems to be repressed, which is lower than that of wild type. When the concentration of tyrosine is relatively low, especially at 3mM, the reaction rate of HpaBC-SMS exceeds that of HpaBC-WT.

*Comparison with experiment data*

In one of our experiment, when constantly given light to prevent the dissociation of photoswitches, that is, the enzyme HpaBC wild type is not dissociated. The comparison bars are plotted below,

(Figure.4 Experiment data and prediction of L-Dopa yielding with substrate with 3mM)

*Limitations*

The unary linear regression for data of HpaBC-SMS may not be so accurate because of its extensive interval distribution. Therefore, we can only see the rough trend of reaction rate of the enzymes.

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