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Model

Kinetics of enzyme catalyzed reactions

Overview

Mathematical model is an essential tool in scientific research to help us understand experimental results, predict the outcome, and improve methods. Through mathematical modeling, we wanted to understand and improve our project by using a model to explain and overcome some of the problems we had encountered during our experiments. We specifically targeted on the modification of MFP series since it is the core of our design and it is most promising to have a robust underwater adhesive property in our co-expressed system.

Several problems occurred when co-expressing our recombinant proteins containing MFP with tyrosinase (mTyr-CNK). Two of the most remarkable difficulties we had encountered was the low modification rate of MFPs and the low yield of the co-expressed proteins.

Overcoming these problems is essential for our project because modified MFP and the protein yield are two of the most important factors that can determine how strong the adhesive property of the recombinant proteins and the hydrogel made from the recombinant proteins can achieve.

Therefore, we wanted to build a model to discuss the modification of tyrosinase in vivo.

Table 1. General assumptions and Symbols used.

Outline and aim

It is well studied that the modification of tyrosine into DOPA by tyrosinase is a crucial step for MFP to become adhesive, therefore, we wanted to have the greatest possible modification rate in order to give our recombinant protein its best performance.

Therefore, a knowledge of the actual mechanism involved in the catalytic reaction is needed for the understanding of tyrosinase modification from tyrosine to DOPA in more detail before we continue to discuss the dynamic model. Here, we want to understand how different concentration of enzyme and substrate can lead to different results of modification rate in a non-changing system, and therefore find out the optimum concentration rate.

The importance of finding out the optimum concentration rate is that the burden inside the cell cannot afford us to overexpress mTyr-CNK. The low yield is probably caused by the high burden, which resulted in inefficient protein expression. Therefore, we wanted to express as little mTyr-CNK as possible, with the premise that the modification rate is high enough.

We applied the Michaelis-Menten enzyme kinematics to predict the optimum concentration of substrate and enzyme to achieve the highest in vivo DOPA modification rate in a system where enzyme concentration is continually changing.

Method

The Michaelis-Menten enzyme kinematics is one of the most useful models to analyze biochemical kinetics. This model provides an insight into the kinetics of tyrosinase modification from tyrosine into DOPA.

The core concept of Michaelis-Menten kinematics is a two-step reaction. The first step is the combination of a substrate with an enzyme active site to form an intermediate product, which then reacts in the second step to form the final product. This can be summarized by the following equation:

Here we assume that the second stage of the reaction is irreversible, and its reaction rate is Kcat. The reaction rate in the first step is denoted as k1 and k2 for forward and backward reactions, respectively. Thus, we can obtain the following reaction rate equations:

Solving these differential equations, we can get:

However, in order to get a more distinct outcome, [ES] is assumed to be constant, which is, d[ES]/dt=0. With this assumption, the relationship of [E], [S], and [ES] can be deduced, where Km is the Michaelis constant:

Therefore, we can rewrite d[P]/dt as:

Notice that Vmax=Kcat[Tyr0], which is when all enzyme is readily modifying a substrate binding with it. Under this condition, the previous formula can be rewritten:

Be aware that [Etot]=[E]+[ES] since the overall enzyme concentration remains unchanged, and [ES]=[E][S]/Km according to the previous equation, we can get the relationship between reaction rate and the substrate concentration:

This is the Michaelis-Menten equation.

We used this equation to calculate the change in the rate of reaction against the substrate concentration when the enzyme concentration is kept constant. The parameters we used for mTyr-CNK are all obtained from previous studies (Do, Kang, Yang, Cha, & Choi, 2017). (Table 2)

Table 2. Parameter used in our model.

At constant enzyme concentration, the reaction rate against substrate concentration is as shown:

Regulated gene expression of mTyr-CNK

However, the Michaelis-Menten enzymatic equation alone is only suitable for most enzyme reactions with a stable enzyme concentration. However, in a co-expression system, the enzyme concentration is continually changing. This requires a dynamic model dependent on the gene expression of mTyr-CNK.

Therefore, this model has the potential to be fully developed using dynamic programming in order to predict and explain a co-expression system.

Reference

Phillips, R. (2013). Physical biology of the cell. New York: Garland Science.

Do, H., Kang, E., Yang, B., Cha, H. J., & Choi, Y. S. (2017). A tyrosinase, mTyr-CNK, that is functionally available as a monophenol monooxygenase. Sci Rep, 7(1), 17267. doi:10.1038/s41598-017-17635-0