*2.1*Introduction

Given that the extra energy consumption of heterologous expression is not conducive to cell growth, we do not want to allow all the expression of first three enzymes to be controlled by strong promoters. Instead, we want to use mathematical modelling to obtain the optimal proportion of enzymes with a higher yield of NAS, and to select the appropriate combination of constitutive promoters according to the modeling results. In a plan of our project, there are two kinds of cells in our design. The first cells can express the first three enzymes in the biosynthesis pathway of melatonin, including TDC, T5H and SNAT. Thus, in this kind of cells, tryptophan is catalyzed into tryptamine by TDC; next, tryptamine is converted into serotonin by T5H; sequentially, N-acetylserotonin (NAS) is synthesized from serotonin by SNAT. The last enzyme of this biosynthesis pathway, COMT which catalyze NAS to melatonin, is expressed in the second cell. The function of each cell is shown in Fig.1.

Fig.1. This figure shows the design of our project. Cell A contains 3 process of enzyme catalysis, and Cell B contains the last part of synthesizing Melatonin.

When it comes to the field of enzyme kinetic, the most famous theory to describe the behavior of enzyme reaction must be the Michaelis-Menten theory. In the following chapters, we will start with traditional M-M theory. However, we will modify this theory since it’s far from reliable when it comes to a complex reaction system. The reasons and results will be shown in the following chapters.

*2.2*Overview of Traditional Michaelis-Menten Theory

Before the improved version, let’s take a quick look at the traditional one. In 1913, Leonor Michaelis and Maud Menten proposed a mathematical model of the reaction. It involves an enzyme, E, binding to a substrate, S, to form a complex, ES, which in turn releases a product, P, regenerating the original enzyme. This may be represented schematically as:

Where k_{1} (forward rate),k
_{2} (reverse rate),
and k_{3} (catalytic rate) denote the rate constants, the double arrows
between S
(substrate) and ES (enzyme-substrate
complex) represents the fact that
enzyme-substrate binding is a reversible process, and the single forward arrow
represents the formation of P (product).

By applying the law of mass action (the rate
of a reaction is proportional to
the product of the concentrations of the reactants), we can write down four
non-linear ordinary differential equations and a conservation condition.

Here, C_{E}, C_{E0},
C_{S}, C_{ES} and C_{P} represent the
concentration of enzyme, total enzyme, substrate, the complex, and the product,
respectively. To solve these equations, the traditional way here is to apply
quasi-steady-state approximation (QSSA) that we assume the substrate is in
instantaneous chemical equilibrium with the complex. Therefore, we can let
equation (2.4) equals to 0 as:

Therefore, we get the reaction velocity:

Where

K_{cat} the turn-over number is the
maximum
number of substrate molecules converted to product per enzyme molecule per
second. The Michaelis constant KM is the CS at which the reaction rate is at
half-maximum. Thus, the concentration of production will be

Both parameters can be measured by measuring a single enzyme reaction with Lineweaver-Burk plot. The idea is to rewrite the equation as:

With equation (2.11) and experimental data,
we can easily get the parameters of
the system by fitting the slope, y-intercept and x-intercept. The y-intercept of
this graph is equivalent to the inverse of V_{max}. The x-intercept of
this graph
represents -1/K_{M}. However, we can only get 2 values from the
Lineweaver-Burk
plot, which is not enough to determine this system. The system contains 3
parameters, and 2 equations cannot help us solve the value of k_{1},
k_{-1} and k_{2}.

*2.3*Derivation of Reversible M-M Theory

Traditional M-M Theory seems OK on a
single enzyme reaction. However, the whole theory fails on complex enzyme
reaction system.

Why is that? Let’s take a look back at
the very basic assumptions of M-M
Theory first. **One of the basic assumption is that the concentration
of
substrate is much larger than the concentration of enzyme, which yields
CS>>CE**. Therefore, the complex can quickly enter the steady state
quickly
enough for us to omit the variation of the complex and use the equation
(2.7).

Hence, what is the restriction of
traditional M-M Theory? In a complex
enzyme reaction, things are different. This restriction might be OK during
the first enzyme reaction of TDC, but it will be poor in the reaction of T5H
and more poor in the reaction of SNAT. **The substrate of the
intermediate
product Tryptamine and Serotonin is zero while the whole reaction
starts**.
Therefore, we cannot use the M-M theory on a complex system.

How do we fix this? Our method is to
abandon this irreversibility character
of the traditional M-M Theory by adding the reversibility between the
product and the complex. We can rewrite the equation (2.1) as:

The difference between equation (2.1) and (2.12) lies in the parameter k-2. This parameter may be small but crucial to the limitation of reaction velocity. Then we can rewrite the equation system (2.2-2.5) as

Although we cannot get a simple expression like equation (2.11) but we can numerically solve the equation system. Here, as is shown in Fig.2. we can see the dynamical difference between equations (2.2-2.5) and equations (2.14-2.17).

*2.4*Reversible M-M Theory in Complex Enzyme System

The chapter 2.3 analyzed the dynamical character of 2 kinds of equations, but that only applies on single-enzyme situation. The reaction in cell A contains 3 enzymes. How do we model this complex enzyme reaction system? In fact, our idea is to firstly write down the system as Fig.3. With Fig.3, we know how to list our equations.

Fig.3. This is the whole reaction system in cell A. E_{1},
E_{2}
and E_{3} are the TDC, T5H and SNAT. S is the substrate Tryptophan.
P_{1}, P_{2} and P_{3}
are the Tryptamine, Serotonin and N-acetyl Serotonin. ES_{i} with
i=1,2,3
represents the enzyme complex. We assume that there is no other inhibition
between P_{i} with E_{j} (i≠j). Therefore, we totally have 12
parameters to be
measured. Note that the main difference with single-enzyme equations is that
P_{1}
is involved in the reaction with E_{2} and P_{2} is involved in
the reaction with
E_{3}

Following chapter 2.3, we can easily derive the equations as

To solve the equations above, we need to get the kinetic constant k in every step of reaction. But how to determine the different parameters? Lineweaver-Burk plot can only yields 2 equations, whereas we have 4 parameters in a single reaction step. Given this situation, we figured out a unique way to measure it. The key to the problem is to draw Lineweaver-Burk plot forwadly and reversely. That is to say, we can reverse the role of S and P, since every enzyme reaction is a reversible reaction. Then, we can get the equation (2.19):

Beyond (2.19), we can even include the chemical equilibrium constant equation (2.20)

In this way, we could be able to determine the whole parameters in one enzyme reaction instead of reduce the equation just to get a simple but not practical equation (2.8). Also, we get an extra equation (2.20) to test the result of our experiment. Aiming to measure the relative parameters (there are totally 12 parameters in this complex system), our team measured the activity of enzymes in vivo quantitatively. Honestly, the result is not satisfying, but we strongly believe that our method is still applicable given enough time and energy. Hence, we would show the numerical result Fig.4. with equation (2.18), and the parameters are set optionally. Still, it’s good way to see the advantages of reversible M-M theory in complex system.scientific supports, bacterial species and other available resources.

*2.5*Summary

With all the above mentioned, we believe that we have derived a unique way of modeling a complex enzyme system. Although the relative parameters are sometime hard to measure (especially reversely measure the reaction), but it still is a much more accurate and effective way compared with traditional way. Besides, there are still some more details to add to the model. For instance, enzymes won’t diffuse through cell membrane, but Tryptamine, Serotonin and NAS do. Therefore, if we add some diffusion terms in equation (2.18) like what we did in the modeling of repressilator, maybe we will get some interesting results. All in all, we hope our work can be helpful in relative enzyme reaction system in cells.

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