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MATHEMATICAL MODELING

SUMMARY

In our mathematical modeling section, we mainly try to simulate the in vivo production of two essential parts, aryl alcohol oxidase (AAO) and manganese peroxidase (MnP) in our engineered yeast Pichia pastoris (P. pastoris). To provide solid coverage of their whole genetic process, we established a systemic dynamic model individually, including the stages of transcription and translation. Moreover, we also examine the production kinetics of two specific mutants modified from the wild-type MnP (wtMnP) protein (PDB: 3M5Q).

ARYL ALCOHOL OXIDASE (AAO) KINETICS

INSTRUCTION

Aryl alcohol oxidase (AAO) is an enzyme, containing FAD. It is originated from Trametes versicolor (T. versicolor). Meanwhile, AAO is categorized into Glucose-Methanol-Choline oxidase/dehydrogenase (GMC) superfamily. Generally speaking, AAO is regarded as an ectoenzyme except in P. chrysosporium^[1].

One significant function of AAO in nature is to oxidize unsaturated alcohol, especially benzyl alcohol. According to documented studies before, after such oxidation hydrogen peroxide ( $H_2 O_2$ ) was produced to activate lignin peroxidase (e.g. LiP, VP, or MnP)^[2]. The mechanism is concluded as a catalytic cycle^[3]: with a FAD experiencing one oxidative (coupled to alcohol to aldehyde oxidation) half-reaction and one reductive (coupled to $O_2$ to $H_2 O_2$ reduction) half-reaction, respectively.

To facilitate the laboratorial, or even industrial production of AAO proteins, we design a single-cell circuit for the production process of AAO. Due to the fact that the production of a natural form of AAO in engineered strain was not carefully studied, we present an ordinary differential equation (ODE) model to simulate the secretion of AAO protein in vivo. Moreover, because there exists the potential of AAO mRNA degradation, we designed a term of its degradation in our formula as well.

Herein, SimBiology, an App of MATLAB, was employed as the platform of AAO expression simulation, and the solver was set as ode23t. We considered the degradation process of AAO mRNA and several scenarios of different AAO mRNA initial concentration. The results showed that the in vivo production of AAO proteins could satisfy our needs and there would be potentials to design a lag time ( $T_{lag}$ ) in our molecular design.

Model Preparation

Michaelis-Menten Equation

Michaelis-Menten equation is one of the most classic mathematical descriptions of the rate of enzymatic reactions in enzymology. With the application of the maximum reaction rate $V_{max}$ and Michaelis constant $K_M$ , we can outline the change of product concentration depended on time.

$v_0 = \frac{V_{max} [S]}{K_M + [S]}$

Law of Mass Action

The rate of the chemical reaction is directly proportional to the activities of the product or concentrations of the reactants, and the proposition is called the law of mass action in chemistry. This law explains and predicts the behaviors of solutions in dynamic equilibrium. Specifically, it implies that for a chemical reaction mixture that is in equilibrium, the ratio between the concentration of reactants and products is constant.

Assumptions

.Assumptions of our model
- We assumed that the transcription rate was 0.066 mol/hour^[4], or 0.0011 M, constantly, and the translation rate of P. pastoris was set in a constant value of 0.024 mol/second.
- In this in vivo single yeast model we assumed the concentration of AAO DNA was in a steady state, and its concentration was 1.0 M.
- According to the BioNumbers Database^[5], the volume of P. pastoris is approximately 1.25 * 10 $^{-10}$ mL. In our model, we saw all the yeast cells as integration and the whole volume was 1.0 L, so the number of P. pastoris was 8.0 * 10 $^{12}$ .
- The total volume of our reaction environment was set as 1.0 L.
Assumptions of Michaelis-Menten equation
- Due to the fact that in most conditions the process of RNA polymerase is a catalytic procedure of RNA polymerase II (Pol II), and its saturation state is worth considering. Therefore we chose Michaelis-Menten equation to simulate the whole mRNA synthesis process.
- In our in vivo single yeast model, we ignored the $k_{cat}$ of the mRNA production reaction for a simplification.
Assumption of law of mass action
- According to Schneider et al.^[6], the degradation rate of mRNA in vivo is independent of neither ribosomes translational process. Therefore we utilized the half-life of mRNA in P. pastoris^[7] to directly calculate the first-order degradation rate constant:
  $k_{at} = \frac{\ln{2}}{t_{\frac{1}{2}}}=\frac{0.693}{27 \ min} =0.026 \ {min}^{-1}$

Model Establishment

In this part we utilized the systems biology MATLAB tool, SimBiology, to construct this single yeast in vivo dynamic model.

Fig. 1 Diagram of AAO protein in vivo production.

Equations

AAO mRNA Production & Degradation

$\frac{d[AAO \ mRNA]}{dt} = \frac{V_{am}[AAO \ DNA]}{K_{am} \ + \ [AAO \ DNA]} \ - \ {\alpha}_{cad}[AAO \ DNA]$

A formula describing the production of AAO mRNA using Michaelis-Menten equation and and degradation of AAO mRNA in the cytosol with law of mass action.

AAO mRNA Translation

$\frac{d[AAO]}{dt} = d_{at}[AAO \ mRNA]$

First-order kinetics was applied to simulate the protein translation stage.

Parameters

Table.1 Parameters used in the AAO model.

Results

Based on our assumptions above, we simulated the production of AAO productions under different conditions with different initial $[AAO \ mRNA]$ values.

A. ${[AAO \ mRNA]}_{in}$ = 0 M

In 10 min the molarity of AAO protein was 0.06 M, and the [AAO] curve was in a hyperbolic shape (Fig. 2).

Fig. 2 Result of AAO protein production when ${[AAO \ mRNA]}_{in}$ = 0 M.

B. ${[AAO \ mRNA]}_{in}$ = 0.1 M or 1.0 M

We altered the initial concentration of AAO mRNA to 0.1 M and 1.0 M respectively, and the corresponding AAO protein concentration curves showed fewer hyperbolic characteristics and $[AAO]$ at 10 min after initial reaction start was 0.84 M (Fig. 3) and 0.30 M (Fig. 4).

Fig. 3 Result of AAO protein production when ${[AAO \ mRNA]}_{in}$ = 0.1 M.

Fig. 4 Result of AAO protein production when ${[AAO \ mRNA]}_{in}$ = 1.0 M.

Summary

From the two curves, we found that different initial concentrations of AAO mRNA in the single yeast cell in vivo system could result in widely divergent results, and the AAO protein concentration curve gradually lost its hyperbolic characteristics with the increasing of AAO mRNA initial concentration.

Due to such findings, we compared the simulation results of three scenarios and could get the conclusion that the production efficacy (0.30 ~ 0.84 M with ${[AAO \ mRNA]}_{in}$ in the range of 0 ~ 1.0 M) was totally acceptable in P. pastoris in practical industry production.

Model Promotion

Protein Production Rate

Here we used the simulated rate of 0.024 molarity/second as the overall translation rate of all the mRNA in P. pastoris. Such estimation might be relatively biased.

However, previous research of Shah et al.^[8] presented an algorithm in a Linux environment to calculate the protein production in yeast cells (in this case Saccharomyces cerevisiae, S. cerevisiae) that is typically limited by the availability of free ribosomes.

Based on numerous parameters (Table. 2) from P. pastoris genome^[9] , and the following algorithm below (Fig. 5), we will try to work out the exact estimated value of the protein production rate in P. pastoris after measuring some of these parameters and transforming the genome data into an acceptable data format.

Table.2 Summary of parameters in this algorithm.

Fig. 5 Pseudocode of this algorithm.

Lag Time between Transcription & Translation

In the Summary part in Results section, we could see that in different conditions with different AAO mRNA initial concentrations, the corresponding production concentrations were distinct. Moreover, the production concentration didn't increase accordingly with the raise of ${[AAO \ mRNA]}_{in}$ .

Therefore, it is necessary for us to modify the vectors we utilized and try to add a period called lag time ( $T_{lag}$ )^[10] in our model to satisfy our most needed production demand.

MANGANESE PEROXIDASE (MNP) KINETICS

Instruction

Manganese peroxidase (MnP) is a glycosylated exocellular enzyme containing heme molecule from Phanerochaete chrysosporium (P. chrysosporium)^[11]. The most common function of MnP is to degrade lignin compounds under natural conditions.

The special catalysis cycle of MnP has been discovered, and this cycle requires the consumption of hydrogen peroxide ( $H_2 O_2$ ). First, one $H_2 O_2$ molecule oxidizes the heme and forms a complex called MnP-I, and then this complex will oxalates one ${Mn}^{2+}$ to ${Mn}^{3+}$ and forms the second complex MnP-II, which can subsequently oxalates one ${Mn}^{2+}$ ion and turn to the initial situation^[12].

Since MnP can MnP can chemically modify and degrade lignin or other difficult-to-degrade chemical substances non-substrate-specifically^[13]. Most of all, previous research has found that some fungal can set colonization on PE thin films^[14], and secrete peroxidase, including MnP, to degrade PE film under nitrogen or carbon stress. With such ideas taken into account, we employed a combination of AAO and MnP in our design (see our Project Design part).

Fig. 6 The catalytic cycle of manganese peroxidase (MnP).

In our simulation we employed SimBiology as well in MATLAB, and simplified the MnP-SpyTag protein into PDB: 3M5Q for convenient. According to the final simulated curve, the production of MnP wasn't in a total balance with AAO production and thus we could invest more on MnP production, and two mutants of PDB: 3M5Q might improve the production efficacy in our future investigation.