Model | Fudan - iGEM 2022

Introduction

When designing the whole manufacturing process, we proposed that the enrichment of aldehyde reductase can be super effective in increasing productivity. Before putting this idea into practice, we wanted to verify the idea. The modeling aims to determine whether the enrichment of the enzyme by phase separation is effective.

The Diffusion model consists of two parts, the Cahn-Hilliard Equation, and the Michaelis-Menten Equation. The Cahn-Hilliard Equation simulates the phase separation process, while the Michaelis-Menten Equation allows us to calculate the reaction rate.

Model design

The first step is based on the Cahn-Hilliard Equation, which describes the differential equation of the phase separation process. During the phase separation process, regions rich in aldehyde reductase will have more and more aldehyde reductase. Thus, although the total concentration of the enzyme in the cell does not change, the number of small regions (fixed size) with high concentrations gradually increases over time.

Next, with the stimulated concentration, we will be able to find out a theoretical speed of enzyme reaction using the Michaelis-Menten Equation. With this equation, the reaction rate can be calculated at different substrate concentrations, which is an essential guide for both production and research.

As discussed above, these two equations are able to describe the whole reaction system from two perspectives. The Cahn-Hilliard Equation is responsible for the kinetic simulation of phase separation, while the Michaelis-Menten Equation calculates the reaction rate according to the concentrations.

Assumption

Without prejudice to the conclusions, we make the following assumptions in order to simplify the model：

The reaction space is a two-dimensional rectangle.
When aldehyde reductase concentrates and aggregates, the other components aggregate accordingly.
A large overdose of substrate exists in the reaction system.
At the beginning of the reaction, the product concentration is so tiny that the inverse reaction can be ignored.
The steady-state assumption, i.e., as the reaction proceeds, the rate of complex formation gradually decreases, and decomposition accelerates, reaching equilibrium at a certain point with a constant concentration of complexes.
The substrate concentration is uniformly distributed within the cell and is not affected by enzyme enrichment.
The enzyme concentration is sufficiently large.

Parameters

Parameter	Meaning
$c$	concentration at a certain point
$t$	time
$M$	mobility
$F$	total free energy
$\sqrt{\gamma}$	the length of the transition regions between the domains
$[S]$	the substrate concentration
$[P]$	the product concentration
$V_{max}$	the maximum reaction rate
$[E]_t$	the total enzyme concentration

Model Establishment

The Cahn-Hilliard Equation

The Cahn-Hilliard Equation is an important class of fourth-order nonlinear diffusion equations for which analytical solutions are difficult to obtain and approximate solutions can only be found by numerical methods.

The Cahn-Hilliard Equation is as follows:

{\frac{\part c_i}{\part t}=\nabla M_{ij}\nabla \frac{\delta F}{\delta c_j(r,t)}} \tag{1.1}

where the mobility $M$ is taken as a constant 1.0 and the total free energy $F$ is expressed as follows:

{F=\int_V [f(c)+\frac {1}{2}\kappa (\nabla c)^2]}dv \tag{1.2}

The chemical free energy uses the conventional double potential well function:

{f(c)=Ac^2(1-c^2)} \tag{1.3}

Then, discrete time using explicit Eulerian time integration method.

{\frac{c^{n+1}_{ij}-c^n_{ij}}{\Delta t}= \nabla ^2M(\frac{\delta F_{ij}}{\delta c})^n} \tag{1.4}

{(\frac{\delta F_{ij}}{\delta c})^n=\mu (c^n_{ij})-\kappa \nabla^2 c^n_{ij}} \tag{1.5}

{\mu (c^n_{ij})=A(2c^n_{ij}(1-c^n_{ij})^2+2(c^n_{ij})^2(1-c^n_{ij}))} \tag{1.6}

{(\nabla ^2u)_{i,j,k}=\frac{u_{i+1,j,k}+u_{i-1,j,k}+u_{i,j+1,k}+u_{i,j-1,k}+u_{i,j,k+1}+u_{i,k,k-1}-6u_{i,j,k}}{h^2}} \tag{1.7}

The Michaelis-Menten Equation

{v=\frac{d[P]}{dt}=V_{max}\frac{[S]}{k_m+[S]}}\tag{2.1}

where $k_m$ is the Michaelis-Menten constant, while $[E]$ , $[S]$ , and $[P]$ are the enzyme, substrate, and product, respectively. And $V_{max}$ is the maximum reaction rate:

{V_{max}=[E]_t \times k_{cat}}\tag{2.2}

When the substrate concentration is lower than $k_m$ , increasing the substrate concentration can significantly improve the reaction speed. When the substrate concentration is more than ten times the $k_m$ value, the increase in substrate concentration will have little effect on the reaction speed, and then the enzyme concentration can be increased effectively.

We queried the relevant parameters of aldehyde dehydrogenase in Bacillus cereus from the Brenda enzyme database, including $k_m$ , $K_{cat}$ , etc. Although the data is not from E. coli, it is the closest to our experimental design. Also, we acquired the concentration of the enzyme in E. coli and the substrate concentration from previous literature.

The functional parameters of the enzyme are listed below:

Parameters of enzyme	Value	Source
$\frac{K_{cat}}{K_m}$	0.015	(Sh et al., 2016)
$k_m$	0.13	(Sh et al., 2016)
$K_{cat}$	0.00195	(Sh et al., 2016)

Table 1. Functional parameters of enzyme Sh et al. are able to collect these functional parameters of the enzyme in 2016. The enzymatic reaction of Bacillus cereus ALDH for the conversion of all-trans-retinal to all-trans-retinoic acid or all-trans-retinol was performed at 37°C in 50 mM PIPES buffer (pH 7.0).

Also, we acquired the concentration of the enzyme in E. coli and the substrate concentration from previous literature. The volume of E. coli is counted as 0.7 $um^3$ , and the concentrations of enzyme and substrate are as follows:

Name	Concentration	Source
aldehyde reductase in E. coli	450 molecules/cell (0.01 mM)	(Li et al., 2014)
substrate (molecular weight: 286.458)	8.7 mg/L (0.107 mM)	(Jang et al., 2011)

Table 2. Possible concentration of enzyme and substrate

Results

Stimulation of enzyme enrichment

Figure 1. The kinematic simulation of phase separation (animation). Each black dot in the figure represents an enzyme particle, while the rectangle stands for the cytoplasm in the bacteria. The code to generate the animation was inspired by https://github.com/nsbalbi/Spinodal-Decomposition

Figure 2. The kinematic simulation of phase separation. a, the figure is captured at the beginning of phase separation (t=0). b,the figure is captured at the end. Each black dot in the figure represents an enzyme particle, while the rectangle stands for the cytoplasm in the bacteria.

It is illustrated in the Figure 1 and Figure 2 that the phase separation can be simulated by solving the Cahn-Hilliard equation. The black part of the image represents the enzyme molecules that gradually cluster over time.

Figure 3. Concentration changes in small areas. The x-axis represents the enzyme concentration in the divided small region, while the y-axis represents the percentage of small regions at that concentration. The different colors of the curves refer to different time points.

We divided the whole reaction area into ten by ten small areas and calculated the percentage of black in each small area. As shown in the graph, more small areas acquired high enzyme concentrations as time passed. In the beginning stage (blue curve), the percentage of black was around 50% in most regions.

Then, we take a percentage of 50 as the average concentration (0.01 mM) and consider 0.011 mM as the maximum solubility concentration of the enzyme. Therefore, we adjust the reasonable concentration of each tiny area according to the threshold value.

Figure 4. The concentration of soluble enzymes in the cell. The x-axis represents the enzyme concentration in the divided small region, while the y-axis represents the percentage of small regions at that concentration. The blue, orange and green curve stands for reaction with no phase separation, phase separation and phase separation with maximum solubility respectively.

After the phase change coalescence occurs, the total amount of enzyme remains unchanged. Since there is an upper limit of soluble enzymes, the concentration of soluble enzymes in the cytoplasm decreases after the phase change occurs. Meanwhile, enzyme concentration in phase change particles is not limited by solubility, thus increasing gradually. In the bigger picture, the total amount of enzyme increases.

The enzyme on the phase change particle can still catalyze the reaction. Since these enzymes of high concentration are wrapped in the particle, they will not stress the cell, ensuring continuous production of the enzymes.

Kinetics of enzyme-catalyzed reactions

Figure 5. Response curve of the Michaelis-Menten Equation. The x-axis represents the substrate concentration, while the y-axis represents the reaction rate. The blue, orange and green curve stands for reaction with phase separation at high enzyme concentration, phase separation at low concentration and with no phase separation.

After calculating the equation using actual data, it is clear from the diagram (Figure 5) that the enzyme is saturated with the substrate under the average concentration. The reaction rate curve with ten times the average concentration (blue curve) far exceeds others, demonstrating that the high enzyme concentration significantly boosts the reaction. The final result implies that enzyme enrichment is essential within the context of our design. The phase separation of enzymes will significantly boost our production rate.

Conclusion

We proved our hypothesis that enzyme enrichment is vital in enhancing reaction rate.

It is important to note that a relatively high enzyme concentration is required to achieve this effect, which involves overexpressing enzyme-related genes. Later in our practical wet experiments, we followed the modeling conclusions and overexpressed the ybbO gene.

Our model gives us deep insight into how phase separation works in the last step of reaction. Thus, it motivates us to use tears in our design to enrich aldehyde reductase by constructing a plasmid inserted with gene ybbO tagged with tandem-dimeric MS2 coat protein. Although it took one month for us to receive the plasmid from Paris, we immediately put it into experiments after receiving the plasmid.

Code Accessibility

Please visit https://github.com/Fudan-iGEM/2022-model

References

Hong, S. H., Ngo, H. P., Nam, H. K., Kim, K. R., Kang, L. W., & Oh, D. K. (2016). Alternative Biotransformation of Retinal to Retinoic Acid or Retinol by an Aldehyde Dehydrogenase from Bacillus cereus. Applied and environmental microbiology, 82(13), 3940–3946. https://doi.org/10.1128/AEM.00848-16

Jang, H. J., Yoon, S. H., Ryu, H. K., Kim, J. H., Wang, C. L., Kim, J. Y., Oh, D. K., & Kim, S. W. (2011). Retinoid production using metabolically engineered Escherichia coli with a two-phase culture system. Microbial cell factories, 10, 59. https://doi.org/10.1186/1475-2859-10-59

Li, G. W., Burkhardt, D., Gross, C., & Weissman, J. S. (2014). Quantifying absolute protein synthesis rates reveals principles underlying allocation of cellular resources. Cell, 157(3), 624–635. https://doi.org/10.1016/j.cell.2014.02.033

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