Team:Tianjin/Model
Heterogenous Degradation By PETase
Model Overview
In our experiment we use engineered bacteria as machines to secrete PETase to degrade PET.At first the bacteria secrete PETase ,and then enzymes diffuse into liquid phase body from the cell surface ,from liquid to the surface of PET successively.PETase adsorbs on PET during which process the substrate binding sites of PETase contact with the surface.Finally PETase finds catalytic sites on plastics and combine them with its active center.Ester bonds are broken and chains in PET are ruptured,resulting in the degradation of PET.
Assumptions
1. There exists feedback inhibition regulation in the growing process of a single bacteria. Unlimited growth of population growth can’t be supported due to the limitation of space and resources in a certain environment. When the population of individual bacterium has too much increased, the environment degrades and the average resource share declines, resulting in reduction in birth rate while mortality rate is increasing. Consequently it is reasonable to assume that there exists feedback inhibition regulation due to the influence of environmental factors.
2. A cell can be roughly considered as a sphere.
3. Enzymes are in the dynamic equilibrium when transferring in the liquid phase.
4. The resistance of enzymes’ spread from liquid body to the surface of PET is much larger than that of from the cells’ surroundings to liquid body.
5. It takes several steps for the enzymes to complete the degradation of PET. Enzymes have the substrate binding sites and active centers. We assume that the enzymes are firstly combined with the polymer substrate through their own substrate binding sites, and then the active centers catalyzed the degradation of the polymers.
6. The degradation of PET takes place on the surface of PET.
7. The mass transfer of enzymes in the liquid phase will cause some of them to stay in the liquid body and the delay of enzyme concentration changes on the PET surface.
Summary
Based on the assumptions above and the description of PET degradation process in a heterogeneous system, we first establish the equation of measuring how much percent the PET degrades and how much the degrading rate is. Then we use the simple but mature Logistic Equation to describe the process of cell growth, and Leudeking-Piret Equation which is a correlation between cell growth rate and product producing rate, to describe the kinetic process of PETase production. Then we use the general method of describing mass transfer process to establish mass transfer rate equation when enzymes diffuse from the cell surface to liquid phase body then from liquid phase body to the PET surface, and the distribution equation of enzyme concentration, respectively. From those equations we can get the total mass transfer rate equation of the enzymes in a heterogeneous system. We analyze and then make a conclusion that the mass transfer diffusion mainly leads to the delay of enzyme concentration change on the PET surface .Finally, taking that MHET, the product of PET degradation, will show competitive inhibition effect into consideration, based on the adsorption equilibrium of PETase on the PET surface, we use the steady state approximation theory and deduce the total kinetic equation of PETase degrading PET. Solve this differential equation we obtain the degradation rate curve of PET under heterogeneous system.
The Formulas for Calculating Biodegradation Percentage and The Degradation Rate
Since our project aims to degrade PET, we need to propose a stable index to measure the degree of degradation of the plastics and another one to measure the degradation rate. Here we choose to describe degradation rate and percentage.
The role of PETase plays in the degradation is to catalyze the cleavage of ester bonds between TPA and EG. This cleavage will directly lead to the decomposition of whole polymer chain and finally the instability of plastics. Thus we choose the ester bond number as an index to evaluate this degradation process.
The calculation formula for calculating the number of moles of ester bonds (namely the number of moles when polymers are completely degraded):
$${n_{EB}} = {n_{EB/{M_{rep}}}} \cdot m/{M_{rep}} (2-1)$$
| nEB | The mole number of the broken ester bonds in theory(μmol) |
| m | Loading amount of the polymer (μg) |
| Mrep | The molar mass of the repeat units in the polymer(μg/μmol) |
| nEB/Mrep | The number of ester bonds in a repeat unit |
The percentage of degradation of ester bonds can be obtained through that ratio.
2-2:
Similarly, the degrading rate can also be described by a series of formulas shown below:
2-3:
The Equation for The Growth Rate of Enzyme Proteins
The kinetics of cell growth
According to the characterisitics of microbial cell growth, Monod Equation is the most commonly used one. Although this equation is considered to be simple and effective when used to describe the growth of bacteria, it is only suitable under the condition that there is no other restrictive substances in the environment. In our system, PETase is secreted by the bacteria to the environment and, more limitation by feedback regulation will arise if it is in a mixed bacteria system. Thus we utilize the Logistic Equation to describe the rhythm of the growth rate.
Logisic model is a typical S-shaped curve that can reflect the inhibition effect caused by the increase of bacteria concentration in the fermentation process. In the early stage, the bacteria concentration is low , namel cx is much lower than cxm, therefore the item of cx/cxm can be neglected. The colony is in the stationary phase after the logarithmic phase and at that time cx is close to cxm. The colony ceases to grow. The whole process can be described in the following equation:
3-1
| Rx | The growth rate of cell growth |
| Cx | The concentration of cells |
| Cxm | The maximum concentration of cells |
| μ | Specific growth rate , namely the growth rate of a unit thalli concentration |
| μm | The maximum specific growth rate |
The definition of μ is :
2-4
When cells are in exponential phase, μ is generally a constant,so
2-5
Under the condition when t=0, cx=cx0, we integrate formula 2.5:
Solve this equation and the growth curve can be obtained:
1. Reporting System
The basis of our reporting system is the part BBa_K339007, Designed by Emily Hicks from Group iGEM10_Calgary. This part can sense the CpxR protein, which will form spontaneously in E.coli when inclusion body and misfolding protein present in the periplasm of E.coli, and then start expressing RFP so that we can detect red fluorescence. As we all know, the inclusion body will inevitably form when we overexpress heterologous protein like PETase in E.coli. Therefore, the emission of red fluorescence can report the overexpression of PETase. What is more, this device can be modified to report overexpression of any heterologous protein only if the PETase gene is replaced by another heterologous gene. After the red fluorescence is detected, we could start the purification of protein.
2. Cell Lysis Based Regulation System
The regulation system consists of two section. The first section is based on the already mentioned reporting system. We change the RFP gene to the novel ddpX (D-alanyl-D-alanine dipeptidase) gene from E.coli genome. The ddpX gene can hydrolyze the D-Ala-D-Ala structure in peptidoglycan molecule and cause damage to the cell wall of E.coli. Under normal condition, this gene only express when the cell is in starvation mode in order to use hydrolysate alanine as carbon source. However, if we overexpress this gene, the cell wall will be dissolved and finally cell lysis will happen. Therefore, in this system, when the PETase is overexpressed, the spontaneously forming inclusion body will induce the expression of ddpX and cause cell lysis. It will provide us with a novel and convenient and way of protein purification when you use E.coli as chassis.
