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):

$${n_{EB}} = {n_{EB/{M_{rep}}}} \cdot m/{M_{rep}} (2-1)$$

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

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:

The formation dynamics of expression product (PETase)

The extreme diversity of metabolites produced by microbial fermentation and the complexity of biosynthesis routes in the cells cause the biosynthetic pathways and metabolic regulation mechanism of those metabolites to show diverse characterisitics. We use the relations between cell growth rate and the production rate to describe the rate of protein production. The universal model can be expressed by Leudeking-Piret Equation:

equtaion

When α≠0 and β=0, the model is growth coupling. When α=0 and β≠0, the model is non-growth coupling. When α≠0 and β≠0, the model is partial-growth coupling.

Plug the cell growth rate into the equation to simplify:

2-6

Integrate 2-6

2-7

Solve this equation and the production curve of PETase can be obtained:

Compared with the growth curve of bacteria, we can see intuitively that at the beginning, the correlation between the enzymes production and the concentration of bacteria is high. And with the time passing by the correlation becomes lower and finally disappears.

The Transfer Process of Enzymes in The Liquid Phase

The secretion of enzymes from cells will inevitably lead to the increase of enzyme concentration nearby. And when enzyme concentrations surrounding the cells are larger than that in the liquid phase body, a driving force will emerge, causing the diffusion of enzymes into liquid phase body. On the other hand, the enzymes in the liquid phase will reach the surface of PET along with the free diffusion of molecules. So there are two transfer processes before the enzymes adsorp on the PET surface.

The mass transfer process of enzymes in the liquid phase body

Cells secrete enzymes to the outside. Here a cell can be regarded as a sphere and there’s a spherical shell composed by the enzymes in the outer surface of the sphere. Inside this shell an unsteady mass transfer process will occur along the radius direction.