# POLYMER MODEL

Oligomerization modeling

Our project is to create new materials via polymerization. Through Flory- Stockmayer theory, we can calculate the gelation and condensation when polymerization. Then, we can adjust our polymerization reaction to meet the properties what we want.

Flory-Stockmayer assumptions

1. All functional groups on a branch unit are equally reactive
2. All reactions occur between two molecules
3. There are no intramolecular reactions

Flory-Stockmayer theory of gel point

The Flory-Stockmayer theory is an ideal prediction for polymers. This theory is mainly to integrate gelation and contraction reactions. Through this theory, we can understand the reaction of polymerization and adjust the conditions of polymerization. Our monomer is Coniferyl alcohol. According to the reaction of the enzyme, our monomer will exhibit three resonance states. We want to make the material more biodegradable and chain-like, so we use oligomer to polymerization. Because coniferyl alcohol has three resonance states, our functional group is three (f=3). Polymerization of LIGGREEN

In the Flory-Stockmayer theory, our reaction belongs to ABg polymerization. LIGGREEN is ABg type of polycondensation, and there is an unreacted A and i(g -1)+1 B group in vivo with degree of polymerization i, and the species can be derived under the assumption of the inner cyclization reaction and the isocratic assumption.
We assume that our reaction only occur in oligomer and monomer. Because we would like to make the material formed in chain structure, we prefer to make oligomerization occur. Through ABg reaction, we are able to achieve oligomerization by adjusting the different conditions. According to the above, our modeling is mainly based on three resonance states and ABg. The initial conditions for the above equation are: Because there is only one monomer group on each species. For the following two types of equations. We can get,  After getting the above formula, we bring in our parameters. With this distribution function, the various molecular parameters of LIGGEEN may be quantitatively known. Figure 1 shows the relationship between the polydispersity index of LIGGEEN under several g parameters and the conversion of the monomer group. g = 1 is the type AB monomer. Linear polycondensation, after the end of the reaction (x = 1), the polydispersity index of the LIGGEEN is only 2. If g > 1, the formation of hyperbranched polymer, the polydispersity index of LIGGEEN becomes very large near the completion of the reaction. After this modeling, we can get conclusion that oligomerization would occur.

References:

1. Stockmayer, W. H. (1943). Theory of molecular size distribution and gel formation in branched‐chain polymers. The Journal of chemical physics, 11(2), 45-55.
2. Flory, P. J. (1941). Molecular size distribution in three dimensional polymers. I. Gelation1. Journal of the American Chemical Society, 63(11), 3083-3090.
3. Carothers, W. H. (1936). Polymers and polyfunctionality. Transactions of the Faraday Society, 32, 39-49.

We assumed our product consists of three bonds of β-5, β-O-4, β-β, and we found that some kinds of enzyme can break β-5& β-O-4 bonds from reference. We try to establish a simple degradation rate model under ideal conditions, predicting the state of degradation and the treatment reference for the product.
In this part, we will use the kinetics of enzymes to build model and first start form Michaelis-Menten kinetics.
From Michaelis-Menten kinetics

Assumed:
(1) Lignin degradation is a one-step reaction
(2) Both enzymes and LIGGREEN are first-order reactions
(3) Value of enzyme activity is a constant.
(4) Reaction solution is 1 unit volume.

Some simulation constants.
1. Initial number of bacteria
2. Bacterial reproductive cycle
3. Enzyme production cycle

In this part, we will use the kinetics of enzymes to build model and first start form Michaelis-Menten kinetics. This model is common and well-known models of enzyme kinetics, and it is takes the form of an equation to describe the rate of enzymatic reactions by reaction rate V and concentration of substrate S. We assume that the balance between E and ES are quickly established to reach steady-state and the concentration of enzyme-substrate [ES] is fixed.

Get the equation and the relation of total enzyme concentration [E_0], free enzyme [E] and enzyme-substrate [ES] is Using both equation can get The reaction rate of enzyme can be expressed as Using equation (4), we can get two relation of reaction rate
(I) at low substrate concentration, (II) at higher substrate concentration, To get more detailed information, we use the equation (4) & (5) to calculate, and Symbolic Math Toolbox in Matlab to solve equation. (S(t):amount of substrate in time t; S(0):initial amount of substrate ) We can know the remaining amount of the substrate from the result, if the value of parameters is known, and we want use this relationship plus our assumptions to estimate biodegradation situation.

Degradation requires the action of enzyme, and assuming that the enzymes are produced by bacteria. The following relationship are used to express the number of bacteria, and the amount of enzyme.  Combine the equation (8) & (9), and the formula become We can use the equation (6), (7) & (10) to estimate degradation rate in initial stage.
(1) at low substrate concentration (2) at higher substrate concentration To get more detailed information, using the equation (4) again and combine with equation (10). (S(t):amount of substrate in time t; S(0):initial amount of substrate)  ω(z) is Wright omega function. This is final result, and we will use this formula to discuss the influence of parameters. \$\$ Figure 1. k_2={10}^{-3}; k_A={10}^{-5};τ = 30;K_m={10}^{-3};S(0)={10}^{-2} ; B_i=25, 20, 15, 10, 5; \$\$

From the results, we found that the more initial bacteria, the faster the degradation, and the increase in the number of benefits will be lower and lower. \$\$ Figure 2. k_2={10}^{-3}; k_A={10}^{-5}; B_i=10;K_m={10}^{-3}; S(0)={10}^{-3}; τ=10, 20, 30, 40, 50 \$\$

From the results, we found that the shorter average reproductive cycle, the faster the degradation. That is, if the material structure is easier reaction by enzyme of bacteria (microorganisms), the time required can be greatly reduced, but if the structure is complicated and difficult to use, it may cause difficulty in degradation, even in the same situation as plastic.

In summary, the simple part of the model is used to explore the degradation situation under ideal conditions. In the future, the data will be continuously collected to improve the model, hoping to make the model more realistic.

References:

1. Chang, Y. C., Choi, D., Takamizawa, K., & Kikuchi, S. (2014). Isolation of Bacillus sp. strains capable of decomposing alkali lignin and their application in combination with lactic acid bacteria for enhancing cellulase performance.
Bioresource technology, 152, 429-436
2. Bugg, T. D., Ahmad, M., Hardiman, E. M., & Singh, R. (2011). The emerging role for bacteria in lignin degradation and bio-product formation. Current
opinion in biotechnology, 22(3), 394-400.
3. Daina, S., Orlandi, M., Bestetti, G., Wiik, C., & Elegir, G. (2002). Degradation of β-5 lignin model dimers by Ceriporiopsis subvermispora.
Enzyme and Microbial Technology, 30(4), 499-505.
4. Buraimoh, O. M., Ilori, M. O., Amund, O. O., Isanbor, C., & Michel Jr, F. C. (2017). The degradation of coniferyl alcohol and the complementary production of chlorogenic acids in the growth culture of Streptomyces albogriseolus KF977548 isolated from decaying wood residues.
Process Biochemistry, 52, 22-29.
5. de Gonzalo, G., Colpa, D. I., Habib, M. H., & Fraaije, M. W. (2016). Bacterial enzymes involved in lignin degradation.
Journal of biotechnology, 236, 110-119.
6. Michaelis Menten Kinetics in bio–physic wiki (http://www.bio-physics.at/wiki/index.php?title=Michaelis_Menten_Kinetics)