M. Watanbe^{1*} and F. Kawai^{2}  
^{1}Graduate School of Environmental Science, Okayama University, Okayama, Japan  
^{2}Center for Nanomaterials and Devices, Kyoto Institute of Technology, Japan  
Corresponding Author :  M. Watanbe Graduate School of Environmental Science Okayama University, Okayama, Japan Email: watanabe@ems.okayamau.ac.jp 
Received March 07, 2012; Accepted April 17, 2012; Published April 19, 2012  
Citation: Watanbe M, Kawai F (2012) Modeling Biodegradation of Polyethylene with Memoryless Behavior in Metabolic Consumption. J Bioremed Biodegrad 3:146. doi: 10.4172/21556199.1000146  
Copyright: © 2012 Watanbe M, et al. This is an opena ccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.  
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Introduction 
Microbial depolymerization processes are classified into exogenous type processes and endogenous type processes. In an exogenous type depolymerization process, molecules reduce in size by liberating monomer units from their terminals. Polymers subject to exogenous depolymerization processes include Polyethylene (PE) and Polyethylene glycol (PEG). PE is a longchain nalkane. Oxidation of an nalkane is started by hydroxylation to produce the corresponding primary (or secondary) alcohol, which is oxidized further to an aldehyde (or ketone) and then to an acid. Carboxylated nalkanes are structurally analogous to fatty acids and subject to âoxidation processes to produce depolymerized fatty acids consisting of two carbon units (acetic acid). Gel permeation chromatography (GPC) analysis of polyethylene wax (PEwax) before and after cultivation of a bacterial consortium KH 12 showed that small molecules are consumed faster than large ones [1]. Figure 1 shows GPC profiles before and after cultivation of fungus Aspergillus sp. AK3 for three and five weeks. 
PE biodegradation involves two essential factors, the gradual size reduction of large molecules due to âoxidation, and the direct consumption or absorption of small molecules by cells. A PE molecule liberate a monomer unit from its terminal in one cycle of âoxidation, and it reduces in size undergoing successive âoxidation processes. A mathematical model based on those two factors was proposed to study PE biodegradation processes [24]. 
PEG is one of the polyethers whose chemical structures are represented by the general structural formula HO(R – O)nH (PEG: R=CH_{2}CH_{2}), and is metabolized by liberating C_{2} compounds [5,6]. Techniques developed for PE biodegradation were extended to exogenous depolymerization processes of PEG [7]. Inverse problems were solved numerically for degradation rates based on the weight distribution with respect to molecular weight before and after cultivation of a microbial consortium E1. Once a degradation rate was found, the transition of the weight distribution was simulated by solving an initial value problem numerically. Dependence of degradation rates on time was also considered in modeling and simulation of microbial depolymerization processes of PEG [811]. A model originally proposed for endogenous type depolymerization processes was applied to an exogenous depolymerization process of PEG [12]. 
Unlike exogenous type depolymerization processes, molecules are cleaved at arbitrary positions in endogenous type depolymerization processes. Example of polymers subject to endogenous type depolymerization processes include Polyvinyl alcohol (PVA). PVA is depolymerized in a succession of two processes: oxidation of a couple of pendant hydroxyl groups, either by oxidase or dehydrogenase, followed by hydrolysis. Cleavage of a carboncarbon chain results from those processes at a carbonyl group and an adjacent methylene group [13]. Matsumura et al. [14] proposed a different metabolism of PVA based on oxidation of a hydroxyl group and aldolase reaction of a monoketone structure, which leads to cleavage of a carboncarbon chain between a methylene group adjacent to a carbonyl group and an adjacent hydroxymethylene group. In either case, PVA is depolymerized by oxidation and the resultant cleavage of a carboncarbon chain. A mathematical model for endogenous depolymerization process was proposed, and an enzymatic depolymerization process of PVA was studied [15,16]. Mathematical model originally proposed for the enzymatic degradation of PVA was applied to enzymatic hydrolysis of polylactic acid (PLA), and the degradability of PVA and PLA was compared [17]. Dependence of degradation rate on time was considered for depolymerization processes of PLA [18]. 
In this study, PE biodegradation is revisited with a novel mathematical theory. A model originally developed for endogenous depolymerization processes is applied to the exogenous depolymerization processes of PE. Memoryless behavior of degraded molecules leads to exponential distribution with respect to the molecular weight. Derivation of the model is described. Experimental results are introduced into analysis and an inverse problem is solved numerically for a degradation rate. An initial value problem is solved numerically to simulate the transition of the weight distribution. Numerical techniques are illustrated and numerical results are presented. 
Modeling Exogenous Depolymerization Processes of Xenobiotic Polymers 
In depolymerization processes, the weight is shifted from classes of large molecules to classes of small molecules. In order to formulate microbial depolymerization processes, let w(t,M) be the weight distribution with respect to the molecular weight M at time t. Suppose that a molecule with molecular weight M is depolymerized in an exogenous type depolymerization process to become a molecule with molecular weight K. The amount MK is metabolized in the depolymerization process. In general, let P_{0}(t,K,M) be the time rate of increase in W(t,K) due to the depolymerization of molecules with molecular weight M, and P1(t,K,M) be the total amount metabolized in production of molecules with molecular weight K from those with molecular weight M. Let p(t,K,M) = p_{0}(t,K,M) + p_{1}(t,K,M). Then p(t,K,M) is the time rate of decrease in w(t,M) to yield the increase P_{0}(t,K,M) in w(t,K). Let C(A,B) be the class of all molecules with molecular weight between A and B. The weight decrease in C(A,B) per unit time is 
(1) 
while the weight increase per unit time is 
(2) 
The total weight in C(A,B) present at time t is 
and its rate of change 
equals the difference between the quantities (1) and (2), 
(3) 
Substituting the expression 
which holds for a function with continuous partial derivatives, equation (3) becomes 
Since this equation holds for an arbitrary interval [A,B], 
and w = w(t,M) is a solution of 
(4) 
[15,16,19] 
In the transition of the weight from W(t,M) to w(t,K), each of molecules with molecular weight M loses the amount MK to become molecules with molecular weight K, while the number of degraded molecules is preserved. The number of degraded molecules is proportional to 
and the increase in w(t,K) is 
This amount equals p_{0}(t,K,M), 
Now the equation (4) becomes 
(5) 
Suppose that γ(t,M) is the loss of amount from w(t,M) per unit time and per unit weight. The amount of loss from w(t,M) per unit time is γ(t,M)w(t,M), and it is expressed in terms of the integral of p(t,K,M), 
(6) 
For Kε[0,M], let q(K,M) denote the decrease in w(t,M) per unit weight in the transition from w(t,M) to w(t,K). Then 
(7) 
The equations (5), (6), and (7) lead to 
(8) 
Given an initial weight distribution f(M), equation (8) and the initial condition 
w(0,M) = f(M) (9) 
form an initial value problem, provided the degradation rate γ(t,M) is given. Given an additional weight distribution g(M) at, t = T(T>0) equation (8), the condition (9), and the condition 
w(0,M) = g(M) (10) 
form an inverse problem to determine the degradation rate γ(t,M), for which the solution of the initial value problem (8) and (9) also satisfies the condition (10). Inputs of the inverse problem are an initial weight distribution f(M) and a final weight distribution g(M), and an outcome is the degradation rate γ(t,M). 
Removal of Time Factor from the Degradation Rate 
In microbial depolymerization processes, time factors of degradability include temperature, dissolved oxygen, and microbial population. Those factors act on polymer molecules evenly regardless of size, and the degradation rate γ(t,M) is a product of a function of t, σ(t), and a function of M, λ(M). Equation (8) becomes 
(11) 
Let 
Then 
and the equation (11) becomes 
(12) 
When λ(M) is given, equation (12) and the initial condition 
W (0,M ) = f (M ) (13) 
form an initial value problem. Equation (12), the initial condition (13) and the condition 
W (T,M ) = g (M ), (14) 
form an inverse problem to find the degradation rate λ(M) for which the solution of the initial value problem (12) and (13) also satisfies the condition (14). The initial value problem (8) and (9) corresponds to the initial value problem (12) and (13), and the inverse problem (8), (9) and (10) corresponds to the inverse problem (12), (13) and (14), provided 
Memoryless Behavior in Microbial Consumption and Exponential Distribution in Liberation of Monomers 
The truncation of monomers in an exogenous depolymerization process is memoryless, which means that the amount truncated from a molecule does not depend on the previous truncation. Let J be the amount truncated from a molecule in an exogenous depolymerization process. The memoryless behavior leads to the exponential distribution [17]. 
(15) 
It is the probablility of the event in which the metabolized amount equals J. In an exogenous depolymerization process, amount MK is truncated from a molecule with molecular weight M to yield a molecule with molecular weight K. Substitution of J = MK leads to the expression of q(K,M) given by 
(16) 
When q(K,M) is given by the expression (16), equation (12) becomes 
(17) 
where 
(18) 
Differentiation of the both sides of the equation (17) results in 
(19) 
and substitution of the expression for the integral obtained from equation (17) leads to 
(20) 
In section 5 the inverse problem is solved numerically to determine the degradation rate λ(M) for which the solution of the initial value problem (20) and (13) also satisfies the condition (14) for c(M) and d(K) given by the expressions (18) and (21). 
The parameter ñ is often referred to as the intensity of the exponential distribution. In order to determine the value of the parameter, liberation of monomer units in the exogenous depolymerization process is considered. In an exogenous type depolymerization process, molecules reduce in size through successive liberation of monomer units. Let L be the molecular weight of a monomer unit truncated from a terminal of a molecule in one cycle of depolymerization process. In case of the âoxidation, L = 28 (CH_{2}CH_{2}). Let TJ be the time it takes for a molecule to reduce in size by the molecular weight J. The liberation of monomer units occurs successively, and T2L = 2TL. In unit of time, the probability for loss of two monomer units is one half of the probability for loss of one monomer unit, and 
q(M − L,M ) = 2q(M − 2L,M ) 
or 
ρe^{−ρ L} = 2ρ^{e−2ρ L} , 
which leads to 
(21) 
Computational Results for an Exogenous Depolymerization Process of Polyethylene 
In previous studies, numerical techniques were developed for the inverse problem to determine the degradation rate λ(M) for which the solution of the initial value problem (20), (13) also satisfies the condition (14) [12]. Those techniques are described in the appendix. The weight distributions before and after cultivation of Aspergillus sp. AK3 for three weeks (Figure 1) were set as the initial and final conditions, and the inverse problem was solved numerically. Figure 2 shows the numerical result. 
\ 
Once the degradation rate is found, transition of the weight distribution is simulated by solving the initial value problem (12), (13). Numerical techniques to solve the initial value problem was developed in previous studies [12,1517]. Those techniques are also illustrated in the appendix. Figure 3 and Figure 4 show numerical results based on the degradation rate. Figure 3 shows the experimental results and numerical results for the transition of weight distribution for three weeks. Figure 4 shows the experimental result and a numerical result for the weight distribution of PEwax after cultivation for five weeks. 
Discussion 
No information on the weight distribution after five weeks was used to obtain the degradation rate shown in Figure 2. Nevertheless an acceptable agreement between the experimental result and the numerical result is demonstrated in Figure 4. The numerical result was obtained using the model with time independent degradation rate. The numerical result shown in Figure 4 indicate that the time factor of the degradation rate was almost constant over the degradation period. The only time factor in this case was the microbial population. The numerical result indicate that the microbial population reached the constant value at an early stage of cultivation, and maintained the constant value for the five weeks of cultivation. 
The mathematical model originally developed for endogenous depolymerization processes is adapted to exogenous depolymerization processes of PE. Some other authors proposed similar models including governing equations for particlesize distribution in simultaneous binary fragmentation and aggregation reactions or population balance equation [2022], equations for mass balance for polymers subject to random chain scission, repolymerization reactions, and chainend scission and related topics [23,24], and an integropartial differential equation of a firstorder bondbreaking process for random scission and a firstorder recombination process [25]. 
In this study, the model based on the exponential distribution of degraded molecules is derived. Numerical results show that the model is widely applicable not only to endogenous depolymerization processes, but also to exogenous depolymerization processes, and that depolymerization processes of mixed type can also be analyzed. 
Acknowledgements 
This work was supported by JSPS KAKENHI 20540118. 
References

Figure 1  Figure 2  Figure 3  Figure 4 
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