Mathematical Modeling and Inverse Analysis for Microbial Depolymerization Processes of Xenobiotic Polymers

Microbial depolymerization processes of xenobiotic polymers are discussed. A mathematical model is formulated and inverse problems for a time factor and a molecular factor of a degradation rate are described. Experimental outcomes are introduced in inverse analyses. Once the time factor and the molecular factor are obtained, the microbial depolymerization process is simulated. Numerical techniques are illustrated and numerical results are presented.


Introduction
Polyethylene (PE) and polyethylene glycol (PEG) are exogenously depolymerizable. In exogenous type depolymerization processes, molecules liberate small units from their terminals. Microbial depolymerization processes of PEG have been documented in detail. Studies of PEG biodegradation include utilization of PEG of average molecular weight 20,000 by Psedonomas aeruginosa [1], degradation of PEG 20,000 by anaerobic bacteria isolated from sludge of a municipal anaerobic digester [2], and confirmation of efficient biodegradation of PEG by Pseudomonas stutzeri [3]. Microbial depolymerization processes of PE and PEG have also been studied mathematically. A mathematical model was proposed in numerical studies of biodegradation of PE [4]. The mathematical techniques devised for PE biodegradation processes were applied to biodegradation processes of PEG [5].
Molecules are broken down randomly in an endogenous type depolymerization process. Polyvinyl alcohol (PVA) and polylactic acid (PLA) are depolymerizable in endogenous type depolymerization processes. A mathematical model was proposed for enzymatic degradation process of PVA [6]. The mathematical techniques devised for PVA were applied to an enzymatic hydrolysis of polylactic acid (PLA) [7]. Techniques originally devised for endogenous type microbial depolymerization processes were applied to exogenous type depolymerization processes of PE and PEG [8].
In this study, mathematical techniques devised in previous studies were reapplied to a depolymerization process of PEG 4000. Weight distribution of PEG 4000 before and after cultivation of Sphingopyxis macrogoltabidus strain 103 were applied in inverse analyses for a molecular factor and a time factor of a degradation rate.

Description of Mathematical Formulation for Exogenous Type Depolymerization Process
In particular, the total weight v(t) of the entire residual polymer at time t is expressed in terms of the integral of w(t,M) over the interval [0,∞) Integral (1) is a reasonable approximation of the integral (2) with suitable values of A and B. In this study, integral (2) over the infinite interval (2) was replaced by an integral over a finite interval (1) with A = 10 3.5 and B = 10 3.75 . Similarly, an integral with the lower limit 0 was replaced by an integral with the lower limit A = 10 3.5 , and an integral with the upper limit ∞ was replaced by an integral with the upper limit B = 10 3.75 .
Suppose that ( ) The parameter L is the molecular weight of a monomer unit, e.g. PE: L = 28 (CH 2 System of Equations (3), (4) for w(t,M) and ( ) t σ is associated with the initial conditions where ( ) 0 f M and 0 σ are the initial weight distribution and the initial microbial population, respectively.

Inverse Problems for Molecular Factor and Time Factor of Degradation Rate and Their Solutions
Once the molecular factor ( ) In view of the expression ( ) equations (3) and (4) respectively. The expression leads to the equation (11) Journal of Materials Science and Chemical Engineering Expressions (5) and (10) and F 2 (M) be the weight distribution for Equation (9), the initial condition (12), and the terminal condition (13) form an inverse problem for ( ) M λ , for which the solution of the initial value problem (9), (12) also satisfies the terminal condition (13). Numerical techniques for the inverse problem (9), (12), (13) were devised in previous studies. Those techniques were applied to weight distributions of PEG 4000 before and after cultivation of Sphingopyxis macrogoltabidus strain 103 for one day, three days, five days, and seven days [15]. Figure 1 shows the molecular factor ( ) M λ of degradation rate based on weight distributions before after cultivation of the microorganism for three days.
Once the inverse problem for ( )

Application of Numerical Techniques to Inverse Problem for Time Factor of Degradation Rate
According to a previous study, function ( ) V τ is well approximated by an exponential function . In this study, function ( ) V τ was replaced with the exponential function (14)

Discussion
Mathematical techniques for depolymerization processes were demonstrated. In particular, exogenous type depolymerization process of PEG 4000 was analyzed.
The weight distributions of PEG 4000 before and after cultivation of Sphingopyxis macrogoltabidus strain 103 for one day, three days, five days, and seven days are introduced into the analysis. Figure 1 shows the molecular factor based on the weight distributions before and after cultivation for three days. The inverse problem for the time factor of the degradation rate was solved (Figure 3).
Once the molecular factor and values of the parameters that determine the time factor of the degradation rate were obtained, the initial value problem (3), (4), (6), (7) was solved numerically, and the exogenous type depolymerization process of PEG 4000 was simulated.