Numerical Study of Microbial Depolymerization Process with the Newton-Raphson Method and the Newton's Method
Abstract
Computational techniques are proposed for a numerical solution of an inverse problem that arises in a study of a microbial depolymerization processes. The Newton-Raphson method in conjunction with the Newton’s method is applied to a time factor that involves three parameters. The Newton-Raphson method reduces a system of three equations for three unknowns to a single variable equation, which is solved with the Newton’s method. Those techniques were applied to a microbial depolymerization process of polyethylene glycol. Introduction of experimental results into analysis leads to simulation of a microbial depolymerization problem.
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Introduction
Petroleum-based polymers have been produced since the early part of the twentieth century. A large amount of those xenobiotic compounds have been accumulated on the surface of the earth, and they are now sources for carbon dioxide emission. Although those macromolecules had been nonexistent until they were invented, some of microorganisms are able to utilize the carbon sources. Mechanisms of microbial depolymerization processes must be clarified for appropriate assessment of the carbon cycle. Microbial depolymerization processes are categorized into exogenous type processes and endogenous type processes. Monomer units are liberated one by one from terminals of molecules in an exogenous type depolymerization process. Polymers degraded in exogenous type depolymerization processes include polyethylene (PE) and polyethylene glycol (PEG). Unlike exogenous type depolymerization processes, primary factor of an endogenous type depolymerization process is arbitrarily scission. Polymers degraded in endogenous type depolymerization processes include polyvinyl alcohol (PVA) and polylactic acid (PLA).
A PE biodegradation process involves two primary factors, gradual weight reduction due to the -oxidation and direct absorption by cells. A PE molecule liberates a two carbon unit from its terminal in one cycle of -oxidation and reduces in size undergoing successive -oxidation processes until it becomes small enough to be absorbed directly into cells. A mathematical model based on those factors was proposed for simulation of PE biodegradation processes [1 - 4]. Polyethylene glycol (PEG) is another polymer depolymerized in exogenous type depolymerization processes. The initial step of liberation from a PEG molecule is oxidation of a terminal unit to produce an aldehyde, which is further oxidized to a monocarboxylic acid. The oxidation is followed by the cleavage of the ether bond, and the molecule reduces by one glycol unit [27]. Numerical techniques developed for PE biodegradation were applied for simulation of an exogenous type depolymerization process of PEG [5]. Temporal factors of degradation rates were taken into consideration [6 - 8].
A mathematical model of endogenous type depolymerization processes was proposed for an enzymatic degradation process of PVA [9, 10]. The model was applied to an enzymatic hydrolysis of polylactic acid (PLA), and degradabilities of PVA and PLA were compared [11]. Temporal dependence of the degradation rate was taken into consideration for the depolymerization process of PLA [14]. The model proposed for endogenous type depolymerization processes was reformulated for exogenous type depolymerization processes of PEG [13] and PE [14]. Numerical techniques developed for PE biodegradation were applied to an exogenous depolymerization process of PEG [15]. Temporal dependence of degradability was taken into consideration for depolymerization processes of PEG [16 – 23, 25, 26].
Conclusion
Application of the Newton-Raphson method to the system (16) was demonstrated in a previous study, while the bisection method was applied to the equation (27) [26]. The techniques were tested by application to model problem in which V(τ) was a constant function. The techniques were also applied to an exponential function V(τ) = e−aτ+b where the least square approximation was applied to V(τ) based on weight distributions before and after cultivation of microbial consortium E1 for one day, three days, five days, and nine days, and the values of the parameters σ0 ≈ 0.14345, k ≈ 1.5690 and h ≈ 9.3784 were obtained [26].
In this study, the application of the Newton's method to the equation (27) was illustrated. The techniques were applied to the residual PEG V(τ) based on the weight distributions before and after cultivation of microbial consortium E1 for two days, four days, and seven days. The values of the parameters σ0 ≈ 0.14345, k ≈ 0.54482 and h ≈ 12.75316 (Table 2) were obtained. Values of σ0 between numerical results in the previous study and this study are almost identical. However there are notable differences in values of k and h.
Equations (16) and (27) were also solved for the solution of the equation (29) with the initial value V0, and the values of the parameters σ0 ≈ 0.14399, k ≈ 0.54052 and h ≈ 8.51592 (Table 3) were obtained. Difference in values of σ0 and k between shown in Table 2 and 3 is negligible. However there is a notable difference between the values of h shown in Tables 2 and 3. Nevertheless, transitions of residual PEG shown in Figures 3 and 5 are indistinguishable, and equation (8) may well be replaced with equation (29). In practical applications, there are cases where amounts of residual polymer is available at discrete points in time instead of weight distributions. Techniques developed in this study will be applicable to those cases.