Computational study of bacterial depolymerization process of xenobiotic polymer

This study shows an efficient applicability of computational techniques to analyses of microbial depolymerisation process. Microorganisms were cultivated in a culture media in which a polymer was a sole carbon source, and weight distributions before and after cultivation were introduced into inverse analysis for a molecular factor and a time factor of a degradation rate. An inverse problem for two parameter values associated with the time factor was solved numerically. Once the molecular factor and the time factor were given, microbial depolymerization process was simulated.


Introduction
In a biodegradation process of Polyethylene (PE), molecules liberate monomer units from their terminals. Such a depolymerization process is called an exogenous type depolymerization process. Polyethylen glycol (PEG) is another exogenously biodegradable polymer. Utilization of PEG of average molecular weight 20000 by Psedonomas aeruginosa was documented [1]. Degradation of PEG 20000 by anaerobic bacteria isolated from sludge of a municipal anaerobic digester was reported [2]. Efficient biodegradation of PEG by Pseudomonas stutzeri was observed [3]. A mathematical model was formulated and numerical techniques were applied to biodegradation of PE [4]. Those techniques were reapplied to a biodegradation process of PEG [5].
Unlike exogenous type processes, random scission is an essential mechanism of endogenous type processes. Polyvinyl alcohol (PVA) and polylactic acid (PLA) are distinctive endogenously depolymerizable polymers. A mathematical model was formulated and numerical techniques were applied to an enzymatic degradation process of PVA [6]. Techniques applied to the enzymatic degradation of PVA was reapplied to an enzymatic hydrolysis of polylactic acid (PLA) [7]. Techniques originally applied to endogenous type processes were replied to exogenous type processes [8].
This study revisited a biodegradation process of PEG. Experimental outcomes before and after cultivation of bacteria in culture media were incorporated into a computational analysis. Numerical solutions of inverse problems for a molecular factor and a time factor of a degradation rate were obtained. Once the inverse problems were solved, biodegradation process was simulated.

Computational model for exogenous type microbial depolymerization process
[mg] is the weight distribution of a polymer with respect to the molecular weight M at time t , and that   t v [mg] is the total weight of polymer molecules with molecular weight between A and B at time t . The total weight   is expressible in terms of the integral of weight distribution with respect to the molecular weight M from A to B , as equation (1) shows, In particular, total weight   t v of the entire residual polymer at time t is the integral of weight distribution with respect to the molecular weight M from 0 to  , as equation (2) shows, Integral (1)  . Similarly, integral with the lower limit 0 was approximated with an integral with the lower limit A , and an integral with the upper limit  was approximated with an integral with upper limit was proposed in previous studies [8 -12].
Here, parameter L is the molecular weight of a monomer unit, e.g. PE: 28 2 2 ). Note that function   M  is the molecular factor of degradation rate, and that the microbial population   t  is the time factor of degradation rate. Note also that equations (2) and (3) lead to equation (5) System of equations (3), (4) forms an initial value problem with initial conditions, where   M f 0 and 0  are an initial weight distribution and an initial microbial population, respectively.

Numerical solutions of inverse problems for molecular factor and time factor of degradation rate
The initial value problem equations (3), (4), (6), (7)  Consider the change of variables from t to  defined by equation and   t v , respectively, where the relation (8) between t and  holds. Note that equation holds, and that equations (10) and (11) (11) hold in view of equations (3) and (4) holds. Equation (10), the initial condition (12), and the final condition (13) form an inverse problem for   M  , for which the solution of the initial value problem of equations (10), (12) also satisfies the final condition as equation (13) shows.
Numerical techniques developed in previous studies were applied to the inverse problem. In particular, weight distributions of PEG before and after cultivation of microbial consortium E-1 for two days, four days, and seven days were introduced into the inverse analysis. Denote the weight distribution before cultivation by Weight distributions after cultivation of the microbial consortium E-1 for two days where . 0 0   Once the initial value problem (10), (14) was solved, equations were solved numerically. A previous study shows that function    V is well approximated by an exponential function     , and iterations were repeated until errors between successive approximations to reduced to a value less than or equal to 12 10  . It took thirty seven iterations for to reduce to a value less than or equal to 10 10  . Final vales of k and h are approximately equal to 0.00623 and 0.489, respectively. Figure 1    is the total amount of monomer units consumed by the microorganisms per unit time. Equation (4) asserts that this amount is converted to the increase of the microbial population. Our numerical results show that the conversion rate per mg and unit population was approximately 0.5.
In a previous study [12], equation (4) was applied to a set of residual PEG values. However in this study, it was applied to the set of weight distributions. The Newton's method in conjunction with the bisection method was applied to the system of equations (16) for two parameters of microbial population. The residual PEG was approximated by an , where the value of  was obtained from a numerical result of initial value problem (10), (14).
Results of this study are summarized in Figures 1 and 2. Figure 1 shows the curve is the population of viable cells. Another factor concerning the optical density was involved in the PEG degradation process. The symbiotic mixed culture E-1 consists of S. Terrae and Rhizomium sp. A study [13] shows that S. terrae is the main degrader in a PEG biodegradation process. It shows that S. terrae cells increase while sufficient carbon sources are available, and that they rapidly decrease after consumption of PEG. The OD 630 conversion of the microbial population ( Figure 1) corresponds to viable S. terrae cells. The numerical result shows the viable S. terrae cells started decreasing after four days. This study demonstrated efficient applicabilities of the Newton's method and the bisection method to inverse problems that arise in studies of microbial depolymerization processes. In particular, equation for two parameters were analyzed numerically with those numerical methods. Applicabilities of those numerical techniques to problems involving three parameters or more will be further investigated in our future study.