Microbial depolymerization processes are classified into exogenous type and endogenous type. Molecules liberate their parts to reduce in an exogenous type depolymerization process. Polyethylene (PE) and polyethylene glycol (PEG) are polymers subject to exogenous type depolymerization processes. Studies showed utilization of PEG of average molecular weight 20,000 by Psedomonas aeruginosa , degradation of PEG 20,000 by anaerobic bacteria isolated from sludge of a municipal anaerobic digester , and efficient biodegradation of PEG by Pseudomonas stutzeri was documented . A mathematical model was proposed and numerical techniques were developed for PE biodegradation . The mathematical techniques were reapplied to a biodegradation process of PEG .
Random breakdown of molecules is the primary factor of endogenous type depolymerization processes. Polyvinyl alcohol (PVA) and polylactic acid (PLA) are polymers subject to endogenous depolymerization processes. A mathematical model was proposed and numerical techniques were developed for an enzymatic degradation of PVA . Those mathematical techniques were reapplied to an enzymatic hydrolysis of polylactic acid (PLA) . Techniques originally developed for endogenous type processes were replied to exogenous type depolymerization processes of PE and PEG .
This study revisits an exogenous type depolymerization process of PEG to demonstrate mathematical techniques. Experimental outcomes before and after cultivation of microbial consortium E-1 in culture media were incorporated into a computational analysis based on a mathematical model. Inverse problems for a molecular factor and a time factor of a degradation rate were formulated. Solutions of those inverse problems allow us to simulate a biodegradation process of PEG.
2. Formulation of Exogenous Type Depolymerization Process
Consider an exogenous type depolymerization process in a culture medium. Let w(t, M) [mg] be the weight distribution of a polymer with respect to the molecular weight M at time t, and v(t) [mg] be the total weight of polymer molecules with molecular weight between A and B at time t. The total weight v(t) over the interval [A, B] is expressed in terms of the integral
The total weight v(t) of the residual polymer at time t is expressed in terms of the integral
Integral (1) well approximates the integral (2) with suitable choices of A and B. In this study, integrals with the lower limit were replaced with integrals with the lower limit A = 103.0, and integrals with the upper limit ∞ were replaced with integrals with upper limit B = 104.2.
Let σ(t) be the population of viable cells at time t. Equations (3) for the weight distribution w(t, M) and the microbial population σ(t) was proposed in previous studies   .
Parameter L is the molecular weight of a monomer unit, e.g. PE: 28 (CH2CH2), PEG: 44 (CH2CH2O), and k and h are positive parameters. The function corresponds to a molecular factor of degradation rate.
This study proposes the equation
where . Note that is the rate of consumption of total weight. System of Equations (3), (4) is associated with the initial conditions
where and are the initial weight distribution and the initial microbial population, respectively.
3. Numerical Solutions of Inverse Problems for Molecular Factor and Time Factor
The initial value problem (3), (4), (5) (6) is solvable for and provided the molecular factor and values of parameters , k, and h are given beforehand. In order to obtain the function and values of parameters, consider change of variables from t to defined by the equation
Denote by by , , and functions that correspond to w(t, M), , and v(t) according to the change of variables (7), respectively.
and that Equations (3) and (4) lead to
respectively, where .
Suppose that F1(M) is the weight distribution for , that is,
and that F2(M) is the weight distribution for , that is,
Equation (8), the initial condition (10), and the final condition (11) form an inverse problem for the molecular factor , for which the solution of the initial value problem (8), (10) also satisfies the final condition (11).
Numerical techniques to solve the inverse problem for were developed in previous studies. Weight distributions of PEG 6000 before and after cultivation of microbial consortium E-1 for one day, three days, five days, seven days, and nine days f0(M), f1(M), f2(M), f3(M), f4(M), and f5(M) were introduced into analysis. Note that w(ti,M) = fi(M) for i = 0, 1, 2, 3, 4, and 5, where t0 = 0, t1 = 1, t2 = 3, t3 = 5, t4 = 7, and t5 = 9. Weight distributions after cultivation of the microbial consortium E-1 for one day and three days were the functions F1(M) and F2(M), respectively, and the inverse problem (8), (10), (11) was solved numerically for T1 = 0 and T2 = 1.0 . Note a positive scalar multiple of a molecular factor is also a molecular factor.
Once the molecular factor was obtained, Equation (10) was solved for with the initial condition
A previous study shows the exponential function
well approximates function so that . In this study, function was approximated by the exponential function (13) with .
Once the function was obtained, Equation (9) was solved with the initial condition
Denote by the solution of the initial value problem (9), (14). In view of the Formula (7), , where
Given m pairs of values of t and τ, , consider the system of equations
where . Here is the value of τ that satisfy . In particular,
Intervals , , were divided into and intervals, respectively, and the initial value problem (9), (14) was solved for
The value of , k, and h that minimize the error
were approximately 0.00156, 0.00518, and 0.404, respectively.
Given and , the initial value problem (3), (5) was solved numerically. Figures 1-5 show some numerical results. Figures 1-4 show comparison between experimental results and numerical results for weight distribution after cultivation of microorganisms for one day, three days, five days, and seven days, respectively. Figure 5 shows the graph of the function .
Figure 1. Weight distribution after cultivation of microbial consortium E-1 for one day. An experimental result and numerical results are shown.
Figure 2. Weight distribution after cultivation of microbial consortium E-1 for three days. An experimental result and numerical results are shown.
Figure 3. Weight distribution after cultivation of microbial consortium E-1 for five days. An experimental result and numerical results are shown.
Figure 4. Weight distribution after cultivation of microbial consortium E-1 for seven days. An experimental result and numerical results are shown.
Figure 5. Transition of microbial population over seven days.
Equation (4) is based on the fact that the increasing rate of the microbial population is proportional to both the consumption rate and the microbial population. Parameter k is the conversion rate from the carbon source to the microbial population per unit population, It is also based on the fact that the decreasing rate of the microbial population is proportional to the microbial population. Parameter h is the rate of loss of active microorganisms. Our analysis shows that the microorganisms convert approximately 0.5% of carbon source consumed per day to their growth, and that approximately 40% of microorganism loses degradability per day. The numerical results show that our model is appropriate.
The authors thank Ms. Y. Shimizu for her technical support.
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