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This study demonstrates mathematical analysis of biodegradation processes of xenobiotic polymers. A model for microbial population is based on the fact that growth rate of microorganisms is proportional to the microbial population and consumption rate of parts of carbon sources. The model is paired with a model for weight distribution. Those models lead to inverse problems for a molecular factor and a time factor of degradation rate. Solution of the inverse problems allows us to simulate the biodegra-dation process.

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 [

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 [

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.

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

v ( t ) = ∫ A B w ( t , M ) d M (1)

The total weight v(t) of the residual polymer at time t is expressed in terms of the integral

v ( t ) = ∫ 0 ∞ w ( t , M ) d M (2)

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 = 10^{3.0}, and integrals with the upper limit ∞ were replaced with integrals with upper limit B = 10^{4.2}.

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 [

∂ w ∂ t = σ ( t ) [ − λ ( M ) w + c ( M ) ∫ M ∞ λ ( K ) d ( K ) w ( t , K ) d K ] (3)

c ( M ) = M e ρ M , d ( K ) = ρ e − ρ K K ( 1 − e − ρ K ) , ρ = log 2 L ,

Parameter L is the molecular weight of a monomer unit, e.g. PE: 28 (CH_{2}CH_{2}), PEG: 44 (CH_{2}CH_{2}O), and k and h are positive parameters. The function λ ( M ) corresponds to a molecular factor of degradation rate.

This study proposes the equation

d σ d t = k u ( t ) σ − h σ , (4)

where u ( t ) = − v ′ ( t ) . Note that u ( t ) is the rate of consumption of total weight. System of Equations (3), (4) is associated with the initial conditions

w ( 0 , M ) = f 0 ( M ) , (5)

σ ( 0 ) = σ 0 , (6)

where f 0 ( M ) and σ 0 are the initial weight distribution and the initial microbial population, respectively.

The initial value problem (3), (4), (5) (6) is solvable for w ( t , M ) and σ ( t ) provided the molecular factor λ ( M ) and values of parameters σ 0 , k, and h are given beforehand. In order to obtain the function and values of parameters, consider change of variables from t to τ defined by the equation

τ = ∫ 0 t σ ( s ) d s . (7)

Denote by by w ( τ , M ) , S ( τ ) , and V ( τ ) functions that correspond to w(t, M), σ ( t ) , and v(t) according to the change of variables (7), respectively.

Note that

∂ W ∂ τ = d t d τ ∂ w ∂ t = 1 σ ( t ) ∂ w ∂ t , d S d τ = d t d τ d σ d t = 1 σ ( t ) d σ d t , d V d τ = d t d τ d v d t = 1 σ ( t ) d v d t

and that Equations (3) and (4) lead to

∂ W ∂ τ = − λ ( M ) W + c ( M ) ∫ M ∞ λ ( K ) d ( K ) W ( τ , K ) d K , (8)

d S d τ = k U ( τ ) S − h , (9)

respectively, where U ( τ ) = − V ′ ( τ ) .

Suppose that F_{1}(M) is the weight distribution for τ = T 1 , that is,

W ( T 1 , M ) = F 1 ( M ) , (10)

and that F_{2}(M) is the weight distribution for τ = T 2 , that is,

W ( T 2 , M ) = F 2 ( M ) . (11)

Equation (8), the initial condition (10), and the final condition (11) form an inverse problem for the molecular factor λ ( M ) , 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 λ ( M ) 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 f_{0}(M), f_{1}(M), f_{2}(M), f_{3}(M), f_{4}(M), and f_{5}(M) were introduced into analysis. Note that w(t_{i},M) = f_{i}(M) for i = 0, 1, 2, 3, 4, and 5, where t_{0} = 0, t_{1} = 1, t_{2} = 3, t_{3} = 5, t_{4} = 7, and t_{5} = 9. Weight distributions after cultivation of the microbial consortium E-1 for one day and three days were the functions F_{1}(M) and F_{2}(M), respectively, and the inverse problem (8), (10), (11) was solved numerically for T_{1} = 0 and T_{2} = 1.0 [

Once the molecular factor λ ( M ) was obtained, Equation (10) was solved for W ( τ , M ) with the initial condition

W ( 0 , M ) = f 0 ( M ) . (12)

A previous study shows the exponential function

V ( τ ) = v 0 e − μ τ ( v 0 = ∫ 0 ∞ f 0 ( M ) d M ) , (13)

well approximates function V ( τ ) so that V ′ ( τ ) = − μ v 0 e − μ τ . In this study, function V ( τ ) was approximated by the exponential function (13) with μ ≈ 0.878 .

Once the function V ( τ ) was obtained, Equation (9) was solved with the initial condition

S ( 0 ) = σ 0 . (14)

Denote by S ( τ , σ 0 , k , h ) the solution of the initial value problem (9), (14). In view of the Formula (7), t = q ( τ , σ 0 , k , h ) , where

q ( τ , σ 0 , k , h ) = ∫ 0 τ d r S ( r , σ 0 , k , h ) .

Given m pairs of values of t and τ, ( t i , τ i ) ( i = 1 , 2 , 3 ) , consider the system of equations

g i ( σ 0 , k , h ) = 0 ( i = 1 , 2 , 3 ) (15)

where g i ( σ 0 , k , h ) = q ( τ i , σ 0 , k , h ) − t i . Here τ i is the value of τ that satisfy v ( t i ) = V ( τ i ) . In particular,

t 1 = 1 , τ i ≈ 0.0 24275 , t 2 = 3 , τ 2 ≈ 0. 288317 , t 3 = 5 , τ 3 ≈ 2 . 62864 .

Intervals [ σ 0 , 0 , σ 0 , 1 ] , [ k 0 , k 1 ] , [ h 0 , h 1 ] were divided into n 1 , n 2 and n 3 intervals, respectively, and the initial value problem (9), (14) was solved for

σ 0 = σ 0 , 0 + i Δ σ 0 , Δ σ 0 = σ 0 , 1 − σ 0 , 0 n 1 , i = 0 , 1 , ... , n 1 ,

k = k 0 + j Δ k , Δ k = k 1 − k 0 n 2 , j = 0 , 1 , ... , n 2 ,

h = h 0 + l Δ h , Δ h = h 1 − h 0 n 3 , l = 0 , 1 , ... , n 3 ,

σ 0 , 0 = 0.0015 , σ 0 , 1 = 0.0016 ,

k 0 = 0.0045 , k 1 = 0.0055 ,

h 0 = 0.35 , h 1 = 0.45 ,

n 1 = 100 , n 2 = 100 , n 3 = 100 .

The value of σ 0 , k, and h that minimize the error

∑ i = 1 5 [ g i ( σ 0 , k , h ) ] 2

were approximately 0.00156, 0.00518, and 0.404, respectively.

Given λ ( M ) and σ ( t ) , 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.

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Watanabe, M. and Kawai, F. (2021) Numerical Study on Microbial Depolymerization Process of Xenobiotic Polymer. Journal of Materials Science and Chemical Engineering, 9, 43-50. https://doi.org/10.4236/msce.2021.91004