STUDY ON EFFECTS OF MICROORGANISM IN

DEPOLYMERIZATION PROCESS OF XENOBIOTIC POLYMERS

BY MODELING AND SIMULATON

Masaji Watanabe

Graduate School of Environmental Science, Okayama University, Okayama, Japan

Fusako Kawai

R & D Center of Bio-based Materials, Kyoto Institute of Technology, Kyoto, Japan

Keywords:

Polymer, Biodegradation, Mathematical modeling, Numerical simulation.

Abstract:

Effects of microorganism in biodegradation process of polyethylene glycol are studied by modeling and simu-

lation. Dynamics of population of microorganism is taken into consideration in modeling of depolymerization

process of exogenous type, and a mathematical model is described. A molecular factor of degradation rate

is obtained by solving an inverse problem, and a time factor of degradation is obtained by analyzing the dy-

namics of population of microorganism. Once the time factor and the molecular factor of degradation rate are

determined, a depolymerization process is simulated by solving an initial value problem.

1 INTRODUCTION

Biodegradation is an essential factor of environmental

protection against accumulation of xenobiotic poly-

mers. It is important not only for water soluble

polymers but also for water-insoluble polymers, be-

cause they are not completely recycled nor inciner-

ated. Microbial depolymerization processes are gen-

erally classiﬁed into exogenous type or endogenous

type. In an exogenous type depolymerization pro-

cess, monomer units are separated from terminals

of molecules stepwise. Examples of exogenous de-

polymerization process include the β-oxidation of

polyethylene (PE). In an endogenoustype depolymer-

ization process, molecules are split at arbitrary po-

sitions. Examples of endogenous depolymerization

process include enzymatic degradation of polyvinyl

alcohol PVA. Mathematical models for those depoly-

merization processes were proposed in previous stud-

ies. Microbial depolymerization processes of xenobi-

otic polymers were studied by analysis based on those

models.

In this study, microbial depolymerization process

of polyethylene glycol (PEG) is analyzed. PEG is

one of polyethers expressed by HO(R-O)

H, for ex-

ample, PEG: R = CH

, polypropylene glycol

(PPG): R = CH

CHCH

, polytetramethylene glycol

(PTMG): R = (CH

)

(Kawai, 1993). They are uti-

lized for constituents of productsincluding lubricants,

antifreeze agents, inks, and cosmetics. PEG is pro-

duced more than any other polyether, and the ma-

jor part of the production is consumed for nonionic

surfactants. PEG is depolymerized by separation of

compounds, either aerobically or anaerobically

(Kawai, 1995; Kawai, 2002a; Kawai, 2002b). High

performance liquid chromatography (HPLC) patterns

were used in analysis as the weight distribution of

PEG with respect to the molecular weight before and

after cultivation of a microbial consortium E1 (Figure

1).

In the previous studies (Watanabe and Kawai,

2004; Watanabe and Kawai, 2005), time indepen-

den degradation rates were assumed. Time depen-

dent degradation rates were considered in a recent

study (Watanabe and Kawai, 2007a). Cubic spline

was used to take the change of microbial population

into consideration (Watanabe and Kawai, 2007b). E

exponential growth in a microbial population was as-

sumed (Watanabe and Kawai, 2009b). The time factor

was also determined by assuming the logistic growth

in the time integral of microbial population (Watan-

abe and Kawai, 2009a). In this study, analysis of

181

Watanabe M. and Kawai F. (2010).

STUDY ON EFFECTS OF MICROORGANISM IN DEPOLYMERIZATION PROCESS OF XENOBIOTIC POLYMERS BY MODELING AND SIMULATON.

In Proceedings of the First International Conference on Bioinformatics, pages 181-186

DOI: 10.5220/0002757601810186

 SciTePress

0.005

0.01

0.015

0.02

0.025

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2

COMPOSITION (%)

LOG M

BEFORE CULTIVATION

AFTER 1-DAY CULTIVATION

AFTER 3-DAY CULTIVATION

AFTER 5-DAY CULTIVATION

AFTER 7-DAY CULTIVATION

AFTER 9-DAY CULTIVATION

Figure 1: Weight distribution of PEG before and after cultivation of a microbial consortium E1 (Watanabe and Kawai, 2009a).

biodegradation of PEG is continued. A change of

variable transforms the time dependent model into a

model for which the degradation rate is time indepen-

dent. Techniques developed in previous studies are

applied to solve an inverse problem to determine the

molecular factor of degradation rate for which the so-

lution of an initial value problem satisﬁes not only the

initial weight distribution but also the weight distribu-

tion after cultivation. The time factor is determined by

assuming change in microbial population due to the

amount of PEG as the sole carbon source. Once the

degradation rate is found, the transition of the weight

distribution was simulated by solving the initial value

problem.

2 MODEL OF EXOGENOUS

TYPE DEPOLYMERIZATION

PROCESS WITH TIME

DEPENDENT DEGRADATION

RATE

In microbial depolymerization of PE, molecules lose

their weight gradually by terminal separations of

monomer units (β-oxidation) until they become small

enough to be absorbed directly into cells. The PE

biodegradation model proposed previously is based

on two essential factors, the gradual weight loss of

large molecules due to β-oxidation and the direct con-

sumption of small molecules by cells. Let t and M

be the time and the molecular weight respectively.

Suppose tha a M-molecule denotes a molecule with

molecular weight M. Let w(t,M) represent the total

weight of M-molecules at time t, and L be the amount

of weight loss due to the β-oxidation. Let the func-

tion ρ(M) represent the direct consumption rate, the

function β(M) represent the β-oxidation rate, which

BIOINFORMATICS 2010 - International Conference on Bioinformatics

182

is the rate of the weight conversion from the class of

M-molecules to the class of (M − L)-molecules due

to the β-oxidation, and α(M) = ρ(M) + β(M). The

equation (1) was proposed to study PE biodegrada-

tion processes (F. Kawai, 2002; M. Watanabe, 2003;

F. Kawai, 2004; M. Watanabe, 2004).

= −α(M) x + β(M + L)

M + L

y, (1)

where x = w(t, M), the total weight of M-molecules at

time t, and y = w(t,M+ L), the total weight of M+L-

molecules at time t.

The mathematical model (1) was originally devel-

oped for the PE biodegradation, but it can be viewed

as a general biodegradation model for exogenous de-

polymerization processes. In an exogenous depoly-

merization process of PEG, a molecule is ﬁrst oxi-

dized at its terminal, and then an ether bond is split.

It follows that L = 44 (CH

O) for the exoge-

nous depolymerization of PEG. PEG molecules stud-

ied here are large molecules, and they can not be ab-

sorbed directly through membrane into cells. Then

ρ(M) = 0, and α(M) = β(M).

Equation (1) is appropriate for the depolymeriza-

tion processes over the period in which the microbial

population has reached a stationary state. On the con-

trary, the change of microbial population should be

taken into account for the period in which it is still

in a developing stage or in a decreasing stage. Then

degradation rate should be time dependent, and the

exogenous depolymerization model is

= −β(t, M)x+ β(t,M + L)

M + L

y. (2)

The solution x = w(t,M) of the equation (2) is asso-

ciated with the initial condition

w(0,M) = f (M) , (3)

where f (M) is the initial weight distribution. Given a

presicribed degradation rate β(t,M), equation (2) and

the initial condition (3) form an initial value problem.

Time factors of the degradation rate such as micro-

bial population, dissolved oxygen, or temperature af-

fect molecules regardless of sizes. Then the degrada-

tion rate should be split into the time dependent part,

which we denote by σ(t), and the molecular depen-

dent part, which we denote by λ(M), and the degra-

dation rate is the product

β(t,M) = σ(t)λ(M) . (4)

The time factor of the degradation rate can be re-

moved by the transformation

τ =

σ(s) ds. (5)

Let

W (τ,M) = w(t, M) ,

X = W (τ,M) ,

Y = W (τ, M + L) .

Then

dτ

σ(t)

and the equation (2) becomes

dτ

= −λ(M) X + λ(M + L)

M + L

Y . (6)

Given the initial weight distribution f (M), solution

the initial value problem is the solution of the equa-

tion (6) subject to the initial condition

W (0, M) = f (M). (7)

Given additional condition

W (T ,M) = g(M) , (8)

solution of the inverse problem is the degradation rate

λ(M) for which the solution of the initial value prob-

lem (6), (7) also satisﬁes the condition (8). When the

solution W (τ,M) of the initial value problem (6), (7)

satisﬁes the condition (8), the solution w(t,M) of the

initial value problem (2), (3) satisﬁes

w(T,M) = g(M) , (9)

where

T =

σ(s) ds. (10)

The inverse problem (6), (7), (8) was solved nu-

merically with techniques developedin previous stud-

ies. Figures 2 shows the degradation rates based on

the weight distribution before and after cultivation for

three days (Watanabe and Kawai, 2009a).

3 TIME FACTOR OF

DEGRADATION RATE BASED

ON MICROBIAL POPULATION

Suppose that the time factor σ(t) represents the mi-

crobial population, and that PEG is the sole carbon

source. The total amount of polymer utilized by mi-

croorganisms per unit time at time t is

A =

∞

σ(t) λ(M) w(t, M) dM.

The microbial population increases under sufﬁcient

carbon supply when the ratio of the microbial pop-

ulation to the total amount is below a certain lavel

STUDY ON EFFECTS OF MICROORGANISM IN DEPOLYMERIZATION PROCESS OF XENOBIOTIC POLYMERS

BY MODELING AND SIMULATON

183

100

120

140

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2

DEGRADATION RATE PER DAY

LOG M

DEGRADATION RATE

Figure 2: Degradation rate based on the weight distribution of PEG before and after cultivation of a microbial consortium E1

(Watanabe and Kawai, 2009a).

σ/A < 1/h It decreases under insufﬁcient carbon sup-

ply when σ/A > 1/h. Then the growth rate of micro-

bial population is propotinal to

1− h

and the microbial population σ(t) is a solution of the

equation

dσ

= k



1− h



σ (11)

Note that equation (11) becomes

dσ

= k



1− h

∞

λ(M)w(t, M) dM



σ. (12)

Equation (12) is associated with the initial condition

σ(0) = σ

. (13)

Equations (2) and (12) are solved simultaneously

to simulate the transition of weight distrubution of

PEG and evolution of microbial population for σ

≈

0.029827, k ≈ 51.6, and h = 500.0. Figure 3 shows

the resulsts of simulation for

4 CONCLUSIONS

In a depolymerization process in which microbial

population is an essential factor, the dependence of

the degradation rate on time becomes signiﬁcant. It

is appropriate to assume that the degradation rate is a

product of a time factor and a molecular factor. In an

environmental setting, the time factor should also de-

pend on other factors such as temperature or dissolved

oxygen. Once these essentials are taken into consider-

ation, the exogenous depolymerization model will be

applicable to assess the biodegradabilityof xenobiotic

polymers in the environment.

Subjects in study of microbial depolymerization

of xenobiotic polymers by modeling and simulation

in next steps includes development of numerical tech-

niques to identify the values of parameters which

appear in the equation (11). Results of simulation

should be compared with experimental results con-

cerning the transition of weight distribution of the

polymer and the microbial population to verify nu-

merical results.

BIOINFORMATICS 2010 - International Conference on Bioinformatics

184

0.005

0.01

0.015

0.02

0.025

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2

COMPOSITION (%)

LOG M

AFTER 3-DAY CULTIVATION

SIMULATION: 3 DAY

AFTER 5-DAY CULTIVATION

SIMULATION: 5 DAY

Figure 3: Numerival simulation of weight distribution and experimental results .

ACKNOWLEDGEMENTS

The authors thank Ms Y. Shimizu for her technical

support. This work was supported by JSPS KAK-

ENHI 20540118.

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