-
ISSN 1330-9862 original scientific paper
(FTB-2434)
Integrated Approach to Mathematical Modelling of Atrazine
Degradation in Different Reaction Systems**
Marijan Bo{njak1*, Nikolina Udikovi} Koli}2, Ines Petri}2, Darin
Cihlar3
and Dubravka Hr{ak21Croatian Academy of Engineering, HR-10000
Zagreb, Ka~i}eva 28, Croatia
2Ru|er Bo{kovi} Institute, Bijeni~ka 54, HR-10000 Zagreb,
Croatia3Infoptim Ltd, HR-10000 Zagreb, Croatia
Received: February 2, 2010Accepted: July 7, 2010
Summary
Based on the known approaches and published mathematical models,
as well as ontheoretical consideration using our experimental data,
the integrated approach to mathe-matical modelling of atrazine
biodegradation processes has been employed and
sophisticaedmathematical models for different reaction systems have
been developed. The applicabilityof these models which take into
account physical, chemical, biochemical and biologicalcomplexity of
atrazine biodegradation was further analyzed in comparison with
mathema-tical models describing simple consecutive reaction systems
using first-order kinetics. Ki-netics of atrazine degradation in
liquid media and soil contaminated with atrazine at thetemperatures
of 10 and 30 °C was assessed and compared. Biodegradation
experiments inliquid media were conducted at atrazine
concentrations ranging from 0.14 to 25 mmol/L,while the experiments
in soil were conducted at atrazine concentration of approx.
0.44mmol/g. Computer simulations were applied to explain
experimental results and test theadequacy of mathematical models.
Detailed analysis of computer simulation data showedthat the
developed integrated mathematical models could be considered as the
most con-venient for describing kinetics of atrazine
biotransformation in both liquid media and con-taminated soil,
although even simple mathematical models are suitable for
explaining someexperimental results, especially when evaluating the
temperature effects on biodegrada-tion efficacy of the applied
mixed bacterial culture.
Key words: atrazine, biodegradation kinetics, mathematical
modelling, computer simula-tion, mixed bacterial cultures
Introduction
Application of pesticides demands detailed studiesof kinetics of
their degradation, which has resulted in aseries of papers
referring to this problem. As pointedout in the report of Soulas
(1), Hill et al. (2) modelled thesoil degradation of several
substituted urea herbicidesby applying simple first-order kinetics.
Applicability ofmathematical models based on fundamental
chemical
392 M. BO[NJAK et al.: Mathematical Modelling of Atrazine
Degradation, Food Technol. Biotechnol. 48 (3) 392–403 (2010)
*Corresponding author; E-mail: [email protected]**Reported
as oral presentation at the 2nd Central European Forum for
Microbiology (CEFORM), Kesztely, Hungary, October 7–9, 2009
theory and on biocatalysis defined in accordance
withMichaelis–Menten kinetics as well as on the theory ofmicrobial
processes was discussed. Discussion was extend-ed to integrated
models taking into account physicochem-ical processes with
suggestions how to better describepesticide degradation processes
in soil. As a continua-tion, Soulas and Lagacherie (3) focused on
the adapta-tion of the models taking into account microbial
interac-
-
tions with soil environments. Methodological and
technicallimitations were mentioned.
Atrazine is the most commonly used herbicide world-wide (4).
However, in European countries, and thereforein Croatia, its use in
agriculture has recently been re-stricted. Remarkable relevance to
the study of atrazinebiodegradation was given in Croatia, where for
the last10 years the studies have been oriented towards the
dis-covery of mixed microbial cultures capable of degradingatrazine
efficiently under different reaction conditionsand towards
describing kinetics relationships of degrad-ation process. The
experimental studies resulted in theacceptance of patent
application and publication of re-search results (5–9). It is also
worth mentioning thatthey contributed to the advances in this area
worldwide.
Park et al. (4) applied three atrazine-degrading bac-terial
strains (Pseudomonas sp. strain ADP, Agrobacteriumradiobacter
strain J14a and Ralstonia sp. strain M91-3) indetailed studies of
sorption equilibria in soil slurries,CO2 production, as well as in
the evaluation of distri-bution coefficients and desorption
parameters. Mathe-matical model describing kinetics of particular
events,consisting of 8 equations (three of them were expressedas
differential equations), was applied to explain experi-mental
results. Experimental results fitted quite well thosetheoretical.
It was established that sorbed atrazine quan-tities for different
sorbents were proportional to aqueousatrazine concentration. High
importance was also givento atrazine and/or other pesticide
sorption-desorption ki-netics in publications of other authors
(10–14).
Wenk et al. (15) studied rapid atrazine minerali-zation applying
an atrazine-degrading Pseudomonas sp.strain. Detailed experimental
investigation of biodegra-dation efficiency depending on the
applied biomassquantity, soil moisture and atrazine
adsorption-desorp-tion rates was performed, but without expressing
anymathematical model. However, their finding that the rateof
atrazine removal from the contaminated soil was pro-portional to
the water content of the soil and the amountof bacteria added to
the soil is relevant for the develop-ment of reliable mathematical
model convenient to ex-plain atrazine biodegradation kinetics.
Kinetics of atrazine biodegradation in the water con-taminated
with this herbicide was studied by Goux et al.(16). The microbial
community designated as COM1 ori-ginating from Belgian maize field
was applied as bio-catalyst. To describe biodegradation kinetics,
two differ-ent mathematical models were used. The first was
theconventional convection dispersion equation, whereas theother
was the equation expressing the first order kine-tics. In the
experiments, media with relatively low initialatrazine
concentration (20 mg/L) were used, and the ap-plied microbial
consortium COM1 showed to be efficientas biocatalyst.
The aim of this work is to demonstrate the advan-tages of
integrated approach to mathematical modellingin explaining atrazine
biodegradation kinetics, and to showhow the developed mathematical
models are suitable fordescribing atrazine biodegradation kinetics.
To this pur-pose computer simulations have been applied.
Materials and Methods
Postulation of a simple mathematical model
At the very beginning of the study of atrazine biode-gradation,
we estimated that experimental results shouldbe evaluated on the
basis of recognised biodegradationkinetics relationships. The
insight into the results of thefirst series of experiments
suggested the application ofmathematical model to the already
obtained experimen-tal data, describing degradation process
kinetics withdifferential equations that express the reaction rates
inaccordance with the first order reaction kinetics and basedon the
reaction scheme:
kAB kBC kCDA —® B —® C —® D
Scheme 1. First order reaction kinetics
where A refers to atrazine, B to hydroxyatrazine, C tocyanuric
acid, and D to the final degradation productsCO2 and NH4
+.
Consequently, the mathematical model MAM1, de-fined by Eqs. 1–3,
is obtained:
dcA/dt=–kAB·cA /1/
dcB/dt=kAB·cA–kBC·cB /2/
dcC/dt=kBC·cB–kCD·cC /3/
where cA, cB and cD are the molar concentrations of A, Band C
substances respectively, while kAB, kBC and kCD arethe
corresponding reaction rate constants.
Development of integrated (structured) mathematicalmodels
Based on the literature data (4,15,16), relatively lowinitial
concentrations of atrazine were applied in the firstseries of
experiments. That is why there was no interestin developing more
adequate mathematical model at thebeginning of experiments.
However, later during re-search the mixed microbial culture capable
of degradingatrazine efficiently even in reaction media with
muchhigher atrazine concentrations provoked an interest
fordeveloping a more adequate mathematical model takinginto account
both the role of microbial culture as a bio-catalytic agent and the
relevance of atrazine solubility inwater. Prior to establishing the
final mathematical model,the model represented by Eqs. 1–3 (MAM1)
was trans-formed into the model describing the growth and
bio-catalytic action of microbial biomass during atrazine
bio-degradation. Therefore, if biomass was expressed in massunits,
the following mathematical model (MAM2) wasobtained:
dgx/dt=mxm·gx·(1–gx/gxm) /4/
dcA/dt=–kABX·gx·cA/(cA+KA) /5/
dcB/dt=kABX·gx·cA/(cA+KA)–kBCX·gx·cB/(cB+KB) /6/
dcC/dt=kBCX·gx·cB/(cB+KB)–kCDX·gx·cC/(cC+KC) /7/
393M. BO[NJAK et al.: Mathematical Modelling of Atrazine
Degradation, Food Technol. Biotechnol. 48 (3) 392–403 (2010)
-
where mxm is maximal specific growth rate of microbialbiomass in
h–1, gx is microbial biomass concentration, gxmis maximal microbial
biomass concentration in g/L, andKA, KB and KC are mean
biocatalytic constants for re-action substances A, B and C,
respectively. When bio-mass decay is observed, instead of logistic
Eq. 4, the mo-dified Eq. 8 is suggested:
dgx/dt=mxm·gx·(1–gx/gxm)·cA/(cA+Kag)–kd·gx /8/
where Kag is biocatalytic constant of biomass growth
withreference to substance A in mmol/L, while kd is specificbiomass
decay rate in h–1.
If biomass was expressed as a number of microbialcells (CFU),
then instead of Eq. 4, Eq. 9 should beapplied:
dNx/dt=mNm·Nx·(1–Nx/Nxm) /9/
where Nx is the number of microbial cells in L–1, Nxm is
maximal number of microbial cells in L–1, mNm is maxi-mal
specific rate of cell number increase in h–1. In thiscase, a
simultaneous transformation of rate constantvalues with respect to
microbial cell units must be done.
In media with higher atrazine concentrations, it isonly partly
dissolved, while the other part is undissolvedor adsorbed on soil
or other carrier particles. In suchcases, mathematical model MAM2,
shown in Eqs. 4–7 or9, should be transformed taking into account
the chang-es of undissolved and total atrazine concentrations.
Re-action system analysis led to the conclusion that the ex-tended
system of differential equations (MAM3) shouldbe applied, i.e. in
addition to Eqs. 4 or 9 and Eqs. 6 and7, the following equations
should be included:
dcA/dt=–kABX·gx·cA/(cA+KA)+ksol·cAins·(cAm–cA) /10/
dcAins/dt=–ksol·cAins·(cAm–cA) /11/
dcAT/dt=dcA/dt+dcAins/dt /12/
where ksol is the specific rate of undissolved atrazine
dis-solved in L/(mmol·h), cAins is the concentration of
undis-solved atrazine in mmol/L, cAm is atrazine solubility
inmmol/L and cAT is total atrazine concentration in mmol/L.The
other conditions for the use of this model are thesame as those in
the model applicable for reaction med-ia where atrazine is
dissolved completely.
The range of applicability of any mathematical modelis very
important. Mathematical models shown in Eqs.1–12 were originally
developed to explain atrazine bio-degradation kinetics in liquid
media. However, litera-ture data suggest that more relevance needs
to be givento the kinetics of atrazine biodegradation in
contaminat-ed soil. Wenk et al. (15) established that the rate of
atra-zine removal from contaminated soil was proportionalto the
water content in the soil and depended on theamount of bacteria
added to the soil. Therefore, hypo-thetically the developed
mathematical model can beadapted for the explanation of kinetics of
atrazine re-moval from contaminated soil. In order to verify
itsapplicability, model parameters were converted withrespect to
the water content in the soil. This can bedemonstrated with the
following example:
If the soil containing 20 % of water is contaminatedwith
atrazine 0.5 mmol/kg, inoculated with 107 cells ofa mixed microbial
culture and then homogenised, thesoil will act as liquid reaction
medium containing (inaqueous phase) at the start: 107/0.2=5·107 of
microbialcells/L, and atrazine 0.5/0.2=2.5 mmol/L. If
atrazinesolubility in the water is estimated to be 0.15 mmol/L,then
the concentration of undissolved atrazine will be2.35 mmol/L.
However, MAM3 cannot be considered as the finalsolution for
describing atrazine biodegradation kinetics.It can be transformed
giving its simpler modifications,or extended by taking into account
other relevant phy-sicochemical and biochemical biodegradation
processeslike those defined by Park et al. (4). Simpler model
modi-fications can be recommended for use in order to supportthe
advantages of more sophisticated mathematical mo-dels. One of the
simpler modifications (MAM3a) can resultfrom fixing the constant
microbial biomass concentra-tion and applying the first order
kinetics (Eqs. 1–3), andthe other (MAM3b) after fixing constant
microbial bio-mass concentration. The extended mathematical
modelMAM4 can be defined by adding Eq. 13 and by modi-fying Eqs. 10
and 12:
dcAim/dt=–kdim·cAim+kim·cA /13/
dcA/dt=–kABX·gx·cA/(cA+KA)++ksol
·cAns·(cAm–cA)+kdim·cAim–kim·cA
/14/
dcAns/dt=–ksol·cAns·(cAm–cA) /15/
dcAT/dt=dcA/dt+dcAns/dt+dcAim/dt /16/
where cAim refers to the sorbed (immobilized)
atrazineconcentration, and cAns to the undissolved atrazine
con-centration, while kim and kdim indicate the rate constantswith
reference to dissolved atrazine (cA) sorption and im-mobilised
atrazine (cAim) desorption, respectively.
Prior to verifying whether the proposed mathema-tical models
could explain the experimental data, it shouldbe mentioned that
microbial processes, especially thosereferring to mixed microbial
cultures, are very complexand that they cannot be described
perfectly with mathe-matical models because of insufficient
information on allprocesses in every moment. However, mathematical
mo-dels can be used to determine the differences in behav-iour
between particular microorganisms, as demonstrat-ed recently (17)
by comparing the behaviour of differentStreptomyces rimosus
strains. Simple systems of differen-tial equations can usually be
solved by known mathe-matical methods with analytical solutions.
There is ananalytical solution also for the system described by
dif-ferential Eqs. 1–3. However, computer simulation canalso be
applied, and it appears as a method of choiceeven for very complex
systems of differential equations.Therefore, it was also applied to
verify the convenienceof mathematical models in this work.
Computer simulation
Due to the previous successful applications (17–21)of Scientist
computer programme (Micromath, St. Louis,MO, USA), it was also
applied in this work. Based on
394 M. BO[NJAK et al.: Mathematical Modelling of Atrazine
Degradation, Food Technol. Biotechnol. 48 (3) 392–403 (2010)
-
the mathematical models developed in this work, ade-quate
computer simulation kinetic models were pre-pared and applied.
Fittings of computer simulation toexperimental data were
statistically validated applyingthe Jacobian matrix, installed as
part of Scientist calcu-lation programme.
Biodegradation experiments and experimental data
The input parameters used for mathematical model-ling are
experimental data of our integrated studies onthe mechanisms and
kinetics of atrazine biodegradationusing mixed and pure bacterial
cultures originating fromthe soil exposed to long-term
contamination with atra-zine and other s-triazine compounds (5–9).
Selected dataof the biodegradation experiments in liquid media
andsoil were used to test the applicability of the
developedmathematical models. The same, well characterized andvery
efficient atrazine-degrading mixed culture was usedin all the
experiments (5).
Biodegradation experiments in liquid media were per-formed at
different initial atrazine concentrations (0.145–22 mmol/L) and two
temperatures (10 and 30 °C) em-ploying shake flask technique as
described previously(6). It is important to note that in all
experiments theculture media were composed of the same mineral
salts– MS medium (22), and that in all cases except one (Fig.1;
computer simulation parameters with reference to allfigures
presented in Table 1) the MS medium was sup-plemented with sodium
citrate (1 g/L), (MS-citrate me-dium). Quantitative measurements of
atrazine and theformed intermediates (hydroxyatrazine and cyanuric
acid)were performed by HPLC analysis as described previ-ously
(6).
The experiments in soil were performed in micro-cosms using soil
that had not previously been exposedto atrazine. The procedure,
including the preparation ofatrazine-contaminated soil, was similar
to that describedelsewhere (9). The experiments were carried out
underaseptic conditions at 10 and 30 °C using a mixed culturegrown
in liquid medium (MS-citrate medium containing0.145 mmol/L of
atrazine) (6). The culture biomass wascentrifuged, resuspended in
MS-citrate medium and add-ed to the soil to the final water content
of 20 %. Quanti-tative determination of atrazine and the formed
interme-diates was performed by using the same HPLC methodas in
liquid culture experiments after soil extraction withmethanol.
Results and Discussion
Series of computer simulations were performed basedon MAM1 and
it was established that this model can beapplied to explain
experimental results when media withrelatively low atrazine
concentrations were investigated.Selected computer simulations
shown in Fig. 1 give evi-dence on the convenience of the applied
mathematicalmodel.
Because the role of microbial biomass cannot beignored in the
process of atrazine biodegradation, aseries of computer simulations
was performed usingMAM2, shown in differential Eqs. 4–9. Data in
Fig. 2
testify that MAM2 model is also convenient for ex-plaining
experimental data shown in Fig. 1.
With respect to the convenience of the application
ofmathematical model for larger range of reaction condi-tions, much
more was expected when applying the ex-tended model MAM3,
represented by Eqs. 9–12. Com-puter simulation data mainly
confirmed the expected, asdemonstrated in Figs. 3–8, especially
when consideringatrazine biodegradation kinetics in soil aqueous
phase(Figs. 7 and 8). However, it should be pointed out thatthe
simplest mathematical model (MAM1) can also beapplied to explain
the same experimental data estab-lished for soil aqueous phase
(Fig. 9). Although thisfinding could be surprising, there is no
doubt that it re-presents useful information, since the simple MAM1
canalso be recommended for use in evaluating the efficien-cy of
atrazine removal from the contaminated soil, iftaking into account
the fact that differences in microbialbiomass in MAM1 were
neglected. Discrepancies be-
395M. BO[NJAK et al.: Mathematical Modelling of Atrazine
Degradation, Food Technol. Biotechnol. 48 (3) 392–403 (2010)
t(degradation)/h
vs.
c(s
ub
sta
nce
)/(m
mo
l/L
)
vs.
vs.
vs.
vs.
t(degradation)/h
c(s
ub
sta
nce
)/(m
mo
l/L
)
vs.
vs.
vs.
vs.
vs.
vs.
Fig. 1. Application of mathematical model MAM1 in
describingatrazine biodegradation kinetics, and fitting computer
simula-tion to experimental data by applying parameters shown in
Ta-ble 1: a) experiment with MS culture medium, b) experimentwith
MS-citrate culture medium. Points: experimental data,curves:
simulated data
b)
a)
-
tween theoretical and experimental data suggest thattheir main
cause appears to be neglecting the differencesin microbial biomass
quantities, which is especially evi-dent when considering the
process kinetics at lower tem-perature, where changes of microbial
growth kineticstake place more slowly. In the process range of
lowerbiomass amount, the experimental values referring toatrazine
amount are higher than those theoretical (simu-lated), whereas in
the range of larger biomass amount,theoretical values are higher
than the corresponding ex-perimental ones, if changes of atrazine
quantities in thecontaminated soil during atrazine biodegradation
are com-pared. Concerning the applicability of simpler
mathema-tical models, it was also established that they could
beapplied in describing the degradation process presentedin Fig. 4.
Computer simulation data (Figs. 4b and c) re-sulted from the
transformation of MAM3 into its modi-fications MAM3a and MAM3b. As
shown, computer si-mulation data are in agreement with experimental
data.The statistically estimated values of model selection
cri-teria for data in Figs. 3 and 4 support such an impres-sion
(Table 2). Since the data in Fig. 3 evidently supportthe
application of MAM3, the same can be said for ex-perimental data
shown in Fig. 4.
More pronounced discrepancies between experimen-tal and
theoretical values can be observed when apply-ing the extended
mathematical model MAM3, as shownin Figs. 5 and 6. Probable reasons
for the observed dis-crepancies could be the exhaustion of some
nutrientduring the later biodegradation phase (Fig. 6), or
differ-ence between experimental and theoretical microbialgrowth
rates during the first process phase (Fig. 5).
On the other hand, good agreement between experi-mental and
theoretical data in cases of atrazine removalfrom contaminated soil
(Figs. 7 and 8, Table 2) suggeststhat the soil enriched with liquid
nutrient medium andinoculated with larger amount of microbial
culture showedto be more convenient for biodegradation than the
cor-responding liquid medium itself for the same processesin shake
flasks when these were started with markedlysmaller amounts of
microbial culture.
Regarding the atrazine biodegradation kinetics in theexperiments
on contaminated soil (Figs. 8 to 10), it canbe concluded that the
obtained results confirmed theobservations of Wenk et al. (15),
giving an emphasis tothe fact that the extended mathematical model
MAM3can be recommended for application in explaining atra-zine
biodegradation kinetics in contaminated soil. The
396 M. BO[NJAK et al.: Mathematical Modelling of Atrazine
Degradation, Food Technol. Biotechnol. 48 (3) 392–403 (2010)
Table 1. Applied values of computer simulation parameters with
reference to particular figures and applied mathematical models
Fig.
Mod
el
Parameter names and values
1
MA
M1 kAB kBC kCD
1a 0.0047 0.00505 0.0051b 0.0235 0.019 0.08
2
MA
M2 kABX kBCX kCDX mm gxm KA KB KC ksol cAm
0.0225 0.0155 0.05 0.35 0.34 0.165 0.165 0.165
3
MA
M3 1.75 1.75 1.25 0.15 0.272 0.14 0.14 0.14 0.90 0.1428
4a 1.9625 4.5 4.5 0.15 0.278 0.14 0.14 0.14 0.90 0.1428
4b
MA
M3a kAB*
6.5kBC*11.5
kCD*11.5
gx
0.28 0.90 0.1428
4c
MA
M3b kABX
1.55kBCX3.75
kCDX3.75 0.28 0.14 0.14 0.14 0.90 0.1428
5a
MA
M3
kABN7.6·10–13
kBCN8.0·10–13
kCDN7.5·10–13 0.222
Nxm1.62·1012 0.15 0.15 0.15 1.5 0.1428
5bkABX0.55
kBCX0.7
kCDX0.6 0.258
gxm
1.65 0.15 0.15 0.15 1.5 0.1428kd
0.0745Kag
0.010
6kABN
5.0·10–13kBCN
1.2·10–11kCDN
1.23·10–13 0.045Nxm
9.8·1010 0.30 0.30 0.50 1.5 0.14287 7.5·10–13 2.5·10–13
2.5·10–13 0.23 5.0·1013 0.14 0.14 0.14 1.0 0.14288 1.6·10–13
1.2·10–13 1.2·10–13 0.0425 5.0·1013 0.25 0.35 0.35 0.5 0.1428
9a
MA
M1 kAB
0.1kBC0.45
kCD0.44
9b 0.0235 0.15 0.1
10a
MA
M4
kABN1.6·10–13
kBCN1.2·10–13
kCDN1.2·10–13
mm
0.0425
Nxm5.0·1013
KA0.25
KB0.35
KC0.35
ksol0.5
cAm0.1428
Nx01.8·1011
cAim00.20
kdim0.01
kim0.1
cAns01.374
10b 1.75·10–13 1.4·10–13 1.2·10–13 0.0425 5.0·1013 0.25 0.35
0.35 0.5 0.1428 1.8·1011 0.00 0.1 0.1 1.57410c 1.6·10–13 1.2·10–13
1.2·10–13 0.0425 5.0·1013 0.25 0.35 0.35 0.5 0.1428 1.8·109 1.00
0.01 0.1 0.57410d 7.5·10–13 2.5·10–13 2.5·10–13 0.23 5.0·1013 0.14
0.14 0.14 1.0 0.1428 1.8·1011 0.00 0.1 0.1 1.574
-
obtained results are in agreement with findings of Parket al.
(4), who found that atrazine quantities sorbed bydifferent sorbents
were proportional to aqueous atrazineconcentrations. The fact that
the majority of other pes-ticides actually being in use in
agriculture are of lowersolubility in water than atrazine implies
that MAM3 couldalso be applicable in describing their
biodegradation ki-
397M. BO[NJAK et al.: Mathematical Modelling of Atrazine
Degradation, Food Technol. Biotechnol. 48 (3) 392–403 (2010)
t(degradation)/h
c(s
ub
sta
nce
)/(m
mo
l/L
)
g(m
icro
bia
lb
iom
ass)/
(g/L
)
vs.
vs.
vs.
vs.
vs.
vs.
vs.
Fig. 2. Application of mathematical model MAM2 in
describingatrazine biodegradation kinetics, and fitting computer
simula-tion to experimental data as those presented in Fig. 1b.
Parame-ter values are shown in Table 1. Points: experimental
data,curves: simulated data
t(degradation)/h
c(s
ub
sta
nce
)/(m
mo
l/L
)
g(m
icro
bia
lb
iom
ass)/
(g/L
)
c(A
,B,C
)/(m
mo
l/L)
vs.
vs.
vs.
vs.
vs.
vs.
vs.
vs.
vs.
vs.
vs.
Fig. 3. Application of mathematical model MAM3 in
describingatrazine biodegradation kinetics, and fitting computer
simula-tion to experimental data by applying parameters shown in
Ta-ble 1. Points: experimental data, curves: simulated data
t(degradation)/h
c( s
ub
sta
nce
)/( m
mo
l/L
)
g(m
icro
bia
lb
iom
ass)/
(g/L
)
c(A
,B,C
)/(m
mo
l/L)
t(degradation)/h
c( s
ub
sta
nce
)/( m
mo
l/L
)
g(m
icro
bia
lb
iom
ass)/
(g/L
)
c(A
,B,C
) /(m
mo
l/L)
t(degradation)/h
c(s
ub
sta
nce
)/( m
mo
l/L
)
g(m
icro
bia
lb
iom
ass)/
(g/L
)
c(A
,B,C
) /(m
mo
l/L)
vs.
vs.
vs.
vs.
vs.
vs.
vs.
vs.
vs.
vs.
Fig. 4. Application of mathematical models MAM3 (a), MAM3a(b)
and MAM3b (c) for description of atrazine biodegradationkinetics,
and agreement of computer simulation with experimen-tal data for
simulation parameters indicated in Table 1. Points:experimental
data, curves: simulated data
a)
b)
c)
-
netics. One of the possibilities is the transformation ofMAM3 by
modifying Eqs. 10–12 and by adding theequation expressing the
kinetics of atrazine sorption anddesorption. The new model, MAM4,
can also be testedfor its applicability with reference to
experimental dataof this work. Chosen examples of computer
simulationsare shown in Fig. 10, which consists of four
diagrams.
Presented data explain well the relevance of
sorption-de-sorption phenomena and the applied inoculum
quantitiesfor atrazine biodegradation efficiency. Data also
suggest
398 M. BO[NJAK et al.: Mathematical Modelling of Atrazine
Degradation, Food Technol. Biotechnol. 48 (3) 392–403 (2010)
t(degradation)/h
g(m
icro
bia
lb
iom
ass)/
(g/L
)
c(s
ub
sta
nce
)/(m
mo
l/L
)
vs.
vs.
vs.vs.
vs.
vs.
vs.
vs.
vs.
vs.
Fig. 5. Application of mathematical model MAM3 in
describingatrazine biodegradation kinetics, and agreement of
computersimulation with experimental data for simulation
parameters(a,b) specified in Table 1. Points: experimental data;
curves:simulated data, applied biodegradation temperature
T/K=303
t(degradation)/h
c(s
ub
sta
nce
)/(m
mo
l/L
)
Mic
rob
ialb
iom
ass/ (
ce
lls/L
10
)´
12
vs.
vs.
vs.
vs.vs.
vs.
vs.
vs.
vs.
vs.
t(degradation)/h
c(s
ub
sta
nce
)/(m
mo
l/L
)
Mic
rob
ialb
iom
ass/(
ce
lls/L
10
)´
9
vs.
vs.
vs.
vs.vs.
vs.
vs.
vs.
vs.
vs.
Fig. 6. Application of mathematical model MAM3 in
describingatrazine biodegradation kinetics, and agreement of
computersimulation with experimental data for simulation
parametersspecified in Table 1. Points: experimental data, curves:
simu-lated data; applied biodegradation temperature T/K=283
t(degradation)/h
c(s
ub
sta
nce
)/(m
mo
l/L
)
Mic
rob
ialb
iom
ass/(
ce
lls/L
10
)´
12
vs.
vs.
vs.
vs.
vs.
vs.
vs.
vs. vs.
Fig. 7. Application of mathematical model MAM3 for descrip-tion
of atrazine removal from the wet soil contaminated withatrazine,
and fitting computer simulation to experimental datafor simulation
parameters specified in Table 1. Points: experi-mental data,
curves: simulated data; applied biodegradationtemperature
T/K=303
a)
b)
-
that the decision whether to apply smaller or larger ino-culums
depends on whether the soil needs to be pro-tected for selected
agriculture or it needs to be efficientlydetoxified. Also, soil
properties (amounts and propertiesof sorbents in the soil reflect
on kdim and kim values) couldmarkedly influence kinetics of
atrazine removal from thecontaminated soil. This observation is in
accordance withliterature data (4,10–14).
The extended mathematical model MAM3 couldalso be improved by
modifying Eqs. 4 and 9. One of the
possibilities is the addition of the term which
expresseskinetics of viable biomass decay (Eq. 8, Fig. 5). There
arealso other possibilities of improving the fitting of com-puter
simulation to the experimental data. Models asthose adapted for
cluster analysis application are re-commended (23). Also, more
sophisticated approachesbased on the structure of mixed microbial
culture andthe roles of individual members of the same culture
canbe suggested for consideration (18–20) prior to definingthe
improved models.
399M. BO[NJAK et al.: Mathematical Modelling of Atrazine
Degradation, Food Technol. Biotechnol. 48 (3) 392–403 (2010)
t(degradation)/h
c(s
ub
sta
nce
)/(m
mo
l/L
)
Mic
rob
ialb
iom
ass/ (
ce
lls/L
10
)´
12
vs.
vs.
vs.
vs.
vs.
vs.
vs.
vs. vs.
Fig. 8. Application of mathematical model MAM3 for descrip-tion
of atrazine removal from the soil contaminated with atra-zine, and
fitting computer simulation to experimental data forsimulation
parameters specified in Table 1. Points: experimentaldata, curves:
simulated data; applied biodegradation tempera-ture T/K=283
Table 2. Agreement of computer simulation with experimentaldata
evaluated through the application of Jacobian matrix. Cal-culated
values for data sets refer to the presented figures
Data setreferringto figure:
CorrelationDetermination
coefficientModel selection
criterion
1a 0.992911110 0.983743322 3.73829909
1b 0.995879887 0.990581313 3.99839293
2 0.991665879 0.977348241 1.95418443
3 0.999926563 0.999831791 6.69030284
4a 0.999998851 0.999996436 9.79457653
4b 0.999997376 0.999993221 9.15166817
4c 0.999999053 0.999997212 10.0402509
5a 1.0000 0.999682868 7.17619325
5b 0.976993968 0.953076535 2.09627444
6 1.0000 1.0000 44.8208048
7 0.999995944 0.984289663 1.95343639
8 0.989976731 0.966118367 1.81345363
9a 0.995577867 0.988252516 3.44411622
9b 0.990292771 0.978580451 3.27202271
10a 0.974996018 0.898684957 0.432377519
10b 0.991432169 0.97266063 1.74248338
10d 0.999995944 0.984289663 1.55343639
t(degradation)/h
c(s
ub
sta
nce
)/(m
mo
l/L
)
vs.
vs.
vs.
vs.
vs.
t(degradation)/h
c(s
ub
sta
nce
)/(m
mo
l /L
)
vs.
vs.
vs.
vs.
vs.
b)a)
Fig. 9. Application of the simple mathematical model MAM1 for
description of atrazine removal from the soil contaminated
withatrazine, and fitting computer simulation to experimental data
for applied simulation parameters specified in Table 1. Points:
experi-mental data referring to Fig. 7 (a) and Fig. 8 (b), curves:
simulated data
-
Based on the comparison of the data presented inFigs. 5–10, it
can be concluded that biodegradation tem-perature had a strong
influence on biodegradation ki-netics. The results are in
accordance with theoreticalexpectation. Comparing Figs. 5 and 6, it
can be observedthat kinetics of atrazine biodegradation was roughly
4 to5 times faster at 30 than at 10 °C. Faster biodegradationrate
was mainly a consequence of much faster biomassgrowth at 30 °C
(about 5 times) than at 10 °C. Since inthe later process phases the
kinetics of particular eventswas not analogous, it can be concluded
that the mixedmicrobial cultures differed in their physiology.
Probablecause of such differences could be inadequate estima-tions
of growth kinetics and/or culture differences withrespect to the
participation of particular microbial popu-lation members present
in the mixed microbial culturesat later process phases. The
differences between the twomicrobial cultures were much less
expressed when theywere applied as inoculums for contaminated soil
(Figs. 7and 8). Therefore, the mean rates of atrazine removalfrom
contaminated soil differed, as expected, more thanfour times (Fig.
9). As it is already known (24), the ap-plication of different
cultivation temperatures can leadto mixed microbial cultures with
different properties,because the temperature affects differently
particularmembers of a mixed microbial culture.
Special advantage of the present work is the use ofmixed
microbial culture capable of degrading atrazineeven when it is
present in the media with relatively highatrazine concentrations
(5). Data shown in Figs. 3 and 4testify that much higher
biodegradation efficiency re-sulted in cases when higher atrazine
quantities werepresent. Further series of experiments should be
de-signed with an aim to give more data on particularprocesses
(especially those referring to microbial bio-mass concentration and
structure, oxygen availabilityand mass transfer) in order to
confirm the applicabilityof mathematical models used in this work.
Reliable pre-dictions of the consequences of different atrazine
appli-cations in protecting different types of soil could be
ex-pected as desired benefit. Therefore, there is need formore
detailed studies of atrazine biodegradation kine-tics in
contaminated soil. The experimental data of this
work enable estimation of some parameters which couldserve as a
tool for useful predictions.
Based on the data shown in Figs. 5–9, activationenergy for
particular biodegradation can be calculated.Also, relationships
relevant for the prediction of conse-quences of different
applications referring to mixed mi-crobial cultures and atrazine
quantities can be found. Theestimated values of activation energies
referring to dif-ferent events are presented in Table 3, which
gives evi-dence that the highest values of activation energy
referto microbial biomass growth. When the sums of activ-ation
energies referring to degradations of atrazine, hydro-xyatrazine
and cyanuric acid are compared, differencescan be observed. These
are a consequence of the appli-cation of different mathematical
models as a basis foractivation energy calculations. This
observation, togeth-er with the finding that the highest activation
energy re-fers to the mixed microbial biomass growth, suggeststhat
the biocatalytic activity of microbial biomass is themost decisive
factor for atrazine biodegradation effici-ency.
When the effects of temperature on atrazine biode-gradation
rates are compared, the half-life times couldbe considered as good
criterion. Computer simulationdata demonstrated in Fig. 9 appear to
be convenient forhalf-life time calculations. The estimated values
(t1/2)30=6.93 h and (t1/2)10=29.50 h indicate that atrazine
biode-gradation at 30 °C was 4.26 times more efficient thanthat at
10 °C. Such a ratio between half-life times sug-gests that
estimations of activation energies and half-lifetimes refer to the
suboptimal temperature range of bio-degradation process
conditions.
On the basis of previous discussion, the applied cul-ture media
and cultivation conditions can be validated.Cultivation media
influenced biodegradation kineticsdepending on carbon and nitrogen
sources (Fig. 1), asexpected on the basis of the already published
data (25).Applied biodegradation conditions possibly differed
withrespect to oxygen availability and mass transfer
rates(especially if comparing shaken liquid cultures withinoculated
soil), and such differences reflected on thebiodegradation
kinetics. The applied values of computer
400 M. BO[NJAK et al.: Mathematical Modelling of Atrazine
Degradation, Food Technol. Biotechnol. 48 (3) 392–403 (2010)
Table 3. Activation energies of atrazine biodegradation present
in the contaminated soil. Calculation based on computer
simulationdata fitted well to the experimental data of atrazine
biodegradation at 10 and 30 °C
Reaction systemTemperaturerange
Activationenergy/
(103 J/mol)Referent kinetic parameter Fig.
Soil samples contaminated withatrazine, and inoculated withmixed
microbial culture
283 to 303 K 60.260 specific growth rate, m 7 and 8
55.133 atrazine (A) degr. rate const., kAB
26.193 hydroxyatrazine (B) degr. rate const., kBC
26.193 cyanuric acid (C) degr. rate const., kCD
51.675 atrazine (A) degr. rate const., kAB 9
39.206 hydroxyatrazine (B) degr. rate const., kBC
52.874 cyanuric acid (C) degr. rate const., kCD
Microbial culture in liquidmedium containing atrazine 283 to 303
K 56.957 specific growth rate, m 5 and 6
-
simulation parameters resulted as a consequence of boththe
applied mathematical models and experimentalconditions.
Finally, the results of this work and the applied ap-proach in
explaining them are in accordance with find-ings of Soulas (1) and
Soulas and Lagacherie (3).
Conclusion
General conclusion which can be drawn from theresults of
mathematical models presented in this workcombined with the applied
methods of evaluation ofexperimental and computer simulation data
suggests
401M. BO[NJAK et al.: Mathematical Modelling of Atrazine
Degradation, Food Technol. Biotechnol. 48 (3) 392–403 (2010)
t(degradation)/h
c(s
ub
sta
nce
)/(m
mo
l/L
)
Mic
rob
ialb
iom
ass/ (
cells
/L1
0)
´1
2
c(A
,B,C
)/(m
mol/L)
vs.
vs.
vs.vs.
vs.
vs.
vs.
vs.
vs.
t(degradation)/h
c(s
ub
sta
nce
)/(m
mo
l /L
)
Mic
rob
ialb
iom
ass/ (
ce
lls/L
10
)´
12
c(A
,B,C
)/(m
mol/L)
vs.
vs.
vs.vs.
vs.
vs.
vs.
vs.
vs.
t(degradation)/h
c(s
ub
sta
nce
)/(m
mo
l/L
)
Mic
rob
ialb
iom
ass/ (
ce
lls/L
10
)´
12
c(A
,B,C
)/(m
mol/L)
vs.
vs.
vs.vs.
vs.
vs.
vs.
vs.
vs.
t(degradation)/h
c(s
ub
sta
nce
)/(m
mo
l/L
)
Mic
rob
ialb
iom
ass/ (
ce
lls/L
10
)´
12
c(A
,B,C
)/(m
mo
l/L
)
vs.
vs.
vs.
vs.
vs.
vs.
vs.
vs.
vs.
vs.
Fig. 10. Application of extended mathematical model MAM4 for
description of atrazine removal from wet soil as a function of
ap-plied microbial culture inoculums quantity and soil properties
referring to atrazine sorption-desorption phenomena. Fitting of
com-puter simulation to experimental data for applied simulation
parameters specified in Table 1. Points: experimental data
referring toFig. 8 (a,b,c) and Fig. 7 (d), curves: simulated
data
c)
b)a)
d)
-
that the integrated approach to methematical modellingof
biodegradation kinetics of atrazine and other pesti-cides can be
recommended for application in furtherstudies.
Symbols
A atrazineB hydroxyatrazineC cyanuric acidD degradation products
CO2 and NH4
+
cA, cB, cC/(mmol/L)concentrations of reaction substancesA, B and
C
cAim/(mmol/L) concentration of immobilised(sorbed) substance A
(atrazine)
cAim0/(mmol/L) initial cAimcAm/(mmol/L) substance A (atrazine)
solubilitycAins/(mmol/L) concentration of undissolved
substance A (atrazine)cAns/(mmol/L) net concentration of
undissolved
substance A (atrazine)cAns0/(mmol/L) initial cAnscAT/(mmol/L)
total substance A (atrazine)
concentrationd mathematical derivation operationEa/(J/mol)
activation energykAB, kBC, kCD/h
–1 reaction rate constants with referenceto the conversion of A
into B, B intoC, and C into D
kABN, kBCN,kCDN/(mmol/h) reaction rate constants with
reference
to concentration of microbial cellsand conversions of A into B,
B intoC, and C into D
kABX, kBCX,kCDX/(mmol/(g·h))
reaction rate constants with referenceto concentration of
microbial biomassand conversion of A into B, B intoC, and C into
D
kAB*, kBC*,kCD*/(L/(g·h)) reaction rate constants with
reference
to the conversions of A into B, B intoC, and C into D catalysed
withconstant biomass concentration
kd/h–1 specific biomass decay rate
kdim/h–1 rate constant with reference to
immobilised atrazine desorptionkim/h
–1 rate constant with reference toimmobilised atrazine
sorption
ksol/(L/(mmol·h)) specific rate of undissolvedsubstance A
(atrazine) dissolving
KA, KB, KC/(mmol/L)biocatalytic constants referring tosubstances
A, B and C
Kag/(mmol/L) biocatalytic constant of biomassgrowth with
reference to substanceA
Nx/L–1 concentration of mixed microbial
population cellsNx0/L
–1 initial NxNxm/L
–1 theoretically maximal concentrationof mixed microbial
population cells
T/K absolute temperature
Greek letters
gx /(g/L) microbial biomass concentration
gxm/(g/L) theoretical maximum of microbialbiomass
concentration
mNm/h–1 maximal specific rate of microbial
population cell number increase
mxm/h–1 maximal specific rate of microbial
biomass growth
Abbreviated computer simulation symbolsA=atrazine, AIM=sorbed
(immobilized) atrazine, AINSand ANS=undissolved atrazine, AT=total
atrazine, B=hydroxyatrazine, C=cyanuric acid, X=microbial
biomass
AcknowledgementsThis work was funded by the Croatian Ministry
of
Science, Education and Sports and by the Croatian In-stitute of
Technology. The authors are indebted to MajaHavriluk for her
valuable engagement in the enrichmentof mixed bacterial cultures as
well as for determinationand evaluation of atrazine-degrading
activitities underdifferent environmental conditions.
References
1. G. Soulas: Modeling of Biodegradation of Pesticides in
theSoil. In: Soil Ecotoxicology, J. Tarradellas, G. Bitton, D.
Ros-sel (Eds.), CRC Press, Boca Raton, FL, USA (1997) pp.
117–138.
2. G.D. Hill, J.W. McGahen, H.M. Baker, D.W. Finnerty,
C.W.Bingeman, The fate of substituted urea herbicides in
agri-cultural soils, Agron. J. 47 (1955) 93–104.
3. G. Soulas, B. Lagacherie, Modelling of microbial degrada-tion
of pesticides in soils, Biol. Fertil. Soils, 33 (2001) 551–557.
4. J.H. Park, Y. Feng, P. Ji, T.C. Voice, S.A. Boyd,
Assessmentof bioavailability of soil-sorbed atrazine, Appl.
Environ. Mi-crobiol. 69 (2003) 3288–3298.
5. D. Hr{ak, M. Havriluk, Mixed bacterial culture for atra-zine
degradation. US patent 0026135 (2009).
6. N. Udikovi} Koli}, D. Hr{ak, A. Begonja Kolar, I. Petri},
S.Stipi~evi}, G. Soulas, F. Martin-Laurent, Combined meta-bolic
activity within an atrazine-mineralizing communityenriched from
agrochemical factory soil, Int. Biodeter. Bio-degr. 60 (2007)
299–307.
7. N. Udikovi}, D. Hr{ak, G. Menda{, D. Filip~i}, Enrichmentand
characterization of atrazine degrading bacterial com-munities, Food
Technol. Biotechnol. 41 (2003) 211–217.
8. N. Udikovi} Koli}, F. Martin-Laurent, M. Devers, I. Petri},A.
Begonja Kolar, D. Hr{ak, Genetic potential, diversity andactivity
of an atrazine-degrading community enriched froma herbicide factory
effluent, J. Appl. Microbiol. 105 (2008)1334–1343.
9. N. Udikovi}-Koli}, D. Hr{ak, M. Devers, V. Klepac-Ceraj,I.
Petri}, F. Martin-Laurent, Taxonomic and functional di-versity of
atrazine-degrading bacterial communities en-
402 M. BO[NJAK et al.: Mathematical Modelling of Atrazine
Degradation, Food Technol. Biotechnol. 48 (3) 392–403 (2010)
-
riched from agrochemical factory soil, J. Appl. Microbiol.109
(2010) 355–367.
10. I.S. Fomsgaard, The influence of sorption on the
degrada-tion of pesticides and other chemicals in soil,
Environmen-tal project No. 902, Danish Environmental Protection
Agen-cy, Denmark (2004).
11. E. Barriuso, W. Koskinen, B. Sorenson, Modification of
atra-zine desorption during field incubation experiments, Sci.Total
Environ. 123/124 (1992) 333–344.
12. L. Guo, W.A. Jury, R.J. Wagenet, M. Flury, Dependence
ofpesticide degradation on sorption: Nonequilibrium modeland
application to soil reactors, J. Contam. Hydrol. 43
(2000)45–62.
13. L. Ma, H.M. Selim, Predicting atrazine
adsorption-desorp-tion in soils: A modified second order kinetic
model, WaterResour. Res. 30 (1994) 447–456.
14. R.D. Wauchope, R.S. Myers, Adsorption-desorption kine-tics
of atrazine and linuron in freshwater-sediment aqueo-us slurries,
J. Environ. Qual. 14 (1985) 132–136.
15. M. Wenk, T. Baumgartner, J. Dobov{ek, T. Fuchs, J. Kucse-ra,
J. Zopfi, G. Stucki, Rapid atrazine mineralisation in soilslurry
and moist soil by inoculation of an atrazine-degra-ding Pseudomonas
sp. strain, Appl. Microbiol. Biotechnol. 49(1998) 624–630.
16. S.J. Goux, M. Ibanez, M. Van Hoorick, P. Debongnie,
S.N.Aghatos, L. Pussemier, Biodegradation of atrazine in
sandsediments and in a sand-filter, Appl. Microbiol. Biotechnol.54
(2000) 589–596.
17. M. Bo{njak, A. Bago Joksovi}, J. Pigac, @. Bo{njak Cihlar,D.
Hranueli, Applicability of mathematical models in defi-ning the
behaviour kinetics distinction among microbialstrains, Chem.
Biochem. Eng. Q. 20 (2006) 375–388.
18. M. Bo{njak, S. Bogdan, Introduction to the theoretical
fun-damentals for increasing the probability of
heterologouscontacts between biological particles, Food Technol.
Biotech-nol. 39 (2001) 183–196.
19. M. Bo{njak, S. Bogdan, Mathematical modelling of
processrelationships in the mixed microbial culture with
probablenew product formation, Annual Croat. Acad. Eng. 2002
(2002)15–21.
20. M. Bo{njak: Introduction to the Kinetics of Microbial
Proces-ses, Graphis, Zagreb, Croatia (2009) (in Croatian).
21. K. Mihaljevi}, M. Bo{njak, Defining of the kinetics of
mi-crobial oxidation process events with reference to L-sorbo-se
formation in a large range of culture conditions, Chem.Biochem.
Eng. Q. 23 (2009) 239–250.
22. R.T. Mandelbaum, L.P. Wackett, D.L. Allan, Mineralizationof
the s-triazine ring of atrazine by stable bacterial mixedcultures,
Appl. Environ. Microbiol. 59 (1993) 1695–1701.
23. V. Topolovec, M. Bo{njak, Cluster analysis and simulationof
microbial population differentiation during the processof
antibiotic biosynthesis Proceedings of the 6th
InternationalSymposium 'Computers at the University', Dubrovnik,
1984,627.1–627.9, SRCE, Zagreb, Croatia (1984) (in Croatian).
24. D. Hr{ak, M. Bo{njak, V. Johanides, Temperature effects
ongrowth kinetics of linear alkyl-benzene sulphonate (LAS)degrading
bacteria in continuous culture, Proceedings of theWorld Surfactant
Congress, Munich, Germany (1984) pp. 249–261.
25. R. Abdelhafid, S. Houot, E. Barriuoso, Dependence of
atra-zine degradation on C and N availability in adapted
andnon-adapted soils, Soil Biol. Biochem. 32 (2000) 389–401.
403M. BO[NJAK et al.: Mathematical Modelling of Atrazine
Degradation, Food Technol. Biotechnol. 48 (3) 392–403 (2010)