integrated approach to mathematical modelling of atrazine … · 2010. 9. 17. · issn 1330-9862...

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


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-


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


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-


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


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.


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


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


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.

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