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Moscow University
Chemistry Bulletin Vol. 57, No.6, pp.79-83, 2002

Vestnik Moskovsko90
Universiteta. Khimiya UDC 579.088;577.158.54

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STUDY OF A PHENOMENOLOGICAL MODEL OF THE KINETICS OF BACTERIAL ADSORPTION ON LOW-ENERGY SURFACES
V. V. Fedorovich, S. V. Kalyuzhnyi, P. van der Meeren., and W. Verstraete..

The mathematical model of the kinetics of bacterial adsorption on polymeric materials with different hydrophobicities proposed earlier was analyzed. The calibration procedure and the algorithm of calculations for the simulation model were described. The model was used for describing the experimental data on the adsorption of E. coli (strain 055) and Listeria monocytogenes bacteria on various polymeric surfaces. The theoretical forecast of the experimental results indicated that the accuracy of the forecast decreased with the lowering of the wetting angle on the surface of the polymer.

In our previous paper [1] we obtained the basic differential equations of the model for the kinetics of bacterial adsorption. The main problem solvedin the first part was the issue of resolution of the motive forces of adsorption in relation to the nature of their origin. The present work concerns the calibration procedure and the theoretical predictions based on this model. In this connection another important problem arises: the problem of comparing the results of different adsorption experiments based on the so-called adhesion protocol covering the experimental conditions of obtaining the data in question. Unfortunately, different authors use dissimilar adhesion protocols. The model developedearlier [1] removesthis uncertainty to some extent by employing a specially introduced parameter. CALIBRATION Experimental OF THE MODEL of the Model

Data for the Calibration

The experimental data used in the calibration of the model were taken from an earlier publication [2]. In this experiment bacteria Escherichia coli (strain 055) were grown on agar within 24 h at 37°C. Listeria monocytogenes microorganisms were cultured on a soy-containing medium within 48 h at 37°C. Further on, the bacteria were precipitated, triply washedin Hank's saline solution by repeated centrifuging, and finally reprecipitated in a liquid medium with specified surface tension (the latter value was varied by changing the concentration of DMSO in Hank's saline solution [2]). After that, 1 ml of a bacterial suspension containing 10s bacteria in a corresponding medium was applied to the surfaces. The bacteria were incubated within 30 min at T = 21°C;subsequently, surfaces the were washedwith Hank's medium to remove the bacteria that were not adhered to the surface. After drying the samples,we counted the number of cells adsorbed on the surface. Information on the wetting angles of bacterial and adsorbing surfaces used in the experiments is presented in Tables 1 and 2.

.. Laboratory of Microbial Ecology, Department of Applied Biological Sciencesand Agriculture, Ghent
University, Ghent B-90000, Belgium. @ 2003 by Allerton Press, Inc.
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Interphase TechnologiesTeam.

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Mo.cow Univer.ity Cherni.try Bulletin

Vol. 57, No.6

Table Experimentally

1

Determined Values of the Wetting Angles for Water on Polymers ('YLv 72.0 mN/m)

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Material Fluorinated copolymer of ethylene and propylene Polystyrene Low density polyethylene Acetal resin Sulfonated polystyrene

Polymer Wetting angle of water (O) 110 :f: 3 95 :f: 2 84 :f: 4 64 :f: 1 24 :f: 3

Surface tension 'YsV (mN/m) 16.7:f: 1.7 25.6 :f: 1.2 32.5 :f: 2.5 44.8 :f: 0.6 66.4:f: 1.3
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Note. The values of the surface tensions were calculated using the equation of state.

Table Experimentally

2

Determined Values of the Wetting Angles on Bacteria in Hank's Saline Solution ('YLv = 72.8 mN/m) Microorganisms Wetting angle of the saline solution (O) 16.7:f: 1.0 26.1:f: 1.2

Strain E. coli 055

Surface tension ")'BV (mN/m) 69.9:f: 0.3 66.3:f: 0.5

L. monocytogenes

Note. The values of the surface tensions were calculated using the equation of state.

THE

METHOD

OF CALCULATION

The calculations were run on an IBM PC (Pentium 200 processor). The computer program was written in Fortran-90. The input parameters were: the value of the wetting angle of the liquid on the surface of the bacterial cell, the value of the wetting angle of the medium on the polymer surface, the surface tension of the medium, the density of the cells in the medium, and the average projected area of the bacterial cell. The output parameter was the number of cells adsorbed on the unit surface within a specified time interval. The trapezoid method [3] was used for the numerical integration of differential equations. The calculations were run in accordance with the algorithm stated in items IA -ID: IA: calculation of 'Ysv based on the experimental values of 'YLV, cos 0SL (Table 1) using Eqs. (5) and (6); IB: calculation of 'YBV based on the experimental values of 'YLV, cos 0BL (Table 1) using Eqs. (7) and (8); Ic: calculation of ")'Bv based on the above-calculated values of ")'BV and ")'BV u~ing a similar set of equations inferred from the Young equation and the equation of state; ID1: input of the current value of ")'Bv; ID2: calculation of 'YLVcos 0SL using 'YLVand 'YLV; ID3: calculation of 'YLv cos 0SL using 'YLVand 'YLv; ID4: calculation of AGh using 'YLV,'YLV,and 'YLv; IDs: numerical integration of differential Eqs. (1) and (3); ID6: return to Step ID1. When analyzing different media for the specified solid surface-bacterial type combination, items from IA to Ic are executed once whereas items from IDl to ID6 are iterated for each value of the surface tension of the medium in question.

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Mo.cow Univer.ity Chemi.try Bulletin

Vol. 57, No.6

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IDENTIFICATION
OF THE PARAMETERS OF THE MODEL

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The model was calibrated on a portion of the data obtained in the foregoing experiment. Used in this experiments were bacteria with th~etting angle not exceeding30° and in concentration Next = 1014l/m3. The 4Jl, 4J2,and 'If values in Eqs. (1) and (2) [1] were identified for L. monocytogenesand a fluorinated ethylene-propylene copolymer (Fig. la). The identification procedure based on the least squaresmethod gave in this case the following values: 4Jl 6 X 10-s (m/s), 4J2 0.205(m/mN), 'lfmonocyt 1.05. The = = accuracy of identification was above 95%. For the E. coli culture and for the same solid material the only parameter to be identified was 'IfE. coli. In the ~Gh > 0 area this parameter was equal to 0.56. For this case the numerical values of 4Jl and 4J2 were taken to be equal to the values obtained for the L. monocytogenes culture. Here we made use of the fact that according to our approach the values of wetting anglesand surface tensions determine unequivocally the rate of adsorption causedby hydrophobic interactions, regardlessof

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the nature of the interacting surfaces. Figure la presents the results of the identification procedureand the corresponding experimental data. It should be noted that the mechanismgoverning the density of bacteria on the surface is of a stochastic nature. Hence, the area of projection of the bacterial cell (SBAC) is a random value. In our calculations we assumedthat the value of this parameter is 1.44x 10-12 m2. This value correlates with the typical geometrical dimensionsof bacteria. In order to simplify the comparison of our results with the experimental data, we present the N(t) function as the number of bacteria per 104 mm2 in all graphs. RESULTS AND DISCUSSION In order to utilize the developedmodel, it is needed to understand the structure of the 'lfmonocyt and lItE. coli parameters becauseno theoretical assumptions about them were made above. These functions were used as constants in the calculation procedure. An analysis of experimental data will make it possible to draw certain inferences. The experimental data characterizing the adsorption of E. coli (Fig. la) contain the region of the surface tension of the liquid (64-69 mN/m) where ~Gh > O. Inside this regionthe value of ~Gh changesconsiderably. Nevertheless,the number of adsorbed cells per unit area inside this interval does not depend on the value of the surface tension of the liquid. If the repulsion within this region has existed, the number of adsorbed cells should have decreased. However, this suggestion is not confirmed by the experiment. This fact was generalizedand used for the determination of the lItmonocyt and lItE. coli valueswithin the ~Gh > 0 region. It is also seenfrom the experimental results that the number of bacteria adsorbed on unit area within the region of lower values of the surface tension of the liquid does not very strongly depend on the type of the adsorbing material. Based on this fact, we assumed that these results are determined by the type of bacteria. Taking the foregoing assumptionsinto account, we used the model for describing the remaining portion of the experimental data. The determination of the 4Jl, 4J2, IItmonocyt, lItE. coli constants in the calibration and procedure for one type of material allowed us to predict all remaining types of materials using the same constants. Figure Ib-e displays the theoretical and experimental curves. It can be seen from Fig. 1 that the accuracy of the prediction of the experimental results decreases with a decline in the wetting angle of the polymer surface. Nevertheless,the theoretical prediction gave acceptable results for the number of bacteria adsorbed on unit area for materials with the wetting angle not exceeding60°. The curves in Fig. Ie demonstrate the lack of coincidence between the model and the experiment. Therefore, in the situation where the adhesion of hydrophilic microorganisms to hydrophilic surfacesis considered, our approach does not give the required degree of accuracy. This can be explained by the inapplicability of the equation of state to hydrophilic surfaces. Figure 2a shows the theoretical prediction of the adsorption kinetics for L. monocytogenes. The calculations were performed for severaltypes of the adsorbing materials at the constant value of the surface tension of the liquid (72 mN/m) on the assumption that the wetting angle may be stable for a long period of time. All curves in Fig. 2a tend to the limiting value representing the situation of a complete filling of the surface. The curves in Fig. 2b show the dynamics of the effective wetting angle of the polymer surface in time expressedby Eq. (9). The curves in Fig. 2b, as well as those in Fig. 2a, tend to the limiting values, which are the wetting angle of the surfaceof the bacterial wall. It should be noted that the existenceof this limiting value is the consequence bacterial repulsion, which is postulated in this work. of [
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Fig. 1 Number of bacteria adsorbed on unit area within 30 min as a function of the surface tension of the liquid (the concentration of cells in the surroundings was 108 ml-l); a: fluorinated copolymer of ethylene and propylene; b: polystyrene; c: low-density polyethylene; d: acetal resin; e: 8ulfonated polystyrene. 1: L. monocytogenes, experiment; p-:L. monocytogenes, the the model; 3: E. coli 055, the experiment; 4: E. coli 055, the model. Based on the results of our study, the following conclusions can be made. The model developed is capable of describing the rate of bacterial adsorption for bacteria with the wetting angle not exceeding300 and for surfaceswith the wetting angle not lower than 600with an acceptable accuracy. The accuracy of the prediction of the number of bacteria adsorbed on unit area per unit time decreases with the lowering of the wetting angle of the adsorbing surface. The existenceof repulsion between bacteria gives rise to the limiting value of the number of the bacteria adsorbed on unit area. The first term in Eq. (1) characterizing the hydrophobic part of the kinetics is similar to the Arrhenius law for the kinetics of the first-order chemical reaction. 82

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ADDENDUM
Notation

III is the value of the contribution of nonhydrophobic interactions to the adsorption process (dimensionless),BLis the bacterial cell-liquid interface, BS is the bacterial cell-polymer interface, BV is the bacterial cell-water vapor interface, AGh is the hydrophobic portion of the free energy of adhesion (mN/m), N(t) is the number of bacteria adsorbed on unit area (11m2), Next is the number of bacteria in the surroundings (11m3), SL is the polymer-liquid interface, SV is the polymer-water vapor interface, saccess is the surface area accessiblefor bacteria (m2), stat is the area of the material participating in the adhesion process (m2), sbac is the projection area of the bacterial cell (m2), 'YLv is the surface tension of the
liquid (mN/m)

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")'BL

is the interphase tension in the bacterial

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system (mN/m),

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interphase tension in the polymer-liquid system (mN/m), 'YBSis the interphase tension in the bacterial cell-polymer system (mN/m), 'YSVis the interphase tension in the polymer-water vapor system (mN/m), ")'Bvis the interphase tension in the bacterial cell-water vapor system (mN/m), 1/>1 a constant (m/s), 1/>2 is is a constant (m/mN), eBL is the wetting angle of the liquid on the surface of the bacterial cell (O), eSL is the wetting angle of the liquid on the surface of the polymer (O),eBs is the effective wetting angle of the liquid on the surface of the polymer partly occupied by the bacteria (O).

REFERENCES

*

1. V. V. Fedorovich,S.V. Kalyuzhnyi, P. van der Meeren, and W. Verstraete, Vestn. Mosk. Univ. Khimiya, vol. 43, p. 416, 2002.

2. D.R. Absolom,F.V. Lamberti, Z. Policova, Zingg, et al., Appl. Environ. Microbiol.,vol. 46, p. 90, W.
1983. 3. W.E. Miln, Numerical Solution of Differential Equations, John Wiley, 1953. 25 October 2002 Chair of Chemical Enzymology E-mail: vfedorovich(!enz.chem.msu.ru

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