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Applied and Environmental Microbiology, March 2007, p. 1742-1752, Vol. 73, No. 6
0099-2240/07/$08.00+0     doi:10.1128/AEM.01521-06
Copyright © 2007, American Society for Microbiology. All Rights Reserved.

Improved Experimental and Computational Methodology for Determining the Kinetic Equation and the Extant Kinetic Constants of Fe(II) Oxidation by Acidithiobacillus ferrooxidans{triangledown}

Sharon Molchanov, Yuri Gendel, Ilya Ioslvich, and Ori Lahav*

Faculty of Civil and Environmental Engineering, Technion, Haifa 32000, Israel

Received 1 July 2006/ Accepted 6 January 2007


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ABSTRACT
 
The variety of kinetics expressions encountered in the literature and the unreasonably broad range of values reported for the kinetics constants of Acidithiobacillus ferrooxidans underscore the need for a unifying experimental procedure and for the development of a reliable kinetics equation. Following an extensive and critical review of reported experimental techniques, a method based on batch pH-controlled kinetics experiments lasting less than one doubling time was developed for the determination of extant kinetics constants. The Fe(II) concentration in the experiments was measured by a method insensitive to Fe(III) interference. Kinetics parameters were determined by nonlinear fitting of the integrated form of the Monod equation to yield a KS of 31 ± 4 mg Fe2+ liter–1 (mean ± standard deviation), a KP of 139 ± 20 mg Fe3+ liter–1, and a µmax of 0.082 ± 0.002 h–1. The corresponding kinetics equation was as follows:

Formula
where S represents the Fe(II) concentration in mg liter–1, P0 represents the initial Fe(III) concentration in mg liter–1, X represents the suspended bacterial cell concentration in cells ml–1, and t represents time in hours. The measured data fit this equation exceptionally well, with an R2 of >0.99. Fe(III) inhibition was found to be of a competitive nature. Contrary to previous reports, the results show that the concentration of Acidithiobacillus ferrooxidans cells has no affect on the kinetics constants. The kinetics equation can be considered applicable only to A. ferrooxidans cells grown under environmental conditions similar to those of the inoculum tested in the study. In contrast, the experimental and computational procedure is completely general and can be applied to A. ferrooxidans irrespective of the culture history.


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INTRODUCTION
 
Acidithiobacillus ferrooxidans is a chemolithotrophic aerobic bacterium capable of oxidizing both ferrous iron to ferric iron and sulfide to sulfate, utilizing the energy derived from the oxidation to support carbon dioxide fixation and cell growth (17).

These characteristics are particularly fitting for the processes of mineral bioleaching, desulfurization of sour and flue gases, and the treatment of Fe(II)-containing acidic solutions, namely, acid mine drainage streams (28, 33).

The importance of these applications is the main incentive behind the intensive studies performed on the kinetics related to biological Fe(II) oxidation. However, despite a considerable effort in the last decades, examination of published kinetics constants reveals large discrepancies, of up to 3 orders of magnitude, between data sets derived from different sources. The large variations in published results may perhaps be explained by the variety of experimental conditions employed (temperature, pH, substrate concentration, etc.), differences in the experimental techniques used for the determination of the kinetics parameters (batch, continuous or "initial rate" experiments), the use of different mathematical models for kinetics data interpretation, the analytical techniques employed, the parameter that was monitored [Fe(II) concentration, Fe(III) concentration, dissolved oxygen concentration, or A. ferrooxidans cell concentration], and also perhaps the fact that different A. ferrooxidans strains were examined. Nevertheless, assuming that A. ferrooxidans strains do not differ significantly from each other and that, therefore, the range of kinetics constants for Fe(II) oxidation obtained in different studies should at least be of the same order of magnitude, it is not unlikely that at least a few of the techniques used for determining these kinetics constants in some of the previously published studies were inadequate. Moreover, it is clear from the diversity of experimental procedures proposed in the literature that a unified methodology for the proper determination of A. ferrooxidans kinetics constants is needed.

As a result of the large differences between kinetics data reported in the various sources, considerable difficulty exists in predicting a priori the Fe(II) oxidation rate under operational conditions in which both the substrate (Fe2+) and the product (Fe3+) are present at various concentrations. Such a situation is not uncommon in processes that utilize A. ferrooxidans for Fe(II) oxidation. For example, processes aimed at H2S removal from biogas in which A. ferrooxidans is used for Fe3+ bioregeneration are often operated at a relatively high Fe(III) concentration (several g Fe3+ per liter) along with a varying Fe(II) concentration, which is a function of both the H2S(g) load into the system and the rate of the concurrent biological Fe(II) oxidation.


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Intrinsic versus extant kinetics data.
 
In a widely cited publication, Grady et al. (14) coined a commonly used nomenclature for the interpretation of kinetics data results. Under this terminology, kinetics constants are divided into "intrinsic" and "extant." Intrinsic kinetics data are defined as "kinetic constants that represent the maximum capability of the members of the microbial community with the fastest growth kinetics." In a long kinetics experiment (multiple duplication times), the composition of the microbial community changes due to an enrichment of the fastest-growing species at the expense of slower-growing species, resulting in "intrinsic" kinetics data (14). In contrast, when a short batch experiment is performed, this leads to the determination of "extant" kinetics constants, i.e., "only limited changes in the protein synthesizing system and synthesis of new enzymes can take place before the substrate is depleted, and changes in the physiological state will be minimal." For practical and design purposes both Grady et al. (14) and Chudova et al. (7) recommended using short batch kinetics tests for the determination of the kinetics data set, because they provide an "insight into the immediate response of the continuous culture from which the cells are obtained" (7).

Accordingly, the goals of the present work were twofold: to determine the appropriate kinetics equation and a credible range for A. ferrooxidans extant kinetics constants and to establish a rigorous empirical/computational procedure to determine the extant kinetics constants for A. ferrooxidans while minimizing the common errors encountered in this respect in the literature. In order to establish the most suitable experimental technique, a comprehensive literature survey was performed.


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Literature survey of kinetics data relating to A. ferrooxidans.
 
Table 1 summarizes the Monod-based kinetics terms found in the literature to describe the growth rate and Fe(II) oxidation rate of suspended cultures of A. ferrooxidans. The models differ from each other by the types of coefficients used, the experimental system and environmental conditions under which they were determined, the computational methods used to interpret the data, the raw parameters monitored, and the analytical methods used to determine them. In the following paragraphs each of these parameters is critically reviewed, and associated advantages/disadvantages are discussed.


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  		TABLE 1. Monod expressions and kinetics constants reported for suspended A. ferrooxidans cultures


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Monod terms suggested for describing the kinetics of Fe(II) oxidation by A. ferrooxidans.
 
Monod's kinetics equation is the most widely used expression to represent both the growth rate of A. ferrooxidans and the rate at which it oxidizes the substrate [Fe(II)]. The simplest equation, which describes the growth rate of A. ferrooxidans solely as a function of the ferrous iron concentration (41, 44), is shown in equation 1.

Formula 1(1)
where µ is the specific growth rate [i.e., (dX/dt)/X, where X = the cell concentration], µmax is the maximum specific growth rate, [Fe2+] is the ferrous iron concentration, and KS–1 is the substrate affinity constant.

The reliability of equation 1 for predicting the Fe(II) oxidation rate is doubtful, since it has been extensively and repeatedly reported that both the Fe(II) oxidation rate and A. ferrooxidans cell growth are sensitive to inhibition caused by both the reaction product [Fe(III)] and also, above a certain Fe(II) concentration, by the substrate itself. To overcome this problem, more complex Monod terms (Table 1) have been introduced to describe the inhibition effect of the substrate (Fe2+), product (Fe3+), bacterial cells, and heavy metal concentrations on the rate of Fe(II) oxidation (or the corresponding dissolved oxygen concentration reduction) and bacterial growth.

With regard to the inhibition caused by the Fe(III) concentration on the rate of Fe(II) oxidation, both "competitive" and "noncompetitive" models have been suggested. Noncompetitive inhibition to Fe(II) oxidation by Fe(III) has been reported by Jones and Kelly (19) and by Nemati and Webb (31); however, most of the works published to date on A. ferrooxidans have reported competitive product inhibition rather than noncompetitive (5, 12, 13, 15, 20, 23, 27, 34).

Substrate inhibition at ferrous iron concentrations higher than 2 to 3 g Fe2+ liter–1 was reported previously (1). Conversely, Jones and Kelly (19) reported an inhibition effect only at Fe(II) concentrations above 5 g Fe2+ liter–1. Another type of competitive inhibition to Fe(II) oxidation, reported to be caused by a high concentration of mine-isolated bacterial cells (but not by laboratory-cultivated cells), was observed by Suzuki et al. (47). Those authors hypothesized that A. ferrooxidans cells have the ability to compete with Fe2+- for Fe2+-binding sites of neighboring A. ferrooxidans cells. They explained this observation by the unusual surface properties of A. ferrooxidans cells, which are responsible for the well-known ability of the bacterium to adsorb to solid surfaces. Subsequent to this observation, Nemati and Webb (29) reported that a linear dependency exists between the value of KS and the biomass concentration, corroborating thus the observation made by Suzuki et al. (47). These authors also reported that the inhibition effect of A. ferrooxidans concentration on the Fe(II) oxidation rate was not restricted to mine-isolated strains.

Other kinetics models that are not based on the Monod model usually describe only particular cases, such as zero-order kinetics at high Fe(II) concentrations (35) or first-order kinetics at low Fe(II) concentrations (38), and will not be discussed further in this paper.


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Assessment of the experimental techniques used for A. ferrooxidans kinetics parameters determination.
 
Three basic techniques have been used in the studies described in Table 1 to determine the kinetics parameters of A. ferrooxidans: initial rate measurements, continuous culture, and batch culture.

In the technique termed "initial rate measurements" (20, 29, 30, 34), the initial biomass concentration and the duration of the batch experiments are chosen in such a way that the oxidation rate of the substrate is negligible relative to the substrate concentration and, thus, the substrate concentration can be considered constant. Cell growth rate and/or Fe(II) oxidation rate are subsequently measured at various initial substrate concentrations, and each experiment constitutes a point on the kinetics graph. Kinetics parameters are either derived from the linear form of Monod's equation (20, 34) or by using nonlinear regression techniques (29, 30). The main drawback of the initial rate measurement technique is that the assumption of constant substrate concentration is bound to be incorrect at low initial substrate concentrations, especially when a relatively high biomass concentration is applied, resulting in high Fe(II) oxidation rates. This drawback is particularly problematic when the value of the substrate affinity constant (KS) is low, as is apparently the case with A. ferrooxidans. Apart from this difficulty, this technique often neglects the possible lag phase in the growth of A. ferrooxidans, which frequently occurs following the introduction of a cell suspension into an oxidation cell. Such neglect may lead to a further error.

A technique based on steady-state chemostat culture was used by Jones and Kelly (19), Nikolov and Karamanev (32), and Gomez and Cantero (13). The biomass concentration in a continuous flow completely mixed bioreactor can be described by the following equations (11):

Formula 2(2)

Formula 3(3)
where D (h–1) is the dilution rate, defined as D = F/V, which is the ratio between the influent flow rate F (in liters h–1) and the volume of the reactor V (in liters), and S0 is the influent substrate concentration (in mg liter–1).

From equations 2 and 3, it follows that at steady state (i.e., dX/dt = 0 and dS/dt = 0), µ = D and X* = Y(S0 – S*), where X* and S* are the biomass and substrate concentrations in the reactor at steady state. By running the reactor at different dilution rates, the corresponding S* values may be measured, the dependency of µ on S* can be obtained, and the kinetics constants may be derived from the Monod equation. Such an approach, however, may and frequently does encounter serious technical problems associated with maintaining a biological reactor at a constant low substrate concentration. This is particularly problematic when the bacteria in question have a high affinity towards a limiting substrate, or in other words, a low KS value (37). When applying the continuous culture technique, there is also a need to eliminate apparatus-related artifacts, such as nonperfect mixing and fluctuations in nutrient medium supply. However, the greatest objection to this technique is the fact that bacteria tend to alter their kinetics properties under the different steady-state conditions applied in a chemostat-type kinetics experiment (i.e., each point in the kinetics graph is obtained under a different set of operational conditions) (37).

A technique for estimating kinetics constants from the growth kinetics of batch cultures was employed by Liu et al. (23), Gomez et al. (12), and Boon et al. (4). In this technique, cells harvested during the logarithmic growth phase are introduced into a fresh medium with a known initial Fe(II) concentration. The substrate (Fe2+) or the biomass concentration are measured as a function of time until Fe(II) is completely oxidized. A chosen kinetics model is subsequently fit (by either a linear or a nonlinear curve fitting technique) to the measured Fe(II) concentrations to determine the kinetics constants. The main disadvantage that has been associated with this technique is that experimental conditions change continuously during a relatively long batch experiment. As mentioned above, changes in community structure and physiological adaptation of the bacteria to the changing environmental conditions during the experiment have been reported to significantly affect the values of the kinetics constants (14). If the duration of the batch experiment is sufficiently long to allow for several cell divisions, a change in physiological state is expected to occur. Moreover, the composition of the microbial community would change due to an enrichment of the fastest-growing species at the expense of slower-growing species. The kinetics measured under such conditions will represent the maximum capability of the members of the microbial community with the fastest growth kinetics (intrinsic kinetics) rather than that of the initial community. However, this problem mainly affects mixed cultures and cultures that are characterized by a high cell yield (such as heterotrophic, aerobic populations). Although the problem tends to be less troublesome in homogenous autotrophic communities, there is still a need to manage the duration of the batch experiment so that population multiplication does not occur and an extant, rather that an intrinsic, kinetics data set is determined.


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Batch tests: linear versus nonlinear data interpretation and differential versus integral curve-fitting techniques.
 
In some of the batch culture and initial rate experiments reported in the literature and shown in Table 1, the interpretation of observed data was based on the linear form of Monod's equation (Lineweaver-Burk's technique) (23, 34, 41). However, the reliability of the linearization method has been seriously and repeatedly criticized (3, 39). It has been shown that this technique suffers from inherent systematic errors, which can be explained as follows. Equation 4 shows Monod's equation to which an explicit error term has been added.

Formula 4(4)
where e is a value that represents a deviation from the theoretical model rather than a true error.

Equation 4 is clearly not linear with respect to [Fe2+]. Although the original Monod equation can be easily transformed into a straight line equation when it is written without the error term (e), this cannot be done when e is included. If an error term is included after linearization, as shown in equation 5, then the result is not a true transformation of equation 4, elin is not the same as e, and minimizing the linear sum of squares (SSlin = Formula 4does not provide the same parameter values as minimization of the true sum of squares, SS (8).

Formula 5(5)
Because of the inherent errors associated with the linear regression technique, the more recent works that have addressed bacterial kinetics determinations have made use of nonlinear regression analysis to interpret kinetics data. This technique is particularly convenient when the monitored parameter reflects the oxidation rate itself, for example, when the oxygen consumption rate is measured. However, usually the rate is not directly measured but rather calculated by differentiating the measured parameter. As a result of numerical differentiation, significant errors may be introduced due to even small fluctuations in the measured parameter. To overcome this problem, an integrated form of Monod's equation has been introduced for estimating the kinetics parameters from a single-substrate depletion curve (40, 42, 43). This technique, which appears to be the most appropriate from the mathematical standpoint, has been applied to determine the kinetics constants of various bacterial populations, such as a methanogenic mixed culture utilizing acetate (44) and Sphingomonas chlorophenolica grown on pentachlorophenol (9). However, to the best of the writers' knowledge, no attempt has been made to obtain the values of the kinetics constants of A. ferrooxidans by applying a nonlinear fitting technique to the integrated form of Monod's equation.


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Choosing the measured experimental parameter.
 
Since a direct measurement of the biomass concentration is cumbersome and has a low accuracy, most of the published kinetics studies rely on the measurement of indirect parameters to represent A. ferrooxidans oxidation activity and growth. Such an indirect approach may include measurement of oxygen uptake rate, measurement of the change in Fe(II) and Fe(III) concentrations with time, or the change with time of the redox potential of the solution. Another option may be to monitor the change in solution turbidity, which has been shown to be correlated with A. ferrooxidans cell concentration (35). The choice between the different monitoring parameter(s), along with the reported use of diverse analytical techniques, is probably one of the main reasons for the large variation between the reported kinetics constants. For example, the extremely high KS value (0.94 g liter–1) reported by Gomez et al. (12) appears to have resulted from the use of the standard o-phenanthroline method for the determination of Fe(II) concentration in the presence of a high Fe(III) concentration. In this regard, Herrera et al. (16) showed that the error in measured Fe(II) concentration is 14% when Fe(II) constitutes 10% of the total iron concentration and may reach 150% when the Fe(II) is 0.5% of the total iron concentration. Moreover, those authors found that the interference of the Fe(III) concentration in the o-phenanthroline analysis is due to direct color formation caused by the reaction of Fe(III) with o-phenanthroline and that the nature of the response of o-phenanthroline to Fe(III) is nonlinear. Therefore, correcting the measurements by comparing the absorbance to a blank to which no o-phenanthroline has been added, as was done, for example, by Boon et al. (4), is clearly incorrect. In another study, Nyavor et al. (34) did not report at all on the Fe(II) analysis method that was used, a fact which tends to detract from the reliability of the results. In order to use the o-phenanthroline method for ferrous iron analysis, one needs to overcome the interference caused by the presence of Fe(III). One way to do this is by adding a Fe(III) complexing agent. One of the well-known chelating agents is nitrilotriacetic acid (10); however, its application is cumbersome. Herrera et al. (16) proposed the use of sodium fluoride as a Fe(III) complexing agent. The procedure is simple and allows an accurate Fe2+ determination (relative errors ranging from 0.0% to 2.4%) at Fe(II) concentrations of ≥5% of the total iron concentration. When Fe(II) amounts to 0.5% of the total iron concentration, the relative error is around 12%.

Another problematic measurement technique that has been proposed for A. ferrooxidans kinetics constant determinations is based on redox readings. Pesic et al. (38) suggested using the redox potential of the solution as a measure of the change in the ratio between Fe(II) and Fe(III) concentrations with time and thus an indirect measure of the Fe(II) oxidation rate. The same method was later used by Nemati and Webb (29, 31) and Harvey and Crundwell (15). However, it appears that this technique may also lead to erroneous results: Nernst's equation is used in this method for calculating the Fe(II) concentration from measured redox potentials and a known total iron concentration:

Formula 6(6)
Where, {alpha}Fe3+ and {alpha}Fe2+ are Fe(III) and Fe(II) activities, F is Faraday's constant (in kJ V–1 equiv–1), n is the number of electrons transferred per molecule, E0 is the standard potential of the Fe3+/Fe2+ couple (in V), and E is the potential of the solution (in V).

However, the redox potential reading depends not only on the ratio between the dissolved Fe(II) and Fe(III) in solution but also on the total iron concentration and on the ionic strength of the solution [which is itself dependent on the ratio between the Fe(III) and Fe(II) concentrations]. Because of the logarithmic scale of the abscissa used in forming the linear calibration curve in this approach, even an apparent small difference between calibration curves may cause a large error in the calculation of the Fe(II) concentration. To overcome this problem, redox calibration curves must be made for each particular total iron concentration used in the kinetics experiments. This was not reported to be carried out in the studies quoted above, a fact which tends to detract from the reliability of their results.


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Experimental conditions chosen for determining the extant kinetics constants of A. ferrooxidans.
 
Based on the discussion thus far and in order to overcome the variety of technical, analytical, and mathematical problems associated with the studies reported in the literature, we opted in the present work to conduct the A. ferrooxidans kinetics experiments under the following experimental conditions: (i) batch pH-controlled experiments lasting less than one doubling time; (ii) measurements of the change in Fe(II) concentration with time by applying the o-phenanthroline method with sodium fluoride as a Fe(III) complexing agent, to avoid Fe(III) interference, as proposed by Herrera et al. (16). Interpretation of the results was performed by applying nonlinear fitting of the integrated form of the Monod equation.

The fundamental Monod equation considered in this study consists only of a substrate affinity term and a product inhibition term. Neither a bacterial cells inhibition term nor a substrate inhibition term was included, as these were found not to affect the Fe(II) oxidation rate under the experimental conditions tested (see Results and Discussion, below). The experiments were conducted at various cell concentrations, and the value of the yield coefficient was incorporated into the kinetics terms. With regard to product inhibition, it was impossible to determine the type of inhibition a priori, and therefore both competitive and noncompetitive models were examined.

It should be emphasized that, by definition, any extant kinetics data set depends on the physiological state of the bacterial population tested. Therefore, the results presented in the paper must be considered particular to the environmental conditions to which the bacteria were subjected prior to the experiments. However, the methodology presented is general and can be applied regardless of the "culture history".


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MATERIALS AND METHODS
 
Preparation of A. ferrooxidans cell suspension.
A pure culture of Acidithiobacillus ferrooxidans was obtained from the German Collection of Microorganisms and Cell Cultures (DSM 14882). The bacteria were grown in a 1-liter Erlenmeyer flask in a medium comprised of 0.4 g liter–1 (NH4)2, 0.4 g liter–1 MgSO4·7H2O, 0.1 g liter–1 K2HPO4, and 12 g liter–1 FeSO4·7H2O (48). Initial pH of the growth medium was adjusted to 1.7 by the addition of 2 N H2SO4. Cell suspensions used in the experiments were taken from the flask and filtered during the logarithmic phase of growth by using a Whatman no. 1 filter paper in order to remove suspended iron-containing solids. Subsequently, 100 ml of the filtrate was refiltered through a 0.45-µm filter. The filter was then washed with 50 ml basal salts solution (pH 1.5) and washed again with 50 ml basal salts solution at pH 2. The cells were then resuspended by backwashing the filter with 10 ml basal salts at pH 2. At the end of the experiments, samples from the growth flask were sent to Hy-labs (Rehovot, Israel) for rRNA gene cloning, sequencing, and identification. All of the clones that were returned exhibited greater than 97% similarity to the original culture (expect value, 0).

Procedure of batch experiments.
For batch experiments, cells (in suspension) were added to a basal salt solution comprising FeSO4 and Fe2(SO4)3 salts to yield a 310-ml solution with known Fe(II), Fe(III), and cell concentrations. Oxygen was supplied in excess by bubbling air through the medium. Dissolved oxygen was maintained at >6 mg/liter throughout the experiments. To improve mixing, a linear shaker at 125 excursions per minute was used. Temperature was maintained at 25 ± 0.1°C by a water bath. Samples of 1 to 2 ml were taken from each batch every few minutes using a syringe, and Fe(II) concentration was determined following 0.22-µm filtration. Since Fe(II) oxidation consumes protons, the pH of the medium was periodically adjusted back to pH 2.00 by the controlled manual addition of 2 N H2SO4.

Analyses.
Fe(II) concentration was determined by the modified o-phenanthroline method proposed by Herrera et al. (16). Absorbance was measured with a Genesys 10 spectrophotometer (Spectronics). Fe(III) concentration was determined by the sulfosalicylic acid technique (48). A. ferrooxidans cells were stained by using a standard Lugol solution and counted in an improved Neubauer counting chamber (0.1-mm depth, 0.0025-mm2 area) with an optical microscope at x400 magnification. Each count was performed at least three times.


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RESULTS
 
Preliminary tests were carried out in order to determine the most appropriate experimental technique for the determination of A. ferrooxidans extant kinetics constants. The goal of this step was to estimate the approximate range of the kinetics parameters by applying the initial rate measurement technique. The results (not shown) showed that the initial Fe(II) oxidation rate is roughly constant at an initial Fe(II) concentration range between 0.2 g Fe2+ liter–1 and 20 g Fe2+ liter–1. Notwithstanding the fact that these results were not useful for accurate determination of the kinetics parameters, they clearly indicate that KS is, in all likelihood, lower than 200 mg Fe2+ liter–1 and also that inhibition caused by Fe(II) is not significant at Fe(II) concentrations below 20 g Fe2+ liter–1.

Based on the knowledge from the preliminary experiments that KS is <200 mg Fe2+ liter–1, batch experiments appeared to be the most appropriate for further investigation. Four sets of batch experiments were performed, each comprising six 310-ml batch containers with an identical initial biomass concentration (X0) and varied initial Fe(II) concentrations (S0). The initial conditions applied in all the experiments are listed in Table 2. In all experiments, air was added in excess and Fe(II) oxidation was continued until at least 97% of the initial Fe2+ was oxidized.


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  		TABLE 2. Batch experiments: initial conditions

Assuming that cell lysis was negligible during the relatively short duration of the experiments, the equation for the change in biomass concentration in a batch culture can be expressed as follows:

Formula 7(7)
Separating the variables in equation 7 and integrating gives the following:

Formula 8(8)
The Monod kinetics expression that includes a term for cell growth and assuming competitive product inhibition is as follows:

Formula 9(9)
where P0 is the initial Fe(III) concentration.

Combining equations 8 and 9 gives:

Formula 10(10)
According to reference 46, equation 10 can be integrated. This integration yields the following:

Formula 11(11)
According to equation 8, a plot of the values of (X – X0) versus (S0 – S) obtained during the batch experiments should yield a straight line with a slope that equals the yield coefficient of A. ferrooxidans (i.e., Y) and an intercept that equals zero. As shown in Fig. 1, a linear curve was indeed obtained (R2 = 0.975), which validates equation 8. The value of the yield coefficient, which was obtained by least-square linear regression analysis, was 2.3 x 1010 cells (g Fe2+)–1. This value is similar to A. ferrooxidans yield values reported in previous works: 2.5 x 1010 cells (g Fe2+)–1 (2), 2.4 x 1010 cells (g Fe2+)–1 (25), 2.23 x 1010 cells (g Fe2+)–1 (6), and (1.7 ± 0.4) x 1010 cells (g Fe2+)–1 (27).


Figure 1
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  		FIG. 1. Data used for determining the value of the yield coefficient of A. ferrooxidans.

Based on the raw results [i.e., Fe(II) concentrations versus time at 25°C and pH 2.0] obtained in the 24 batch experiments and the knowledge of the cell concentration calculated based on Y and the initial cell concentration, the parameters KS, KP, and µmax were obtained by applying nonlinear least-squares data fit using the Gauss-Newton algorithm with Levenberg-Marquardt modifications for global convergence. Calculations were made using the function "nlinfit" within the Statistics Toolbox of the Matlab software (26). The 95% confidence intervals for parameter estimates were calculated using the statistical Matlab function "nlparci" (26). The values (means ± standard deviation) of the calculated extant kinetics constants and associated errors estimates were as follows: KS, 31 ± 4 mg Fe2+ liter–1; KP, 139 ± 20 mg Fe3+ liter–1; µmax, 0.082 ± 0.002 h–1.

The results from 16 out of 24 experiments were used for calibration, while the remaining 8 were used for validation. R2 values were 0.9913 for the calibration procedure and 0.9917 for the validation. The parameters KS and KP were found to be highly correlated with each other, with a correlation coefficient of –0.99. The linear combination of these parameters was obtained from a Fisher matrix: {Delta}KP = 5.38{Delta}KS, where {Delta} is the difference between the value of a parameter and its optimal value. This information means that for each value of KS inside the confidence interval, there is a value of KP that produces with equation 11 and the measured data approximately the same sum square errors (18).

To assess the accuracy of the kinetics constants obtained in the study, as well as to extend the upper Fe3+ concentration limit within which the calibrated equation 10 can be used, a series of batch experiments were performed with an identical initial Fe(II) concentration (1.2 g liter–1 of Fe2+), an identical biomass concentration (17.9 x 107 cells ml–1), and an initial Fe(III) concentration varying from 0 to 1.2 g liter–1 of Fe3+. Because of space limitations, only the results of two of these experiments are presented in Fig. 2. However, for all six runs the model predicted the measured data extremely well (R2 of all six runs was between 0.98 and 0.99).


Figure 2
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  		FIG. 2. Change in Fe(II) concentration with time in batch culture experiments with different initial ferric iron concentrations: •, 0 mg liter–1 of Fe3+; {blacksquare}, 1,200 mg liter–1 of Fe3+. Solid lines represent model predictions.

To determine whether the bacterial culture used in the present study was sensitive to the experimental procedure applied, two tests were performed. First, two identical batch culture experiments were carried out on different days to test the reproducibility of the results. The results are shown in Fig. 3. Second, six identical runs were performed with starvation times (defined as the time interval between cell inoculation and ferrous iron addition to the culture) varying from 0 to 6 h. The results of these experiments are shown in Fig. 4.


Figure 3
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  		FIG. 3. Results of two pairs of identical batch culture tests held on different days to check the effect of "culture history" on the reproducibility of the kinetics results. x, run I; {blacksquare}, run II.


Figure 4
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  		FIG. 4. Effect of starvation period (defined as the time interval between cell inoculation and ferrous iron addition to the culture) on the rate of Fe(II) oxidation of A. ferrooxidans.


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DISCUSSION
 
Using the values of the extant kinetics constants derived from the kinetics data by nonlinear fitting to an integrated form of the Monod equation, the following equation was developed to predict the rate of ferrous iron oxidation by a cell suspension of A. ferrooxidans:

Formula 12(12)
where S is the Fe(II) concentration in mg liter–1, X is the suspended bacterial cell concentration (in cells ml–1), P0 is the initial Fe(III) concentration in mg liter–1, and t is time in hours.

Note that strictly speaking equation 12 can be considered applicable only to A. ferrooxidans cells grown under conditions similar to those of the inoculum tested in this study. In contrast, the experimental and computational procedures are completely general and can be applied irrespective of the culture history.

In all 30 runs carried out in this study, equation 12 fit the empirical results exceptionally well. Equation 12 does not include a Fe(II) concentration inhibition term because it was found in "initial rate" experiments that the Fe(II) concentration does not inhibit the oxidation rate up to a concentration of (at least) 20 g Fe2+ liter–1.

In order to assess the effect on the oxidation of Fe(II) when the initial Fe(III) concentration was other than zero, ferric iron was added to the reaction mixture in six batch experiments at an initial concentration ranging from 0 to 1,200 mg liter–1. The very high prediction accuracy obtained in these experiments corroborated both the correctness of equation 12 and the accuracy of each of the three kinetics constants obtained in the study and also showed that equation 12 can be considered applicable up to (at least) Fe(III) concentrations of around 2.4 g Fe3+ liter–1.

Since one set of parameters was found to fit exceptionally well the data from all the experiments (which were run with different bacterial cells concentrations), it was concluded that the value of KS does not depend on the A. ferrooxidans cell concentration. This conclusion can be considered valid for the cell concentration range used in the experiments, i.e., from 2.77 x 107 cells ml–1 to 3.1 x 108 cells ml–1. This finding totally contradicts the conclusion made in reference 29 that a linear dependency exists between the values of KS and the biomass concentration (those authors applied a cell concentration range between 3.25 x 107 cells ml–1 and 4.47 x 108 cells ml–1, i.e., practically the same cell concentration range as in the current study).

A comparison between the kinetics constants obtained in our study with previously published constants, some of which were obtained under entirely different experimental conditions or interpreted using different models, is clearly problematical. The maximum specific growth rate (µmax) values found in the literature vary between 0.047 h–1 (36) and 1.78 h–1 (19), i.e., a difference of 2 orders of magnitude. Excluding values derived from the application of linear regression techniques (which can be assumed to be inherently inaccurate), the range narrows to between 0.047 h–1 and 0.23 h–1 (32). The µmax value found in the study (0.082 h–1) lies inside this range. Note that the µmax value found in the study corresponds to a doubling time of around 10 h.

The values of KS obtained in the literature by applying the competitive Fe(III) inhibition models vary from 0.028 g liter–1 (34) to 0.09 g liter–1 (23), if one excludes the exceptionally high value of 0.94 g liter–1 reported by Gomez et al. (12). This value is probably incorrect, because Gomez et al. (12) performed batch experiments without pH control, the Fe(II) oxidation was measured by the standard phenanthroline method without compensation for Fe(III) interference, and the experiments were completed at a high Fe(II) concentration of 0.5 g liter–1, i.e., well above the expected KS value. As was the case with the maximum specific growth rate, the KS value found in the present study (0.031 g liter–1) conforms well to the range of previously reported values.

With respect to A. ferrooxidans kinetics constants reported in the literature, the largest discrepancy appears between the values of KP (product inhibition coefficient). In the models that assume competitive inhibition, KP was reported to be as low as 0.06 g liter–1 (19) and as high as 4.38 g liter–1 (13), i.e., a difference of 2 orders of magnitude. The value of KP (0.37 g liter–1) found in the present study lies within this range. However, considering the differences in the methods and the problematic nature of many of the techniques used (as explained before), a considerable number of the reported KP values cannot be regarded as reliable.

Competitive or noncompetitive inhibition?
A disagreement exists in the literature regarding the use of competitive versus noncompetitive inhibition models to describe the inhibiting effect of the Fe(III) concentration on A. ferrooxidans kinetics. To determine the type of inhibition, the common technique is to compare the values of KS and µmax that are determined in the absence and presence (at a constant total iron concentration) of the inhibiting substance. According to this technique, if competitive inhibition dominates, KS, but not µmax, should change. In contrast, in the case of noncompetitive inhibition, µmax should decrease and KS should remain unchanged. However, such a procedure cannot be carried out in the course of batch experiments, since the concentration of the inhibiting substance [Fe(III) in this case] changes constantly during the experiment. Therefore, to establish the type of Fe(III) inhibition in the current investigation, the integrated form of the noncompetitive inhibition model shown in equation 13 was also applied to the raw results, and the most appropriate inhibition model was chosen by comparing (i) the R2 values, which quantify the goodness of fit of the prediction equations to the measured data, and (ii) the standard deviations of the estimated constants.

Formula 13(13)
Combining equations 8 and 13 gives the following:

Formula 14(14)
Integration of equation 14 yields the following:

Formula 15(15)
When applying the raw data from the kinetics experiments to equation 15, the following results were obtained: KS, 151.4 ± 26.7 mg Fe2+ liter–1; KP, 3,839 ± 898 mg Fe3+ liter–1; µmax, 0.11 ± 0.01241 h–1. The R2 values derived from applying equation 15 to the raw data were 0.9549 and 0.9565 for the calibration and validation procedures, respectively, i.e., the "noncompetitive" model fit the experimental data much less well than the "competitive" model (R2 of 0.9913 and 0.9917, respectively). In addition, the standard deviations of the parameters fit by the noncompetitive model (8.8%, 11.7%, and 5.6% for KS, KP, and µmax, respectively) were considerably larger than those obtained by fitting the competitive model (6.4%, 7.2%, and 1.2% for KS, KP, and µmax, respectively).

Figure 5 shows measured versus predicted results of four individual runs (run numbers 3, 11, 18, and 20 in Table 2). It can be seen that the prediction of the noncompetitive inhibition model did not fit well the experimental results (the R2 values for these runs were 0.8873, 0.8908, 0.9320, and 0.8629, respectively), whereas the R2 values for the same runs when applying the competitive model were much higher: 0.9862, 0.9686, 0.9805, and 0.9944, respectively. Further examples for the inadequacy of the noncompetitive model to the empirical results are not shown in order to conserve space, but the inevitable conclusion is that for A. ferrooxidans kinetics the competitive inhibition model results in a much better prediction of measured data than the noncompetitive model. Moreover, extrapolation of the noncompetitive model by applying equation 15 to the results from the experiments in which Fe3+ was added initially to the reaction mixtures resulted in very inaccurate prediction with the respect to the measured data. For example, the prediction of the experiment with an initial Fe(III) concentration of 1,200 mg liter–1 gave an R2 of 0.795, whereas the same data, when applied to the competitive inhibition model, yielded an R2 of 0.9887.


Figure 5
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  		FIG. 5. Comparison between competitive (solid lines) and noncompetitive (dashed lines) inhibition model predictions. (a) Run 3; (b) run 11; (c) run 18; (d) run 20 (see Table 2).

Repeatability of the kinetics results.
Several studies concerned with determining bacterial kinetics constants have reported that a change in "culture history" was one of the major reasons for the large variations observed in the values of reported Monod kinetics parameters (14, 21, 22). The accepted hypothesis is that the way in which a culture is grown often determines the nature of the enzymatic systems that are expressed and also the physiological state, which is the sum total of a cell's macromolecular composition. The physiological state may further determine how rapidly the bacteria can synthesize enzymes, as well as how rapidly these enzymes would react. In this respect, Sommer et al. (45) showed poor reproducibility of biodegradation experiments (Pseudomonas cepacia with toluene as the only carbon and energy source) and concluded that the reason was that they were performed on different days.

Figure 3 shows that the duplicates of two similar experiments (each experiment was initiated with a different initial Fe2+ concentration, and both experiments were repeated on two different days) were practically identical, the slight variation being attributable to normal analytical fluctuations. Taking into account that it is almost impossible to prepare two solutions with an identical biomass concentration (X0 differed slightly between the runs, i.e., 4.41 x107 cells ml–1 in run I versus 4.39 x 107 cells ml–1 in run II), it can be safely concluded that at least in the case of A. ferrooxidans, kinetics experiments carried out on different days can be very well reproduced, provided that prior to the kinetics experiment the inocula are prepared in an identical fashion (see Materials and Methods), resulting in populations with closely similar physiological conditions.

Since the time between the preparation of cell suspensions (by filtration) and the beginning of the experiments varied in the different experiments and sometimes even between the various runs within the same experiment, a second experiment was conducted to validate that the time lag within the experimental procedure itself [defined here as a "starvation period," because during this time no Fe(II) was supplied to the bacteria] did not have an effect on the kinetics constants determined. Figure 4 clearly shows that a starvation period, within the 6 hours tested, had no effect on the value of the kinetics constants. This finding has obvious practical implications on A. ferrooxidans kinetics-related laboratory procedures.


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ACKNOWLEDGMENTS
 
This research was supported by research grant IS-3522-04 from BARD, the United States-Israel Binational Agricultural Research and Development Fund.


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FOOTNOTES
 
* Corresponding author. Mailing address: Faculty of Civil and Environmental Engineering, Technion, Haifa 32000, Israel. Phone: 972 4 9292191. Fax: 972 4 9228898. E-mail: agori{at}tx.technion.ac.il. Back

{triangledown} Published ahead of print on 19 January 2007. Back


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Applied and Environmental Microbiology, March 2007, p. 1742-1752, Vol. 73, No. 6
0099-2240/07/$08.00+0     doi:10.1128/AEM.01521-06
Copyright © 2007, American Society for Microbiology. All Rights Reserved.




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