Kinetics of Substrate Biodegradation under the Cumulative Effects of

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Kinetics of Substrate Biodegradation under the Cumulative Effects of Bioavailability and Self-Inhibition Mehdi Gharasoo,*,†,‡ Florian Centler,† Philippe Van Cappellen,§ Lukas Y. Wick,† and Martin Thullner† †

Department of Environmental Microbiology, UFZ - Helmholtz Centre for Environmental Research, Permoserstraße 15, 04318 Leipzig, Germany ‡ Institute of Groundwater Ecology, Helmholtz Zentrum München, Ingolstädter Landstr. 1, 85764 Neuherberg, Germany § Department of Earth and Environmental Sciences, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada ABSTRACT: Microbial degradation is an important process in many environments controlling for instance the cycling of nutrients or the biodegradation of contaminants. At high substrate concentrations toxic effects may inhibit the degradation process. Bioavailability limitations of a degradable substrate can therefore either improve the overall dynamics of degradation by softening the contaminant toxicity effects to microorganisms, or slow down the biodegradation by reducing the microbial access to the substrate. Many studies on biodegradation kinetics of a selfinhibitive substrate have mainly focused on physiological responses of the bacteria to substrate concentration levels without considering the substrate bioavailability limitations rising from different geophysical and geochemical dynamics at pore-scale. In this regard, the role of bioavailability effects on the kinetics of self-inhibiting substrates is poorly understood. In this study, we theoretically analyze this role and assess the interactions between self-inhibition and mass transfer-limitations using analytical/numerical solutions, and show the findings practical relevance for a simple model scenario. Although individually self-inhibition and mass-transfer limitations negatively impact biodegradation, their combined effect may enhance biodegradation rates above a concentration threshold. To our knowledge, this is the first theoretical study describing the cumulative effects of the two mechanisms together.

1. INTRODUCTION Microbial degradation processes are highly relevant for many environmental disciplines such as contaminant hydrogeology, nutrient recycling or global biogeochemical cycling of elements. Knowledge of the kinetics of such degradation processes and their dependency on environmental factors is needed for a quantitative understanding of many environmental systems. A multitude of factors control microbial degradation kinetics in a particular environment, but the substrate abundance often has the most dominant effect among all. Even small changes in the concentration level of a substrate in the vicinity of a microbial cell can trigger the physiological response of the cell in form of changes in its metabolic activity, such as specific growth rate, consumption rate, respiration rate or chemotaxis. While it is assumed that the increasing of substrate concentration would promote biodegradation activity, two additional mechanisms may intervene this: the inhibitory effect of substrate at high concentrations1 and the substrate bioavailability.2 Although the two mechanisms are entirely different in nature, the separate impact of each of them is considered detrimental for the efficiency of bioremediation and shown to reduce substrate degradation rates. Over the past decades, substrate inhibition has been investigated experimentally and analytically (e.g., refs 3−5), © XXXX American Chemical Society

and a vast number of substrates have been shown to be inhibitory at high concentration levels to many of the organisms metabolizing them, for example, inhibition of Pseudomonas putida PpG7 (NAH7) by vapor-phase naphthalene,6 and Pseudomonas cepacia G4 by phenol.7 When the concentration of a degradable contaminant is continuously varied from very low to very high values in an aqueous suspension of metabolizing microorganisms, particular patterns of concentration dependency can be observed. No metabolic activity is usually noticed at extremely low concentrations. Increasing the substrate concentration will initiate and subsequently increase the degradation activity of the microorganisms. Eventually the increase of substrate concentration reaches to a point at which further concentration increases do not increase the culture metabolic activity and at concentrations above this point the metabolic activity of culture even decreases (substrate inhibition). Therefore, for every self-inhibitive substrate there is an optimum concentration level at which the maximum degradation rate is obtained.3,8 The physiological consumption rates of microReceived: December 1, 2014 Revised: March 23, 2015 Accepted: April 1, 2015

A

DOI: 10.1021/es505837v Environ. Sci. Technol. XXXX, XXX, XXX−XXX

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Environmental Science & Technology

where t (T) denotes time, c (ML−3) the contaminant bulk concentration in the system, u ⃗ (LT−1)the fluid velocity vector, and D (L2 T−1) the diffusion coefficient of the contaminant. The consumption rate of the substrate R(c) (ML−3T−1) is a function of its bioavailable concentration and the degradation capacity of the bacteria inhabiting the system, and is defined by kinetics of the reactive processes (i.e., in the context of this study by one of the rate expressions explained below). 2.1. Kinetics of Biodegradation. The presented kinetics consider biodegradation rates to be limited by the presence of a single compound. This can be the organic carbon substrate, the associated electron acceptor/donor or any other compound consumed during the degradation process. For the sake of readability the term substrate is used for this unspecified ratelimiting compound. All other compounds potentially affecting the degradation rate are implicitly assumed to be present at concentrations sufficiently high (but nontoxic) to avoid any additional rate limitation. 2.1.1. Substrate Consumption without Inhibitory/Bioavailability Effects. In the absence of bioavailability restrictions, the bioavailable concentration of a substrate cb (ML−3) is equal to its bulk concentration c (ML−3) and the reaction rate is described by MichaelisMenten kinetics,20,21 cb R = qmax cb + K m (2)

organisms are reduced at substrate concentrations higher (due to inhibition) and lower (due to substrate limitation) than this point. The majority of microorganisms are known to show this type of behavior in response to the changes of concentration of an inhibitive substrate which is mechanistically very important for the metabolic activities of microorganisms in various natural ecosystems and their industrial applications, such as food processing fermentation and wastewater treatment.3,4 Apart from the cells’ physiological responses to the changes of substrate concentration which kinetically affect the biodegradation process, some environmental factors can limit the physical accessibility of substrate to the cells and thus directly influence the biodegradation kinetics.9,10 Such mass-transfer limitations of substrate access to the microbial cells is generally referred to as bioavailability limitations and is a well-known type of limitation, typically observed at small scales, and has obvious importance for the metabolic activities of microorganisms.9−13 In porous environments, such as soils, aquifers or aquatic sediemnts, the bioavailability of a substrate at pore-scale , next to transport processes, is additionally controlled by a number of other physiochemical processes such as sorption or dissolution (see, e.g., refs 14−16). In developing a more mechanistic framework for future biogeochemistry models considering both mechanisms of inhibition and bioavailability together is critical. While a large fraction of the literature in biogeochemistry and biotechnology has been exclusively dedicated either to bioavailability limitations (e.g., ref 9,10, and 12) or to substrates inhibition (e.g., refs 3, 4, and 17−19), and their mechanisms of influence are already wellexplained by mathematical models, a link between these two important mechanisms and the formulation of their combined effect is still missing. Up to now, except for a few qualitative or semiquantitative analyses of this complex issue (see, e.g., ref 8), there has been no specific mathematical method to establish the tie between these two mechanisms and to provide a discrete definition for degradation kinetics of a single rate-limiting substrate which is both, bioavailability restricted and self-inhibitive. The purpose of this research is to provide a solution for filling this gap, determining the relationship between substrate self-inhibition and its bioavailability limitations such that the combined kinetics can be theoretically expressed. The attempt to explore these processes in a more quantitative fashion is facilitated by tractable and well-grounded mathematical models (both analytical and numerical) relating the physiological limitations of the microorganisms to the bioavailability limitations inside their habitat. Despite the seemingly suppressive effect of both mechanisms on substrate degradation rates, the cumulative effect was shown to improve degradation rates at concentrations higher than a certain threshold. The potential relevance of the obtained results for environmental systems is demonstrated using a numerical simulation example of substrate transport and degradation in a simple porous medium.

with Km (ML−3) as substrate half-saturation constant, qmax (ML−3T−1) as maximum volumetric degradation rate, and cb as bioavailable concentration of the substrate. It should be pointed out that the empirical Monod expression22 is identical in form to Michaelis−Menten kinetics and eq 2 sometimes referred to as Monod equation. 2.1.2. Consumption of a Substrate with Limited Bioavailability. In the subsurface, substrates and bacteria are usually distributed differently. The microbial uptake thus depends on the mass transfer of the substrate to the bioavailable vicinity of the bacterial cells. In presence of bioavailability restrictions, the bioavailable concentration cb is not equal to the bulk concentration c and a linear exchange model is suggested to describe the link between them:8−11,13,23 rex = k tr(c − cb)

where ktr (T ) is the limiting mass transfer coefficient controlling substrate bioavailability. The system is at (quasi-) steady state condition as long as the relation between c and cb in the system is maintained by eq 3. The rate of substrate exchange rex is then equal to its degradation rate for all the situations fulfilling the (quasi-) steady state condition. For the cases where the degradation rate of substrate is given by Michaelis−Menten kinetics (eq 2), the expression for mass transfer (eq 3) can be combined with it to form the following equation (known as Best9 kinetics),

2. THEORETICAL DESCRIPTIONS The fate of a substrate inside an aqueous environmental system is usually determined by transport processes (e.g., advection and diffusion/dispersion) and reactive transformations (e.g., biodegradation), and is described by the advection-diffusionreaction equation: ∂c = −∇·(cu ⃗) + D∇2 c − R(c) ∂t

(3)

−1

R=

k tr (c + K m + qmax /k tr) 2

⎛ ⎛ ⎞1/2 ⎞ 4cq max /k tr ⎜ ⎜ ⎟ ⎟ ⎜1 − ⎜1 − (c + K + q /k )2 ⎟ ⎟ ⎝ ⎠ ⎠ m tr max ⎝

(4)

which describes the substrate degradation rate as a function of its bulk concentration in the presence of small-scale bioavailability restrictions.

(1) B


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2.1.3. Consumption of a Substrate with Self-Inhibitory Effects. For microorganisms, organic substrates behave as nutrients at low concentrations while they may induce toxicity at higher concentrations. There are currently many modeling approaches to describe the inhibitory effect of a substrate on microbial kinetics. The variation of suggested equations for different inhibition kinetics (competitive, noncompetitive, selftoxicity, mixed-toxicity, etc.) have been already reviewed extensively in, for example, Andrews,3 Chambon et al.,26 Han and Levenspiel,27 Mulchandani and Luong,28 and Neufeld et al.29 Many of these equations are a modified version (or adjusted according to the experimental observations) of the classical inhibition model

Lower values of ktr promote higher bioavailability limitations and eventually lead to lower degradation rates (in comparison to Michaelis−Menten kinetics). The solution of eq 4 for different values of ktr is illustrated in Figure 1: Top. In general, there are no specific constraints for definition of ktr and any upscaled, linear description of ktr can be used. For example, pore network case studies of Gharasoo et al.24,25 used the suggested expression by Hesse et al.11 to describe the intrapore limitations of mass transfer inside pores.

R = qmax

cb k i (cb + K m)(cb + k i) 5

(5) −3

suggested by Haldane with ki (ML ) as inhibition coefficient. Haldane eq 5, though empirical, is merely based on Michaelis− Menten kinetics and was proposed to best represent the noncompetitive inhibition of substrate.1,23,29,30 A comparison between the solutions of eqs 5 and 2 (Figure 1: Middle) shows that opposite to the continuous increase of consumption rates in Michaelis−Menten kinetics, for inhibitory kinetics the presence of an inflection point at SmaxI (ML−3) = (ki Km)1/2 causes the rate to be inversely proportional to the substrate concentration at concentrations higher than SmaxI.3,8 In the presence of inhibitory effects, the maximum specific degradation rate Rmax (ML−3T−1), shown with a horizontal line in Figure 2, can be therefore achieved at SmaxI (shown with a vertical line in Figure 2), and calculated by using SmaxI in eq 5: qmax R max = (1 + K m/k i )2 (6) It is worth mentioning that independent to the values of ki and Km, the inhibited rates never exceed the original Michaelis− Menten rates (Figure 1: Middle). 2.1.4. The Mixed Kinetics of Bioavailability and Inhibition. Until now, inhibition equations have been mostly derived based on Michaelis−Menten kinetics and excluded the limitations caused by mass-transfer related mechanisms on substrate bioavailability. To include the mass-transfer limitations for a self-inhibitive substrate eqs 3 and 5 must be merged assuming the (quasi-) steady state conditions (see Section 2.1.2). Through an algebraic rearrangement of terms, one can obtain the following cubic polynomial, k i qmax ⎞ 2c + k i + K m 2 ⎛ R3 − R + ⎜(c + k i)(c + K m) + ⎟ 2 k tr ⎠ k tr k tr ⎝ R − k i qmax c = 0

(7)

the solution of which determines the substrate consumption rate under both effects of mass-transfer limitations and selfinhibition.31 Unlike the combination of Michaelis−Menten kinetics (eq 2) with the linear exchange model (eq 3) which resulted into a quadratic polynomial, the solution of which was relatively easy to express analytically (eq 4), it is difficult to derive an analytical expression for the solution of eq 7. It is comparatively easier to solve it numerically assuming all parameter values are known. The solution of eq 7 has either three real roots or one real and two imaginary roots. After eliminating the imaginary, negative or out of the feasible range roots from the solution of eq 7, there is always a real root left

Figure 1. Top: Bioavailability-restricted degradation rates following Best kinetics eq 4 with different values of the mass-transfer limiting term (ktr) in comparison to (nonrestricted) Michaelis−Menten kinetics. Middle: Substrate self-inhibition affected degradation rates following the Haldane relationship eq 5 with different values of the inhibition coefficient (ki) in comparison to (noninhibited) Michaelis−Menten kinetics. Bottom: Degradation rate simultaneously affected by bioavailability restrictions and substrate self-inhibition following the new, mixed-kinetic expression eq 7 with different values of the inhibition coefficient (ki) in comparison to the reference (noninhibited) Best kinetics, all considering the same constant mass transfer coefficient (ktr = 0.01). C


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simulations of a simple one-dimensional porous medium were performed. Using the numerical platform BRNS (Biogeochemical Reaction Network Simulator32) the fate of a biodegradable substrate was simulated according to eq 1. The BRNS is a flexible tool for the simulation of reaction kinetics of arbitrary size and complexity in which the ODEs describing the kinetics of reactions can be numerically solved inside a one-dimensional transport system. The BRNS uses a fully implicit finite difference scheme for the transport and an iterative Newton−Raphson method for the reactions where both are linked by a sequential noniterative operator splitting technique in every time step. In the simulations, an influx of a substrate using a boundary concentration of ci = 8Km was considered with the substrate degradation rate described by one of the kinetic expressions described above. Transient simulations were performed until steady state was reached.

3. RESULTS AND DISCUSSION 3.1. The Analytical Relations between Bioavailability Restrictions and Inhibition. The combination of the degradation kinetics of self-inhibitive substrates following Haldane (eq 5) and the mass-transfer limitation of substrate bioavailability given by a linear exchange term (eq 3) allowed for the introduction of a new kinetics expressing the cumulative effect of both mechanisms together. Although this new kinetics depends on a cubic term which can not be expressed in closed mathematical form, the numerical solution of such term can be easily obtained using standard mathematical softwares. The resulting solution allows the comparison of degradation rates of substrates subject to either bioavailability restrictions or selfinhibition, or to both effects simultaneously. The rates comparison showed the expected result that independent to the existence of bioavailability restrictions, substrate degradation rates in the presence of inhibition are always lower than the equivalent noninhibited kinetics (Figure 1: Middle and Bottom). A similar reduction of substrate degradation rates due to bioavailability restrictions can be observed for a noninhibitive substrate. It was however revealed that in the presence of substrate self-inhibition a reduced bioavailability of the substrate leads to the counterintuitive observation of increased degradation rates at a certain range of substrate concentrations (Figure 2). Figure 2 introduces two concentration thresholds: cgh ′ at the intersection point of the Haldane relationship (eq 5) with the Best (eq 4), and cgh at the intersection point of the Haldane relationship (eq 5) with the mixed kinetics (eq 7). The Haldane (eq 5) rates are higher for a self-inhibitive nonlimited substrate at concentrations c < c′gh, while the monotonically increasing rates of the Best eq 4 for a noninhibitive rate-limited substrate at c > c′gh become in comparison higher. For a self-inhibitive substrate at concentrations c > cgh, higher degradation rates are achieved in the presence of bioavailability limitations, that is, by the mixed kinetics (eq 7). For c < cgh, the limitation of substrate bioavailability leads to an additional reduction of the rates, that is, higher rates are obtained from Haldane kinetics (inhibition only, eq 5). The value of cgh is therefore critical to know indicating the threshold concentration of a self-inhibitory contaminant between the two mechanisms of inhibition-only (classical) and inhibition in the presence of bioavailability limitations (derived in this study). Here we present the mathematical ways to compute c′gh and cgh for any self-inhibitive substance of interest with any combination of Km, ki and ktr values. It is shown in Figures 2 and 4 that

Figure 2. Top: Comparison between nonrestricted Michaelis−Menten kinetics (eq 2), the kinetics of self-inhibited degradation (eq 5), bioavailabilty-limited (eq 4), and mixed bioavailability-inhibitionlimited degradation (eq 7) for identical values of the inhibition coefficient (ki = 1.5Km) and of the mass-transfer coefficient (ktr = 0.01). The left and right vertical lines indicate SmaxI = (ki Km)1/2, and SmaxIB (eq 8), respectively, and the horizontal line shows the maximum specific rate Rmax (eq 6). Bottom: A closer look at intersections of eqs 4 and 5 ′ ), and 5 and 7 (cgh). The exact values of cgh ′ and cgh are listed in Table (cgh 1

which is the desired solution. Figure 1 (Bottom) shows the solution of eq 7 in comparison to a reference Best equation (ktr = 0.01) for different values of ki. In presence of both mass-transfer limitations and substrate inhibition, the maximum degradation rate Rmax is attained at the substrate concentration SmaxIB (ML−3), shown with a vertical line in Figure 2, and derived from the first derivative of eq 7 as S maxIB = S maxI +

qmax /k tr (1 +

K m/k i )2

(8)

Substrate consumption rate increases with increasing substrate concentrations as long as c < SmaxIB, and decreases when c > SmaxIB. SmaxIB values are always greater than SmaxI independent to the values of Km, ktr, and ki. Regardless of the chosen parameter values, rates from eq 7 (derived in this study) never exceed the Best rates (noninhibited reference eq 4; Figure 1: Bottom). It should be noted that in the presence of bioavailability limitations the rates never exceed Rmax but higher rates are shifted toward higher concentrations. 2.2. Numerical 1D Reactive Transport Simulations. To assess the effect of different microbial degradation kinetics on the fate of a substrate in an environmental system, numerical D


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Environmental Science & Technology independent to the actual values of ki and ktr, the values of cgh and c′gh are very close and in the same order of magnitude with c′gh slightly smaller than cgh. 3.1.1. Finding the Threshold Concentration cgh ′ . This concentration threshold defines which rate-limiting process would be more restrictive for a given environmental system. For example, two microbial species with similar metabolic potentials coexist: one with access limitation to the substrate but noninhibited (eq 4), and one without access limitations but inhibited at high substrate concentrations (eq 5). Certainly, for a single microbial species determining the value of cgh ′ has only limited practical relevance as a specific substrate cannot be both inhibitive and noninhibitive to a certain microorganism. Since calculating the value of c′gh is relatively easier than of cgh, we propose an iterative Bisection method33 to first determine the ′ and then to use it as an initial guess to find cgh. Finding value of cgh c′gh needs the rates from both equations of Haldane (eq 5) and Best (eq 4) to be equal. Combining these two equations and the rearrangement of the terms according to c gives k tr c 3 − (qmax + 2k tr K m + k tr ki)c 2

(

+ k tr K m k i + k tr K m(k i + K m) + qmax (k i + K m) −

⎞ 2 q k i⎟c + (k tr K m2 k i + qmax K m k i) = 0 Φ́ max ⎠

(9a)

⎛ ⎛ ⎞1/2 ⎞ 4cqmax /k tr ⎜ ́ ⎜ ⎟ ⎟ Φ = ⎜1 − ⎜1 − 2⎟ (c + K m + qmax /k tr) ⎠ ⎟ ⎝ ⎝ ⎠

Figure 3. Flowchart of the algorithm used to calculate the values of c′gh (eq 9a and 9b) and cgh (method 2, eqs 10a and 10b). Note that 0 < B < 1 and (b ∈ ) and their actual value changes depending on the stability of the solution.

(9b)

Expanding the term Φ́ (eq 9b) and including it into eq 9a, even if it was possible, would have made the final expression very difficult to solve. With the current setup we propose an algorithm to solve the cubic equation (eq 9a) by estimating first the value of Φ́ from eq 9b. In this method, the value of Φ́ for a concentration ci ∈ (SmaxI,SmaxIB) is calculated and then eq 9a is solved for the computed value of Φ́. Elimination of the imaginary roots from the solution of eq 9a gives either three or one real root. After the elimination of the negative real roots, the closest root csol to ci is chosen to estimate the new ci, ci,new = ci ± (ci − csol)/b where b is an integer number (b ∈ ) and its value depends on the stability of solution and the accuracy tolerance (see Bisection method33). In rare cases, the solution of eq 9a has no positive real root and the ci,new has to be estimated by reduction or promotion of ci depending on its initial value. In this study, we used SmaxIB as the

initial guess and since both cgh ′ and cgh are obviously smaller than it, a deduction order was used to estimate the next ci,new = ci − Bci. B is a real number (0 < B < 1) and its value depends on the distance between the values of SmaxI and SmaxIB. By doing the above iterative procedure, cgh ′ is found when one of the roots of eq 9a is equal to the concentration value ci which was initially used to estimate Φ́. The flowchart in Figure 3 illustrates the algorithm of the above procedure. The values of c′gh for an arbitrary selfinhibited contaminant with half-saturation constant Km = 0.261 μM and a set of ki and ktr value combinations is listed in Table 1 and graphically shown in Figure 4 for a specific case of ktr = 0.01. 3.1.2. Finding the Threshold Concentration cgh. To determine the value of cgh one needs to solve the set of eqs 5 and 7 together,

⎧ c ki (5) R = qmax ⎪ + ( c K m)(c + k i) ⎪ ⎨ ⎞⎛ ⎞ ki qmax ⎞ ⎪ R3 2c + k i + K m 2 ⎛⎛ ⎟⎟R − k i q c = 0 (7) R + ⎜⎜⎜c + ki⎟⎜c + K m⎟ + ⎪ 2 − max k tr ⎠ k tr ⎠⎝ ⎠ ⎝⎝ ⎩ k tr

the value of c′gh is marginally increased (e.g., increments about 2 orders of magnitude smaller than the original value of c′gh) and substituted into the Haldane eq 5 to obtain a rate which is then used to evaluate eq 7. This procedure continues with the slight increase of estimated concentration until the value of cgh is found and eq 7 equals zero. Method 2: This method is similar to the method which was described earlier for finding the value of cgh ′ . First, a rigorous

Direct substitution of R from eq 5 into eq 7 results into a polynomial of degree 9 which is computationally elaborated and difficult to solve. In the following, two different methods are presented for calculating cgh: Method 1: Since the value of cgh is slightly higher than c′gh (Figure 2: Bottom), one way to find cgh is through finding first the value of cgh ′ and to use its value as an initial guess in an iterative procedure similar to the Bisection method.33 In this procedure E


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Table 1. Concentration Values at the Intersection of Eqs 5 and 4 (or Values of cgh ′ ), and at Intersection of Eqs 5 and 7 (or Values of cgh) Calculated for Different Combinations of ktr and ki Values and a Half-Saturation Constant Km = 0.261 × 10−6 M ktr (s−1) ki(M) = Km ki(M) = 1.5Km ki(M) = 5Km ki(M) = 10Km a

0.001

0.005

0.01

0.05

0.1

2.694 × 10−6 2.698 × 10−6 3.295 × 10−6 3.296 × 10−6 5.851 × 10−6 5.853 × 10−6 7.991 × 10−6 7.993 × 10−6

1.085 × 10−6 1.093 × 10−6 1.324 × 10−6 1.334 × 10−6 2.287 × 10−6 2.302 × 10−6 3.022 × 10−6 3.049 × 10−6

7.130 × 10−7 7.348 × 10−7 8.709 × 10−7a 8.963 × 10−7a 1.49 × 10−6 1.531 × 10−6 1.942 × 10−6 2.006 × 10−6

2.525 × 10−7 3.560 × 10−7 3.128 × 10−7 4.349 × 10−7 5.544 × 10−7 7.65 × 10−7 7.382 × 10−7 1.043 × 10−6

1.592 × 10−7 3.070 × 10−7 2.003 × 10−7 3.755 × 10−7 3.711 × 10−7 6.712 × 10−7 5.072 × 10−7 9.312 × 10−7

c′gh(M) cgh(M) cgh ′ (M) cgh(M) c′gh(M) cgh(M) cgh ′ (M) cgh(M)

Values of cgh ′ and cgh for the example shown in Figure 2.

the descriptions of inhibition can also be extended to other types of inhibition (see Appendix (B)) as long as the final degree of the inhibition equation is not higher than the reference eq 5, suggested by Haldane.5 For example the inhibition equations: eq 13 suggested by Andrews3 for uncompetitive inhibition, eq 14 proposed by Neufeld et al.29 for competitive inhibition, and eq 15 by Webb34 have an identical degree as the Haldane eq 5. The inclusion of bioavailability restrictions to more complicated inhibition formulas of higher degree is however more complex and the presented analytical method cannot be used for those equations, including, for example, eq 16 by Yano and Koga,35 eq 17 by Teissier,36 eq 18 by Aiba et al.,17 eq 19 by Han and Levenspiel27 when n > 2 or m > 2, and the earlier version of eq 19 by Luong.37 It is noteworthy to mention that the numerical implementation of the coupling between the bioavailability restrictions with the above, or any other complex inhibitory equation, is possible, for instance by using the BRNS32 or its enhancements.24,38,39 The quasi steady-state assumption is an important criteria for derivation of analytical expressions presented here. This assumption holds as long as temporal changes in substrate bulk concentration are sufficiently slow to allow a quasi-instantaneous adaptation of the mass transfer to these changes. Such mass transfer processes might be slow in general (e.g., slow bleed-out of substrate from a secondary domain or slow desorption from surfaces of a solid matrix) but the quasi steady-state assumption may still hold as long as the bulk concentration changes at even slower time scales. For some natural biodegradation problems the temporal changes of the system might be too fast to ignore the adaptation of the mass transfer dynamics. For such cases, using a general-purpose numerical simulator allows relaxation of the assumption of quasi steady-state. The presented concepts are derived based on the mass transfer limitation of a single inhibitive compound. Although this applies for many environmental and man-made systems (at least for the defined parts of them), the combined rate-limiting effects of several compounds are also common especially in mixing zones and/or highly transient systems. In case the degradation rate is simultaneously limited by two or more compounds, the presented concept still holds if only one compound is inhibitive and mass transfer limited. In this case, the extra reduction of the rate by other compound(s) can be accounted by implementing their spatiotemporal variabilities. If however more than one compound is significantly mass transfer limited or inhibitive, numerical solutions are only able to determine whether the effect of the reduced bioavailability would still be beneficial or not. 3.3. The Reactive Transport Case Study. To demonstrate the practical implication of the combined occurrence of substrate

Figure 4. Comparison between the solution of the inhibition-only (Haldane) kinetics (eq 5, thin lines) with the mixed bioavailabilityinhibition kinetics (eq 7 with ktr = 0.01, thick lines) for some arbitrary values of the inhibition coefficient (ki).

order-reduction technique was applied to reduce the degree of the final expression. Theoretically, this means both eq 5 and 7 have one root equal to zero at zero concentration that can be reduced. Appendix (A) shows the derivation steps of obtaining the following expressions, the solution of which gives the value of cgh. ⎛ 2c 3 + 3(K m + k i)c 2 + ⎜(K m + k i)2 + 2 K m k i ⎝ −

2 qmax k i ⎞ ⎟c + K m k i(K m + k i) = 0 Φ k tr ⎠

⎛ Φ = ⎜⎜1 − ⎝

1−

⎞ ⎟ k tr(2c + k i + K m) ⎟⎠

(10a)

4 qmax ki

2

(10b)

Equation 10a is solved by calculating the value of Φ from eq 10b using the previously presented algorithm in Figure 3. The values of cgh for some arbitrary self-inhibited substrates with halfsaturation constant Km = 0.261μM and different ki and ktr value combinations are listed in Table 1 and also graphically shown in Figure 4 for ktr = 0.01. 3.2. Including Bioavailability Restrictions to Other Types of Rate Expressions. The theoretical concepts presented here for including the bioavailability restrictions to F


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be achieved respectively by Haldane and mixed kinetics. Therefore, they are observed in Figure 5 as the inflection points of their respective kinetics, that is, at these concentrations the functions concavity (or the sign of the curvature) changed. To indicate the potential implication of determining cgh, we assumed a change of the porous medium along the flow path of a biodegradable self-inhibitive substrate: first a medium with bioavailability limitations as above (following the mixed kinetics eq 7) followed by a second medium allowing full bioavailability (following the Haldane kinetics eq 5). Given that bioavailability limitations soften the inhibition effects at c > cgh, a transition between these two media at a flow path length corresponding to this concentration (approximately at length of 22x (Figure 5: Bottom)) would allow for the most effective removal of the substrate leading to the smallest penetration depth along such flow path. 3.4. General Findings and Their Practical Implications. The current study bridges the gap between bioavailability limitations (mass-transfer obstacles, contaminant diffusion, intrapore limitations, etc.) and biological limitations (substrate self-inhibition), and explores their interactions when both limitations are present. A quick review of the available empirical equations describing the substrate inhibition (e.g., refs 27, 28, and 30) reveals that the bioavailability restrictions of an inhibitive substrate were often overlooked in literature. Instead, in the presence of substrate inhibition, the concerns have been mainly centered on Haldane kinetics,5 or on its modifications. The present study is a step forward toward extending both descriptions of bioavailability and inhibition and to formulate them together. The presented results are not restricted to a specific type of environmental system and applies to the systems in which the individual processes are well described by linear exchange model eq 3, Michaelis−Menten eq 2, or Haldane kinetics eq 5. Seeking an analytical model able to capture the combined mechanisms of substrate self-inhibition together with the physical limitations of its bioavailability, we found that bioavailability limitations do not always restrict biodegradation rates and depending on the degree of substrate toxicity and its concentration they can act as a stimulating factor. Apart from determining (and formulating) the toxic dose of substrate to bacteria in the presence of mass-transfer limitations, we were able to show that in substrate concentrations higher than a threshold (cgh) the combination of the two actively limiting factors together (bioavailability and inhibition) is less severe than the presence of one of them (only inhibition). Given the high occurrence of bioavailability limitations in environmental systems and high number of toxic but biodegradable compounds, the presented effects may take place in several natural or artificial systems. While beyond the scope of this study, future research must show how often biodegradation rates in specific real world system are affected by this combination of limiting factors. The current findings not only allow the theoretical description of these two mechanisms together, but also provide a new perspective on the practical consequences of substrate bioavailability. In particular the presented finding may lead to a new perspective for the management of contaminated sites subject to in situ bioremediation and/or for the design of biofilters. For contaminants which have an inhibitory effect on their biodegradation rate the skillful use of mass-transfer limitations can mitigate the toxicity/inhibition effect of the contaminants and can increase the biodegradation rates. For porous media it has been shown that mass transfer and thus

self-inhibition and bioavailability restrictions, the different biodegradation kinetics were numerically solved for a onedimensional reactive transport example describing, for example, a groundwater flow path or a bioreactor with a granular matrix (Figure 5). For the given arbitrary example each rate limiting

Figure 5. Top: Concentration profiles along a transport pathway (e.g., a groundwater flow path). x and t are arbitrary length and time units, Cin = 8 Km; qmax = 0.125Km/t, ktr= 0.00625/t, ki = Km, flow velocity v = 0.125 x/ t, longitudinal dispersivity αL = 1.25 x. Results depend only on the bioavailability number Bn = ktrKm/qmax = 0.05,10 but not on the value of the individual rate parameter. Values for SmaxI and SmaxIB (the concentrations of an inhibitive substrate at which the biodegradation rates are reaching their maximum in the absence and the presence of bioavailability restrictions, respectively) are shown by horizontal dashed lines. Bottom: Concentration profiles of an inhibitive substrate along a transport pathway where a change of medium occurs from bioavailability restricted to fully bioavailable. The medium replacement at cgh = 3.66Km resulted into the most effective substrate degradation and the shortest substrate penetration length.

process, substrate self-inhibition or bioavailability restriction, alone reduced the degradation of substrate significantly, and led to an increased penetration of substrate along the transport pathway. In presence of bioavailability restrictions, the concentration of a self-inhibitive substrate initially reduced faster along the transport pathway in comparison to the case where only self-inhibition was present. The rate of reduction however turned slower for concentrations lower than SmaxIB and eventually reached to a point (cgh = 3.66 Km, calculated using the analytical methods above) where the absence of bioavailability restrictions resulted in higher degradation rates. For this example, SmaxI=Km and SmaxIB = 6 Km (calculated by eq 8) are the concentrations at which the maximum degradation rate can G


Article

Environmental Science & Technology bioavailability depend on the pores size distributions and their heterogeneity.11,24 This implies that more heterogeneous and/or coarse media (exhibiting lower bioavailability) might be more suitable for the treatment of contaminants at high concentrations especially if only a moderate reduction of the concentration levels is required. The inhibition effects become less relevant with further decrease of concentration while the limitations of bioavailability remain still influential. Therefore, if the target values of the bioremediation require major concentration reductions, the bioavailability of the contaminants should increase at lower concentration levels, which requires a gradual transition to a more homogeneous and/or finer medium (exhibiting higher bioavailability). Furthermore, while the usual solution of dealing with toxic aqueous concentrations of contaminants includes the dilution of the mixture, which in turn results in the pollution and eventually treatment of a larger volume of water, the practical use of the presented findings can lead to the design of more sophisticated systems in which an advanced use of mass-transfer limitation mechanisms leads to more efficient bioremediation techniques. Another practical implication of the above findings is relevant for the systems allowing the experimental determination of SmaxIB, that is, the concentration of a self-inhibiting substrate at which the maximum degradation rate (Rmax) is observed in the presence of bioavailability restrictions. When in the absence of bioavailability restrictions all biodegradation parameters of this specific self-inhibitive substrate is known including the value of SmaxI, the difference between SmaxI and SmaxIB provides a measure for the calculation of the mass transfer limiting coefficient ktr. Determining the optimum value of ktr is required for the design of systems in which the highest possible degradation rate (Rmax) is desired at a given toxic concentration of a substrate. Besides, for many systems value of ktr is difficult to obtain experimentally and an additional approach for its determination would provide a more quantitative estimation of mass transfer obstacles and their implication on overall bioavailability of substrate in a given system, and allows in general a better understanding of degradation processes within the system. In addition to the contents relevance for contaminant bioremediation or other biotechnological applications, the presented findings also have ecological implications. While the usual assumption is that the inhibitory effect of a substrate and the limitation of its bioavailability are acting as two additive ratelimiting factors, the results of this study show that such assumptions are only true if the substrate concentrations are below a certain level. At concentrations above this level, it was found that the bioavailability restrictions dampen the substrate inhibitory effects and increase its biodegradation. In another word, the two negative mechanisms compensate each others effect at particular conditions. Predicting/determining the functional response of microbial (or biological) systems to multiple stressors is a complex task, and the effect of two stressors potentially counteracting or assisting each other would be of major relevance for such tasks.

Replacing the last term of eq 7 by the right-side of the above equation, and a simple algebraic rearrangement of the terms followed by an order reduction, gives the quadratic equation: R2 − (2c + k i + K m)R + k i qmax = 0 k tr

(11)

Solution of eq 11 is straightforward and given by, Φ

 ⎞ ⎛  4 qmax k i k tr ⎜ ⎟ (2c + k i + K m) ⎜1 ± 1 − R= 2 ⎟ 2 k (2 c k K ) + + tr i m ⎝ ⎠ (12)

This equation is very similar to the Best eq 4 and a similar technique which was used to find c′gh value can again be employed to find cgh. The rate from eq 12 should be equal to the rate from Haldane eq 5 at cgh. Combining the two equations and rearranging the terms for c gives eq 10a. It may be of interest to note that eq 12 gives both roots of the quadratic eq 11 and for the upcoming calculations the root with the Φ value defined by eq 10b is only valid (with the minus sign). B. Different Equations for Substrate Inhibition. Andrews:3 qmax R= 1 + K m/c + c /k i (13) Neufeld et al.:29 qmax c k i

R=

K mk i + c(K m + k i)

(14)

Webb:34

R=

qmax c(1 + c /k i) K m + c + c 2/k i

(15)

35

Yano and Koga: qmax c R= K m + c + c 3/k i2 Teissier:

(16)

36

R = qmax (exp( −c /k i) − exp( −c /K m))

Aiba et al.: R=

qmax c exp( −c /k i) Km + c

Han and Levenspiel: R = qmax

■

(17)

17

(18)

27

c(1 − c /c°)n c + K m(1 − c /c°)m

(19)

AUTHOR INFORMATION

Corresponding Author

*Phone: +49 89 3187 2560; fax: +49 89 3187 3361; e-mail: [email protected].

■

Notes

APPENDICES A. Derivation of eqs 10a and 10b. Equation 5 can be rewritten as,

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research was accomplished with financial support from the Marie Curie Early Stage Research Training (EST), grant MC-

ki qmax c = R(c + K m)(c + k i) H


Article


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EST-20984 (RAISEBIO). Additional funds were provided by the Helmholtz Association (GReaT MoDE, VG-NG-338). This work benefited from the complementary suggestions of Pierre Regnier (ULB).

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Kinetics of Substrate Biodegradation under the Cumulative Effects of

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