Extending the Rayleigh Equation to Allow Competing Isotope

Apr 27, 2007 - The Rayleigh equation relates the change in isotope ratio ... distributed over several metabolic pathways each potentially having a dif...
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Environ. Sci. Technol. 2007, 41, 4004-4010

Extending the Rayleigh Equation to Allow Competing Isotope Fractionating Pathways to Improve Quantification of Biodegradation BORIS M. VAN BREUKELEN* Department of Hydrology and Geo-Environmental Sciences, Faculty of Earth and Life Sciences, VU University Amsterdam, De Boelelaan 1085, NL-1081 HV Amsterdam, The Netherlands

The Rayleigh equation relates the change in isotope ratio of an element in a substrate to the extent of substrate consumption via a single kinetic isotopic fractionation factor (R). Substrate consumption is, however, commonly distributed over several metabolic pathways each potentially having a different R. Therefore, extended Rayleigh-type equations were derived to account for multiple competing degradation pathways. The value of R as expressed in the environment appears a function of the R values and rate constants of the various involved degradation pathways. Remarkably, the environmental or apparent R value changes and shows non-Rayleigh behavior over a large and relevant concentration interval if Monod kinetics applies and the half-saturation constants of the competing pathways differ. Derived equations were applied to previously published data and enabled (i) quantification of the share that two competing degradation pathways had on aerobic 1,2-dichloroethane (1,2-DCA) biodegradation in laboratory batch experiments and (ii) calculation of the extent of methyl tert-butyl ether (MTBE) biodegradation shared over aerobic and anaerobic degradation at a field site by means of an improved solution to two-dimensional (carbon and hydrogen) compound-specific isotope analysis (CSIA).

Introduction Compound-specific isotope analysis (CSIA) allows determination of the occurrence of (bio)transformation processes in the environment including nitrate-reduction (1), sulfatereduction (2), methane-oxidation (3), biodegradation of BTEX (benzene, toluene, ethylbenzene, xylenes) compounds (47) and methyl tert-butyl ether (MTBE) (8-10), as well as microbial (11-16) and abiotic degradation (17) of chlorinated ethenes. Moreover, the extent of (bio)transformation as expressed in increasing isotope ratios of elements in the substrate can be quantified by use of the Rayleigh equation (18, 19) that assumes occurrence of a single degradation process associated with a constant kinetic isotopic fractionation factor (R). Kinetic isotopic fractionation factors were determined for a range of substrates transformed via different metabolic pathways by a number of microbial species under various environmental conditions. The variation in R can be considerable among metabolic pathways or among microbial * Corresponding author phone: +31-20-5987393; fax: +31-205989940; e-mail: [email protected]. 4004

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species even if environmental conditions are similar. For example, aerobic degradation of 1,2-dichloroethane (1,2DCA) may occur via two metabolic pathways having isotopic enrichment factors ( ) 1000 × (R-1)) differing by 1 order of magnitude (20). Isotopic enrichment factors for sulfatereduction ranged between 2 and 42‰ depending on the type of sulfate-reducing prokaryote reducing the sulfate (2). Sulfate reducers that oxidize the carbon source completely to CO2 showed greater fractionation (>15-42‰) than sulfate reducers that released acetate as the final product of carbon oxidation (2-19‰) (2). The type of metabolic pathway expressed or the local prokaryote community present thus potentially affects the isotope fractionation factor observed during bacterial sulfate reduction in a particular habitat (2). The large variation observed for R (or ) underpins the difficulty of selecting the appropriate value to accurately quantify the progress of the degradation process in the environment and, furthermore, suggests that the Rayleigh equation allowing only a single R value associated with substrate consumption is rather limited if several degradation pathways having different R values are of importance. The objective of this paper is to derive and apply extended Rayleigh-type equations to previously published data (8, 20) to (i) quantify the share competing metabolic pathways have to biotransformation as shown for aerobic 1,2-DCA biodegradation in laboratory batch experiments and (ii) calculate the precise extent of degradation shared over aerobic and anaerobic biodegradation via an improved solution on twodimensional isotope analysis (e.g., combining two isotope pairs such as 13C/12C and 2H/1H) as shown for MTBE in groundwater below an industrial landfill site.

Isotopic Fractionation Expressed During Substrate Consumption Via Competing Pathways Derivation of an Equation Assuming First-Order Kinetics. If a substrate is being degraded via two pathways, the change in concentration of a light isotope (e.g., 1H, 12C, 14N, 32S) of an element in substrate, S, is described as follows:

dL ) ( - Lk1L - Lk2L)dt

(1)

where L is the concentration of the light isotope, and Lk1 and Lk are the first-order rate constants for L degradation via the 2 first and second degradation pathway, respectively. Since the first-order reaction rates for L and the heavy isotope, H (e.g., 2H, 13C, 15N, 34S), are related via the kinetic isotopic fractionation factors of both processes, R1 ) Hk1/Lk1 for pathway 1, and R2 ) Hk2/Lk2 for pathway 2, the change in H concentration can be described as follows:

dH ) ( - Lk1R1H - Lk2R2H)dt

(2)

Integration of eqs 1 and 2 gives

Lt ) L0e-( k1 + L

Lk

2)t

(3)

and

Ht ) H0e-( k1R1 + L

Lk

2R2)t

(4)

where Ht (and Lt) and H0 (and L0) are the concentrations of H (and L) at t ) t and t ) 0, respectively. Dividing eq 4 by eq 3, using R ) H/L gives

RSt ) RS0e(-( k1 + L

Lk

2)t)([

Lk

10.1021/es0628452 CCC: $37.00

1R1

+ Lk2R2]/[Lk1 + Lk2]-1)

(5)

 2007 American Chemical Society Published on Web 04/27/2007

where RSt is the isotope ratio (H/L) of (an element in) substrate S at time is t and RS0 is the initial isotope ratio. Since Lk is approximately equal to the overall first-order degradation rate constant, k, of a degradation pathway at low natural abundance level of the heavy isotope and at low degree of fractionation (21), we can rewrite eq 5 to

RSt ) RS0 f

([k1R1+k2R2]/[k1+k2]-1)

) R S0 f

(RA-1)

(6)

where f is the fraction of S (H + L) remaining (St/S0 ) e-(k1+k2)t ), and RA is the apparent isotopic fractionation factor as observed for the overall degradation process. If we define the ratio of the first-order rate constant for the first process and the overall first-order rate constant for degradation of S, as F ) k1/(k1+ k2), we obtain the following:

RSt ) RS0 f (FR1+(1-F)R2-1) ) RS0 f (RA-1)

(7)

where RA ) FR1 + (1 - F)R2. Note that eq 7 equals the Rayleigh equation (RSt ) RS0 f (R-1) (18, 19)) if only one degradation pathway occurs (e.g., F ) 0 (RA ) R2) or F ) 1 (RA ) R1)). Equation 7 shows that if two processes degrading the same substrate occur simultaneously, Rayleigh fractionation behavior will be observed where RA is a function of the isotopic fractionation factors of the individual processes (R1 and R2) and their share to the overall degradation rate. Interestingly, RA may take every value in between the range defined by the R values of the individual pathways, depending on the share of the pathways to overall degradation. The general equation for isotope fractionation via n multiple pathways follows from extending eq 6: n(k

RSt ) RS0 f ([Σ1

1R1..knRn)]/[Σ1

n(k

1...kn)]-1)

(8)

Another unit of isotopic fractionation used further in this paper is the kinetic isotopic enrichment factor, , and relates to R as  ) (R - 1) × 1000. Equation 9 is derived from the definition of RA (in eq 7), and shows that the rate ratio, F, of two competing degradation pathways can be calculated from the observed RA or A, and the R or  values associated with the two pathways. Note that for three or more pathways unique solutions are not possible.

F)

RA - R2 A - 2 ) R1 - R2 1 - 2

(9)

Derivation of an Equation Assuming Monod Kinetics. Biodegradation commonly does not follow strictly first-order kinetics but rather Monod kinetics (22):

-

dS S S ) kX ) Vm dt Ks + S Ks + S

(10)

where k is the maximum rate per unit of biomass, X is the biomass concentration, S is the substrate concentration, and Ks is the half-saturation constant. Assuming that the biomass remains constant, kX can be combined to Vm, the maximum rate constant. Derivation of a Rayleigh-type equation considering multiple degradation pathways following Monod kinetics is as not straightforward as that for first-order kinetics. Therefore, simpler approach was chosen. Multiplication of RA as defined in eq 6 with the substrate concentration S, gives the following:

RA )

k1SR1 + k2SR2 rate1‚R1 + rate2‚R2 ) k1S + k2S rate1 + rate2 or A )

rate1‚1 + rate2‚2 rate1 + rate2

(11)

Thus, besides as a function of the rate constants, A may also be described as a function of the reaction rates of the competing reactions. Likewise, for Monod kinetics, eqs 10 and 11 can be combined to form

(

V1 A )

)

( ) (

) )

S S ‚ + V2 ‚ K1 + S 1 K2 + S 2 S S V1 + V2 K1 + S K2 + S

(

(12)

where the subscript of V (Vm), K (Ks), and  denotes the number of the degradation pathway. If [S] . Ks, then S/(Ks + S) approaches 1, and therefore, for pseudo zero-order conditions, eq 12 can be simplified to

A )

V1‚1 + V2‚2 V1 + V2

(13)

while if [S] , Ks, the rate may be approximated as (V/Ks)‚S, since Ks + S ∼ Ks, and consequently for pseudo first-order conditions, eq 12 can be simplified to

V2 V1 ‚ + ‚ K1 1 K2 2 A ) V1 V 2 + K1 K2

(14)

Equations 13 and 14 show that at high or at low substrate concentrations, A reaches a constant value as a function of the Vm, Ks, and  values of the two pathways. However, at intermediate concentrations, A changes from the A value expressed at high concentration to the A value expressed at low concentration. The A value will change as function of the substrate concentration if the Ks values of the two competing pathways are dissimilar as will be shown next. Potential Effects of Monod Kinetics on Isotopic Fractionation. Figure 1 shows A values calculated using eq 12 as function of the substrate concentration for various combinations of Vm and Ks ratios of two pathways consuming the same substrate. The calculated A values were confirmed by numerical model simulations using the approach described in van Breukelen et al. (16). Model results are only presented for one simulation (Figure 1). Figure 1 shows that a change in A value occurs, if Ks values differ, from approximately 2 orders above, down to 1 order below the minimum Ks of the (Ks min) two pathways. At high concentrations, A is determined by the Vm ratio (Figure 1; eq 13), while at low concentrations, both the Vm and Ks ratio affects A (Figure 1; eq 14). The degradation pathway having the lowest Ks value controls the A value at lower concentrations because the lower the Ks value the less the degradation rate becomes impeded at lower concentrations. For example, A of the most extreme case (K1/K2 ) 92 ) 81 and V1/V2 ) 9) approaches a mixture of 90% 1 and 10% 2 at high concentration because pathway one has a 9 times higher rate than pathway two under those conditions. However, if substrate concentrations become decreased during degradation, A changes and steadily shifts toward 2 because the degradation rate of pathway two decreases much less than the rate of pathway one as its substrate affinity is much higher (K2 , K1) and, thereby, becomes the dominant pathway at low concentrations. Figure 1 also suggests a relation between the overall degradation rate and A if Ks values differ. The logarithm of the degradation rate indeed very well correlates to A around intermediate concentration levels (0.5 < log(S/Ks min) < 2; results not shown). Equation 12 illustrated in Figure 1 confirms the hypothesis of Elsner et al. (23) that non-Rayleigh type of isotope evolution can be observed in cases where the VOL. 41, NO. 11, 2007 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 1. Change in apparent kinetic isotopic enrichment factor (EA) as function of the substrate concentration, S, and Vm and Ks ratios (ratio ) 1 if not indicated) of two degradation reactions (numbered 1 and 2) consuming the same substrate, having E1 and E2 as enrichment factors, respectively. The dots are results of a numerical model simulation (K1/K2 ) 9 × 9 ) 81; V1/V2 ) 9). degradation may include parallel pathways whose proportion changes as contaminant degradation proceeds, e.g., due to differences in affinity of the different enzymes. Slater et al. (24) also suggested variation in reaction pathways over the course of degradation as an explanation for the systematic change in isotopic enrichment factor observed during a perchloroethylene (PCE) biodegradation experiment. A change in A during degradation as result of different Ks values of the degradation pathways involved may be a common situation in the environment, since Ks values for many substrates reported in the literature range from 1 to over 3 orders of magnitude (25). For aerobic benzene degradation, for instance, the ratio in Ks varied an order of magnitude between three pure cultures (4.5 mg/L for Pseudomonas aeruginosa, 4.7 mg/L for P. putida, but 47.1 mg/L for P. fluorescens) in experiments conducted under similar conditions (26). The half-saturation constant for benzene degradation is shown in the literature to range from 0.12 mg/L (27) to around 270 mg/L (28). Large ranges in Ks have also been reported in the literature for other compounds including perchloroethylene (PCE; 0.02 (29) to 0.74 mg/L (30)), vinylchloride (VC; 0.22 (29) to 52 mg/L (30)), and 1,2dichloroethane (1,2-DCA; 2.4 to 26 mg/L (31)). Considering (i) the detection limit for CSIA of around 0.01 mg/L, (ii) the usual range in substrate concentration of 10-2 to 103 mg/L, (iii) Ks values in the 10-1 to 102 mg/L range, and (iv) the potentially large Ks ratios among degradation pathways, nonRayleigh behavior may be occurring at the usual concentration levels in the environment if degradation is controlled by two or more degradation pathways having different halfsaturation constants. Note that differences among competing pathways in inhibition constants or half-saturation constants for oxidants consumed in the degradation process may likewise result in non-Rayleigh behavior. Determination of the Distribution of Two Competing Degradation Pathways in Laboratory Microcosm Experiments. Hirschorn et al. (20) observed a significant variation 4006

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in the magnitude of stable carbon isotope fractionation during aerobic biodegradation of 1,2-dichloroethane (1,2-DCA) in a combined dataset on experiments involving in-situ microcosms, enrichment cultures, and microbial strains (Figure 2). Figure 2 shows that the outer limits in  value were determined by the two separate enzymatic pathways that may be involved in microbial degradation of 1,2-DCA: the monooxygenase pathway (SN1 reaction) breaking a C-H bond, and the hydrolytic dehalogenase pathway (SN2 reaction) breaking a C-Cl bond (20). In-situ microcosm (IM) experiments and enrichment cultures (EC) showed intermediate  values with respect to microbial strain endmembers. Interestingly, enrichment cultures from the East Louisiana (EL) site where the SN1 reaction dominated showed much higher  values during the repetition of the experiment 1 year later (EC (2001) versus EC (2000) in Figure 2). Hirschorn et al. (20) suggested that a change in dominant degradation mechanism occurred over time in the enrichment cultures and, furthermore, hypothesized that, based on the  values identified for the pure cultures with known degradation pathways, the degradation mechanisms operative in the microcosms and the enrichment cultures can be deduced. This hypothesis was tested by means of applying eq 9 to the A values of the IM and EC experiments, under the assumption of the simultaneous occurrence of the SN1 (1 ) -3.0 ( 0.2‰), and the SN2 (2 ) -32.3 ( 1.8‰) pathway. Error propagation was applied to calculate 95% confidence intervals of the estimated rate fractions of the SN1 versus the SN2 pathway. The results shown in Figure 2 reveal that IMs from the EL site were nearly fully dominated by reaction SN1, but still had a contribution of 2-11% of reaction SN2, while IMs from the WL site were dominated by reaction SN2, but had a substantial contribution of 15-30% of reaction SN1. Enrichment cultures from the EL site only had a contribution of maximum 3% of the SN2 pathway in 2000. Interestingly, these enrichment cultures experienced a strong shift over time toward pathway SN2 as recorded in 2001, and expressed

FIGURE 2. Isotopic enrichment factors (E) for aerobic 1,2-DCA oxidation as determined by Hirschorn et al. (20) for the SN1 and SN 2 reaction (Pw), enrichment cultures (EC), and in-situ microcosms (IM). The circles are the calculated rate fractions for the SN1 versus the SN 2 pathway for IM and EC experiments. EL is the East Louisiana site, WL is the West Louisiana site. Error bars indicate 95% confidence intervals. at that moment in time a shared distribution over both pathways in a ratio of 0.25-0.38 to 0.62-0.75 for reaction SN1 to SN2 based on the combined dataset of the three replicates. This application shows that eq 9 allows estimation of the distribution of competing pathways at a particular location from the A value provided that the  values associated with the pathways are known. Vice versa, if the distribution of the pathways could be established independently via molecular genetic techniques such as analysis of mRNA levels for particular genes that link to specific metabolic processes, e.g., ref 32, eq 9 allows us to estimate A. Competition among different reaction pathways for the same substrate and its effect on isotopic fractionation has been described for other cases as well. VanStone et al. (17) found a significant negative correlation between the  value and the logarithm of the surface area normalized rate constant during abiotic reduction of cis-dichloroethylene (cDCE) and vinyl chloride (VC) on Fe(0). Besides the possibility of reduced fractionation reflecting reduced mass discrimination that may occur during accelerated reaction rates in kinetically controlled reactions, the authors raised the possibility that these differences in reaction rate and  value are a function of the relative importance of hydrogenolysis versus β-elimination pathways on iron surfaces. Another example is reductive dechlorination of trichloroethylene (TCE) that may be branched to three reaction products: cDCE, trans-dichloroethylene (tDCE) and 1,1-DCE. Although cDCE is usually the main daughter product during reductive dechlorination of TCE, dominance of tDCE (33) or 1,1-DCE (34) has been observed as well. Isotopic fractionation associated with tDCE production is likely considerably larger than for cDCE production, because considerably more depleted tDCE than cDCE was observed during enhanced bioremediation of TCE via lactate addition (12). Variation in branching ratios during TCE degradation may, therefore, possibly explain the large range in  observed for TCE degradation: -2.5 to -13.8‰ (16).

Improved Two-Dimensional Isotope Analysis Selection of the appropriate site-specific A value to enable CSIA based quantification of biodegradation by means of the Rayleigh equation may be subject to large uncertainty since its value may differ considerably depending on the controlling degradation pathway even within one redox regime as seen before for 1,2-DCA. Acquiring  values for each site via lab experiments is time-consuming and multiple

experiments must be performed if the distribution of competing pathways varies spatially. An elegant solution to this problem is the combination of CSIA from two isotope pairs (e.g., 13C/12C and 2H/1H), called “two-dimensional isotope analysis”, and it has been applied before to determine whether methyl tert-butyl ether (MTBE) degradation in the field occurred under either aerobic or anaerobic conditions (8, 10). The current two-dimensional isotope analysis approach, however, cannot cope with biodegradation being shared over two pathways like aerobic fringe and anaerobic core degradation within pollution plumes. Derivation of an Analytical Equation for Improved TwoDimensional Isotope Analysis. This section shows the derivation of an equation that calculates the distribution of two competing pathways based on two-dimensional CSIA, and subsequently, how the extent of biodegradation can be precisely quantified by means of a modified version of the Rayleigh equation. The subsequent equation is obtained after ln transformation of the Rayleigh equation and using  instead of R (21):

1000ln

(

10-3δSt + 1

10-3δS0 + 1

)

) ∆ )  ln f

(15)

where δSt and δS0 are the ratios of the heavy isotope to the light isotope of an element in substrate S at time t ) t and t ) 0, respectively, expressed in the δ notation (δsample (‰) ) ((Rsample - Rreference)/Rreference) × 1000, where Rreference is the isotope ratio of the international standard). The left-hand term equals the isotopic shift, ∆ (‰), at a (downgradient) location with respect to the source. Therefore, by combining ∆13C ) C ln f and ∆2H ) H ln f we can derive the following equation:

∆ 2H H ) ∆13C C

(16)

In the case of two simultaneously occurring metabolic pathways we can combine eqs 16 and 9 to

HA FH1 + (1 - F)H2 ∆ 2H ) )Φ) 13 CA FC + (1 - F)C ∆ C 1 2 VOL. 41, NO. 11, 2007 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

(17)

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FIGURE 3. Improved two-dimensional isotope analysis based on δ13C and δ2H CSIA of MTBE at an industrial landfill site previously investigated by Zwank et al. (8). Black circles are observations, arrows indicate the EH/EC slopes (average, full lines; minimal and maximal, dashed lines) as expected based on literature values (Table 1), shaded areas indicate the distribution, F, of aerobic and anaerobic degradation, and also the extent of biodegradation (%) based on eqs 19-20 is shown as black lines.

TABLE 1. Isotopic Enrichment Factors (EC and EH) for Aerobic and Anaerobic MTBE Degradation aerobic degradation

anaerobic degradation

C

-1.5 ( to -2.4 ( 0.3 ‰ (35, 36); µb ) -1.8 ‰

H

-29 ( 4 to -66 ( 3 ‰ (36); µ ) -41.1 ‰ 12 to 44; µ ) 23

-9.2 ( 5 (9), -13 ( 1.1 (10), -15.6 ( 4.1 ‰ (37); µ ) -12.6 ‰ -16 ( 5 ‰ (10)

a0.1

H/C a

95% confidence interval.

b

1 to 1.7; µ ) 1.3

µ ) average.

and rearranging to isolate F:

F)

ΦC2 - H2 (H1 - H2) - Φ(C1 - C2)

(18)

Thus, the distribution, F, of two degradation pathways transforming the same substrate is a function of the ratio of the isotopic shifts of the two isotope pairs, Φ, and the  values of both isotope pairs associated with the two pathways. Subsequently, the fraction of substrate remaining due to biodegradation, f, can be uniquely determined by use of a modified version of the Rayleigh equation given below (eqs 9 and 15 combined):

f)

(

)

δSdowngradien t + 1000 δSsource + 1000

1000/(F1+(1-F)2)

(19)

The extent of biodegradation (B) follows from

B ) (1 - f ) × 100 (%) 4008

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(20)

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Application of Improved Two-Dimensional Isotope Analysis. To test the improved two-dimensional isotope analysis, eqs 17-20 were applied to the carbon and hydrogen isotope signatures of MTBE in groundwater below an industrial landfill site earlier investigated by Zwank et al. (8), using the isotope signature of the source well (δ13C ) -26.35‰, δ2H ) -71.3‰), the isotope signatures of the downgradient wells (δ13C up to 40.0‰, δ2H up to 60.3‰), and selecting the range and average C and H values from the literature (Table 1). Figure 3 shows that biodegradation at the site is unlikely exclusively associated with anaerobic degradation as previously concluded by Zwank et al. (8) based on an evaluation scheme that in a roundabout way reproduces the slope of the regression between ∆2H and ∆13C ()1.8, R2 ) 0.92; excluding data points with ∆ < 0). The correlation becomes poor (R2 ) 0.20) if the three samples showing the highest extent of degradation following dominantly anaerobic degradation are left out of the regression analysis, indicating that the H/C slope followed by each sample individually, and accordingly, the sharing over aerobic and anaerobic degradation is highly variable. Figure 3 shows that actually only one out of 19 samples was certainly fully subjected to anaerobic degradation, and biodegradation must have occurred for at least 10% under aerobic conditions in all other samples assuming average H/C ratios. Even if the maximum H/C ratios for both pathways were assumed (Table 1, Figure 3), thereby maximizing the contribution of anaerobic degradation, only around a third of the samples were dominantly (>90%) anaerobically degraded. Aerobic degradation was the major pathway (>50% degradation) for 32% (16-63%) of the samples, and the dominant pathway (>90% degradation) for 16% of the samples. The outcome of the current improved two-dimensional isotope analysis, where MTBE must have been considerably degraded aerobically, is not surprising. Although the ground-

water was depleted in oxygen ( H anaerobic, while C aerobic < C anaerobic (Table 1). The improved two-dimensional isotope analysis method presented here gives equal results independent of the isotope pair selected for quantification. Implications for Contaminant Hydrogeology. The present paper shows that, rather than assuming dominance of one out of several possible pathways and selecting the isotopic enrichment factor of that particular pathway, the simultaneous or phased occurrence of several pathways must be considered. Consequently, A for the overall biodegradation process may also vary in time and space if dominance of the one over the other pathway varies, or if microorganisms associated with each pathway are differently distributed in space. Improved two-dimensional isotope analysis is shown to enable accurate prediction of the extent of biodegradation if two competing pathways occur at a field site either simultaneously or subsequently. Note though that CSIAbased quantification of biodegradation by means of (any form of) the Rayleigh equation always slightly underestimates the true extent of biodegradation in physically heterogeneous environments (38, 39); therefore, its estimates may be regarded as being conservative. To enable two-dimensional isotope analysis, a substrate should follow sufficiently different H/C (or for example Cl/

C) slopes associated with each competing pathway. The larger the difference in slope or the wider the “isotope window”, the more accurate shared degradation can be quantified if the extent of biodegradation is low. Note that, at least for one isotope pair, the difference in  among the competing pathways must be considerable, while it may be similar for the other isotope pair. Based on C and H values reported in the literature, two-dimensional isotope analysis must be feasible for other contaminants as well, including 1,2-DCA, to discriminate the monooxygenase from the hydrolytic dehalogenase pathway during aerobic degradation as suggested by Hirschorn et al. (20), and benzene to discriminate among redox conditions (H/C: ∼3.1-8.2 for aerobic degradation (35); ∼12.1-17.5, nitrate-reducing (40); 21.9, sulfatereducing (40); and 28.1-31.6, methanogenesis (40)). Many more organic contaminants may qualify for two-dimensional isotope analysis. However, while C values are usually available, H values (or Cl values for chlorinated hydrocarbons (11), and N values for nitroaromatic compounds (41)) have mostly not yet been determined for all relevant degradation pathways. The varying H/C slopes among the different redox conditions for benzene underline the limitations of twodimensional isotope analysis: the method only applies if the number of relevant pathways occurring at a specific location can be narrowed down to two. Interestingly, the number of isotope pairs measured should equal the (maximum) number of competing degradation pathways that may be uniquely quantified under optimal conditions via multidimensional isotope analysis. For three-dimensional isotope analysis (e.g., δ2H, δ13C, and δ37Cl of chlorobenzenes), each of the three combinations of two isotope pairs defines the distribution of two of the total of three pathways. Since the sum of the distribution of all pathways must equal one, the share of each of the three pathways can, therefore, be uniquely determined and used to calculate the extent of biodegradation as occurred via the three pathways.

Acknowledgments I thank four anonymous reviewers for their suggestions and comments which improved the statistical aspects of this manuscript. The research was supported by the VU University Amsterdam via direct university funding.

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Received for review December 1, 2006. Revised manuscript received February 25, 2007. Accepted March 19, 2007. ES0628452