The author would like to draw the reader’s attention to the following article which is republished in its entirety:

Sawada K, Funakawa S, Kosaki T 2010: Simulating short-term dynamics of non-increasing soil respiration rates by a model using Michaelis-Menten kinetics. Soil Sci. Plant. Nutr., 4, 570-578.

The publisher apologize for any confusion this may have caused.

Original Article

Simulating short-term dynamics of non-increasing soil respiration rates by a model using Michaelis-Menten kinetics

Kozue SAWADA1, Shinya FUNAKAWA^{1} and Takashi KOSAKI^{2}

^{1}Graduate School of Agriculture, Kyoto University, 606-8502 and ^{2}Graduate School of Urban, Environmental Science, TokyoMetropolitan University, Tokyo 192-0364, Japan

correspondence: K. SAWADA, Soil Science Laboratory, Graduate School of Agriculture, Kyoto University, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan. Email: kozue.do@gmail.com

Received 26 November 2009. Accepted for publication 7 May 2010.

Abstract

Short-term dynamics of soil respiration rates over time measured at hourly intervals during less than 12 h after microbial substrates were added to soils can be classified into first-order, zero-order and growth-associated types. To simulate the zero-order type respiration rates, a model using Michaelis-Menten kinetics is proposed, because this kinetics model includes a maximum respiration rate but no increase in microbial biomass. In this model, soil respiration by microorganisms was assumed to be the sum of the mineralization of easily available substrates (R), which include both added glucose (G) and substrates released by disturbance such as a mixing treatment (D) and constant mineralization under steady state conditions. By analyzing the short-term dynamics of previously published respiration rates, which show no increase with time, for a Kazakh forest, a Japanese forest and a Japanese arable soils, the parameter values of z_{r} and D_{0}, which indicate the ratio of respired to utilized R and the initial concentration of D, respectively, were estimated. This allowed simulation of the decreasing concentrations of R and estimation of the parameters V_{max} and K_{M} in the Michaelis-Menten equation. Simulations using the obtained parameter values matched the measured data well. Correlation coefficients (r^{2}) and root mean square errors (RMSE) indicated that the simulations usually matched the measured data, which included not only zero-order respiration rates but also first-order respiration rates. Therefore, the proposed model using Michaelis-Menten kinetics can be used to simulate the short-term dynamics of respiration rates, which show no increase over time, when easily available substrates would be added in soils.

Soil respiration has long been used as an indicator of microbial activity and the biodegradability of organic substrates. In addition, recently, it has become important to simulate soil respiration accurately, because soil respiration measures the release of CO_{2} from soil, CO_{2} being one of the greenhouse gases associated with global warming. Therefore, several mathematical models on soil respiration have been described. In these models, first-order kinetics is often used to simulate the respiration in surface soils (eg. Lundquist et al. 1999). First-order kinetics has also been used in most soil organic matter models (McGill 1996). The respiration rate (v) immediately after the addition of various concentrations of several substrates (S) can be described by Michaelis-Menten kinetics as follows:

where V_{max} is a maximum respiration rate and K_{M} is the Michaelis constant, which is the concentration of S at which half the maximum respiration rate occurs (eg. van Hees et al. 2005). The first-order kinetics can be derived from the Michaelis-Menten kinetics only when the concentrations of S are much less than the concentrations of K_{M} as follows:

Therefore, the utility of first-order kinetics in simulating the dynamics of soil respiration rates is implicitly based on the assumption that the real respiration rate is much less than the maximum respiration rate. On the other hand, respiration arising from microbial growth because of the high substrates for microbial biomass, which would occur in subsoils (Jia et al. 2007) and in rhizoshere (Toal et al. 2000), have been often simulated by models using the Monod kinetics, which have been used frequently in pure culture experiments.

However, the short-term dynamics of soil respiration rates over time measured at hourly intervals can be classified into not only first-order and growth-associated types but also zero-order type (Anderson and Domsch 1985; Sawada et al. 2008, 2009; Stenström et al. 2001). Zero-order soil respiration rates initially remain constant at a maximum respiration rate and then decrease later as the substrate is depleted. Sawada et al. (2008, 2009) suggested that substrates would not be used for biosynthesis in the zero-order type, but would be incorporated into cytoplasm as storage compounds. The short-term dynamics of zero-order type respiration rates cannot be described by Monod kinetics models because microbial growth cannot occur. At the same time, it is not appropriate to apply the first-order kinetics to zero-order type respiration rates, because the initial respiration rate achieves its maximum because of higher substrate concentrations than K_{M} in Michaelis-Menten kinetics. Therefore, it would be reasonable to test the possibility for utilizing the Michaelis-Menten kinetics to simulate zero-order type respiration rates because this equation can include maximum respiration rates but no increase in microbial biomass, unlike the first-order and the Monod kinetics.

Although higher concentrations of easily available substrates than K_{M} do not usually occur in the bulk soil under field conditions, they would be often added in the soils when soils experience temporal C ‘flushes’ in the rhizosphere or around plant residues and/or when manure and compost are applied to arable soils. In addition, a fraction of microbial biomass would be decreased and changed to the easily available substrates when soils are exposed to disturbance such as drying-rewetting, freezing-thawing, cultivation and application of soil disinfectant. Some easily available substrates added would have extremely short mean residence times of 1–10 h (van Hees et al. 2005). Therefore, it would be important to examine the short-term dynamics of respiration rates measured at hourly intervals after the addition of substrates for accurate estimations of soil respiration. In addition, the dynamics of respiration rates would be possible to show the zero-order type after addition of substrates. Therefore, the objective of the present study is to propose a model using Michaelis-Menten kinetics to simulate the short-term dynamics of zero-order type respiration rates over time.

Materials and methods

Data on short-term dynamics of respiration rates

We used the data of Sawada et al. (2008, 2009), which show no increase with time when measured at hourly intervals, for short-term respiration rates without added glucose and after various concentrations of glucose (which were regulated to give the zero-order type respiration rates) were added. These concentrations of added glucose were equivalent to less than 50% of microbial biomass C present in these soils and less than 70 mmol L^{−1} in the soil solution, which were almost similar to those being found in root cells. The data are rewritten in Figure 1(a–c), for the Kazakh forest, the Japanese forest and Japanese arable soils, respectively. The respiration rates used were those until the time when the added glucose was all used by microbes. This time is defined as t_{e}, which is the time of the final measurement. These times were 9, 12 and 6 h for the Kazakh forest, Japanese forest and Japanese arable soils, respectively (Sawada et al. 2008, 2009).

Equations and parameters used in the proposed model

Following Lundquist et al. (1999), the carbon pool utilized by microbes for respiration is divided into R and S, which represent the rapidly and slowly utilized C pools, respectively. In the model proposed in the present study, it is assumed that S is utilized at a steady state, and thus is mineralized constantly. In addition, it is assumed that R is the sum of added glucose (G) and easily available substrate (D) released by disturbance such as a mixing treatment with a spatula before starting measurements, which is not found under steady state conditions and is mineralized based on Michaelis-Menten kinetics. Since R fluctuates with time, we specify the concentration of R at t hours after starting measurements as R_{t} (μg C g^{−1} soil). Therefore, respiration rates are described by,

(1)

where P is CO_{2}-C produced (μg CO_{2}-C g^{−1} soil) and thus dP/dt is the respiration rate (μg CO_{2}-C g^{−1} soil h^{−1}), V_{max} is a maximum respiration rate (μg CO_{2}-C g^{−1} soil h^{−1}), K_{M} is the Michaelis constant, which is the concentration of R at which half the maximum respiration rate occurs (μg C g^{−1} soil), and C is a constant respiration rate assuming constant mineralization of S (μg CO_{2}-C g^{−1} soil h^{−1}). In this equation, it is assumed that V_{max}, K_{M}, and C are constants with time and dP/dt, R_{t} are variables with time.

The concentration of R immediately after starting measurements (R_{0}) is the sum of the added glucose (G_{0}) (μg C g^{−1} soil) and the easily available substrates released by disturbance (D_{0}) (μg C g^{−1} soil),

(2)

In addition, z_{r} is defined as the ratio of cumulative respired to utilized R.

Parameterization of the proposed model

The first step in the parameter estimation for the three soils requires determining the parameter value of C, to separate the constantly mineralized S pool and the rapidly mineralized C pool (R). This is directly determined as the respiration rate in the without added glucose treatment, which becomes constant under steady-state conditions. The respiration rates at 12 and 6 h were used as the parameter values of C for the Japanese forest and arable soils, respectively (Fig. 1b,c and Table 1). The average respiration rate without added glucose from 2 to 9 h was used for the Kazakh forest soil since the respiration rates did not decrease after 2 h, although they varied widely (Fig. 1a and Table 1).

Table 1. Each parameter estimated in the proposed model

The second step estimates the parameter values of z_{r} and D_{0}, which are the ratio of cumulative respired to utilized R, and the initial concentration of R released by disturbances, respectively. The cumulative respiration at t hours (P_{t}) is calculated by accumulating the respiration rates in Figure 1 (Fig. 2). The cumulative amounts of respiration derived from S are calculated as C multiplied by t and increase linearly. Therefore, the cumulative amounts of respiration derived from R are calculated as (P_{t} − C t) (Fig. 2). The fraction z_{r} of R_{0} is all respired until t_{e} hours, which is the time of the final measurements, and can thus be described by,

Since (P_{te} − C t_{e}) is calculated using the respiration data shown in Figure 2 and G_{0} is the concentration of added glucose, a linear regression of (P_{te} − C t_{e}) against G_{0} is obtained (Fig. 3a for example for the Japanese forest soil and Table 1). Since the slope of this linear regression represents z_{r} and the intercept represents z_{r}D_{0}, the values of z_{r} and D_{0} can be estimated for the three soils (Table 1).

The third step estimates the parameter values of V_{max} and K_{M} in the Michaelis-Menten kinetics. Since (P_{t} − C t) is the cumulative amount of respiration derived from R until t hours, (P_{t} − C t)/z_{r} is the concentration of R utilized until t hours. The remaining R at t hours (R_{t}) can be described by,

(5)

Since the parameter values of z_{r} and D_{0} can be estimated as described above, G_{0} is the concentration of added glucose, and (P_{t} − C t) is calculated using the respiration data shown in Figure 2, R_{t} can be calculated (Fig. 3b for example for the Japanese forest soil). The following modification to Eq. 3 is used,

(6)

Since respiration rates derived from R (dP/dt − C) and R_{t} can be obtained (Figs 1b and 3b, respectively, for the Japanese forest soil), (dP/dt − C) plotted against R_{t} can be approximated by the Michaelis-Menten kinetics shown in Eq. (6) and the parameter values of V_{max} and K_{M} can be estimated for the three soils (Fig. 4 and Table 1).

Parameterization of the first-order kinetics

The cumulative respiration after addition of glucose has often been fitted using an integrated form of first-order kinetics (e.g. Saggar et al. 1999). In our study, the cumulative amounts of respiration shown in Figure 2 can be also fitted by the integrated form of the first-order kinetics,

where A is the amount of the C pool converted to CO_{2}-C (μg C g^{−1} soil) and k is a rate constant (h^{−1}). The parameter values in this kinetics can be estimated for the three soils as shown in Table 2.

Table 2. Each parameter estimated by the first-order kinetics

Soil

Added glucose (μg C g^{−1})

A

k

r^{2}

***P < 0.001.

Kazakh forest

300

118 (2.9)***

0.269 (0.017)***

0.995***

400

148 (3.8)***

0.231 (0.013)***

0.997***

500

209 (9.8)***

0.140 (0.011)***

0.997***

600

238 (12.8)***

0.140 (0.013)***

0.996***

Japanese forest

100

141 (3.7)***

0.117 (0.005)***

0.999***

150

152 (2.6)***

0.120 (0.003)***

1.000***

200

182 (4.2)***

0.105 (0.004)***

0.999***

250

200 (6.8)***

0.100 (0.005)***

0.999***

300

246 (13.7)***

0.080 (0.006)***

0.998***

Japanese arable

30

25.5 (0.82)***

0.177 (0.009)***

0.999***

40

30.0 (0.67)***

0.166 (0.006)***

1.000***

50

35.8 (1.83)***

0.141 (0.011)***

0.999***

60

47.9 (4.46)***

0.105 (0.013)***

0.999***

Statistics

The value of C was calculated as the mean and standard errors of triplicate measurements. Other parameter values were estimated using the mean value of triplicate measurements. The linear and nonlinear regression analyses were conducted by a least-squares method to estimate the parameter values in Eq. 4 and 6 and the first-order kinetics, respectively, using Sigmaplot 2002 for Windows Version 8.0 (SPSS Inc 2002).

Two statistical indices were used to evaluate the performance of simulation: the correlation coefficient (r^{2}), which represents the degree of association, and the root mean square error (RMSE), which represents the degree of coincidence. The statistical significances of RMSE were assessed by comparing with the values obtained assuming a deviation corresponding to the 95% confidence interval of the measurements using the standard errors of the triplicate measurements (Smith et al. 1996).

Results and discussion

Use of Michaelis-Menten kinetics to simulate respiration rates

Several authors have applied Michaelis-Menten kinetics to the respiration rates immediately after the addition of various concentrations of substrates to estimate the parameter values of V_{max} and K_{M} (eg. van Hees et al. 2005). However, in the present study, since the initial respiration rates in the zero-order type were very similar for all added glucose concentrations (Fig. 1), Michaelis-Menten kinetics could not be applied to the initial respiration rates. Therefore, not only the initial respiration rates but also all the respiration rates, which were measured until all the added glucose was utilized, were analyzed. To achieve this, the parameter z_{r}, was introduced. It was also assumed that R_{0} includes not only the added glucose (G_{0}) but also easily available substrate released by disturbance (D_{0}). Estimating z_{r} and D_{0} enabled us to obtain the dynamics of decreasing concentrations of R (Fig. 3c for example for the Japanese forest soil), and to decipher the relationship between the respiration rates derived from R (dP/dt − C) and the remaining concentrations of R (R_{t}) using all of the measured respiration rates (Fig. 4). This relationship fitted the Michaelis-Menten kinetics well as shown in Eq. 6, and the parameter values of V_{max} and K_{M} were estimated for all three soils (Fig. 4 and Table 1).

The estimated parameter values of z_{r} were about 20% in all three soils, which was already shown in zero-order type by Sawada et al. (2008, 2009). The estimated D_{0} value was highest in the Japanese forest soil among three soils used in the present study. This is probably because easily available substrates associated with clay minerals and/or occluded within aggregate were released more by the mixing treatment in the Japanese forest soils with higher clay content (45%) than in other two soils (28% and 8% in clay contents). However, actual form and quality of the D_{0} are still unclear, although these would include the low molecular weight compounds, such as organic acids, amino acids and sugars. Therefore, a further study on the nature of the D_{0} would be necessary. The estimated values of V_{max} were approximately proportional to the substrate-induced respiration rates (SIR), which were 27.4, 17.4 and 4.96 μg CO_{2}-C g^{−1} soil h^{−1} in the Kazakh forest, Japanese forest and Japanese arable soils, respectively (Sawada et al. 2009), because the SIR is the initial maximum respiration rate after the addition of sufficient concentration of substrate. The shape of simulating curve by the Michaelis-Menten kinetics, as shown in Figure 4, is determined by the K_{M} divided by the V_{max}. The estimated values of K_{M} were roughly proportional to the estimated V_{max}, although the higher K_{M}/V_{max}, which leads to the gentler curve, was observed in the Japanese forest soil than in the other soils (i.e. calculated values of K_{M}/V_{max} were 3.63, 5.99 and 3.76 h in the Kazakh forest, Japanese forest and Japanese arable soils, respectively). Calculated K_{M}/V_{max} in previous reports were the range from 1 to 10 h, which were the same order as our results, although the majority of these values was obtained using the data on the rate of not respiration but glucose uptake at a few hours after glucose addition (Anderson and Domsch 1986; Anderson and Gray 1990; Coody et al. 1986).

Simkins and Alexander (1984) measured the rate of mineralization of benzoate by an induced population of Pseudomonas sp. at various substrate concentrations in pure culture, and showed that the dynamics of these rates over time are sometimes fitted by the Michaelis-Menten kinetics derived from the Monod kinetics when the initial cell density is much greater than the number of new organisms which could be produced from the substrate since the growth of the population during the course of an experiment becomes insignificant. Although in the pure culture experiment by Simkins and Alexander (1984) it is assumed that the substrates incorporated by cell are utilized by the growth of the population, Sawada et al. (2008, 2009) suggested that soil microbes would not use substrates for biosynthesis but would incorporate those into cytoplasm as storage compounds when the dynamics of the respiration rates showed the zero-order type. Therefore, it is considered that the dynamics of the respiration rates over time after the wider range of substrate concentrations are added would be fitted by the Michaelis-Menten kinetics in soils than in pure culture.

Simulations by the proposed model

The short-term dynamics of respiration rates shown in Figure 1 could be simulated using the estimated constant parameters V_{max}, z_{r}, C, and K_{M} and the variable parameter R_{t} in which initial values of R_{0} were set to the sum of G_{0} and the estimated D_{0}. In Figure 1 the measured and simulated dynamics of the respiration rates over time are compared. The measured data appeared to be well represented by the model simulation (Fig. 1). Table 3 summarizes two statistical indices used to evaluate the performance of the proposed model. The r^{2} values showed that all datasets except for the Kazakh forest soil without added glucose showed high positive correlations between the measured and simulated data (P <0.01) (Table 3). There is no significant correlation between the measured and simulated data in the Kazakh forest soil without added glucose, because the respiration rates were almost constant with time and there is no clear positive or negative trend in the measured and simulated data (Fig. 1a). The RMSE values in all datasets were within the 95% confidence interval of the measured data. The two statistical indices showed that the measured data were adequately represented by the simulation, except for the Kazakh forest soil without added glucose.

Table 3. Statistical indices to evaluate the performance of simulation using the proposed model

Soil

Added glucose (μg C g^{−1})

r^{2}

RMSE

**P < 0.01, ***P < 0.001, * within the 95% confidence interval. NS, non-significant; RMSE, root mean square errors.

Kazakh forest

0

0.104NS

10.9*

300

0.947***

16.4*

400

0.964***

12.7*

500

0.983***

14.7*

600

0.993***

7.7*

Japanese forest

0

0.882***

9.9*

100

0.970***

6.9*

150

0.989***

4.5*

200

0.991***

6.1*

250

0.989***

4.9*

300

0.973***

7.4*

Japanese arable

0

0.873**

6.1*

30

0.982***

8.2*

40

0.971***

9.8*

50

0.981***

8.6*

60

0.966***

10.1*

Comparison between the first-order kinetics and the proposed models

The short-term dynamics of respiration rates were simulated by the differential form of the first-order kinetics using the parameter values shown in Table 2 (Fig. 5). Table 4 summarizes the two statistical indices used to evaluate the performance of the differential form of the first-order kinetics. The r^{2} values were lower than those for the proposed model in all 13 datasets (Tables 3 and 4), indicating a better simulation by the proposed model than by the differential form of the first-order kinetics. The RMSE values for nine datasets out of 13 in simulations by the differential form of the first-order kinetics were higher than in the proposed model (Tables 3 and 4). A possible reason for having some lower RMSE values when using the first-order kinetics than using the proposed model is because one set of parameter values must be used to simulate the respiration rates after addition of various concentrations of glucose for each soil in the proposed model, while 13 different sets of parameter values must be used to simulate the 13 datasets in the first-order kinetics.

Table 4. Statistical indices to evaluate the performance of simulation by the differential form of the first-order kinetics

Soil

Added glucose (μg C g^{−1})

r^{2}

RMSE

**P < 0.01, ***P < 0.001, * within the 95% confidence interval. RMSE, root mean square errors.

Kazakh forest

300

0.895***

23.0

400

0.909***

20.9

500

0.911***

13.2*

600

0.903***

14.8*

Japanese forest

100

0.929***

9.9*

150

0.962***

7.8*

200

0.955***

7.8*

250

0.941***

9.0*

300

0.887***

10.5*

Japanese arable

30

0.938**

9.2*

40

0.965***

7.2*

50

0.945**

8.2*

60

0.869**

8.9*

The parameter values of k in the first-order kinetics decreased with increasing concentrations of added glucose in all soils, which indicated that the turnover rates of glucose decreased with increasing concentrations (Table 2). In addition, the concentrations of R_{0} (= G_{0} + D_{0}) were higher than K_{M} when all concentrations of glucose were added to soils (Table 1). These indicated that the respiration rates after the addition of glucose were determined not only by the concentrations of substrates but also by the amounts and activity of the microbial biomass. In this case, the first-order kinetics should not be used in principle, even if the lower RMSE values were sometimes obtained using the first-order kinetics compared with the proposed model.

Possible cases that might be used by the proposed model

In the proposed model, the concentration of easily available substrate (R) was not constant under steady state conditions but decreased with time. Therefore, the proposed model can be used when high concentrations of easily available substrates are added to the soils. For example, a temporal C ‘flush’ may often occur in the rhizosphere or around plant residues. Rapid increase of substrates is possibly brought, when manure and compost are applied to arable soils, and/or when soils are exposed to rapid disturbance such as drying-rewetting, freezing-thawing, cultivation and application of soil disinfectant. In these cases, the short-term measurements at hourly intervals would be necessary to accurately understand the fate of easily available substrates, which have the rapid turnover rates. In addition, Hess and Schmidt (1995) proposed that it would be better to analyze non-cumulative, rate-based soil respiration data to estimate accurate parameters, because cumulative data accumulates errors while dampening noise. Our results also confirm that analysis of rate-based respiration data are better for selecting the appropriate models and understanding detailed mechanisms of soil microbial respiration. And then, if the short-term dynamics of respiration rates show both the zero-order and the first-order types, the proposed model can be used to simulate these respiration rates and the fate of the easily available substrates.

However, if much higher concentrations of easily available substrates are added to soil and the microbial growth occurs because of sufficient concentrations of substrates for microbial growth, the proposed model cannot be used since it does not simulate the microbial growth. Such situations could be resulted when less humified manure and compost which contain high concentrations of easily available substrates are applied to soil, and/or when the large fraction of microbial biomass is seriously decreased and changes to easily available substrates by catastrophic soil disturbance. Therefore, detailed information is needed to know in what situations microbial growth does not occur. For example, as Sawada et al. (2008, 2009) reported, when glucose was added at rates below 23.6%, 18.7%, and 46.7% of biomass C in Japanese arable and forest soils and a Kazakh forest soil, respectively, where microbial growth did not occur and the proposed model should be used. In addition, although it has often been observed that the decomposition rates of a wide variety of easily available substrates can be described by the Michaelis-Menten kinetics (e.g. van Hees et al. 2005), the form and quality of substrates may influence some parameters in the proposed model. Further study on the form and quality of easily available substrates may be needed to use the proposed model as a practical matter.

Conclusions

In spite of several limitations described above, the use of the proposed model using the Michaelis-Menten kinetics can cover a wide range of situations of microbial respiration when comparable to less concentrations of substrates, relative to microbial biomass, are added to soil. Especially, when the respiration rates show the zero-order type, in which these were determined not only by the concentrations of substrates but also by the amounts and activity of the microbial biomass, and therefore, the first-order kinetics should not be used, the use of the proposed model would be effective. At the same time, parameters obtained by analysis for the proposed model greatly help us to understand the intensity of disturbance or specific behaviors of soil microbes when substrates are added. Further study is needed to apply this model to the possible situations mentioned above.