### Abstract

- Top of page
- Abstract
- Introduction
- Models for microbial maintenance
- Relationship between three models
- Overall maintenance coefficient and sensitivity analysis
- Implications for microbial ecology modeling
- Conclusions
- Acknowledgements
- References

We attempted to reconcile three microbial maintenance models (Herbert, Pirt, and Compromise) through a theoretical reassessment. We provided a rigorous proof that the true growth yield coefficient (*Y*_{G}) is the ratio of the specific maintenance rate (*a* in Herbert) to the maintenance coefficient (*m* in Pirt). Other findings from this study include: (1) the Compromise model is identical to the Herbert for computing microbial growth and substrate consumption, but it expresses the dependence of maintenance on both microbial biomass and substrate; (2) the maximum specific growth rate in the Herbert (*μ*_{max,H}) is higher than those in the other two models (*μ*_{max,P} and *μ*_{max,C}), and the difference is the physiological maintenance factor (*m*_{q} = *a*); and (3) the overall maintenance coefficient (*m*_{T}) is more sensitive to *m*_{q} than to the specific growth rate (*μ*_{G}) and *Y*_{G}. Our critical reassessment of microbial maintenance provides a new approach for quantifying some important components in soil microbial ecology models.

### Introduction

- Top of page
- Abstract
- Introduction
- Models for microbial maintenance
- Relationship between three models
- Overall maintenance coefficient and sensitivity analysis
- Implications for microbial ecology modeling
- Conclusions
- Acknowledgements
- References

Maintenance requirements of microbial biomass represent the additional consumption of energy and carbon for purposes other than the production of biomass (Marr *et al*., 1963; Anderson & Domsch, 1985a). The early terminology ‘endogenous metabolism’ postulated by Herbert is thought to be equivalent to the maintenance energy requirements (Dawes & Ribbons, 1962, 1964, 1965; Pirt, 1965; Mason *et al*., 1986). Although the concept of maintenance energy is largely studied in starving cells, growing cells should also be included (Dawes & Ribbons, 1962, 1964). Some studies presume that growth is a secondary feature of energy utilization after maintenance purposes (Dawes & Ribbons, 1964). van Bodegom (2007) summarized eight nongrowth components for microbial maintenance: (1) cell motility, (2) osmoregulation, (3) proofreading, synthesis and turnover of macromolecular compounds, (4) defense against O_{2} stress, (5) shifts in metabolic pathways, (6) energy spilling reactions, (7) changes in stored polymeric carbon, and (8) extracellular losses of compounds not involved in osmoregulation. The first four components were classified as the physiological maintenance (van Bodegom, 2007).

Mathematical modeling of the growth of microbial biomass and consumption of substrate usually follows (Tempest & Neijssel, 1984):

- (1)

- (2)

- (3)

where *x* and *s* are the concentrations (contents) of microbial biomass and substrate, respectively; *μ*(*s*) is the observed specific growth rate of microbial biomass (h^{−1}); *q*(*s*) is the observed specific consumption rate of substrate (h^{−1}); and *Y* denotes the apparent growth yield coefficient.

It is noted that Eqn (3) is a general formula relating the growth of microbial biomass to the consumption of substrate (van Bodegom, 2007). However, the expressions for Eqns (1) and (2) are specific and can be different based on different assumptions (Beeftink *et al*., 1990). Two models have been widely used and their major difference is associated with the maintenance component. One is the Herbert model specified by the specific maintenance rate (*a*) in Eqn (1), which may be regarded as an endogenous metabolism rate resulting in consumption of maintenance energy and decrease in the biomass (Beeftink *et al*., 1990). The other is the Pirt model characterized by the maintenance coefficient (*m*) in Eqn (2) representing the consumption of substrate for nongrowth functions (Pirt, 1965).

The two models have caused the debates on the relationship between the two rate constants, that is, *a* and *m*. Most of the studies postulated or derived that the true growth yield (*Y*_{G}) was the key connecting the two parameters (Schulze & Lipe, 1964; Pirt, 1965; Nagai *et al*., 1969; Van de Werf & Verstraete, 1987; Beeftink *et al*., 1990). Another interpretation was that the apparent yield coefficient (*Y*) rather than *Y*_{G} served as the link, and the overall maintenance coefficient was insensitive to the variation in physiological maintenance (van Bodegom, 2007). In addition, Beeftink *et al*. (1990) put forward a model called the ‘Compromise’, based on mechanistic considerations that combined the features of previous models. However, the maximum specific growth rates were regarded as the same for all models, and the solutions for *μ* and *q* from the Compromise model were thought to be between the solutions from the Herbert and Pirt models.

A complete analysis of all the maintenance components is beyond the scope of this study. The present contribution attempts to reconcile the models describing microbial maintenance through clarifying the relationships between the three models and deriving a new equation for the overall maintenance coefficient (*m*_{T}). We also aim to elucidate implications for the modeling of soil organic carbon (SOC) decomposition based on our reassessment of microbial maintenance.

### Relationship between three models

- Top of page
- Abstract
- Introduction
- Models for microbial maintenance
- Relationship between three models
- Overall maintenance coefficient and sensitivity analysis
- Implications for microbial ecology modeling
- Conclusions
- Acknowledgements
- References

Two forms of relationship have been proposed to relate *m* to *a*: one is the commonly used *a* = *Y*_{G}·*m* (Pirt, 1965), the other is *a* = *Y*·*m* (van Bodegom, 2007). Although the first one has been widely used, no rigorous and clear derivations are available. We agree that the first form (*a* = *Y*_{G}·*m*) is correct and the proof is shown as follows.

Two assumptions are made for the derivations: (1) *Y*_{G} is a constant and identical for the three models; and (2) the observed maximum specific growth rate [i.e., maximum *μ*(*s*) in Eqns (4), (6), or (11), denoted by *μ*_{m}] at *s *≫ *K*_{s} for the three models should be equal, as well as the observed maximum specific consumption rate [i.e., maximum *q*(*s*) in Eqns (5), (7), or (8), denoted by *q*_{m}].

When *s *≫ *K*_{s}, *g*(*s*)1, from Eqns (4) and (6), one can derive that

- (12)

Similarly, from Eqns (5) and (7), it follows that

- (13)

Combination of Eqns (12) and (13) shows that

- (14)

Equations (13) and (14) also imply that *μ*_{max,H} > *μ*_{max,P} since both *a* and *m* are greater than 0.

Substitution of Eqns (6) and (7) into (3) shows that

- (15)

where *μ*_{P} = *μ*_{max,P}·*g*(*s*).

Substituting Eqn (14) into (15) gives

- (16)

Similarly, from Eqns (3), (4), and (5), we can derive

- (17)

where *μ*_{H} = *μ*_{max,H}·*g*(*s*).

Equations (16) and (17) imply that *Y* depends on substrate concentration (*s*) since *Y*_{G} has been assumed constant and both *μ*_{H} and *μ*_{P} are *s* dependent.

Previous studies did not correctly or convincingly show the validity of Eqn (14). It seemed that Schulze & Lipe (1964) derived Eqn (14), but they mixed *Y* with *Y*_{G} and presumed that *Y* = *Y*_{G}. Regardless of this equality, there were sign errors in Eqns (31a) and (32) of Schulze & Lipe (1964). The same sign error occurred in Eqn (2) of Marr *et al*. (1963). Pirt (1965) directly defined the relation between *m* and *a* using the same expression as Eqn (14) and then worked out the same formula as Eqn (16) (Pirt, 1982). van Bodegom (2007) thought that *a* = *Y*_{G}·*m* was wrong, but his derivation process was incorrect. Eqn (8b) in van Bodegom (2007) was correct (i.e., the same as Eqn (17) of this study); however, he misrepresented ‘*μ*’ in his Eqn (8b), as equivalent to the ‘*μ*_{H}’ in our study, but the ‘*μ*’ in his Eqn (8a) should be the ‘*μ*_{P}’ in our study. Because of the confusion between *μ*_{H} and *μ*_{P}, his Eqn (9) relating *a* = *Y*·*m* was incorrect.

We show [Eqn (12)] that the maximum specific growth rates in the Herbert and Pirt models are not equal (i.e., *μ*_{max,H} > *μ*_{max,P}). This was not realized by Beeftink *et al*. (1990). In his Eqns (6)-(9) and Figs 1 and 2, he indicated that the maximum specific growth rates (*μ*_{max,H} and *μ*_{max,P}) for the two models were the same, which could result in different observed maximum specific growth rate (*μ*_{m}) or consumption rate (*q*_{m}) using different models. This is inconsistent with our knowledge that the maximum observations (i.e., *μ*_{m} or *q*_{m}) in a given experiment should be the same.

Comparing the Compromise model with the models of Herbert and Pirt, we found that the Compromise model is identical to the Herbert model for *μ* and *q* from the mathematical perspective. However, the Compromise model explicitly expresses that the microbial maintenance is associated with both microbial biomass and substrate and decomposes the overall maintenance into two components in Eqns (8) and (11), respectively. As illustrated in Fig. 1, the solutions of *μ* and *q* from the Herbert and the Compromise model are identical given the additional constraints that *Y*_{G} is the same in all model formulations. The values of *μ* and *q* by the Pirt model are higher than the *μ* and *q* by the Herbert and Compromise models at low substrate concentration. However, with the increasing of substrate concentration, the *μ* and *q* by the three models approach the same *μ*_{m} and *q*_{m}, respectively. Our analysis indicates that the illustrations of Figs 1 and 2 in Beeftink *et al*. (1990) were incorrect.

### Overall maintenance coefficient and sensitivity analysis

- Top of page
- Abstract
- Introduction
- Models for microbial maintenance
- Relationship between three models
- Overall maintenance coefficient and sensitivity analysis
- Implications for microbial ecology modeling
- Conclusions
- Acknowledgements
- References

The overall maintenance coefficient (*m*_{T}) derived by van Bodegom (2007) was based on the relation of *a* = *Y*·*m*. This incorrect relation resulted in an incorrect expression for *m*_{T} [see Eqn (18) of his article]. In this study, we did not consider the partitioning of microbial biomass into inactive/reactive fractions. According to the Compromise model described by Eqns (8) and (11), we can add the two maintenance items (*m*_{q}·*h*(*s*)/*Y*_{G} and *m*_{q}·[1− *h*(*s*)]) to give

- (20)

where *μ*_{G} = *μ*_{max,C}·*h*(*s*) denotes the specific growth rate.

Equation (20) shows that *m*_{T} depends on the physiological maintenance factor (*m*_{q}), the specific growth rate (*μ*_{G}), and the true growth yield (*Y*_{G}) with the assumption of *μ*_{max,C} being a constant. It is evident that *m*_{T} is not a constant with a range of *a* = *m*_{q} ≤ *m*_{T} ≤ *m*_{q}/*Y*_{G} = *m* when 0 < YG < 1. Therefore, the overall maintenance coefficient from the Compromise model is between the values from the Herbert model (i.e., *a*) and the Pirt model (i.e., *m*).

Defining the sensitivity index (Lenhart *et al*., 2002):

- (24)

where *X* represents *m*_{q}, *μ*_{G}, or *Y*_{G}, *I* denotes the sensitivity of *m*_{T} to *X* at *X* = *X*_{0}.

We used the variable ranges *m*_{q} ∈ (0.01, 0.3) h^{−1} and *μ*_{G} ∈ (0.01, 0.85) h^{−1} from Pirt (1982), and *Y*_{G} ∈ (0.2, 0.7) from Devevre & Horwath (2000). Assuming that *m*_{p} and *μ*_{G} follow log-uniform distributions and *Y*_{G} follows a uniform distribution, we computed the median values as 0.06, 0.09, and 0.45 for *m*_{q}, *μ*_{G}, and *Y*_{G}, respectively. Using these medians as *X*_{0}, we finally calculated that the values of *I* were 1.0, 0.1, and 0.2 for *m*_{q}, *μ*_{G}, and *Y*_{G}, respectively. According to Lenhart *et al*. (2002), the sensitivities of *m*_{T} to *m*_{q}, *μ*_{G}, and *Y*_{G} were classified as very high, medium, and high.

We also conducted a sensitivity analysis using the multi-parameter sensitivity analysis (MPSA) method. Different from the above single-point (i.e., medians) analysis, MPSA assesses the parameter sensitivity in the entire parameter space based on the Monte Carlo simulations (Wang *et al*., 2009). The procedure of MPSA is summarized as follows (Wang & Chen, 2012): (1) Select the parameters and determine their value ranges/distributions. (2) Randomly generate a series of parameter values from certain probability distributions within their ranges. (3) Run the model using these parameter sets and compute the objective function values (OBFs). The OBFs are defined as the sum of squared errors between observed and simulated values. In particular, observed values achieve the OBF using the median of the characteristic range for each parameter. (4) Identify which parameter sets are acceptable or unacceptable by comparing the OBFs to a given criterion, for example, the 50th percentile of the sorted OBFs. A parameter set with its OBF less than the criterion is classified as an acceptable one, otherwise it is classified as unacceptable. (5) Evaluate the sensitivity of each parameter by comparing the degree of difference between two cumulative distribution curves for acceptable and unacceptable parameter values. A greater discrepancy between the two curves means higher parameter sensitivity. From the results of MPSA shown in Fig. 2, the sensitivity of *m*_{q} was much higher than that of *μ*_{G} and *Y*_{G}.

Based on the sensitivity analysis from the two methods, we can conclude that the physiological maintenance factor (*m*_{q}) is the most sensitive parameter. In contrast, van Bodegom (2007) found that the overall maintenance coefficient was insensitive to the physiological maintenance, however, this results from an incorrect analysis of *m*_{T}.

### Implications for microbial ecology modeling

- Top of page
- Abstract
- Introduction
- Models for microbial maintenance
- Relationship between three models
- Overall maintenance coefficient and sensitivity analysis
- Implications for microbial ecology modeling
- Conclusions
- Acknowledgements
- References

In the microbial-enzyme modeling of SOC decomposition, dissolved organic carbon (DOC) serves as a substrate for microbial biomass (MBC) (Chapman & Gray, 1986; Conant *et al*., 2011). Regarding the DOC-MBC system, we need to take into account specific growth respiration rate (*R*_{g}), specific maintenance respiration rate (*R*_{m}), specific enzyme synthesis/production rate (*P*_{E}), and specific microbial mortality rate (*r*_{M}) (Ryan, 1990; Blagodatsky *et al*., 2000; Jin & Bethke, 2003; Blagodatskaya *et al*., 2011; Franklin *et al*., 2011). Our critical reassessment of microbial maintenance provides a clear diagram (Fig. 3) for quantifying these components:

- (25)

- (26)

- (27)

Equations (25)-(27) were derived from a re-analysis of the Compromise model with 0 < YG < 1. The overall maintenance coefficient (*m*_{T}) is resolved into two components: the first component accounts for the synthesis and turnover of macromolecular compounds majorly including enzyme synthesis and microbial mortality and the second is the maintenance respiration. It is evident that the notation for the specific maintenance respiration rate (*R*_{m}) in Eqn (26) is similar to *R*_{g} with *m*_{q} instead of *μ*_{max,C} in Eqn (25). Different from the existing models where the maintenance respiration rate is a constant (e.g., Schimel & Weintraub, 2003; Lawrence *et al*., 2009), *R*_{m} derived herein also depends on the concentration of substrate that is included in the function of *h*(*s*). The allowance of specific maintenance respiration rate varying with substrate is consistent with the experimental observations of maintenance energy depending on the specific growth rate (van Verseveld *et al*., 1984), since the specific growth rate is controlled by the substrate concentration for a given maximum specific growth rate.

Equations (26), (27) and Fig. 3 indicate that (1) the parameter *m*_{q} represents a combined specific maintenance rate for both enzyme synthesis and microbial mortality, which are dominant microbial maintenance components (Mandelstam, 1958; van Bodegom, 2007); and (2) a combination of *m*_{q} and *Y*_{G}, that is, (1/*Y*_{G} − 1)·*m*_{q}, denotes the specific maintenance respiration rate under sufficient substrate conditions. The synthesis of enzymes and the mortality of microbial biomass are presumed to be independent of substrate, whereas the maintenance respiration depends upon both microbial biomass and substrate.

It is noted that the values of *m*_{q} and *μ*_{G} used in the above sensitivity analysis come from the continuous-flow cultures (Kuhn *et al*., 1980). In soil conditions, these reaction rates could be one to several orders of magnitude lower than the values shown above (Anderson & Domsch, 1985b). The discrepancy of *m*_{q} and *μ*_{G} in the soil from those in pure-culture conditions might be explained by the facts that the substrate discontinuity or occlusion within soil aggregates and the existence of inactive biomass fractions could slow the reaction rates between microorganisms and substrates (Anderson & Domsch, 1985b; Conant *et al*., 2011; Wang *et al*., 2012b).

### Conclusions

- Top of page
- Abstract
- Introduction
- Models for microbial maintenance
- Relationship between three models
- Overall maintenance coefficient and sensitivity analysis
- Implications for microbial ecology modeling
- Conclusions
- Acknowledgements
- References

Our theoretical reassessment of microbial maintenance provided a rigorous proof that *a* = *Y*_{G}·*m*. Comparison of the three models indicates that the Compromise model is identical to the Herbert model for computing microbial growth and substrate consumption from the mathematical perspective, but the Compromise model is capable of decomposing microbial maintenance into two components [see Eqns (8) and (11)] depending on both biomass and substrate. In contrast to the illustration of Beeftink *et al*. (1990), we proposed a new one (Fig. 1) to show that the maximum specific growth rate (*μ*_{max,H}) in the Herbert model is higher than those (*μ*_{max,P} and *μ*_{max,C}) in the other two models, with the difference attributed to the specific maintenance factor (*m*_{q} or *a*). From the Compromise model, we derived a new expression for the overall maintenance coefficient (*m*_{T}) and found that *m*_{T} was more sensitive to *m*_{q} than to *μ*_{G} and *Y*_{G}. Finally, we proposed an approach to quantify the specific growth respiration rate (*R*_{g}), specific maintenance respiration rate (*R*_{m}), enzyme synthesis rate (*P*_{E}) plus microbial mortality rate (*r*_{M}) in the microbial ecology model. Although the Compromise model was derived on a mechanistic basis and is a compromise between the Herbert and the Pirt model (Beeftink *et al*., 1990), the inability to describe the competition between the growth and maintenance energy requirements is a limitation of this model. Further modifications of the maintenance concepts and models are needed to solve this issue.