Scientific Publications - Work Done by Microbiology Reader Bioscreen C

Journal of Microbiological Methods, 2001, 43, 183-196

Comparison of  maximum specific growth rates  and lag times estimated  from absorbance and  viable count data  by different mathematical models

Paw Dalgaard^a and Kostantinos Koutsoumanis<sup>b

a</sup> Danish Institute for Fisheries Research, Department of Seafood Research, Søltofts Plads, Technical University of Denmark, Building 221, DK-2800, Kgs. Lyngby, Denmark
^b Agricultural University of Athens, Laboratory of Microbiology and Biotechnology of Foods, Iera Odos 75, Athens 11855, Greece

Received 15 May 2000; revised 9 September 2000; accepted 25 September 2000. Available online 13 December 2000.

ABSTRACT

Maximum specific growth rate (_max) and lag time () were estimated from viable count and absorbance data and compared for different microorganisms, incubation systems and growth conditions. Data from 176 growth curves and 120 absorbance detection times of serially diluted cultures were evaluated using different mathematical growth models. Accurate estimates of _max and were obtained from individual absorbance growth curves by using the Richard model, with values of the parameter m fixed to 0.5, 1.0 or 2.0 to describing different degrees of growth dampening, as well as from absorbance detection times of serially diluted cultures. It is suggested to apply the two techniques complementarily for accurate, rapid and inexpensive estimation of microbial growth parameter values from absorbance data. In contrast, considerable limitations were demonstrated for the ability of the Exponential, the Gompertz and the Logistic models to estimate _max and values accurately from absorbance data. Limitations of these models were revealed due the wide range of growth conditions studies.

Author Keywords: Automated turbidimetry; Growth models; Growth parameters; Predictive microbiology
 

1. INTRODUCTION

Accurate estimation of microbial growth parameters, particularly maximum specific growth rate (_max) and lag time (), is essential in many areas of microbiology for example, to characterize effects of anti-microbials, optimize microbial media and to develop kinetic models for use in food and fermentation technology. Estimation of microbial growth parameters from absorbance measurements has the advantages of being rapid, non-destructive, inexpensive and relatively easy to automate as compared to many other techniques and particularly when compared to classical viable counts methods. Absorbance measuring devices typically have high detection thresholds in the range ~10⁶–10⁷ bacteria/ml. Consequently, growth rates determined directly from changes in absorbance can only be calculated for relatively dense microbiological cultures (Koch, 1994). Clearly, it is interesting to determine if specific growth rates (_t) estimated from changes in absorbance (ABS) of dense cultures corresponds to values determined by more sensitive measurements, particularly the classical viable count methods (VC). This question is, e.g., very important in predictive food microbiology where growth parameters estimated from absorbance data are used increasingly for development of models. Those models are used to predict the time required for very low levels of spoilage and pathogen microorganisms to reach critical limits under various growth conditions.

Numerous techniques and mathematical growth models have been used for estimation of growth rates and lag times from absorbance data. Most early studies determined ABS values from the linear part of log-transformed absorbance growth curves (Koch, 1994). Later, absorbance data measured directly, corrected for non-linearity, or transformed into equivalent viable counts were fitted to sigmoidal growth models, e.g., the Logistic and the Gompertz equations ( Corman et al., 1986; McClure et al., 1993; Dalgaard et al., 1994; Hudson and Mott, 1994; Begot et al., 1996; Chorin et al., 1997). Combinations of absorbance data and viable counts were also suggested for estimation of microbial growth parameters ( Bréand et al., 1997; Stephens et al., 1997; Augustin et al., 1999). For aerobic cultures and high inoculation levels, Hudson and Mott (1994) found ABS_G values (see nomenclature in Table 1) and VC values to be similar but the absorbance technique underestimated lag time. However, for aerobic cultures and low inoculation levels, ABS_G and ABS_E values were substantially lower than VC (Dalgaard et al., 1994; Neumeyer et al., 1997). In studies of different microorganisms and atmospheres, ABS_L values, obtained by using different forms of the Logistic model, were similar to VC(Dalgaard et al., 1994; Dalgaard and Dalgaard; Nerbrink et al., 1999). Furthermore, absorbance detection times (DT) of serially diluted cultures allowed accurate estimation of ABS_DT values and a method for estimation of lag time from such data was recently suggested (Cuppers and Smelt, 1993; Baranyi and Pin, 1999). As indicated above, kinetic parameters can be estimated accurately from absorbance data but it remains unclear if the various methods and growth models used are specific, e.g., to microbiological species or are broadly applicable to different types of microorganisms and growth conditions.

 

 

Table 1. Nomenclature
 

 

The objective of the present study was to compare _max and values estimated from viable count and absorbance data by different mathematical growth models. Growth of five groups of bacteria, including fermentative and non-fermentative species, were studied by using different (i) incubation systems, (ii) atmospheres, (iii) temperatures and (iv) media with various levels of carbon source, electron acceptor, sodium chloride and buffer. Growth parameters were estimated from 176 viable count and absorbance growth curves as well as from 120 absorbance detection times of serially diluted cultures. The ratio of growth rates R(VC/ABS) and the percentage difference between lag times were used to compare _max and values estimated from viable count and absorbance data.

2. MATERIALS AND METHODS

Mixtures of five to eight strains of Brochothrix thermosphacta, lactic acid bacteria, Photobacterium phosphoreum, Pseudomonas spp. and Shewanella putrefaciens were studied using incubation systems, media and temperatures as shown in Table 2. To obtain a wide range of growth yields and _max values, cultures were incubated under aerobic and anaerobic conditions with and without glucose and the electron acceptor trimethylamine-oxide (TMAO) (Table 2). Due to expected absence of growth under anaerobic conditions, Pseudomonas spp. was only studied with aerobic incubation. S. putrefaciens is highly CO₂ sensitive (Dalgaard, 1995b) and therefore it was not incubated with 100% CO₂.

 

 

Table 2. Microorganisms, incubation systems, growth media and temperatures studied

 

2.1. Bacterial strains

All strains were isolated from spoiled seafood. Isolation and identification of P. phosphoreum (NCIMB 13476-83) and S. putrefaciens (S0, S15, S30, S50 and S100) from Danish seafood were previously described (Dalgaard, 1995a; Dalgaard et al., 1997b). Brochothrix thermosphacta, lactic acid bacteria (LAB), and Pseudomonas spp were isolated in Greece from Boque and Gilt head seabream (Koutsoumanis et al., 1998; Koutsoumanis and Nychas, 1999) by using streptomycin sulphate thallous acetate cycloheximide (actidione) agar (STAA, Oxoid, CM881, supplemented with SR 151), M.R.S. agar (Oxoid, CM 361) and cetrimide–fusidin–cephaloridine (CFC) agar ( Mead and Adams, 1977), respectively. The following tests were used to confirm that the latter three groups of strains belonged to the expected taxonomic groups: shape, size, motility, Gram test, catalase, oxidase and glucose fermentation. Isolates from STAA and M.R.S. agar were further tested for growth on acetate agar, production of gas from glucose and gluconate, final pH in La-broth, NH₃ production and pH increase from arginine metabolism (aerobically and anaerobically with 0.1 and 2.0% glucose), and for production of polymer from sucrose. These tests were carried out as previously described (Dalgaard and Jørgensen, 2000). Isolates from CFC agar were further tested for reduction of TMAO ( Gram et al., 1987) and fluorescence on King agar B ( King et al., 1954).

2.2. Growth experiments

For each combination of microorganism and growth condition, duplicate growth experiments were carried out at constant temperatures. A total of 176 individual growth curves were generate by using both viable counts and absorbance measurements. In addition absorbance detection times in microplates were determined for 120 serially diluted cultures. The incubation systems and media used are shown in Table 2. Precultures were prepared as previously described ( Dalgaard, 1995b) by using APT broth for B. thermosphacta, LAB, and Pseudomonas spp. and GMB for P. phosphoreum and S. putrefaciens. Precultures for the individual growth curves were diluted to provide inoculation levels of 10²–10⁴ cfu/ml for all incubation systems. Precultures to be used for determination of absorbance detection times were diluted to provide an inoculation level of 10⁵–10⁶ cfu/ml followed by four successive 10-fold dilutions.

Flask cultures were incubated with agitation (~100 rpm). Media in Hungate tubes were saturated with 100% N₂ or 100% CO₂ (Dalgaard et al., 1997a) and cultures were incubated without agitation. Ninety-six-well microplates (Nunc™, Roskilde, Denmark) with 250 l of medium per well were used for incubation at 0°C whereas the automated Bioscreen C system with 100-well-microplates containing 300 l per well (Labsystems, Helsinki, Finland) were used at 15 and 25°C.

2.3. Growth measurements

Cultures were appropriately diluted in physiological saline (0.85% NaCl) with peptone (0.1%) before enumeration by spread plating. Cultures incubated in the Bioscreen C were sampled without removing the microplates from the instrument. Holes were carefully melted in microplate lids by using a hot wire. Typically, 12–15 viable count data points were produced for each growth curve. B. thermosphacta, LAB, and Pseudomonas spp. were enumerated on APT agar plates (25°C, 3 days). Long and Hammer agar (van Spreekens, 1974) plates were used for enumeration of P. phosphoreum (15°C, 5 days) and S. putrefaciens (25°C, 3 days). Changes in absorbance of all cultures were measured at 540 nm. At each sampling time, 1.0 ml of medium from agitated flask cultures was dispensed into disposable microcuvettes (Müller ratiolab^®, Dreieich, Germany) and absorbance measured. Hungate tubes were vortexed and absorbance measured directly without opening the tubes. Absorbance was measured with air as the blank with a simple spectrophotometer (Novaspec II, Pharmacia Biotech, Allerød, Denmark). Microplates were agitated for 10 s prior to measurement of absorbance by using a Multiscan RC microplate reader (Labsystems, Helsinki, Finland) or by the automated Bioscreen C system. Absorbance growth curves with at least 20 data points were generated for all incubation systems but in most cases a higher number of data points were obtained (see, e.g., Fig. 3 and Fig. 4). Apparent growth yields were expressed as the difference between the initial and final absorbance of cultures. Non-linearity of the absorbance response was not corrected for by dilution of cultures or mathematically.

2.4. Maximum-specific growth rate (_max)

_max values were estimated from viable counts data (VC) by the log-transformed four-parameter Logistic model (Eq. (1)). From absorbance growth curves, _max values were estimated by the non-transformed four-parameter Logistic model (Eq. (2)), the Richards model ( Eq. (3)), the modified Gompertz model ( Eq. (4)), and the Exponential model ( Eq. (5)) ( Pruitt et al., 1979, Zwietering et al., 1990; Dalgaard et al., 1994). In addition, _max values were estimated from absorbance detection times of 10-fold serially diluted cultures in microplates by using (i) Eq. (6) with data from a single series of detection times (ABS_DT) and (ii) the ANOVA method of Baranyi and Pin (1999) with data from duplicate series of detection times (ABS_BP).

See Table 1 for description of parameters in growth models ( (1), (2), (3), (4), (5) and (6)). The suffix ‘i’ in N_i and DT_i (Eq. (6)) indicate cell levels and absorbance detection times corresponding to different 10-fold serially diluted cultures, respectively. Fig. P( Anonymous, 1999) or Statgraphics ( Anonymous, 1998) were applied for fitting of models including the non-linear regression required to estimate parameter values in several of the growth models indicated above.

The ratio R(VC/ABS) of _max values estimated from viable counts and absorbance data was calculated for each growth experiment and growth model. Only one model (Eq. (1)) was used for estimation of _max values and lag time from viable count data. The reason is that the modified Gompertz model (Gibson et al., 1988; Zwietering et al., 1990) is known to overestimate _max values of typical microbial growth cultures by approximately 10–20%(Baranyi et al., 1993; Dalgaard et al., 1994; Dalgaard, 1995b; Membré et al., 1999) and that the Baranyi model ( Baranyi and Roberts, 1995), also popular in predictive microbiology, provides _max values which are practically identical to those obtained from the less complicated Logistic model (Eq. (1)) and the exponential model ( Dalgaard et al., 1994; Dalgaard, 1995b). For estimation of lag time from viable count data, very similar values have been obtained by the Baranyi model and the Logistic model whereas the modified Gompertz models may provide negative lag time estimates ( Dalgaard, 1995b).

2.5. Lag time ()

values were estimated from viable count growth data (VC) by using Eq. (7) and parameter values obtained from Eq. (1) ( Dalgaard, 1995b).

From absorbance growth curves, values (ABS) were estimated as shown in Fig. 1. Firstly, the time (t_ΔABS) and viable counts (N_ΔABS) corresponding to a 0.05 units increase in absorbance were determined from fitted model parameter values. Lag time (ABS) was then calculated by using Eq. (8). Values of N₀ and N_ΔABS were determined from viable counts for each growth curve. For practical estimation of lag time from absorbance growth curves it is important to determine if N_ΔABS values depends on growth condition for different microorganisms and this was evaluated. In addition, lag time (ABS_BP) was estimated from duplicate series of absorbance detection times by the ANOVA method of Baranyi and Pin (1999). Values of ABS_BP and ABS_BP were estimated by using a Microsoft Excel spreadsheet and Solver add-in (Baranyi and Pin, 1999).

Fig. 1. Bacterial growth curve showing the general relationship between changes in viable count and absorbance of cultures. Significance of the parameters t_ΔABS, N_ΔABS, and N₀, used for estimation of lag time from absorbance growth curves, is indicated.

Lag times determined from viable count and absorbance data were compared by their difference expressed in percent of the inflection point in the Logistic model (t_i in Eq. (1)) as shown in Eq. (9) below. The inflection point t_i in the Logistic model, corresponds to the time when N_t=N_max/2, i.e., the time when log(cfu/ml) is 0.3 units below Log(N_max). Lag times and _max values are frequently used to predict times required for food-related microorganisms to reach high numbers, e.g., numbers corresponding to food spoilage. For this reason, the inflection point (t_i) was chosen to express discrepancy between lag times estimated from absorbance data and viable counts.

3. RESULTS AND DISCUSSION

3.1. Maximum specific growth rate (_max)

3.1.1. Effect of mathematical models and growth yield on R(VC/ABS)

Values of _max and growth yield varied from 0.009 to ~1.0 h⁻¹ and from 0.1 to ~2.2 absorbance units, respectively. Within this range of growth conditions, ABS_L values were independent of growth yield but the average R(VC/ABS_L) value of 1.28±0.74 (AVG±SD, n=176) indicated that some ABS_L and VC values differed considerably as also shown in Fig. 2a. The highest R(VC/ABS_L) values were obtained for the non-fermentative micro-organisms, Pseudomonas spp. and S. putrefaciens, growing in microplate cultures. Omitting these data, resulted in an average R(VC/ABS_L) value of 1.14±0.43 (n=162). Previously, average R(VC/ABS_L) values of 1.00–1.12 were found for various microorganisms growing under different atmospheres, temperatures, pH, NaCl, lactate and acetate levels (Dalgaard and Dalgaard; Augustin et al., 1999; Nerbrink et al., 1999). Clearly, the Logistic model has been appropriate for accurate estimation of _max values in some studies but variability of R(VC/ABS_L) values, as found in the present study, suggests this model (Eq. (2)) as inappropriate for estimation of _max values from absorbance growth curves in general.

 

 

Fig. 2. Ratios, R(VC/ABS), between maximum specific growth rates determined from viable counts by the log-transformed Logistic model (VC) and from individual absorbance growth curves by the Logistic model (A), the Gompertz model (B), the Exponential model (C) and the Richards model (D). For the Richards model fixed values of the parameter ‘m’ of 0.5, 1.0 or 2.0 was used and for each growth curve the ‘m’ value that provide the lowest residual mean square (rms) value was selected. One hundred and seventy-six growth curves with different yields were evaluated by the different models.

 

The Gompertz and the Exponential models ((4) and (5)) underestimated _max values determined from absorbance growth curves resulting in average R(VC/ABS) values of 2.9±2.2 and 3.2±2.2, respectively, (Fig. 2b,c). It is noteworthy that ABS_G and ABS_E values depended strongly on growth yield. In fact, _max values were underestimated by as much as 10–20-fold for some cultures with low growth yield (Fig. 2b,c). Clearly, these models cannot be used in general for estimation of _max values from absorbance growth curves. In agreement with the present study, the Gompertz and the Exponential models were found previously to under estimate _max values resulting in an average R(VC/ABS) value of ~1.6 for aerobic microbiological cultures, with high growth yields (Dalgaard et al., 1994; Neumeyer et al., 1997). A strong effect of growth yield on R(VC/ABS) values was not documented in those previous studies due to the limited range of environmental conditions studied. Growth yields of microbial cultures are little effected by relatively wide ranges of temperatures and water activities, at least for some microorganisms (Krist et al., 1998). However, parameters like pH, carbon substrates, electron acceptors, extreme growth conditions, incubation systems and many other factors influence microbial growth yields substantially. For studies of such growth conditions, the Gompertz and the Exponential models should be avoided because estimated _max values will not reflect actual growth rates but the combined effect of growth conditions on growth rates and growth yields. In fact, the effect of growth yield on ABS_G and ABS_E may explain the substantial differences in max values previously estimated from absorbance of cultures in agitated flasks, microplates and fermentors at similar environmental conditions (Begot et al., 1996; Potvin et al., 1997). The present study does not support the recommendation of using the Gompertz model ( Eq. (4)) for direct estimation of growth rates from absorbance growth curves ( Begot et al., 1996). Furthermore, Fig. 2b shows that calibration factors of 1.5–1.6 as previously used to correct for the difference between VC and ABS_G (Dalgaard et al., 1994; Neumeyer et al., 1997) only can be applied successfully, for a limited range of growth conditions resulting in high growth yields.

The pronounced differences in ABS values obtained by the Logistic model as compared to the Gompertz and the Exponential models arose because the two later models determine ABS values as slopes of log-transformed absorbance growth curves. In contrast, the parameter ABS_L in Eq. (2) does not correspond to any particular slope of an absorbance growth curve as is shown by the differential form of this model ( Eq. (10)). The differential model shows the specific growth rate at time t (_t) to differ from the maximum specific growth rate (ABS_L) when absorbance of a culture (ABS_t) approaches the maximal absorbance (ABS_max). Consequently, the maximum specific growth rates (ABS_L) may be accurately estimated even in situation where changes in absorbance are only determined at times when growth is dampened. Of course, the Logistic model only provides accurate estimates of _max values where dampening of growth curve corresponds to the growth dampening dictated by this model (Eq. (10)).

In the present study, shapes of absorbance growth curves clearly influenced R(VC/ABS_L) values (Fig. 3). A slow dampening of growth observed for some aerobic and microplate cultures resulted in R(VC/ABS_L) values >1.0, whereas abrupt dampening of growth, seen, e.g., with limitation of carbon substrate, provided R(VC/ABS_L) values <1.0. (Fig. 3). The Richards model includes a parameter (m in (3) and (11)) that allows this model to simulate growth curves with different degrees of dampening. However, the integrated form of the Richards model ( Eq. (3)) provided stable parameters estimates only for 61 out of the 176 absorbance growth curves studied (results not shown). This confirmed the Richards model to have poor statistical properties and simple reparametrization is unlikely to overcome the problem as previously reported (Ratkowsky, 1983, pp. 73–75). Nevertheless by using fixed values of 0.5, 1.0 or 2.0 for the parameter m, stable ABS_RF estimates were obtained and the Richards model then provided substantially more accurate estimates of _max values from absorbance growth curves than the Logistic model (Fig. 2 and Fig. 3). As seen by comparison of (2) and (3) as well as (10) and (11) the Logistic model is a special case of the Richards model with m=1.0. The average R(VC/ABS_RF) value was 1.07±0.34 (n=176) as compared to 1.28±0.74 for R(VC/ABS_L) determined by the Logistic model. The most appropriate of the fixed values of 0.5, 1.0 or 2.0 for the parameter m in the Richards model can in most cases be determined simply by visual inspection of growth curves (Fig. 3). Furthermore, the m value that provided the closest ABS_RF and VC values also resulted in the lowest residual mean square (rms) value (Fig. 3). Thus appropriate fixed m value for estimation of ABS_RF values from absorbance growth curves can be determined easily.

 

 

Fig. 3. Effect of growth dampening on the ratio R(VC/ABS). VC was estimated from viable counts by Eq. (1) and ABS from absorbance growth curves by using the Richards models (Eq. (3)) with fixed values of the parameter ‘m’ of 0.5, 1.0 and 2.0. For each absorbance growth curve the solid line shows the fit of data to the Richards model with the m value providing the lowest residual mean squares (rms) value. () S. putrefaciens grown aerobically at 25°C; () S. putrefaciens grown in microplate cultures at 25°C; () B. thermosphacta grown aerobically 15°C in APT broth with 2% glucose; () B. thermosphacta grown under 100% N₂ at 15°C in APT broth with 1% glucose; and () lactic acid bacteria grown under 100% N₂ at 15°C in APT broth without glucose.

 

3.1.2. Effects of incubation systems, physiology of microorganisms and media on R(VC/ABS_RF)

The average R(VC/ABS_RF) value was close to 1.0 but values for microplate cultures of the non-fermentative microorganisms, Pseudomonas spp. and S. putrefaciens, differed from the remaining data (Table 3 and Fig. 2). The high R(VC/ABS_RF) values observed for this combination of incubation system and microbial physiology showed that _max values cannot always be accurately estimated from absorbance growth curves. Limited diffusion of oxygen into culture media in microplates most likely explain the high R(VC/ABS_RF) values for non-fermentative microorganisms. As an example, VC values were similar for Pseudomonas spp. growing in agitated flasks and in microplates cultures but ABS_RF values differed for the two incubation systems (Fig. 4). Growth yields of the non-fermentative microorganisms depended strongly on oxygen availability and slow increases in absorbance of microplate cultures (Fig. 4b) most likely reflected a limited diffusion of oxygen into the culture medium rather than the growth rate potential of the non-fermentative microorganisms. Omitting data for the non-fermentative microorganisms growing in microplates, resulted in an average R(VC/ABS_RF) value of 1.01±0.19 (n=162) as compared to 1.14±0.43 for R(VC/ABS_L).

Table 3. Effect of incubation systems and atmospheres on the ratio R(VC/ABS_RF) of growth rates determined from viable counts by Eq. (1) and from absorbance data by using the Richards model ( Eq. (3)) or absorbance detection times of serially diluted cultures ( Eq. (6))

 

Fig. 4. Growth at 25°C of Pseudomonas spp. in agitated Erlenmeyer flask (A) and in Bioscreen C microplates (B). () Viable counts; () absorbance measurements. Due to a very high number of measurements in the Bioscreen experiments individual data points cannot be discriminated.

 

Low inoculation levels of 10²–10⁴ cfu/ml were used in the present study and VC values therefore corresponded to _max, i.e., the growth rate potential of cultures as the conditions studied (Eq. (10)). Consequently, ABS_RF values, as determined from absorbance growth curves by the Richards model with fixed m values, were accurate estimates of _max values, except for non-fermentative organisms growing in microplates (Table 3). It is worth noting that the Richards model estimated _max values accurately despite of the fact that non-linearity of absorbance data had not been corrected for by dilution of cultures or by mathematical transformation of the absorbance data. Non-linearity of absorbance data can be very substantial but, in agreement with the present study, the effect on ABS_L values was previously quantified and found to be insignificant (Dalgaard et al., 1994).

The cell morphology of B. thermosphacta changes between coccobacilli and chains of rods depending on growth phase. Rattanasomboon et al. (1999) concluded that turbidimetry, for this reason, overestimated the specific growth rate. The present study was unable to confirm these observations. In fact, the Richards model estimated _max values and lag times of B. thermosphacta accurately (Table 3 and Table 4). It has also been pointed out that calibration functions were needed for estimation of _max values from absorbance growth curves when growth conditions influence the relation between absorbance and viable counts (Baranyi and Roberts, 1995; Chorin et al., 1997). However, estimation of _max values from absorbance data by the Richards model, as suggested in the present study and discussed above, was sufficiently robust to overcome effects of growth conditions on cell size and absorbance non-linearity (Table 3).

 

 

Table 4. D%(VC−ABS) values and cell levels corresponding to an increase in absorbance of 0.05 units (N_ΔABS)
 

 

Absorbance detection times of 10-fold serially diluted cultures provided an average R(VC/ABS_DT) value of 0.96±0.14 (SD), when simple linear regression was applied for estimation of ABS_DT values (Eq. (6)). The more complicated ANOVA procedure suggested by Baranyi and Pin (1999) resulted in a similar R(VC/ABS_BP) value of 0.97±0.16 (S.D.). The dilution methods provided accurate estimates of _max values for all microorganisms studied including non-fermentative microorganisms growing in micro-plates (Table 3). This supported the hypothesis that only the upper part of absorbance growth curves for these cultures were influenced by oxygen limitation.

3.2. Lag time ()

3.2.1. Estimation of from individual absorbance growth curves

Lag times estimated from viable count growth curves varied between 0 and 175 h in the 176 growth experiments. Lag times estimated from absorbance growth curves by Eq. (9) depend on values of t_ΔABS, N_ΔABS, N₀ and ABS (see Fig. 1 and Table 1). The time until absorbance of a culture increases by 0.05 units (t_ΔABS) is easily determined but estimation of N_ΔABS values can be more problematic as growth conditions may influence the size of microbiological cells and thereby N_ΔABS. With N₀ and N_ΔABS determined from individual viable count growth curves, average D%(VC−ABS) values were close to zero (Table 4). Thus, lag times were accurately estimated from individual absorbance growth curves, except for micro-plate cultures of non-fermentative microorganisms, particularly Pseudomonas spp., where large negative lag time estimates were obtained (Table 4). Omitting these data, that resulted from poorly estimated _max values, provided an average D%(VC−ABS) value of −0.9±10.3 (SD) (n=162). This showed VC and ABS to be similar but there was a very close correlation (r=0.97) between R(VC/ABS_RF) and D%(VC−ABS) values showing the small differences between VC and ABS to be caused, almost exclusively, by differences between VC and ABS_RF.

For practical estimation of lag time from absorbance growth curves, N_ΔABS values obviously, cannot be determined by viable counts for each individual growth curve. With average N_ΔABS values, determined for each combination of microorganism and incubation system, D%(VC−ABS) became −0.9±13.9 (SD) (n=162). In this case, the correlation between R(VC/ABS_RF) and D%(VC−ABS) values was reduced (r=0.92), indicating that both ABS and N_ΔABS values influenced lag time estimation. Variation in Log(N_ΔABS) values, in the present study (Table 4), could not be related to systematic effects of growth conditions on cell size, possibly with the exception of S. putrefaciens where cell size increase with temperature of incubation (results not shown). Previously, glycerol was shown to influence cell size of Bacillus cereus in such a way that cultures without glycerol and with 7.74 Log(cfu/ml) had the same absorbance as cultures of 7.93 Log(cfu/ml) with 20% glycerol (Chorin et al., 1997). Growth conditions that influence the size if microbiological cells clearly reduce precision of lag times estimated by Eq. (8) from absorbance growth curves where N_ΔABS in practice must be assumed to be constant. Nevertheless, if growth conditions like glycerol only change N_ΔABS by 0.2 Log(cfu/ml) lag time estimates in many cases will be sufficiently accurate to be useful in practice.

As shown in the previous section, ABS values estimated by using the Logistic, the Gompertz and the Exponential models were substantially less accurate and less precise than values obtained by the Richards model. Consequently, lag times determined from ABS_L, ABS_G, and ABS_E values will often be inaccurate and of insufficient precision for use in practice. In agreement with this conclusion, lag time of micro-plate cultures of Listeria monocytogenes could not be determined accurately from absorbance data by using the Logistic model (Augustin et al., 1999). These authors suggested lag time could be estimated by combined use of absorbance and viable count measurements. In contrast, the present study showed that lag time could be estimated accurately and with a reasonable precision from absorbance growth curves by using the Richards model ( Table 4).

Estimation of N₀ values is required for calculation of lag time by Eq. (8). However, when a single pre-culture is used for inoculation of a larger number of absorbance cultures, as in factorially designed experiments, costs and efforts required to determine N₀, e.g., by viable counting or direct microscopy are modest.

3.2.2. Estimation of from detection times of serially diluted cultures

Detection times for duplicated series of 10-fold diluted cultures provided an average D%(VC−ABS) value of −5.6±10.9 (S.D.) when calculated by the ANOVA procedure of Baranyi and Pin (1999). Baranyi and Pin (1999) in their study evaluated a higher number of replications of serially diluted cultures but presented no experimental data to show if kinetic parameters obtained by their new method corresponded to values from classical methods like viable count. To estimate lag time from absorbance data, the present study indicates individual absorbance growth curves, in most cases, will be as accurate and more cost efficient than the ANOVA procedure requiring replicated series of diluted cultures. However, for lag time estimation of non-fermentative microorganisms in microplate cultures, dilution methods are more appropriate than individual absorbance growth curves ( Table 4). Microplate systems are often convenient but can be difficult to operate at low temperatures and under modified atmosphere conditions. It seems logical to use individual growth curves and serially diluted cultures complementarily and thereby allow lag times to be estimated from absorbance data for wide ranges of microorganisms and growth conditions.

4. CONCLUSIONS

The Richards model, with values of the parameter m fixed to 0.5, 1.0 or 2.0, allowed _max values to be estimated accurately from absorbance growth curves. Accurate _max values were obtained independently of the growth yield of cultures and for wide ranges of growth conditions. In contrast, the Gompertz and the Exponential models were inappropriate for estimation of _max values under conditions that influenced growth yields. The Richards model estimated _max values more precisely than the Logistic model and this enabled lag times to be determined from individual absorbance growth curves. Accuracy and precision of _max values and lag times obtained by the Richards model, corresponded to values estimated from absorbance detection times of serially diluted cultures.

The application of absorbance measurements, as suggested here, is useful in many areas of microbiology where accurate as well as rapid and inexpensive estimation of microbial growth parameters is required. Clearly, turbidimetry is limited by high detection thresholds of measuring devices and to study growth of pathogenic microorganisms, sometimes important in even very low levels, these techniques are restricted to conditions where high cell densities are reached. In contrast, spoilage bacteria are only important in foods when then growth to high levels occurs. For such microorganisms the absorbance techniques evaluated here will most often be of considerable practical importance.

ACKNOWLEDGEMENTS

This study was supported by the European Commission through the projects ‘Predictive modelling of shelf-life of fish and meat products’ (AIR2-CT93-1251), ‘Development, modelling and application of time temperature integrator systems to monitor chilled fish quality (FAIR-CT95-1090)’ and ‘Predictive models of microbial growth in food (COST 914)’. We thank Ole Mejlholm and Rene Poulsen from DIFRES for skilful technical assistance and Tom Ross, University of Tasmania in Australia, for valuable comments on the manuscript. Józef Baranyi, Institute of Food Research, Norwich, UK, kindly provided a Microsoft Excel spreadsheet to facilitate application the Baranyi and Pin (1999) variance ratio method for lag time estimation.

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