Question

 

Imagine you are working for the state Department of Health and you are charged with monitoring bacteria at the local reservoir, which is used not only for hydroelectric production but also recreational activities. The bacteria concentration, measures as colony forming units (CFUs), is the following after a storm event:

Choose a linear regression model for this data. The model results should never go below zero, so choose the regression approach carefully. In your model consideration, you want to also ponder whether bacteria can increase in concentration at any point. Use the data to make this judgement.

a. Estimate the bacteria concentration at time = 0.

b. Estimate the time the bacteria concentration will reach 150 CFUs.

Answer 1

ABSTRACT

We developed a new numerical method to estimate bacterial growth parameters by means of detection times generated by different initial counts. The observed detection times are subjected to a transformation involving the (unknown) maximum specific growth rate and the (known) ratios between the different inoculum sizes and the constant detectable level of counts. We present an analysis of variance (ANOVA) protocol based on a theoretical result according to which, if the specific rate used for the transformation is correct, the transformed values are scattered around the same mean irrespective of the original inoculum sizes. That mean, termed the physiological state of the inoculum, αˆ, and the maximum specific growth rate, μ, can be estimated by minimizing the variance ratio of the ANOVA procedure. The lag time of the population can be calculated as λ = −ln αˆ/μ; i.e. the lag is inversely proportional to the maximum specific growth rate and depends on the initial physiological state of the population. The more accurately the cell number at the detection level is known, the better the estimate for the variance of the lag times of the individual cells.

Automated measures are commonly used to estimate bacterial growth parameters. Unfortunately, little information is obtained on the lag phase because the change in the physical properties of a culture (turbidity, conductance, etc.) is detectable only at high cell concentrations. This problem is serious, for example, in food microbiology, where predicting the end of the lag phase is of great importance (5).

Microbiologists traditionally divide bacterial growth curves into lag, exponential, and stationary phases. The maximum specific growth rate, denoted by μ, can be estimated by the slope of the tangent drawn to the inflexion of the sigmoid curve which is fitted to the data representing the natural logarithm of the cell concentration against time. (If log10 is used instead of the natural logarithm, the slope of that tangent is ln 10 ≈ 2.3 times smaller than the real specific rate).

The period of lag, λ, is usually interpreted as the time elapsed from the inoculation to the intercept of the tangent with the level of inoculum (6). Figure ​Figure11 demonstrates these definitions, concentrating on the lag and exponential phases. The most popular functions suitable to fit viable-count growth curves are listed, for example in reference 5. Generating viable-count data, however, is a laborious task, and consequently there is a great interest in finding alternative approaches to estimate bacterial growth parameters.

FIG. 1

The intercept of the inoculum level with the tangent drawn to the exponential phase of the growth curve marks the end of the lag phase. The detection time, Tdet, depends linearly on the lag, λ, if the detection level is in the exponential phase. ...

The parameter α = exp (−μλ) was introduced in reference 2 to quantify the physiological state of the initial population. As shown previously (3), the lag parameter of a bacterial growth curve, also termed population lag, is not a simple arithmetical average of the lag times of the individual cells, τi (i = 1…x0, where x0denotes the initial cell number). The physiological state of the inoculum, however, is equal to the arithmetical mean of the physiological states of the individual cells, the αi = exp (−μτi) quantities. We refer to this as physiological-state theorem. This theorem is valid irrespective of the actual distribution of the individual lag times. The proof of the theorem relies upon a rather geometrical definition of population lag based on the logarithmic representation of bacterial growth.

Consider this biological interpretation of the physiological state of the inoculum. ln x0 is the natural logarithm of the inoculum level, from where growth starts after the lag period, λ. Find another, hypothetical growth curve which will be identical to the previous “real” growth curve in its exponential phase but has no lag (Fig. ​(Fig.1).1). Denote the initial point of this hypothetical growth curve ln x0(hyp). During the lag period of the real growth curve, the hypothetical growth curve increases, on the logarithmic scale, by μλ; hence, α = exp (−μλ) is identical to the α = x0(hyp)/x0 factor. In other words, α expresses the potential fraction of the initial counts which, without lag, could “catch up” with the real growth curve, which does have lag. The extreme values of this fraction are 0 and 1, corresponding to the situations that the real growth curve has “infinitely long lag” and “no lag,” respectively. The α physiological state is a dimensionless parameter quantifying the “suitability” of the culture to the actual environment. It is, in fact, an initial value, just like the inoculum level, from which the lag parameter was derived in reference 3 by λ = −ln α/μ, expressing the idea that the lag is inversely proportional to the maximum specific growth rate and depends on the physiological state of the inoculum as well as on the actual environment.

In this paper, we highlight a useful feature of the physiological-state parameter. We develop a new method, based on the physiological-state theorem and an analysis of variance (ANOVA) procedure, to estimate the maximum specific growth rate and the lag time of a homogeneous bacterial population. The advantage of the method is that it uses detection times, which are the first data available when recording bacterial growth, and allows for the estimation of the within-population variance of lag times.

Go to:

THEORY

First we summarize the mathematical consequences of the physiological-state theorem that we use for our ANOVA procedure.

Suppose that the initial culture consists of x0 cells. Let the individual lag times be denoted by τi (i = 1…x0), and suppose that they are identically distributed, independently of each other. It was demonstrated in reference 3 that if the variance of the generation times is not much larger than that of the individual lag times, the variance of the maximum specific growth rate is negligible. Therefore, we consider μ to be constant in this study.

Let αi = exp (−μτi) (i = 1…x0) denote the individual physiological states. If λ(x0) denotes the population lag, then α(x0) = exp [−μλ(x0)] is the physiological state of the inoculum consisting of x0 cells. According to the physiological state theorem (see the proof in reference 3),

1

Therefore, as is known from mathematical statistics: (i) the expected value of the physiological state of the initial population is the same as the (common) expected value, αˆ, of the individual physiological states: E[α(x0)] = E(αi) = αˆ; (ii) with higher initial counts, the physiological state of the initial population approaches αˆ more closely; and (iii) denoting the (common) variance of the individual physiological states by ν, the rate of the above convergence can be estimated by the relation that the variance of the physiological state of the initial population is x0 times smaller than the variance of the individual physiological states: Var[α(x0)] = Var(αi)/x0 = ν/x0.

ANOVA protocol.

We use the population state theorem to develop an ANOVA procedure for our method. We use the indices i, j, and k to differentiate between x0 cells of an inoculum (i = 1…x0), between n detection times generated by x0(j) initial cells (j = 1…n), and between m groups of identical inoculum levels (k = 1…m).

Suppose that a culture, growing from x0 initial counts, reaches a certain detection level, Xdet, at time Tdet, while still in the exponential phase. As can be seen from Fig. ​Fig.1,1,

2

From this equation, it follows that the detection time does not depend on both Xdet and x0 independently but only on the ratio r = x0/Xdet, which we call the dilution ratio. If the variance of the dilution ratio is negligible, the distribution of the lag times, apart from a constant additive term, is identical to the distribution of the detection times.

Suppose that we measure the T(j) detection times for some subcultures generated by x0(j) initial counts (j = 1,2,…n). Denote α(j) = α[x0(j)]. Then, from equation 2,

3

where

4

Besides, as we have seen,

5

where ν is the common variance of the exp (−μτi) individual physiological states (see above).

For the expected value of the physiological state, an efficient estimation is the weighted average of the α(j)values, where the weights are proportional to the reciprocals of the respective variances. By means of equation 5, after simplification, we obtain

6

where

7

The variance of the ᾱ estimator is

8

Suppose that we dilute a culture from the detection level, Xdet, and obtain n = n1 +…+ nm subcultures, where nk subcultures belong to group k, characterized by the r(k) = rk,1 =…= r1,nk dilution ratio (denote their detection times by Tm,1 ,…, Tm,nk) (k = 1…m).

Let ᾱ(k) be the mean of the kth group of the physiological state observations, and let

9

Then, from equation 3,

10

According to equation 8, the variance of ᾱ(k) is

11

Define the V value as the variance ratio

12

Following the standard ANOVA technique,

As can be checked,

Therefore

13

Calculate the variances in the denominator of equation 12:

14

After substitution, we obtain:

15

The distribution of the V variance ratio, if completed with the respective degrees of freedom, is very close to that of the F-distribution, except that the differences in the summations are not normally distributed. Even so, assuming that the F-statistics are sufficiently robust, the maximum specific growth rate, μ, can be estimated by minimizing the V variance ratio. The advantage of this is that V is dimensionless and independent of the Xdet and x0(j) values and depends only on the dilution ratios and the detection times.

The procedure can be followed in an example given as an Excel sheet in Tables ​Tables11 and ​and2,2, with m = 2 groups (see above).

TABLE 1

Sample data and their organization given in an Excel sheet to follow the ANOVA procedurea

TABLE 2

Respective Excel statements generating the numerical results given in Table 1a

Go to:

MATERIALS AND METHODS

Bacterial strains and turbidity measurements.

We used Pseudomonas putida NCFB 754 (from spoiled milk), Pseudomonas fragi NCFB 2902 (from beef), and Pseudomonas lundensis NCFB 2908 (from minced beef). Frozen strains were grown in tryptone soy broth (TSB, Oxoid/Unipath) at 25°C in three successive 24-h subcultures immediately prior to the experiments. Equal volumes of the cultures were combined in the inoculum. Dilutions were made in TSB to obtain appropriate cell concentrations.

The turbidity of the cultures was measured at 600 nm by Bioscreen (Labsystems, Basingstoke, United Kingdom) in TSB. Microtiter plates containing with 300 μl/well were incubated at 25°C. The initial optical density (OD) was usually 0.11 to 0.12 (due to the media). The Bioscreen was set to record the detection times needed to reach ODdet = 0.15, equivalent to Xdet ≈ 107 cells/well. This estimate was checked by making a series of dilutions from a culture grown in TSB at 25°C for 24 h. The ODs of the dilutions were monitored by the Bioscreen, while bacterial counts were estimated by plating on tryptone soy agar (Oxoid/Unipath).

A culture whose turbidity was equivalent to ODdet = 0.15 was used to produce a total of m = 7 groups of subcultures with different inoculum levels. The groups k = 1…7 were characterized by r1…r7 dilution ratios, where r1 = 10−3 · 2−6 and rk = rk − 1/2 (k = 2…7) because of consecutive binary dilutions. Note that the exact value of Xdet belonging to ODdet is not necessary for our method; it is enough to know that Xdet is reached in the exponential phase.

The data were collected in a Microsoft Excel spreadsheet, and the Solver add-in of the software was used to minimize the calculated V variance ratio with respect to the maximum specific growth rate, μ. Sample data and program are given in Tables ​Tables11 and ​and2,2, respectively.

Go to:

RESULTS

The observed detection times belonging to seven groups of initial counts (k = 1…7) are shown in Fig. ​Fig.2.2. The groups are characterized by the r1 = 10−3 · 2−6 … r7 = 10−3 · 2−12 dilution ratios. Using the Xdet = 107cells/well detection level, corresponding to a turbidity equivalent to ODdet = 0.15, the initial counts in the wells of the lowest inoculum level were around x0 = r7Xdet = 10−3 · 2−12Xdet = 2.44 cells/well. As mentioned above, however, the actual values of x0 or Xdet were not used to estimate the maximum specific growth rate or the population lag.

FIG. 2

Detection times from different inoculum levels obtained by a series of binary dilutions. The lower the inoculum level, the larger is the scatter of the detection times. x, measured detection times; –––, detection time predictions ...

The ANOVA procedure described above estimated μ = 1.07 h−1 for the maximum specific growth rate of the pseudomonads at 25°C. From μ, the mean physiological state of the inoculum was estimated as αˆ = 0.27 (the grand mean is shown in Fig. ​Fig.3).3). The population lag was calculated as λ = −ln αˆ/μ = 1.21 (h).

FIG. 3

Physiological-state values, obtained by transforming the detection times by means of the respective dilution ratios and the estimated specific growth rate. x, physiological-state data; ■, group mean of the physiological-state data generated by ...

To demonstrate, how robust the technique is, Fig. ​Fig.44 shows the scatter and trend of the physiological-state values at two specific growth rates which were obtained by perturbing the calculated μ value. If the specific rate is chosen about 10% lower or higher, the (group means of the) physiological states show an obvious downward or upward tendency, accordingly.

FIG. 4

Demonstration of the robustness of the method. If the maximum specific growth rates are slightly perturbed (μ = 0.95 h−1 [A] or μ = 1.2 h−1 [B] instead of the correct μ = 1.07 h−1 ...

Go to:

DISCUSSION

Detection times, i.e. the times, Tdet(j), taken to reach a detectable population size, Xdet, from different x0(j)initial levels, have been used by other authors to estimate bacterial growth parameters (see for example, reference 4). Unfortunately, the variance of the observed detection times increases as the inoculum size decreases. We overcome this problem by applying the physiological-state theorem of reference 3. An important consequence of this theorem is that the variance of the α = exp (−μTdet)/r value is inversely proportional to the r = x0/Xdet dilution ratio. This relationship was used to develop an ANOVA procedure.

To apply our method, the detection level should be in the exponential phase. If, for example, Xdet is close to the stationary phase, the method underestimates the real specific rate. Another source of error is the possible error in the r dilution ratio.

The physiological-state theorem is valid irrespective of the distribution of the lag times, τi, of the individual cells. An important case, however, when these are exponentially distributed, deserves special attention. In that case, as shown in reference 3, the mean individual lag time is

with the same τ variance, while the variance of the individual physiological states is

Note that the mean individual lag time is larger than the population lag.

Applying the above formulae to our numerical results, the average of the lag times of the individual cells was 2.5 h, with v = 0.084 variance (ca. 0.29 h standard deviation). By using v and the estimated Xdet = 107detection level, the standard deviations of the ᾱ(k) group means can be calculated (assuming an exponential distribution for the individual lag times) from equation 11. These estimated standard deviations are represented by the differences between the dotted lines and the grand mean of the α values in Fig. ​Fig.3.3. The fact that they are close to the standard deviations of the groups (which can be calculated simply from the raw data, irrespective of the exponential assumption) suggests that the distribution of the lag times of the individual cells is, indeed, close to exponential.

An important point in the applicability of the method is that, as follows from the assumptions of the physiological state theorem, the total number of cells in a homogeneous living space should be considered for the inoculum, as well as for the detection level (cells/well), and not just the density of the inoculum. Therefore, a population of, say, 1 cell/ml in a 1-liter volume (1,000 cells altogether) should produce the same lag as a 103-cell/ml concentration in a 1-ml volume. This relationship does not hold in practice because the cells do not grow independently but exchange chemical signals (1) whose effectiveness is dependent on the actual size of the living space. It is beyond the scope of this paper to take this complication into account.

As noted by Renshaw (7), stochastic approaches should be used to study the dynamics of bacterial growth at low population levels, an area of great interest in, for example, food microbiology. The distribution of the detection times of cultures with small initial numbers has not been previously examined in detail and has the potential to be used in the development of stochastic approaches.Automated measures are commonly used to estimate bacterial growth parameters. Unfortunately, little information is obtained on the lag phase because the change in the physical properties of a culture (turbidity, conductance, etc.) is detectable only at high cell concentrations. This problem is serious, for example, in food microbiology, where predicting the end of the lag phase is of great importance (5).

ABSTRACT

We developed a new numerical method to estimate bacterial growth parameters by means of detection times generated by different initial counts. The observed detection times are subjected to a transformation involving the (unknown) maximum specific growth rate and the (known) ratios between the different inoculum sizes and the constant detectable level of counts. We present an analysis of variance (ANOVA) protocol based on a theoretical result according to which, if the specific rate used for the transformation is correct, the transformed values are scattered around the same mean irrespective of the original inoculum sizes. That mean, termed the physiological state of the inoculum, αˆ, and the maximum specific growth rate, μ, can be estimated by minimizing the variance ratio of the ANOVA procedure. The lag time of the population can be calculated as λ = −ln αˆ/μ; i.e. the lag is inversely proportional to the maximum specific growth rate and depends on the initial physiological state of the population. The more accurately the cell number at the detection level is known, the better the estimate for the variance of the lag times of the individual cells.

Microbiologists traditionally divide bacterial growth curves into lag, exponential, and stationary phases. The maximum specific growth rate, denoted by μ, can be estimated by the slope of the tangent drawn to the inflexion of the sigmoid curve which is fitted to the data representing the natural logarithm of the cell concentration against time. (If log10 is used instead of the natural logarithm, the slope of that tangent is ln 10 ≈ 2.3 times smaller than the real specific rate).

The period of lag, λ, is usually interpreted as the time elapsed from the inoculation to the intercept of the tangent with the level of inoculum (6). Figure ​Figure11 demonstrates these definitions, concentrating on the lag and exponential phases. The most popular functions suitable to fit viable-count growth curves are listed, for example in reference 5. Generating viable-count data, however, is a laborious task, and consequently there is a great interest in finding alternative approaches to estimate bacterial growth parameters.

FIG. 1

The intercept of the inoculum level with the tangent drawn to the exponential phase of the growth curve marks the end of the lag phase. The detection time, Tdet, depends linearly on the lag, λ, if the detection level is in the exponential phase. ...

The parameter α = exp (−μλ) was introduced in reference 2 to quantify the physiological state of the initial population. As shown previously (3), the lag parameter of a bacterial growth curve, also termed population lag, is not a simple arithmetical average of the lag times of the individual cells, τi (i = 1…x0, where x0denotes the initial cell number). The physiological state of the inoculum, however, is equal to the arithmetical mean of the physiological states of the individual cells, the αi = exp (−μτi) quantities. We refer to this as physiological-state theorem. This theorem is valid irrespective of the actual distribution of the individual lag times. The proof of the theorem relies upon a rather geometrical definition of population lag based on the logarithmic representation of bacterial growth.

Consider this biological interpretation of the physiological state of the inoculum. ln x0 is the natural logarithm of the inoculum level, from where growth starts after the lag period, λ. Find another, hypothetical growth curve which will be identical to the previous “real” growth curve in its exponential phase but has no lag (Fig. ​(Fig.1).1). Denote the initial point of this hypothetical growth curve ln x0(hyp). During the lag period of the real growth curve, the hypothetical growth curve increases, on the logarithmic scale, by μλ; hence, α = exp (−μλ) is identical to the α = x0(hyp)/x0 factor. In other words, α expresses the potential fraction of the initial counts which, without lag, could “catch up” with the real growth curve, which does have lag. The extreme values of this fraction are 0 and 1, corresponding to the situations that the real growth curve has “infinitely long lag” and “no lag,” respectively. The α physiological state is a dimensionless parameter quantifying the “suitability” of the culture to the actual environment. It is, in fact, an initial value, just like the inoculum level, from which the lag parameter was derived in reference 3 by λ = −ln α/μ, expressing the idea that the lag is inversely proportional to the maximum specific growth rate and depends on the physiological state of the inoculum as well as on the actual environment.

In this paper, we highlight a useful feature of the physiological-state parameter. We develop a new method, based on the physiological-state theorem and an analysis of variance (ANOVA) procedure, to estimate the maximum specific growth rate and the lag time of a homogeneous bacterial population. The advantage of the method is that it uses detection times, which are the first data available when recording bacterial growth, and allows for the estimation of the within-population variance of lag times.

Go to:

THEORY

First we summarize the mathematical consequences of the physiological-state theorem that we use for our ANOVA procedure.

Suppose that the initial culture consists of x0 cells. Let the individual lag times be denoted by τi (i = 1…x0), and suppose that they are identically distributed, independently of each other. It was demonstrated in reference 3 that if the variance of the generation times is not much larger than that of the individual lag times, the variance of the maximum specific growth rate is negligible. Therefore, we consider μ to be constant in this study.

Let αi = exp (−μτi) (i = 1…x0) denote the individual physiological states. If λ(x0) denotes the population lag, then α(x0) = exp [−μλ(x0)] is the physiological state of the inoculum consisting of x0 cells. According to the physiological state theorem (see the proof in reference 3),

1

Therefore, as is known from mathematical statistics: (i) the expected value of the physiological state of the initial population is the same as the (common) expected value, αˆ, of the individual physiological states: E[α(x0)] = E(αi) = αˆ; (ii) with higher initial counts, the physiological state of the initial population approaches αˆ more closely; and (iii) denoting the (common) variance of the individual physiological states by ν, the rate of the above convergence can be estimated by the relation that the variance of the physiological state of the initial population is x0 times smaller than the variance of the individual physiological states: Var[α(x0)] = Var(αi)/x0 = ν/x0.

ANOVA protocol.

We use the population state theorem to develop an ANOVA procedure for our method. We use the indices i, j, and k to differentiate between x0 cells of an inoculum (i = 1…x0), between n detection times generated by x0(j) initial cells (j = 1…n), and between m groups of identical inoculum levels (k = 1…m).

Suppose that a culture, growing from x0 initial counts, reaches a certain detection level, Xdet, at time Tdet, while still in the exponential phase. As can be seen from Fig. ​Fig.1,1,

2

From this equation, it follows that the detection time does not depend on both Xdet and x0 independently but only on the ratio r = x0/Xdet, which we call the dilution ratio. If the variance of the dilution ratio is negligible, the distribution of the lag times, apart from a constant additive term, is identical to the distribution of the detection times.

Suppose that we measure the T(j) detection times for some subcultures generated by x0(j) initial counts (j = 1,2,…n). Denote α(j) = α[x0(j)]. Then, from equation 2,

3

where

4

Besides, as we have seen,

5

where ν is the common variance of the exp (−μτi) individual physiological states (see above).

For the expected value of the physiological state, an efficient estimation is the weighted average of the α(j)values, where the weights are proportional to the reciprocals of the respective variances. By means of equation 5, after simplification, we obtain

6

where

7

The variance of the ᾱ estimator is

8

Suppose that we dilute a culture from the detection level, Xdet, and obtain n = n1 +…+ nm subcultures, where nk subcultures belong to group k, characterized by the r(k) = rk,1 =…= r1,nk dilution ratio (denote their detection times by Tm,1 ,…, Tm,nk) (k = 1…m).

Let ᾱ(k) be the mean of the kth group of the physiological state observations, and let

9

Then, from equation 3,

10

According to equation 8, the variance of ᾱ(k) is

11

Define the V value as the variance ratio

12

Following the standard ANOVA technique,

As can be checked,

Therefore

13

Calculate the variances in the denominator of equation 12:

14

After substitution, we obtain:

15

The distribution of the V variance ratio, if completed with the respective degrees of freedom, is very close to that of the F-distribution, except that the differences in the summations are not normally distributed. Even so, assuming that the F-statistics are sufficiently robust, the maximum specific growth rate, μ, can be estimated by minimizing the V variance ratio. The advantage of this is that V is dimensionless and independent of the Xdet and x0(j) values and depends only on the dilution ratios and the detection times.

The procedure can be followed in an example given as an Excel sheet in Tables ​Tables11 and ​and2,2, with m = 2 groups (see above).

TABLE 1

Sample data and their organization given in an Excel sheet to follow the ANOVA procedurea

TABLE 2

Respective Excel statements generating the numerical results given in Table 1a

Go to:

MATERIALS AND METHODS

Bacterial strains and turbidity measurements.

We used Pseudomonas putida NCFB 754 (from spoiled milk), Pseudomonas fragi NCFB 2902 (from beef), and Pseudomonas lundensis NCFB 2908 (from minced beef). Frozen strains were grown in tryptone soy broth (TSB, Oxoid/Unipath) at 25°C in three successive 24-h subcultures immediately prior to the experiments. Equal volumes of the cultures were combined in the inoculum. Dilutions were made in TSB to obtain appropriate cell concentrations.

The turbidity of the cultures was measured at 600 nm by Bioscreen (Labsystems, Basingstoke, United Kingdom) in TSB. Microtiter plates containing with 300 μl/well were incubated at 25°C. The initial optical density (OD) was usually 0.11 to 0.12 (due to the media). The Bioscreen was set to record the detection times needed to reach ODdet = 0.15, equivalent to Xdet ≈ 107 cells/well. This estimate was checked by making a series of dilutions from a culture grown in TSB at 25°C for 24 h. The ODs of the dilutions were monitored by the Bioscreen, while bacterial counts were estimated by plating on tryptone soy agar (Oxoid/Unipath).

A culture whose turbidity was equivalent to ODdet = 0.15 was used to produce a total of m = 7 groups of subcultures with different inoculum levels. The groups k = 1…7 were characterized by r1…r7 dilution ratios, where r1 = 10−3 · 2−6 and rk = rk − 1/2 (k = 2…7) because of consecutive binary dilutions. Note that the exact value of Xdet belonging to ODdet is not necessary for our method; it is enough to know that Xdet is reached in the exponential phase.

The data were collected in a Microsoft Excel spreadsheet, and the Solver add-in of the software was used to minimize the calculated V variance ratio with respect to the maximum specific growth rate, μ. Sample data and program are given in Tables ​Tables11 and ​and2,2, respectively.

Go to:

RESULTS

The observed detection times belonging to seven groups of initial counts (k = 1…7) are shown in Fig. ​Fig.2.2. The groups are characterized by the r1 = 10−3 · 2−6 … r7 = 10−3 · 2−12 dilution ratios. Using the Xdet = 107cells/well detection level, corresponding to a turbidity equivalent to ODdet = 0.15, the initial counts in the wells of the lowest inoculum level were around x0 = r7Xdet = 10−3 · 2−12Xdet = 2.44 cells/well. As mentioned above, however, the actual values of x0 or Xdet were not used to estimate the maximum specific growth rate or the population lag.

FIG. 2

Detection times from different inoculum levels obtained by a series of binary dilutions. The lower the inoculum level, the larger is the scatter of the detection times. x, measured detection times; –––, detection time predictions ...

The ANOVA procedure described above estimated μ = 1.07 h−1 for the maximum specific growth rate of the pseudomonads at 25°C. From μ, the mean physiological state of the inoculum was estimated as αˆ = 0.27 (the grand mean is shown in Fig. ​Fig.3).3). The population lag was calculated as λ = −ln αˆ/μ = 1.21 (h).

FIG. 3

Physiological-state values, obtained by transforming the detection times by means of the respective dilution ratios and the estimated specific growth rate. x, physiological-state data; ■, group mean of the physiological-state data generated by ...

To demonstrate, how robust the technique is, Fig. ​Fig.44 shows the scatter and trend of the physiological-state values at two specific growth rates which were obtained by perturbing the calculated μ value. If the specific rate is chosen about 10% lower or higher, the (group means of the) physiological states show an obvious downward or upward tendency, accordingly.

FIG. 4

Demonstration of the robustness of the method. If the maximum specific growth rates are slightly perturbed (μ = 0.95 h−1 [A] or μ = 1.2 h−1 [B] instead of the correct μ = 1.07 h−1 ...

Go to:

DISCUSSION

Detection times, i.e. the times, Tdet(j), taken to reach a detectable population size, Xdet, from different x0(j)initial levels, have been used by other authors to estimate bacterial growth parameters (see for example, reference 4). Unfortunately, the variance of the observed detection times increases as the inoculum size decreases. We overcome this problem by applying the physiological-state theorem of reference 3. An important consequence of this theorem is that the variance of the α = exp (−μTdet)/r value is inversely proportional to the r = x0/Xdet dilution ratio. This relationship was used to develop an ANOVA procedure.

To apply our method, the detection level should be in the exponential phase. If, for example, Xdet is close to the stationary phase, the method underestimates the real specific rate. Another source of error is the possible error in the r dilution ratio.

The physiological-state theorem is valid irrespective of the distribution of the lag times, τi, of the individual cells. An important case, however, when these are exponentially distributed, deserves special attention. In that case, as shown in reference 3, the mean individual lag time is

with the same τ variance, while the variance of the individual physiological states is

Note that the mean individual lag time is larger than the population lag.

Applying the above formulae to our numerical results, the average of the lag times of the individual cells was 2.5 h, with v = 0.084 variance (ca. 0.29 h standard deviation). By using v and the estimated Xdet = 107detection level, the standard deviations of the ᾱ(k) group means can be calculated (assuming an exponential distribution for the individual lag times) from equation 11. These estimated standard deviations are represented by the differences between the dotted lines and the grand mean of the α values in Fig. ​Fig.3.3. The fact that they are close to the standard deviations of the groups (which can be calculated simply from the raw data, irrespective of the exponential assumption) suggests that the distribution of the lag times of the individual cells is, indeed, close to exponential.

An important point in the applicability of the method is that, as follows from the assumptions of the physiological state theorem, the total number of cells in a homogeneous living space should be considered for the inoculum, as well as for the detection level (cells/well), and not just the density of the inoculum. Therefore, a population of, say, 1 cell/ml in a 1-liter volume (1,000 cells altogether) should produce the same lag as a 103-cell/ml concentration in a 1-ml volume. This relationship does not hold in practice because the cells do not grow independently but exchange chemical signals (1) whose effectiveness is dependent on the actual size of the living space. It is beyond the scope of this paper to take this complication into account.

As noted by Renshaw (7), stochastic approaches should be used to study the dynamics of bacterial growth at low population levels, an area of great interest in, for example, food microbiology. The distribution of the detection times of cultures with small initial numbers has not been previously examined in detail and has the potential to be used in the development of stochastic approaches.Automated measures are commonly used to estimate bacterial growth parameters. Unfortunately, little information is obtained on the lag phase because the change in the physical properties of a culture (turbidity, conductance, etc.) is detectable only at high cell concentrations. This problem is serious, for example, in food microbiology, where predicting the end of the lag phase is of great importance (5).

ABSTRACT

We developed a new numerical method to estimate bacterial growth parameters by means of detection times generated by different initial counts. The observed detection times are subjected to a transformation involving the (unknown) maximum specific growth rate and the (known) ratios between the different inoculum sizes and the constant detectable level of counts. We present an analysis of variance (ANOVA) protocol based on a theoretical result according to which, if the specific rate used for the transformation is correct, the transformed values are scattered around the same mean irrespective of the original inoculum sizes. That mean, termed the physiological state of the inoculum, αˆ, and the maximum specific growth rate, μ, can be estimated by minimizing the variance ratio of the ANOVA procedure. The lag time of the population can be calculated as λ = −ln αˆ/μ; i.e. the lag is inversely proportional to the maximum specific growth rate and depends on the initial physiological state of the population. The more accurately the cell number at the detection level is known, the better the estimate for the variance of the lag times of the individual cells.

Microbiologists traditionally divide bacterial growth curves into lag, exponential, and stationary phases. The maximum specific growth rate, denoted by μ, can be estimated by the slope of the tangent drawn to the inflexion of the sigmoid curve which is fitted to the data representing the natural logarithm of the cell concentration against time. (If log10 is used instead of the natural logarithm, the slope of that tangent is ln 10 ≈ 2.3 times smaller than the real specific rate).

The period of lag, λ, is usually interpreted as the time elapsed from the inoculation to the intercept of the tangent with the level of inoculum (6). Figure ​Figure11 demonstrates these definitions, concentrating on the lag and exponential phases. The most popular functions suitable to fit viable-count growth curves are listed, for example in reference 5. Generating viable-count data, however, is a laborious task, and consequently there is a great interest in finding alternative approaches to estimate bacterial growth parameters.