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Applied and Environmental Microbiology, February 1999, p. 732-736, Vol. 65, No. 2

Estimating Bacterial Growth Parameters  by Means of Detection Times

József Baranyi^* and Carmen Pin

Institute of Food Research, Reading Laboratory, Reading RG6 6BZ, United Kingdom

Received 8 June 1998/Accepted 15 October 1998

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, , 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  = ln /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, , is usually interpreted as the time elapsed from the inoculation to the intercept of the tangent with the level of inoculum (6). Figure 1 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.

The parameter  = exp (µ) 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, _i (i = 1...x₀, where x₀ denotes 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 _i = exp (µ_i) 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, . Find another, hypothetical growth curve which will be identical to the previous "real" growth curve in its exponential phase but has no lag (Fig. 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 µ; hence,  = exp (µ) is identical to the  = x₀^(hyp)/x₀ factor. In other words,  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  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  = ln /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.

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 _i (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 _i = exp (µ_i) (i = 1...x₀) denote the individual physiological states. If (x₀) denotes the population lag, then (x₀) = exp [µ(x₀)] is the physiological state of the inoculum consisting of x₀ cells. According to the physiological state theorem (see the proof in reference 3),

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, , of the individual physiological states: E[(x₀)] = E(_i) = ; (ii) with higher initial counts, the physiological state of the initial population approaches more closely; and (iii) denoting the (common) variance of the individual physiological states by , 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[(x₀)] = Var(_i)/x₀ = /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, X_det, at time T_det, while still in the exponential phase. As can be seen from Fig. 1,

From this equation, it follows that the detection time does not depend on both X_det and x₀ independently but only on the ratio r = x₀/X_det, 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 ^(j) = [x₀^(j)]. Then, from equation 2, 

where

Besides, as we have seen,

where  is the common variance of the exp (µ_i) individual physiological states (see above).

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

where

The variance of the estimator is

Suppose that we dilute a culture from the detection level, X_det, and obtain n = n₁ +...+ n_m subcultures, where n_k subcultures belong to group k, characterized by the r^(k) = r_k,1 =...= r_1,nk dilution ratio (denote their detection times by T_m,1 ,..., T_m,nk) (k = 1...m).

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

Then, from equation 3, 

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

Define the V value as the variance ratio

Following the standard ANOVA technique,

As can be checked,

Therefore

Calculate the variances in the denominator of equation 12:

After substitution, we obtain:

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 X_det 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 1 and 2, with m = 2 groups (see above).

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 OD_det = 0.15, equivalent to X_det  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 OD_det = 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 = 10³ · 2⁶ and r_k = r_k  1/2 (k = 2...7) because of consecutive binary dilutions. Note that the exact value of X_det belonging to OD_det is not necessary for our method; it is enough to know that X_det 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 1 and 2, respectively.
 

The observed detection times belonging to seven groups of initial counts (k = 1...7) are shown in Fig. 2. The groups are characterized by the r₁ = 10³ · 2⁶ ... r₇ = 10³ · 2¹² dilution ratios. Using the X_det = 10⁷ cells/well detection level, corresponding to a turbidity equivalent to OD_det = 0.15, the initial counts in the wells of the lowest inoculum level were around x₀ = r₇X_det = 10³ · 2¹²X_det = 2.44 cells/well. As mentioned above, however, the actual values of x₀ or X_det were not used to estimate the maximum specific growth rate or the population lag.

The ANOVA procedure described above estimated µ = 1.07 h¹ for the maximum specific growth rate of the pseudomonads at 25°C. From µ, the mean physiological state of the inoculum was estimated as  = 0.27 (the grand mean is shown in Fig. 3). The population lag was calculated as  = ln /µ = 1.21 (h).

To demonstrate, how robust the technique is, Fig. 4 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.

Detection times, i.e. the times, T_det^(j), taken to reach a detectable population size, X_det, 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  = exp (µT_det)/r value is inversely proportional to the r = x₀/X_det 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, X_det 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, _i, 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  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 X_det = 10⁷ detection level, the standard deviations of the ^(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  values in Fig. 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.

J.B. thanks the U.K. Ministry of Agriculture Fisheries and Food for support under project FS 3202.

^* Corresponding author. Mailing address: Institute of Food Research Reading Laboratory, Earley Gate, Whiteknights Rd., Reading RG6 6BZ, United Kingdom. Phone: (44)118 9357000. Fax: (44)118 9357222. E-mail: jozsef.baranyi@bbsrc.ac.uk.

Permanent address: Departamento de Nutricion y Bromatologia III, Facultad de Veterinaria, Universidad Complutense de Madrid, 28040 Madrid, Spain.

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