INTRODUCTION

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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.

View larger version (10K): [in this window] [in a new window]   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, , if the detection level is in the exponential phase. The parameter µ can be used to define a hypothetical inoculum level from which a growth curve, without lag, is able to catch up with the real curve with lag.

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.

	THEORY

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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),

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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).

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	MATERIALS AND METHODS

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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.

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	RESULTS

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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.

View larger version (11K): [in this window] [in a new window]   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 obtained by the new method.

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).

View larger version (15K): [in this window] [in a new window]   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 the same inoculum level; ---, grand mean, , the estimate for the mean physiological state of the initial population; ···, the expected theoretical deviation from the grand mean, calculated with Xdet = 107 detection level and assuming an exponential distribution for the individual lag times.

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.

View larger version (13K): [in this window] [in a new window]   FIG. 4.   Demonstration of the robustness of the method. If the maximum specific growth rates are slightly perturbed (µ = 0.95 h1 [A] or µ = 1.2 h1 [B] instead of the correct µ = 1.07 h1), the physiological-state values show a strong downward and upward tendency, respectively. For explanations of symbols, see the legend to Fig. 3.

	DISCUSSION

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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.