Skip to main page content

• HOME
• Current Issue
• Archive
• Alerts
• About ASM
• Contact us
• Tech Support
• Journals.ASM.org

This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Reprints and Permissions Copyright Information Books from ASM Press MicrobeWorld Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Malakar, P. K. Articles by Barker, G. C. Search for Related Content PubMed PubMed Citation Articles by Malakar, P. K. Articles by Barker, G. C. Agricola Articles by Malakar, P. K. Articles by Barker, G. C.

 Previous Article  |  Next Article 

Applied and Environmental Microbiology, November 2008, p. 7098-7099, Vol. 74, No. 22
0099-2240/08/$08.00+0     doi:10.1128/AEM.01277-08
Copyright © 2008, American Society for Microbiology. All Rights Reserved.

Estimating Single-Cell Lag Times via a Bayesian Scheme

P. K. Malakar^* and G. C. Barker

Institute of Food Research, Norwich Research Park, Colney, Norwich NR4 7UA, United Kingdom

Received 9 June 2008/ Accepted 9 September 2008

Top ABSTRACT INTRODUCTION REFERENCES

 
Network models offer computationally efficient tools for estimating the variability of single-cell lag phases. Currently, optical methods for estimating the variability of single-cell lag phases use single-cell inocula and are technically challenging. A Bayesian network model incorporating small uncertain inocula addresses these limitations.

Top ABSTRACT INTRODUCTION REFERENCES

 
It is possible to measure the variability of growth from single cells by microscopy (4, 5), but these methods are laborious. Optical density measurements provide a rapid method for generating growth data from small inocula. However, the cell density for which a significant optical measurement is obtained is quite high. The detection threshold for this optical method is typically 10⁷ cells ml^–1. Conclusions concerning growth from a single cell to a population of 10⁷ cells must be extrapolated from growth observed after the detection threshold (3). This extrapolation adds to measurement uncertainty when trying to estimate the variability of growth from single cells.

There is a consensus among microbiologists that bacterial cells undergo a period of adjustment (lag phase) when placed in a new environment. After adjustment, an exponentially growing population is established which, ultimately, reaches an upper limit (stationary phase). This population limit is caused by the depletion of nutrients or by the accumulation of waste products of metabolism. Growth is not observed during the lag phase. The lag phase and subsequent growth before the stationary phase can be modeled using a biphasic linear function,

(1)

where N_t is the population size at time t, n is the inoculum size, µ is the specific growth rate, and L is the population lag time.

It has been shown that the population lag phase, under the assumptions of the biphasic linear model, is related to the lag phase of the individual cells which make up the inoculum (1). A bacterial population at time t, grown from an inoculum consisting of n cells, can be represented by

(2)

where µ is the specific growth rate for cells (we assume that this is constant within the cell inoculum). Note that the lag phases of individual cells in the inoculum, L_i, are identically and independently distributed random variables. The natural logarithm of the cell population for a sufficiently long time (when t is greater than the maximum of L_i) is

(3)

Then, from the biphasic growth model, the population lag time K_n, arising from an initial inoculum of size n, is

(4)

and the time it takes to establish a population with size N_b in the exponential phase is

(5)

The Bioscreen (Labsystems, Finland) system is an automated optical density reader. Bacterial suspensions are dosed into wells arranged in a honeycomb pattern in a plate. Each well can hold 400 µl of cell suspension, and the turbidity of the suspension increases as the density of bacterial cells in the suspension increases. The increase of turbidity can be correlated with cell numbers. There are 200 wells in a Bioscreen plate, and the measurement of population growth using this system can generate sufficient data for estimating the variability of single-cell lag phases. If each well initially holds one cell, then for each well

(6)

This can be interpreted as the distribution of times, t_b, for a population of wells and is equivalent to a shifted form of the distribution, L, for individual cell lag times (2).

It is a challenge experimentally to place exactly one cell in all of the wells in a Bioscreen honeycomb plate. If it is possible to introduce one cell into 50% of the wells and two cells into the rest of the wells, then t_b has a mixture distribution

(7)

where the terms in brackets are symbolic representations for distributions that correspond to monodisperse inocula. Generalizing to an uncertain number of cells in each well gives

(8)

where p(n) is the distribution of inoculum sizes. With this construction, efficient statistical methods combined with Bayesian methodology can be used to establish the distribution of lag times of single cells, p(L_i), for small n. The scheme computes the Bayesian posterior for p(L_i) based on prior beliefs and on observed information for t_b from the Bioscreen system. Prior beliefs concerning N_b and µ can be obtained from independent calibration experiments, so these variables include only slight uncertainty.

Practical schemes introduce an uncertain number of cells into each well of a Bioscreen plate by serial dilution of a volume containing a known density of cells. This inoculation leads to a Poisson distribution for the number of cells in a well, and a significant number of wells will be empty. In this case, a well that does not show any signs of growth during an experimental period could still contain a cell (or cells) if this cell has a long lag time. Equation 8 is not sufficient for estimating the lag-phase distribution of cells unless the wells in a Bioscreen plate are certainly not empty (i.e., n is >0).

To address this problem, we can consider the Bioscreen experiment a combination of two events. The first event is the initial inoculation of a well with an uncertain number of cells which is determined by a Poisson distribution and parameterized by the expectation of the cell number . The second event is uncertain growth from cells in the well which is quantified by the specific growth rate µ and the distribution of single-cell lag times (L); a finite distribution parameterized by , within the range (0, d), is a convenient representation for prior belief concerning L. Note that can be a set of parameters; for example, if L has a one-sided truncated exponential distribution (0 < L d), then is the set containing both d and the exponential rate. We reexpress equation 1 as

(9)

where L₀ is a delta function at a t_b of d {p[L₀(t_b = d)] = 1}. We introduce t_c as the duration of the Bioscreen experiment, and if a population of size N_b had been established before or at t_c, in each well, the observed time taken to establish this population is equal to t_b. If a population with size N_b has not been established at time t_c, then t_b is uniformly distributed between t_c and d.

This provides a complete prescription for the estimation of the Bayesian posterior distribution for L based on a Bioscreen observation (i.e., values for t_b). The posterior distribution is suitable for inclusion in quantitative risk assessments. Note that equations 2 to 9 are derived from equation 1.

 
We acknowledge support from BBSRC United Kingdom.

We thank J. Baranyi and A. Metris for their helpful discussions during the preparation of the manuscript.

 
* Corresponding author. Mailing address: Institute of Food Research, Norwich Research Park, Colney, Norwich NR4 7UA, United Kingdom. Phone: 44 1603 255141. Fax: 44 1603 507723. E-mail: pradeep.malakar{at}bbsrc.ac.uk

Published ahead of print on 19 September 2008.

Top ABSTRACT INTRODUCTION REFERENCES

 

1
1. Baranyi, J. 1998. Comparison of stochastic and deterministic concepts of bacterial lag. J. Theor. Biol. 192:403-408.[CrossRef][Medline]
2
2. Baranyi, J., and C. Pin. 1999. Estimating bacterial growth parameters by means of detection times. Appl. Environ. Microbiol. 65:732-736.[Abstract/Free Full Text]
3
3. Francois, K., F. Devlieghere, A. R. Standaert, A. H. Geeraerd, J. F. van Impe, and J. Debevere. 2006. Effect of environmental parameters (temperature, pH and a_w) on the individual cell lag phase and generation time of Listeria monocytogenes. Int. J. Food Microbiol. 108:326-335.[Medline]
4
4. Stringer, S. C., M. D. Webb, S. M. George, C. Pin, and M. W. Peck. 2005. Heterogeneity of times required for germination and outgrowth from single spores of nonproteolytic Clostridium botulinum. Appl. Environ. Microbiol. 71:4998-5003.[Abstract/Free Full Text]
5
5. Wu, Y., M. W. Griffiths, and R. C. McKellar. 2000. A comparison of the Bioscreen method and microscopy for the determination of lag times of individual cells of Listeria monocytogenes. Lett. Appl. Microbiol. 30:468-472.[CrossRef][Medline]

---

Applied and Environmental Microbiology, November 2008, p. 7098-7099, Vol. 74, No. 22
0099-2240/08/$08.00+0     doi:10.1128/AEM.01277-08
Copyright © 2008, American Society for Microbiology. All Rights Reserved.

This article has been cited by other articles:

• Malakar, P. K., Barker, G. C. (2009). Estimating Risk from Small Inocula by Using Population Growth Parameters. Appl. Environ. Microbiol. 75: 6399-6401 [Abstract] [Full Text]  

This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Reprints and Permissions Copyright Information Books from ASM Press MicrobeWorld Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Malakar, P. K. Articles by Barker, G. C. Search for Related Content PubMed PubMed Citation Articles by Malakar, P. K. Articles by Barker, G. C. Agricola Articles by Malakar, P. K. Articles by Barker, G. C.