Maximum Likelihood based comparison of the specific growth rates for P. aeruginosa and four mutator strains K.R. Philipsen a, ⁎, L.E. Christiansen a , L.F. Mandsberg b , O. Ciofu b , H. Madsen a a DTU Informatics, Technical University of Denmark, Richard Petersens Plads, Building 321, DK-2800, Kgs. Lyngby, Denmark b Department of International Health, Immunology and Microbiology, University of Copenhagen, Blegdamsvej 3B, DK-2200, Copenhagen, Denmark abstract article info Article history: Received 26 June 2008 Received in revised form 28 August 2008 Accepted 29 August 2008 Available online 5 September 2008 Keywords: Bacterial growth Likelihood-ratio test Maximum Likelihood Specific growth rate Parameter estimation The specific growth rate for P. aeruginosa and four mutator strains mutT, mutY, mutM and mutY–mutM is estimated by a suggested Maximum Likelihood, ML, method which takes the autocorrelation of the observation into account. For each bacteria strain, six wells of optical density, OD, measurements are used for parameter estimation. The data is log-transformed such that a linear model can be applied. The transformation changes the variance structure, and hence an OD-dependent variance is implemented in the model. The autocorrelation in the data is demonstrated, and a correlation model with an exponentially decaying function of the time between observations is suggested. A model with a full covariance structure containing OD-dependent variance and an autocorrelation structure is compared to a model with variance only and with no variance or correlation implemented. It is shown that the model that best describes data is a model taking into account the full covariance structure. An inference study is made in order to determine whether the growth rate of the five bacteria strains is the same. After applying a likelihood-ratio test to models with a full covariance structure, it is concluded that the specific growth rate is the same for all bacteria strains. This study highlights the importance of carrying out an explorative examination of residuals in order to make a correct parametrization of a model including the covariance structure. The ML method is shown to be a strong tool as it enables estimation of covariance parameters along with the other model parameters and it makes way for strong statistical tools for inference studies. © 2008 Elsevier B.V. All rights reserved. 1. Introduction Proper estimation of growth parameters is essential in many areas, for instance in determining the effect of antimicrobial treatment (Dalgaard and Koutsoumanis, 2001) or when modelling growth of bacteria in food processing and storage (Juneja et al., 2007; Shama et al., 2005). Furthermore, it is very important to be able to tell whether the growth of different bacteria strains is identical. This can form the basis of in vivo or in vitro experiments, such as competition experiments (Montanari et al., 2007), where two or more bacteria are competing to survive and overtake the population. If the growth rates of bacteria strains are not identical in a normal unstressed environ- ment, this will affect the result of a competition experiment carried out in a stressed environment, e.g. by adding antibiotics. Thus, it is very important to correctly determine whether the growth rates are identical. Bacterial growth is typically classified by the maximum growth rate μ max and the lag time (Baty and Delignette-Muller, 2004), when the growth rate is considered to be time dependent. Alternatively the growth is described by a Monod expression (Monod, 1949), which depends on the substrate content and contains the para- meters μ max and the OD value were half the maximum growth is reached, κ 50 . The Monod model should be considered when not enough substrate is available to reach intolerable numbers of bacteria before the growth rate decreases due to substrate depletion (Zwietering et al., 1990). The objective of the current study is to determine whether the growth of P. aeruginosa and four mutator strains mutT, mutY, mutM and mutY–mutM can be regarded as identical. For this study optical density, OD, measurements are available for each strain growing in LB media. The study is motivated by a competition experiment between P. aeruginosa and each of the four mutator strains, for which interpretations of the results rely on the growth rates being identical. Examination of identical growth rates is relevant, as mutator strains are often considered to have lower fitness and thereby growth rate due to a higher mutation rate and thus more deleterious mutations. The mutation rates of the bacteria considered are listed in Table 1 . OD measurements are used in stead of CFU count, as this method demands less resources, and it is also the choice of measurement Journal of Microbiological Methods 75 (2008) 551–557 ⁎ Corresponding author. Tel.: +45 45253363. E-mail address: krp@imm.dtu.dk (K.R. Philipsen). URL: http://www.imm.dtu.dk/~krp (K.R. Philipsen). 0167-7012/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.mimet.2008.08.009 Contents lists available at ScienceDirect Journal of Microbiological Methods journal homepage: www.elsevier.com/locate/jmicmeth