Individual-based modelling of bacterial cultures to study the microscopic causes of the lag phase
Introduction
The lag phase in bacterial cultures is of great importance in food microbiology. It is an area of study in predictive microbiology, which attempts to understand the microscopic causes of the lag phase, so as to predict the evolution of a microbial culture under given conditions. In consequence, the cell mechanisms of growth and division, as well as the cell adaptation mechanisms in a new environment, must be understood.
Mathematical descriptions of the lag phase already exist. Some of them (Baranyi and Roberts, 1994; Hills and Wright, 1994; McKellar, 1997) are based on a dynamic description of growth, by means of differential equations. Later models introduced stochastic treatments (Buchanan et al., 1997; McKellar, 2001; Baranyi, 2002), with more realistic features. Recent studies have also tried to relate individual cells’ lag times with population lag time (Wu et al., 2000; Métris et al., 2003; Kutalik et al., 2005). All these models have been used to study different aspects of the lag phase, such as the influence of the medium (temperature, pH, etc.) and the initial conditions of the inoculum. Nevertheless, in order to develop existing models, more knowledge of the underlying biological mechanisms causing the lag must be incorporated, which requires a more complete knowledge of the behaviour of individual cells (Buchanan et al., 1997; McKellar et al., 2002). Modelling and simulation at an individual level are good tools to explore the biological mechanisms of the bacteria. This is also a good way to study the mechanistic constraints of the individual cells in depth, something not taken into account by the aforementioned mathematical and stochastic treatments (Swinnen et al., 2004).
Furthermore, experimental techniques have been developed to obtain results from individual cells (Smelt et al., 2002; Elfwing et al., 2004). Modelling and simulation at the individual level can be useful to understand the results obtained.
Moreover, this method can be used to design new experiments to improve understanding of individual and population behaviour.
For these reasons, it seems to be appropriate to introduce Individual-based Modelling (IbM) into this research field (Kreft et al., 1998; Grimm, 1999; Ginovart et al., 2002a; McKellar and Knight, 2000; Swinnen et al., 2004). The principles of IbM are explained by Bernaerts et al. (2003). Actually, two papers which use an IbM approach to study microbial lag have recently been published (Dens et al., 2005a, Dens et al., 2005b). However, these studies concern a specific system: the lag phenomena induced by temperature shifts in Escherichia coli cultures.
Our group has developed, coded and used the IbM simulator we named INDividual DIScret SIMulation (INDISIM) (Ginovart et al., 2002a). INDISIM permits the study of the evolution of a bacterial culture. It is based on the individual behaviour of the bacteria, over a period of time in a specific environment, and subject to appropriate boundary conditions, in which space and time are discrete.
It shares some ideas with other population models, concerning the model of the microorganism. For instance, Fredrickson and Mantzaris (2002) present a set of population balance equations for microbial cultures which introduces cellular diversity. Nevertheless, the use of continuous mathematics (differential equations) makes the introduction of greater complexity in the model very difficult. The use of a discrete model makes it possible to achieve similar objectives in a methodologically easier way. It also facilitates the gradual introduction of complexities such as spatial diversity and heterogeneity (Ginovart et al., 2002c) and metabolic reactions (Ginovart et al., 2005). Moreover, INDISIM allows the study of the effects of individual diversity (Wagensberg et al., 1988).
The use of this IbM has another relevant advantage. It offers the possibility of identifying and studying separately different contributions to the lag phase. For instance, in this work the relationship between the evolution of the cellular biomass distribution and the lag phase will be isolated and studied, as well as the effect of the enzyme synthesis rate.
The main concern of this work is to validate our simulator as a tool for studying this kind of phenomena. This is the first in a series of studies on the lag phase that we are planning to carry out. Therefore, we have started working on a generic system by making a simple hypothesis. In Section 2 we introduce the simulator INDISIM, which has been adapted for the study of the lag phase. In this way, we present the model of bacteria implemented in the program, the simulated culture conditions, and some mathematical methods used in this study.
Section 3 contains some results related to the study of the lag phase. First, several microscopic causes of the lag phase are isolated and studied separately: the biomass limitation and evolution, and the enzymatic adaptation. Next, the qualitative variability of the lag phase with different factors is presented. The relationship between the individual and the population lag phase is studied at the end of the section. The discussion and several conclusions are presented in Section 4.
Section snippets
The INDISIM simulator
INDISIM is designed to simulate the growth and behaviour of microbial cultures (Ginovart et al., 2002a). It was developed using Compaq Visual Fortran Professional Edition 6.1.0, and has been successfully used in previous studies (Bermúdez et al., 1989; Ginovart et al., 2002b, Ginovart et al., 2002c, Ginovart et al., 2005).
INDISIM is discrete in space and time, and monitors a group of cells at each time step, using a set of random time-dependent variables for each individual (Fig. 1).
One of our
Microscopic causes of the lag phase
Two microscopic causes of the lag phase are assumed and studied separately in this section, although both of them are probably simultaneous and inseparable in real systems. Firstly, the results related to the culture biomass evolution as a microscopic cause of the lag are presented. Secondly, the enzyme generation is considered as another microscopic cause.
Discussion and conclusions
Two microscopic causes of the lag phase have been described and discussed:
- (i)
The effects of the cellular mean mass and the biomass distribution have been studied through two new parameters: MMD and MDD. With these new parameters we have shown the relationship between the lag phase and the mass evolution of the culture to the exponential growth characteristics.
- (ii)
The enzymatic adaptation has also been studied, for both intracellular and extracellular enzymes. The latter has been shown to facilitate
Acknowledgements
We thank Dr. J. Baranyi and Dr. M. Silbert for their critical reading of and critical comments on this work. We gratefully acknowledge the financial support of the DURSI Generalitat de Catalunya 2003ACES00064 and 2005FI-00364, and the Plan Nacional I+D+i of the Ministerio de Educación y Ciencia CGL 2004-01144.
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