Individual-based modelling of bacterial cultures to study the microscopic causes of the lag phase

Abstract

The lag phase has been widely studied for years in an effort to contribute to the improvement of food safety. Many analytical models have been built and tested by several authors. The use of Individual-based Modelling (IbM) allows us to probe deeper into the behaviour of individual cells; it is a bridge between theories and experiments when needed.

INDividual DIScrete SIMulation (INDISIM) has been developed and coded by our group as an IbM simulator and used to study bacterial growth, including the microscopic causes of the lag phase. First of all, the evolution of cellular masses, specifically the mean mass and biomass distribution, is shown to be a determining factor in the beginning of the exponential phase. Secondly, whenever there is a need for an enzyme synthesis, its rate has a direct effect on the lag duration. The variability of the lag phase with different factors is also studied. The known decrease of the lag phase with an increase in the temperature is also observed in the simulations.

An initial study of the relationship between individual and collective lag phases is presented, as a complement to the studies already published. One important result is the variability of the individual lag times and generation times. It has also been found that the mean of the individual lags is greater than the population lag.

This is the first in a series of studies of the lag phase that we are carrying out. Therefore, the present work addresses a generic system by making a simple set of assumptions.

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.