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Journal of Theoretical Biology 241 (2006) 939–953

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

Clara Prats�, Daniel Lopez, Antoni Giro, Jordi Ferrer, Joaquim Valls

Escola Superior d’Agricultura de Barcelona, Departament de Fısica i Enginyeria Nuclear, Campus del Baix Llobregat,

Universitat Politecnica de Catalunya, Av. del Canal Olımpic s/n, 08860 Castelldefels, Barcelona, Spain

Received 27 June 2005; received in revised form 24 January 2006; accepted 25 January 2006

Available online 9 March 2006

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.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Lag phase; Bacterial growth; Individual-based Modelling; Growth rate; Mathematical modelling

1. Introduction

The lag phase in bacterial cultures is of great importancein food microbiology. It is an area of study in predictivemicrobiology, which attempts to understand the micro-scopic causes of the lag phase, so as to predict the evolutionof a microbial culture under given conditions. In con-sequence, the cell mechanisms of growth and division, aswell as the cell adaptation mechanisms in a new environ-ment, must be understood.

Mathematical descriptions of the lag phase already exist.Some of them (Baranyi and Roberts, 1994; Hills andWright, 1994; McKellar, 1997) are based on a dynamicdescription of growth, by means of differential equations.

e front matter r 2006 Elsevier Ltd. All rights reserved.

i.2006.01.029

ing author. Tel.: +3493 5521144; fax: +34 93 5521001.

ess: clara.prats@upc.edu (C. Prats).

Later models introduced stochastic treatments (Buchananet al., 1997; McKellar, 2001; Baranyi, 2002), with morerealistic features. Recent studies have also tried to relateindividual cells’ lag times with population lag time (Wuet al., 2000; Metris et al., 2003; Kutalik et al., 2005). Allthese models have been used to study different aspects ofthe lag phase, such as the influence of the medium(temperature, pH, etc.) and the initial conditions of theinoculum. Nevertheless, in order to develop existingmodels, more knowledge of the underlying biologicalmechanisms causing the lag must be incorporated, whichrequires a more complete knowledge of the behaviour ofindividual cells (Buchanan et al., 1997; McKellar et al.,2002). Modelling and simulation at an individual level aregood tools to explore the biological mechanisms of thebacteria. This is also a good way to study the mechanisticconstraints of the individual cells in depth, something not

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taken into account by the aforementioned mathematicaland stochastic treatments (Swinnen et al., 2004).

Furthermore, experimental techniques have been devel-oped to obtain results from individual cells (Smelt et al.,2002; Elfwing et al., 2004). Modelling and simulation at theindividual level can be useful to understand the resultsobtained.

Moreover, this method can be used to design newexperiments to improve understanding of individual andpopulation behaviour.

For these reasons, it seems to be appropriate tointroduce Individual-based Modelling (IbM) into thisresearch field (Kreft et al., 1998; Grimm, 1999; Ginovartet al., 2002a; McKellar and Knight, 2000; Swinnen et al.,2004). The principles of IbM are explained by Bernaerts etal. (2003). Actually, two papers which use an IbMapproach to study microbial lag have recently beenpublished (Dens et al., 2005a, b). However, these studiesconcern a specific system: the lag phenomena induced bytemperature shifts in Escherichia coli cultures.

Our group has developed, coded and used the IbMsimulator we named INDividual DIScret SIMulation(INDISIM) (Ginovart et al., 2002a). INDISIM permitsthe study of the evolution of a bacterial culture. It is basedon the individual behaviour of the bacteria, over a periodof time in a specific environment, and subject to appro-priate boundary conditions, in which space and time arediscrete.

It shares some ideas with other population models,concerning the model of the microorganism. For instance,Fredrickson and Mantzaris (2002) present a set ofpopulation balance equations for microbial cultures whichintroduces cellular diversity. Nevertheless, the use ofcontinuous mathematics (differential equations) makesthe introduction of greater complexity in the model verydifficult. The use of a discrete model makes it possible toachieve similar objectives in a methodologically easier way.It also facilitates the gradual introduction of complexitiessuch as spatial diversity and heterogeneity (Ginovart et al.,2002c) and metabolic reactions (Ginovart et al., 2005).Moreover, INDISIM allows the study of the effects ofindividual diversity (Wagensberg et al., 1988).

The use of this IbM has another relevant advantage. Itoffers the possibility of identifying and studying separatelydifferent contributions to the lag phase. For instance, inthis work the relationship between the evolution of thecellular biomass distribution and the lag phase will beisolated and studied, as well as the effect of the enzymesynthesis rate.

The main concern of this work is to validate oursimulator as a tool for studying this kind of phenomena.This is the first in a series of studies on the lag phase thatwe are planning to carry out. Therefore, we have startedworking on a generic system by making a simplehypothesis. In Section 2 we introduce the simulatorINDISIM, which has been adapted for the study of thelag phase. In this way, we present the model of bacteria

implemented in the program, the simulated culture condi-tions, 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 phaseare isolated and studied separately: the biomass limitationand evolution, and the enzymatic adaptation. Next, thequalitative variability of the lag phase with different factorsis presented. The relationship between the individual andthe population lag phase is studied at the end of thesection. The discussion and several conclusions arepresented in Section 4.

2. Methods

2.1. The INDISIM simulator

INDISIM is designed to simulate the growth andbehaviour of microbial cultures (Ginovart et al., 2002a).It was developed using Compaq Visual Fortran Profes-sional Edition 6.1.0, and has been successfully used inprevious studies (Bermudez et al., 1989; Ginovart et al.,2002b, c, 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 premises is to describe the system in the

simplest possible way. Different levels of complexity can beintroduced, as and when required (Ginovart et al., 2002c,2005). Therefore, the generic and simplest form ofINDISIM has been used in the present work, with somespecific features described below. Its complexity will beincreased as the work proceeds.The culture’s growth simulations take place in a two-

dimensional (2D) space divided into spatial cells. Becausewe consider an agitated and spatially homogeneousculture, the results in a 3D space would be the same.Periodic boundary conditions are used. There is a series ofindividual rules for each bacterium in the culture at eachtime step. These take into account motion, uptake,metabolism, reproduction and viability (Fig. 1). The useof probabilistic arguments (normal distribution aroundmean values) is essential in order to reproduce the diversityof the real cultures. In Fig. 2, a simulation that correctly fitsexperimental data from McKellar and Lu (2003) is shown.A more detailed explanation of INDISIM can be found inthe appendices to this paper.In this work, a standard Pentium IV computer (mini-

mum 512Mb) has been used, with a maximum number of2.5� 105 simulated cells. This number of cells is largeenough, as periodic boundary conditions are used. In thiswork, the simulations have been run for periods of between10 and 30min.

2.2. Modelling the bacteria

The model works with generic bacteria with someparticular properties.

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Fig. 1. Flow chart of the computer code (INDISIM). Note that there are three main parts: (a) the initialisation, where the input parameters are read (1),

the spatial structure is defined according to the cellular method and periodic boundary conditions (Allen and Tildesley, 1987) (2) and the environment and

initial population are configured (3); (b) the main loop, where at each time step the actions described on the right and detailed in Appendix B are taken for

every bacteria of the medium (4), the actions over the medium (nutrient diffusion or redistribution, enzyme action over polymers, extracellular enzyme

diffusion, etc.) are done (5) and a new configuration is obtained (6); and (c) the final part of the code, where the necessary calculations are made to obtain

the results (7).

Fig. 2. Simulated growth curve (straight line) that fits experimental points

from McKellar and Lu (2003). The calculation of the lag time by means of

the geometrical method is also shown (dotted lines).

C. Prats et al. / Journal of Theoretical Biology 241 (2006) 939–953 941

2.2.1. Individual mass

Each bacterium has its own mass, which increases ordecreases depending on different conditions (maintenanceenergy, nutrient availability, etc.). The initial mass is givenaccording to the culture we want to simulate.

2.2.2. Maintenance energy

The energy necessary for cellular maintenance isproportional to the individual mass, so one importantconstant is that which determines this proportionality.

2.2.3. Individual reproduction

The reproduction process starts when an individualreaches a certain cellular mass, mR, which is given by themean characteristic mass mR of the bacterial class with atypical deviation (Ginovart et al., 2002a). The relationshipbetween the initial mass and mR will be an important factorfor the existence of a lag phase. Once the reproductionprocess has started, the period before the duplication isfixed (tR).

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2.2.4. Individual minimum mass and bacterial death

The shortage of nutrient particles causes a lack of energysources. As a consequence, the bacteria must obtain themaintenance energy from the digestion of biomass(Nystrom, 2004) which, in turn, produces a decrease inthe individual bacterial mass. In this work, the viability ofthe cell is determined only by its minimum mass; if thebacterial mass falls below the minimum mass, it becomesinactive.

2.2.5. Enzyme

If we take into account a metabolic adaptation, theenzyme synthesis rate is controlled, as well as the energycost of this synthesis. The amount of intracellular enzymeis also monitored, when it is considered the limiting factor.

2.2.6. Spatial position

Each bacterium has a spatial position and a movementradius.

It is important to point out that the culture’s lag phase is aphenomenon that takes place during the first stages of itsgrowth, when there is no nutrient limitation. Moreover, weare simulating the growth of a culture under spatiallyhomogeneous conditions (batch culture and agitated med-ium). Therefore, we have chosen a simple individual bacterialbehaviour model, which is valid under these conditions.

Further work on the lag phase requires a morecomprehensive individual model for the bacterial behaviour.For instance, it is known that the stationary phase of aculture has an influence on the subsequent lag phase of thatculture, when it is inoculated into a new medium. The studyof stationary phase requires a more complete inactivationand death model, which is currently under development.

2.3. Modelling the enzyme effects

In Section 3.1, some results regarding the microscopiccauses of the lag phase are presented. In the first part ofthis section no enzymatic limitation is considered. In thesecond part (Section 3.1.2) a metabolic adaptationmodelled as an enzymatic synthesis is taken into account,so the enzyme presence and activity must be controlled. Inthis case two different situations have been considered:first, a simulated culture in which the bacteria need tosynthesize intracellular enzymes to catabolize the uptakennutrient; and second, a simulated culture medium withpolymers that are not assimilable by the bacteria, soextracellular enzymes must be synthesized by the cells anddropped into the medium, to hydrolyse the polymers intoassimilable monomers.

2.3.1. Intracellular enzyme

As the enzyme limits nutrient use, it will indirectlyrestrict the cellular uptake rate. Therefore, in our model theintracellular enzyme will limit the maximum number of

nutrient particles that can be consumed per unit of timeand per unit of cellular surface (Umax).Four new parameters are introduced, with the corre-

sponding typical deviation: a maximum enzyme quantityper cell per unit of mass; an initial quantity of enzyme percell (equal to zero in the simulations presented here); themean synthesis rate per unit of mass and time step; and theenergy cost of this synthesis per simulated unit of generatedenzyme. If the bacterial cell does not have any intracellularenzyme, it has to synthesize some enzymes before beingable to take up nutrient from the medium.In our model, at each time step each bacterial cell can

synthesize a certain quantity of enzyme. This synthesistakes place while the intracellular enzyme quantity is underthe maximum and once the cellular maintenance require-ments are satisfied, according to the bacterial mass, thesynthesis rate and the bacterial energy availability. Theintracellular enzyme quantity limits the cellular uptakewith a factor (intracellular enzyme quantity)/(maximumenzyme quantity). Once the intracellular enzyme quantityreaches the fixed maximum value, the bacterial metabolismis adjusted to the available nutrient and uptake goes on asusual. When reproduction takes place, the enzyme quantityis distributed proportionally to the new cells’ masses.

2.3.2. Extracellular enzyme

The nutrient particles are considered to be polymers thatcannot be incorporated through the cellular membrane.Therefore, extracellular enzymatic activity is necessary.Enzymes must be synthesized by the bacteria, hydrolysingthe polymers into assimilable monomers. In this situationthe monomer concentration is the limiting factor for thegrowth rate of the culture.The parameters for describing this process (with the

corresponding typical deviations) are the mean synthesisenzyme rate per unit of mass and time step; the energy costof this synthesis per simulated unit of generated enzyme;the number of monomers contained in a polymer; theinitial enzyme quantity per spatial cell; and the mean timeduration of the simulated enzyme particles.At each time step and for each individual cell, if there are

polymers in the culture medium and a lack of monomersaround the bacterium, the cell synthesizes enzymesaccording to its own mass, the synthesis rate and its energyavailability. These enzymes are dumped into the spatial cellwhere the bacterium is located. At each time step, eachsimulated enzyme particle breaks down a certain numberof polymers into the fixed number of monomers.Extracellular enzymes are diffused within the medium.

At each time step a fraction of extracellular enzyme isdenaturalized, according to the mean time durationparameter.

2.4. Simulated culture conditions

The culture is supposed to grow in an agitated liquidmedium. We worked with batch cultures, i.e. the initial

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nutrient is fixed and no input or output is programmed. Ateach time step the nutrient particles are redistributed, eitheruniformly or by means of diffusion (see Appendix A).

It should be noted again that during the lag andexponential phases there is no nutrient limitation. There-fore, for the purposes of this work the effect of redistribu-tion is the same as the effect of diffusion; it has no effecton the results whether one or the other is chosen.Unpublished simulations carried out by the group (Pratset al., 2005), showed that the nutrient redistribution andthe bacteria movement have an observable effect only inthe transition from the exponential phase to the stationaryphase.

We worked with inoculums of 100 bacteria, except whenwe wanted to study an individual cell inoculum. The initialmean mass and biomass distribution vary with eachsimulation. We speak of homogeneous inoculum if thebacteria have an initial mass equal to a fixed value, and wecall it heterogeneous inoculum when we use a samplepopulation from a previous simulation, taken from theexponential or the final phase, among which bacteria havedifferent initial individual masses (Fig. 3). The former caseis useful to perfectly control the initial conditions and theirrelationship with the growth, but the second one is morerealistic.

2.5. Mathematical methods

2.5.1. Lag time calculus

We took the lag parameter definition of Baranyiand Roberts (1994). We made a logarithmic regression ofan interval in the exponential growth. Specifically, if N0 isthe initial number of bacteria and Nf the final one, wetook the interval between ½lnN0 þ 0:5ðlnNf � lnN0Þ� and½lnN0 þ 0:9ðlnNf � lnN0Þ�. The intersection of the pro-longation of this straight line with the lnN0 line gives thelag time (Fig. 2).

2.5.2. Biomass distribution

To study the distribution of cell biomass, 25 discreteintervals of biomass with the same width were considered

Fig. 3. Initial cellular biomass distribution of a homog

and identified with km; the number of bacteria in eachinterval was counted to obtain the relative frequency ðpkm

Þ.Among other parameters, we evaluated the biomassdistribution evolution during the simulated growth.

3. Results

3.1. Microscopic causes of the lag phase

Two microscopic causes of the lag phase are assumedand studied separately in this section, although both ofthem are probably simultaneous and inseparable in realsystems. Firstly, the results related to the culture biomassevolution as a microscopic cause of the lag are presented.Secondly, the enzyme generation is considered as anothermicroscopic cause.

3.1.1. Biomass evolution and the lag phase

Now we need to consider inoculums that have a lowinitial cellular mean mass. The growth of an homogeneousinoculum and the growth of a heterogeneous inoculumhave both been studied. One important aspect is that inboth cases a lag phase in the number of individuals isobserved, but it is not observed in the total biomassevolution (Fig. 4). This is common behaviour of thecultures under some specific conditions (Cooper, 2004),and a direct consequence of our reproduction model, as itconsiders a threshold in mass in order to initiate thedivision process, which has a fixed duration. In this case, asthe only limitation is the initial low mean mass and nometabolic adaptation is necessary, the biomass increasesfrom the outset, while the number of individuals begins togrow after a while.INDISIM reproduces the typical culture biomass evolu-

tion (Fig. 5). As time goes by, there is an evolution of thebiomass distribution in two senses.On the one hand, the mean mass increases. Let us call

mðtÞ the instantaneous bacterial mean mass. During theexponential growth, there is a constant mean bacterialbiomass, mexp. We define the mean mass distance (MMD)(Eq. (1)) in order to evaluate this change and study its

eneous (left) and a heterogeneous (right) inoculum.

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relationship with the lag phase

DmðtÞ ¼jmðtÞ � mexpj

mexp. (1)

Fig. 5. Evolution of the culture biomass distribution from a homogeneous inoc

diversity distribution is observed. During the exponential phase the distribut

backward movement of the distribution.

Fig. 4. Typical simulated curves, using INDISIM, representing the

evolution of lnN (straight line) and the logarithm of total biomass

(dotted line). The same slope is observed in the two curves.

However, the distribution shape changes. The instanta-neous cell’s biomass relative frequency is represented bypkmðtÞ. There is a characteristic distribution for the

exponential phase that corresponds to its maximumdiversity (Wagensberg et al., 1988). We estimate pkm ;exp

by calculating the mean of the instantaneous relativefrequencies of the exponential phase, in the same timeinterval used for the calculation of the lag parameter.We define the mass distribution distance (MDD) asfollows:

DpkðtÞ ¼

X25

km¼1

jpkmðtÞ � pkm ;expj. (2)

As we presume that both evolutions are related to the lagphase, we define the product distance (PD) (Eq. (3)), whichtakes into account the two distances defined below:

DðtÞ ¼ DmðtÞDpkðtÞ ¼

mðtÞ � mexp

�� ��mexp

X25

km¼1

jpkmðtÞ � pkm;expj.

(3)

ulum. In the first stages the adaptation of the distribution to the maximum

ion remains constant. At the end, when the nutrient runs out, there is a

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Fig. 6. Evolution of mean mass distance, mass distribution distance and product distance for a homogeneous (A) and a heterogeneous inoculum (B). The

dotted lines point out the lag parameter calculated in the simulation (Fig. 3).

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We have studied the evolution of these distances in twodifferent cultures:

(A)

a homogeneous inoculum (Fig. 3) with initialmt¼0=mR ¼ 0:233;

(B)

a heterogeneous inoculum (Fig. 3) with initialmt¼0=mR ¼ 0:275:

Fig. 7. Typical simulated curves, using INDISIM, corresponding to the

adaptation with intracellular enzymes. The lag phase on lnN evolution

(straight curve) and the initial decrease in the biomass (dotted curve) can

be observed.

where mt¼0 is the initial mean mass.The graphical representation of the three distances

(Fig. 6) shows that in both cases there is an importantdecrease in values during the lag phase. Although MMDand MDD seem to be good indicators in some cases, thebest results are provided by PD; the exponential phase ofthe growth begins when the PD value is around 0.04.

There is no constant exact value for PD that fits in withthe lag parameter, so PD cannot be used as a lag measure.Nevertheless, these results show two of the phenomena thattake place during the lag phase. There is an increase in themean mass up to the typical exponential value. At the sametime, there is an adaptation of the mass distribution to themaximum diversity. Both are necessary in order to bringthe lag phase to a close and begin exponential growth.

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3.1.2. Enzyme generation and the lag phase

Sometimes the inoculum has to adapt to a new medium.In these cases, the bacteria need to synthesize new enzymesto be able to take up the new substances. Unlike the casestudied in Section 3.1.1, the metabolic adaptation causes aninitial decrease in the biomass which is observed in thesimulations (Fig. 7). This is a phenomenon observed inbacteria under stress conditions (Nystrom, 2004). Thebacteria degrade some endogenous material to survive,with the consequent volume (and mass) reduction, whileadapting themselves to the new medium.

The enzyme particles may be intracellular or extracel-lular enzymes. Probably both of them are necessary, but wehave studied them separately. Two series of simulationshave been done, one for each type of enzyme. Specifically,our simulations relate the lag phase duration to the meanenzyme synthesis rate (Figs. 8 and 9). In these simulations,the initial biomass distribution is taken from the end phaseof a previous simulation. Enzyme and nutrient particlediffusion is considered in the extracellular case, instead ofthe redistribution. A reference simulation with no enzyme

Fig. 8. Growth curves (lnN) of simulations performed with different intrace

enzyme generation rate (b), and maximum growth rate versus intracellular enzy

correspond to results from the same simulations. The dotted line corresponds

Fig. 9. Growth curves (lnN) of simulations performed with different extrace

enzyme generation rate (b), and maximum growth rate versus extracellular enz

correspond to results from the same simulations. The dotted line corresponds

effects has been carried out for each series, in order toisolate the low initial mass effects studied in Section 3.1.1.This will be a constant effect which is present in all thesimulations with enzyme synthesis.It has been shown that in both cases an increase in the

enzyme synthesis rate produces a shorter lag phase, as weexpected (Figs. 8b and 9b). The main difference is found inthe maximum growth rate mmax. In the intracellular enzymecase, when the synthesis rate is too low the culture cannotreach the maximum growth rate permitted by the medium(Fig. 8c). If we consider extracellular enzymes, cooperationbetween bacteria is possible. This means that it is easier forthe culture to reach the maximum growth rate permitted bythe medium in all simulations (Fig. 9c).

3.2. Variability of the lag phase with different factors

In order to validate the suitability of our simulator, westudied the relationship of the lag phase of a homogeneousinoculum with the temperature (uptake), the initial meanmass of the inoculum and the maintenance energy.

llular enzyme generation rates (a). Lag time duration versus intracellular

me generation rate (c). The cardinal numbers (1–11) in the different graphs

to the reference curve, without the enzyme effect.

llular enzyme generation rates (a). Lag time duration versus extracellular

yme generation rate (c). The cardinal numbers (1–8) in the different graphs

to the reference curve, without the enzyme effect.

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This is a conceptual study to validate the correctperformance of the simulator and progress in thetheoretical understanding of real systems.

3.2.1. Temperature

The maximum uptake constant U is related to theaccessible nutrient particles around the cell. These particlesare contained in a spherical volume with radius r as follows:

U / r3. (4)

This accessible volume given by r is fixed by the meankinetic energy of the nutrient particles, which is directlyrelated to the temperature (Allen and Tildesley, 1987), asfollows:

r /ffiffiffiffiffiffiEk

p, (5)

Ek / T . (6)

From Eqs. (5) and (6), a relationship between the uptakeconstant U and the temperature T can be deduced

U / T3=2. (7)

It must be stressed that Eq. (7) is a microbial levelrelationship. The results of the simulations show that theglobal culture behaviour is in agreement with Ratkowsky’sobservations (Ratkowsky et al., 1981, 1983; Ginovart et al.,2002a).

It has been seen that a change in the uptake constant inthe simulator means a change in the simulated tempera-ture. It can be observed that the simulator gives a goodresponse to the changes in that parameter (Fig. 10): whenthe temperature (uptake constant) increases, the lagdecreases and the maximum growth rate increases, asobserved experimentally by Zwietering et al. (1994).Specifically, considering that in this work the enzymedenaturalisation temperature has not been taken intoaccount, the simulation results fit well with the results ofRatkowsky (2003).

Another important result has been obtained withINDISIM. The product lmmax remains approximatelyconstant for cultures having the same initial state and

Fig. 10. Results of the series of simulations performed with different temp

temperature (b) and the product lm (c).

growing under different (but constant) temperatures(Fig. 10c). This fact has been observed in severalexperiments and included in different models (Baranyiand Roberts, 1994). In our work the cultures follow anadaptation process from the initial mass distribution to thecharacteristic mass distribution of the exponential phase.In other words, the cultures must cover the distance

between the aforementioned mass distributions (see Section3.1.1). This distance divided by the lag phase parameter l isthe mean velocity of this adaptation process. When there isno lag phase in biomass growth (there is no physiologicaladaptation) but there is a lag in the number of bacteria(Fig. 4), the characteristic parameter to evaluate thevelocity of the culture’s growth is the maximum growthrate mmax. Therefore, lmmax is directly related to the distance

to be covered by the biomass distribution.This is an important result of INDISIM because it

reproduces a macroscopic observation without havingintroduced any specific microscopic hypothesis. It is aproof of the consistency of the methodology used.

3.2.2. Initial cellular mean mass

The growth of inoculums with different initial cellularmean mass has been simulated. It has been noted that,while the lag phase clearly decreases when the initial meanmass increases, the maximum growth rate remains con-stant, which is consistent with the experimental results(Fig. 11).

3.2.3. Maintenance energy

We assume that the maintenance energy is proportionalto the bacteria biomass. The proportionality constant isgiven as a parameter. If we increase this parameter, the lagphase increases and the maximum growth rate decreases(Fig. 12).

3.3. Individual and population lag phase

As INDISIM controls each bacterium in the medium, itis easy to study the individual lag times. We have studied

eratures: lag time versus temperature (a), maximum growth rate versus

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Fig. 11. Results of the series of simulations performed with different initial cellular mean masses: growth curves (lnN) of the different simulations (a), lag

time duration versus initial cellular mean mass (b), and maximum growth rate versus initial cellular mean mass (c).

Fig. 12. Results of the series of simulations done with different maintenance energy constants: growth curves (lnN) of the different simulations (a), lag

time duration versus maintenance energy constant (b), and maximum growth rate versus maintenance energy constant (c).

C. Prats et al. / Journal of Theoretical Biology 241 (2006) 939–953948

the distribution of individual lag times, as well as thedistribution of the subsequent generation times (Fig. 13).The first important result is that, as described by Baranyiand Pin (2001), the mean of the individual lag times ismuch longer than the collective lag phase. The control thatthe first cells to reproduce have over the culture behaviouris the cause of this effect.

The distribution of the individual generation timesconverges in a Gaussian manner during the exponentialphase. Nevertheless it has a lower limit, which isestablished by the minimum reproduction cycle time. Thisis the reason why the first column is longer than expected.

4. Discussion and conclusions

Two microscopic causes of the lag phase have beendescribed and discussed:

(i)

The effects of the cellular mean mass and the biomassdistribution have been studied through two newparameters: MMD and MDD. With these new para-

meters we have shown the relationship between the lagphase and the mass evolution of the culture to theexponential growth characteristics.

(ii)

The enzymatic adaptation has also been studied,for both intracellular and extracellular enzymes. Thelatter has been shown to facilitate cooperation betweencells.

The effect of environmental factors has shown what wasalready known to experimental researchers: an increase intemperature causes a decrease in the lag phase, while anincrease in the maintenance constant causes an increase inthe lag duration.This simulator has been shown to be especially useful in

the study of individual lag times. In this sense, one of theresults demonstrated by Baranyi (1998) has been matched:the mean value of the individual lags is greater than thecollective lag phase. Many different thought experimentscan be carried out in this way. INDISIM is a very potenttool in the field of low-population cultures, where realexperiments are still rather limited.

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Fig. 13. Distribution of individual generation times for the first six generations of a homogeneous inoculum. The first generation corresponds to the

distribution of the individual lag times.

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We believe that INDISIM is a potent tool and a goodcomplement for studying the lag phase. The suitability ofthis simulator to study the lag phase has been checked, and

its potential to improve and extend the results here presentedhas also been examined. INDISIM builds a bridge betweenindividual behaviour and collective observations, and

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Fig. A1. Matter transport: diffusion between neighbouring cells. A factor

1=ffiffiffi2p

is considered in the diagonals (dotted arrows).

C. Prats et al. / Journal of Theoretical Biology 241 (2006) 939–953950

between theoretical modelling and experimental results. It isalso a helpful tool in acquiring deeper understanding of thecell behaviour during the lag phase.

Further refinements will contribute to the developmentand improvement of predictive food microbiology. Amongother things, we need to address specific experimental datato quantitatively reproduce experimental results, so thatthe simulator may be used to design new experiments or tosupport new mathematical models. We are also trying toincorporate into the model elements of spatial hetero-geneity, since most foods present a heterogeneous environ-ment (Fleet, 1999).

A simplified demo version of INDISIM can be down-loaded from:http://mie.esab.upc.es/mosimbio/lag-demo.htm.

Acknowledgements

We thank Dr. J. Baranyi and Dr. M. Silbert for theircritical reading of and critical comments on this work. Wegratefully acknowledge the financial support of the DURSIGeneralitat de Catalunya 2003ACES00064 and 2005FI-00364, and the Plan Nacional I+D+i of the Ministerio deEducacion y Ciencia CGL 2004-01144.

Appendix A. Modelling the environment

INDISIM is discrete in space. It was developed to study 2Dor 3D spaces. In this work, the growth of cultures takes placein a 2D square divided into 300� 300 spatial cells. Each cell isidentified by its coordinates x and y, and its nutrient particlecontent is controlled. Periodic boundary conditions are usedin this work. In most of the examples here presented, weassume that the motion of the nutrient particles is due to thehomogeneous redistribution of these particles in the media.This assumption is made in the cases where nutrient motion isnot important for the results. The nutrient redistribution takesplace at every time step, after the bacterial activity.

In the case of extracellular enzymes, the enzyme andnutrient motion is significant for the simulated culturegrowth. In this case discrete enzyme and nutrient diffusionare considered. Discrete Fick’s law (Bormann et al., 2001)is considered between neighbouring spatial cells (Fig. A1).Diffusion takes place instead of redistribution, after thebacterial activity.

Appendix B. Modelling the bacteria

The main characteristic of the IbM is that the cells areconsidered one by one. Therefore, at each time step thebacteria are subjected, one by one, to the actions describedbelow (Ginovart et al., 2002a).

B.1. Motion

During the simulation, the position of a bacterial cell i

may change to a new position in space, according to a

given probability pmov. The new position is randomlychosen from the available sites within a radius dmax.

B.2. Nutrient uptake

Each bacterium takes up nutrient particles from themedium, according to the following parameters:

(i)

Umax, the maximum number of nutrient particles thatcan be consumed per unit of time and per unit ofcellular surface.

(ii)

Dmax determines the maximum uptake range. Fromthe position of a given bacterium, Dmax defines thenumber of sites, or spatial cells, that may be reachedfor the purpose of nutrient particle consumption.

(iii)

k is a given percentage of the n(Dmax) availablenutrient particles within the given range Dmax. It takesinto account probabilistic considerations regarding theentry of nutrient particles into the bacterial cell,through the cellular membrane. Therefore, the numberof available nutrient particles is kn(Dmax).

(iv)

U ¼ Z1ðcmaÞ is the maximum number of nutrientparticles that can be taken up by the bacteria. Z1 is arandom variable with mean Umax and standarddeviation s, m is the mass of the cell, a is a parameterrelated to its geometry and c is a normalisationconstant whose value depends on the value of a. Fora spherical bacterial cell, if the uptake is considered tobe proportional to the bacterial surface, a ¼ 2=3.

At each time step, the bacteria take up the minimumquantity between U and kn(Dmax). This model is based on

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Blackman’s kinetics for its simplicity, in spite of knownlimitations. It is important to stress that the chosen kineticsapplies to each individual. However, when each individualfollows Blackman’s kinetics, the culture behaviour fits wellwith the experimental results (Ginovart et al., 2002a;Bermudez et al., 1989), allowing the study of the effect oftemperature (Ginovart et al., 2002a; Bermudez et al., 1989).However, the widely used Michaelis–Menten’s (or Monod’s)kinetics does not fit correctly with the experimental results.Other kinetics with three or more parameters have beensuggested, developed and tested over the years, but they arecase-specific and require re-fitting of these parameters to thesystem under study (Koch, 2005; Kovarova-Kovar and Egli,1998; Button, 1998; Dabes et al., 1973).

B.3. Metabolism

Metabolism converts nutrient particles into maintenanceenergy and biomass. The following parameters areintroduced:

(i)

I denotes a prescribed number of nutrient particles perunit of biomass that a bacterial cell needs in order tomaintain its viability.

(ii)

Y denotes a constant required to model the metabolicefficiency that accounts for the synthesized biomassunits per metabolized nutrient particle.

By using the above parameters, and recalling themeaning of U, we set the following control rules:

(a)

Maintenance energy for the viability of a bacterial cell, mI. (b) A control relation to check whether the nutrient

particles absorbed by a bacterial cell are sufficient forits maintenance, UXmI?

(c)

If there is no possibility of covering the maintenancerequirements of a bacterial cell, check the possibility ofits lyses or inactivation.

(d)

Once the viability of the bacterial cell is achieved, allowfor the increase of its mass from m to (m+B), whereB ¼ ðU2mIÞY .

(e)

Allow for reserves within each bacterial cell, which canbe used up in the above processes whenever the localand external level of nutrient particles is too low for thesupply of sufficient maintenance requirements.

B.4. Reproduction

The steps described below are followed:

(i)

To initiate the reproduction cycle, the mass of thebacterial cell must reach a specific mass mR4mR;min

identified by the individual reproduction mass; ob-tained from the normal random distribution withmean mass MR and standard deviation sR.

(ii)

When the microorganism reaches the mass mR, thebacterial cell must wait for a fixed period of time, tR,

before the physical separation into two new bacterialcells begins to take place.

(iii)

When the duplication is completed, two new bacteriaappear in the medium with characteristics similar tothe original bacterial cell, with masses m1 andm2 ¼ m�m1, each, approximately, half the mass ofthe original bacterial cell.

(iv)

The new bacterial cells are allowed to take up newpositions in the physical lattice, or one may remain inthe original spatial cell.

Note that when the reproduction process begins, thebacterium is still active; that is, it can change position in themedium and continue to consume nutrient particles andmetabolize them. Hence the biomass synthesis rate duringtR will determine the mean mass of the culture.

B.5. Viability

Whenever the environmental conditions become unfa-vourable (the nutrient runs out), the bacterium may lose itsviability. As a result, when the bacterial mass drops belowa certain value (a fraction of its mR), the inactivation of thebacterium takes place.

B.6. Enzyme synthesis

The enzyme synthesis takes place during the metabolism.After meeting the cell maintenance requirements, andbefore the biomass synthesis, we check whether there isenough energy to synthesize some enzyme particles. Thisquantity is fixed by the enzyme synthesis rate.

B.7. Statistical averages

At the end of the process we relate the ‘microscopic’description of the bacterial colony to its ‘macroscopic’, orglobal, description, namely the properties observed byexperiment. Hence, once the single-cell variables areobtained we take a simple arithmetic average to obtainthe population-averaged properties.

Appendix C. Simulation parameters

In this section, the simulation parameters for Fig. 2 arepresented. In Table C1 the physical parameters (regardingspace and time equivalences) are specified. In Table C2 thedifferent parameters in simulation units (s.u.) are shown. Inorder to provide an understanding of these s.u., the valuesof three parameters in a specific point of the exponentialgrowth (t ¼ 202:12 h) are specified in Table C3. Finally, theobtained results of this simulation (lag phase andmaximum growth rate) are presented in Table C4.As mentioned in Section 2.4, in order to study the

biomass distribution, 25 discrete intervals are defined. InTable C5 the values for establishing these intervals arespecified.

ARTICLE IN PRESS

Table C2

Simulation parameters in simulation units (s.u.)

Parameter Value

Initial mean cellular mass, m0 (s.u.) 2663.4

Maximum uptake constant, Umax (s.u.) 315

Standard deviation in Umax, s(Umax) 0.3

Mean mass to begin the reproduction cycle, MR (s.u.) 15,000

Standard deviation in MR, s(MR) 0.4

Minimum reproduction mass, mR,min (s.u.) 5000

Initial nutrient particles per spatial cella, Nutxy,0 (s.u.) 145,000

Nutrient particles per unit of biomass for maintenance, I (s.u.) 0.0070

Biomass synthesis efficiency, Y 1

Maximum enzyme per mass unit, Emax (s.u.) 10

Standard deviation in Enzmax, s(Emax) 0.5

Enzyme generation rate per unit of mass and time step, Venz

(s.u.)

1

aOne nutrient particle of the simulation is not equivalent to one real

nutrient particle.

Table C5

Values for study of the biomass distribution

Result Value

Minimum mass (s.u.) 1000

Maximum mass (s.u.) 31,000

Mass interval (s.u.) 1200

Table C3

Some values at a specific point during the exponential growth

(t1 ¼ 202:12 h)a

Parameter Value

Mean mass, mðt1Þ (s.u.) 9731.6

Mean energy for the cellular maintenance, m � Iðt1Þ (s.u.) 67.41

Mean number of uptaken particles, Uðt1Þ (s.u.) 96.98

aOne nutrient s.u. can be used to generate a biomass s.u. or an energy

s.u. for the cellular maintenance.

Table C4

Results of the simulation

Result Value

Lag-parameter, l (h) 30.12

Mean number of uptaken particles, mmax (10�2 h�1) 4.39

Table C1

Physical parameters of the simulation

Parameter Value

Simulated system volume, VS (10�10m3) 1.13

Spatial cell volume, VC (10�15m3) 1.26

Maximum number of bacteria per spatial cell, IC (bacterial cells) 4

Time step equivalence, tseq (min) 4.19

Initial bacterial cells concentration, C0 (104 cell/ml) 4.42

Maximum movement radiuma, dmax (10�5m) 2.1

Maximum uptake range, Dmax (10�5m) 1.0

Fixed period of time for the reproduction, tR (min) 41.9

aIn this case it does not represent the real bacterial motion. It is used to

represent the culture agitation.

C. Prats et al. / Journal of Theoretical Biology 241 (2006) 939–953952

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