Modeling Bacterial Population Growth from Stochastic Single-CellDynamics

Antonio A. Alonso,a Ignacio Molina,a Constantinos Theodoropoulosb

Process Engineering Group, IIM-CSIC Spanish Council for Scientific Research, Vigo, Spaina; School of Chemical Engineering and Analytical Science, University ofManchester, Manchester, United Kingdomb

A few bacterial cells may be sufficient to produce a food-borne illness outbreak, provided that they are capable of adapting andproliferating on a food matrix. This is why any quantitative health risk assessment policy must incorporate methods to accu-rately predict the growth of bacterial populations from a small number of pathogens. In this aim, mathematical models havebecome a powerful tool. Unfortunately, at low cell concentrations, standard deterministic models fail to predict the fate of thepopulation, essentially because the heterogeneity between individuals becomes relevant. In this work, a stochastic differentialequation (SDE) model is proposed to describe variability within single-cell growth and division and to simulate populationgrowth from a given initial number of individuals. We provide evidence of the model ability to explain the observed distribu-tions of times to division, including the lag time produced by the adaptation to the environment, by comparing model predic-tions with experiments from the literature for Escherichia coli, Listeria innocua, and Salmonella enterica. The model is shown toaccurately predict experimental growth population dynamics for both small and large microbial populations. The use of stochas-tic models for the estimation of parameters to successfully fit experimental data is a particularly challenging problem. For in-stance, if Monte Carlo methods are employed to model the required distributions of times to division, the parameter estimationproblem can become numerically intractable. We overcame this limitation by converting the stochastic description to a partialdifferential equation (backward Kolmogorov) instead, which relates to the distribution of division times. Contrary to previousstochastic formulations based on random parameters, the present model is capable of explaining the variability observed in pop-ulations that result from the growth of a small number of initial cells as well as the lack of it compared to populations initiatedby a larger number of individuals, where the random effects become negligible.

Often bacterial contamination of foods starts with a small numberof bacteria that are capable of adapting and proliferating by re-

peated divisions on a given food matrix. At low cell concentrations,standard deterministic models fail to predict the variability of thebacterial population. This is so because at low initial cell numbers,heterogeneity between individuals and its influence on the divisiontimes become relevant and have a net influence on the population.Consequently, the behavior of individual bacteria cannot be ne-glected when assessing possible health risks along the food chain,either during storage or during distribution.

Recently, attention has been drawn to the need for modeling andsimulation methods to observe and describe the variability of single-cell behavior and small populations (1, 2) in order to produce realisticestimations of safety risks along the food chain, for instance, duringstorage and distribution or during food processing.

In this paper, connections are established between individualbacterial growth and division, the corresponding distributions oftimes to division, and population growth curves that in the longterm can aid the quantification, on a probabilistic basis, of micro-bial risk and product shelf life.

A number of modeling approaches for bacterial populationdynamics have been built around the concept of times to divisionand particularly the first lag time to division of cell populations(3–6). The development of analytical techniques capable of mea-suring single-cell parameters such as cell length renewed interestin modeling single-cell growth. Experimental techniques for sin-gle-cell studies include turbidimetry (see for example, references 7and 8), lithographic techniques (9), and flow cytometry (10). Re-cently, time-lapse microscopy has been successfully applied to ob-tain quantitative information on colonial growth dynamics orig-

inated from single cells (2). Based on the flow chambermicroscopy technique proposed in reference 10, models for indi-vidual cell growth have been developed as described in references4 and 11. They are essentially adaptations of the now classicalmodel proposed in reference 12 to describe the growth of bacterialpopulations before attaining the stationary phase. It consists oftwo ordinary differential equations that can be written as

dy

dt� �a (1)

da

dt� va(1 � a) (2)

where y in equation 1 represents the natural logarithm of the pop-ulation size and � the maximum specific growth rate. Variablea(t) in equations 1 and 2 is known as the adjustment function andrelates to a certain physiological state of the population. This vari-able has been introduced to describe the gradual adaptation of thecells to the new environment. Initially, it takes a small value which

Received 6 May 2014 Accepted 8 June 2014

Published ahead of print 13 June 2014

Editor: D. W. Schaffner

Address correspondence to Antonio A. Alonso, antonio@iim.csic.es.

Supplemental material for this article may be found at http://dx.doi.org/10.1128/AEM.01423-14.

Copyright © 2014, American Society for Microbiology. All Rights Reserved.

doi:10.1128/AEM.01423-14

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increases at a rate proportional to � (in equation 2) up to a max-imum of one. This variable induces a delay in the growth of thepopulation size which depends on the inverse of v (the largerthe rate, the smaller the delay becomes) and is employed to modelthe observed initial lag time.

In the context of a single-cell growth, the state variable y, whichrelates to the size of the population in reference 12, is identified asa critical variable that determines cell growth and characterizesdivision when it reaches a particular threshold. In previous works,it has been interpreted as cell length (see, e.g., references 4 and 11)or cell DNA content (13). Similarly, the adjustment function inreference 12 is used in references 4 and 11 to model the adaptationof a given cell to the environment.

Stochastic fluctuations in gene transcription and translationwithin a cell or in response to cell-to-cell disturbances are consid-

ered the main sources of heterogeneity among individual cellswithin a population (14, 15). In order to capture such “behavioralnoise” (2), previous approaches combined deterministic equa-tions for single-cell or population dynamics of the form of equa-tions 1 and 2, with random parameters and/or initial randomconditions (both described by appropriate probability distribu-tion functions).

In reference 11, the random nature of cell division is modeledby imposing a uniformly distributed random length threshold atwhich each cell divides. This seems to be also the case in reference4, although parameter estimation for cells subject to different heatshock treatments is based on a deterministic model. However, theconnections between such models and the observed distributionsof division times have not been clearly discussed yet.

Inspired by reference 3, a model of cell population dynamics is

FIG 1 Distribution of times to division for an ensemble of 10,000 realizations obtained from the SDE (equations 3 and 4) (bars) and from equation 8 (lines).Model parameters are � of 0.6060 h�1 and � of 0.0821. Panel a represents distributions for cells under adaptation (a0 � 0.1000). Panel b represents cellscompletely adapted (a0 � 1.0000). The corresponding parity plots comparing the cumulative distributions computed from the SDE (QSDE) and the � distribu-tions (equation 8) (Q�) are presented in panels c and d for both scenarios.

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suggested in reference 16 that explicitly includes a lag time distri-bution function. This is in agreement with other well-known ap-proximations such as the Weibull or gamma probability distribu-tion functions (8, 17). Recently, a model similar to that proposedin reference 12 with parameters following random distributionshas been employed as suggested in reference 2 to simulate popu-lation growth rates.

In reference 5, a population dynamics stochastic model hasbeen proposed for Escherichia coli that makes use of time-to-divi-sion distribution functions to compute the time (the stochasticvariable) at which cell division occurs. This information is pro-vided to the algorithm either directly from experimental data (e.g.,flow chamber microscopy or turbidimetry) or from a gamma dis-tribution, which has been previously fitted to experimental data.

In this work, connections between individual bacterial growthvariables and distribution of times to division are established by astochastic version of the models proposed in reference 4 and inreferences 11 and 18. The underlying premise is that cell growth isthe result of a large number of biochemical reactions taking placeon a microscopic domain (thus involving a relatively small num-ber of molecular species). Standard assumptions can then be in-voked to relate a chemical (or biochemical) microscopic masterequation to its mesoscopic (chemical Langevin) counterpart (19),which is nothing but a stochastic differential equation (SDE) (20).

Hence, heterogeneity between individuals or fluctuationswithin each individual (e.g., due to behavioral noise) is repre-sented by an SDE that will result from adding a stochastic compo-nent to the specific growth rate. The introduction of stochasticityon the growth rate has a strong biological interpretation: SDEs areusually employed in a systems biology context to collect the ran-dom effects of fluctuations on the system. Gillespie (19) gives con-vincing arguments to show how the accumulation of stochasticevents can be cast into SDEs, stating that the aggregated effect ofmany events at the cell level can be captured and described by SDEmodels. Models based on SDE systems have been used previouslyto describe cell population growth and division for plankton (13)and bacteria (21). This approach has also been adopted in the

study of bacterial systems under the action of bacteriophage (22)or antibiotics (23).

In the context of cell growth and division, such representation,which seeks the aggregation of the undergoing biochemical pro-cesses during the cell cycle, is shown to reproduce reasonably wellthe time to division distributions observed and reported in theliterature.

It is well known from stochastic systems theory (24) that thecollective effect of an SDE system, namely, the evolution of theprobability distribution associated with the random state vari-ables, can be computed as the solution of two partial differentialequations (PDE): the so-called Kolmogorov equations, forward orbackwards in time.

We made use of one such equation, the backward Kolmogorov,to characterize time to division distributions (TTD)—includingfirst time to division—and to efficiently estimate parameters ofthe underlying stochastic dynamics from the experimental distri-butions. It is important to emphasize here that from a computa-tional point of view, this way of approaching the model calibrationproblem is particularly efficient. This is so because it only requiresthe solution of a partial differential equation, as opposed to ob-taining a complete ensemble of realizations by repeatedly solvingthe SDE to reconstruct the density function associated to the dis-tribution of times to division (e.g., by Monte Carlo methods).

Similar approaches have been employed previously, althoughin the context of gene expression networks to extract kinetic pa-rameters associated with individual cells from protein distribu-tions obtained by cell population measurements (25). The use ofstochastic methods related to Kolmogorov equations in predictivemicrobiology has been suggested previously (26), albeit in thecontext of population growth dynamics, as a means to describevariability of the environment as well as uncertainty due to limi-tations of the measurement equipment. However, to the best ofour knowledge, these methods have not been employed so fareither to model single-cell growth kinetics or to estimate param-eters based on experimental TTD. Our model assumes that bacte-rial division is subject to stochastic fluctuations which integrate

FIG 2 Comparison between the experimental distribution (bars) of times to first division for Listeria innocua (4) and the theoretical distribution of times todivision (equation 8) (lines). Estimated model parameters are presented in Table 2. (a) No heat shock; (b) 5-min heat shock.

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the effect of transcription at the level of each individual. From thatpoint of view our approach follows the assumptions implicit inreference 25, in which a master equation is employed to connectparameters associated with a stochastic process (a signaling net-work) to observations obtained at the cell population level. In thiswork, we used observations obtained at the population level(time-to-division distributions) to estimate parameters of the sto-chastic process that describes division.

MATERIALS AND METHODSStochastic model for single-cell growth and division. In order to char-acterize the variability of single-cell kinetics within a population, a sto-chastic (SDE-based) version of the single-cell growth model discussed inreference 4 is proposed. The model is formally similar to the one repre-sented by equations 1 and 2, although y in our case relates with size(length) of a single cell instead of number of individuals, and a with its

corresponding adjustment function reflecting the physiological state ofthe cell. Growth starts at a given initial length, x(0) (equal to x0), and a(0)is equal to a0,with a0 being a small quantity provided that the cell under-goes a first division. The adjustment function induces a delay in x whichmimics the initial adaptation of the cell to the new environment (the lagphase). After this period, growth proceeds exponentially up to a thresholdvalue, x(T), equal to xdiv, which determines the division of the cell in twodaughter cells. The time, T, at which such value is reached defines the timeto division, which when it occurs for the first time also includes the lagphase period.

The model we propose assumes that the specific cell growth rate, �, issubject to a stochastic fluctuation �W characterized by a Wiener process(see, for example, reference 20). For the sake of completeness, the maincharacteristics of the Wiener process and its role in the statistical proper-ties of the corresponding random variable Y(t) are discussed in the sup-plemental material. Accordingly, the dynamics for single-cell growth iswritten as a linear time-dependent stochastic differential equation:

FIG 3 TTD distributions (1st to 4th times to division) for Escherichia coli at 25°C (5). (a) 1st division; (b) 2nd division; (c) 3rd division; (d) 4th division. In eachplot, experimental distributions (bars) and fittings to gamma distributions (dashed lines) reported in reference 5 are compared with the corresponding modelestimations (solid lines).

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�Y � �a(t)�t � �a(t)�W (3)

where {Y(t)�R; t 0} denotes the random variable which takes values y(y � ln x) at t with a probability P(y,t y0,0). Function P(y,t y0,0) repre-sents the conditional probability of Y(t) being equal to y given Y(0) beingequal to y0. Function a(t) is the particular solution of equation 2 for a(0)(equal to a0), being of the form

a(t) �a0

a0 � (1 � a0)exp(�vt)(4)

The first term at the right-hand side of equation 3 is known as the driftand collects the deterministic size growth dynamics. Stochasticity is addedin the second term on the right-hand side of equation 3, where a newparameter, �, is postulated that expresses the intensity of the stochasticfluctuation.

Although in practice it is accepted that the size (length) to division(and thus the size [length] of the resulting daughter cells as well) is ran-domly distributed (11), a minimum critical length seems essential to trig-ger the process (4). In the formulation we present in this paper, this ran-dom effect is aggregated together with other sources of variability withinand between individuals, into the stochastic part of the growth dynamics(the second term on the right-hand side of equation 3).

Similarly to what has been proposed in reference 26 in the context ofstochastic population dynamics, more elaborated formulations of the sto-chastic model 3 are possible, which can incorporate distinct sources ofstochasticity. These might include, in addition to Y, a random equivalentof the adjustment function, which defined as A(t) would take values a(t) att with probability PA (a,t a0,0). Adding this new state, the system wouldresult into the following set of time-independent SDEs:

�Y � �A�t � �A�W (5)

�A � vA (1 � A)�t � �A(1 � A)�W (6)

where v and ε are the corresponding parameters associated with the sto-chastic adjustment function.

In a quite similar way, other stochastic variables (states) describing thedifferent sources of biological variability, such as the critical size to initiatedivision, fluctuations in the initial size of the daughter cells, or cell-to-cellinteractions, can be included in the description. Such extensions, how-ever, are not considered in the present work, as they involve extra param-eters which call for additional experimental information, difficult to col-lect or unavailable in the existing literature.

In this paper, cell growth and division are modeled by equations 3 and4. Note, however, that under appropriate experimental design conditions,the method we propose in this work can be extended in a straightforwardmanner to estimate those parameters.

Model calibration. Single-cell parameters � and � in equation 3 andthe initial value of the adjustment function a0 in equation 4 can be esti-mated from experimental time-to-division (TTD) distributions [which

FIG 4 TTD distributions (1st to 3rd times to division) for Escherichia coli at32°C (5). (a) 1st division; (b) 2nd division; (c) 3rd division. In each plot,experimental distributions (bars) and fittings to gamma distributions (dashedlines) reported in reference 5 are compared with the corresponding modelestimations (solid lines).

TABLE 1 Gamma function parameters given by reference 5a

Temp (°C)

Avg time and SD (h)

1st division 2nd division 3rd division 4th division

25 3.30, 1.02 1.04, 0.31 0.98, 0.29 0.92, 0.2932 1.01, 0.48 0.67, 0.22 0.53, 0.20a Distributions of times to division for Escherichia coli observed in experiments at 25and 32°C were fitted in reference 5 to gamma functions.

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we refer to hereafter as �(T)] as those reported in references 4 and 5), forinstance, by a least-squares method. Essentially it aims at the minimiza-tion of the differences between �(T) and a theoretical TTD �(T;�), de-fined as a function of the parameters. Formally the problem can be statedas follows:

min

J() with J() � �0

��(T ;) � �(T)�2dT (7)

where J(�) is the objective function to be minimized (i.e., the integral overtime to division of the square errors between data and model) and �represents the set of parameters to be estimated.

Assuming that growth and division accept a description based onequations 3 and 4, the theoretical TTD distribution can be computed bythe formula

��T� �P(d, T�y0, 0)

�0

P(d, S�y0, 0)dS

for 0 T (8)

where P(d,T y0,0) corresponds to the probability of the cell length reach-ing for the first time a value exp(d) at t equal to T, given the cell initial sizebeing x0 [so that y0 is equal to ln(x0)]. Such probability is obtained fromthe solution of a partial differential equation (PDE) (equation A-16 in the

TABLE 2 Model parameters estimated for the data provided by references 5 and 4 for Escherichia coli and Listeria innocua, respectivelya

Division � (h�1) � a0 SSqED—model SSqED—gamma

1stE. coli at 25°C 0.2400 0.1960 0.6130 1.80 � 10�5 2.00 � 10�5

E. coli at 32°C 0.5940 0.5577 0.6045 1.55 � 10�5 2.90 � 10�5

L. innocua,K0

0.4500 0.7746 0.3826 6.50 � 10�6

L. innocua,K5

0.0720 0.7669 0.1089 8.19 � 10�7

2ndE. coli at 25°C 0.6060 0.0821 1.0000 3.95 � 10�5 1.89 � 10�4

E. coli at 32°C 0.8100 0.1875 1.0000 1.16 � 10�6 1.18 � 10�4

3rdE. coli at 25°C 0.5700 0.1433 1.0000 9.57 � 10�6 8.43 � 10�5

E. coli at 32°C 0.9360 0.1572 1.0000 1.40 � 10�5 3.50 � 10�4

4thE. coli at 25°C 0.5880 0.0922 1.0000 4.42 � 10�5 2.08 � 10�4

a For Escherichia coli the data sets were collected at 25 and 32°C. The last two columns indicate the summation of the square errors (SSqED) between experimental data and themodel or the gamma functions proposed in reference 5 (see also Table 1). Data taken from reference 4 correspond with bacteria under no heat shock (K0) and subjected to a 5-minheat shock (K5).

FIG 5 Objective function J in expression 7 is plotted on a three-dimensional plot for different values of � and � to show the presence of a minimum. ExperimentalTTD data employed to compute the objective function correspond with the second division times at 32°C for Escherichia coli (5).

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supplemental material), known in stochastic calculus as the backwardsKolmogorov equation, with appropriate boundary and final conditionsfor every T in the interval (0, ).

In this work, the partial differential equation A-16 with the corre-sponding boundary conditions has been solved with a finite differencediscretization scheme (see http://www.matmol.org/) to approximate theoriginal partial differential equation (PDE) (27). For all cases, a meshconsisting of 501 elements was considered enough to accurately approx-imate the equation. Time integration of the resulting set of ordinary dif-ferential equations has been performed in Matlab with a standard ODEsolver (ode15s).

During first division, cells undergo adaptation, so a0 is a positive num-ber (smaller than 1) to be estimated. After the second, third, and fourthdivision times, cells are assumed to be adapted to the environment so thata0 is equal to 1 and the estimation reduces to the computation of thegrowth rate (�) and the intensity of the stochastic fluctuation (�).

Optimizers fminsearch and fmincon from the Matlab optimizationtoolbox have been employed to solve the least-squares minimizationproblem (equation 7), leading both methods to the same results for thecases considered.

In order to test the performance of the proposed model to reproducetime to division distributions, two sources of experimental data have beenemployed. One is taken from reference 4 and corresponds to the distribu-tions of times to first division for Listeria innocua under different heatshock durations ranging from no shock to a 5-min shock duration. Theother source comes from reference 5 and corresponds to the distributionsof times to first division and successive division times (up to the fourthdivision) for Escherichia coli at 25 and 32°C.

Simulation of bacterial population growth. Population growth froma given number of colony formation individuals over a given time horizonhas been simulated by assigning to each bacterium (including its corre-sponding offspring) equations 3 and 4 to be solved from its initial size toits size of division. The process is then repeated for each new offspringover the time horizon. A number of numerical solution methods for solv-ing the stochastic differential equations are available (see reference 28). Inthis work, the Euler-Maruyama algorithm has been selected for its sim-plicity. For convenience, simulation of population dynamics from theproposed single-cell stochastic model has been performed on a clustercomposed of 12 processing nodes (openSUSE 11.0 Linux with 23.5 GB ofRAM) and 160 processors in total, using the SGE task manager to distrib-ute the calculations between them.

RESULTS AND DISCUSSIONTheoretical TTD distributions versus SDE realizations. A com-putational experiment has been performed to show that equation8 provides a precise representation of the TTD distributions ob-tained by a number of individual bacteria growing according toequations 3 and 4 up to a critical length or size. Theoretical argu-ments that support such equivalence can be found in the frame-work of Ito calculus. For the interested reader, these are summa-rized in the supplemental material.

Cells with initial length (x0) of 8 were assumed to divide atdouble their length, i.e., to attain a division length (xdiv) of 16. Thetime to division of each realization (cell) is calculated as the timethe cell, which grows according to equation 3, first reaches the xdiv.

FIG 6 Growth curves at 32°C for Escherichia coli measured by viable counts (dots) for different inoculum size versus those generated by simulation (lines).Experimental data are taken from reference 5.

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To simulate cell growth, equation 3 is solved numerically with theEuler-Maruyama method (28) until the first time to division,namely, the time variable Y attains a value (d) that is identical toln(xdiv). The corresponding distribution of times to division(TTD) is constructed by repeating the simulation for a sufficientnumber of cells (each cell constitutes a realization) in order toproduce a representative ensemble. In the present study, ensem-bles comprised around 10,000 realizations.

Parameters employed in the simulation (� � 0.6060 h�1 and� � 0.0821) are on the order of those obtained from data takenfrom the literature for E. coli (see next subsection and Table 2 forfurther details). In order to evaluate the effect of the adjustmentfunction on the resulting TTD, two values were considered: an a0

of 0.1, which would correspond to a cell on its first division time,and an a0 of 1 for a cell completely adapted that is growing at itsmaximum specific rate.

Figure 1a and b present the resulting TTD distributions ob-tained from the SDE (bars) and from the proposed theoreticaldistribution 8 (lines) for a cell population undergoing adaptationand completely adapted, respectively. Parity plots in Fig. 1c and dcomparing cumulative distributions obtained from the SDE(QSDE) and the proposed theoretical distribution (Q�) prove thata perfect match exists between the two approaches. This is inagreement with the Kolmogorov-Smirnov tests applied with a0.05 significance level to check coincidence between the samplesof times to division and the cumulative distribution associatedwith � (equation 8).

It must be noted here that apart from the fact that the backwardKolmogorov PDE (equation A-16 in the supplemental material)offers direct information of the statistical properties of the sys-tems, its solution is by far much more efficient from a computa-tional point of view than is computing a significant number ofrealizations by means of the SDE, which usually demands a suffi-ciently populated ensemble to be constructed beforehand in orderto be representative. For the scenarios considered in Fig. 1, theensemble must be on the order of thousands of realizations. As anindication, solving the PDE in Matlab on a standard personalcomputer takes on the order of seconds and never more than a fewminutes. On the other hand, computing just one representativeensemble for any of the two cases considered requires around 1 hof computing time. For this reason, the use of the backward Kol-mogorov equation is preferred to Monte Carlo methods for sto-chastic model calibration purposes, which usually require re-peated evaluations of objective function 7.

Model performance to describe experimental TTD. We in-vestigated the capability of our stochastic model of single-cellgrowth to describe the distribution of times to division observedin experiments. The proposed model consists of SDE 3 togetherwith adjustment function 4. Model parameters include the growthrates, (� and �), the intensity of the stochastic fluctuation (�), theinitial length (x0 [thus b � ln x0]), the length at division (xdiv) (sothat d � ln xdiv), and the initial value of the adjustment function(a0). Following reference 4, it will be assumed that � is equal to �.As we have shown above for the computational experiment, theTTD distribution produced by our model can be precisely com-puted by means of equation 8.

Based on cell length data reported in reference 4, an averagevalue for cell length at an xdiv of 16 (in the units of pixels as theauthors report) was selected in all simulations. Division is as-sumed to result in two daughter cells of similar length, so that x0 is

FIG 7 Comparison between experimental TTD distributions for Salmonellaenterica taken from reference 2 (bars) and the model for the 1st, 2nd, and 3rddivision times (panels a, b, and c, respectively).

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equal to 8 pixels. Cell viability is taken into account in the modelby setting a minimum cell length below which it is assumed thatthe cell will die.

In order to compute the theoretical distribution of times todivision (equation 8), we set y0 as equal to ln x0 and solve the PDEA-16 discussed in the supplemental material for different timehorizons (T). In this study, a minimum length of 4 pixels wasselected, so that the domain for variable x lay in the interval be-tween 4 and 16 pixels. Hence, boundary conditions A-14 with cbeing equal to ln(4) and d being equal to ln(16) must be imposed(see the supplemental material).

A comparison of model predictions with experimental data ispresented in Fig. 2 for the first division times of Listeria innocuaunder no heat shock (Fig. 2a) and for a 5-min shock duration (Fig.2b) (4). The comparison of the predictions of our model and thecorresponding fittings provided by gamma functions for Esche-richia coli data at different division times and temperatures (5) ispresented in Fig. 3 and 4. Table 1 summarizes the gamma functionparameters given by reference 5, in the form of mean time andstandard deviation of the distributions.

Plots in Fig. 3 and 4 show the experimental data (bars) togetherwith the corresponding fittings to gamma functions provided inreference 5 (dashed lines), as well as model estimations (solidlines). Parameter values obtained from model calibration are pre-sented in Table 2, including the resulting summation of squareerrors (SSqED—model), which corresponds with the final valueattained by the objective function (denoted as J in equation 7). Aquantitative measure of the agreement between gamma distribu-tions and the experimental data is given in the last column of Table2, through the corresponding summations of square errors(SSqED— gamma).

As can be seen in the figures, there is quite good agreement(corroborated by Kolmogorov-Smirnov tests) between our modeland the experimental data for all division times at both tempera-tures. In fact, our model is capable of fitting experimental data forall divisions at 25 and 32°C, better than gamma distributions: asshown in Table 2, the values of the summation of square errors(SSqED—model) for 2nd to 4th division times are 1 order ofmagnitude smaller than those corresponding to the gamma dis-tributions (SSqED— gamma). For the 1st division time, modelfittings to the corresponding experimental data are only slightlybetter than those provided by gamma distributions (values forSSqED—model and SSqED— gamma are of the same order ofmagnitude), probably due to larger errors or uncertainties in theexperimental measurements.

A sensitivity analysis test was performed on the computed pa-rameters to calculate how their changes affect the objective func-tion. A typical shape of the objective function J for different pa-rameter sets is presented in Fig. 5, showing in all cases a clearminimum. This implies that just one optimal set of parameterscomplies with a given TTD curve, which suggests that the modelparameters are identifiable.

Finally, it is worth noting how similar the values are for thegrowth rates for the second to fourth times to division (� � 0.6 to0.7 h�1 at 25°C or � � 0.8 to 1.0 h�1 at 32°C) to the valuescorresponding to the first time to division (� � 0.24 h�1 at 25°Cor � � 0.60 to 0.70 h�1 at 32°C). The intensities of the stochasticfluctuations, on the other hand, remain quite uniform in the rangeof 0.1 to 0.2 for � at 25°C and 0.2 to 0.3 for � at 32°C for 1st to 4thtimes to division.

Stochastic modeling of population dynamics. To demon-strate the capability of the proposed model to describe populationgrowth, we followed reference 5 and made use of the parametersobtained from the TTD at 32°C for the first, second, and thirddivisions to simulate growth curves for Escherichia coli. As in ref-erence 5, the results of the simulation are compared with experi-mental data measured by viable counts and different inoculumsizes, showing excellent agreement, as can be seen in Fig. 6. Theproposed stochastic model can therefore be used to provide aneffective link between single cell growth and cell population dy-namics.

It can be argued that the model described by equation 3 is notparticularly superior to other constructions fitted to the experi-mental data, although it has been shown already to produce amore consistent fit (see Fig. 3 and 4). In this regard, it must be saidthat the proposed model has a clear biological interpretation,whereas a gamma function approximation is a mere empiricalfitting. The model makes use of a critical variable which might berelated to length or size (e.g., via the total amount of DNA) ofsingle cells, evolves during their cell cycle, and determines divisionin a way that incorporates cell variability within a bacterial popu-lation.

Moreover, the use of a differential formulation instead of aparticular solution makes the model predictive even underchanges in environmental variables (such as temperature or pH,for instance), provided that experimental data are available to cal-ibrate secondary models that relate stress variables to parametersof the single-cell growth model. From this point of view, the ap-proach presented can be particularly useful to predict populationgrowth under variable environmental conditions.

TABLE 3 Model parameters estimated for the data provided byreference 2 for Salmonella enterica at 25°C

Division � (h�1) � a0 Objective function

1st 0.4223 0.3077 0.4302 6.36 � 10�3

2nd 0.4839 0.1876 1.0000 1.31 � 10�3

3rd 0.6237 0.1686 1.0000 3.38 � 10�4

FIG 8 Detailed representation of a few realizations for population growthstarting from one single bacterium.

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Finally, it must be remarked that although alternative stochas-tic methods, and particularly the one proposed in reference 5 tosimulate population growth from a given TTD, constitute legiti-mate approaches, the computational efficiency is inferior to theone based on the use of equation 3. The direct use of the TTDdistribution (as in reference 5) requires one to obtain and keeptrack of the set of random division times, which becomes quiteinefficient as the size of the population increases. Handling divi-sion by means of stochastic differential equations as in our ap-proach becomes more efficient both from a mathematical andfrom a computational point of view.

Effect of initial number of cells. In order to test the model’sability to reproduce the behavior for small populations and topredict population growth as a function of the initial populationsize, we made use of the experimental data reported in reference 2for Salmonella enterica at 25°C. Data were obtained by the authors

using time-lapse microscopy and comprise time-to-division dis-tributions for first, second, and third division times (presented inthe plots in Fig. 7 as bars) as well as plots of colonial growthevolution starting from single individuals.

Stochastic model 3 was calibrated according to the proceduredescribed in Materials and Methods. Model parameters for thedifferent division times are presented in Table 3. Comparison be-tween experimental and estimated TTD distributions are plottedin Fig. 7, showing a quite reasonable agreement, especially whenone takes into account the relatively scarce experimental informa-tion available.

Figure 8 presents a few simulations of population growthcurves initiated from one cell. As it can be seen in the figure, theresemblance with the experimental observations reported in ref-erence 2 (Fig. 4 in that article) is remarkable.

Our model is also able to explain the effect of the initial number

FIG 9 Ten thousand simulated realizations for population growth curves for Salmonella enterica starting from different initial cell numbers: 1 single bacterium(a), 2 single bacteria (b), 10 individuals (c), and 100 individuals (d).

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of cells on the variability of population growth. Figure 9 presentssimulations starting from 1 to 100 individuals, showing that vari-ability in population growth is reduced as the number of cellsinitiating the population increases.

From this point of view, the proposed model is in agreementwith deterministic growth descriptions involving large populationsizes and with the fact that random effects become negligible as thesize of the population increases. Note, however, that this is not thecase when models with random parameters are employed insteadto simulate population growth curves.

To illustrate this fact, a model similar to that proposed in ref-erence 12, with lag time and specific growth rate parameters fol-lowing random distributions, has been employed (as suggested inreference 2) to simulate population growth rates. Simulations aredepicted in Fig. 10 for different numbers of initial cells using the

random distributions for model parameters reported in reference2. This model predicts larger variabilities as the initial number ofcells decreases (Fig. 10), due to the random nature of its parame-ters; however, given sufficient time, variability spreads out nomatter the number of initial cells, which is in contradiction to theobserved experimental behavior of large microbial populations(variability becomes negligible through the law of large numbers).In contrast, the model we propose achieves a constant (stationary)variability range which reduces with population size and becomesnegligible for populations initiated with large number of individ-uals, which is in agreement with classical bacterial growth kinetics(in this example, for numbers above 100 individuals, as can beseen in Fig. 9d). This effect can also be seen in Fig. 11a and b, whichdepict probability distribution functions at different times forpopulations starting from 1 and 100 individuals, respectively. As

FIG 10 Ten thousand simulated realizations for population growth curves for Salmonella enterica using an ordinary differential equation with randomparameters for different initial cell numbers. Parameter distribution is taken from reference 2. (a) One single bacterium; (b) 2 single bacteria; (c) 10 individuals;(d) 100 individuals.

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observed in Fig. 11a and b, distribution width shrinks as the initialnumber of individuals increases. Note that in both situations, dis-tributions evolve to a constant shape function which travels rightin the x axis (representing log of cell numbers).

Finally, it must be remarked that our model allows the compu-tation of probability (and cumulative probability) distributionsassociated to the population as a function of time in a straightfor-ward manner (Fig. 11). As an example, let us suppose that for thecase considered (Salmonella enterica at 25°C), 1,000 individualsdefine a risk threshold. Figure 11c and d indicate that the proba-bility of a bacterial population being larger than 1,000 individualsis negligible before 10 days if growth is initiated with 1 cell, whilesuch time is reduced to more than half (less than 5 days) if theinitial number of cells is 10. From this perspective, the plots of theresulting cumulative probability for scenarios involving different

initial bacteria can give indications on expected shelf life, whichmay be of help for risk assessment and shelf life evaluations.

Conclusions. A stochastic version of an individual bacterialgrowth model has been proposed to describe cell heterogeneity fora given population and to predict time-to-division distributions.Distributions are reconstructed by solving the backward Kolm-ogorov equation, which is a partial differential equation establish-ing the functional relationship of probability density with bacte-rial length (size) and time.

The method is computationally efficient since it only requiresthe solution of a partial differential equation (which takes only afew seconds on a standard personal computer) to reconstruct thedensity function of the distribution and thus the distribution oftimes to division while still maintaining information about thestochastic nature of the process. Evidence of the model’s ability to

FIG 11 Probability and cumulative probability distribution functions for Salmonella enterica populations computed with the proposed model (parameters arein Table 3) starting from one individual (a and c) and from 100 individuals (b and d). Panels c and d depict the cumulative probability for the population to attain1,000 individuals as a function of time for 1 and 100 bacteria, respectively.

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cope with the observed behavior is given for the distributions ofdivision times of Escherichia coli and Listeria innocua reported inthe literature.

In addition, the capability of the model to describe populationgrowth has been demonstrated, explaining population variabilityincrease as the initial number of cells is reduced, thus providing abridge that links individual bacterial growth and the growth ofbacterial populations and opens the door for applications in riskassessment and prediction of shelf life.

ACKNOWLEDGMENTS

This work has been funded by the Spanish Ministry of Science and Inno-vation through mobility grant Salvador de Madariaga (PR2011-0363) andby project ISFORQUALITY (AGL2012-39951-C02-01).

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