Modeling Bacterial Population Growth from Stochastic Single Cell Dynamics

Transcription

1 Modeling Bacterial Population Growth from Stochastic Single Cell Dynamics Antonio A. Alonso, Ignacio Molina, and Constantinos Theodoropoulos Process Engineering Group, IIM-CSIC, Spanish Council for Scientific Research, Eduardo Cabello, Vigo, Spain School of Chemical Engineering and Analytical Science, University of Manchester, Manchester M PL, UK June, Abstract Few bacterial cells may be sufficient to produce a food-borne illness outbreak, provided that they are capable of adapting and proliferating on a food matrix. This is why any quantitative health risk assessment policy must incorporate methods to accurately predict the growth of bacterial populations from a small number of pathogens. In this aim, mathematical models have become a powerful tool. Unfortunately, at low cell concentrations, standard deterministic models fail to predict the fate of the population, essentially because the heterogeneity between individuals becomes relevant. In this contribution a stochastic differential equation (SDE) model is proposed to describe variability within single cell growth and division and to simulate population growth from a given initial number of individuals. We provide evidence of the model ability to explain the observed distributions of times to division, including the lag-time produced by the adaptation to the environment, by comparing model predictions with experiments from the literature for Escherichia coli, Listeria innocua and Salmonella enterica. The model is shown to accurately predict experimental growth population dynamics both for small and large microbial populations. Author to whom correspondence should be addressed. antonio@iim.csic.es

2 The use of stochastic models for the estimation of parameters to successfully fit experimental data is a particularly challenging problem. For instance if Monte Carlo methods are employed to model the required distributions of times to division, the parameter estimation problem can become numerically intractable. We overcome this limitation by converting the stochastic description to a partial differential equation (backward Kolmogorov) instead, which relates to the distribution of division times. Contrary to previous stochastic formulations based on random parameters, the present model is capable of explaining the variability observed in populations that result from the growth of a few number of initial cells as well as the lack of it when compared to populations initiated by a larger number of individuals, where the random effects become negligible.

3 Introduction Often bacterial contamination of foods starts with a small number of bacteria that are capable of adapting and proliferating by repeated divisions on a given food matrix. At low cell concentrations, standard deterministic models fail to predict the variability of the bacterial population. This is so because at low initial cell numbers, heterogeneity between individuals and its influence on the the division times becomes relevant and has a net influence on the population. Consequently, the behavior of individual bacteria cannot be neglected when assessing possible health risks along the food chain, either during storage or distribution. Recently, attention has been drawn on the need of modeling and simulation methods to observe and describe the variability of single cell behavior and small populations [, ] in order to produce realistic estimations of safety risks along the food chain, for instance during storage and distribution, or during food processing. In this paper connections will be established between individual bacteria growth and division, the corresponding distributions of times to division, and population growth curves that in the long term can aid the quantification, on a probabilistic basis, of microbial risk and product shelf-life. A number of modeling approaches for bacteria population dynamics has been built around the concept of times to division and particularly the first lag time to division of cell populations [,,, ]. The development of analytical techniques capable of measuring single cell parameters such as cell length, renewed interest in modeling single cell growth. Experimental techniques for single cell studies include turbidimetry [see for example,, ], lithographic techniques [] or flow cytometry []. Recently, time-lapse microscopy has been successfully applied to get

4 quantitative information of colonial growth dynamics originated from single cells []. Based on the flow chamber microscopy technique proposed in [], models for individual cell growth have been developed in [] and []. They are essentially adaptations of the now classical model proposed by [] to describe the growth of bacteria populations before attaining the stationary phase. It consists of two ordinary differential equations that can be written as: dy dt da dt = µa, () = νa( a), () where y represents the natural logarithm of the population size and µ the maximum specific growth rate. Variable a(t) is known as the adjustment function and relates to a certain phys- iological state of the population. This variable has been introduced to describe the gradual adaptation of the cells to the new environment. Initially, it takes a small value which increases at a rate proportional to ν up to a maximum of one. This variable induces a delay in the growth of the population size which depends on the inverse of ν (the larger the rate, the smaller the delay becomes) and is employed to model the observed initial lag time. In the context of a single cell growth, the state variable y which relates to the size of the population in [] is identified as a critical variable that determines cell growth and characterizes division when it reaches a particular threshold. In previous works, it has been interpreted as cell length (e.g. [] or []) or cell DNA content []. Similarly, the adjustment function in [], is used in [] and [] to model the adaptation of a given cell to the environment. Stochastic fluctuations in gene transcription and translation within a cell, or in response to cellto-cell disturbances are considered as the main sources of heterogeneity among individual cells within a population [, ]. In order to capture such behavioral noise [], previous approaches combined deterministic equations for single cell or population dynamics of the form () and

5 (), with random parameters and/or initial random conditions (both described by appropriate probability distribution functions). In [] the random nature of cell division is modeled by imposing a uniformly distributed random length threshold at which each cell divides. This seems to be also the case in [], although parameter estimation for cells subject to different heat shock treatments is based on a deterministic model. However, the connections between such models and the observed distributions of division times have not been clearly discussed yet. Inspired by [], a model of cell population dynamics is suggested in [] that explicitly includes a lag time distribution function. This is in agreement with other well known approximations such as the Weibull or Gamma probability distribution functions [, ]. Recently, a model similar to that proposed by [] with parameters following random distributions has been employed as suggested in [] to simulated population growth rates. In [] a population dynamics stochastic model has been proposed for Escherichia coli that makes use of time to division distribution functions to compute the time (the stochastic variable) at which cell division occurs. This information is provided to the algorithm either directly from experimental data (e.g flow chamber microscopy or turbidimetry) or from a Gamma distribution, which has been previously fitted to experimental data. In this work, connections between individual bacterial growth variables and distribution of times to division will be established by a stochastic version of the models proposed in [] and [, ]. The underlying premise is that cell growth is the result of a large number of biochemical reactions taking place on a microscopic domain (thus involving a relatively small number of molecular species). Standard assumptions can then be invoked to relate a chemical (or biochemical) microscopic master equation to its mesoscopic (chemical Langevin) counterpart [] which is

6 nothing but a stochastic differential equation (SDE) []. Hence, heterogeneity between individuals or fluctuations within each individual (e.g. due to behavioral noise) will be represented by a SDE that will result from adding a stochastic component to the specific growth rate. The introduction of stochasticity on the growth rate has a strong biological interpretation: SDEs are usually employed in a systems biology context to collect the random effects of fluctuations on the system. Gillespie [] gives convincing arguments to show how the accumulation of stochastic events can be cast into SDEs, stating that the aggregated effect of many events at the cell level can be captured/described by SDE models. Models based on SDE systems have been used previously to describe cell population growth and division for plankton [] and bacteria []. This approach has been also the one adopted in the study of bacterial systems under the action of bacteriophage [] or antibiotics []. In the context of cell growth and division, such representation which seeks the aggregation of the undergoing biochemical processes during the cell cycle, will be shown to reproduce reasonably well the time to division distributions observed and reported in the literature. It is well known from stochastic systems theory [] that the collective effect of a SDE system, namely the evolution of the probability distribution associated to the random state variables, can be computed as the solution of two partial differential equations (PDE): the so called Kolmogorov equations, forward or backwards in time. We make use of one such equation, the backward Kolmogorov, to characterize time to division distributions (TTD) -including first time to division- and to efficiently estimate parameters of the underlying stochastic dynamics from the experimental distributions. It is important to emphasize here that from a computational point of view, this way of approaching the model calibration problem is particularly efficient. This is so because it only requires the solution of

7 a partial differential equation as opposed to obtaining a complete ensemble of realizations by repeatedly solving the SDE to reconstruct the density function associated to the distribution of times to division (e.g. by Monte Carlo methods). Similar approaches have been employed previously, although in the context of gene expression networks to extract kinetic parameters associated to individual cells from protein distributions obtained by cell population measurements []. The use of stochastic methods related to Kolmogorov equations in predictive microbiology has been suggested previously [], albeit in the context of population growth dynamics, as a means to describe variability of the environment as well as uncertainty due to limitations of the measurement equipment. However, to the best of our knowledge, these methods have not been employed so far to model either single cell growth kinetics, or to estimate parameters based on experimental TTDs. Our model assumes that bacteria division is subject to stochastic fluctuations which integrate the effect of transcription at the level of each individual. From that point of view our approach follows the assumptions implicit in [] where a master equation is employed to connect parameters associated to a stochastic process (a signalling network) to observations obtained at cell population level. Here, we use observations obtained at the population level (time to division distributions) to estimate parameters of the stochastic process that describes division. MATERIALS AND METHODS Stochastic model for single cell growth and division In order to characterize the variability of single cell kinetics within a population, a stochastic (SDE-based) version of the single cell growth model discussed in [] is proposed. The model is formally similar to the one represented by equations () and () although y in our case relates with size (length) of a single cell instead of number of individuals, and a with its corresponding

8 adjustment function reflecting the physiological state of the cell. Growth starts at a given initial length x() = x, for which a() = a, with a being a small quantity provided that the cell undergoes a first division. The adjustment function induces a delay in x which mimics the initial adaptation of the cell to the new environment (the lag phase). After this period, growth proceeds exponentially up to a threshold value x(t ) = x div which determines the division of the cell in two daughter cells. The time T at which such value is reached defines the time to division, which when it occurs for the first time also includes the lag-phase period. The model we propose assumes that the specific cell growth rate µ is subject to a stochastic fluctuation δw characterized by a Wiener process [see for example, ]. For the sake of completeness, the main characteristics of the Wiener process and its role in the statistical properties of the corresponding random variable Y (t) are discussed in the Supplemental Material. Accordingly, the dynamics for single cell growth is written as a linear time dependent stochastic differential equation: δy = µa(t)δt + ξa(t)δw Y () = y () where {Y (t) R; t > } denotes the random variable which takes values y (y = ln x) at t with a probability P(y, t y, ). Function P(y, t y, ) represents the conditional probability of Y (t) = y given Y () = y. Function a(t) is the particular solution of () for a() = a, being of the form: a(t) = a a + ( a )exp( νt) () The first term at the right hand side of () is known as the drift and collects the deterministic size growth dynamics. Stochasticity is added in the second term on the right hand side of () where a new parameter ξ is postulated that expresses the intensity of the stochastic fluctuation. Although in practice it is accepted that the size (length) to division (and thus the size/length

9 of the resulting daughter cells as well) is randomly distributed [], a minimum critical length seems essential to trigger the process []. In the formulation we present in this paper this random effect is aggregated together with other sources of variability within and between individuals, into the stochastic part of the growth dynamics (the second term at the right hand side of equation ). Similarly to what has been proposed in [] in the context of stochastic population dynamics, more elaborated formulations of the stochastic model () are possible, which can incorporate distinct sources of stochasticity. These might include, in addition to Y, a random equivalent of the adjustment function, which defined as A(t) would take values a(t) at t with a probability P A (a, t a, ). Adding this new state, the system would result into the following set of time independent SDEs: δy = µaδt + ξaδw () δa = νa( A)δt + ϵa( A)δW () where ν and ϵ are the corresponding parameters associated to the stochastic adjustment function. In a quite similar way, other stochastic variables (states) describing the different sources of biological variability such as the critical size to initiate division, fluctuations in the initial size of the daughter cells or cell-to-cell interactions, can be included in the description. Such extensions however will not be considered in the present work as they involve extra parameters which call for additional experimental information, difficult to collect or unavailable in the existing literature. In this contribution, cell growth and division will be modeled by equations () and (). Note however that under appropriate experimental design conditions, the method we propose in this work can be extended in a straightforward manner to estimate those parameters.

10 Model calibration Single cell parameters µ and ξ in (), and the initial value of the adjustment function a in () can be estimated from experimental time to division (TTD) distributions ( we will refer to as ˆΛ(T )) as those reported in [] or [] for instance, by a least squares method. Essentially it aims at the minimization of the differences between ˆΛ(T ) and a theoretical TTD Λ(T ; θ), defined as a function of the parameters. Formally the problem can be stated as follows: min θ J(θ) with J(θ) = [Λ(T ; θ) ˆΛ(T )] dt () where J(θ) is the objective function to be minimized (i.e. the integral over time to division of the square errors between data and model) and θ represents the set of parameters to be estimated. Assuming that growth and division accepts a description based on equations () and (), the theoretical TTD distribution can be computed by the formula: Λ(T ) = P(d, T y, ) P(d, S y, )ds for T () where P(d, T y, ) corresponds to the probability of the cell length reaching for the first time a value exp(d) at t = T, given the cell initial size being x (so that y = ln(x )). Such probability is obtained from the solution of a partial differential equation (PDE) (equation A- in the Supplemental Material), known in stochastic calculus as the backwards Kolmogorov equation, with appropriate boundary and final conditions for every T in the interval (, ). In this work, the partial differential equation (A-) with the corresponding boundary conditions has been solved with a finite differences discretization scheme (see to approximate the original PDE (Partial Differential Equation) []. For all the cases considered a mesh consisting of elements was enough to accurately approximate the equation. Time integration of the resulting set of ordinary differential equations has been performed in Matlab

11 with a standard ODE solver (odes). During first division, cells will undergo adaptation so that a is a positive number (smaller than ) to be estimated. After the second, third and fourth division times cells are assumed to be adapted to the environment so that a = and the estimation reduces to the computation of the growth rate µ and the intensity of the stochastic fluctuation ξ. Optimizers fminsearch and fmincon from the Matlab optimization toolbox have been employed to solve the least squares minimization problem (), leading both methods to the same results for the cases considered. In order to test the performance of the proposed model to reproduce time to division distri- butions, two sources of experimental data have been employed. One is taken from [] and corresponds to the distributions of times to first division for Listeria innocua under different heat shock durations ranging from no shock to a minutes shock duration. The other source comes from [] and corresponds to the distributions of times to first division and successive division times (up to fourth division) for Escherichia coli at C and C. Simulation of bacterial population growth Population growth from a given number of colony formation individuals over a given time horizon has been simulated by assigning to each bacteria (including its corresponding offsprings) equations () and () to be solved from its initial size to its size of division. The process is then repeated for each new offspring over the time horizon. A number of numerical solution methods for solving the stochastic differential equations are at hand [see ]. In this work the Euler-Maruyama algorithm has been selected for its simplicity. For convenience, simulation of population dynamics from the proposed single cell stochastic model has been performed on a

12 cluster composed of processing nodes (opensuse. Linux with. GB of RAM) and processors in total, using the SGE task manager to distribute the calculations between them. RESULTS AND DISCUSSION Theoretical TTD distributions versus SDE realizations A computational experiment has been performed to show that equation () provides a precise representation of the TTD distributions obtained by a number of individual bacteria growing according to equations () and () up to a critical length/size. Theoretical arguments that support such equivalence can be found in the framework of Ito calculus. reader, these are summarized in the Supplemental Material. For the interested Cells with initial length x = were assumed to divide at double their length i.e. x div =. The time to division of each realization (cell) is calculated as the time the cell, which grows according to equation (), first reaches the division length x div. To simulate cell growth, equation () is solved numerically with the Euler-Maruyama method [] until the first time to division, namely the time variable Y attains a value d ln(x div ). The corresponding distribution of times to division (TTD) is constructed by repeating the simulation for a sufficient number of cells (each cell constitutes a realization) in order to produce a representative ensemble. In the present study ensembles comprised around realizations. Parameters employed in the simulation (µ =.hr and ξ =.) are in the order of those obtained from data taken from the literature for E. coli (see next subsection and Table for further details). In order to evaluate the effect of the adjustment function on the resulting TTD, two values were considered: a =. which would correspond to a cell on its first division time and a = for a cell completely adapted that is growing at its maximum specific rate.

13 Figures a and b represent the resulting TTD distributions obtained from the SDE (bars) and from the proposed theoretical distribution () (continuous lines) for a cell population undergoing adaptation and completely adapted, respectively. Parity plots in Figures c and d comparing cumulative distributions obtained from the SDE (Q SDE ) and the proposed theoretical distribution (Q Λ ) prove that a perfect match exists between the two approaches. This is in agreement with the Kolmogorov-Smirnov tests applied with a. significance level to check coincidence between the samples of times to division and the cumulative distribution associated to Λ (equation ()). It must be noted here that apart from the fact that the backward Kolmogorov PDE (equation A- derived in the Supplemental Material) offers direct information of the statistical properties of the systems, its solution is by far much more efficient from a computational point of view than computing a significant number of realizations by means of the SDE, which usually demands a sufficiently populated ensemble to be constructed beforehand in order to be representative. For the scenarios considered in Figure, the ensemble must be in the order of thousands of realizations. As an indication, solving the PDE in Matlab on a standard PC takes in the order of seconds and never more than a few minutes. On the other hand computing just one representative ensemble for any of the two cases considered requires around one hour of computing time. For this reason the use of the backward Kolmogorov equation is preferred to Monte Carlo methods for stochastic model calibration purposes which usually require repeated evaluations of the objective function (). Model performance to describe experimental TTD We investigate the capability of our stochastic model of single cell growth to describe the distri- bution of times to division observed in experiments. The proposed model consists of the SDE

14 () together with the adjustment function (). Model parameters include the growth rates, µ and ν, the intensity of the stochastic fluctuation ξ, the initial length x (thus b = ln x ), the length at division x div (so that d = ln x div ) and the initial value of the adjustment function a. Following [] it will be assumed that ν = µ. As we have shown above on the computational experiment, the TTD distribution produced by our model can be precisely computed by means of equation (). Based on cell length data reported in [], an average value for cell length at division x div = (in the units of pixels as the authors report) was selected in all simulations. Division is assumed to result in two daughter cells of similar length, so that x = pixels. Cell viability is taken into account in the model by setting a minimum cell length below which it is assumed that the cell will die. In order to compute the theoretical distribution of times to division (), we set y = ln x and solve the PDE (A-) discussed in the Supplemental Material for different time horizons T. In this study a minimum length of pixels was selected, so that the domain for variable x lies in the interval between and pixels. Hence, boundary conditions (A-) for c = ln() and d = ln() must be imposed (see Supplemental Material). A comparison of model predictions with experimental data is presented in Figure for the first division times of Listeria innocua under no heat shock (Figure a), and for a minutes shock duration (Figure b) []. The comparison of the predictions of our model and the corresponding fittings provided by Gamma functions for Escherichia coli data at different division times and temperatures [] is presented in Figure and Figure. Table summarizes the Gamma function parameters given by [], in the form of mean time and standard deviation of the distributions. In Figures and, plots show the experimental data (bars) together with the corresponding

15 fittings to Gamma functions provided by [] (dashed lines), and model estimations (continuous lines). Parameter values obtained from model calibration are presented in Table, including the resulting summation of square errors (SSqED-Model) which corresponds with the final value attained by the objective function (denoted as J in ()). A quantitative measure of the agreement between Gamma distributions and the experimental data is given in the last column of Table, through the corresponding summations of square errors (SSqED-Gamma). As it can be seen in Figures and there is a quite good agreement (corroborated by Kolmogorov- Smirnov tests) between our model and the experimental data for all division times at both temperatures. In fact, our model is capable of fitting experimental data for all divisions at C and C, better than Gamma distributions: As shown in Table, the values of the summation of square errors SSqED-Model for nd to th division times are one order of magnitude smaller than those corresponding to the Gamma distributions (SSqED-Gamma). For the st division time, model fittings to the corresponding experimental data are only slightly better than those provided by Gamma distributions (values SSqED-Model and SSqED-Gamma are of the same order of magnitude) probably due to larger errors/uncertainties in the experimental measurements. A sensitivity analysis test was performed on the computed parameters to calculate how their changes affect the objective function. A typical shape of the objective function J for different parameter sets is presented in Figure, showing in all cases a clear minimum. This implies that just one optimal set of parameters complies with a given TTD curve, which suggests that the model parameters are identifiable. Finally, it is worth noting how similar the values are for the growth rates for the second to fourth times to division (µ..hr at C or µ..hr at C) compared to the

16 values corresponding to the first time to division (µ.hr at C or µ..hr at C). The intensities of the stochastic fluctuations on the other hand remain quite uniform in the range of ξ.. at C, and ξ.. at C, for st to th times to division. Stochastic modeling of population dynamics To demonstrate the capability of the proposed model to describe population growth, we follow [] and make use of the parameters obtained from the TTD at C for the first, second and third divisions to simulate growth curves for Escherichia coli. As in [], the results of the simulation are compared with experimental data measured by viable counts and different inoculum sizes, showing excellent agreement as it can be seen in Figure. The proposed stochastic model can therefore be used to provide an effective link between single cell growth and cell population dynamics. It can be argued that the model described by () is not particularly superior to other constructions fitted to the experimental data, although it has been shown already to produce a more consistent fit (see Figures and ). In this regard, it must be said that the proposed model has a clear biological interpretation whereas a Gamma function approximation is a mere empirical fitting. The model makes use of a critical variable which might be related to length or size (e.g via the total amount of DNA) of single cells, evolves during their cell cycle and determines division in a way that incorporates cell variability within a bacterial population. Moreover, the use of a differential formulation instead of a particular solution makes the model predictive even under changes in environmental variables (such as temperature or ph, for in- stance), provided that experimental data are available to calibrate secondary models that relate stress variables to parameters of the single cell growth model. From this point of view the approach presented can be particularly useful to predict population growth under variable en-

17 vironmental conditions. Finally, it must be remarked that although alternative stochastic methods, and particularly the one proposed in [] to simulate population growth from a given TTD, constitute legitimate approaches, the computational efficiency are inferior than the one based on the use of (). The direct use of the TTD distribution (as in []) requires to obtain and keep track of the set of random division times what becomes quite inefficient as the size of the population increases. Handling division by means of stochastic differential equations as in our approach becomes more efficient both from a mathematical and from a computational point of view. The effect of the initial number of cells In order to test model ability to reproduce the behavior for small populations and to predict population growth as a function of the initial population size, we make use of the experimental data reported in [] for Salmonella enterica at C. Data were obtained by the authors using time-lapse microscopy and comprise time to division distributions for first, second and third division times (presented in the plots in Figure as bars) as well as plots of colonial growth evolution starting from single individuals. The stochastic model () was calibrated according to the procedure described in Section. Model parameters for the different division times are presented in Table. Comparison be- tween experimental and estimated TTD distributions are plotted in Figure a-c showing a quite reasonable agreement especially when one takes into account the relatively scarce experimental information available. Figure presents a few simulations of population growth curves initiated from one cell. As it can be seen in the Figure, the resemblance with the experimental observations reported in []

18 (Figure in that article) is remarkable. Our model is also able to explain the effect of the initial number of cells on the variability of population growth. Figure presents simulations starting from to individuals, showing that variability in population growth reduces as the number of cells initiating the population increases. From this point of view the proposed model is in agreement with deterministic growth descriptions involving large population sizes, and with the fact that random effects become negligible as the size of the population increases. Note however that this is not the case when models with random parameters are employed instead to simulate population growth curves. To illustrate this fact a model similar to that proposed in [] with lag time and specific growth rate parameters following random distributions, has been employed (as suggested in []) to simulate population growth rates. Simulations are depicted in Figure for different numbers of initial cells using the random distributions for model parameters reported in []. This model predicts larger variabilities as the initial number of cells decreases (Figures a-d), due to the random nature of its parameters, however given sufficient time, variability spreads out no matter the number of initial cells, which is in contradiction with the observed experimental behavior of large microbial populations (variability becomes negligible through the law of large numbers). By contrast, the model we propose achieves a constant (stationary) variability range which reduces with population size and becomes negligible for populations initiated with large number of individuals, which is in agreement with classical bacterial growth kinetics (in this example, for numbers above individuals as it can be seen in Figure d). This effect can be shown also in Figures a-b which depict probability distribution functions at different times for populations starting from and individuals, respectively. As observed in the Figures, distribution width

19 shrinks as the initial number of individuals increases. Note that in both situations, distributions evolve to a constant shape function which travels right in the x-axis (representing log of cell numbers). Finally it must be remarked that our model allows the computation of probability (and cumulative probability) distributions associated to the population as a function of time in a straightforward manner (see Figure ). As an example, let us suppose that for the case considered (Salmonella enterica at C) individuals define a risk threshold, Figures c-d indicate that the probability of a bacterial population being larger than individuals is negligible before days if growth is initiated with cell, while such time reduces to more than half (less than days) if the initial number of cells is. From this perspective, the plots of the resulting cumulative probability for scenarios involving different initial bacteria can give indications on expected shelf-life, what may be of help for risk assessment and shelf-life evaluations Conclusions A stochastic version of an individual bacterial growth model has been proposed to describe cell heterogeneity on a given population and to predict time to division distributions. Distributions are reconstructed by solving the backward Kolmogorov equation which is a partial differential equation establishing the functional relationship of probability density with bacteria length/size and time. The method is computationally efficient since it only requires the solution of a partial differential equation (which takes only a few seconds on a standard PC) to reconstruct the density function of the distribution and thus the distribution of times to division, while it still maintains information about the stochastic nature of the process. Evidence of the model ability to cope with the observed behavior is given for the distributions of division times of Escherichia coli and Listeria

20 innocua reported in the literature. In addition, it has been demonstrated the capability of the model to describe population growth, explaining population variability increase as the initial number of cells reduces, thus providing a bridge that links individual bacteria growth and the growth of bacteria populations, and opens the door for applications in risk assessment and prediction of shelf-life. Acknowledgements This work has been funded by the the Spanish Ministry of Science and Innovation through the mobility grant Salvador de Madariaga (PR-), and by project (ISFORQUALITY, AGL--C-). References [] J. Baranyi. Stochastic modelling of bacterial lag phase. Journal of Food Microbiology, ():,. [] K. Koutsoumanis and A. Lianou. Stochasticity in colonial growth dynamics of individual bacteria cells. Applied and Environmental Microbiology, ():,. [] J. Baranyi. Comparison of stochastic and deterministic concepts of bacterial lag. Journal of Theoretical Biology, ():,. [] Z. Kutalik, M. Razaz, A. Elfwing, A. Ballagi, and J. Baranyi. Stochastic modelling of individual cell growth using flow chamber microscopy images. Food Microbiology, ():,. International Journal of [] C. Pin and J. Baranyi. Kinetics of single cells: observation and modeling of a stochastic process. Applied and Environmental Microbiology, ():,.

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24 Tables Table : Distributions of times to division for Escherichia coli observed in experiments at T = C and T = C were fitted in [] to Gamma functions. Function parameters are average time (τ) and standard deviation (σ), expressed in hours. Temperature Time to st div. nd div. rd div. th div. T = C τ =., σ =. τ =., σ =. τ =., σ =. τ =., σ =. T = C τ =., σ =. τ =., σ =. τ =., σ =. -

25 Table : Model parameters estimated for the data provided by [] and [] for Escherichia coli and Listeria innocua, respectively. For Escherichia coli the data sets were collected at C and C. Last two columns indicate the summation of the square errors (SSqED) between experimental data and the model or the Gamma functions proposed in [] (see also Table ). Data taken from [] correspond with bacteria under no heat shock (K), and subject to a minutes heat shock (K). st division µ (hr ) ξ a SSqED (Model) SSqED (Gamma) E. coli at T = C..... E. coli at T = C..... L. innocua, K.... L. innocua, K.... nd Division µ (hr ) ξ a SSqED (Model) SSqED (Gamma) E. coli at T = C..... E. coli at T = C..... rd Division µ (hr ) ξ a SSqED (Model) SSqED (Gamma) E. coli at T = C..... E. coli at T = C..... th Division µ (hr ) ξ a SSqED (Model) SSqED (Gamma) E. coli at T = C..... Table : Model parameters estimated for the data provided by [] for Salmonella enterica at C. st division µ (hr ) ξ a Obj. Func..... nd Division µ (hr ) ξ a Obj. Func..... rd Division µ (hr ) ξ a Obj. Func.....

26 (a) (b).. Probability density. Probability density Time to division (h) (c) Time to division (h) (d) Q SDE. Q SDE Q Λ Q Λ Figure : Distribution of times to division for an ensemble of realizations obtained from the SDE () and (bars) and from equation () (continuous lines). Model parameters are µ =.hr and ξ =.. Figure (a) represents distributions for cells under adaptation (a =.). Figure (b) represents cells completely adapted (a =.). The corresponding parity plots comparing the cumulative distributions computed from the SDE (Q SDE ) and the Λ distributions () (Q Λ ) are presented in Figures (c) and (d) for both scenarios.

27 (a) (b).. x Probability density... Probability density Time to division (min) Time to division (min) Figure : Comparison between the experimental distribution (bars) of times to first division for Listeria innocua [] and the theoretical distribution of times to division ()(continuous lines). Estimated model parameters are presented in Table. (a) No heat shock (b) minutes heat shock.

28 (a) (b)... Probability density.... Probability density... Time to division (h) (c)... Time to division (h) (d) Probability density Time to division (h) Probability density Time to division (h) Figure : TTD distributions (st to th times to division) for Escherichia coli at T = C []. (a) First division, (b) nd division, (c) rd division, (d) th division. In each plot, experimental distributions (bars) and fittings to Gamma distributions (dashed lines) reported in [] are compared with the corresponding model estimations (continuous lines).

29 (a)... Probability density Time to division (h) (b). Probability density..... Time to division (h) (c).. Probability density..... Time to division (h) Figure : TTD distributions (st to rd times to division) for Escherichia coli at T = C []. (a) First division, (b) nd division, (c) rd division. In each plot, experimental distributions (bars) and fittings to Gamma distributions (dashed lines) reported in [] are compared with the corresponding model estimations (continuous lines).

30 J.. ξ.... µ.... Figure : Objective function J in expression () is plotted on a D plot for different values of µ and ξ to show the presence of a minimum. Experimental TTD data employed to compute the objective function correspond with the second division times at T = C for Escherichia coli [].

31 log(cfu) log(cfu) Time (h) Time (h) log(cfu) log(cfu) Time (h) Time (h) Figure : Growth curves at T = C for Escherichia coli measured by viable counts (dots) for different inoculum size vs those generated by simulation (continuous lines). Experimental data are taken from [].

32 (a).. Probability density.... Time to division (h) (b). Probability density.... Time to division (h) (b).. Probability density.... Time to division (h) Figure : Comparison between experimental TTD distributions for Salmonella enterica taken from [] (bars) and the model for the st, nd and rd divisions times, in plots (a), (b) and (c), respectively.

33 .. Ln (Cell number)... Time (h) Figure : A detailed representation of a few realizations for population growth starting from one single bacteria.

34 (a) (b) Ln (Cell Number) Ln (Cell Number) Time (h) (c) Time (h) (d) Ln (Cell Number) Ln (Cell Number) Time (h) Time (h) Figure : simulated realizations for population growth curves for Salmonella enterica starting from different initial cell numbers: (a) one single bacteria, (b) two single bacteria, (c) individuals, (d) individuals.

35 (a) (b) Ln (Cell Number) Ln (Cell Number) Time (h) (c) Time (h) (d) Ln (Cell Number) Ln (Cell Number) Time (h) Time (h) Figure : simulated realizations for population growth curves for Salmonella enterica using an ordinary differential equation with random parameters for different initial cell numbers. Parameter distribution is taken from []. (a) one single bacteria, (b) two single bacteria, (c) individuals, (d) individuals.

36 (a) (b)... h h h h h h h h h h Probability density.. Probability density... log Cell number..... log Cell number (c) (d).... Cumulative probability Time (h) Cumulative probability Time (h) Figure : Probability and cumulative probability distributions functions for Salmonella enterica populations computed with the proposed model (parameters are in Table ) starting from one individual (a, c) and from individuals (b, d). (c)-(d) plots depict the cumulative probability for the population to attain individuals as a function of time, for and bacteria respectively.