Probabilistic Model of Microbial Cell Growth, Division, and Mortality

ABSTRACT After a short time interval of length δt during microbial growth, an individual cell can be found to be divided with probability Pd(t)δt, dead with probability Pm(t)δt, or alive but undivided with the probability 1 − [Pd(t) + Pm(t)]δt, where t is time, Pd(t) expresses the probability of division for an individual cell per unit of time, and Pm(t) expresses the probability of mortality per unit of time. These probabilities may change with the state of the population and the habitat's properties and are therefore functions of time. This scenario translates into a model that is presented in stochastic and deterministic versions. The first, a stochastic process model, monitors the fates of individual cells and determines cell numbers. It is particularly suitable for small populations such as those that may exist in the case of casual contamination of a food by a pathogen. The second, which can be regarded as a large-population limit of the stochastic model, is a continuous mathematical expression that describes the population's size as a function of time. It is suitable for large microbial populations such as those present in unprocessed foods. Exponential or logistic growth with or without lag, inactivation with or without a “shoulder,” and transitions between growth and inactivation are all manifestations of the underlying probability structure of the model. With temperature-dependent parameters, the model can be used to simulate nonisothermal growth and inactivation patterns. The same concept applies to other factors that promote or inhibit microorganisms, such as pH and the presence of antimicrobials, etc. With Pd(t) and Pm(t) in the form of logistic functions, the model can simulate all commonly observed growth/mortality patterns. Estimates of the changing probability parameters can be obtained with both the stochastic and deterministic versions of the model, as demonstrated with simulated data.

[1]  Mark D. Normand,et al.  The logistic (Verhulst) model for sigmoid microbial growth curves revisited , 2007 .

[2]  A. Cazemier,et al.  Effect of sporulation and recovery medium on the heat resistance and amount of injury of spores from spoilage bacilli , 2001, Journal of applied microbiology.

[3]  Christopher J Doona,et al.  A quasi-chemical model for the growth and death of microorganisms in foods by non-thermal and high-pressure processing. , 2005, International journal of food microbiology.

[4]  M. Peleg Microbial survival curves: Interpretation, mathematical modeling and utilization , 2003 .

[5]  F. Feeherry,et al.  Thermal inactivation and injury of Bacillus stearothermophilus spores , 1987, Applied and environmental microbiology.

[6]  S. Isobe,et al.  Prediction of pathogen growth on iceberg lettuce under real temperature history during distribution from farm to table. , 2005, International journal of food microbiology.

[7]  S. Leibler,et al.  Bacterial Persistence as a Phenotypic Switch , 2004, Science.

[8]  A. Kai,et al.  A new logistic model for Escherichia coli growth at constant and dynamic temperatures , 2004 .

[9]  M. Peleg,et al.  A Weibullian model for microbial injury and mortality. , 2007, International journal of food microbiology.

[10]  C. Pin,et al.  Kinetics of Single Cells: Observation and Modeling of a Stochastic Process , 2006, Applied and Environmental Microbiology.

[11]  Micha Peleg,et al.  On modeling and simulating transitions between microbial growth and inactivation or vice versa. , 2006, International journal of food microbiology.

[12]  Tom Ross,et al.  Predictive Microbiology : Theory and Application , 1993 .

[13]  J. V. Van Impe,et al.  Towards a novel class of predictive microbial growth models. , 2005, International journal of food microbiology.

[14]  F. Busta Practical Implications of Injured Microorganisms in Food , 1976 .

[15]  Micha Peleg,et al.  A model of microbial growth and decay in a closed habitat based on combined Fermi's and the logistic equations , 1996 .

[16]  F. E. Feeherry,et al.  A Quasi-Chemical Kinetics Model for the Growth and Death of Staphylococcus aureus in Intermediate Moisture Bread , 2003 .

[17]  R. F. Mcfeeters,et al.  Energy-Based Dynamic Model for Variable Temperature Batch Fermentation by Lactococcus lactis , 2002, Applied and Environmental Microbiology.

[18]  James M. Jay,et al.  Modern food microbiology , 1970 .

[19]  I. Taub,et al.  The mathematical properties of the quasi-chemical model for microorganism growth-death kinetics in foods. , 2005, International journal of food microbiology.

[20]  S. Holdsworth Thermal processing of packaged foods , 1997 .

[21]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[22]  J. Baranyi Stochastic modelling of bacterial lag phase. , 2002, International journal of food microbiology.

[23]  M Peleg,et al.  Reinterpretation of microbial survival curves. , 1998, Critical reviews in food science and nutrition.

[24]  M. Peleg,et al.  The non-linear kinetics of microbial inactivation and growth in foods. , 2007 .

[25]  J Baranyi,et al.  Validating and comparing predictive models. , 1999, International journal of food microbiology.

[26]  Baranyi Comparison of Stochastic and Deterministic Concepts of Bacterial Lag. , 1998, Journal of theoretical biology.

[27]  R. Zwanzig,et al.  Generalized Verhulst laws for population growth. , 1973, Proceedings of the National Academy of Sciences of the United States of America.

[28]  Marcel H. Zwietering,et al.  Modelling microorganisms in food , 2007 .

[29]  J Baranyi,et al.  A dynamic approach to predicting bacterial growth in food. , 1994, International journal of food microbiology.

[30]  Micha Peleg,et al.  Advanced Quantitative Microbiology for Foods and Biosystems: Models for Predicting Growth and Inactivation , 2006 .

[31]  Stanley Brul,et al.  Microbial systems biology: new frontiers open to predictive microbiology. , 2008, International journal of food microbiology.