Microbial cell modelling is a fascinating subject and one that deserves special attention. Over the years a number of different models have been described to explain cell growth, substrate use and product formation. It can also be about designing a fermentation process or knowing the lethality of a thermal preservation process so that you can design a microbiologically safe method for killing bacteria is one of the applications for microbial cell modelling.
Before anything is said, a model should be as simple as possible to apply and interpret. So many become extremely complicated but then bacterial populations are not simple and straightforward.
The Role of Mathematical Models In Fermentation
Mathematical models form the basis of microbial cell modelling. A mathematic model is a collection of mathematical relationships used to describe a process. Using mathematical models invariably produces a simplification but with good reason because studying microbial populations is complicated. Modelling however does offer us insights into how microbes behave in fermentations for example. We can use different models to describe experimental data and to explore various parameters that might prove useful in examination of microbial performance. Models also prove useful in optimizing fermentations and in the control of processes.
Types Of Mathematical Models Used In Microbial Cell Growth
From the outset its worth just considering two categories of models because they underpin the type of modelling that is conducted today. These models are either segregated or non-segregated.
The Non-Segregated Models
These are models which consider the entire population of cells to be identical. Such a system is easy to examine mathematically because they are described by one cell concentration. The early history of cell modelling exploits the non-segregated models.
The Segregated Models
Segregated models are those which consider there are differences between cells within a population. These ‘different’ cells are categorised into individual ‘compartments’. Such models rely on an ability to distinguish one or more characteristics that make such cells in a population differ. They are mathematically more complex to understand but have helped in the exploration of cell growth modelling.
Having established two basic models, its possible to look at other types:-
Unstructured Models – the cell is viewed as a single entity that interacts with its environment
Structured Models – this is a type which considers individual reactions occurring in a cell or groups of reactions in a cell.
The Stochastic models are statistical models. They consider characteristics of cell behaviour which have a probability distribution. There is a probability associated with each level of characteristic.
The obverse of the stochastic models are deterministic models. These have outputs completely determined by the model inputs, without consideration of any random variation.
Early Microbial Models
A number of modelling methods have been available since the 1920s which are built on microbial kinetics and their dependency on kinetic data. Bigelow came up with one of the earliest predictive microbial models (Bigelow, 1921). Here inactivation is treated like a chemical 1st-order reaction but the assumption is made that all the spores in the population have identical sensitivity to heat.
Leudeking-Piret Model
The Leudeking-Piret model is one model often cited and still used in the measurement of product formation.
More advanced linear models include Baranyi’s and Robert’s models (Baranyi & Roberts, 1994), and Gompertz’s equation (Gougouli and Koutsoumanis, 2013). The more advanced models are based on sigmoidal functions and are used purely to fit microbial growth data and predict maximum growth rates along with other kinetic parameters. Fujikawa et al. (2004) developed a new logistic model for bacterial growth at dynamic temperatures using a numerical solution with the fourth-order Runge–Kutta method and demonstrated that the newly developed model could successfully predict Escherichia coli and Salmonellae growth curves for various patterns of the temperature history.
In more recent times, nonlinear kinetic models have come to the fore because they are more accurate representations of microbial growth (Heldman and Newsome, 2003).
It is now possible to put together data to help in the predictions using free software written in various models (Geeraerd et al., 2005). The non-linear curve often show downward concavity as in the presence of a shoulder or an upward concavity which is the presence of a tail.
The presence of a shoulder in death kinetics is due to four possible reasons:-
- Microorganisms become organised into clumps. The shoulder is a time before all but one organism in such a clump has been killed.
- Cells tend to counter the effects of a lethal treatment. The shoulder is the period when cells are able to resynthesize a critical component and death only occurs when the rate of destruction exceeds the rate of synthesis.
- A shoulder could describe the protective effect of the medium or some nutritional components such as fats and proteins which prevent cell death.
- A shoulder can also describe a type of cumulative injury that occurs before cell inactivation.
Non-linear survival models are either mechanistic or pseudo-mechanistic. To avoid having to gather lots of data means applying probabilistic methods to cut out so much effort. Among these kinds of models, the logistic regression type is a useful tool for defining the combination of factors that helps to prevent the growth of microorganisms within food systems (Sosa-Morales et al., 2009). Ratkowsky and Ross (1985) developed a logistic regression model that was proposed to model the boundary between growth and no growth for bacteria controlled by temperature, pH and preservatives. This particular model integrated probability with the kinetic aspects of predictive microbiology.
The Weibull model is one of the least known but most applicable statistical probability method for life cycle analysis. We see it used for describing the thermal inactivation of microbial cells (van Boekel, 2002). It is best described as a statistical modelling tool for analysing the distribution of a parameter such as inactivation time.
It is a suitable alternative to the Bigelow model of first-order kinetics and the classical models. There is a fine review of the various models used in predictive microbiology (Bevilacqua et al., 2015).
Part of the power of the Weibull model is that it can take into account biological variation when thermal death of bacteria or indeed any microbe is considered.
The model relies on two parameters, a scale diameter α which denotes time and a dimensionless shape parameter β. The model conveniently accounts for the way microorganisms often die following a period of exponential growth. There is an oft observed nonlinearity in the semi-logarithmic survivor curves that are constructed. Likewise, the classical first-order approach is a special case of this Weibull model.
If you look at the various parameters, the shape parameter β accounts for upward concavity of a survival curve (β<1), a linear survival curve (β=1), and downward concavity (β>1). Although the Weibull model is of an empirical nature, a link can be made with physiological effects. β<1 indicates that the remaining cells have the ability to adapt to the applied stress, whereas β>1 indicates that the remaining cells become increasingly damaged.
References
Baranyi, J., and Roberts, T. A. (1994). A dynamic approach to predicting bacterial growth in food. Int. J. Food Microbiol. 23, pp. 277–294. doi: 10.1016/0168-1605(94)90157-0
Baranyi, J., and Roberts, T.A. (2000) Principles and applications of predictive modeling of the effects of preservative factors on microorganisms. In: Lund, B.M., Baird-Parker, T.C., Gould, G.W., editors. The microbiological safety and quality of food, Vol 1. Gaithersburg Md. : Aspen Publishers. pp 342–58.
Bevilacqua, A., Speranza, B., Sinigaglia, M., Corbo, M.R. (2015) A Focus on the Death Kinetics in predictive Microbiology: benefits and Limits of the Most Important Models and Some Tools Dealing with their Application in Foods. Foods 4 pp. 565-580
Bigelow, W.D. (1921) The logarithmic nature of thermal death time curves. J Infect Dis. 29 pp. 528–36
Buchanan, R.L. (1992) Predictive microbiology: mathematical modeling of growth in foods. In: Finley, J.W., Robinson, S.F., Armstrong, D.J., editors. American Chemical Society Symposium Series. Food Safety Assessment. 484: pp. 250–60
Buchanan, R.L., Golden, M.H., Whiting, R.C. (1993) Differentiation of the effects of pH and lactic or acetic acid concentration on the kinetics of Listeria monocytogenes inactivation. J. Food Prot. 56(6) pp. 474–8
Fujikawa, H., Kai, A., and Morozumi, S. (2004). A new logistic model for Escherichia coli growth at constant and dynamic temperatures. Food Microbiol. 21, pp. 501–509. doi: 10.1016/j.fm.2004.01.007
Geeraerd, A.H., Valdramidis, V.P., Van Impe, J.F. (2005) GInaFiT, a freeware tool to assess non-log-linear microbial survivor curves. Int. J. Food Microbiol. 102 pp. 95–105. doi:10.1016/j.ijfoodmicro.2004.11.038.
Gougouli, M., Koutsoumanis, K.P. (2013) Primary models for fungal growth. P. Dantigny, E.Z. Panagou (Eds.), Predictive Micology, Nova Science Publishers Inc., New York, USA.
Heldman, D.R., Newsome, R.L. (2003) Kinetic models for microbial survival during processing. Food Technol. 57(8) pp. 40–7
Mafart, P., Couvert, O., Gaillard, S., & Leguérinel, I. (2002). On calculating sterility in thermal preservation methods: application of the Weibull frequency distribution model. International Journal of Food Microbiology, 72(1), pp. 107-113
McMeekin, T.A., Olley, J.N., Ross, T., Ratkowsky, D.A. (1993) Predictive microbiology: theory and application. Sharpe, A.N., editor. Somerset : Research Studies Press LTD. 582 p.
Peleg, M. (2006). Advanced quantitative microbiology for foods and biosystems: models for predicting growth and inactivation. CRC Press.
Ratkowsky, D.A., Ross, T. (1995) Modeling the bacterial growth/no-growth interface. Lett. Appl. Microbiol. 20: pp. 29–33.
Sosa-Morales, M.E., García, H.S., López-Malo, A. (2009) Colletotrichum gloeosporioides growth-no growth interface after selected microwave treatments. J. Food Prot., 72, pp. 1427-1433
van Boekel, M.A.J.S. (2002) On the use of the Weibull model to describe thermal inactivation of microbial vegetative cells. Int. J. Food Microb. 74 (1-2) pp. 139-159 https://doi.org/10.1016/S0168-1605(01)00742-5
I would like to see some mathematical modelling that brings in quorum sensing as I think this factor affects the growth of cells especially in biofilms. In the open environment, cell growth modelling is nowadays straightforward but what about the complexities of of the biofilm ? I read recently that cells use potassium ion channels to generate electrical signals across the membrane and I think that is an interesting subject to look at.