Microbial Population Growth, Mortality, and Transitions between Them

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This Demonstration simulates different patterns in the evolution of microbial populations. It can describe unlimited growth, growth reaching equilibrium, growth ending in mortality, pure mortality, and mortality ending in new growth. It is based on a deterministic version of a probabilistic model originally developed for microbial cell growth, where a member of the population, after any time increment, can be in one of three states: alive but undivided, divided, or dead, with probability rates that vary with time.

Contributed by: Mark D. Normand and Micha Peleg (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Snapshot 1: logistic ("sigmoid") growth

Snapshot 2: exponential growth

Snapshot 3: pure mortality

Snapshot 4: mortality followed by renewed growth

This Demonstration describes the evolution of a microbial population. A viable cell at time can be in one of three states at time : divided with a probability of , dead with a probability of , or alive but undivided with a probability of , where and are the division and mortality probability rate functions, respectively. These two functions are assumed to be logistic, that is, governed by three parameters each: or , or , and or , all of which have their values entered with sliders. They represent these probability rate functions' asymptotic level, rate of ascent, and the time location of the inflection point, respectively, that is, and . The Demonstration then calculates and plots the number of cells alive at time , , descendent of the specified initial number, . The calculation is based on a version of the model for the limiting case where tends to zero, making it continuous and fully deterministic.

You can modify the probability rate functions to create curves depicting growth, mortality, and transitions between them. You can also enter the initial number of cells, , to calculate the number of cells at a chosen time , , and you can set the maximum value of the time axis in the plots. The top plot in the panel shows the chosen and functions and the bottom plot shows the corresponding growth and/or mortality curve, .

References:

J. Horowitz, M. D. Normand, M. G. Corradini, and M. Peleg, "Probabilistic Model of Microbial Cell Growth, Division and Mortality," Applied and Environmental Microbiology, 76(1), 2010 pp. 230–242.

M. G. Corradini, M. D. Normand, and M. Peleg, "A Stochastic and Deterministic Model of Microbial Heat Inactivation," Journal of Food Science, 75, 2010 pp. R59–R70.

Three Mathematica 7 notebooks and three PDF files containing probabilistic growth and mortality models.



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