The time required for a complete fission cycle—from parent cell to two new daughter cells—is called the generation, or doubling, time. The term generation has a meaning similar to that for humans. It is the period between an individual’s birth and the time of producing off spring. In bacteria, each new fission cycle or generation increases the population by a factor of 2, or doubles it. Thus, the initial parent stage consists of 1 cell, the first generation consists of 2 cells, the second 4, the third 8, then 16, 32, 64, and so on. As long as the environment remains favorable, this doubling effect can continue at a constant rate. With the passing of each generation, the population will double, over and over again.
The length of the generation time is a measure of the growth rate of an organism. Compared with the growth rates of most other living things, bacteria are notoriously rapid. The average generation time is 30 to 60 minutes under optimum conditions. The shortest generation times average 5 to 10 minutes, and longest generation times require days. For example, Mycobacterium leprae, the cause of Hansen’s disease, has a generation time of 10 to 30 days—as long as in some animals. Most pathogens have relatively short doubling times. Cultures of Salmonella enteritidis and Staphylococcus aureus, bacteria that cause food-borne illness, double in 20 to 30 minutes, which ex plains why leaving food at room temperature even for a short period has caused many a person to be suddenly stricken with an attack of food-borne disease. In a few hours, a population of these bacteria can easily grow from a small number of cells to several million.
Figure 1 illustrates growth: (1) The cell population size can be represented by the number 2 with an exponent (21, 22, 23, 24); (2) the exponent increases by one in each generation; and (3) the number of the exponent also gives the number of the generation. This growth pattern is termed exponential. Because these populations often contain very large numbers of cells, it is useful to express them by means of exponents or logarithms (see “Exponents,” Online Appendix 3). The data from a growing bacterial population are graphed by plot ting the number of cells as a function of time. The cell number can be represented logarithmically or arithmetically. Plotting the logarithm number over time provides a straight line indicative of exponential growth. Plotting the data arithmetically gives a constantly curved slope. In general, logarithmic graphs are preferred because an accurate cell number is easier to read, especially during early growth phases.
Fig1. The mathematics of population growth. (a) Starting with a 219 single cell, if each product of reproduction goes on to divide by binary fission, the population doubles with each new cell division or generation. This process can be represented by logarithms (2 raised to an exponent) or by simple numbers. (b) Plotting the logarithm of the cells produces a straight line indicative of exponential growth, whereas plotting the cell numbers arithmetically gives a curved slope. *Note that the left scale is logarithmic and the right scale is arithmetic.
Predicting the number of cells that will arise during a long growth period (yielding millions of cells) is based on a relatively simple concept. One could use the method of addition 2 + 2 = 4; 4 + 4 = 8; 8 + 8 = 16; 16 + 16 = 32, and so on, or a method of multiplication (for example, 25 = 2 × 2 × 2 × 2 × 2), but it is easy to see that for 20 or 30 generations, this calculation could be very tedious. An easier way to calculate the size of a population over time is to use an equation such as:
Nf = (Ni)2n
In this equation, Nf is the total number of cells in the population at some point in the growth phase, Ni is the starting number, the exponent n denotes the generation number, and 2n represents the number of cells in that generation. If we know any two of the values, the other values can be calculated. Let us use the example of Staphylococcus aureus to calculate how many cells (Nf) will be present in an egg salad sandwich after it sits in a warm car for 4 hours. We will assume that Ni is 10 (number of cells deposited in the sandwich while it was being prepared). To derive n, we need to divide 4 hours (240 minutes) by the generation time (we will use 20 minutes). This calculation comes out to 12, so 2n is equal to 212. Using a calculator we find that 212 is 4,096.
Final number (Nf) = 10 × 4,096
= 40,960 bacterial cells in the sandwich
This same equation, with modifications, is used to determine the generation time, a more complex calculation that requires knowing the number of cells at the beginning and end of a growth period. Such data are obtained through actual testing by a method discussed in the following section.
