Appendix B Some Mathematical Considerations in Microbiology

Appendix B Some Mathematical Considerations in Microbiology Scientific Notation Generation Time Scientific notation (also called exponential notation) is a mathematical device developed to express numbers, particularly very small and very large numbers, in a manner that is convenient and readable. For example, 1.234 * 10 -8 is much less cumbersome than 0.00000001234. Similarly, 5.67 * 1017 is more easily read than 567,000,000,000,000,000. A number expressed in scientific notation is composed of a coefficient, which is always a number with only one whole number digit to the left of the decimal place, times 10 to some expo­ nential power. In the examples above, the coefficients are 1.234 and 5.67, respectively. The exponents are -8 and 17, respectively. To write a number in scientific notation, move the decimal point either right or left so that there is only one nonzero number to the left of it. For example, When bacteria and other microbes reproduce by binary fission, the number in the population doubles with each division cycle (generation). Such growth is called logarithmic (or exponential) growth. The number in a population can be calculated as 2 (because each cell produces two offspring) multiplied by itself as many times as there are generations. In other words, the number of cells in a population arising from a single individual is expressed mathematically as 454 becomes 4.54 In this case, we moved the decimal point (which is understood to be at the end of a whole number) two places to the left. Since we moved two places left, the exponent will be positive 2, and the scientific notation is 4.54 * 102 Similarly, 4,500,264.75 becomes 4.50026475 * 106 since we moved the decimal six places to the left. As you would expect, when we have to move the decimal point to the right, as in 0.00562 (which becomes 5.62), then the exponent will be a negative number, in this case -3; therefore, 0.00562 = 5.62 * 10 -3. 2number of generations A cell dividing for five generations would produce a population of 32 (25 = 2 * 2 * 2 * 2 * 2 = 32). When the population begins with more than one individual, then the final population equals original number of cells * 2number of generations For example, seven cells reproducing for five generations would produce a population of 224 cells (7 * 25 = 7 * 2 * 2 * 2 * 2 * 2 = 224). In most cases, microbiologists are not concerned with the number of generations required to produce a given population, but they do want to know the generation time, that is, the time required for a bacterial cell to grow and divide. This is also the time required for a population of cells to double in number. To calculate generation time, scientists must first calculate how many generations have been produced, knowing the beginning and ending population sizes. When population sizes are converted to logarithms, the number of generations is calculated as Logarithms A logarithm is the number of times a number, called the base number, must be multiplied by itself to get a certain number. For example, the number 10 must be multiplied by itself three times to get 1000 (10 * 10 * 10); therefore, the logarithm of 1000 in base 10 is 3. Similarly, the logarithm of 100,000 is 5. Though any number can be used as the base number, in microbiology, base 10 is commonly used. Microbiologists use base 10 logarithms to express pH values and the size of bacterial populations in culture. When a number is expressed in scientific notation, the logarithm is the exponent when the coefficient is exactly 1. When the coefficient is a number other than 1, the logarithm must be calculated using the logarithm function on a calculator. number of = generations logarithm (log) number of cells log number of at the end of reproduction cells initially log 2 Log 2 is used in the formula because each cell produces two offspring each time it divides. Log 2 = 0.301. The number of generations is used to calculate generation time: generation time 60 minutes * number of hours = (min>generation) number of generations A-12 Z03_BAUM7206_05_SE_APPB.indd 12 12/1/16 3:44 PM Appendix B     A-13 For example, if 100 bacteria multiply to produce a population of 3.28 * 106 cells in 7 hours, then the generation time for this bacterium is calculated as follows: number of = generations = logarithm (log) number of cells - log number of at the end of reproduction cells initially log 2 generation time (min>generation) = = 60 minutes * hours number of generations 60 min * 7 h 15 = 28 minutes/generation log (3.28 * 106) - log (100) log 2 = 15 generations Z03_BAUM7206_05_SE_APPB.indd 13 12/1/16 3:44 PM