Microbial Growth and Biofilms

Exponential growth calculation

During the exponential (logarithmic) phase, each cell divides at a fixed interval, so the population doubles once per generation. The generation time g is the time needed for one doubling. If n generations occur in time t, then n = t/g and N = N_0 \times 2^{n}, where N_0 is the initial number and N is the final number. Solving for the number of generations gives n = \log_{2}(N/N_0) = (\log N - \log N_0)/\log 2, and the mean generation time is g = t/n. In logarithmic form, \log_{10} N = \log_{10} N_0 + n\log_{10} 2, with \log_{10} 2 \approx 0.301. This form is useful when a question asks for the logarithm of the cell number rather than the cell number itself. Working backward, the initial population is N_0 = N/2^{n}, so a known final count and a known number of generations give the starting inoculum. When the fold increase is not a power of two, n is not an integer and must be kept as a fractional value to calculate g, rounding only at the end. The mean generation time over a period is simply the elapsed time divided by the number of generations, g = t/n, which is why a fold increase of 2^{5} over 480 minutes corresponds to g = 96 min. Lytic phage add a complication: uninfected cells keep dividing until infected cells lyse, so the population must be tracked generation by generation rather than with a single exponential formula. Students should not confuse generation time (time per doubling) with the specific growth rate \mu = \ln 2/g, which is a rate constant per unit time.

Microbial growth yield

Growth yield describes how much biomass is produced per unit of energy source consumed. The molar growth yield Y_{\text{ATP}} is expressed as grams of dry cell mass produced per mole of ATP generated, and it is calculated by dividing the grams of cell mass formed per mole of substrate by the moles of ATP produced per mole of substrate. Under fermentative conditions, the pathway determines the ATP yield per glucose: the Embden-Meyerhof pathway in homofermentative bacteria such as Streptococcus faecalis gives 2 mol ATP per mole of glucose, whereas the Entner-Doudoroff pathway in Zymomonas mobilis gives 1 mol ATP per mole of glucose. Therefore, the same molar biomass yield per glucose corresponds to a higher Y_{\text{ATP}} in the organism with the lower ATP yield. Thus, the ATP-based yield is calculated as Y_{\text{ATP}} = Y_{\text{glucose}}/\text{ATP per glucose}, which is a standard relation in microbial energetics. A related counting problem appears in the same group of items: a circular plasmid cut at n sites by a restriction enzyme yields n linear fragments, because each cut opens the circle once. For one mole of plasmid, the moles of fragments are therefore n times the moles of plasmid. This is a restriction-mapping fact rather than a growth-yield calculation, but it is tested alongside the yield items.