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Generation Time (hrs)

Figure 11.19 Representative microbial generation times under optimal conditions.

their environments, so that bacteria living in systems where substrates become available slowly will tend to be slow growing. Also, bacteria utilizing substrates that release little energy may have minimum doubling times of several hours or even days. Such substrates can include hard-to-degrade organic substances as well as some inorganic energy sources. For example, the chemolithotroph Nitrobacter winogradskyi derives only ~17 kcal of energy per mole of NO2 -N that it oxidizes to nitrate, as compared to the ^600 kcal/ mol potentially available to E. coli from the oxidation of glucose. This is reflected in the ^18-hour minimum generation time of N. winogradskyi (Figure 11.19).

Modeling Exponential Growth If one cell begins growing exponentially, and cell number (N) is plotted vs. time, initially N increases in discrete steps: 1-2-4-8-16-32-64- • • • (Figure 11.20). After a period of time, the cell divisions would no longer be synchronized precisely, so that the intermediate numbers (such as 50) would be present for brief periods. Still, only whole numbers of cells can occur (not 50.2), making the plot a series of small steps. However, once high enough numbers are reached, the rate of increase would appear to be smooth because the steps are so small.

On the other hand, biomass increases as a continuous function (except on the molecular level). Thus, biomass is theoretically more amenable than cell number to mathematical expression and is the focus of our discussion. In practice, however, it is also usually perfectly acceptable to use the same sorts of expressions with N, except when counts are very low.

Rather than biomass itself, more commonly it is biomass concentration (X) that is considered. This typically might be on a mass per volume basis for liquids (e.g., mg/L in water or media) and gases (mg/m3 of air), or on a mass per mass basis for solids (mg/kg of soil or sludge).

Time

Figure 11.20 Theoretical stair-stepped batch bacterial growth curve.

Time

Figure 11.20 Theoretical stair-stepped batch bacterial growth curve.

If the logarithm of X is plotted vs. time (t) for an exponentially growing culture, a straight line results (Figure 11.21). Using natural logarithms (loge, or ln), the equation for a straight line then gives ln Xt = mt + ln X0

where Xt is the value of X at time t, m is the slope of the line, and X0 is the starting value of X (at time 0). Differentiating equation (11.1) (assuming that m is constant):

Figure 11.21 Similar plot of exponential growth.

From equation (11.2) it can be seen that m is equal to the rate at which the biomass concentration is growing, divided by the biomass concentration at that time. Therefore, m is referred to as the specific growth rate and has units of time-1 (inverse time, or "per time''). Growth rates are typically expressed per minute or per hour for bacteria in the laboratory, but more often per day in environmental systems and for other microorganisms, since growth rates are lower.

Returning to equation (11.1) and rearranging yileds ln Xt — ln X0 = mt

Exponentiating each side gives eln(X, /Xo) = em t

Xt = Xoemt

There are thus three forms of the exponential growth equation: the differential form [equation (11.3)]; the logarithmic form [equation (11.1), which gives a straight line on a semilog plot]; and the exponential form [equation (11.4), useful for calculating the biomass concentration at a time in the near future].

Equation (11.3) is the most fundamental form of the three. This is because, unlike the other two, it is not dependent on an assumption that m is constant. It simply states that the rate of biomass increase is proportional to the specific growth rate and to the current amount of biomass.

Assuming again that m is constant, suppose that we look at Xt after one doubling time, when it would have doubled from its starting value to 2X0:

Thus, the relationship between specific growth rate and doubling time is ln 2 , m = — (11.5)

At the minimum doubling time, the maximum specific growth rate will be achieved. The symbol (1 (read "mu-hat") or mmax is commonly used for this term. Using E. coli's minimum td value of 20 min, for example:

Example 11.6 What is the meaning of the specific growth rate, m?

Answer It is the rate of change of biomass concentration in a batch culture (one with no inflow or outflow) at any instant, divided by the concentration at that point in time [equation (11.2)]. Suppose that m = 0.05h" in a culture that at a certain time has a concentration of 100 mg/L. From equation (11.3):

In other words, at that instant, the concentration would be increasing at the rate of 5 mg/L per hour. At a later time, the concentration might reach 200 mg/L, and then dX = mX = ^ x 200mg = 10-mg j* 1 u t i dt h L L • h

So, as the concentration increased, the rate of increase also increased, even though the specific growth rate itself did not change.

The doubling time is easily computed from m by rearranging equation (11.5):

ln2 0.693

Using equation (11.4), the biomass concentration at a future time for an exponentially growing culture can be calculated based on the present concentration and the specific growth rate. As an example, starting with a single cell with a mass of 10"12 g in 1 L of medium, and using the maximum specific growth rate of E. coli, after 10 hours the biomass concentration would be

Xt = X0em = 10"12 g/L x e(2 08/h«10h) « 1 mg/L

However, this equation predicts that after 2 days, the biomass of the same culture would be ~ 2 x 1031 g, or 2 x 1025 metric tons. This is about 4000 times the mass of Earth; obviously, this exponential growth model has its limits! In the next section we show how an improved model can be developed.

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