Mathematics of Microbial Growth and Chemostat Operation Exponential Growth Although many analyses of microbial growth can be done graphically or algebraically, as discussed in Chapter 9, for some purposes it is convenient to use a differential equation that expresses the quantitative relationships of growth : dX / dt =μX (1) where X may be cell number or some specific cellular component such as protein and μ is the instantaneous growth-rate constant. If this equation is integrated, we obtain a form which reflects the activities of a typical batch culture population : InX = InXo + μ(t) (2)

where In refers to the natural logarithm (logarithm base e) , X0 is cell number at time 0. X is cell number at time t, and is elapsed time during which growth is measured. This equation fits experi- mental data from the exponential phase of bacteria cultures very well, such as those in Figure 9.2. Taking the antilogarithm of each side gives X = Xo е u t (3) This equation is useful because it allows prediction of population density at a future time by knowing the present value and μ, the growth-rate constant. As discussed in Section 9.2, an important constant parameter for an exponentially growing populations the doubling(or generation) time. Doubling of the population has occurred when X / X0 = 2. Rearranging and substituting this value into equation 3 gives 2 = e u (t gen ) (4)

Taking the natural logarithm of each side and rearranging gives μ = In2 / t gen = 0.693 / t gen (5) The generation time t gen may be used to define another growth parameter, k, as follows : k = 1 / t gen (6) where k is the growth-rate constant for a batch culture. Combining Equations 5 and 6 shows that the two growth-rate constants, μ and k, are related : μ = 0.693 k (7) It is important to understand that μ and k are both reflections of the same growth process of an exponentially increasing population. The difference between them may be seen in their derivation above : μ is an average value for the population over a finite period of time.

This distinction is more than a mathmatical point. As was emphasized in Chapter 9, microbial growth studies must deal with population phenomena, not the activities of individual cells. The constant k reflects this averaging assumption. However, the k reflects this averaging assumption. However, the constant μ, being instantaneous, is a closer approximaiton of the rate at which individual activities are occurring. Further, the instantaneous constant μ allows us to consider bacteria growth dynamics in a theoretical framework separate from the traditional batch culture. Mathematical Relationships of Chemostats An especially important application of the instantaneous growth- rate constant μ is, of course, the chemostat, a culture device (Section 9.6) in which population size and growth rate may be maintained at constant values of the experimenter’s choosing over a wide range of values, As we saw in Section 9.5, the rate of bacterial growth is a function of nutrient concentration. This function represents a saturation process which may be described by the equation

μ = μ max [ S / ( Ks + S ) ] (8) where μ max is the growth rate at nutrient saturation, S is the concentration of nutrient, and Ks is a saturation constant which is numerically equal to the nutrient concentration at which μ = ½ u max. Equation 8 is formally equivalent to the Michaelis- Menten equation used in enzyme kinetic analyses. Estimates of the unknown parameters μ max and Ks can be made by plotting 1 / μ on the y axis versus 1 / s on the x axis. A straight line will be obtained which intercepts the y axis, and the value at the intercept is 1 / μ. The slope of the line is Ks / μ max, and since μ max is known, Ks can then be calculated. Values for Ks as a rule are very low. A few examples are 2.1*10-5M for glucose (Escherichia coli), 1.34*10-6M for oxygen (Candida utilis), 3.5*10-8M for phosphate (Aspergillum spp.), and 4.7*10-13M for thiamine (Cryp- tococcus albidus). The key to the operation of the chemostat is that, in a steady-state population, the nutrient concentration is very low, usually very near 0. Therefore, each drop of fresh medium entering the chemostat supplies nutrient which is consumed almost instantaneously by the population and the

experimenter has direct control over the growth rate : if nutrient is added more rapidly, μ decreases ; if it is added more slowly, μ decreases. In a chemostat the rate at which medium enters is equal to the rate at which spent medium and cell exit through the overflow. This rate is called the dilution rate, D, and is defined : D = F / V ( 9 ) where F is the flow rate ( in units of volume per time, usually ml/ hour) and V is chemostat volume (in ml). Therefore, D has units of time-1 (usually hour-1) as does μ. In fact, at steady state, μ = D. That μ must equal D may be seen by consideration of Equation 8 and 9 and the discussion above. When S is low, μ is a direct function of S. But since nutrient is instantly consumed upon addition, the value of S at any moment is controlled by the rate of medium addition, that is, the dilution rate D. A further crucial consequence of the relation μ = D is that a chemostat is a self- regulating system within broad limits of D. Consider a steady-state chemostat functioning with μ = D.

If μ were to increase momentarily due to some variation in the culture, S would necessarily decline, as the increased growth caused increased nutrient consumption. The lower value of S would in turn cause u to decrease back to its stable level. Conversely, if μ were to decrease momentarily, then S would increase as “extra” nutrient went unconsumed. This rise in S would lead to higher values of μ until steady state is again reached. This biological feedback system functions well in practice, as it does in theory. Chemostat populations may be maintained at constant growth rates (μ) for long periods of time : months or even years. There is another important aspect to chemostat studies and that is population density. The relation between cell growth and nutrient consumption may be defined : Y = dX / dS (10)

where Y is the yield constant for the particular organism on the particular medium and dX / dS is the amount of cell increase per unit or substrate consumed. The yield constant is in units of Y = weight of bacteria formed / weight of substrate consumed and is a measure of the efficiency with which cells convert nutrient to more cell material. The composition of the medium in the reservoir is control to the successful steady-state operation if the chemostat. All components in this medium are present in excess, except one, the growth-limiting substrate. The symbol for the concentration of that substrate in the reservoir is SR, whereas that for the same substrate in the culture vessel is S. Taking into account Equations 1, 8, and 10, the following considerations can be made. When a chemostat is inoculated with a small number of bacteria, and the dilution rate does not exceed a certain critical value(Dc), the number of organisms will start to increase. The increase is given by

dX / dt = growth – output or dX / dt = μX –DX (11) since initially μ >D, dX / st is positive. However, as the population density increases, the concentration of the growth-limiting substrate decreases, causing a decrease of μ (Equation 8). As discussed above, if D is kept constant, a steady state will be reached in which μ = D and dX / dt = 0 Steady states are possible only when the dilution rate does not exceed a critical value Dc. This value depends on the concentration of the growth-limiting substrate in the reservoir(SR) : Dc = μ max [ SR / ( Ks + SR) ] (12) Thus, when SR >> Ks (which is usually the case), steady states can Be obtained at growth rates close to μ max . The change in S is given By dS / dt = SRD - sD - μo X / Y ( 13 )

When a steady state is reached, dS / dt = 0. from Equation 11 and 13, the steady-state values of organism concentration ( X ) and growth-limiting substrate ( S ) can be calculated : X = Y ( SR - S ) (14) S = Ks [ D / (u max – D)] (15) These equations have the constants, SR, Y, Ks and μ max . Once their values are known, the steady-state values of and can be predicted for any dilution rate. As can be seen from Equation 15, the steady-state substrate concentration (S ) depends on the dilution rate (D) applied, and is independent of the substrate concentration in the reservoir(SR) Thus K determines S, and determines. A specific example of the use of the equations to predict chemostat behavior can be obtained by study of Figure 9.10 In that figure, the constants are u max = 10 hour – 1 ; y = 0.5 ; Ks = 0.2 g / liter ; SR = 10g / liter. With these constants, the curves in Figure 9.10 can be generated.

Experimental Uses of the Chemostat One of the major theoretical advantages of a chemostat is that this device allows the experimenter to control growth rate (u) and population density ( ) independently of each other. Over rather wide ranges (see above and Section 9.6) any desired value of u can be obtained by alteration of D. In practice this means changing the flow rate, since volume is usually fixed (Equation 9). Similarly, the population density may be determined by varying nutrient concentration in the reservoir (Equation 14). This independent control of these two crucial growth parameters not possible with conventional batch cultures. A practical advantage to the chemostat is that a population may be maintained in a desire growth condition ( u and ) for long periods of time. There fore, experiments can be planned in detail and performed whenever most convenient. Comparable studies with batch cultures are essentially impossible, since specific growth conditions are constantly changing.

Mathematics of the Stationary Phase The differential equation (Equation 1) expressing the exponential growth phase can be modified to include the trend toward the stationary phase, which occurs at high population density. A second term is added so the equation that expresses the maximum population attainable for that organism under the environmental conditions specified, here called Xm. The equation then assumes a form often called the logistics equation : dX / dt = uX – (u/Xm) X2 (16) The second term in this equation essentially expresses self-crowding Effects, such as nutrient depletion or ingibitor buildup. In integrated Form, this equation will graph as typical microbial growth curve, Showing exponential and stationary phases. The length of the exponential phase will be determined by the size of Xm. When X is small, the second term will have little (dX/dt) will approach zero.

The logistics equation is widely used by ecologists studying higher organisms to express the growth rate and carrying capacity of the habitat for animals and plants. The equation is equally applicable to microbial situations. Mathematics of Disinfection An equation can also be developed expressing the death rate of unicellular organisms upon treatment with a disinfectant or other lethal agent. If Xd represents the number of dead cells at time t, then X0-Xd will represent the survivors. The death rate is proportional to the survivors : dX / dt = k (X0 – Xd) (17) Upon integration, this equation gives a typical first-order relationship ; kt = In [X0 / (X0 – Xd) ] ( 18 )

and when the logarithm of X0-Xd (that is, the number of survivors) is plotted against time, a straight line will be obtained. This equation is useful when determining the length of time necessary to allow a disinfectant to completely sterilize something ( for example, the contact time for chlorine gas in a water purification plant)