Biochemical Engineering Fundamentals Part 1 - Bailey J., Ollis D. 1989

Kinetics of substrate utilization, metabolite and biomass production processes in cell cultures
Kinetics of thermal death of cells and spores

The activity of Cells, spores, and Viruses in air or liquid media can be reduced through their destruction (caused by heat, radiation, chemical agents, or mechanical forces), mechanical Separation (via filtration or centrifugation), or inhibition (resulting from supercooling, drying, dehydration, or exposure to chemical Reagents). Liquids are sterilized primarily by heating (in industrial settings) or chlorination (for domestic use), whereas air sterilization relies mainly on filtration. Mass-transfer effects play a major role in the latter two processes and will be discussed in the next chapter; this section focuses on The kinetics of Cell and spore inactivation.

Figure 7.33 illustrates the results of experimental studies on the effects of elevated temperatures on vegetative cells and spores. In general, cells are killed by heat much faster than spores. This is readily understood considering that endospore formation, as noted earlier, serves as a defense mechanism allowing certain cells to survive adverse conditions. Thermal Treatment also inactivates viruses and Bacteriophages; consequently, thermal sterilization used in the microbiological industry reduces both the viable microbial population and the concentration (titer) of viruses in the nutrient feed stream entering the Reactor.

Before examining the equations that describe The rate of population decline during sterilization, several Preliminary Remarks must be made. The death of any individual cell is presumably caused by the thermal Denaturation of one or more vital cellular Proteins, such as Enzymes. The kinetics of such cooperative transformations in complex molecules can vary over time in highly unpredictable ways. Furthermore, the rate of molecular processes ultimately leading to cell death depends on the medium composition, including solvent concentration. For instance, as shown in Table 7.9, the coagulation Temperature (denaturation followed by extensive irreversible cross-linking of denatured protein molecules) of egg albumin increases as the Water content in the system decreases.

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FIG. 7.33. Results of experimental Determination of the thermal death rate of E. coli and the thermal inactivation rate of vegetative spores of Bacillus subtilis and vegetative spores of B. stearothermophilus. [Reproduced from: a) Aiba S., Humphrey A., Millis N., Biochemical Engineering, 2nd ed., University of Tokyo Press, Tokyo, 1973; b, c) Burton H., Jayne-Williams D., Sterilized Milk, in Recent Advances in Food Science, vol. 2: Processing, Hawthorn J., Leitch J. M. (eds.), p. 107, Butterworths & Co., Publishers Ltd., London, 1962.] a — death rate of E. coli in buffer solution; b — inactivation rate of Bacillus subtilis spores; c — inactivation rate of Bacillus stearothermophilus spores.

These data, along with various other experimental findings, indicate that for relatively dehydrated organisms and structures (such as viruses or spores), inactivation is best viewed as a two-step process involving initial Hydration followed by potential denaturation. This mechanism aligns with the dependence of the death rate on relative humidity observed in bacteriophages [34]. The combined effect of multiple inactivating factors is not necessarily additive; for example, simultaneous dehydration and thermal treatment may be less effective than might be inferred from the efficacy of each factor applied individually.

Table 7.9. Dependence of Albumin Coagulation Temperature on Water Content^a

^a Data from: Frobisher M., Fundamentals of Microbiology, p. 259, W. B. Saunders Company, Phila., 1968.

As mentioned above, a population consists of a multitude of cells of varying ages. The Nature of The Cell wall and the relative importance of any metabolic pathway depend on the age of both the individual cell and the culture as a whole. Consequently, the resistance of cells to thermal or otherwise induced inactivation will depend on their history—a rather indefinite factor that cannot always be easily quantified. In particular, cells in an exponentially growing population possess relatively permeable walls, which facilitate efficient solute exchange between the intracellular volume and the environment. If these solutes affect protein stability, their presence is expected to influence cell growth in the exponential phase as well. Therefore, the most reliable data on population behavior can be obtained only through experiments conducted under conditions closely approximating those of the intended practical process.

We begin our analysis of cell death kinetics by noting that a linear relationship between the logarithm of the surviving fraction of cells and time (Fig. 7.33) indicates a first-order process for the decline of the viable cell population n:

It follows that at a constant k_d

Here n₀ is the concentration of spores or vegetative cells at t = 0. The slope of the semilogarithmic plot is equal to −k_d; the plot of lnk_d versus reciprocal temperature is also linear, meaning that the temperature dependence of k_d can be expressed by an Arrhenius-type equation:

For many spores and vegetative cells, the parameter E_d ranges from 50 to 100 kcal/mol. Another previously used parameter, D_r, defined as 2.303/k_d and termed the decimal reduction time, represents the time required to reduce the viable population by a factor of 10.

The kinetic equations given above are valid only provided that the number of spores or cells in the population is statistically large enough (Section 7.1). As the number of cells or spores (n) decreases, the likelihood of deviations from the values predicted by these equations increases because, for example, under a normal distribution, the standard deviation increases in proportion to 1/n. The deterministic model's preceding equation implies that the surviving fraction of the population (assuming no cell growth occurs under the given lethal conditions) is determined by

If The Fate of each Organism is independent of the others, if the organisms do not self-replicate, and if the lethal conditions are identical for every organism, stochastic Analysis of the sterilization process shows that the probability of finding N viable organisms in the population at any time t is given by [33]

Here N₀ is the concentration of viable organisms in the liquid immediately prior to sterilization.

In this equation, the parameter k_d has the same physical meaning as the rate constant in equation (7.124); within the stochastic model, k_d can be interpreted as the reciprocal of the organism's mean lifespan. As the number of organisms drops to a relatively small value, the assumption of a homogeneous population adequately described by a single parameter k_d becomes increasingly less valid. For instance, the data shown in Fig. 7.34 for Staphylococcus aureus in a neutral phosphate buffer exhibit a positive deviation from the behavior predicted by the stochastic distribution; consequently, a small fraction of the population possesses a higher resistance to thermal treatment.

Sometimes, a relatively large Number of viable organisms is permitted to remain after sterilization (for instance, premium-grade pasteurized milk must not contain more than 30,000 live Bacteria per mL). In other situations, such as the cultivation of pure cultures, sterilization requirements are significantly more stringent. In such cases, it is crucial to estimate the probability of population extinction—that is, the probability of inactivating all organisms. Substituting N = 0 into equation (7.128) yields

It follows that the probability of survival of at least one organism is

FIG. 7.34. Thermal inactivation of S. aureus. [Reprinted by permission from: Walker G. C., Harmon L. G., Thermal Resistance of Staphylococcus aureus in Milk, Whey, and Phosphate Buffer, Appl. Microbiol., 14, 584 (1966).]

Typically N₀ ≫ 1, therefore

where

The value (1—P₀) can be interpreted as the fraction of sterilization cycles that should not result in the complete destruction of all organisms.

Having examined cellular METABOLISM, we found that living cells often possess multiple alternative pathways for energy assimilation and Biosynthesis. The presence of such backup systems increases an organism's chances of survival under adverse conditions. Statistical analysis shows that, accounting for the existence of these backup systems, the population size should be

where ω is the number of types of vital subcellular structures; r is the number of units of each Structure per organism; is the average specific degradation rate for each type of structure.

Note that this equation can be expressed as a sum of decaying exponential Functions and, consequently, in semi-logarithmic coordinates, large segments of the corresponding graphical dependence should not form a straight line.

The simplest approach to analyzing a population of m distinct cell types (spores or viruses) also relies on the assumption that the organism species act independently; the total population of viable organisms will then equal the sum of the individual populations described by equations of type (7.125) with their respective individual coefficients k_di:

Again, the graph of this dependence in semi-logarithmic coordinates should not be a straight line. Furthermore, The values of the individual parameters k_di are not always known; certain Methods for estimating the upper and lower bounds of n(t)/n₀ based on initial data are discussed by Hutchinson and Lasso*. The reliability of such estimates depends on the accuracy of the initial population inactivation data and requires determining the second derivative d²n/dt² at t = 0.