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

Kinetics of substrate utilization, metabolite production, and biomass formation in cell cultures
Segregated models for cell growth kinetics and metabolite production

Cell populations are typically heterogeneous in the sense that individual Cells within a population vary in size, age, growth rate, and other properties. For a cell population growth kinetics model to closely approximate real-world conditions, it must also account for the distribution of cells across various categories with differing biochemical activities. Another advantage of this modeling approach is that it bridges the gap between the CHARACTERISTICS OF THE Cell Cycle AND Other properties of an isolated cell, on the one hand, and the overall population traits, on the other. The primary drawback of segregated models stems from their mathematical complexity. Consequently, this section focuses primarily on age distribution and correlated distributions in cell populations reproducing by binary fission. The derivation of population material balance equations used in segregated models, along with the solutions of these model equations for certain microbial systems, can be found in studies [29–31].

The Cell age frequency function W (a) allows for the Determination of the relative number of cells with an age between a and a + da as W (a) da. In this section, we use the symbol a to denote cell age, assuming that a zero age corresponds to newly divided cells, whereas dividing cells have an age t_D equal to the duration of the cell cycle or the doubling time of an individual cell. For balanced exponential growth of a population, the cell-age-based population balance equation takes the following form:

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Here, μ represents the overall specific growth rate of the population. Since all cells in the population range in age from 0 to t_D, W

must also satisfy the condition

Cell Division does not always yield two viable cells. This can be caused by cell Aging or death, or by the growth of a population containing unstable Plasmids in a selective medium. If Θ denotes the fraction of viable daughter cells produced by the binary fission of a viable parent cell, the balance equation relating the number of dividing cells to new viable cells can be written as follows:

Solving equation (7.108) subject to conditions (7.109) and (7.110) yields

Equations (7.110) and (7.111) lead to the following expression for the relationship between the growth rate μ and the cell generation time:

Since the mean population doubling time (the average cell cycle duration) is equal to

then

Thus, if 0 is non-zero, t_D is less than This relatively simple relationship illustrates a very important general principle: the overall kinetic Properties of a Population do not necessarily correspond to the kinetic parameters at the single-cell level.

In many cases, calculating various frequency Functions and averaged population properties with concrete physical meaning relies on the age density function, which cannot be measured directly. Suppose, for instance, that the synthesis of certain cellular components or metabolic products occurs at a constant rate during a specific period of the cell cycle from a₁ to a₂. Specifically, DNA Synthesis in eukaryotes is known to occur exclusively during the S phase. It has also been reported that certain organisms synthesize specific Enzymes only within a very narrow time window of the cell cycle. Under such circumstances, The rate of cellular product formation can be expressed by the equation

For the scenario discussed above, we obtain

For any quantitative parameter of cell growth x (such as cell mass, protein content, volume, or potentially the concentration of a specific metabolic product) directly linked to cell-cycle age a (in other words, if a can be expressed or defined as a function of x), the frequency function for a can be transformed into a frequency function for x According to the equation

Suppose, for example, that the mass of an individual cell increases linearly with time over the course of the cell cycle:

m(a) = m₀ + ak      (7.118)

so that

Substituting The values of W from (7.111) and a from (7.119) into equation (7.117), we obtain

Here again, It is important to emphasize the distinction between the growth kinetics of a single cell and that of a cell population. The mass of an individual cell may increase in accordance with equation (7.118) (such a relationship has also been observed experimentally for certain organisms), whereas the total mass of a population of the same cells increases exponentially—for instance, in accordance with equation (7.37), which describes The kinetics of balanced population growth in a batch process. One of the advantages of a structured (segregated) approach to Modeling cell growth kinetics lies in the possibility of establishing a direct link between the kinetic and regulatory parameters of individual cell growth and the growth kinetics of the cell population.

Assuming that the density of cellular matter remains constant, equation (7.120) can also be interpreted as a cell volume Distribution Function, which can be determined experimentally using a Coulter counter or flow cytometry light-scattering Methods (see Chapter 10). In practice, however, researchers more frequently measure the quantitative characteristics of Selection/30.html">The population as a whole. These characteristics can be calculated using the appropriate frequency functions and the equation

Other modeling approaches allow population heterogeneity to be addressed from different Perspectives. Several Mathematical models of metabolite formation in microbial populations have been proposed that take into account long-term aging effects. In the Ping Shu model, the specific rate of cellular product formation is expressed as

where θ is the total residence time of the cells in the Reactor [32]. The concentration of the metabolic product at time t in a batch process can then be calculated using the equation

Here, M(θ,t) is the frequency function at time t for the concentration of cell mass whose residence time in the reactor is equal to θ. Since cellular matter with an age of 0 at time t is generated over the time interval t − θ, it follows that

M(θ,t) = r_x(t - θ)      (7.123)

This model is notable for its considerable versatility and is well-suited for describing various mechanisms of metabolite accumulation observed in batch microbiological processes.