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

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

The simplest structured models involve the compartmentalization of biomass into a small number of components. Sometimes these components have an approximate biochemical interpretation, such as a synthetic component (RNA and precursors) and a structural component (DNA and Proteins). Compartments can also be defined as an assimilatory component and a synthetic component. Alternatively, the formulation of simple compartmental models relies on METABOLISM/2.html">THE CONCEPT OF metabolic bottlenecks.

The Use of structured models with a small number of variables is also justified from the standpoint of certain systems dynamics principles. Each type of reaction and transport phenomenon occurring within a Cell population (molecular collisions, Chemical Reactions, diffusion, RNA turnover, Protein Synthesis, Cell Division, completion of a batch process, and spontaneous Mutations) is characterized by its own relaxation time—that is, the time required to return to a steady state following a perturbation (recall Section 3.1). Relaxation times can range from fractions of a second to several hours. From the perspective of systems dynamics, the most crucial factor in modeling is the relationship between the time scale of environmental changes (t_E) and THE SPECTRUM OF relaxation times for intracellular processes. It can be assumed that cellular processes characterized by a rapid response to environmental changes (i.e., a small ratio of relaxation time to t_E) remain in a quasi-steady state (Section 3.2.1). In the opposite extreme case, where the relaxation time of certain cellular processes is large compared to t_E, these processes can be considered "frozen" in their initial state. An example is profound genetic changes, which virtually never occur during a single batch process.

It has been established that for many complex systems, only two or three relaxation times fall within the same time scale as environmental changes; this further Supports the validity of approximating the dynamics of complex systems using a simplified model with two or three variables. At the same time, it is often difficult or even impossible to relate some of these variables to measurable physical quantities of the system. Below, we will examine several Examples of simple compartmental kinetic models of cell population growth.

Williams proposed a two-compartment model that describes certain aspects of cell growth dynamics in a batch process with remarkable accuracy [16]. The fundamental postulates of this model are formulated as follows:

1. The synthetic component (1) is produced As a result of the uptake of an external nutrient S with a yield coefficient Y (The ratio of the mass of component 1 to the mass of the substrate). The formation reaction of the synthetic cellular component is first-order with respect to the total cell density x (the ratio of cell mass to culture volume) and the mass concentration of the nutrient (the ratio of substrate mass to culture volume).

2. The structural-genetic component of The Cell (2) is produced from component 1 at a rate proportional to p₁p₂ (where p_i is the ratio of the mass of i to unit cell volume).

3. Cell division requires and is sufficient upon the doubling of component 2. Consequently, the numerical cell density is proportional to the concentration of component 2 in the culture.

4. Biomass is constructed exclusively from components 1 and 2.

For a well-mixed batch Reactor, these same postulates can be expressed mathematically:

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Taking into account that

μ = k₁s      (7.72)

p₁ + p₂ = p_c = const      (7.73)

equation (7.71) can be written in the following form:

where f₂ = p₂/р_с represents the fraction of cell mass consisting of component 2. Using the two equations of this model, s can be expressed in terms of x. Transforming equation (7.69) in this manner yields an expression identical in form to equation (7.51). Consequently, changes in x will follow a logistic equation of type (7.52).

The approach proposed by Williams has been used to simulate cell population growth in a batch process using an inoculum from the stationary phase in a nutrient-depleted medium, which corresponds to p₁ = 0. The results obtained in this way are shown in Fig. 7.22; they successfully reproduce several commonly observed practical features of microbial culture growth in batch processes, including:

1. A lag phase during which cell size increases.

2. An exponential growth phase in which cell size reaches its maximum.

FIG. 7.22. Simulation of cell population growth using a stationary-phase inoculum, according to Williams. The ordinates show the time variation of dimensionless parameters: Substrate Concentration (s/s₀), biomass concentration (x/Ys₀), and population number (f₂x/Ys₀). The time variation of the relative cell mass (-1 + 1/f₂) is also depicted. (Parameters: k₁ = 0.0125; k₂рс = 0.025; Y = 0.5; x₀ = 0.05; s₀ = 1; f₂(0) = 1.)

3. Changes in cellular composition during the growth cycle. Since such changes evidently occur even in the exponential phase, it follows that cell growth is not perfectly balanced in any of the growth phases.

4. A stationary phase characterized by relatively small Cells.

Special attention should be paid to the fact that time-dependent changes in population density evaluated by mass and by cell number are non-proportional; in other words, the mass of an individual cell depends on the growth rate and even on the previous History of the growth rate. Similar results were obtained during detailed studies of the growth of E. coli and S. cerevisiae populations [30, 31]. These data indicate that the dynamics of cell mass changes can often be described with a fairly good approximation using relatively simple mathematical models; conversely, calculations of population number dynamics are more complex.

Fig. 7.23 schematically depicts several modified Variants of the model described above. In the two-compartment model of Harder and Roels, component G corresponds to Enzymes that catalyze The conversion of substrate into intermediates utilized in the synthesis of Biopolymers; cellular building blocks are represented by component K, which is formed from the nutrient S at a rate dependent on The amount of G [17]. In this simple two-component cell, the interconversion of G and K via polymerization and depolymerization accounts for maintenance metabolism. The constituents of the three-component cell (Fig. 7.23) have a more clearly defined biochemical nature; here, K denotes RNA, G denotes protein, and R denotes other biomass components [17]. In the latter case, maintenance metabolism is also represented as the turnover of components K and G.

Compared to some of the more complex structured models discussed below, these small compartmental models are mathematically relatively simple and contain a small number of kinetic parameters. They have been applied with moderate success to describe various states of unbalanced cellular growth. On the other hand, the lack of a clear biochemical Definition of the components in several cases makes Structure/47.html">Model Evaluation and parameter estimation difficult. When working with such models, we cannot utilize our knowledge of cellular metabolic pathways, The regulation of molecular processes, and Cell Cycle patterns. These data can only be applied to model cell growth kinetics if we have a deeply structured cell model that accounts for a significantly larger number of components, intracellular reactions, and interactions.

FIG. 7.23. Schemes of the two- and three-compartment models described by Harder and Roels [17].