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

Kinetics of substrate utilization, metabolic product formation, and biomass accumulation in cell cultures
Structured models of cell growth kinetics
Metabolic models

In this section, we will review and examine the features of two distinct Structured models of Cell growth kinetics that incorporate various aspects of cellular METABOLISM. Using these models as Examples, we will explore some of the inherent advantages and drawbacks common to all more detailed models. These examples will also demonstrate that as a model incorporates a greater number of biological details, it becomes increasingly specific to a particular Organism or process. When developing a metabolic kinetic model for a specific system, it is essential to account for key metabolic details, which are sometimes gathered from scientific data or the biochemical engineering literature. In a sense, the more detailed our model becomes, the more a priori knowledge we must have about the organism under study. Otherwise, the task of selecting kinetic equations and parameter values becomes overly ambiguous, and the unknown model parameters cannot be determined from the available (limited) experimental data. Although some of the highly structured models described here and discussed later apply to individual Cells, the deterministic principle typically adopted assumes the Description of the behavior of an average cell within a large cell population.

First, we will examine the metabolic model developed by Bijkerk and Hall [18], as well as by Pamment, Hall, and Barford [19], for the aerobic growth of budding Yeast S. cerevisiae. This model is based on the following assumptions, most of which align with known data on the patterns of the Cell Cycle, metabolism, and the Introduction/15.html">Regulation of enzyme expression in this organism.

Assumptions Adopted in the Model

1. Individual isolated cells are not considered.
2. The growth-limiting substrate S serves as both the carbon and energy source; ethanol is denoted by the symbol E.
3. Biomass consists of two components, A and B.
4. Mass A provides for substrate uptake and Cellular energy supply, while mass B is responsible for cell synthesis and division.
5. The accumulation of energy and metabolic products occurs during the G₁ phase of The Cell cycle, is mediated by mass A, and is described by The conversion of mass A into mass B [see equations (7.75) and (7.76) below].
6. DNA Replication, mitosis, and Cell Division (the S, G₂, and M phases) occur over a defined period, are mediated by mass B, and are described by the conversion of mass B into mass A [see equation (7.77) below].
7. All Enzymes participating in Fermentation are grouped together and designated by the symbol E_f. Similarly, all Respiratory Chain enzymes are grouped together and designated by E_r.
8. In each enzyme system, enzyme Biosynthesis is regulated via two distinct pathways. First, enzymes are produced at rates proportional to the metabolite flux passing through that enzyme system. Second, enzymes of each system are also produced during cellular adaptation to changing conditions; here, The rate of their biosynthesis is proportional to the difference between the current Enzyme Concentration and a "control" value (e_F and e_R for fermentation and Respiration processes, respectively). The "control" enzyme concentrations are proportional to the metabolite flux that would be achieved at non-limiting enzyme concentrations.
9. Glycolysis and respiration rates depend linearly on e_f/e_F and e_r/e_R, respectively.
10. Glycolytic enzymes are produced As a result of respiratory pathway activity (this pathway generates enzymes for Gluconeogenesis, biosynthesis, and the utilization of intracellular storage CARBOHYDRATES).

Expressed in this manner, cell growth can be described by the following stoichiometric equations [hereinafter, the subscript W denotes variables expressed in mass units; in other cases, moles or activity units (for enzymes) are implied]:

Fermentation:

Respiration:

Division:

The rates r_A, r_B, and r_C can be expressed by the following equations:

The "control" enzyme concentrations are given by

The rates of enzyme biosynthesis for the fermentation and respiration systems are assumed to be equal to

In these equations, e_r and e_f denote the enzyme activities per unit culture volume. Consequently, as previously mentioned, this structured model is not based on "intrinsic" or intracellular concentrations.

By comparing simulation results with experimental data on batch yeast culture growth, the model parameters were determined, with their values listed in Table 7.3. Applying this kinetic model to steady-state continuous culture growth experiments required a slight modification of only a single parameter, a₃—the growth yield coefficient due to respiration—shifting from 1.79 in the batch process to 1.50 in the continuous process. This change was partly due to the known physiology of this organism, specifically the requirement for gluconeogenesis during batch culture growth on ethanol and the absence of such a requirement during continuous culture growth on glucose (in chemostat cultures).

Table 7.3. Parameters of the mathematical model for S. cerevisiae growth under aerobic conditions [19]

^a The values of k_f and k_h were calculated from other model parameters using equations given in [19].

FIG. 7.24. (a) Simulation-calculated dependencies of cell mass density x (solid curve) and specific enzyme activities e_f/x and e_r/x (dashed curves) on the dilution rate D (h^-1) for steady-state continuous culture growth of S. cerevisiae in a chemostat. Solid squares indicate experimental determinations of x; (b) experimental results for the specific activities of fermentation enzymes, NADP⁺-Glutamate dehydrogenase (GluDH — NADP⁺) and Alcohol dehydrogenase (ADH); (c) experimental results for the specific activities of malate dehydrogenase (MDH) and isocitrate lyase (ICL), which are involved in respiration. [Reprinted with permission from: Pamment N. B., Hall R. J., Barford J. R., MATHEMATICAL MODELING OF Lag Phases in Microbial growth, Biotech. Bioeng., 20, 345 (1978).]

Figure 7.24a shows graphs of cell mass density x (x = a + b) and the specific activities of fermentation enzymes (e_f/x) and respiration enzymes (e_r/x) as Functions of the dilution rate for steady-state continuous culture growth of S. cerevisiae, calculated using the parameters listed in Table 7.3. Solid squares in this figure denote experimental determinations of x, which are in good agreement with the calculated data. Note that here, both experimental and calculated dependencies of x on D differ sharply from the analogous simplified dependence obtained using the unstructured Monod model (Fig. 7.6). Figure 7.24b presents experimental results for the specific activity dependence of two fermentation-involved enzymes (NADP⁺-Glutamate Dehydrogenase and alcohol dehydrogenase) on the dilution rate in a chemostat. These dependencies are in qualitative agreement with the model's calculated data for e_f/x. A maximum was experimentally observed in the curve representing the specific activity of two respiration-related enzymes (malate dehydrogenase and isocitrate lyase) as a function of the dilution rate (Fig. 7.24c); the model correctly predicts the presence of a maximum in the e_r/x versus D dependence.

These results, along with several others presented in [18], convincingly demonstrate some of the advantages inherent in a more structured description of population growth. The model under consideration has been successfully applied to describe fed-batch growth of S. cerevisiae (see Section 9.1.1); it also accurately captures the lag phase in experimental batch processes.

As a final example, let us examine the model of an individual E. coli cell developed by Shuler and co-workers [24]; it is the most successful and highly structured microorganism growth model proposed to date. Figure 7.25 schematically illustrates the metabolites, Biopolymers, and reactions accounted for in this model, which are assumed to occur within an E. coli B/r A cell. Dashed lines indicate information flows that regulate reaction kinetics. The model includes reactions for cell envelope formation; consequently, the duration of the cell cycle is a model-predicted value rather than an input parameter, unlike in most other structured models. Because this model was specifically designed to describe cell growth under carbon- or nitrogen-limited conditions, it incorporates many structural cellular elements involved in the transport and assimilation of these nutrients. The reactions shown in Fig. 7.25 also account for ATP synthesis and utilization.

Figure 7.25 omits important details of the model related to The regulation of DNA replication initiation. Over a very brief period ("burst synthesis"), the repressor protein RP is synthesized; it is neutralized by the anti-repressor protein ARP, which is synthesized at a rate proportional to the rate of cell envelope construction. Once the RP concentration drops sufficiently, Transcription is initiated, producing the short RNA segment required to trigger replication.

Unfortunately, The Scope of this book does not permit a sufficiently detailed description of this model; readers can find additional information in [20]. This model includes approximately 100 stoichiometric and kinetic parameters, almost all of which can be determined from published biochemical studies of E. coli. The wealth of information embedded in this model is matched only by its remarkable capabilities.

FIG. 7.25. Diagram of a highly structured individual cell model describing the growth of E. coli B/r A on a medium containing glucose and ammonium salts. [Reprinted with permission from: Shuler M. L., Domach M. M., Mathematical models of the Growth of Individual Cells, in Foundations of Biochemical Engineering, Blanch H. W., Papoutsakis E. T., Stephanopoulos G. (eds.), p. 101, American Chemical Society, Washington, 1983.]

A₁ — ammonium ion; A₂ — glucose (and compounds derived from it within the cell); W — End products of Energy Metabolism (CO₂, H₂O, and acetate) excreted during aerobic growth; P₁ — Amino Acids; P₂ — ribonucleotides; P₃ — deoxyribonucleotides; P₄ — cell envelope precursors; M₁ — Proteins (in both the Cytoplasm and cell envelope); M_2RT1 — immature "stable" RNA; M_2RTM — mature "stable" RNA (rRNA and tRNA; in all cases, rRNA is assumed to comprise 85%); M_2M — Messenger RNA; M₃ — DNA; M₄ — non-protein fraction of the cell envelope (assumed to contain 16.7% peptidoglycans, 47.6% Lipids, and 35.7% Polysaccharides); M₅ — Glycogen; PG — ppGpp; E₁ — enzymes involved in the conversion of P₂ to P₃; E₂, E₃ — compounds involved in cell envelope formation and the regulation of its component synthesis; GLN — glutamine; E₄ — Glutamine Synthetase; * — extracellular substances.

The model accurately describes the time of Chromosome replication initiation and other key Features of the cell cycle over a wide range of growth rates. Figure 7.26 shows double-reciprocal plots of the specific growth rate of an individual cell versus the concentration of the growth-limiting nutrient (in this case, the ammonium ion), obtained experimentally (points) and calculated using the described model (solid curve). Note that both the simulation results and the experimental data indicate the existence of multiple ammonium ion utilization mechanisms; this is manifested as A change in slope at very high nutrient concentrations (small 1/s values). The model-predicted trends for cellular growth rate, intracellular glycogen content, and cell size (Fig. 7.27) are in good agreement with experimental results. The Williams model discussed earlier predicts an increase in cell size at high growth rates; in contrast, the highly structured model described here, which accounts for a significantly greater number of biochemical features and metabolic details, shows that cell size depends not only on the growth rate but also on which nutrient is growth-limiting.

FIG. 7.26. Dependence of the specific growth rate of nitrogen-limited E. coli B/r A on ammonium ion concentration (solid line represents model calculations; bold dots represent experimental data). [Reprinted with permission from: Shuler M. L., Domach M. M., Mathematical Models of the Growth of Individual Cells, in Foundations of Biochemical Engineering, Blanch H. W., Papoutsakis E. T., Stephanopoulos G. (eds.), p. 101, American Chemical Society, Washington, 1983.]