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

Kinetics of substrate utilization, metabolite production, and biomass formation in cell cultures
Kinetics of metabolite production
Modeling the kinetics of metabolite production based on molecular transformation mechanisms; genetically structured models

The more accurately the kinetic description in a model reflects the actual chemical events taking place within a Cell, the more universal our model becomes. Here, by “universality,” we mean The ability to yield satisfactory results under conditions different from those used during model development and parameter estimation. The Mechanisms of Protein synthesis, which is of great importance in The production of Enzymes, Hormones, and other practically relevant Polypeptides, have been studied in such depth that it is now possible to develop models at the level of molecular events and interactions. Such models should be highly versatile and useful for optimizing environmental parameters and the genetic CHARACTERISTICS OF THE Organism.

The material balance for the mRNA produced during the METABOLISM/31.html">Transcription of a given Gene G can be expressed by the following equation (here, square brackets denote the molar concentrations of the respective component in The Cell, expressed in moles per unit of cell volume) [27]:

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In this equation, k_p is the overall transcription rate constant, and k_d is the rate constant for mRNA inactivation and/or degradation; both of the latter processes are assumed to be first-order. Equation (7.98) also accounts for the promoter utilization efficiency coefficient η, which reflects the modulation of transcription by the operator. In many cases of practical interest, estimating η in terms of the concentrations of the effector protein, inducer, or repressor is the most crucial step in the entire modeling process.

Similarly, the material balance equation for the intracellular Translation product, i.e., the protein of interest, can be written as:

Here, The rate of Protein Synthesis is proportional to the concentration of the mRNA encoding that protein and the efficiency of its utilization on Ribosomes (it is quite reasonable to assume, for instance, that A change in The nucleotide sequence within the ribosome-binding site will lead to a change in the value of I). In this case, we again assume that the protein inactivation reaction is of the first order.

Under balanced growth, the time derivatives of [mRNA] and [P] are equal to zero, and under this condition, it follows from equations (7.98) and (7.99) that

This equation directly expresses the intracellular protein concentration in terms of the parameters characterizing gene and mRNA expression, as well as protein stability within the cell.

Table 7.6. Values³ of parameters for the Gene Expression kinetics model in E. coli^a

^a The table lists average values that may require refinement when calculating the expression rate of a specific gene and the synthesis of its corresponding protein. The specific growth rate μ is given in units of h^-1.

“Data from: Lee S. B., Bailey J. E., Analysis of Growth Rate Effects on Productivity of Recombinant Escherichia coli Populations, Biotech. Bioeng., 26, 66 (1984).

Table 7.6 presents typical average parameter values for the described model of Protein synthesis in E. coli, reflecting the observed dependences of transcription and translation rate parameters on the growth rate.

Current data indicate that in many cases, the rate of GENE EXPRESSION IN E. coli is limited by the transcription step. Consequently, the expression rate of an operator-regulated gene depends on the transcription efficiency of that gene, which in turn is determined by the combined influence of modulating agents on the operator and RNA polymerase binding sites. As an example, let us consider the lac promoter-operator system of E. coli, focusing primarily on the combined effect of the lac repressor, inducer, and DNA on gene expression and, consequently, on ß-galactosidase activity [28].

The amount of enzyme produced is proportional to the promoter utilization efficiency η, which in turn is proportional to the probability that the operator site O is not bound to the repressor protein R. Thus,

Here, [O]₀ is the total concentration of operator sites in the E. coli cell. The fractional expression on the right-hand side of equation (7.101) can be determined by taking into account the interactions among R, O, other non-specific DNA binding sites (D), and the inducer I; all these interactions are assumed to be at equilibrium:

The operator O can exist in three different forms, whose concentrations can be related by the following material balance equation:

Similarly, the material balance equation for R can be written as

Since the concentration [D] of non-specifically bound DNA sites exceeds the total concentration of R, we can assume that

Using approximation (7.105), the equilibrium nature of reactions (7.102), the material balance equation (7.104), and neglecting terms with the factor [O]₀/[R]₀, we can readily derive the following equation:

Experimental data available in the literature make it possible to estimate the parameters of this equation approximately, with an accuracy of at least within an order of magnitude [28]:

Estimating the parameters of this mathematical model requires extensive experimental data, yet molecular-level modeling also offers distinct advantages. Specifically, its parameters have a rigorous physicochemical interpretation and can often be determined in separate in vitro experiments. Furthermore, a model of this level possesses a special feature known as genetic Structure. In this context, genetic structure refers to a clear dependence of certain model parameters on the genetic makeup; thus, a change in the nucleotide sequence of the lac operator must be accompanied by Changes in the parameters K_a and K_D that characterize the interactions between the lac operator and the repressor protein. As our understanding of the mechanisms of DNA–Protein Interactions advances and deepens, we will likely be able to calculate The Effect of specific nucleotide sequence alterations on the corresponding DNA–protein binding constants. Consequently, a kinetic model developed at THE MOLECULAR LEVEL will enable nucleotide sequence mapping that reflects cell population productivity. The potential of this approach for the rational, quantitative optimization of organism genetic modification (or, at least, of cloned DNA segments) is self-evident.

As an example, let us consider the application of equation (7.106) to calculate the effect of repressor concentration on the rates of induced and uninduced Transcription of the lac promoter-operator in the E. coli chromosome. The necessary experimental data, including intracellular repressor concentrations for each of the mutants, are known for the i^-, i^q, and i^sq E. coli mutant strains. These values can be used directly in the simulation and do not affect other model parameters. As shown in Table 7.7, the model satisfactorily captures the effect of intracellular repressor concentration both in the presence and absence of an inducer. Additional Examples of molecular-level regulation modeling can be found in the References at the end of the chapter.