Unstructured Kinetic Models

An unstructured kinetic representation provides a simplistic global (with respect to cell mass) view of the net result of metabolic rate processes occurring in the living cells. In brief, the unstructured representation can be described as

more Cell mass + Non-biomass Products(^~^Pj)

This representation then involves conservation equations for cell mass, key nutrients, and metabolites of interest (target products), the rates of generation/consumption of the individual species being expressed in general in terms of concentrations of nutrients in abiotic phase, Ni (JVj = Ci for nutrient Ni as per the notation commonly used in literature in biochemical engineering and biotechnology), cell mass concentration, X (X = Cx), concentrations of metabolites of interest (Pj, Pj = Cj for product Pj for an extracellular metabolite and Pj = CjCx for an intracellular metabolite as per the notation commonly used in literature in biochemical engineering and biotechnology), and other parameters such as culture pH and temperature (T). For biomass (cell mass), the specific cell growth rate is therefore expressed as ¡1 = ii{N\. N2, ..., Pi, P2, ■ ■., X, pH, T). Consumption of a nutrient Ni implies that the rate of its generation in the biotic phase, rfen, is negative [Eq. 2.15], with the consumption rate usually being expressed as (- rfen) = <7j(iVi, N2, ..., Pi, P2, X, fi, pH, T) and being referred to as the cell mass-specific uptake rate of nutrient TVj. Similarly, for a target metabolite Pj, the rate of its generation in the biotic phase, r®en (whether or not the metabolite is excreted) [Eqs. 2.14a and 2.15], is expressed as r|en — £j(Ni, N2,..., Pi, P2, ..., X, pH, T), with £3 being referred to as the cell mass-specific production rate of metabolite Pj.

2.5.1 Rate Expressions for Cell Growth

More commonly, the dependence of specific cell growth rate on concentrations of various nutrients, cell mass and target metabolites is expressed in uncoupled, multiplicative form as per the relation

M = ^oMNMN2)..MN)..^i{Pi)^2{P2)...Vj{Pj)...ip{X). (2.17)

with Ho being constant characteristic of a particular strain. The popular forms of 4>i{Ni) and ipj(Pj) are based on the following common experimental observations. Cell growth is usually promoted by increased presence of some nutrients iVj (i.e., with increasing concentrations of these) at least up to some threshold levels and may be discouraged at high concentrations of some of these (the so-called "substrate inhibition"). Such nutrients are called (growth) rate-limiting nutrients. Nutrients which do not influence cell growth are not rate-limiting [<^t(Ar,) == 1 for these nutrients]. Cell growth may be unaffected by the presence of a target metabolite [<Pj{Pj) = 1 for a metabolite Pj\ or may be discouraged as the product Pj accumulates in the culture [i.e., fj(Pj) decreases with increasing Pj}. The former is often the case when the amount of Pj in the culture is significantly less than that of cell mass (X), an example of this situation being production of many antibiotics and enzymes. The latter is often the case when the amount of Pj in the culture is comparable or greater than that of cell mass (X), an example of this situation being production of alcohols (such as ethanol and butanol) and organic acids (such as acetic, citric, formic, lactic and succinic acids) by a variety of microbial species. One of the determinants of cell proliferation is accessibility of nutrients in the abiotic phase to cells. This accessibility is reduced with increasing biomass concentration in the culture and as a result, cell growth may be discouraged as the biotic fraction of the culture is increased [i.e., ip(X) may decrease with increasing X, <p{X) < 1]. The popular forms of cj>i(Ni), fj(Pj) and <p{X) are provided in Table 2.1.

In the classical chemical literature, rate expressions for homogeneous (fluid-based) reactions are of the power-law type, i.e., the rate of a reaction is proportional to some power of concentration of a reactant or a product for that reaction, the power (exponent) being referred to as the order of the reaction with respect to that species. For the large number of expressions available in the literature for rate of cell growth (see [35, 426, 545] for several examples of these), the orders of reactions with respect to nutrients are less than unity and positive those with respect to end-products are non-positive (not surprising since synthesis of building blocks for cellular material and synthesis of end-products are competing processes as concerns utilization of nutrient and energy resources within the living species) (Table 2.1).

The activity of each cell is a net result of thousands of molecular level chemical reactions occurring inside the cell that are promoted by a large number of enzymes (biological catalysts). These reactions are therefore surface-based reactions. Following the classical literature in chemistry on catalytic reactions, the rate expressions for individual reactions are usually of the Langmuir-Hinshelwood type, Michaelis-Menten expression being one example [126]. Depending on the positive (activation, induction) and negative (inhibition, repression) effects of various chemicals on the activity of an enzyme and the rate of the reaction it catalyzes, expressions with varying degrees of complexity have been proposed in the literature, all of these bearing a strong resemblance to the Langmuir-Hinshelwood type rate expressions used for chemical catalytic reactions. Due to enzyme-based

Table 2.1. Dependence of the specific cell growth rate ¡i on concentrations of nutrients, products and cell mass [35, 447, 451]

Function

Form

Reference

(1 + KjN~X,)~l

[19, 190, 600] [8, 9, 35] [35] [35] [7, 453]

<Pj(Pj)

[47, 190, 239, 326, 338, 395] [544, 600]

<P(X)

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