Structured Kinetic Models

Structured kinetic representations are warranted in situations involving significant changes in composition of the biotic phase and the kinetics of eel-

lular rate processes is significantly sensitive to changes in cell composition. Since it is not practical to account for variations in every component of the biotic phase, the structured kinetic model for a particular bioprocess must focus on carefully selected key components and rate processes of major interest for that bioprocess. Depending on a particular application of interest, a variety of structured kinetic representations have been reported in the literature. These can be classified as (1) morphologically structured models, (2) chemically structured models, (3) genetically structured models, and (4) metabolically structured models, and some binary and higher combinations of (l)-(4). Since antibiotic (such as penicillin) production is the example industrial bioprocess considered throughout this book, the illustrations provided below will pertain more often to antibiotic production processes. In these illustrations, which are imported from the literature, the notation used in the source will be followed as much as possible so that interested readers will have little difficulty in accessing additional details in the source references.

2.6.1 Morphologically Structured Models

The kinetics of nutrient utilization and product (for example, an antibiotic) formation by filamentous microorganisms and molds is usually quite complex. A characteristic of growth of these living species is cellular differentiation, with different cell types differing from one another in terms of functions (cell growth, uptake of different nutrients, and synthesis of a target metabolite such as an antibiotic) they perform. The illustration provided here pertains to penicillin production. Morphologically structured models for penicillin production by P. chrysogenum have been proposed earlier by Megee et al. [382], Nielsen [423], and Paul and Thomas [459]. The models due to Megee et al. [382] and Paul and Thomas [459], although very comprehensive, involve large number of parameters, which make their identification and validation difficult. In comparison, the model proposed by Nielsen [423, 424, 425] is simplified and flexible and is used here as an illustration. This is not to say that we prefer one structured model over the others. Three cell types are considered in the Nielsen [423, 424, 425] model, apical cells, subapical cells, and hyphal cells (denoted as a, s and h, respectively). Uptake of nutrients and formation of biomass occurs only in apical and subapical compartments of an hyphal element (a multicellular unit). In an hyphal element, the apical compartment is located between a tip and the first septum. The cells in the interior (with respect to apical compartment) have an intracellular composition similar to that of apical cells and form the subapical compartment. Three metamorphosis reactions are considered in the Nielsen [423, 424, 425] model: branching, tip exten sion and differentiation. During tip extension, some apical cells become subapical cells. Branching refers to formation of new apical compartments from the cells in the subapical compartment. The subapical cells further away from the tip become more and more vacuolated as their age increases. As a result, cells further away from the tip contain large vacuoles. These cells, which form the hyphal compartment, play an important role in transport of protoplasm toward the tip section. Formation of vacuolated hyphal cells from the subapical cells is referred to as differentiation. The transition from active subapical cells to completely vacuolated hyphal cells takes place gradually. The hyphal cells located in the vicinity of the subapical compartment are therefore assumed to retain the metabolic activity and ability to grow as do the subapical cells. This has been accounted for in the formulation of the model by considering that a fraction fh, of the hyphal cells is metabolically active ([423, 424, 425]).

The kinetic expressions for branching, tip extension and differentiation are considered to be first order in cell type being transformed, which leads to the following rate expressions for the metamorphosis reactions under consideration. Branching (1):

Extension (2):

Differentiation (3):

In Eqs. 2.22 - 2.24, Za, Zs, and Zh represent the mass fractions of apical, subapical, and hyphal cells, respectively, in the total cell population, Uj s (j = 1, 2, 3) the rates of the three metamorphosis reactions and kU] 's (j = 1, 2, 3) the kinetic coefficients for these. Differentiation is assumed to be inhibited by the carbon source (usually glucose, S = glucose concentration in the abiotic phase). The form of <f>3(S) in Eq. 2.24 is a special case of the form of <i>i{Ni) (Ni = S) in Table 2.1. The specific growth rates of each cell type have been represented by the Monod kinetics, viz.,

In Eq. 2.25, kj's (j = a, s, h) represent the maximum values of specific growth rate of each cell type. The mass balances for the three cell types then can be described by Eqs. 2.14a and 2.14b (cj = Zj, i = a, s, h, rJrans = 0) with rfen (i = a, s, h) being rfn =ui -u2 + fiaZa, rfen = u2 - tii - u3 + (2.26)

Via addition of Eq. 2.14b for the three cells types and in view of the identity Za + Zs + Zh — 1, the expression for the specific cell growth rate ¡i can be deduced to be f-l — l^a^a + HsZs +fhtihZh. (2.27)

Cell death and cell lysis have not been considered in the model by Nielsen ([423, 424, 425]). One must note that the specific cell growth rate in this structured kinetic representation is dependent not only on the extracellular glucose concentration (S), but also on the fractions of the three cell types (intracellular variables as concerns the total cell population). The following expressions have been employed for the other two key processes, viz., glucose (5) uptake and penicillin (P) synthesis (specific rates denoted as as and ep, respectively).

ers = otin + ms + a2eP, eP = k2{Zs + fhZh)x{S), (2.28)

In Eq. 2.28, the parameters ai, a2, k2, K2, and Kj are independent of S, X and Zi s (i = a,s,h). The rate expression for glucose uptake in Eq. 2.28 is similar to Eq. 2.19, with a\ being the reciprocal of the cell mass yield with respect to glucose {Yx/s)- Before leaving this illustration, the functional segregation of the three cell types must be commented upon. While all three cell types are considered to be capable of growth [Eq. 2.25], only the subapical cells and a fraction (fa) of the hyphal cells are capable of synthesizing penicillin [Eq. 2.28], and the three cell types participate in different metamorphosis reactions [Eqs. 2.22-2.24], The conservation equations for glucose (5) and the extracellular product (penicillin, P) are provided by Eq. 2.15 with N, = 0 and i?fn - 0 (?: - S, P, products of hydrolysis of penicillin pooled together with penicillin), r|en = - as, r®en = ep, and Cj = J, J = P, S, X. Conservation equations for oxygen in the gas phase and the abiotic phase have not been considered in the Nielsen model [423, 424, 425], since it is assumed that the culture is not oxygen limited (with dissolved oxygen level assumed to be in excess of 45% saturation). The supply of sufficient amounts of oxygen may indeed become a critical problem at high biomass concentrations. The state variables [Eq. 2.1] in this model therefore are x = [X S P Zs Zh]T for batch and continuous cultures and x = [X S P Zs Zh V}T for a fed-batch culture. The identity Za + Zs + Zh = 1 implies that only two of the three fractions of the cell population are independent state variables.

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