The effects of key chemicals on the key rate processes are accounted for in the chemically structured models. All viable cells in the cell population are considered to be functionally similar, with conservation equations in the abiotic and biotic phases being considered for those species that are present in both phases. For such species, generation in the abiotic and biotic phases and transport across the interface between the abiotic and biotic phases must be fully accounted for. Synthesis of several antibiotics and other secondary metabolites by a host of microorganisms is inhibited by high concentrations of phosphate. Since cell growth is promoted by phosphate and the production of the secondary metabolite depends on both the cell mass concentration and production of the secondary metabolite per unit cell mass (specific production of the target metabolite), an optimum phosphate level which leads to maximum production of the secondary metabolite exists, as has been shown in previous experimental studies (see [461]). The illustration provided here pertains to a structured model for alkaloid production by Claviceps purpurea [461]. Let p and p\nt denote the concentrations of extracellular and intracellular phosphate (KH2PO\), respectively, the dimensions for the respective variables being g phosphate/L abiotic phase and g phosphate/g biomass.

Phosphate is considered to be the rate-limiting nutrient as concerns cell growth. Following expressions have been employed for specific cell growth rate (fj,) and specific cell lysis rate (r^) [Eq. 2.10].

In the relations above, k\, k2, and K\ are the kinetic parameters. The dependence of /z on puA is expressed by the Tessier equation. Cell lysis releases phosphate into the abiotic phase in quantity proportional to the cell mass phosphate content (Yp/x) and the intracellular phosphate concentration (pi). This release must be accounted for in the mass balance for extracellular phosphate, which is described by Eq. 2.12 with Cp = p and Np = 0, and RfTl and r*rans being

Rfn = (Yp/x + pint)k2X2V/Va, 7"praris = -j-^p- (2-30)

III Eq. 2.30, and Kn are kinetic parameters. The conservation equation for intracellular phosphate is provided by Eq. 2.14b with c, = cp — pmt and r®en = - Yp/x /J . The specific phosphate utilization rate for generation of biomass is therefore considered to be proportional to specific cell growth rate. It should be noted that the expression for Tprans in Eq. 2.30 is similar to the Monod expression. The transport of many species from the abiotic phase to the biotic phase and vice versa is facilitated by transport proteins (such as permeases). The rate expressions for transport of species across the outer cell membrane therefore bear close resemblance to the rate expressions for enzyme-catalyzed reactions (such as the Michaelis-Menten expression). Finally, the mass balance for the alkaloid (target metabolite) is provided by Eq. 2.15 with Cx = Ca = a, Na = 0, Rfn = 0, and rfn being provided by rr=£a = kiK3/(K3+p>), (2.31)

with and being the kinetic parameters. It is evident from Eqs. 2.29 and 2.31 that while increasing intracellular phosphate content is conducive for cell growth, it inhibits alkaloid synthesis due to repression of phosphatase activity. In this chemically structured model, the rates of all key kinetic activities, viz., cell growth, phosphate consumption, (- r|en), and alkaloid production are expressed in terms of the conditions prevailing in the biotic phase, viz., pi in the present case. The state variables [Eq. 2.1] in this model therefore are x = [X p pi a]T for batch and continuous cultures and -X. = [X p pi a V]T for a fed-batch culture.

2.6.3 Chemically and Morphologically Structured Models

This illustration pertains to production of the antibiotic cephalosporin C (CPC) by the mold Cephalosporin acremonium. As in the case of penicillin production (section 2.6.1), experimental observations have revealed that the cell population is comprised of three different morphological types, viz., hyphae (h), swollen hyphal fragments (s), and arthrospores (a). These three cell types undergo two metamorphosis reactions shown below.

Transformation of hyphae into swollen hyphal fragments involves assimilation of glucose (a carbon source, denoted as g) and methionine (a nitrogen source, denoted as m). The uptake of these two nutrients is confined mainly to hyphae and swollen hyphal fragments. Of the three cell types, only the swollen hyphal fragments are primarily capable of synthesizing CPC. Experimental studies have indicated that the rate of CPC synthesis is directly related to the activity of enzymes responsible for this. These enzymes are induced by intracellular methionine and are repressed by glucose. Only hyphae are capable of replication (growth). The rate of reaction (i) is expressed as a function of concentrations of hyphae, glucose and methionine, while the rate of reaction (ii) is expressed as a function of concentrations of glucose and swollen hyphal fragments. Let Zh, Zs and Za denote the mass fractions of hyphae, swollen hyphal fragments, and arthrospores, respectively, in the total cell population. Then the conservation equations for the three cell types can be expressed as in Eq. 2.14a with ct = Zit rtrans = 0) { = h, s, a. The net rates of generation of the three cell types, ri (i = h, s, a), are expressed as ([35, 374])

rh = (n'-0- kD)Zh, rs = (3Zh - (7 + kD)Zs, ra = -yZs - kDZa (2.33)

with kfj being the kinetic coefficient for cell death or cell lysis and the specific rates fi', ¡3 and 7 being expressed as

H' = /¿m(f>i{g), ¡3 = (fcn + k12m/{Km + m))<j>2(g), 7 = hi + k22(f>i{g), Mg) = g/(Kg + g), Ma) = g/(KG+g), (2.34)

with g and m denoting concentrations of glucose and methionine, respectively, in the abiotic phase, and fim, &11, &12, ^21,^22, Kg, Kg and Km being the kinetic parameters. In view of the identities Zh + Zs + Za = 1 and Xh/Zh = Xs/Zs = Xa/Za — X, it can be deduced from Eqs. 2.14a and 2.33 that

The structured model by Matsumura et al. [374] incorporates conservation equations for glucose and methionine in the abiotic phase. A separate mass balance for intracellular glucose is not considered. Hence, one needs to employ Eq. 2.15 for glucose (i = g) and Eq. 2.12 for methionine (i — m) with JVj = 0 and fifen = 0 (i = g, m). Glucose uptake occurs predominantly in hyphae and swollen hyphal fragments and as a result

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