with ¡j, denoting the specific cell growth rate, r^ the specific rate of cell loss due to cell death or cell lysis, and /¿net the net specific growth rate, respectively, the basis for each being unit biomass amount. It must be kept in mind that C\ (later referred to also as X) represents the concentration of viable cell mass. The mass fraction of dead cells in the total cell population is considered to be negligible. In view of Eqs. 2.8 and 2.10, the temporal variation in Cx can be described as

with D denoting the dilution rate for the culture. The mass balance above applies for all three reactor operations under consideration, with D being trivial for a batch operation.

The conservation equation for a specie i in the abiotic phase in its general form can be expressed as

with CiF denoting the concentration of specie i in the nutrient feed, Rfen the rate of generation of specie i due to any reactions in the abiotic phase, rtrans j-j^ biomass-specific rate of transport of specie i from the biotic phase to the abiotic phase, and Qa being trivial in batch and fed-batch operations. When the specie is supplied in the feed gas (as is the case with oxygen), CiF is trivial and Ni is non-trivial. For species which are not transported into the biotic phase (for example, macromolecules like starch), r'rans is trivial. Although bulk of the chemical transformations occur in the biotic phase, some species may undergo reactions in the abiotic phase and for these species i?fen is non-trivial. Two examples of this situation are acidic or enzymatic hydrolysis of starch to generate readily metabolizable carbohydrates and degradation of antibiotics and enzymes in abiotic phase. This sets the stage for accounting for the intracellular components (components of the biotic phase).

The conservation equation for a specie i in biotic phase can be succinctly stated as d{CiCd*V) = (ft - rrns)CxV ~ QctCx (2.13)

with Cj denoting intracellular concentration of specie i in the biotic phase (mass of i per unit biomass, i.e., mass fraction of specie i in the biotic phase) and r; the net rate of generation of specie i in the biotic phase. In view of the biomass balance [Eq. 2.10], Eq. 2.13 can be restated as

It should be observed that Eq. 2.14a is valid irrespective of the mode of operation of the bioreactor and the last term on the right side represents the effect of dilution due to net cell growth. For species which are retained inside the cells (e.g., large macromolecules), rj,rans is trivial. The net rate of generation of specie i in the biotic phase, r,, includes the rate of loss of specie i from the biotic phase due to cell death or cell lysis. In view of Eq. 2.10, Eq. 2.14a can be stated alternately as

with rfen being the net rate of generation of specie i in the biotic phase exclusive of the rate of its loss from the biotic phase due to cell death or cell lysis. In the case of cell lysis, specie i will be introduced into the abiotic phase at the rate equal to rdCxV and this must be accounted for in Rfen in Eq. 2.12.

Conservation equations for intracellular species (and therefore temporal variations in quantities of these) are not accounted for in an unstructured representation of kinetics of cellular processes (the so-called unstructured kinetic models). For species that are present in both abiotic and biotic phases, examples of which include readily metabolizable nutrients and metabolites that are excreted by the cells, no differentiation is (can be) made between rfen (specie synthesis in the biotic phase) and r-rans (specie transport across the outer cell membrane into the abiotic phase). The conservation equation for a specie i in the abiotic phase is then based on Eq. 2.12 and is expressed as

For a target product metabolite (specie i) that is retained in the cells, an unstructured kinetic model must still include the conservation equation for the intracellular product, viz., Eq. 2.13, with r-rans of course being trivial.

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