Bioreaction Engineering Principles pp 359-381 | Cite as

# Population Balance Equations

## Abstract

In Chap. 7, cell population balances are written in terms of a distribution of mass fractions of the total biomass. This allows a direct combination of intracellularly structured models and population models. However, the population balances based on mass fractions do not permit the incorporation into the model of specific events in the cell cycle. Since there are numerous examples that show a direct influence of certain specific events in the cell cycle on the overall culture performance, e.g., the distribution of plasmids to daughter cells on cell division in recombinant cultures, we need to derive a population balance based on cell number to obtain a correct description of these processes.

### Keywords

Biomass Fermentation Carbohydrate Macromolecule Agglomeration### References

- Cazzador, L. (1991). Analysis of oscillations in yeast continuous cultures by a new simplified model,
*Bull Math. Biol.***5**, 685–700.Google Scholar - Christiansen, T., Spohr, A., Nielsen, J. (1999). On-line study of growth kinetics of single hyphae of
*Aspergillus oryzae*in a flow-through cell.*Biotechnol. Bioeng*.**63**, 147–153.CrossRefGoogle Scholar - Hjortso, M. A. and Bailey, J. E. (1982). Steady-state growth of budding yeast populations in well-mixed continuous-flow microbial reactors,
*Math. Biosci.***60**, 235–263.CrossRefGoogle Scholar - Hjortso, M. A. and Bailey, J. E. (1983). Transient responses of budding yeast populations,
*Math. Biosci.***63**, 121–148.CrossRefGoogle Scholar - Hjortso, M. A. and Bailey, J. E. (1984a). Plasmid stability in budding yeast populations: Steady state growth with selection pressure.
*Biotechnol. Bioeng.***26**, 528–536.CrossRefGoogle Scholar - Hjortso, M. A. and Bailey, J. E. (1984b). Plasmid stability in budding yeast populations: Dynamics following a shift to nonselective medium.
*Biotechnol. Bioeng.***26**, 814–819.CrossRefGoogle Scholar - Kothari, I. R., Martin, G. C., Reilly, P. J., Martin, P. J., and Eakman, J. M. (1972). Estimation of parameters in population models for
*Schizosaccharomyces pombe*from chemostat data.*Biotechnol. Bioeng.***14**, 915–938.CrossRefGoogle Scholar - Eakman, J. M., Fredrickson, A. C., and Tsuchiya, H. M. (1966). Statistics and dynamics of microbial cell populations.
*Chem. Eng. Prog. Symp. Ser.***62**, 37–49.Google Scholar - Krabben, P., Nielsen, J., Michelsen, M. L. (1997). Analysis of single hyphal growth and fragmentation in submerged cultures using a population model.
*Chem. Eng. Sci*.**52**, 2641–2652.CrossRefGoogle Scholar - Kreyszig, E. (1988).
*Advanced Engineering Mathematics*, 6th ed., John Wiley & Sons, New York.Google Scholar - Lievense, J. C. and Lim, H. C. (1982). The growth and dynamics of
*Saccharomyces cerevisiae*,*Ann. Report Ferm. Proc.***5**, 211–262.Google Scholar - Metz, B. (1976).
*From Pulp to Pellet*, Ph.D. thesis. Technical University of Delft, Delft.Google Scholar - Metz, B., Bruijn, E. W., and van Suijdam, J. C. (1981). Method for quantitative representation of the morphology of molds.
*Biotechnol. Bioeng.***23**, 149–162.CrossRefGoogle Scholar - Nielsen, J. (1993). A simple morphologically structured model describing the growth of filamentous microorganisms.
*Biotechnol. Bioeng.***41**, 715–727.CrossRefGoogle Scholar - Ormerod, M. G. (ed.), 2000 (3
^{rd}edition). Flow cytometry. BIOS Scientific Publishers, Oxford University Press.Google Scholar - Ramkrishna, D. (1979). Statistical models for cell populations.
*Adv. Biochem. Eng*.**11**, 1–48.Google Scholar - Ramkrishna, D. (1985). The status of population balances.
*Rev. Chem. Eng*.**3**, 49–95.CrossRefGoogle Scholar - Seo, J. H. and Bailey, J. E. (1985). A segregated model for plasmid content on growth properties and cloned gene product formation in
*Escherichia coli*.*Biotechnol. Bioeng.***27**, 156–166.CrossRefGoogle Scholar - Singh, P. N. and Ramkrishna, D. (1977). Solution of population balance equations by MWR.
*Comp. Chem. Eng.***1**, 23–31.CrossRefGoogle Scholar - Spohr, A. B., Mikkelsen, C. D., Carlsen, M., Nielsen, J., Villadsen, J. (1998). On-line study of fungal morphology during submerged growth in a small flow-through cell.
*Biotechnol. Bioeng*.**58**, 541–553.CrossRefGoogle Scholar - Subramanian, G. and Ramkrishna, D. (1971). On the solution of statistical models of cell populations.
*Math. Biosci.***10**, 1–23.CrossRefGoogle Scholar - van Suijdam, J. C. and Metz, B. (1981). Influence of engineering variables upon the morphology of filamentous molds.
*Biotechnol. Bioeng*.**23**, 111–148.CrossRefGoogle Scholar