A different perspective of microbial growth

Simon Brown


The logistic growth model involves two constants: one (k) reflecting an upper limit to growth and the other (β) related to the specific growth rate (μ). In the batch culture of microbes, both k and μ depend on the initial substrate concentration (s0), but this is not apparent from the logistic model. One standard resolution of this is to assume that μ is a function of s, rather than s0. A simple model is described that rectifies this discrepancy and provides explicit expressions for k and β. The model is consistent with experimental data, as is illustrated using data obtained during the growth of Candida utilis on glucose.


batch culture; Candida utilis; growth: logistic model; substrate utilisation


Ahmad, M.N. & Holland, C.R. (1995). Growth kinetics of single-cell protein in batch fermenters. Journal of Food Engineering 26, 443-452. (https://dx.doi.org/10.1016/0260-8774(94)00066-I)

Briggs, G.E. & Haldane, J.B.S. (1925). A note on the kinetics of enzyme action. Biochemical Journal 19, 338-339. (http://dx.doi.org/10.1042/bj0190338)

Brown, S. (2007). Two implications of common models of microbial growth. ANZIAM Journal 49, C230-C242. (http://dx.doi.org/10.21914/anziamj.v49i0.340)

Brown, S. (2014). An estimate of the duration of the lag phase of the logistic growth curve. Annals of West University of Timişoara, Series of Biology 17, 25-32. (https://doaj.org/article/8e567c9c594b4ff4bd5308af7d98e3d5)

Brown, S., Muhamad, N., Pedley, K.C. & Simcock, D.C. (2014). The kinetics of enzyme mixtures. Molecular Biology Research Communications 3, 19-30. (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5019218)

Brown, S. & Simcock, D.C. (2013). Confidence bands for rational rate equations. International Journal of Emerging Sciences 3, 15-27. (https://researchonline.jcu.edu.au/35821/)

Brown, S. & Simcock, D.C. (2014). On series approximations of Michaelis-Menten kinetics. Scientia 126, art01. (http://dx.doi.org/10.12969/Scientia.Vol126.Sect2.Art01)

Buchner, H., Longard, K. & Riedlin, G. (1887). Ueber die Vermehrungsgeschwindigkeit der Bacterien. Centralblatt für Bacteriologie und Parasitenkunde 2, 1-7. (https://www.biodiversitylibrary.org/page/51600787)

Dabes, J.N., Finn, R.K. & Wilke, C.R. (1970). The growth rate of microorganisms as a function of the concentration of a single limiting substrate, Lawrence Radiation Laboratory Report UCRL-19959, University of California, Berkeley. (https://escholarship.org/uc/item/07d3z367)

Dabes, J.N., Finn, R.K. & Wilke, C.R. (1973). Equations of substrate-limited growth: the case for Blackman kinetics. Biotechnology and Bioengineering 15, 1159-1177. (https://dx.doi.org/10.1002/bit.260150613)

Edwards, V.H. (1970). The influence of high substrate concentrations on microbial kinetics. Biotechnology and Bioengineering 12, 679-712. (https://dx.doi.org/10.1002/bit.260120504)

Fujimoto, Y. (1963). Kinetics of microbial growth and substrate consumption. Journal of Theoretical Biology 5, 171-191. (https://dx.doi.org/10.1016/0022-5193(63)90058-4)

Grant, D.J.W. (1967). Kinetic aspects of the growth of Klebsiella aerogenes with some benzenoid carbon sources. Journal of General Microbiology 46, 213-224. (https://dx.doi.org/10.1099/00221287-46-2-213)

Herbert, D., Elsworth, R. & Telling, R.C. (1956). The continuous culture of bacteria; a theoretical and experimental study. Journal of General Microbiology 14, 601-622. (http://dx.doi.org/10.1099/00221287-14-3-601)

Holmberg, A. (1982). On the practical identifiability of microbial growth models incorporating Michaelis-Menten type nonlinearities. Mathematical Biosciences 62, 23-43. (https://dx.doi.org/10.1016/0025-5564(82)90061-X)

Ihaka, R. & Gentleman, R. (1996). R: a language for data analysis and graphics. Journal of Computational and Graphical Statistics 5, 299-314. (http://dx.doi.org/ 10.1080/10618600.1996.10474713)

Kargi, F. & Shuler, M.L. (1979). Generalized differential specific rate equation for microbial growth. Biotechnology and Bioengineering 21, 1871-1875. (https://dx.doi.org/10.1002/bit.260211014)

Konak, A.R. (1975). An equation for batch bacterial growth. Biotechnology and Bioengineering 17, 271-272. (https://dx.doi.org/10.1002/bit.260170211)

La Motta, E.J. (1976). Kinetics of continuous growth cultures using the logistic growth curve. Biotechnology and Bioengineering 18, 1029-1032. (https://dx.doi.org/10.1002/bit.260180715)

Lee, J., Chang, H.–L., Parulekar, S.J. & Hong, J. (1991). An alternate method for estimation of cell growth kinetics from batch cultures. Biotechnology and Bioengineering 37, 26-34. (https://dx.doi.org/10.1002/bit.260370106)

Lobry, J.R. & Flandrois, J.P. (1991). Comparison of estimates of Monod’s growth model parameters from the same data set. Binary 3, 20-23. (https://www.simiode.org/resources/1016)

Lobry, J.R., Flandrois, J.P., Carret, G. & Pavé, A. (1992). Monod’s bacterial growth model revisited. Bulletin of Mathematical Biology 54, 117-122. (https://dx.doi.org/10.1007/BF02458623)

Marić, V., Einsele, A. & Fiechter, A. (1979). Respiratory activity and growth kinetics of Candida yeasts related to carbon sources and available energy. European Journal of Applied Microbiology and Biotechnology 8, 157-165. (https://dx.doi.org/10.1007/BF00506179)

Milota, J. (1982). Differential growth models for microbial populations. Aplikace Matematiky 27, 1-16. (http://dml.cz/dmlcz/103941)

Monod, J. (1937). Ration d’entretien et ration de croissance dans les populations bactériennes. Comptes Rendus hebdomadaire des séances de l’Académie des Sciences 205, 1456-1457. (https://gallica.bnf.fr/ark:/12148/bpt6k3157c/f1455)

Monod, J. (1941). Croissance des populations bactériennes en fonction de la concentration de l’aliment hydrocarboné. Comptes Rendus hebdomadaire des séances de l’Académie des Sciences 212, 771-773. (https://gallica.bnf.fr/ark:/12148/bpt6k3164q/f771)

Monod, J. & Tessier, G. (1936). La concentration de l'aliment, facteur quantitatif de l'accroissement des populations d'infusoires. Comptes Rendus hebdomadaire des séances de l'Académie des Sciences 202, 162-164. (https://gallica.bnf.fr/ark:/12148/bpt6k3154f/f162)

Monod, J., Wyman, J. & Changeux, J.-P. (1965). On the nature of allosteric transitions: a plausible model. Journal of Molecular Biology 12, 88-118. (http://dx.doi.org/10.1016/S0022-2836(65)80285-6)

Morel, M. & Monod, J. (1946). Sur l’utilisation du saccharose par Proteus vulgaris. Annales de l’Institut Pasteur 72, 647-651. (https://gallica.bnf.fr/ark:/12148/bpt6k6496737z/f650)

Morin, F. & Monod, J. (1942). Sur l’expression analytique de la croissance des populations bactériennes. Revue Scientifique 89, 227-229. (https://gallica.bnf.fr/ark:/12148/bpt6k65708678/f231)

Moser, H. (1958). The dynamcs of bacterial populations maintained in the chemostat, Carnegie Institution of Washington Publication 614, Washington. (https://catalog.hathitrust.org/Record/001639505)

Pang, S.–Y.M., Tristram, S. & Brown, S. (2010). Salicylhydroxamic acid inhibits the growth of Candida albicans. International Journal of Biology and Life Sciences 6, 40-46. (http://ecite.utas.edu.au/75733)

Pang, S.–Y.M., Tristram, S. & Brown, S. (2010). The contribution of growth rate to the pathogenicity of Candida spp. International Journal of Medicine and Medical Sciences 1, 80-86. (https://www.researchgate.net/publication/277137216)

Pang, S.-Y. M., Tristram, S. & Brown, S. (2011). Inhibition of the growth of pathogenic Candida spp. by salicylhydroxamic acid. International Journal of Biological and Life Sciences 7, 1-7. (https://www.researchgate.net/publication/277137283)

Phelps, A. (1936). Growth of protozoa in pure culture. II. Effect upon the growth curve of different concentrations of nutrient materials. Journal of Experimental Zoology 72, 479-496. (https://dx.doi.org/10.1002/jez.1400720307)

Postma, E., Kuiper, A., Tomasouw, W.F., Scheffers, W.A. & van Dijken, J.P. (1989). Competition for glucose between the yeasts Saccharomyces cerevisiae and Candida utilis. Applied and Environmental Microbiology 55, 3214-3220. (http://dx.doi.org/10.1128/AEM.55.12.3214-3220.1989)

Ramkrishna, D. & Song, H.–S. (2008). A rationale for Monod’s biochemical growth kinetics. Industrial and Engineering Chemistry Research 47, 9090-9098. (https://dx.doi.org/10.1021/ie800905d)

Rutgers, M., Teixeira de Mattos, M.J., Postma, P.W. & van Dam, K. (1987). Establishment of the steady state in glucose-limited chemostat cultures of Klebsiella pneumoniae. Journal of General Microbiology 133, 445-451. (https://dx.doi.org/10.1099/00221287-133-2-445)

Schaefer, W. (1948). Recherches sur la croissance du Mycobacterium tuberculosis en culture homogène. Annales de l'Institut Pasteur 74, 458-463. (https://gallica.bnf.fr/ark:/12148/bpt6k62022801/f26)

Slininger, P.J., Branstrator, L.E., Bothast, R.J., Okos, M.R. & Ladisch, M.R. (1991). Growth, death, and oxygen uptake kinetics of Pichia stipitis on xylose. Biotechnology and Bioengineering 37, 973-980. (https://dx.doi.org/10.1002/bit.260371012)

Teissier, G. (1936). Les lois quantitatives de la croissance. Annales de Physiologie et de Physicochimie Biologique 12, 527-586. (https://gallica.bnf.fr/services/image/highlighter/ark:/12148/bpt6k42230319/f554)

Teissier, G. (1942). Croissance des populations bactériennes et quantité d’aliment disponsible. Revue Scientifique 80, 209-214. (https://gallica.bnf.fr/ark:/12148/bpt6k65708678/f213)

Tobajas, M. & Garcia-Calvo, E. (2000). Comparison of analysis methods for determination of the kinetic parameters in batch cultures. World Journal of Microbiology and Biotechnology 16, 845-851. (https://dx.doi.org/10.1023/A:1008971708358)

van Urk, H., Postma, E., Scheffers, W.A. & van Dijken, J.P. (1989). Glucose transport in Crabtree-positive and Crabtree-negative yeasts. Journal of General Microbiology 135, 2399-2406. (https://dx.doi.org/10.1099/00221287-135-9-2399)

Verhulst, P.–F. (1838). Notice sur la loi que la population suit dans son accroissement. Correspondance Mathématique et Physique 10, 113-121. (https://archive.org/details/correspondancem07unkngoog)

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SCIENTIA International Identifiers: ISSN: 2282-2119 . DOI prefix: 10.12969 . EAN: 977-2282-211-00-9 . Handle (hdl) prefix: 11167

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