Growth curves of microorganisms and cell cultures

Senescence is normally a feature of the post-mitotic phase of the cell life-cycle and is immediately preceded by (and sometimes partially overlaps with) the growth period. Typically the pattern of growth in biological systems is density-dependent, beginning slowly when cell mass is small, reaching a maximal rate when density is optimal with respect to metabolic and environmental constraints, declining as limiting external and internal factors become increasingly influential and finally asymptotic. Such sigmoidal behaviour is usually described mathematically by some variation of the logistic equation (McCullagh and Nelder 1989). The generalised, archetypal logistic-type function is:



To fit this kind of function is virtually impossible without computers, and even then it took until the late sixties to crack the problem of fitting the generalised logistic with statistical rigour. Nowadays anyone can do it with absolute confidence that the error estimates all conform to the mind-bending laws of non-linear estimation.

The asymptote of the fitted curve corresponds to the size of the stationary-phase culture (or fully-grown organ). By putting together coefficients in various combinations we can get mean weighted absolute and relative growth-rates, duration of growth, inflexion point of the S-shaped curve and, of course, estimated values of G, instantaneous absolute growth rate and relative growth rate for each value of t. All these are computed with standard errors, and you can get a plot and tables in any format you choose (Thomas and Potter 1985, Gay and Thomas 1995).

References

• Gay AP, Thomas H (1995) Leaf development in Lolium temulentum: photosynthesis in relation to growth and senescence. New Phytologist 130: 159-168.
• McCullagh P, Nelder JA (1989) Generalized Linear Models, 2nd edition. London: Chapman and Hall.
• Thomas H, Potter JF (1985) Fitting logistic type curves to extension growth data for leaves of grass species by means of the Maximum Likelihood Program: analysis of leaf extension in Lolium temulentum at optimal and chilling temperatures. Environmental and Experimental Botany 25: 157–163.

Further reading and external links

• Online curve-fitting: Interactive 2- and 3-D data modelling
• Bacterial growth Explanation and movie
• Cui Y, Zhu J, Wu R (2006) Functional mapping for genetic control of programmed cell death. Physiological Genomics 25: 458-469. Modelling cell growth in relation to cell death