Characterization of the Time Variations of Reproductive Processes in Living Cells

SINCE the first published account of the duration of some phase in the mitotic cycle of living cells by Mitscherlich1 in 1848, there have been more than 800 publications on the in vivo and in vitro kinetics of normal and neoplastic animal cells. Implicit in most calculations, based on these observations and in mathematical models used in the analysis of the life cycle of mammalian cells, is the assumption that the generation time (that is, mitotic cycle time, replication time) is constant2–5, or so nearly so that variations can be ignored. The potential danger of assuming a mean or “average cell” in models of growth, normal and neoplastic cell intercomparisons, and chemotherapy schedules has not been generally recognized. Likewise, the assumption of a normal distribution or other distribution functions of mathematical convenience contributes little to our insight into the physical or biological process which results in a particular distribution of, for example, DNA replication time, cell duplication, time and mitotic time6. The danger of using mean times or even appropriate frequency functions to describe cell replication without knowing the magnitude of circadian rhythm in cell physiology7 and DNA synthesis rates8 has not been emphasized by most workers concerned with mammalian, normal and abnormal cell physiology and cancer chemotherapy. The great variability in cell replication time for individual cell types in constant environmental conditions is a biological fact of importance in understanding cell behaviour and effecting optimum chemotherapy. This communication presents a technique for characterizing cell replication times and associated variability. Previous attempts to graduate cell variability were initiated with the work of Rahn9, Kendall10, Powell11,12, and more recently Kubitschek13,14. About eight frequency functions which fit the distribution of generation time or the reciprocal of generation time, “generation rate”, have been presented. These functions fit the positively skewed bell-shaped curve with about the same degree of “good-ness-of-fit”6 but tell us little about the underlying biophysical processes. We, in common with Powell and Errington15, do not believe that a normal distribution can be used to describe the reciprocal of generation time advocated by some13,14,16. We have examined a model which not only characterizes quantitatively the kinetic parameters of cell growth but also allows inclusion of a circadian rhythm function. The derivation follows from the assumption that the number of cells which are dividing at time t is directly related to the time which has elapsed since some threshold, t0, which is necessary for the essential events to have occurred before replication. Assuming that the number of cells dividing is also directly proportional to the number of unfissioned cells of those which were originally present in the environment, then where k2 is some positive coefficient and N0 is the number of cells in the environment at t = 0. Furthermore, it is assumed that all N0 cells were newly born at t = 0. If k2 = 2α, then . The constant can be evaluated from initial conditions to get This equation gives a bell-shaped curve skewed to the right when (dN/dt) is plotted against time. It is important to note that there is no remarkable difference between the fit of this function and that of others10–12 to experimental data. The probability frequency function is This relation was fit to the experimental data on the bacterium, E. coli13; the “functional mitotic interval” from ascites tumour Epithelioma 25517; the protozoan Euglena gracilis18 in two environmental conditions; human amnion cells16, and HeLa cells19, using a method of least squares with computing machines. The results are shown in Fig. 1 as the cumulative frequency curves (integrated form of equation 2). Equation 2 has been evaluated for nineteen sets of experimental results of generation time measurements on a wide variety of cells and in all cases a good fit to the observed data was found (see Table 1), but it should be emphasized that many different distribution functions can be made to fit these experimental data. Furthermore, although the same function may fit all experimental data, one cannot argue that this implies that the same processes control the phenomena being observed—it is a necessary but not a sufficient condition.