The time-pattern of rises and falls in proliferation fades with senescence of mortal lines and is perpetuated in immortal rat hepatoma Fao cell line

Unmasking chaotic attributes in time series of living cell populations

Background

Long-range oscillations of the mammalian cell proliferation rate are commonly observed both in vivo and in vitro. Such complicated dynamics are generally the result of a combination of stochastic events and deterministic regulation. Assessing the role, if any, of chaotic regulation is difficult. However, unmasking chaotic dynamics is essential for analysis of cellular processes related to proliferation rate, including metabolic activity, telomere homeostasis, gene expression, and tumor growth.

Methodology/Principal Findings

Using a simple, original, nonlinear method based on return maps, we previously found a geometrical deterministic structure coordinating such fluctuations in populations of various cell types. However, nonlinearity and determinism are only necessary conditions for chaos; they do not by themselves constitute a proof of chaotic dynamics. Therefore, we used the same analytical method to analyze the oscillations of four well-known, low-dimensional, chaotic oscillators, originally designed in diverse settings and all possibly well-adapted to model the fluctuations of cell populations: the Lorenz, Rössler, Verhulst and Duffing oscillators. All four systems also display this geometrical structure, coordinating the oscillations of one or two variables of the oscillator. No such structure could be observed in periodic or stochastic fluctuations.

Conclusion/Significance

Theoretical models predict various cell population dynamics, from stable through periodically oscillating to a chaotic regime. Periodic and stochastic fluctuations were first described long ago in various mammalian cells, but by contrast, chaotic regulation had not previously been evidenced. The findings with our nonlinear geometrical approach are entirely consistent with the notion that fluctuations of cell populations can be chaotically controlled.

The concept of telomeric non-reciprocal recombination (TENOR) applied to human fibroblasts grown in serial cultures: concordance with genealogical data

Since the discovery of the limited life span of human fibroblasts some 50 years ago, many genealogical studies have been undertaken to describe growth kinetics of fibroblasts in serial cultures by their individual division behavior. It is now accepted that proliferation capacities of human fibroblasts strongly depend on their telomere lengths and integrity. Telomeres shorten with each replication round, and there is a direct correlation between cell division capacity and telomere lengths; that is, the consumption of disposable telomeric DNA repeats during cell divisions progresses until critically short telomeres determining the replicative senescence of the cells are present. Recently, we have suggested that telomeres in fibroblasts can also become elongated during DNA replication by telomeric non-reciprocal recombination (TENOR). Here we discuss genealogical data collected over the last decades as well as more recent findings on the telomere-driven replicative senescence process, and we summarize both to give an integrated picture.

Prolonged perturbation of the oscillations of hepatoma Fao cell proliferation by a single small dose of methotrexate

The proliferation rate of various cell types in vitro, including hepatoma Fao cells, displays aperiodic oscillations. The frequency of these oscillations is about one every 3-5 weeks, and there are variations in cell functions and polarity. Topological analysis has showed that these oscillations in growth rate are determined, and presumably chaotic. One characteristic of complex chaotic systems is that their dynamics can be persistently modified by a small external perturbation. We show that treatment with a single small dose of the anticancer drug methotrexate causes long-term stable alteration of the oscillatory dynamics of Fao cell proliferation. The oscillations of growth rate are shifted, and their mean level decreased according to a fractal pattern.

Deterministic dynamics control oscillations of bone marrow cell proliferation

OBJECTIVE

The production of blood cells in vivo, both normal and tumoral, displays oscillatory dynamics. Many cells in long-term cultures also show large amplitude oscillations of proliferative rate. Therefore we examined the proliferation dynamics of mouse bone marrow cells (MBM) and their clonogenic progenitor production (BMP), in order to characterize these dynamics.

METHODS

Five Dexter-type cultures of MBM cells and their clonogenic BMP production were examined for up to seven-months periods of time. The recorded time series exhibited a complex pattern of oscillations with variable amplitudes. We previously reported a method that allowed analysis of such nonlinear dynamics of hepatoma cell proliferation. We applied this method, based on the two-dimensional recurrent representation of data, to analyze the fluctuations of bone marrow cells proliferation.

RESULTS

The proliferation rate of mouse bone marrow cells shows large amplitude oscillations every 2 to 3 weeks. Mathematical analysis revealed a deterministic mechanism that controls all proliferation local maxima of MBM cells. Dynamics for progenitor production resembled that of parental cells. This reflects a predominant negative feedback on bone marrow cell proliferation.

CONCLUSION

These dynamics were opposite of that previously described for hepatoma cells where the dominant control is applied to the local minima (troughs of proliferation). Therefore, the complex system of cell proliferation is controlled by a bipolar mechanism, with a predominant dampening command depending on the cell type. We propose that the dominant dampening control of local maxima in bone marrow cells protects the stock of stem cells.

Evidence for deterministic chaos in aperiodic oscillations of proliferative activity in long-term cultured Fao hepatoma cells

The proliferative activity of long-term cultured mammalian cells exhibits traits of a complex dynamic system, with a succession of spontaneous rises and falls in proliferation rate. We analyzed three successive series of proliferation data for the Fao hepatoma cell line in long-term cultures. In the three series the proliferation rate displayed apparently disordered oscillations, which each lasted about 3–5 passages, with variable amplitude and were therefore unpredictable. Such non-linear kinetics raises the major issue of whether these fluctuations are random, or determined and coordinated. We used a graphical method of analysis of the data, which demonstrated that all troughs of proliferation were mathematically related to a common value in each series. This common value was itself related to the maximum level of proliferation of the cell line. Non-linear analysis thus confirmed that the fluctuations in proliferation rate of tumoral Fao cells are, at least in part, determined. This pattern evokes chaotic dynamics and is evidence for the flexible coordination of the complex system linking positive and negative growth regulators in long-term cultured cells.