Unmasking chaotic attributes in time series of living cell populations

Abstract

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.

Figure 1. Method of map construction and analysis.

A: segment of a curve of the proliferation rate of rat liver cancer cell line Fao: 27 consecutive 6-day passages in culture. X-axis: passage number; Y-axis: proliferation rate, expressed as population doublings/passage (PD/passage). B: the corresponding map is constructed by displaying the proliferation rate data in the recurrent form x_i+1 versus x_i (e.g. the green lines x_i+1 to x_i that construct the point corresponding to the segment p13-p14). The successive points on the map are joined together, as a succession of vectors. When x_i is a local peak (i.e. if x_i−1<x_i>x_i+1), then the vector points south-east (highlighted in red). Whereas when x_i is a trough, (i.e. if x_i−1>x_i<x_i+1), the vector points north-west in the plane (highlighted in blue). The bisecting line, i.e. the line perpendicular to the mid-point of the vector, is drawn for each trough (blue dotted arrows) and peak (red dotted arrows) vector. Coordination, if any, of the bisecting lines defines a fixed point, i.e. a point on the diagonal where x_i = x_i+1 (which is therefore a stable level of cell growth), as shown here for convergence of the bisecting lines of trough vectors on coordinates 6.25/6.25. Note that there is no such coordination of the peak vectors in these cancer cells. (The complete analysis of the cell line was published in [14]).

Figure 2. Lorenz oscillator: oscillatory patterns and phase-space representation.

Lorenz attractor standard values for the constants were set as follows: Left panel: Ordinate: oscillatory behavior of variables x (top) and z (bottom). Abscissa: time. Right panel: phase-space representation of the attractor.

Figure 3. Rössler system: oscillatory patterns and phase-space representation.

Rössler system standard values for the constants were set as follows:with a = 0.398, b = 1, c = 3. Left panel: Ordinate: oscillatory behavior of variables x (top) and z (bottom). Abscissa: time. Right panel: phase-space representation of the attractor.

Figure 4. Verhulst system: oscillatory pattern and phase-space representation.

Verhulst system standard values for the constants were set as follows:with r = 3.72. Left panel: Ordinate: oscillatory behavior of variable x. Abscissa: time. Right panel: phase-space representation of the attractor. Chaotic (top) and birhythmic (bottom) conditions are compared.

Figure 5. Duffing oscillator: oscillatory patterns and phase-space representation.

Duffing oscillator standard values for the constants were set as follows:with . Left panel: Ordinate: oscillatory behavior of variables x (top) and y (bottom). Abscissa: time. Right panel: phase-space representation of the attractor.

Figure 6. Lorenz oscillator: analysis of the map x_i vs x_i+1.

Left panel: the bisecting lines of the trough vectors (top, red lines) and of the peak vectors (bottom, green lines) for variable x. Right panel: the bisecting lines of the trough vectors (top, red lines) and of the peak vectors (bottom, green lines) for variable z. Note the focal concentration of maxima bisecting lines (at coordinates 12.5/12.5) and minima bisecting lines (at coordinates 40/40) for variable z The focal concentration of bisecting lines for the variable x is slightly less precise.

Figure 7. Rössler system: analysis of the map x_i vs x_i+1.

Top: bisecting lines (red) for the trough vectors. Bottom: bisecting lines for the peak vectors (green lines). There is a characteristic convergence of the bisecting lines of local maxima on the map. In contrast, the local minima are not coordinated.

Figure 8. Verhulst system: analysis of the map x_i vs x_i+1.

Left panel displays the maps for chaotic conditions. Top: bisecting lines of the peak vectors (red lines). Bottom: bisecting lines of the trough vectors (green lines). Note the convergence of bisecting lines of local minima. The local maxima were not coordinated with this set of parameters. Right panel: maps for birhythmic dynamics. Top: note the two narrow bundles of superimposed bisecting lines of the two types of peak vectors (red lines). Bottom: note the two narrow bundles of the superimposed bisecting lines of the two types of peak vectors (green lines).

Figure 9. Duffing oscillator: analysis of the map x_i vs x_i+1.

Top: map for the variable y. Bottom: map for the variable x. Bisecting lines for the vectors of peaks are in green, bisecting lines for the vectors of troughs are in red. Note dual control of local maxima and minima for the variable y, with convergence of bisecting lines of peaks on a low fixed-point (coordinates about −2/−2) and convergence of local minima on a high-fixed point (coordinates about 2/2). In contrast the bisecting lines of the peak and trough vectors of the variable x are disordered.

Figure 10. Comparison with other dynamics.

Left panel: return maps, Right panel: oscillatory behaviors. Top: sinusoidal oscillations. The vectors for local minima and local maxima are superimposed on one line perpendicular to the diagonal, and their bisecting lines are superimposed on the diagonal, and oriented upward for the local minima, and downward for the local maxima. Middle: birhythmic oscillations. There are two vectors representing all local maxima, the bisecting lines of which intersect the diagonal at a low fixed point, and two vectors representing all local minima, the bisecting lines of which intersect the diagonal at a high fixed point. Bottom: random numbers; the bisecting lines of the vectors are dispersed.