Application of time-series regularity metrics to ion flux data from a population of pollen tubes

Figures & data

Table 1. Nicotiana tabacum L. pollen tubes. Analysis of the extracellular ion fluxes at a critical temperature and beyond – dynamic metrics

Temperature	Spectral signature ($\beta$)*	Corrected R/S Hurst exponent**	Largest Lyapunov exponent (emb. dims)	Approximate entropy ($S_a$)	Sample entropy ($S_s$)
25.9 ± 0.5°C (T = Tc)	1.0067 ± 0.024	0.9696445	−15.85 (12)	0.2490113 Low value Deterministic	0.1622784
19.8 ± 0.5°C (T ≠ Tc)	0.632 ± 0.025	0.6743406	−16.24 (12)	0.8849474 High value Random	0.7860483

*) Data from [4], showing the canonical case ($\frac{1}{{\mathrm{f}}^{\beta }}$, $\beta =1$]) of pink (flicker) noise at the optimum temperature of 25.9 ± 0.5°C. Pink noise is a signal or process in the frequency spectrum of whose PSD is inversely proportional to frequency. The PSD of pink noise drops 10 dB per decade. The value of $\beta$ at 19.8 ± 0.5°C though still at a pink noise range (in broad sense) is nearing the white noise, which is a random signal having equal intensity at different frequencies, giving it a constant PSD.

**) Calculated with the R [12] Practical Numerical Math Functions package. The hurstexp(x) function calculates the Hurst exponent of a time series x using R/S analysis [5,6].

Figure 1. Double resonance curve of the reciprocal of the calculated (approximate) dynamic entropy of the extracellular ion fluxes as a function of temperature for the pollen tubes of Nicotiana tabacum L. The determined germination temperature was T_c^ger = 16.37 ± 0.24°C, while for the optimal growth, it was T_c^opt = 25.25 ± 0.03°C. The solid royal blue line corresponds to the multi-peak Lorentz fit (R² = 0.69), which is interpolated by the B-spline. The errors (empirical: x_err = ± 0.5°C; from Lorentz fit: y_err = ± 0.125) are represented by the drone-like objects in the plot. The arrow indicates the peak of reverse entropy (corresponding to the local minimum of dynamic entropy) that is associated with germination. The approximate entropy does reflect closely what was obtained for the spectral exponent before [compare with Fig. 5 in 4]

Pietruszka M, Olszewska M. Extracellular ionic fluxes suggest the basis for cellular life at the 1/f ridge of extended criticality. Eur Biophys J. 2020;49:239–252.

 PubMed Web of Science ®Google Scholar

R Core Team. R: a language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing; 2020. Available from: htpps://www.R-project.org/

 Google Scholar

Hurst HE. Long-term storage capacity of reservoirs. Trans Am Soc Civil Eng. 1951;116:770–808.

 Google Scholar

Weron R. Estimating long range dependence: finite sample properties and confidence intervals. Phys A. 2002;312:285–299.

 Web of Science ®Google Scholar