An effective formula describing expansive plant growth is derived from the modified Lockhart/Ortega-type equation. Its applicability is demonstrated on selected experimental data extracted from available literature. Quantitative information about the ''diffusion rate'' (k 2 ) of the growth factors is obtained for two different model species in plant science: Arabidopsis thaliana L. belongs to the dicots and Zea mays L. belongs to the monocots. It is shown that the value of the diffusion rate may be useful in comparing different datasets and serve as a measure of reproducibility of standard measurements. Analysis of the formula and fits allows to identify and suggest a set of criteria for reporting future experiments, which would improve comparability and reproducibility of the results. Keywords Auxin Á Arabidopsis thaliana L. Á Juglone Á Fusicoccin Á Garlic extract Á Zea mays L. Á Lockhart equation Á Ortega equation Communicated by U. Feller.

We investigated the growth-temperature relationship in plants using a quantitative perspective of a recently derived growth functional. We showed that auxininduced growth is achieved by the diffusion rate, which is almost constant or slowly ascending in temperature, while the diffusion rate of fusicoccin (FC)-induced growth increases monotonically with temperature for the entire temperature range (0-45)°C, although for some concentrations of indole-3-acetic acid (IAA), ''super-diffusion'' takes place for unperturbed growth. We also calculated the cross-correlations and the derivative of cross-correlations for elongation growth (rate) and pH as a function of time delay (lag) parameterised by temperature for artificial pond water (APW) control conditions (endogenous growth) and exogenous IAA and FC that were introduced into the medium. Dimensionality analysis revealed that discontinuities in the cross-correlation derivative corresponded to H ? ion kinetics, which attained definite numerical values that were approximately proportional to the (logarithm of) proton secretion rates (or relative buffer acidification). Furthermore, three types of experiments were compared: for abraded coleoptiles, coleoptile segments and intact growing seedlings. From the cross-correlation analysis, it was found that the timing of IAA and FC-induced proton secretion and growth matched well. Unambiguous results concerning the canvas constituting acid growth hypothesis were obtained by cross-and auto-correlation analysis: (1) for abraded coleoptiles, because of the lowering of the cuticle potential barrier, auxin-induced cell wall pH decreased simultaneously with the change in growth rate; no advancement or retardation of pH (proton efflux rate) or growth rate took place, (2) exogenous protons were able to substitute for auxin thus causing wall loosening and growth, (3) although the underlying molecular mechanisms differ vastly, a potent stimulator of proton secretion, the fungal toxin FC, promoted growth that was similar to auxin, although of an elevated intensity; as for auxin-no advancement or retardation took place.

We report on our results concerning growth rate and oscillation modes of the individual pollen tube apex. The observed volumetric growth and growth rate periodicity in the longitudinal (axial) direction are accompanied by transverse oscillations with similar frequencies but higher energies than the axial modes. Examination of the time-domain coherence between oscillations in mutually perpendicular directions revealed minimal energy dissipation in the unperturbed (isotonic) case, opposite to the two remaining cases (hypertonic, hypotonic) with notable correlations. We conjecture that the minimal energy loss is therefore optimal in the natural growth conditions. The longitudinal growth velocity is also found to be the fastest in the unperturbed case. As a result, the isolated system (pollen tube tip) is conserving energy by transforming it from elastic potential energy of extending apical wall to the kinetic energy of periodical motion. The energy dissipation is found to be about 20 % smaller in axial direction than in lateral one, indicating that the main energy consumption is dedicated to the elongation. We further observe that the hypertonic spectrum is shifted towards lower and the hypotonic towards higher frequencies with respect to the isotonic spectrum. In consequence, the turgor pressure inside the growing cell influences monotonically the frequency of both modes of oscillations. The calculated power spectrum seen as a measure of the overall energy efficiency of tip growth under hypertonic, hypotonic and isotonic conditions implies that the biochemistry has been fine tuned to be optimal under normal growth conditions, which is the developmental implication of this work. A simple theoretical extension of the Ortega equation is derived and analysed with respect to its contribution to power spectrum. We show that the plastic term, related to the effective turgor pressure, with maximum contribution at frequency f = 0 is responsible for the steady growth. In turn, the elastic contribution dependent on the time derivative of pressure fluctuations tends to move the system into oscillatory mode (f > 0). None of those mechanisms is privileged over another. The coupling mechanism is naturally generated through a convolution of those two terms and will decide about the overall character of the growth for each particular case.Electronic supplementary materialThe online version of this article (doi:10.1007/s00425-014-2083-5) contains supplementary material, which is available to authorized users.

The augmented growth equation introduced by Ortega is solved for the apical portion of the pollen tube as an oscillating volume, which we approach in the framework of a two-fluid model in which the two fluids represent the constant pressure and the fluctuating features of the system. Based on routine Fourier analysis, we calculate the energy spectrum of the oscillating pollen tube, and discuss the resonant frequency problem of growth rate oscillations. We also outline a descriptive model for cell wall thickness fluctuations associated with small, yet regular variations (~ 0.01 MPa) observed in turgor pressure. We propose that pressure changes must lead to the sliding of wall layers, indirectly resulting in a wave of polarization of interlayer bonds. We conclude that pollen tube wall thickness may oscillate due to local variations in cell wall properties and relaxation processes. These oscillations become evident because of low amplitude/high frequency pressure fluctuations δP being superimposed on turgor pressure P. We also show that experimentally determined turgor pressure oscillates in a strict periodical manner. A solitary frequency f0 ≈ 0.066 Hz of these (~ 0.01 MPa in magnitude) oscillations for lily pollen tubes was established by the discrete Fourier transform and Lorentz fit.

It has been interesting that nearly all of the ion activities that have been analysed thus far have exhibited oscillations that are tightly coupled to growth. Here, we present discrete Fourier transform (DFT) spectra with a finite sampling of tip-growing cells and organs that were obtained from voltage measurements of the elongating coleoptiles of maize in situ. The electromotive force (EMF) oscillations (~ 0.1 μV) were measured in a simple but highly sensitive resistor–inductor circuit (RL circuit), in which the solenoid was initially placed at the tip of the specimen and then was moved thus changing its position in relation to growth (EMF can be measured first at the tip, then at the sub-apical part and finally at the shank). The influx- and efflux-induced oscillations of Ca2+, along with H+, K+ and Cl- were densely sampled (preserving the Nyquist theorem in order to ‘grasp the structure’ of the pulse), the logarithmic amplitude of pulse spectrum was calculated, and the detected frequencies, which displayed a periodic sequence of pulses, were compared with the literature data. A band of life vital individual pulses was obtained in a single run of the experiment, which not only allowed the fundamental frequencies (and intensities of the processes) to be determined but also permitted the phase relations of the various transport processes in the plasma membrane and tonoplast to be established. A discrete (quantised) frequency spectrum was achieved for a growing plant for the first time, while all of the metabolic and enzymatic functions of the life cell cycle were preserved using this totally non-invasive treatment.

Plant growth and intracellular H+ ion kinetics are known to be strictly correlated, although the history of this discovery, which is known as the acid-growth hypothesis, has faced many difficulties and provoked a long-lasting discussion. Simultaneous measurements of the plant cell extension and pH of the incubation medium helped to defend the theory and together with some of the newest physics-based models, offered a new insight at the molecular level. This article focuses on both the biological and physical aspects of plant growth in the presence of endogenous auxin. Our aim was to circumvent the experimental and conceptual pitfalls associated with the standard use of cut and/or abraded coleoptile segments. Therefore, we simultaneously investigated the growth of intact seedlings of maize (Zea mays L.) and pH of the incubation medium. The growth rates were measured by applying a non-invasive technique that records time-lapse images of the macroscopic elongation of the coleoptiles, while changes in the pH were monitored using a pH/Ion meter. In the experiments, we intentionally introduced growth stimulators: indole-3-acetic acid (IAA), fusicoccin (FC), gibberellic acid (GA3), and a growth inhibitor cadmium chloride (CdCl2), in order to analyse the resultant effect of both exogenous and endogenous factors.

In the ''Conclusions'' section an accidental erroneous occurrence of ''influx rate'' should be replaced by ''efflux rate''. The corrected sentence reads: ''Recently, the presented approach was successfully applied to quantitatively describe effective diffusion rates and correlations between growth and proton efflux rates in Zea mays L. (Pietruszka and Haduch-Sendecka 2016).'' The online version of the original article can be found under

This work presents the application of several our own novel methods of analysing the kinetics of plant growth, which create, among others, a common platform for the comparison of experimental results. A relatively simple formula is used to parameterize the wide range of data that has been obtained for Zea mays L. in the literature, though it can also be used for different species. A biophysical/biochemical interpretation of the parameters was obtained from a theoretical model that is based on a modified Lockhart equation. The derived formula, which was extended for practical use in Zajdel et al. (Acta Physiol Plant 38:5, 2016), and which was implemented in the attached computer program (ibid.), allowed the data that was obtained from the growth-related problems to be parameterized in a simple way. As a working example that shows the robustness of our approach, we comment in detail on the qualitative assessments of the impact of chloride ions on auxin-induced growth. We note that calculated continuous curves (fits), which are rooted in the growth functional that was introduced by Pietruszka (J Theor Biol 315:119–127, 2012), were in a perfect agreement (R2 ~ 0.99998) with the raw experimental data that was published recently by Burdach et al. (Ann Bot 114:1023–1034, 2014). This fact justified the use of this strict technique, which allows for the determination of kinetic coefficients, to critically evaluate the results and suppositions (claims) therein. Moreover, we calculated the time-delay derivative of elongation growth—pH cross-correlations, and validated the “acid growth hypothesis” in figures by considering, amongst others, the magnitude of the H+-activity of elongation growth (per μm). An empirical constant (field strength), EH+ = Em/(log10 1/aH+ ∙ μm) = 0.157 ± 0.009 [V/mm] was obtained, where Em [mV] is the membrane potential in the perenchymal coleoptile cells of Zea mays L. When this relation is known, the membrane potential can not only be determined for intact growth, but also for different intervening substances exclusively from growth (or growth rate) and pH measurements, i.e. without performing electrophysiological measurements. However, the question of whether this constant is universal remains open.Electronic supplementary materialThe online version of this article (doi:10.1186/s40064-016-3626-y) contains supplementary material, which is available to authorized users.