A programming toolbox for calculating beta-Euler shape exponents from plant growth data.

Since the acid growth theory was introduced in plant physiology and mainframe computers became more widely available in the mid-20th century, there has been a growing need to accurately predict plant cell morphological parameters during growth. This article presents a computer program that uses an original numerical method to solve a highly nonlinear growth equation. The program is written in Python, a popular open-source scientific software environment called CoCalc or SAGE. This program can be used to determine the growth of an individual plant cell or multicellular organ, such as a coleoptile or hypocotyl segment, at the non-meristemic limit. This standalone program is designed to be user-friendly and accessible to all readers, without barriers. With only a few key parameters, including pH and temperature, this program provides a practical set of tools for comparing growth-related experimental data across various areas of plant biology. Additionally, it could be useful in predicting plant growth during assisted migration, particularly in the face of climate change.

Proton-polarized states in DNA.

We observed signatures of a phase transition in the double-stranded DNA fragment of known length and sequences using a non-invasive semiconductor-electrolyte interface technique and statistical physics methods. Observations revealed a coherence peak in the electromotive force and a significant decline in calculated dynamic entropy at a critical temperature and pH. This behavior may arise from the dynamic interaction of proton (H+) pairs with opposite momentum and spin, carrying a charge q=2+ under critical conditions.

Phase coherent quasi-particle formation in biological systems.

The problem of the origin of canonical and aberrant DNA mutations and the contribution of protons to genetic stability is an essential topic in molecular biology. Based on the empirical results, we reconsidered canonical and tautomeric mutations under the two-fluid model of quantum physics. We assumed that the pressure exerted by protons (H+) in the DNA environment, through changes in pH, could alter the concentration ratio of canonical and tautomeric base pairs, which were found to be different at and beyond the criticality level, respectively. We anticipate that the deviation of the cellular system from a specific (critical) temperature at which dynamic entropy reaches a minimum and a critical pH occurs could result in tautomerization and point mutations.

Collective excitations of germinating pollen grains at critical points.

In plants, the germinating pollen grain (pollen tube) is a single, elongated cell that serves as a conduit through which gametes pass. Pollen tubes display a fast growth rate, which under certain conditions, changes periodically and is accompanied by ion exchange with the growth environment. Therefore, pollen tubes exposed to various abiotic conditions may adversely affect or improve their reproductive performance and fertility. We examined a collection of live pollen tubes of tobacco (Nicotiana tabacum L.) and hyacinth (Hyacinthus orientalis L.) using a non-invasive semiconductor-electrolyte interface technique in the vicinity of the germination temperature or optimum growth temperature of a pollen grains/tubes. The time series measurements and numerical calculations, performed using information theory methods, represent signatures of collective dynamics in living cells at critical-molecularly encoded-germination and growth temperatures. This method (and soil pH data) can facilitate assisted plant migrations from one ecosystem to another as the Earth faces climate change.

Collective excitations at non-equilibrium phase transition in metabolically active red blood cells.

Collective excitations of superconductors and superfluids have been extensively studied in condensed matter physics, while recent experimental advances have made it possible to study the non-equilibrium dynamics of human blood. Here, we show that some dynamic quantitative metrics calculated for the ion fluxes of two isolated peripheral blood droplets that were spatially separated by the presence of a semiconductor exhibited the characteristic features of a quasi-particle (or collective excitation) at a critical point. In the experiment, the spontaneous peak, which indicates order, appeared at a physiological (hereafter: critical) temperature of 36 °C in the human blood. The ordering effect, which was still present in the weak magnetic field of 350 mT, disappeared above the critical magnetic field of approximately 500 mT, suggesting a dynamic Meissner effect in the system (henceforth "dynamic" means derived from the "time series" - a series of real numbers). Moreover, a superconducting gap ratio of approx. 2.91 was found below the upper limit (4) of the BCS theory for weak coupling. Both these effects indicate the existence of a "superconducting" (ion) environment that is conducive to the emergence of quasiparticles. While the dynamic structure of the time series is substantially isotropic at temperatures beyond the phase transition, the system undergoes symmetry breakdown and non-equilibrium phase transition at a critical state. The designated series of dynamic variables can be used in medicine, inter alia, in screening tests as new indicators describing the patient's health.

Non-equilibrium phase transition at a critical point of human blood.

Blood is the basic medium in the existence, evolution and physiological balance of animals and represents the biochemical "Internet" of the body; at least human blood exhibit the presence of an emergent phase that is highly unusual. Homeostasis, the state of the optimal functioning of the body, is maintained in living organisms by many chemical and physical conditions, particularly temperature. However, no regulatory mechanism has been identified that has led to a predetermined (molecularly encoded) optimal, individually variable, very specific temperature of around 36 °C. Additionally, the homeostatic temperature range, which is kept within predetermined limits, is merely an empirical fact. In the following, I will show that the reference temperature that is necessary to achieve homeostasis can be established, and a preset homeostatic range can be determined, using an original experimental method and refined tools of mathematical physics related to the nonlinear measures of the complexity of human blood. Moreover, signatures of a macroscopic coherent state in a non-equilibrium system at a critical temperature are obtained.

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

Detecting the presence of an irregularity/regularity or chaos in the ion flows of an evolving plant cell is an important task that can be unraveled by performing the analyses by different metrics. Here I show that the results of the advanced fluctuation estimation methods that are obtained from the time series that is generated by the extracellular ion fluxes of tobacco pollen tubes (Nicotiana tabacum L.) have long-range correlations at critical temperatures. Further experimental evidence has been found to support the claim that the autonomous growth organization of extreme plant cell expansion is accomplished by self-organizing criticality (SOC), which is an orchestrated instability that occurs in an optimally evolving cell. The temperature-induced synchronous action of the ionic fluxes that are manifested, inter alia, by minimal dynamic entropy enabled the molecularly encoded information about germination and optimal growth temperatures of tobacco pollen tubes to be determined.

Dynamic entropy of human blood.

Temperature control is a process that is used by biological systems to maintain a stable internal state for survival. People have an individually variable physiological temperature of about 36.6 °C, which can be modified by many undesirable factors. Based on an analysis of a time series of extracellular ionic fluxes that were obtained using the non-invasive solute-semiconductor interface technique, I show that this extremely specific (critical) temperature is encoded by a local minimum in the dynamic entropy of an isolated drop of human blood. Moreover, a dynamic zeroth-order normal fluid/"superfluid" nonequilibrium phase transition, which was reflected by a spontaneous symmetry breaking that occurred in the phase space, was revealed. The critical scaling of the dynamic measures for the covariates such as the spectral signature and Lyapunov exponent was also determined.

Extracellular ionic fluxes suggest the basis for cellular life at the 1/f ridge of extended criticality.

The criticality hypothesis states that a system may be poised in a critical state at the boundary between different types of dynamics. Previous studies have suggested that criticality has been evolutionarily selected, and examples have been found in cortical cell cultures and in the human nervous system. However, no one has yet reported a single- or multi-cell ensemble that was investigated ex vivo and found to be in the critical state. Here, the precise 1/f noise was found for pollen tube cells of optimum growth and for the physiological ("healthy") state of blood cells. We show that the multi-scale processes that arise from the so-called critical phenomena can be a fundamental property of a living cell. Our results reveal that cell life is conducted at the border between order and disorder, and that the dynamics themselves drive a system towards a critical state. Moreover, a temperature-driven re-entrant state transition, manifest in the form of a Lorentz resonance, was found in the fluctuation amplitude of the extracellular ionic fluxes for the ensemble of elongating pollen tubes of Nicotiana tabacum L. or Hyacintus orientalis L. Since this system is fine-tuned for rapid expansion to reach the ovule at a critical temperature which results in fertilisation, the core nature of criticality (long-range coherence) offers an explanation for its potential in cell growth. We suggest that the autonomous organisation of expansive growth is accomplished by self-organised criticality, which is an orchestrated instability that occurs in an evolving cell.

Determination of selected dynamic quantities of growing intact seeds of maize.

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.

Single measurement detection of individual cell ionic oscillations using an n-type semiconductor - electrolyte interface.

Pollen tubes are used as models in studies on the type of tip-growth in plants. They are an example of polarised and rapid growth because pollen tubes are able to quickly invade the flower pistil in order to accomplish fertilisation. How different ionic fluxes are perceived, processed or generated in the pollen tube is still not satisfactorily understood. In order to measure the H+, K+, Ca2+ and Cl- fluxes of a single pollen tube, we developed an Electrical Lab on a Photovoltaic-Chip (ELoPvC) on which the evolving cell was immersed in an electrolyte of a germination medium. Pollen from Hyacinthus orientalis L. was investigated ex vivo. We observed that the growing cell changed the (redox) potential in the medium in a periodic manner. This subtle measurement was feasible due to the effects that were taking place at the semiconductor-liquid interface. The experiment confirmed the existence of the ionic oscillations that accompany the periodic extension of pollen tubes, thereby providing - in a single run - the complete discrete frequency spectrum and phase relationships of the ion gradients and fluxes, while all of the metabolic and enzymatic functions of the cell life cycle were preserved. Furthermore, the global 1/fα characteristic of the power spectral density, which corresponds to the membrane channel noise, was found.

Frequency-associated transition from single-cell asynchronous motion to monotonic growth.

This paper presents a Fourier analysis of the Ortega equation that examines the growth dynamics of plants, specifically the pollen tubes or non-meristematic zones of elongating coleoptiles. A frequency-induced transition from highly nonlinear (periodical) growth-like the one observed in pollen tubes-to monotonically ascending and asymptotically saturated (sigmoid-like) growth, which is anticipated within the framework of a 'two-fluid model', is shown. A dynamic phase diagram is calculated and presented in the form of a live video clip.

A quantitative report on the impact of chloride on the kinetic coefficients of auxin-induced growth: a numerical contribution to the "acid growth hypothesis".

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.

Correction: Ion Frequency Landscape in Growing Plants.

[This corrects the article DOI: 10.1371/journal.pone.0138839.].

Ion Frequency Landscape in Growing Plants.

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.

Pressure-induced wall thickness variations in multi-layered wall of a pollen tube and Fourier decomposition of growth oscillations.

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.

Power spectrum, growth velocities and cross-correlations of longitudinal and transverse oscillations of individual Nicotiana tabacum pollen tube.

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.

Pressure-induced cell wall instability and growth oscillations in pollen tubes.

In the seed plants, the pollen tube is a cellular extension that serves as a conduit through which male gametes are transported to complete fertilization of the egg cell. It consists of a single elongated cell which exhibits characteristic oscillations in growth rate until it finally bursts, completing its function. The mechanism behind the periodic character of the growth has not been fully understood. In this paper we show that the mechanism of pressure--induced symmetry frustration occurring in the wall at the transition-perimeter between the cylindrical and approximately hemispherical parts of the growing pollen tube, together with the addition of cell wall material, is sufficient to release and sustain mechanical self-oscillations and cell extension. At the transition zone, where symmetry frustration occurs and one cannot distinguish either of the involved symmetries, a kind of 'superposition state' appears where either single or both symmetry(ies) can be realized by the system. We anticipate that testifiable predictions made by the model (f is proportional to √P) may deliver, after calibration, a new tool to estimate turgor pressure P from oscillation frequency f of the periodically growing cell. Since the mechanical principles apply to all turgor regulated walled cells including those of plant, fungal and bacterial origin, the relevance of this work is not limited to the case of the pollen tube.

Persistent symmetry frustration in pollen tubes.

Pollen tubes are extremely rapidly growing plant cells whose morphogenesis is determined by spatial gradients in the biochemical composition of the cell wall. We investigate the hypothesis (MP) that the distribution of the local mechanical properties of the wall, corresponding to the change of the radial symmetry along the axial direction, may lead to growth oscillations in pollen tubes. We claim that the experimentally observed oscillations originate from the symmetry change at the transition zone, where both intervening symmetries (cylindrical and spherical) meet. The characteristic oscillations between resonating symmetries at a given (constant) turgor pressure and a gradient of wall material constants may be identified with the observed growth-cycles in pollen tubes.

A biosynthesis/inactivation model for enzymatic WLFs or non-enzymatically mediated cell evolution.

We perform the analysis of influence of a 'wall-loosening factor' (hereafter: WLF) activity in cases of isotropic or anisotropic growth of a plant cell/organ. We further explore a generalized form of the Lockhart/Ortega type of equation and make the 'extensibility' Φ (and the yield stress Y) a time and space dependent parameter, able to report on changing (location-dependent) viscoelastic cell wall properties. This procedure results in scalar and tensor equations, which model WLF-mediated isotropic/anisotropic loosening of polymers composing plant cell walls, thereby allowing pressure-driven polymer creep and plant cell expansion growth. An application to six empirical situations, which temporally and spatially vary the amount of WLFs in the cell wall, is anticipated. Combining the resulting explicit formulae with a curve fitting routine provides a new analytical tool that may relate to physiology and biochemistry of the growth process. It is shown, that the regression lines calculated for the derived growth functions perfectly fit (R(2)~/=0.99998) the experimental data.