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. "Symbols are very adept at hiding the truth." (Dan Brown)

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We investigate theoretically and next experimentally a new possibility to detect critical temperatures of solids by means of a very simple electrical circuit consisting of an analyzed sample (exhibiting phase transitions) and a contact electrode (hereafter reference electrode) where the constant voltage is applied only to the latter one. The measured system is placed into a thermostat and the electric current flow through the reference electrode is measured as function of temperature. By assuming a model Hamiltonian for the probed sample describing ferromagnetic, superconducting or reentrant phase transitions and a one-band model for the contact electrode we calculate d.c. conductivity of the reference electrode. The temperature dependence of the conductivity of this electrode clearly indicates (in the form of kinks) the transition temperatures connected with phase transitions occurring in the investigated material. This is due to the fact that the chemical potential of the whole system in contact should equal at equilibrium. Our considerations suggest straightforward application of such a circuit in a direct laboratory praxis, especially because (beyond simplicity) the applied method possesses unlimited temperature range and can be considered as noninvasive with respect to the investigated sample. To verify the effect experimentally we have used as an investigated sample an antiferromagnetic Cr material and Cu as the reference electrode. The measurements of the resistivity R(Cr + Cu) and R(Cu) alone as functions of temperature made a possibility to plot the difference R(Cr + Cu) -R(Cu) vs temperature. This plot enabled to identify the critical Neel temperature of the Cr sample corresponding to the profound minimum in this curve.

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The pH/T duality of acidic pH and temperature (T) action for the growth of grass shoots was examined in order to derive the phenomenological equation of wall properties for living plants. By considering non-meristematic growth as a dynamic series of state transitions (STs) in the extending primary wall, the critical exponents were identified, which exhibit a singular behaviour at a critical temperature, critical pH and critical chemical potential (μ) in the form of four power laws: $$f_{\pi } \left( \tau \right) \propto \left| \tau \right|^{\beta - 1}$$ f π τ ∝ τ β - 1 , $$f_{\tau } (\pi ) \propto \left| \pi \right|^{1 - \alpha }$$ f τ ( π ) ∝ π 1 - α , $$g_{\mu } (\tau ) \propto \left| \tau \right|^{ - 2 - \alpha + 2\beta }$$ g μ ( τ ) ∝ τ - 2 - α + 2 β and $$g_{\tau } (\mu ) \propto \left| \mu \right|^{2 - \alpha }$$ g τ ( μ ) ∝ μ 2 - α . The indices α and β are constants, while π and τ represent a reduced pH and reduced temperature, respectively. The convexity relation α + β ≥ 2 for practical pH-based analysis and β ≡ 2 “mean-field” value in microscopic (μ) representation were derived. In this scenario, the magnitude that is decisive is the chemical potential of the H+ ions, which force subsequent STs and growth. Furthermore, observation that the growth rate is generally proportional to the product of the Euler beta functions of T and pH, allowed to determine the hidden content of the Lockhart constant Ф. It turned out that the pH-dependent time evolution equation explains either the monotonic growth or periodic extension that is usually observed—like the one detected in pollen tubes—in a unified account.

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A simple electrical set-up to detect phase transitions is proposed and applied to a series of investigated samples: Gd, Cr, TiNi and CuZnSn, all exhibiting different types of phase transitions. The sample is glued to a contact electrode (here Ag) and immersed into a thermostat. We measure the electrical resistivity of the contact electrode as function of temperature. Due to the fact that the chemical potentials of the sample and the contact electrode should be equal (µ s = µ e ) the electron gas of the contact electrode "feels" a phase transition taking place in the investigated sample. Therefore we observe kinks in the resistivity plot of the contact electrode which easily localize the proper critical temperatures: T C (Gd), T N (Cr) and T struct (TiNi, CuZnSn). The proposed method visualizes the prevailing role of the chemical potential at phase transitions and provides a completely new ("remote") tool to detect critical points in solids. IntroductionThe chemical potential related to the electron gas of a system can be considered as a global quantity able to reflect all critical (or characteristic) points connected with different types of phase transitions (transformations) possible to appear in solids with varying temperature, pressure, concentration, etc. . Direct observation of the chemical potential critical behaviour has been performed by the use of electrochemical cells [21,22]. A quite new approach anticipated theoretically [25] is based on the temperature dependence of the resistivity of the contact electrode connected to a sample exhibiting phase transitions. Due to this method the critical temperatures of the sample should be observed as subtle but measurable kinks in the resisivity plot of the reference electrode. Astonishigly enough, the effect really exists. Here we report on some direct observations which allow to localize critical temperatures connected with different types of phase transitions taking place in Gd, Cr, TiNi and CuZnSn materials.