Chemical Potential-Induced Wall State Transitions in Plant Cell Growth

Abstract

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 \u2009+\u2009 β \u2009≥\u20092 for practical pH-based analysis and β \u2009≡\u20092 “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.