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# Chemical Potential-Induced Wall State Transitions In Plant Cell Growth

Mariusz A. Pietruszka

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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 + 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.