pH/$T$ duality - wall properties and time evolution of plant cells

TL;DR

This paper explores the duality between pH and temperature in plant cell growth, proposing a phase transition framework with critical exponents and scaling laws to describe the growth dynamics and wall properties.

Contribution

It introduces a novel phase transition model for plant cell growth, linking pH and temperature with critical exponents and power laws, bridging biology with physical phase transition theory.

Findings

Identified critical exponents for plant growth related to pH and temperature.

Derived scaling relations and convexity inequalities for growth analysis.

Proposed a chemical potential-driven equation of state for living plants.

Abstract

We examined the pH/$T$ (or $\mu$/$T$) duality of acidic pH and temperature ($T$) for the growth of grass shoots in order to determine the equation of state (EoS) for living plants. By considering non-meristematic growth as a dynamic series of 'state transitions' (STs) in the extending primary wall, we identified the critical (read: optimum) exponents for this phenomenon, which exhibit a singular behaviour at a critical temperature, critical pH and critical chemical potential ($\mu$) in the form of four power laws: $F(\tau)_\pi\propto|\tau|^{\beta-1}$, $F(\pi)_\tau\propto|\pi|^{1-\alpha}$, $G(\tau)_\mu\propto|\tau|^{-2-\alpha+2\beta}$ and $G(\mu)_\tau\propto|\mu|^{2-\alpha}$. The power-law exponents $\alpha$ and $\beta$ are numbers, which are independent of pH (or $\mu$) and $T$ that are known as critical exponents, while $\pi$ and $\tau$ represent a reduced pH and reduced temperature, respectively. Various 'scaling' predictions were obtained - a convexity relation $\alpha + \beta \ge 2$ for practical pH-based analysis and a $\beta \equiv 2$ identity in microscopic representation. In the presented scenario, the magnitude that is decisive is the chemical potential of H$^+$ ions (protons), enforcing subsequent STs and growth. The EoS span areas of the biological, physical, chemical and Earth sciences cross the borders with the language (adapted formalism) of phase transitions.