We examined the pH/Tduality of acidic pH and temperature (T) for the growth of grass shoots in order to determine the phenomenological equation of wall properties (‘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 exponents’ (read: optimum) for this phenomenon, which exhibit a singular behaviour at a critical temperature, critical pH and critical chemical potential (μ) in the form of four power laws:Fπ(τ)∝|τ|β−1,Fτ(τ)∝|π|1−α,Gμ(τ)∝|τ|−2−α+2βandGτ(μ)∝|μ|2−α. The power-law exponents α and β are numbers that are independent of pH (or μ) and T, which are known as critical exponents, while π and τ represent a reduced pH and reduced temperature, respectively. Various scaling predictions were obtained – the convexity relation α + β ≥ 2 for practical pH-based analysis and a β ≡ 2 identity in a ‘microscopic’ representation. In the presented scenario, the magnitude that is decisive is the chemical potential of the H+ions (protons), which force subsequent STs and growth. Furthermore, we observed that the growth rate is generally proportional to the product of the Euler beta functions ofTand pH, which are used to determine the hidden content of the Lockhart constant Ф. It turned out that the evolution equation, when expressed in terms of the same dynamic set of variables, explains either the monotonic growth or periodic extension that is usually observed – like the one detected in pollen tubes – in a unified account. We suggest that cell growth evolves along the path with the least activity, thereby optimising growth under any physiological conditions. The pH dynamics in close-to-natural conditions appears to essentially be responsible for this extreme trajectory, thereby providing a highly nonlinear pH(t),transformation. Moreover, the drops in pH that are induced by auxin or fusicoccin, when next converted by the augmented Lockhart equation, are enough to explain a significant fraction of the increase in the growth rate. A self-consistent recurring model is proposed to embrace the inherent complexity of such a biological system, in which several intricate pathways work simultaneously, in order to reconcile the conflicting views of plant cell extension and growth. Eventually, we pose the question: Is the chemical potential of protons a master regulator for tip-growing cells?Author summary
In plant development, sudden changes such as cell expansion or pollen tube oscillations seem to depend on a correlative group of events rather than on slow shifts in the apex. Hence, in order to understand or to control the processes in the extending cell wall, we need to unravel the general principles and constraints that govern growth. The quest for these principles has primarily focused on the molecular, though merely descriptive, level. Here, we show that it is possible to analyse oscillatory state changes computationally without even requiring knowledge about the exact type of transition. Our results suggest that the cell wall properties and growth of plant cells can be accurately and efficiently predicted by a set of physical and chemical variables such as temperature, pressure and the dynamic pH of the growing plant, which build a scaffold for more specific biochemical predictions. In this context, we observed that cell growth evolves along the path the least action, thereby optimising growth under any physiological conditions. The model equations that we propose span the fields of the biological, physical, chemical and Earth sciences. The common denominator that ties the growth factors together is the chemical potential of protons, which is possibly a central core-controlling mechanism that is able to produce a macroscopic outcome, i.e. structurally and temporally organised apical growth.