Expansive Growth vs. pH Reflects a Poisson Point Process of Binding/Unbinding Events in Plant Cell Walls

The paramount role of $$\mathrm{pH}$$ and temperature $$\left(T\right)$$ in the expansive growth of a plant coleoptile/hypocotyl non-meristematic zone or plant and fungal cells was examined within the framework of the underlying chemical bond statistics in order to reproduce an experimental plot of growth vs. $$\mathrm{pH}$$ . Here, according to the definition, $$\mathrm{pH}=\mathrm{pH}\left({\mu }_{{\mathrm{H}}^{+}}\left(T\right), T\right)$$ is considered as a function of the chemical potential of the H+ (hydronium) ions ( $${\mu }_{{\mathrm{H}}^{+}})$$ , as well as an implicit and explicit function of $$T$$ . The derivation of the $$\mathrm{pH}$$ and $$T$$ dependent expansive growth distribution from the Poisson statistics of the “tethers” that reproduce the chemical bonds between microfibrils was determined. The probability distribution for the attachment/detachment/reattachment events of the tethers that are connected to the microfibrils in the elongation zone was obtained. The two distinct but interrelated modes of the expansive growth, which are known as “acid growth” and “auxin growth” were distinguished in the analytic model, while the acid growth hypothesis was verified and confirmed at the semi-empirical and microscopic levels for the first time. Moreover, further perspectives, in which the macroscopic variables $$\left(P, V, T\right)$$ with $$P$$ standing for the turgor pressure and $$V$$ for the cell volume, and the microscopic variables, $${E}^{{\varvec{d}},{\varvec{r}}}$$ , which represent the binding energies of the detachment/reattachment events at the expense of ATP energy, and $${\mu }_{{\mathrm{H}}^{+}}$$ can occur simultaneously, were identified. With a few assumptions that are partly based on experimental data it was possible to synthesise a link between the microscopic, explicit statistical explanation of bond dynamics and the macroscopic rheological properties of the cell wall at a given $$\mathrm{pH}$$ and temperature. A statistical description that predicted the importance of $$\mathrm{pH}$$ and temperature-dependent chemical potential of the H+ ions in microscopic events that result in growth would be supposedly applicable across scales.