Special Solutions to the Ortega Equation

Abstract

The previously established augmented growth equation (Cosgrove, Plant Physiol 78:347–356, 1985; Ortega, Plant Physiol 79:318–320, 1985), suitable for description of pressure relationships in the growing plant cell, is revised with respect to the inclusion of changing cell wall properties hitherto represented by two constants, Φ and ε, connected with viscoelastic behavior. This phenomenological equation in the modified form is capable of appropriate description of volumetric extensibility, growth rate, and pressure changes in growing plant cells. This concerns deposition of new material in the polymer cell wall intercalation process, but it can also be used successfully for induced cell wall loosening, for example, by expansin EXPA (EXPB) proteins. In this context, a specific shape of the proposed equations, armed with a small number of physiologically explained parameters, opens up an experimental perspective for determining vital numbers connected with interactions at the nanoscale (polymer bonds “dilution” of an extending cell wall), or even at the molecular level in the cell wall (calculating numbers proportional to the expansin molecule’s active surface area). A systematic survey of ready-to-use deterministic solutions originating from the Ortega equation, reporting on both reversible (elastic) and irreversible (plastic) features of a growing plant cell, is presented. These findings also provide a quantitative cytomechanical model able to account for the important role of mechanical properties of the cell wall in cellular growth processes. An important feature of the analytical approach is that all model equations, after calibrating on existing data, allow new results to be inferred without further experimental work.