General and analytic solutions of the Ortega equation.

Ortega (1985) proposed a simple first-order differential equation to describe plant cell extension and, unlike the Lockhart equation (Lockhart, 1965), it takes into account elastic deformation of the cell wall. Cosgrove (1985) and Ortega (1985) solved the equation for the particular case of a previously growing plant cell that is deprived of its water source; thus, GR 5 0. The results (confirmed by experiments) show that P(t) decreases exponentially to the turgor threshold Y due to the cell wall-loosening process. In this article, their results are extended to the more general case, where water absorption is included; thus, GR 61⁄4 0. The problem of the growth rate change in the course of time is also considered here. For the large-scale period (days), growth is well described by the sigmoid curve and consists of three phases: acceleration, linear growth with maximal velocity, and cessation of cell elongation (Fogg, 1975; Schopfer and Mohr, 1995). Then, from the mathematical point of view, the growth rate is of the type of ;t exp(2t) (a time derivative of the sigmoid curve). This function, however, leads to nonanalytical solutions with no clear interpretations. In this study, the time-dependent growth rate GR(t) is modeled with a mathematical function that approximates the exact function very accurately. Importantly, the new approximate function leads to analytical solutions with new predictions for the behavior of P(t) and valuable interpretations of the parameters within the solutions.