Solutions for a local equation of anisotropic plant cell growth: an analytical study of expansin activity

Abstract

This paper presents a generalization of the Lockhart equation for plant cell/organ expansion in the anisotropic case. The intent is to take into account the temporal and spatial variation in the cell wall mechanical properties by considering the wall 'extensibility' (F), a time-and space-dependent parameter. A dynamic linear differential equation of a second-order tensor is introduced by describing the anisotropic growth process with some key biochemical aspects included. The distortion and expansion of plant cell walls initiated by expansins, a class of proteins known to enhance cell wall 'extensibility', is also described. In this approach, expansin proteins are treated as active agents participating in isotropic/anisotropic growth. Two-parameter models and an equation for describing a-and b-expansin proteins are proposed by delineating the extension of isolated wall samples, allowing turgor-driven polymer creep, where expansins weaken the non-covalent binding between wall polysaccharides. We observe that the calculated halftime (t 1/2 ¼ eF 0 log 2) of stress relaxation due to expansin action can be described in mechanical terms.

A Statistical Model of Expansive Growth in Plant and Fungal Cells: The Case of Phycomyces

“…7 and 8). The stress relaxation halftime t 1/2 is shown to be related to the average bond dissociation rate k d , consistent with previous work on relating these two quantities (9). These results are of paramount importance because 1) stress relaxation is central to the process of expansive growth in cells and 2) experimental stress relaxation reveals macroscopic constitutive relationship for viscoelastic materials.…”

“…The halftime of exponential decay, i.e., the time taken for the wall force to relax to half of its initial value, can be calculated from Eq. 16 as t 1/2 ¼ ln2/k d , similar to the conclusions of previous work (9). The halftime is often useful in quantifying relaxation in experimental measurements and provides a timescale that can be compared for different species.…”

“…Lockhart's equations have been used in many subsequent experimental investigations (5)(6)(7)(8) that study the relationship between expansive growth rate, turgor pressure, and mechanical properties of the cell wall. They have also been used in their local form as constitutive equations to model complex growth situations (9), including sporangiophore development (10) and ''tip'' growth morphogenesis (11). Though Lockhart's equations for cell wall extension are attractive because of their simplicity, their applicability is limited because they do not consider elastic (reversible) wall deformations.…”

A Statistical Model of Cell Wall Dynamics during Expansive Growth

“…Lockhart's equations have been used in many subsequent experimental investigations [5][6][7][8] that study the relationship between expansive growth rate, turgor pressure, and mechanical properties of the cell wall. They have also been used in their local form as constitutive equations to model complex growth situations 9 , including sporangiophore development 10 and "tip" growth morphogenesis 11 . Though Lockhart's equations for cell wall extension are attractive due to their simplicity, their applicability is limited because they do not consider elastic (reversible) wall deformations.…”

Multiscale Systems Analysis of Root Growth and Development: Modeling Beyond the Network and Cellular Scales

“…However, while being appealingly simple, the Lockhart equation does not contain the level of detail required in continuum mechanical models to describe bodies undergoing bending or twisting deformations in two or three coordinate directions. Even for simple unidirectional expansion, numerous refinements to the Lockhart equation have been suggested (Ortega, 1990;Passioura and Fry, 1992;Dyson and Jensen, 2010;Pietruszka, 2011) that capture specialized features of cell and tissue expansion.…”