Multiscale modelling of auxin transport in the plant-root elongation zone

Abstract

In the root elongation zone of a plant, the hormone auxin moves in a polar manner due to active transport facilitated by spatially distributed influx and efflux carriers present on the cell membranes. To understand how the cell-scale active transport and passive diffusion combine to produce the effective tissue-scale flux, we apply asymptotic methods to a cell-based model of auxin transport to derive systematically a continuum description from the spatially discrete one. Using biologically relevant parameter values, we show how the carriers drive the dominant tissue-scale auxin flux and we predict how the overall auxin dynamics are affected by perturbations to these carriers, for example, in knockout mutants. The analysis shows how the dominant behaviour depends on the cells' lengths, and enables us to assess the relative importance of the diffusive auxin flux through the cell wall. Other distinguished limits are also identified and their potential roles discussed. As well as providing insight into auxin transport, the study illustrates the use of multiscale (cell to tissue) methods in deriving simplified models that retain the essential biology and provide understanding of the underlying dynamics.

Dynamic regulation of auxin oxidase and conjugating enzymes AtDAO1 and GH3 modulates auxin homeostasis

“…18. We treated auxin concentrations as uniform within each cell because of the small size of cells in this region and relatively rapid auxin diffusion within the cytoplasm compared with within the apoplast (18)(19)(20). The small cells near the root tip are in contrast with the larger cells farther from the root tip, where subcellular variations in auxin have previously been considered (21,22).…”

“…In the epidermal, cortical, and endodermal cells, we assume that the auxin concentrations have reached their far-field asymptotic values, hence setting them equal to those in neighboring (rootward) cells of the same type. This assumption implies an appropriate shootward flux of auxin through the outer tissue layers (19). Starting from an initial condition in which all remaining concentrations equal zero, we simulated the ODEs until the concentrations and fluxes reached a steady state.…”

A multi-scale analysis of drug transport and response for a multi-phase tumour model

“…These techniques allow the derivation of effective ordinary or partial differential equations at the tissue scale, that directly incorporate explicit information regarding the microscale in a mathematically precise manner. Examples employing these methods across a range of biological applications include Band and King (2012), Fozard et al (2010), King (2011, 2012), Ptashnyk and Chavarría-Krauser (2010), Roose (2010), andTurner et al (2004). However, of particular interest to the current work is , in which flow and transport equations in a solid tumour and its vasculature are homogenized to obtain a macroscale description of drug transport; and the more recent works of O'Dea et al (2014) and Penta et al (2014) which consider the homogenization of models for growing tissues.…”

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

“…These methods aim to relate tissue-level descriptions to processes that occur at a finer scale, for example, showing how a quantity such as the tissuelevel hormone velocity depends on the detailed cell-level transport processes (Band and King, 2012), and thus can be used to derive simpler descriptions of complex models, essentially moving from a cell-based description to a continuum approximation. While such tissue-level (continuum) descriptions have traditionally been used in top-down approaches, these descriptions have had essentially phenomenological components: Relating the quantities that appear within them to the cellular and subcellular properties is a crucial and challenging aspect of multiscale mechanistic modeling.…”

“…By relating each compartment to its spatial position, one can simplify a system of hundreds of ODEs governing a cell-based model to a few partial differential equations describing the evolving hormone concentration in space and time. Continuum approaches have been explored by Band and King (2012) to analyze how cell-scale PIN and AUX1 distributions affect the auxin velocity through root outer layers. Similar homogenization techniques can also capture auxin transport through subcellular compartments, as implemented by Chavarria-Krauser and Ptashnyk (2010) in analyzing auxin transport through a uniform array of cells.…”