Multiscale Asymptotic Analysis Reveals How Cell Growth and Subcellular Compartments Affect Tissue-Scale Hormone Transport.

Determining how cell-scale processes lead to tissue-scale patterns is key to understanding how hormones and morphogens are distributed within biological tissues and control developmental processes. In this article, we use multiscale asymptotic analysis to derive a continuum approximation for hormone transport in a long file of cells to determine how subcellular compartments and cell growth and division affect tissue-scale hormone transport. Focusing our study on plant tissues, we begin by presenting a discrete multicellular ODE model tracking the hormone concentration in each cell's cytoplasm, subcellular vacuole, and surrounding apoplast, represented by separate compartments in the cell-file geometry. We allow the cells to grow at a rate that can depend both on space and time, accounting for both cytoplasmic and vacuolar expansion. Multiscale asymptotic analysis enables us to systematically derive the corresponding continuum model, obtaining an effective reaction-advection-diffusion equation and revealing how the effective diffusivity, effective advective velocity, and the effective sink term depend on the parameters in the cell-scale model. The continuum approximation reveals how subcellular compartments, such as vacuoles, can act as storage vessels, that significantly alter the effective properties of hormone transport, such as the effective diffusivity and the induced effective velocity. Furthermore, we show how cell growth and spatial variance across cell lengths affect the effective diffusivity and the induced effective velocity, and how these affect the tissue-scale hormone distribution. In particular, we find that cell growth naturally induces an effective velocity in the direction of growth, whereas spatial variance across cell lengths induces effective velocity due to the presence of an extra compartment, such as the apoplast and the vacuole, and variations in the relative sizes between the compartments across the file of cells. It is revealed that hormone transport is faster across cells of decreasing lengths than cells with increasing lengths. We also investigate the effect of cell division on transport dynamics, assuming that each cell divides as soon as it doubles in size, and find that increasing the time between successive cell divisions decreases the growth rate, which enhances the effect of cell division in slowing hormone transport. Motivated by recent experimental discoveries, we discuss particular applications for transport of gibberellic acid (GA), an important growth hormone, within the Arabidopsis root. The model reveals precisely how membrane proteins that mediate facilitated GA transport affect the effective tissue-scale transport. However, the results are general enough to be relevant to other plant hormones, or other substances that are transported in a similar way in any type of cells.

A biomechanical model of anther opening reveals the roles of dehydration and secondary thickening.

Understanding the processes that underlie pollen release is a prime target for controlling fertility to enable selective breeding and the efficient production of hybrid crops. Pollen release requires anther opening, which involves changes in the biomechanical properties of the anther wall. In this research, we develop and use a mathematical model to understand how these biomechanical processes lead to anther opening. Our mathematical model describing the biomechanics of anther opening incorporates the bilayer structure of the mature anther wall, which comprises the outer epidermal cell layer, whose turgor pressure is related to its hydration, and the endothecial layer, whose walls contain helical secondary thickening, which resists stretching and bending. The model describes how epidermal dehydration, in association with the thickened endothecial layer, creates forces within the anther wall causing it to bend outwards, resulting in anther opening and pollen release. The model demonstrates that epidermal dehydration can drive anther opening, and suggests why endothecial secondary thickening is essential for this process (explaining the phenotypes presented in the myb26 and nst1nst2 mutants). The research hypothesizes and demonstrates a biomechanical mechanism for anther opening, which appears to be conserved in many other biological situations where tissue movement occurs.

A model of crosslink kinetics in the expanding plant cell wall: yield stress and enzyme action.

The plant primary cell wall is a composite material containing stiff cellulose microfibrils that are embedded within a pectin matrix and crosslinked through a network of hemicellulose polymers. This microstructure endows the wall with nonlinear anisotropic mechanical properties and allows enzymatic regulation of expansive cell growth. We present a mathematical model of hemicellulose crosslink dynamics in an expanding cell wall incorporating strain-enhanced breakage and enzyme-mediated crosslink kinetics. The model predicts the characteristic yielding behaviour in the relationship between stress and strain-rate seen experimentally, and suggests how the effective yield and extensibility of the wall depend on microstructural parameters and on the action of enzymes of the XTH and expansin families. The model suggests that the yielding behaviour encapsulated in the classical Lockhart equation can be explained by the strongly nonlinear dependence of crosslink breakage rate on crosslink elongation. The model also demonstrates how enzymes that target crosslink binding can be effective in softening the wall in its pre-yield state, whereas its post-yield extensibility is determined primarily by the pectin matrix.

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

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.

Modelling crystal aggregation and deposition in the catheterised lower urinary tract.

Urethral catheters often become encrusted with crystals of magnesium struvite and calcium phosphate. The encrustation can block the catheter, which can cause urine retention in the bladder and reflux into the kidneys. We develop a mathematical model to investigate crystal deposition on the catheter surface, modelling the bladder as a reservoir of fluid and the urethral catheter as a rigid channel. At a constant rate, fluid containing crystal particles of unit size enters the reservoir, and flows from the reservoir through the channel and out of the system. The crystal particles aggregate, which we model using Becker-Döring coagulation theory, and are advected through the channel, where they continue to aggregate and are deposited on the channel's walls. Inhibitor particles also enter the reservoir, and can bind to the crystals, preventing further aggregation and deposition. The crystal concentrations are spatially homogeneous in the reservoir, whereas the channel concentrations vary spatially as a result of advection, diffusion and deposition. We investigate the effect of inhibitor particles on the amount of deposition. For all parameter values, we find that crystals deposit along the full length of the channel, with maximum deposition close to the channel's entrance.