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 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 and transport. 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

Gibberellin and abscisic acid transporters facilitate endodermal suberin formation in Arabidopsis

The plant hormone gibberellin (GA) regulates multiple developmental processes. It accumulates in the root elongating endodermis, but how it moves into this cell file and the significance of this accumulation are unclear. Here we identify three NITRATE TRANSPORTER1/PEPTIDE TRANSPORTER (NPF) transporters required for GA and abscisic acid (ABA) translocation. We demonstrate that NPF2.14 is a subcellular GA/ABA transporter, presumably the first to be identified in plants, facilitating GA and ABA accumulation in the root endodermis to regulate suberization. Further, NPF2.12 and NPF2.13, closely related proteins, are plasma membrane-localized GA and ABA importers that facilitate shoot-to-root GA translocation, regulating endodermal hormone accumulation. This work reveals that GA is required for root suberization and that GA and ABA can act non-antagonistically. We demonstrate how the clade of transporters mediates hormone flow with cell-file-specific vacuolar storage at the phloem unloading zone, and slow release of hormone to induce suberin formation in the maturation zone

Modelling removal of sulphur dioxide from flue gas in purification devices

Many chemical filters contain reactive components where harmful substances are removed or transformed. In this thesis, we consider the problem of removal of sulphur dioxide from flue gas using filters made from a porous catalytic medium. This is inspired by a filter, designed by W. L. Gore and Associates, Inc., which converts gaseous sulphur dioxide into liquid sulphuric acid via a chemical reaction that occurs on the surface of microscopic catalytic pellets. During the device operation, the liquid sulphuric acid accumulates within the filter and reduces its efficiency. We derive a series of mathematical models that explore various aspects of this problem and ultimately serve to predict the performance of the device. We begin by considering a fundamental fluid dynamics problem of spreading of a thin film under surface tension with liquid injection due to the chemical reaction. We then formulate a microscale model for the gas and liquid transport within the porous filter material and, separately, present a radially symmetric model for a single catalytic pellet, which we use to estimate the unknown reaction rate constant based on real observations. Although we do not explicitly account for the porous scaffold of the filter, the model set-up is appropriate for the case of hydrophobic material. We move on to upscale the microscale equations to a set of device-scale equations using homogenisation techniques. We obtain numerical solutions and asymptotic predictions for various limits that compare well, and also explore the effect of changing the system parameters, such as the gas speed, on the effective cleansing of flue gas. In addition, we develop a model for two neighbouring pellets, one of which is completely submerged by sulphuric acid, which is useful to understand the long-term behaviour of the system once a continuous layer of liquid forms near the surface of the filter sheets. We also study a simplified problem where the filter is made from hydrophilic material and identify the key differences in the resulting liquid transport. Finally, we present a model for the hygroscopy of sulphuric acid, which helps evaluate the effect of water absorption on the liquid growth within the filter. All the models we develop retain generality and can be applied to other physical and industrial processes.</p

Data for paper "A homogenised model for a reactive filter"

Datasets for the graphs of numerical solutions. The data were created using MATLAB and COMSOL

Data for paper "Maxwell-type models for the effective thermal conductivity of a porous material with radiative transfer in the voids"

Datasets for the graphs of numerical solutions. The data was created using Mathematica

A simple model for the desulphurisation of flue gas using reactive filters

Desulphurisation of flue gas is essential before it can be released safely into the atmosphere. One way of removing sulphur dioxide is to use a purification device incorporating a reactive filter, in which the flue gas stream passes in front of a porous-catalyst-filled structure which converts the gaseous sulphur dioxide into liquid sulphuric acid. In this paper, we build and solve a simple mathematical model to describe the operation of a paradigm reactive filter. Our model captures the transport of sulphur dioxide through the device via advection in the main “outer” flow and diffusion through the catalyst structure along with the production of sulphuric acid. This sulphuric acid gradually accumulates in the filter rendering it less efficient. We determine the clogging time for an individual channel (that is, the time at which the entrance to the channel becomes completely filled with liquid) and explore how the concentrations of sulphur dioxide and oxygen and the thickness of the sulphuric acid layer change as the key dimensionless parameters are varied, comparing numerical and asymptotic results where appropriate. We then turn our attention to the device scale and solve our model numerically to determine the overall lifetime of the device. We vary the key dimensionless parameters and explore how they affect the efficiency of the device. In the physically relevant parameter regime, we find an explicit solution to the outer flow problem which agrees well with numerical solutions and provides a formula for the lifetime of the device. Finally, we propose a formula for determining the catalyst reaction rate, given data on the concentration of sulphur dioxide exiting the device

Data for paper "Simple Model for the Hygroscopy of Sulfuric Acid"

Datasets for the figures with numerical solutions, asymptotic predictions and experimental data. The data was created using Mathematica

Data for paper "Surface-tension- and injection-driven spreading of a thin viscous film"

Datasets for the graphs of numerical solutions. The data was created using MATLAB

A homogenised model for a reactive filter

Many chemical filters contain reactive components where harmful substances are removed or transformed. In this paper, we derive a homogenised model for a flue-gas filter that converts sulphur dioxide into liquid sulphuric acid. We consider a microscale domain, focused on a single catalytic pellet, and homogenise over both the gaseous and the liquid phase to obtain macroscale equations for the concentration of sulphur dioxide and the thickness of the liquid sulphuric acid layer that grows around the pellets. We explore two interesting limits of the homogenised model, in which the reaction rate at the pellet surface is small, and where the mass transfer across the gas–liquid interface is small, respectively. We then couple the macroscale equations to an equation governing the external gas flow through the filter. We solve the resulting model and consider asymptotic reductions based on the filter geometry. We consider two distinguished limits and, for one of them, obtain an explicit solution for the sulphur dioxide concentration and the void fraction in the filter. We vary parameters such as the gas speed and establish the operating regimes for effective cleansing of flue gas

A model for the lifetime of a reactive filter

The data was created using MATLAB to simulate the mathematical models derived in the associated journal article: A model for the lifetime of a reactive filter K.B. Kiradjiev, C.J.W. Breward & I.M. Griffiths 2022 J. Eng. Math. (in press)