Mathematical Modelling of Auxin Transport in Plant Tissues:Flux Meets Signalling and Growth

Plant hormone auxin has critical roles in plant growth, dependent on its heterogeneous distribution in plant tissues. Exactly how auxin transport and developmental processes such as growth coordinate to achieve the precise patterns of auxin observed experimentally is not well understood. Here we use mathematical modelling to examine the interplay between auxin dynamics and growth and their contribution to formation of patterns in auxin distribution in plant tissues. Mathematical models describing the auxin-related signalling pathway, PIN and AUX1 dynamics, auxin transport, and cell growth in plant tissues are derived. A key assumption of our models is the regulation of PIN proteins by the auxin-responsive ARF-Aux/IAA signalling pathway, with upregulation of PIN biosynthesis by ARFs. Models are analysed and solved numerically to examine the long-time behaviour and auxin distribution. Changes in auxin-related signalling processes are shown to be able to trigger transition between passage and spot type patterns in auxin distribution. The model was also shown to be able to generate isolated cells with oscillatory dynamics in levels of components of the auxin signalling pathway which could explain oscillations in levels of ARF targets that have been observed experimentally. Cell growth was shown to have influence on PIN polarisation and determination of auxin distribution patterns. Numerical simulation results indicate that auxin-related signalling processes can explain the different patterns in auxin distributions observed in plant tissues, whereas the interplay between auxin transport and growth can explain the `reverse-fountain' pattern in auxin distribution observed at plant root tips

Localised auxin peaks in concentration-based transport models for plants

We study the existence and bifurcation structure of stationary localised auxin spots in concentration-based auxin-transport models posed on one- and two-dimensional networks of plant cells. In regular domains with small active transport coefficient and no diffusion, the geometry of the cellular array encodes the peaks' height and location: asymptotic calculations show that peaks arise where cells have fewer neighbours, that is, at the boundary of the domain. We perform numerical bifurcation analysis for a concrete model available in literature and provide numerical evidence that the mechanism above remains valid in the presence of diffusion in both regular and irregular arrays. Using the active transport coefficient as bifurcation parameter, we find snaking branches of localised solutions, with peaks emerging from the boundary towards the interior of the domain. In one-dimensional regular arrays we observe oscillatory instabilities along the branch. In two-dimensional irregular arrays the snaking is slanted, hence stable localised solutions with peaks exist in a wide region of parameter space: the competition between active transport and production rate determines whether peaks remain localised or cover the entire domain

Fluctuations in auxin levels depend upon synchronicity of cell divisions in a one-dimensional model of auxin transport

Auxin is a well-studied plant hormone, the spatial distribution of which remains incompletely understood. Here, we investigate the effects of cell growth and divisions on the dynamics of auxin patterning, using a combination of mathematical modelling and experimental observations. In contrast to most prior work, models are not designed or tuned with the aim to produce a specific auxin pattern. Instead, we use well-established techniques from dynamical systems theory to uncover and classify ranges of auxin patterns as exhaustively as possible as parameters are varied. Previous work using these techniques has shown how a multitude of stable auxin patterns may coexist, each attainable from a specific ensemble of initial conditions. When a key parameter spans a range of values, these steady patterns form a geometric curve with successive folds, often nicknamed a snaking diagram. As we introduce growth and cell division into a one-dimensional model of auxin distribution, we observe new behaviour which can be explained in terms of this diagram. Cell growth changes the shape of the snaking diagram, and this corresponds in turn to deformations in the patterns of auxin distribution. As divisions occur this can lead to abrupt creation or annihilation of auxin peaks. We term this phenomenon ‘snake-jumping’. Under rhythmic cell divisions, we show how this can lead to stable oscillations of auxin. We also show that this requires a high level of synchronisation between cell divisions. Using 18 hour time-lapse imaging of the auxin reporter DII:Venus in roots of Arabidopsis thaliana, we show auxin fluctuates greatly, both in terms of amplitude and periodicity, consistent with the snake-jumping events observed with non-synchronised cell divisions. Periodic signals downstream of the auxin signalling pathway have previously been recorded in plant roots. The present work shows that auxin alone is unlikely to play the role of a pacemaker in this context

Localised auxin peaks in concentration-based transport models for plants

We study the existence and bifurcation structure of stationary localised auxin spots in concentration-based auxin-transport models posed on one- and two-dimensional networks of plant cells. In regular domains with small active transport coefficient and no diffusion, the geometry of the cellular array encodes the peaks' height and location: asymptotic calculations show that peaks arise where cells have fewer neighbours, that is, at theboundary of the domain. We perform numerical bifurcation analysis for a concrete model available in literature and provide numerical evidence that the mechanism above remains valid in the presence of diffusion in both regular and irregular arrays. Using the active transport coefficient as bifurcation parameter, we find snaking branches of localised solutions, with peaks emerging from the boundarytowards the interior of the domain. In one-dimensional regular arrays we observe oscillatory instabilities along the branch. In two-dimensional irregular arrays the snaking is slanted, hence stable localised solutions with peaks exist in a wide region of parameter space: the competition between active transport and production rate determines whether peaks remain localised or cover the entire domain

Chemical Potential‑Induced Wall State Transitions in Plant Cell Growth

The pH/T duality of acidic pH and temperature (T) action for the growth of grass shoots was examined in order to derive the phenomenological equation of wall properties for living plants. By considering non-meristematic growth as a dynamic series of state transitions (STs) in the extending primary wall, the critical exponents were identified, which exhibit a singular behaviour at a critical temperature, critical pH and critical chemical potential (μ) in the form of four power laws: f ( ) ∝ −1 , f ( ) ∝ 1− , g ( ) ∝ −2− +2 and g ( ) ∝ 2− . The indices α and β are constants, while π and τ represent a reduced pH and reduced temperature, respectively. The convexity relation α + β ≥ 2 for practical pH-based analysis and β ≡ 2 “meanfield” value in microscopic (μ) representation were derived. In this scenario, the magnitude that is decisive is the chemical potential of the H+ ions, which force subsequent STs and growth. Furthermore, observation that the growth rate is generally proportional to the product of the Euler beta functions of T and pH, allowed to determine the hidden content of the Lockhart constant Ф. It turned out that the pH-dependent time evolution equation explains either the monotonic growth or periodic extension that is usually observed—like the one detected in pollen tubes—in a unified account

Modelling of self-organizaton of microtubules in plant cells

Microtubules are ubiquitous elements of any eucaryotic cell, serving many functions at different stages of its life. In plant cells they form so-called plant cell cortex, where they are organized into parallel arrays. These arrays serve as a matrix of synthesis of a plant cell wall, defining the direction of growth. Microtubule arrays are sensible to tropic stimuli. However, the nature of such sensibility is still not well established. We provide a computational analysis of this phenomenon. Using both kinetic Monte-Carlo simulations and theoretical investigation, we show that compression due to mechanical stress may cause orientation of microtubules along major stress lines. We also show that anisotropic distribution of chemical agents interacting with microtubule-associated proteins may also cause orientation of microtubules. Such mechanisms are primarily connected with gravitropism but similar reorientations of microtubules in response to light may suggest that these mechanisms can also be relevant for other tropisms.Los microtúbulos son elementos ubicuos de cualquier célula eucariota, que poseen distintas funciones en diferentes etapas de su vida. En células de plantas ellos forman la estructura llamada corteza célula vegetal, donde se organizan en conjuntos paralelos. Estos conjuntos sirven como una matriz de síntesis de una pared celular vegetal, definiendo la dirección del crecimiento de la célula. Los conjuntos de microtúbulos son sensibles a estímulos trópicos. Sin embargo, la naturaleza de dicha sensibilidad todavía no está bien establecida. Les ofrecemos un análisis computacional de este fenómeno. Usando simulaciones cinéticas Monte-Carlo y la investigación teórica, nos mostramos que la compresión debida al estrés mecánico puede causar la orientación de los microtúbulos a lo largo de las principales líneas de tensión. También se muestra que la distribución anisotrópica de agentes químicos que interactúan con las proteínas asociadas a los microtúbulos también puede provocar orientación de los microtúbulos. Estos mecanismos están conectados principalmente con gravitropismo pero reorientaciones similares de microtúbulos en respuesta a la luz puede sugerir que estos mecanismos también pueden ser relevantes para otros tropismos

Transport phenomena in plant-internal processes: growth and carbon dioxide transport

Aim of the here presented work was the quantitative modeling of plant-internal processes. Growth of cells and tissues was one of the central themes, although the lateral transport of carbon dioxide (CO2 ) was also treated. These processes depend strongly on fluxes of water, hormones and/or CO2 . Thus, suitable transport equations were sought for to describe these processes. Using the Lockhart-Equations, which are well known in biology to describe the growth of a whole cell, local formulations of energy and mass conservation were obtained. These formulations can be used to determine local growth patterns in cells. This was shown through a numerical example of a spherical cell. Finally, the conservation equations found, were shown to be consistent with the empirical Lockhart-Equations. Plant organs, such as roots and hypocotyls, have spatial and temporal growth patterns. For example, the spatial distributions of growth in primary roots is given by a bell-shaped distribution along the organ axis. This particular one dimensional growth pattern was modeled here through the transport of two hypothetical phytohormones and using the Lockhart-Equations as the underlying growth equations. Because the hypothetical hormones were chosen to have auxin and cytokinin (two of the most important plant hormones) properties, the model stays in a plant physiological context. Not only one dimensional growth patterns are found in roots and hypocotyls. These tend to have organ curvature and torsion, as becomes clear particularly in tropisms (e.g. gravitropism, hydrotropisms and phototropism). Although these processes are known for a long time in biology, no suitable measures to characterize the production of curvature and torsion have been defined. Using a curvature and torsion conservation equation, a measure for their production was found here. These measures were then exemplified in a simple model of the root gravitropic reaction, and applied in the characterization of the gravitropic reaction of Arabidopsis thaliana (L.) Heynh. wild-type and pin3 mutant roots. The gravitropic reaction is believed to be regulated by the hormone auxin. pin3 mutants are deficient in the PIN3 protein, which is essential in the transport of auxin in the root tip. Through comparison of the reaction of wild-type and pin3 roots, it was shown here that the gravitropic reaction is not solely regulated by auxin, so that other regulation mechanisms need to exist. Finally, transport equations were found, which describe the transport and assimilation of CO2 in leaves. Using gas-exchange and chlorophyll fluorescence measurements, the homogenized lateral diffusion coeffcient of leaves was determined. Moreover, the strategy behind the existence of lateral diffusion in leaves was discussed (plants differ in the porosity of their leaves). Throughout the work presented here, it became clear how fructiferous the application of transport equations in biology is. The importance of a quantitative description in biology became also clear. Everyday new questions arise in biology. An answer to these may only be found using an interdisciplinary approach

Cellular patterns during leaf development

A major problem in developmental biology is to understand how the behaviour of individual cells creates reproducible biological shapes. Moreover, the reproducibility of form also happens at the cellular level and cellular patterns are evident in the temporal as well as the spatial scales. Understanding the principles underlying these cellular patterns will contribute to linking the individual cell dynamics to the collective phenomenon of morphogenesis. The relatively flat shape, the absence of cell migration and apoptosis makes the leaf of Arabidopsis thaliana an excellent system to study morphogenesis at the cellular level. However the information about the cellular dynamics that is available has only been inferred indirectly or restricted to few cells. In this thesis, I present some methods that permitted the characterisation of cellular dynamics over long time periods at the organ level. I present the Lobe Contribution Elliptical Fourier Analysis (LOCO-EFA), that enabled the quantification of complex cell geometries and provided a shape profile to evaluate populations of cells. In turn, the shape profile of individual cells was used to develop a tracking algorithm that, integrated with a segmentation algorithm, resulted in a powerful tool to recognize cells in a succession of images and extract cell shape, cell area, cell position and cell lineages. The synergy between in vivo imaging and computational tools permitted the study of cellular patterns at unprecedented resolution. Interestingly, the dynamics of cell growth and cell shape are highly influenced by the cell age and to a limited extend, by their position. Moreover, direct measurement of cell division shows that the division zone is restricted towards the base of the leaf, but is not constant in length and that the frequency of divisions decreases over time in a rather gradual fashion. At single cell level, new events of lobe formation were identified, suggesting that the intracellular patterning underlying the multi-polar pavement cell shape is dynamic and lobes are newly formed rather than being specified at a single time point. The cellular dynamics of growth, shape and divisions using long time-lapse and imaging enabled me to revisit previous hypothesis and propose new ones about the regulation of cellular behaviour during leaf morphogenesis. (The CD-ROM referred to in the thesis contains Movies in AVI format. However these were submitted as separate files which could not be uploaded to the repository. Please contact the author for more information.