Localized auxin peaks in concentration-based transport models of the shoot apical meristem

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.

Scaling relations for auxin waves

We analyze an 'up-the-gradient' model for the formation of transport channels of the phytohormone auxin, through auxin-mediated polarization of the PIN1 auxin transporter. We show that this model admits a family of travelling wave solutions that is parameterized by the height of the auxin-pulse. We uncover scaling relations for the speed and width of these waves and verify these rigorous results with numerical computations. In addition, we provide explicit expressions for the leading-order wave profiles, which allows the influence of the biological parameters in the problem to be readily identified. Our proofs are based on a generalization of the scaling principle developed by Friesecke and Pego to construct pulse solutions to the classic Fermi-Pasta-Ulam-Tsingou model, which describes a one-dimensional chain of coupled nonlinear springs.

Computational Models of Auxin-Driven Patterning in Shoots

Auxin regulates many aspects of plant development and behavior, including the initiation of new outgrowth, patterning of vascular systems, control of branching, and responses to the environment. Computational models have complemented experimental studies of these processes. We review these models from two perspectives. First, we consider cellular and tissue-level models of interaction between auxin and its transporters in shoots. These models form a coherent body of results exploring different hypotheses pertinent to the patterning of new outgrowth and vascular strands. Second, we consider models operating at the level of plant organs and entire plants. We highlight techniques used to reduce the complexity of these models, which provide a path to capturing the essence of studied phenomena while running simulations efficiently.

Modeling Plant Tissue Development Using VirtualLeaf

Cell-based computational modeling and simulation are becoming invaluable tools in analyzing plant development. In a cell-based simulation model, the inputs are behaviors and dynamics of individual cells and the rules describing responses to signals from adjacent cells. The outputs are the growing tissues, shapes, and cell-differentiation patterns that emerge from the local, chemical, and biomechanical cell-cell interactions. In this updated and extended version of our previous chapter on VirtualLeaf (Merks and Guravage, Methods in Molecular Biology 959, 333-352), we present a step-by-step, practical tutorial for building cell-based simulations of plant development and for analyzing the influence of parameters on simulation outcomes by systematically changing the values of the parameters and analyzing each outcome. We show how to build a model of a growing tissue, a reaction-diffusion system on a growing domain, and an auxin transport model. Moreover, in addition to the previous publication, we demonstrate how to run a Turing system on a regular, rectangular lattice, and how to run parameter sweeps. The aim of VirtualLeaf is to make computational modeling more accessible to experimental plant biologists with relatively little computational background.

The nonlinear initiation of side-branching by activator-inhibitor-substrate (Turing) morphogenesis

An understanding of the underlying mechanism of side-branching is paramount in controlling and/or therapeutically treating mammalian organs, such as lungs, kidneys, and glands. Motivated by an activator-inhibitor-substrate approach that is conjectured to dominate the initiation of side-branching in a pulmonary vascular pattern, I demonstrate a distinct transverse front instability in which new fingers grow out of an oscillatory breakup dynamics at the front line without any typical length scale. These two features are attributed to unstable peak solutions in 1D that subcritically emanate from Turing bifurcation and that exhibit repulsive interactions. The results are based on a bifurcation analysis and numerical simulations and provide a potential strategy toward also developing a framework of side-branching for other biological systems, such as plant roots and cellular protrusions.

Catechol, a major component of smoke, influences primary root growth and root hair elongation through reactive oxygen species-mediated redox signaling

Nicotiana attenuata germinates from long-lived seedbanks in native soils after fires. Although smoke signals have been known to break seed dormancy, whether they also affect seedling establishment and root development remains unclear. In order to test this, seedlings were treated with smoke solutions. Seedlings responded in a dose-dependent manner with significantly increased primary root lengths, due mainly to longitudinal cell elongation, increased numbers of lateral roots and impaired root hair development. Bioassay-driven fractionations and NMR were used to identify catechol as the main active compound for the smoke-induced root phenotype. The transcriptome analysis revealed that mainly genes related to auxin biosynthesis and redox homeostasis were altered after catechol treatment. However, histochemical analyses of reactive oxygen species (ROS) and the inability of auxin applications to rescue the phenotype clearly indicated that highly localized changes in the root's redox-status, rather than in levels of auxin, are the primary effector. Moreover, H2 O2 application rescued the phenotype in a dose-dependent manner. Chemical cues in smoke not only initiate seed germination, but also influence seedling root growth; understanding how these cues work provides new insights into the molecular mechanisms by which plants adapt to post-fire environments.