Gravity profoundly influences plant growth and development. Plants respond to changes in orientation by using gravitropic responses to modify their growth. Cholodny and Went hypothesized over 80 years ago that plants bend in response to a gravity stimulus by generating a lateral gradient of a growth regulator at an organ's apex, later found to be auxin. Auxin regulates root growth by targeting Aux/IAA repressor proteins for degradation. We used an Aux/IAAbased reporter, domain II (DII)-VENUS, in conjunction with a mathematical model to quantify auxin redistribution following a gravity stimulus. Our multidisciplinary approach revealed that auxin is rapidly redistributed to the lower side of the root within minutes of a 908 gravity stimulus. Unexpectedly, auxin asymmetry was rapidly lost as bending root tips reached an angle of 408 to the horizontal. We hypothesize roots use a "tipping point" mechanism that operates to reverse the asymmetric auxin flow at the midpoint of root bending. These mechanistic insights illustrate the scientific value of developing quantitative reporters such as DII-VENUS in conjunction with parameterized mathematical models to provide high-resolution kinetics of hormone redistribution.environmental sensing | systems biology R oot gravitropism has fascinated researchers since Knight (1) and Darwin (2). More recently, reorientation of Arabidopsis seedlings has been shown to trigger the asymmetric release of the growth regulator auxin from gravity-sensing columella cells at the root apex (Fig. 1A) (3-5). The resulting lateral auxin gradient is hypothesized to drive a differential growth response, where cell expansion on the lower side of the elongation zone is reduced relative to the upper side, causing the root to bend downward (6-8). Despite representing one of the oldest hypotheses in plant biology, key questions about auxin-regulated root gravitropism remain to be experimentally determined. How rapidly does the lateral auxin gradient form? Is this timescale consistent with the theory that auxin redistribution drives root bending? How long does the lateral auxin gradient persist? What triggers auxin redistribution to return to equal levels?Our understanding of gravity-induced auxin redistribution has been limited by the tools available to monitor auxin concentrations at high spatiotemporal resolution. Currently, the most widely used tools to follow auxin distribution in tissues are auxin-inducible reporters such as DR5::GFP (3, 4). However, as an output of the auxin response pathway (Fig. 1B), the activity of the DR5 reporter does not directly relate to endogenous auxin abundance, but also depends on additional parameters including local auxin signaling capacities and rates of transcription and translation (Fig. 1B). In practice, these intermediate processes confer a time delay of ∼1.5-2 h between changes in auxin abundance and DR5 reporter activity (9, 4), making it difficult to quantify the speed and magnitude of fold changes in auxin distribution during a root gravitropic response.Auxi...

Vascular development and homeostasis are underpinned by two fundamental features: the generation of new vessels to meet the metabolic demands of under-perfused regions and the elimination of vessels that do not sustain flow. In this paper we develop the first multiscale model of vascular tissue growth that combines blood flow, angiogenesis, vascular remodelling and the subcellular and tissue scale dynamics of multiple cell populations. Simulations show that vessel pruning, due to low wall shear stress, is highly sensitive to the pressure drop across a vascular network, the degree of pruning increasing as the pressure drop increases. In the model, low tissue oxygen levels alter the internal dynamics of normal cells, causing them to release vascular endothelial growth factor (VEGF), which stimulates angiogenic sprouting. Consequently, the level of blood oxygenation regulates the extent of angiogenesis, with higher oxygenation leading to fewer vessels. Simulations show that network remodelling (and de novo network formation) is best achieved via an appropriate balance between pruning and angiogenesis. An important factor is the strength of endothelial tip cell chemotaxis in response to VEGF. When a cluster of tumour cells is introduced into normal tissue, as the tumour grows hypoxic regions form, producing high levels of VEGF that stimulate angiogenesis and cause the vascular density to exceed that for normal tissue. If the original vessel network is sufficiently sparse then the tumour may remain localised near its parent vessel until new vessels bridge the gap to an adjacent vessel. This can lead to metastable periods, during which the tumour burden is approximately constant, followed by periods of rapid growth.

2Plants can acclimate by using tropisms to link the direction of growth to 41 environmental conditions. Hydrotropism allows roots to forage for water, a process 42 known to depend on abscisic acid (ABA) but whose molecular and cellular basis 43 remains unclear. Here, we show that hydrotropism still occurs in roots after laser 44 ablation removed the meristem and root cap. Additionally, targeted expression 45 studies reveal that hydrotropism depends on the ABA signalling kinase, SnRK2.2, and 46 the hydrotropism-specific MIZ1, both acting specifically in elongation zone cortical 47 cells. Conversely, hydrotropism, but not gravitropism, is inhibited by preventing 48 differential cell-length increases in the cortex, but not in other cell types. We conclude 49 that root tropic responses to gravity and water are driven by distinct tissue-based 50 mechanisms. In addition, unlike its role in root gravitropism, the elongation zone 51 performs a dual function during a hydrotropic response, both sensing a water 52 potential gradient and subsequently undergoing differential growth. 53 3 Tropic responses are differential growth mechanisms that roots use to explore the 54 surrounding soil efficiently. In general, a tropic response can be divided into several steps, 55 comprising perception, signal transduction, and differential growth. All of these steps have 56 been well characterized for gravitropism, where gravity sensing cells in the columella of the 57 root cap generate a lateral auxin gradient, whilst adjacent lateral root cap cells transport 58 auxin to epidermal cells in the elongation zone, thereby triggering the differential growth that 59 drives bending [1][2][3][4] . In gravi-stimulated roots, the lateral auxin gradient is transported 60 principally by AUX1 and PIN carriers [3][4][5] . 61Compared with gravitropism, the tropic response to asymmetric water availability, i.e., 62 hydrotropism, has been far less studied. Previously, it was reported that surgical removal or 63 ablation of the root cap reduces hydrotropic bending in pea [6][7][8] and Arabidopsis thaliana 9 , 64suggesting that the machinery for sensing moisture gradients resides in the root cap. It has 65 also been reported that hydrotropic bending occurs due to differential growth in the 66 elongation zone 7,10 . However unlike gravitropism, hydrotropism in A. thaliana is independent 67 of AUX1 and PIN-mediated auxin transport 11,12 . Indeed, roots bend hydrotropically in the 68 absence of any redistribution of auxin detectable by auxin-responsive reporters 13,14 . 18,19 . 83However it is unclear whether this broad expression pattern is necessary for MIZ1's function 84 in hydrotropism or whether ABA signal transduction components in general have to be 85 expressed in specific root tip tissues for a hydrotropic response. The present study describes 86 a series of experiments in A. thaliana designed to identify the root tissues essential for a 87 hydrotropic response. We report that MIZ1 and a key ABA signal-transduction component 88SnRK2....

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Abstract. In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of system parameters, including the speed and strength of synaptic coupling and the speed of axonal signals. The theory is illustrated with the construction and stability analysis of front solutions to a scalar neural field model, and a limiting case is shown to recover recent results of Zhang [Differential Integral Equations, 16 (2003), pp. 513-536]. Traveling fronts and pulses are considered in more general models possessing either a linear or piecewise constant recovery variable. We establish the stability of coexisting traveling fronts beyond a front bifurcation and consider parameter regimes that support two stable traveling fronts of different speeds. Such fronts may be connected, and depending on their relative speed the resulting region of activity can widen or contract. The conditions for the contracting case to lead to a pulse solution are established. The stability of pulses is obtained for a variety of examples, in each case confirming a previously conjectured stability result. Finally, we show how this theory may be used to describe the dynamic instability of a standing pulse that arises in a model with slow recovery. Numerical simulations show that such an instability can lead to the shedding of a pair of traveling pulses. Key words. traveling waves, neural networks, integral equations, Evans functionsAMS subject classification. 92C20 DOI. 10.1137/0406059531. Introduction. Traveling waves in neurobiology are receiving increased attention from experimentalists, in part due to their ability to visualize them with multielectrode recordings and imaging methods. In particular, it is possible to electrically stimulate slices of pharmacologically treated tissue taken from the cortex [19], hippocampus [31], and thalamus [28]. For cortical circuits such in vitro experiments have shown that, when stimulated appropriately, waves of excitation may occur [12,40]. Such waves are a consequence of synaptic interactions and the intrinsic behavior of local neuronal circuitry. The propagation speed of these waves is of the order cm s −1 , an order of magnitude slower than that of action potential propagation along axons. The class of computational models that are believed to support synaptic waves differ radically from classic models of waves in excitable systems. Most importantly, synaptic interactions are nonlocal (in space) and involve communication (space-dependent) delays (arising from the finite propagation velocity of an action potential) and distributed delays (arising from neurotransmitter release and dendritic processing). In many continuum models

We investigate a three-dimensional multiscale model of vascular tumour growth, which couples blood flow, angiogenesis, vascular remodelling, nutrient/growth factor transport, movement of, and interactions between, normal and tumour cells, and nutrient-dependent cell cycle dynamics within each cell. In particular, we determine how the domain size, aspect ratio and initial vascular network influence the tumour's growth dynamics and its long-time composition. We establish whether it is possible to extrapolate simulation results obtained for small domains to larger ones, by constructing a large simulation domain from a number of identical subdomains, each subsystem initially comprising two parallel parent vessels, with associated cells and diffusible substances. We find that the subsystem is not representative of the full domain and conclude that, for this initial vessel geometry, interactions between adjacent subsystems contribute to the overall growth dynamics. We then show that extrapolation of results from a small subdomain to a larger domain can only be made if the subdomain is sufficiently large and is initialised with a sufficiently complex vascular network. Motivated by these results, we perform simulations to investigate the tumour's response to therapy and show that the probability of tumour elimination in a larger domain can be extrapolated from simulation results on a smaller domain. Finally, we demonstrate how our model may be combined with experimental data, to predict the spatio-temporal evolution of a vascular tumour.

Cancer is a complex disease in which a variety of factors interact over a wide range of spatial and temporal scales with huge datasets relating to the different scales available. However, these data do not always reveal the mechanisms underpinning the observed phenomena. In this paper, we explain why mathematics is a powerful tool for interpreting such data by presenting case studies that illustrate the types of insight that realistic theoretical models of solid tumour growth may yield. These range from discriminating between competing hypotheses for the formation of collagenous capsules associated with benign tumours to predicting the most likely stimulus for protease production in early breast cancer. We will also illustrate the benefits that may result when experimentalists and theoreticians collaborate by considering a novel anti-cancer therapy.

Observations on Mount St Helens indicate that the spread of recolonizing lupin plants has been slowed due to the presence of insect herbivores and it is possible that the spread of lupins could be reversed in the future by intense insect herbivory [Fagan, W. F. and J. Bishop (2000). Trophic interactions during primary sucession: herbivores slow a plant reinvasion at Mount St. Helens. Amer. Nat. 155, [238][239][240][241][242][243][244][245][246][247][248][249][250][251]. In this paper we investigate mechanisms by which herbivory can contain the spatial spread of recolonizing plants. Our approach is to analyse a series of predatorprey reaction-diffusion models and spatially coupled ordinary differential equation models to derive conditions under which predation pressure can slow, stall or reverse a spatial invasion of prey. We focus on models where prey disperse more slowly than predators. We comment on the types of functional response which give such solutions, and the circumstances under which the models are appropriate.