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- Materials and Methods
- Supporting Information
The growth of elongated plant organs is typically controlled by competition between turgor pressure and targeted wall softening, allowing cells and tissues to expand in a highly regulated manner. Studies of individual cells have demonstrated how cellulose microfibrils embedded in the cell wall can promote axial elongation with minimal radial expansion, at a rate determined by the properties of the wall's pectin matrix and hemicellulose crosslinks (Cosgrove, 2000, 2005; Baskin, 2005; Boyer, 2009). In a tissue, cells adhere strongly to their neighbours, even though the mechanical properties of neighbouring cell types may differ. Elongation of an organ such as the primary root of Arabidopsis is therefore determined by the integrated effect of multiple cells, and will be mediated by geometric as well as biomechanical factors.
As cells traverse the elongation zone (EZ) of the Arabidopsis root, their growth rates change: measurements show a dramatic increase in the cell's relative elongation rate (RER) on entering the EZ; this RER is then maintained at a high level before reducing to zero as cells progress to the mature zone (van der Weele et al., 2003; Basu et al., 2007; Chavarría-Krauser et al., 2008). Because the root's overall elongation rate depends on the rate at which mature cells are produced and their length, the duration and growth of cells within the EZ critically affects root growth. Many genetic mutants with reduced root length have reduced cell growth within the EZ (Benfey et al., 1993; Band et al., 2012b; Wen et al., 2013). The regulation of growth by phytohormones is of particular significance. For example, it is recognised that different hormones target different cell layers (Swarup et al., 2005; Úbeda-Tomás et al., 2008; Hacham et al., 2011), with auxin and brassinosteroid targeting the epidermis and gibberellin targeting the endodermis. This raises the question of how signals acting on different cell layers together regulate the shape of the growing root, and why particular hormones come to have a dominant influence on specific cell layers (Úbeda-Tomás et al., 2012).
In building systems-level descriptions of plant organs, it is necessary to integrate the action of multiple individual components acting across disparate time and length scales (Band et al., 2012a). In modelling growth of the Arabidopsis root, a number of these components have recently been put into place. At the level of an individual cell wall, chemo-mechanical models have addressed the turnover of pectin (Rojas et al., 2011) and of hemicellulose cross-links that bind to cellulose microfibrils (Dyson et al., 2012), showing in the latter case how a stretch-dependent breakage rate can explain yielding behaviour of the wall. At the level of a single cell, a model describing the reorientation of microfibrils as a cell elongates has revealed a potential biomechanical mechanism for the suppression of cell elongation as cells leave the EZ (Dyson & Jensen, 2010). These studies demonstrate how variants of the Lockhart equation (Lockhart, 1965; Ortega, 1985) (in which cell-wall material is characterised by yield and extensibility parameters) provide a practical description of plant materials at different scales. These descriptions have been integrated into a two-dimensional representation of a multicellular plant root (Fozard et al., 2013), illustrating how differential expansion generates bending and microfibril reorientation inhibits growth. The value of this approach is that simulations can capture detailed biomechanical properties of cell walls and a realistic representation of multicellular tissue geometry, while being coupled to descriptions of hormone transport and signalling pathways between and within individual cells.
In the development of simulations of this kind, techniques from multiscale modelling enable us to connect representations of a system across different spatial scales, providing mechanistic insights in addition to significant computational advantages. Here we pursue such an approach, seeking to understand how the mechanical properties of individual cells over the cross-section of an elongating organ such as a root contribute to the properties of the tissue as a whole, particularly in driving morphometric changes such as gravitropic bending. While a Lockhart-style description applies at both the cell and tissue levels, we show how geometric factors play an increasingly important role at larger scales. In particular, we present and exploit measurements of cell-wall lengths and thicknesses in characterising mechanical properties of the whole tissue. Our model demonstrates the geometric advantage possessed by epidermal cells, relative to other cell layers, in influencing elongation and bending properties, which we quantify for the Arabidopsis root. The model also reveals a fundamental relationship between RER and curvature growth rate, providing new insights into existing observations (Chavarría-Krauser et al., 2008), which we exploit to derive predictions of gravitropic bending angles.
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- Materials and Methods
- Supporting Information
The concept of the epidermis taking a dominant role in controlling the elongation rate of plant organs is well established, particularly given the observation of inner tissues of a stem elongating when the outer layers are removed (Kutschera & Niklas, 2007). This demonstrates that, in situ, the inner tissues are under net compression and the outer layers under tension, with the epidermis appearing to restrain rapid growth. In aerial organs, the associated gradients of tissue stress (the stress field averaged over multiple cells, but not necessarily the whole cross-section) may contribute to the organ's structural stability (Vandiver & Goriely, 2008). The gradient of tissue stress can be explained through gradients in material properties, although active stress generation is also a candidate mechanism (Baskin & Jensen, 2013). In root systems, however, and Arabidopsis in particular, it is less clear that substantial gradients in tissue stress exist across a cross-section. However, as our model demonstrates, the epidermis still maintains its predominant role in regulating elongation.
There is an intuitive explanation for this observation: epidermal cells have much greater net perimeter in the root cross-section than any other cell layer. To quantify this advantage, we took detailed measurements of cell-wall lengths and thicknesses across a set of cross-sections. We adopted the widely accepted Lockhart model for cell and tissue elongation, and derived the relationship between tissue-level mechanical parameters (yield and extensibility) in terms of properties of individual cells. Incorporating geometric data, we showed how the epidermis has at least a six-fold influence compared with any internal layer in determining tissue-level growth parameters ϕeff and Yeff (Fig. 4a,b). At present, however, the contributions of these individual parameters to patterns of root elongation across the EZ remain a matter of debate: the initiation of growth at the distal end of the EZ can be expected to arise through a drop in yield, given our measurements showing that turgor is uniform along the root (Fig. 3a); however, the inhibition of growth at the proximal end can be explained either by an increase in effective yield, or a substantial drop in effective extensibility (possibly associated with reorientation of microfibrils in highly elongated cells (Dyson & Jensen, 2010)).
Naturally, asymmetries across the root can generate curvature. As shown in (Eqn 3), three independent mechanisms emerge: a gradient of yield across the cross-section; a gradient of extensibility; and material asymmetry. The last of these may always be present to some extent, and the effect has been recognised previously in simulations (Fozard et al., 2013); however, in normal roots it is likely to be compensated by tropic responses (or possibly a mechanism of proprioception (Bastien et al., 2013)), so we do not consider it further. Geometrical factors play a crucial role in determining the relative magnitudes of the remaining contributions to the CGR and in determining the net direction of bending, with the epidermis once more taking a dominant role (Fig. 4c).
According to our simple model, the tension in a cell wall is T = Y + (RER/ϕ) when an organ is elongating (with RER > 0). Thus, the tension in a peripheral cell layer may be elevated relative to inner tissues by having larger Y or smaller ϕ, leading to gradients of tissue stress. During bending, the tissue stress is inherently asymmetric, although the total stress and its moment must vanish when integrated across the root cross-section (in the absence of external forces).
Significantly, the component of CGR in (Eqn 3) generated by ϕ is proportional to the RER, unlike that arising from yield or geometry. Fortunately, distributions of CGR and RER have previously been measured along Arabidopsis roots during gravitropic bending by Chavarría-Krauser et al. (2008). (Note that the curvature growth rate reported by Chavarría-Krauser et al. (2008) is the spatial gradient of a Lagrangian time derivative of root angle; this differs from the Lagrangian time derivative of a spatial gradient of root angle used here. We assume the difference may be neglected in the argument that follows.) Their data show that, 3 or more hours after the gravity stimulus in the central EZ, the CGR and RER have similar distributions, with closely aligned maxima, supporting the hypothesis that ϕ generates curvature in this region. However, within the first hour after the stimulus, in the distal EZ, the CGR is large where the RER is small (Chavarría-Krauser et al., 2008), making it more plausible that Y, rather than ϕ, generates curvature in this region. Significantly, curvature generation in the distal EZ was observed both in wild-type and in a pin3 auxin transport mutant, whereas that in the central EZ was seen in wild-type but not the mutant. PIN3 is thought to be a key player in creating the asymmetric auxin fluxes from the root tip during a gravitropic response, therefore these data suggest that the PIN3-dependent auxin asymmetry generates the curvature in the central EZ but not in the distal EZ (whether the curvature in the distal EZ is due to auxin asymmetries created by a different process, or a different mechanism entirely, remains unresolved).
A potential explanation of this observation is that distinct structural elements of the cell wall are targeted in the two regions. In previous models, we have shown how yield can arise from the action of hemicellulose crosslinks (Dyson et al., 2012), while extensibility can be characterised primarily by properties of the pectin matrix (Dyson & Jensen, 2010). Thus, we can hypothesise that the PIN3-dependent auxin asymmetry regulates the matrix, but that the pathway targeting crosslinks is PIN3-independent.
Given evidence that extensibility gradients can generate curvature, we computed the bending angle arising from a constant transverse gradient of extensibility propagating along the root. Because CGR is then proportional to RER, the curvature acquired by cells as they move through the EZ is proportional to logeβ (see (Eqn 6)), where β is the factor by which cells elongate through the EZ. The bending angle is then determined by the length of root exposed to the extensibility gradient, which in our model was expressed as the speed V at which the cells leave the EZ times a time of exposure ta, giving the simple expression (Eqn 7). Remarkably, the predicted net bending angle is insensitive to patterns of growth within the EZ, depending only on β. This prediction neglects additional bending driven by gradients of yield, and takes no account of the gravitropic control system that initiates the bending response, factors which will be addressed in future studies.
There are numerous additional weaknesses in our modelling approach that remain to be addressed: the validity of the underlying Lockhart model (which can be replaced with much more sophisticated constitutive assumptions (Huang et al., 2012), and by employing simulations that capture the anisotropic viscoelastic properties of individual cells (Fozard et al., 2013)); the assumption of quasi-stationarity (noting for example that the RER distribution may change after a gravitropic stimulus (Chavarría-Krauser et al., 2008)); the neglect of environmental forces that enter the force and moment balances that we employed; the incorporation of cell division (Chavarría-Krauser & Schurr, 2004); and three-dimensional effects such as twisting and torsion.
In summary, we have shown how, for a highly organised tissue such as the primary root of Arabidopsis, cell-level properties can be integrated to determine properties at the tissue level. Our approach provides an efficient strategy to incorporate the properties of individual cell walls in multiscale models for root gravitropism. Our model predicts that the parameters determining root elongation and curvature generation are most sensitive to the material properties of the epidermis, which is targeted by auxin (Swarup et al., 2005). Hormones targeting internal layers must exert a greater influence on wall mechanical properties in order to influence growth and curvature rates. We have shown how geometric data can be used to quantify this difference, and demonstrated how to predict resulting bending angles.