The advent of user-friendly instruments for measuring force/deflection curves of plant surfaces at high spatial resolution has resulted in a recent outpouring of reports of the ‘Young's modulus’ of plant cell walls. The stimulus for these mechanical measurements comes from biomechanical models of morphogenesis of meristems and other tissues, as well as single cells, in which cell wall stress feeds back to regulate microtubule organization, auxin transport, cellulose deposition, and future growth directionality. In this article I review the differences between elastic modulus and wall extensibility in the context of cell growth. Some of the inherent complexities, assumptions, and potential pitfalls in the interpretation of indentation force/deflection curves are discussed. Reported values of elastic moduli from surface indentation measurements appear to be 10- to >1000-fold smaller than realistic tensile elastic moduli in the plane of plant cell walls. Potential reasons for this disparity are discussed, but further work is needed to make sense of the huge range in reported values. The significance of wall stress relaxation for growth is reviewed and connected to recent advances and remaining enigmas in our concepts of how cellulose, hemicellulose, and pectins are assembled to make an extensible cell wall. A comparison of the loosening action of α-expansin and Cel12A endoglucanase is used to illustrate two different ways in which cell walls may be made more extensible and the divergent effects on wall mechanics.
Introduction
Interest in cell wall mechanics has received a major boost from recent elegant models of plant morphogenesis in which mechanical stresses in growing cell walls guide a dynamic supracellular system that involves microtubule reorganization, oriented cellulose deposition, and auxin transport. The models simulate a highly choreographed process regulating the location and directionality of cell growth and cell wall reinforcement in the shoot apical meristem or other tissues or single cells such as the trichome (Laufs et al., 2009; Kierzkowski et al., 2012; Robinson et al., 2013; Bassel et al., 2014; Sampathkumar et al., 2014b; Boudon et al., 2015; Yanagisawa et al., 2015). These ideas, stimulated by advances in fluorescent tagging of cell components and confocal imaging of living shoot apical meristems (Meyerowitz et al., 2005; Bainbridge et al., 2008; Hamant et al., 2008), have revived the concepts of biophysical control of meristem dynamics pioneered by Green (1999) and extended by subsequent advances in cell biology and genetics (Landrein and Hamant, 2013; Braidwood et al., 2014). Additional technical advances include methods based on micro-indentation and atomic force microscopy (AFM) to assess wall stiffness at cellular and subcellular resolution in living meristems as well as in other tissues and single cells (Milani et al., 2013; Bonilla et al., 2015; Braybrook, 2015; Weber et al., 2015). Mechanical measures of cell walls are most informative when interpreted in terms of the structure of cell walls, which are comprised of relatively stiff cellulose microfibrils embedded in a hydrated matrix of pectins, hemicellulose, and structural protein. Concepts of how these components are linked to one another have changed considerably since early views of the primary cell wall as a fiber-reinforced polymer composite, and concepts of cell wall structure are currently in a state of flux (Cosgrove, 2014b).
Thus it is perhaps timely to review the notion of cell wall extensibility and how it relates to cell growth, cell wall structure, the action of wall-loosening processes, and purely mechanical measures of wall viscoelasticity such as wall stiffness. I use ‘viscoelasticity’ here rather loosely to encompass the range of material properties of cell walls that include elastic, viscous, plastic, and retarded elastic deformations, but not to include enzyme-dependent, or chemorheological, flows. Although integration of these inter-related concepts into a comprehensive theory of plant cell growth is still some way off, recent movement in this direction is refining our appreciation of plant growth as a subtle and dynamic process. Complementing these developments are genetic studies of ‘cell wall integrity sensing’ (Wolf et al., 2014; Hamann, 2015; Hofte, 2015), which have identified signal transduction elements in this defense response, but leave unanswered questions such as what ‘wall integrity’ means at the structural level and what is actually being sensed. This topic is also linked to a resurgence of interest in pectins and their methylation state in wall structure and wall sensing (Peaucelle et al., 2008, 2011; Palin and Geitmann, 2012; Wolf and Greiner, 2012; Braybrook and Peaucelle, 2013).
In this review, I use the term ‘wall extensibility’ to mean the ability of the cell wall to increase in surface area irreversibly during growth. Unlike growing bacterial cells, where an increase in wall surface is directly coupled to addition of new wall components, plant cells grow in surface area by a spreading movement of cellulose microfibrils and associated matrix components in the plane of the wall without necessary addition of new wall polymers (Cosgrove, 2014a). It involves a coupled process of wall stress relaxation and water uptake, described below. In the long term, addition of new polymers is needed to maintain wall integrity, but wall synthesis and surface expansion are separable processes, at least in cells with diffuse growth patterns. In tip-growing cells such as the pollen tube (Rojas et al., 2011; Sanati Nezhad et al., 2014), wall synthesis and surface expansion are more tightly coupled and thus a modified conceptual framework may be needed, which is beyond the scope of this review, but many of the concepts discussed here also apply in modified form to tip growth.
Wall extensibility, as used here, is characteristic of growing cell walls and differs profoundly from elastic (=reversible) extensibility, which is a property of both growing and non-growing cells, for example when cell turgor pressure is modulated or when an external force, such as from an AFM tip, suddenly impinges on a cell. As reviewed two decades ago (Cosgrove, 1993), wall extensibility depends on the rate of wall-loosening processes acting on the cell wall to induce stress relaxation, which creates the conditions for sustained cell water uptake and irreversible physical enlargement of the growing cell. A central message of this update is that wall extensibility and wall elasticity are not the same, and in some instances are not correlated with each other. A second message is that methods to measure non-elastic properties such as extensibility of cell walls at cellular resolution in complex tissues need to be developed to probe deeper into the physical, cellular, and genetic control of plant morphogenesis.
What is wall extensibility and how is it measured?
Early concepts of wall extensibility and its relationship to cell growth were summarized by Heyn (1940) and updated by Cleland (1971). These ideas, in which cell growth was thought to result from yielding of a plastic wall to the forces generated by cell turgor pressure (P), pre-dated the first detailed molecular model of a growing cell wall (Keegstra et al., 1973) and did not offer a specific molecular concept of wall extensibility other than the general notion that the physical viscosity or plasticity of the wall matrix controlled wall extensibility. Although ample evidence has accrued in the meantime that this purely viscoelastic concept of cell wall growth falls short of reality (Taiz, 1984; Cosgrove, 1993), it seems to be the assumption of many recent studies that use micro-mechanical mapping of surface stiffness to infer morphogenetic control mechanisms for meristems and single cells.
A major (but often misunderstood) conceptual advance in the field was taken by Lockhart (1965) who worked out a consistent biophysical theory connecting cell water uptake and yielding of the cell wall, eventually leading to recognition of wall stress relaxation as the seminal biophysical event underlying cell growth (Ray et al., 1972; Cosgrove, 1985; Hamant and Traas, 2010). Wall stress relaxation results from rearrangements (yielding) of the load-bearing polymers in the cell wall, consequently reducing cell turgor pressure and creating the impetus for simultaneous water uptake into the cell, volumetric expansion of the cell wall chamber, and restoration of wall stress (Fig. 1A). The molecular nature of stress relaxation is key to understanding how plants regulate their growth and will be discussed below, but a complete understanding of the mechanism still eludes us, largely because the stress distribution and load-bearing components of the growing cell wall are not well understood.
Biophysics of plant cell growth. (A) Diagram illustrating the concept of wall stress relaxation and its connection with turgor pressure and induced water flows. (1) In a well-hydrated non-growing cell, the cell reaches osmotic equilibrium, with wall stresses counter-balancing the outward force of turgor pressure against the wall. (2) Growing cell walls are loosened, resulting in a reduction in cell wall stress and turgor pressure. This means that elastic elements in the cell wall shrink as a result of the slippage or bond breakage caused by wall loosening. (3) In response to the reduced cell turgor and water potential, water flows into the cell, elastically expanding the wall and restoring turgor and wall stress. This process is illustrated as discrete steps, but relaxation, water influx, volume enlargement, and turgor restoration occur simultaneously. (B) Time course for stress relaxation measured, for example, with the pressure probe in a growing tissue that is isolated from an external water supply. Turgor pressure decays gradually to a yield threshold Y with a rate constant given by the cell wall extensibility coefficient ϕ times the cell volumetric elastic modulus εv. Based on Cosgrove (1987a). (C) Graph of growth rate as a function of turgor, idealized in the equation growth rate=ϕ (P–Y), where ϕ is wall extensibility, P is turgor pressure. and Y is yield threshold. From such a plot, estimates of Y and ϕ may be made.
Wall stress relaxation in vivo has been measured by use of the pressure probe, psychrometer, or pressure chamber to monitor the decay in water potential or turgor pressure in growing tissues isolated from an external supply of water, a condition that blocks the uptake of water needed to restore turgor and wall stress following wall relaxation (Cosgrove, 1987a, b). Under these conditions, turgor pressure ideally decays to a steady value, the yield threshold, with a rate constant that depends on the wall extensibility coefficient (Fig. 1B). Hence the general notion of wall extensibility is defined in this formulation with two parameters, the yield threshold and the extensibility coefficient. These parameters are also revealed in plots of growth rate as a function of turgor pressure (Fig. 1C). Actual plots of growth rate versus turgor are not so linear and display a more gradual transition at the yield threshold, as does creep of isolated cell walls subjected to a series of tensile stresses (Takahashi et al., 2006). Thus the idealized equation representing the turgor dependence of the rate of cell wall enlargement [GR=ϕ(P–Y); see Fig. 1C legend] is only a rough approximation of the actual behavior of growing tissues. Turgor dependence of plant growth is an inevitable physical consequence of the facts that (i) wall stress is essential for stress relaxation (and ensuing water uptake) and (ii) the stress relaxation rate is a function of the wall stress itself. When growth is measured as % h−1, extensibility (ϕ) has units of % h−1 MPa−1, while the yield threshold (Y) is usually measured in units of cell turgor pressure (MPa). Living cells may complicate this relatively simple ideal behavior by modulating wall-loosening processes dynamically in response to reduced turgor, reduced growth rate, or other conditions (Green et al., 1977; Nakahori et al., 1991).
Unfortunately, these methods for measuring wall relaxation do not currently lend themselves to micromechanical mapping of meristems. Kierzkowski et al. (2012) and Nakayama et al. (2012) took a step in this direction by measuring osmotic inflation and deflation of cells on the surface of the shoot apical meristem, but did not map extensibility or yield thresholds. Their approach might be extended by use of a series of osmotic solutions to calculate growth rate versus turgor for each surface cell (as in Fig. 1C) and plotting extensibility coefficient and yield threshold values across the meristem surface.
Contrary to early ideas that wall growth and extensibility arise from passive polymer motions of a viscoelastic wall, various observations indicate that there is more to it and that it depends on continual wall loosening.
(i) When isolated cell walls are heat treated so as to inactivate wall-bound enzymes and are clamped at constant force (i.e. in wall creep assays), they undergo only a transient viscoelastic extension that decays with a half-time of a few minutes or less, in contrast to the walls of living cells that extend steadily for many hours or days (Taiz, 1984). The viscoelastic extension is a combination of elastic and retarded elastic deformations that are distinct from wall loosening related to growth (Cosgrove, 1993; Nolte and Schopfer, 1997; Schopfer, 2006). Likewise, the stress relaxation behavior of inactivated cell walls differs from that of living cells; compare Cosgrove (1987b) with Yamamoto et al. (1970).
(ii) When isolated walls that are not inactivated are clamped at constant force at acidic pH, they extend steadily for many hours under the continued action of endogenous expansins bound to the cell wall (Cleland et al., 1987; Cosgrove, 1989; McQueen-Mason et al., 1992). Extensions of 50–100% occur before the walls thin to the point of breakage. Such breakage does not occur in walls of living cells, presumably because the wall is continually reinforced by newly incorporated polymers.
(iii) When respiration of living cells is inhibited by cold or metabolic inhibitors, growth ceases almost immediately but the viscoelastic properties of the isolated cell wall do not change [i.e. as measured by stress/strain assays (Taiz, 1984)]. In a similar vein, when growth is rapidly inhibited by light (Cosgrove, 1988b) or rapidly stimulated by auxin (Cleland, 1984), wall viscoelastic properties change only slowly after the change in growth.
Thus, while growing cell walls clearly have viscoelastic properties (Taiz, 1984; Schopfer, 2006) and viscoelasticity sometimes tracks changes in wall extensibility (with some delay), they are distinctive properties. Ray (1987) proposed extensibility to be the result of wall loosening in series with subsequent viscoelastic extension, each governed by separate rate constants k1 and k2, respectively (Fig. 2). The observations cited above suggest that k2 is larger than k1. In such a case, the wall deformation following sudden changes in wall stress will be (transiently) dominated by purely viscoelastic extensions (k2) whereas under steady-state conditions wall enlargement is controlled by loosening (k1). This formulation may oversimplify the behavior of growing cell walls, but it provides a handy conceptual framework for thinking of loosening and viscoelasticity as distinctive aspects of the growing cell wall.
Conceptual model of Ray (1987) linking cell wall loosening and wall viscoelasticity with rate constants k1 and k2 for biochemical and viscoelastic steps. This concept was formulated for wall extension but can be extended for parallel hydraulic steps (shown in the bottom line), with stress relaxation originating from wall loosening and subsequent water uptake driving the passive viscoelastic extension.
How does wall stiffness relate to wall extensibility?
With the advent of user-friendly atomic force microscopes and other micro-indentation devices coupled to microscopes, reports of cell wall stiffness or modulus for shoot apical meristems, leaf epidermal layers, single cells, and other plant tissues have become common (Milani et al., 2013; Routier-Kierzkowska and Smith, 2013). The principle of the indentation method is deceptively simple: a cantilever or other mechanical device is maneuvered to the surface of a cell or tissue, and a force–deflection curve is measured as the tip of the device makes contact with the surface, pushes against it, and then retracts. Probes vary from relatively sharp, pyramidal shapes of AFM tips as small as ~4nm diameter (Xi et al., 2015) to spherical beads or cylinders with contacts of 1–10 μm diameter (Braybrook, 2015). The deformation is typically fast, in the millisecond range for some AFM-based measurements to seconds for other types of indentation protocols, so wall loosening during the measurement is unlikely to affect the force–deflection response. There is a trade-off in speed: too fast and hydraulic flows of the bathing fluid and within the wall may influence the deflection rate; too slow and water outflows from the compressed cells can influence the process, if the wall is pressed into the cell as in ball tonometry (Cappella and Dietler, 1999; Lintilhac et al., 2000; Forouzesh et al., 2013; Beauzamy et al., 2015).
While the appearance of such force–deflection curves may resemble the curves generated by uniaxial stress–strain assays of hypocotyl or coleoptile wall specimens (i.e. in an ‘Instron’ tester) (Cleland, 1967), such resemblance belies the many complications inherent in micro-scale indentation of living plant surfaces. One major difference is the direction of the applied force: a tensile force in the plane of the cell wall, as used in a uniaxial tensile tester, is resisted by cellulose microfibrils in the direction of the force, whereas this is not the case for an indentation force normal to the plane of the wall for indentation assays. Some of the assumptions and complicating issues of the indentation measurements have been discussed in the literature (McKee et al., 2011; Milani et al., 2013; Braybrook, 2015). Tip size, shape, and penetration depth can influence whether the probe penetrates into the cell wall, deforms the cell wall locally, or changes the whole cell shape (and adjacent cells) more globally. Usually it is unclear how the deflection is partitioned into these three different aspects of wall displacement. Extracting information about cell wall material properties (usually quantified as a modulus or its reciprocal, a compliance) from indentation measurements always involves fitting the force–deflection data to some kind of model and making assumptions about wall structure, cell shape, and the influence of turgor pressure, which stretches the cell wall and applies a pre-stress to the load-bearing wall polymers.
Turgor pressure is an important factor to be considered because it increases the effective stiffness of the cell, much as air pressure in a bicycle tire stiffens the tire (stiffness measures the resistance to deformation). There are at least three potential components to this effect: (i) non-linearity in the stress–strain curve (‘strain hardening’), which generally increases with wall strain; (ii) resistance to indentation due to pre-stresses in the plane of the wall (‘tightening of the drum head’); and (iii) resistance stemming from the outward force of turgor pressure, when the wall is perceptibly pressed into the cell (the latter is the basis for tonometry assessments of turgor). Non-linear stress–strain behavior has long been known in the field of plant wall relations, where the cell volumetric elastic modulus generally increases as turgor pressure increases (see, for eample, Buchner et al., 1981; Tomos and Leigh, 1999). Separating turgor effects from inherent wall properties can be done, but may be complicated, involving osmotic treatments, finite element modeling of the cell, and many assumptions for tractable mechanical analysis (Hayot et al., 2012; Forouzesh et al., 2013; Bonilla et al., 2015; Digiuni et al., 2015; Weber et al., 2015). For surface indentation measurements, the tensioning effect of turgor is likely to have a large effect on the force–deflection curves, even at small deflections. It is for this reason that some AFM-based studies have plasmolyzed the shoot apical meristem before measurement of wall mechanics (e.g. Peaucelle et al., 2011). While this procedure removes complications due to turgor, it also alters the pattern of wall stresses across the meristem and the mechanical state of the cell walls.
It is also worth noting at this point that there are many different kinds of moduli in the world of plant cell walls, and they measure different properties. Accessible introductions to this topic include Wainwright et al. (1976), Niklas (1992), and Burgert (2006). Generally speaking, a modulus is a measure of the stress/strain ratio where stress is conventionally defined as force per unit area and strain as relative change in length, width, thickness, volume, etc. There are many different ways to apply a stress and to measure the concomitant strain, and consequently there are many different kinds of moduli; they may all be expressed in the same units, but they measure different things and ought not to be mixed up. In the case of the cell volumetric elastic modulus mentioned above, the modulus (εv) is defined as dP/dV where dP is a change in cell turgor pressure (MPa) and dV is the relative change in cell volume (dimensionless). This modulus measures the elastic stretchability of the cell wall chamber and is not the bulk modulus (=1/compressibility) used in mechanics, with which it has sometimes been confused (Cosgrove, 1988a). When cells lose water, cell volume and turgor pressure decrease and εv quantifies this relationship. In cells with very stiff walls (high modulus), turgor pressure falls steeply as cells lose water. Values of εv for non-lignified plant cells typically range from 1MPa to 50MPa (Zimmermann and Steudle, 1978; Cosgrove, 1988a; Tomos and Leigh, 1999). With an εv of 10MPa, a cell would lose 1% of its volume for a 0.1MPa reduction in turgor pressure. In contrast, the bulk modulus of water (the main component of plant cells) is ~2000MPa, so an increase in atmospheric pressure by 0.1MPa would compress the cell negligibly (0.005%).
A more commonly cited modulus is the Young's modulus of elasticity (E), an engineering unit that quantifies the stress/strain ratio in a uniaxially extended strip of ‘Hookean’ material, defined as material displaying linear elasticity (i.e. strain is proportional to stress and is fully and immediately reversible upon reduction of stress). Growing plant cell walls—with their hierarchical, multilamellate, and anisotropic arrangement of stiff cellulose microfibrils bonded to each other and to hydrated matrix polymers—exhibit elastic deformation plus complicated, non-linear, and time-dependent strains that fall outside the Hookean ideal. The non-linear character of plant cell walls during mechanical deformation was highlighted in a recent study (Digiuni et al., 2015), but it has been recognized for many years.
Engineers generally avoid non-linear behavior by working at small strains (say, <0.1%) where non-linearities become inconsequential. This is in contrast to the larger strains often imposed during micro-indentation studies of cell walls and whole-tissue stress–strain assays. Non-linearity becomes a troublesome issue when modulus values are calculated from different parts of a stress–strain curve. The anisotropic construction of growing cell walls necessitates recognition of at least three distinct elastic moduli (Fig. 3): EL for changes in length, EW for changes in width, and ET for changes in thickness of a cell wall strip [this is an incomplete description of the full mechanics of the wall, which is beyond the scope of this review; see Niklas (1992) and Weber et al. (2015) for more detailed treatment]. Cellulose microfibrils, which are deposited in the plane of the cell wall, greatly influence EL and EW, whereas pectins probably exert dominant control of ET (Ha et al., 1997). The complex structure of plants cell walls means that these moduli are first-order approximations of the actual elastic behavior of cell walls. Additionally, the propensity of growing cell walls to deform irreversibly in a time-dependent manner under the tensile stresses generated by cell turgor or external forces complicates their mechanical properties even further. It is, of course, exactly this non-linear, time-dependent, irreversible behavior of growing cell walls that is of greatest relevance for growth and morphogenesis. At low strains (<0.1% deformation) and rapid measurements, elastic behavior probably dominates, but at larger strains (>0.1%) and longer time scales the non-elastic behaviors may become significant. For an example of the issues involved, see Hansen et al. (2011) who have reviewed the background for the time dependence of viscoelastic measurements of cell walls and described a procedure of dynamic testing for living, but not growing, potato explants.
Schematic diagram of the anisotropic construction of plant cell walls, made of multiple lamellae (only two are shown), each with aligned cellulose (longitudinal in the top lamella, transverse the bottom lamella), and embedded in a hydrated matrix of pectins and hemicellulose. The three major elastic moduli (EL, EW, ET) govern the stress–strain responses of walls to forces applied in the three orthogonal axes of the wall. Note that cellulose microfibrils contribute to EL and EW, but have less influence on ET. Wall indentation measurements may be most sensitive to ET.
Notwithstanding the foregoing complications, Young's elastic moduli of cell walls based on indentation assays are commonly reported. From the force–deflection curves generated by micro-scale indentations of plant cell surfaces, one may estimate an elastic modulus by fitting the data to an appropriate model. The type of model depends on the size of the tip, the depth of probe penetration into the wall, and the kind of wall deformation that is envisaged.
For nanometer-scale measurements made with a typical atomic force microscope using a sharp probe (~4–40nm diameter), a common assumption is that wall deformation is local, that large-scale bending of the cell wall is minimal, and that the influence of turgor pressure on cell wall stiffness is negligible. The force–deflection curve is then fitted to a mechanical model, usually a variant of the Hertz–Sneddon model (Sneddon, 1965) or the DMT model (Derjaguin et al., 1975), to estimate an elastic modulus of the deformed material (Routier-Kierzkowska and Smith, 2013; Braybrook, 2015). Complications in the analysis and interpretation of the complex curves obtained with living cells are outlined by Bonilla et al. (2015) who describe an empirical approach to distinguish different modes of wall deformation that occur during whole-cell indentation with small tips.
In a pioneering report of micro-mechanical mapping of the shoot apical meristem, Milani et al. (2011) observed that reduction of cell turgor pressure by addition of 0.25M NaCl had no effect on mechanical measurements, at least within the error of the measurement. This seems a perplexing result because tensioning of the wall by turgor pressure should make the wall appear stiffer. The authors estimated the wall deformation of their measurements to be ~300nm in diameter. This is ~20-fold larger than the spaces between cellulose microfibrils, which are believed to bear much of the turgor-generated wall stress; thus I would expect a pronounced turgor dependence of the indentation curves. The lack of turgor dependence in their measurements suggests to me that the AFM tip did not probe the mechanically relevant (load-bearing) portion of the cell wall. Because the outer part of the cell wall is the oldest region, it has undergone very large extensions in the course of its lifetime, possibly fragmenting its structure and uncoupling it from turgor-generated tensile stresses in the plane of the wall. This ought to be the case for the outer epidermal wall in the shoot apical meristem, but probably not for internal cell walls within the meristem that are relatively young (i.e. formed during a relatively recent cell division). This interpretation would be consistent with the concept summarized by Taiz (1984) that the inner half of the cell wall bears turgor-generated wall stresses and that the outer half may not be important for wall enlargement. If true, it casts doubt on the interpretation of AFM measurements that mechanically probe only the outermost part of the epidermal wall. Differences in the young and old regions of the wall could lead to differential contraction and shear stresses across the wall when turgor pressure is reduced, but apparently this possibility has not been studied in a quantitative manner.
Turgor dependence of plant cell stiffness is reflected in the everyday experience that carrots become limp upon dehydration and has been documented many times: 50 years ago as measurements of whole-tissue modulus (Virgin, 1955; Nilsson et al., 1958), a generation later as measurements of single-cell εv (Huesken et al., 1978; Buchner et al., 1981), and more recently in micro-indentation studies (Wang et al., 2004; Hayot et al., 2012; Forouzesh et al., 2013; Weber et al., 2015). In the later set of studies, large-scale wall deformations were employed and the deflection data were fit to mechanical models of whole-cell deformation. A detailed example illustrating the modeling that is required and the assumptions involved in such models is the analysis of tobacco cell suspension cells indented with a cylindrical probe of ~3 μm diameter by Weber et al. (2015). Two elastic moduli for these cylindrical cells were estimated: EL (153MPa) for changes in cell length and EW (571MPa) for changes in cell circumference (the third modulus, for cell wall thickness, was not estimated). Using these elastic moduli and assuming the cells are ideal cylinders with a turgor pressure of 0.6MPa, I estimated εv for these cells to be ~15MPa, which is within the range of values reported in the literature. Change in length made the largest contribution to the change in volume for the εv estimate. This calculation serves as a check that the wall moduli reported in this study are consistent with the expected pressure–volume relations of such cells and also gives a point of comparison for values of EL, EW, and εv. Wang et al. (2004) provide another point of comparison for spherical tomato suspension cells compressed between two surfaces. They report a Young's modulus of 2.3GPa and an estimated εv of 7MPa. They considered the wall to be isotropic and 126nm thick and they also took into account the outflow of water from the cell when it is compressed in this way (during the 1s compression). A theoretical sensitivity analysis performed by Weber et al. (2015) showed that their calculations were particularly sensitive to estimates of cell radius, indentation depth, and cell wall thickness, assumed to be 1250nm, which is ~10-fold thicker than estimates from other studies of suspension cells (Samuels et al., 1995; Sabba et al., 1999; Wang et al., 2004). Re-parameterization of geometrical quantities reduced some of these dependencies. One has to wonder whether the ~10-fold difference in Young's moduli in these two studies is a reflection of different assumptions of the thickness of the cell walls rather than differences in the material properties of the walls.
Compared with these estimates of elastic moduli in the plane of the plant cell wall, elastic moduli based on indentation using a small probe (10nm to 1 μm) yield values that are much smaller, by factors of 10- to >1000-fold (Milani et al., 2011; Peaucelle et al., 2011; Radotic et al., 2012; Ma et al., 2013; Zdunek and Kurenda, 2013; Sampathkumar et al., 2014a; Wang et al., 2014; Braybrook, 2015). The reason for the wide divergence in indentation values does not seem to have been explored, but it appears too large to be based on real differences in cell wall structure. The systematic difference between in-plane tensile moduli and values from indentation assays arises in part from the nature of the nano-indentation method, which applies a force that is normal to the plane of the cell wall. In other words, they are measuring different properties of an anisotropic wall (note that cell walls, which are traditionally called isotropic because of a random direction of cellulose in the plane of the wall, are still highly anisotropic when the third dimension, wall thickness, is considered). The calculation of a modulus from force–deflection curves by micro-indentation typically assumes that the indented material (i) is isotropic; (ii) shows linear elasticity; and (iii) is >10 times thicker than the depth of indentation. The construction of epidermal walls deviates from these assumptions, and mechanical contributions due to cell turgor pressure add to the difficulty. As noted by Milani et al. (2013), ‘it is unclear which properties of the cell wall are measured, because the force is roughly perpendicular to cellulose microfibrils’. In cross-sections of dry woody cell walls, where the nano-indentation method was first applied to plant cell walls, experimental complications and problems of interpretation were previously identified (Gindl and Schöberl, 2004). The mechanical situation for indentations of growing tissues presents a different set of problems, but I hazard a guess that the method at its best measures a type of wall compressive modulus, dominated by pectin properties, but with some (poorly defined) influence of the microfibrils at right angles to the principle force vector. The term Young's modulus or even ‘apparent’ Young's modulus for these measurements seems misleading. The term ‘indentation modulus’ (Eder et al., 2013) may be a clearer way of describing these values and avoiding confusion with Young's moduli in the plane of the wall.
From the foregoing discussion, it is evident that the modulus calculated from micro-indentation measurements differs from the in-plane tensile moduli of the cell wall. This disparity may not affect some conclusions based on relative changes in stiffness across a surface, but it could have significant ramifications for dynamical models of plant morphogenesis that depend on use of correct values for in-plane stresses and tensile moduli of the wall. Chickarmane et al. (2010) and Sampathkumar et al. (2014a) review some of these models. Additionally, recent studies indicate that the pectin-rich middle lamella can make a substantial contribution to the mechanical behavior of the multicellular epidermal surface (Zamil et al., 2014, 2015; Kim et al., 2015). Hence, the mechanical behavior of a subcellular patch of epidermal cell wall may not fully account for the mechanics of a multicellular sheet of epidermal cell wall. A related point was made by Chan et al. (2011) and Crowell et al. (2011) in studies of cellulose reinforcement of epidermal cell walls in Arabidopsis hypocotyls. They inferred from microscopic observations (but not mechanical measurements) that the outer epidermal wall is not the major determinant of the directionality of hypocotyl growth, which instead was linked to the anisotropic orientation of cellulose in the side walls and inside walls. Previous work supporting this concept was summarized by Baskin (2005). Hence, the contribution of internal cell walls to morphogenesis of other growing organs and meristems is worth further scrutiny (Baskin and Jensen, 2013), perhaps using the combination of mechanical measurements and modeling employed for shoot apical meristems.
Extensibility, cell wall structure, and wall loosening
A grand challenge in this field is to relate cell wall structure to the mechanics of cell walls and the action of cell wall-loosening agents that induce wall stress relaxation and provoke water uptake needed for cell growth. For more than two decades the dominant concept of the growing cell wall has been of a scaffold of well-separated cellulose microfibrils mechanically tethered together by xyloglucans and embedded in a pliant pectin gel (Hayashi, 1989; Pauly et al., 1999; Scheller and Ulvskov, 2010). Evidence for and against the tethered-network concept was summarized by Scheller and Ulvskov (2010), but recent years have witnessed mounting dissatisfaction with the concept (Cosgrove, 2014b). It falls short in view of the mild phenotype of xyloglucan-deficient mutants (Cavalier et al., 2008; Park and Cosgrove, 2012a), nuclear magetic resonance (NMR) analysis of matrix–cellulose interactions in the wall (Dick-Perez et al., 2011; Wang et al., 2015), and enzymatic probes of cell wall mechanics (Park and Cosgrove, 2012b). In addition, recent AFM imaging of never-dried primary cell walls shows that cellulose microfibrils are not evenly separated, as commonly depicted, but frequently make close contact with each other, forming junctions or bundles that may be important for wall mechanics (Zhang et al., 2014). Cellulose bundles are also seen in cellulose mats made by Acetobacter xylinus where they may contribute to the mechanics of these cellulosic hydrogels (Whitney et al., 1999; Martinez-Sanz et al., 2015). Thompson (2005) challenged the tethered-network concept by a calculation that hydrogen bonding between xyloglucans and cellulose was too weak to resist turgor-generated wall stresses (however, I believe the calculation made questionable assumptions); instead, he offered a purely polymer-based view of wall extensibility as controlled by hydration, spacing, and entanglement of matrix polymers. Although cell wall hydration contributes greatly to primary wall mechanics (Edelmann, 1995; Tang et al., 1999; Evered et al., 2007; Kennedy et al., 2007; White et al., 2014; Kim et al., 2015), the concept seems unlikely to provide a mechanism by which a cell regulates its growth.
A recent alternative to the tethered-network concept proposes that wall mechanics and wall extensibility are controlled at limited sites where cellulose microfibrils come into close contact (Park and Cosgrove, 2012b). In agreement with experimental results, most of the xyloglucan is bound to cellulose but does not serve a load-bearing role under static conditions. This concept accounts for lack of wall loosening by xyloglucan-specific endoglucanase and cellulose-specific endoglucanase, alone or in combination, in contrast to the effectiveness of bifunctional endoglucanases that cut both of these wall components (Park and Cosgrove, 2012b). It also accounts for the lack of appreciable loosening by xyloglucan endotransglucosylase (Saladie et al., 2006). The revised model proposes limited, inaccessible junctions (dubbed biomechanical hotspots) where cellulose and xyloglucan chains are tightly adherent to each other (Fig. 4). Such sites may also be the target of expansin (Wang et al., 2013). Suggestions of direct cellulose–cellulose junctions in primary walls may also be found in earlier reports (Boyd and Foster, 1975; Scheller and Ulvskov, 2010).
An artistic rendition of the arrangement of cellulose, xyloglucan and pectins based upon recent results (Cosgrove, 2014b). Cellulose microfibrils (~3nm diameter) are shown in blue and their arrangement is traced from an AFM image of the most recently deposited surface of an onion epidermal cell wall (Zhang et al., 2014). Xyloglucans (green) are shown as random coil structures interspersed with pectins and with limited contacts with cellulose (based on NMR results). Pectins (yellow) are shown as two forms, a rigid form in contact with cellulose microfibrils and a more mobile form filling the spaces between microfibrils. Potential cellulose–cellulose junctions are highlighted in red. Image adapted from Current Opinion in Plant Biology, 22C, Cosgrove DJ. Re-constructing our models of cellulose and primary cell wall assembly, 122–131, Copyright (2014), with permission from Elsevier.
The biomechanical hotspot model was tested by atomistic simulation (Zhao et al., 2014), which indicated that the strength of cellulose–xyloglucan–cellulose junctions may contribute substantially to the mechanical strength of the cell wall. This was a nanometer-scale test, but the more interesting mechanical properties of the hotspot model may emerge from simulation of larger scale behavior of the wall, much as the mechanical properties of a fishnet depend not just on the strength of the individual junctions between two ropes but on the spacing between junctions as well as the thickness and flexibility of the ropes. A nominal attempt in this direction was a finite element model (FEM) that simulated a single lamella of nearly parallel cellulose microfibrils connected by limited xyloglucan linkers (Nili et al., 2015). The linkers were implemented in the model as short tethers between microfibrils rather than as the monolayer of xyloglucan adhesive suggested by the enzyme results of Park and Cosgrove (2012b) and modeled by Zhao et al. (2014). The study concluded that relatively few xyloglucan linkers per cellulose microfibril were needed to maintain mechanical stability, but that additional load-bearing elements were needed to maintain realistic elastic moduli. This elastic weakness might be eliminated by including multiple lamellae with cellulose oriented in different directions in the lamellae. The authors speculated that pectins might contribute to mechanical strength of the wall, but pectins were not included in their simulation and likewise were missing in another FEM of a multilamellate cell wall (Kha et al., 2010). One problem is that the mechanism by which pectins may interact with cellulose is still unclear, an issue that merits deeper study, as discussed next.
Pectins
The recent re-evaluation of the role of xyloglucans in cell wall models coincides with renewed attention to the role of pectins (Palin and Geitmann, 2012; Peaucelle et al., 2012; Wang et al., 2012; Wolf and Greiner, 2012; Braybrook and Peaucelle, 2013). There are diverse views on the structure and the mechanical role of pectins in growing cell walls. The dominant and traditional view imagines that they form a soft, gel-like, viscous matrix in which the cellulose–hemicellulose network is embedded, but otherwise not binding cellulose (Carpita and Gibeaut, 1993; Cosgrove, 2005). In this view, pectins may contribute to cell wall stiffness during rapid deformations, but their viscosity would cause pectin-borne stresses to decay rapidly, displacing forces onto more rigid components of the cell wall. If the gel were to become sufficiently rigid, for example by calcium-cross-linking of unesterified regions of homogalacturonan, it could in theory lock the wall into an inextensible state; however, it is unclear whether this actually occurs. Pectin de-esterification is likely to affect many physico-chemical properties of the wall. A different view, first proposed in the pioneering molecular model of the cell wall by the Albersheim group (Keegstra et al., 1973; Albersheim, 1975), considers pectins to be part of a massive macromolecular matrix that contains covalently linked domains of pectin, hemicellulose, and protein and that extensively binds to cellulose surfaces through xyloglucan domains. Although this model was largely abandoned in the 1980s in favor of the tethered network concept (Fry, 1989; Hayashi, 1989; McCann et al., 1990), linkages between pectin and xyloglucan or pectin and proteins have been reported recently (e.g. Popper and Fry, 2008; Tan et al., 2013). A recent review (Park and Cosgrove, 2015) concludes that the abundance of pectin–xyloglucan linkages is low in most plant tissues, with the notable exception of cell suspension cultures. Low abundance, however, does not necessarily mean they are insignificant for wall mechanics. This point remains an open question.
Unlike xyloglucans, which bind tightly and irreversibly to cellulose (Hayashi, 1989), pectins bind to cellulose relatively weakly or not at all, at least during binding assays in vitro (Zykwinska et al., 2005, 2008). Lin et al. (2015b) reported weak, reversible pectin binding to bacterial cellulose pellicles, also called hydrogels because they are made of a loose, random entanglement of cellulose fibrils containing ~99% water. The pectin interactions with cellulose were weak and included components of mere inclusion and entrapment within the large free space of the entangled cellulosic mat. Neutral pectins (arabinan, galactan) showed stronger interactions than homogalacturonan (Lin et al., 2015a), consistent with the in vitro binding results cited above. Pectin contributed to the compression resistance of the cellulose–pectin composite by limiting hydraulic flows out of the composites, whereas xyloglucan interactions with the cellulosic pellicles were more stable and altered the pellicle compressibility in different ways (Lopez-Sanchez et al., 2015). Mobility-resolved 13C-NMR indicated that the pectin molecules were highly mobile, not tightly bound to cellulose.
These results with artificial composites are consistent with the common view that pectins are held in the wall only weakly, but are in contrast to 13C-NMR studies of native pectin in plant primary cell walls, which show abundant pectin–cellulose cross-peaks in 2D cross-polarization measurements, indicating extensive, stable pectin–cellulose interactions (Wang et al., 2012, 2015). As judged by solid-state NMR, cellulose contacts with pectins were more abundant than contacts with xyloglucan. Moreover, although pectins are the most mobile polymers in the primary cell wall, nearly half of the pectin was in a rigid form, with a 13C T1 relaxation time similar to that of cellulose. These results, supported by previous observations (e.g. Ryden and Selvendran, 1990; Foster et al., 1996), were interpreted to mean that a substantial pectin component in native cell walls is stably bound to cellulose surfaces or entrapped within cellulose bundles. However, the molecular nature of the cellulose–pectin interaction still is not clear and needs further investigation. Moreover, these results do not establish whether or not pectins have a significant static load-bearing role in cell wall mechanics.
In biomechanical assays (creep assays), wild-type Arabidopsis cell walls showed a relatively modest loosening effect when treated with pectin-degrading enzymes, with a somewhat larger effect in a xyloglucan-deficient mutant (Park and Cosgrove, 2012a). The results indicated that pectins do not bear static loads in wild-type cell walls, but may take on a larger mechanical role in the absence of xyloglucan. One limitation of this enzymatic approach is that if pectins are entrapped among microfibril bundles, they might not be accessible to enzymatic digestion, potentially missing a cryptic load-bearing component. Moreover, the essential role of pectins in cell–cell adhesion (i.e. the middle lamella) limits the extent of pectin weakening treatments feasible for these creep experiments where the sample is held under constant tensile load (multicellular wall samples fall apart when pectins are significantly degraded). This technical limitation may be circumvented by micro-indentation assays with isolated cell walls. In such assays with onion epidermal walls, Xi et al. (2015) used a sharp pyramidal probe and a Hertz model to calculate an elastic modulus from the force–deflection curves. Removal of calcium with a chelator reduced wall stiffness, decreasing the calculated indentation modulus from 23MPa to 16MPa, presumably as a result of reduced ionic cross-linking of homogalacturonan within the wall (the middle lamella was not probed). This result confirms that pectins contribute to the compression resistance of cell walls to fast deformations. This still leaves open the question of their contribution to static tensile loads and to slow creep in the plane of growing cell walls. For comparison, the tensile (Young's) modulus of comparable wall material was estimated as 400MPa (Kim et al., 2015)—a clear example of the difference between the indentation modulus and tensile modulus of the wall.
In an elegant study of fast and reversible polymer motions in onion epidermal walls stretched in the plane of the wall, Wilson et al. (2000) combined dynamic mechanical analysis with polarized infrared (IR) spectroscopy. They concluded that homogalacturonan motions were uncoupled in time from cellulose and xyloglucan motions. To explain this temporal shift, they suggested that homogalacturonan motions were dominated by strains in the middle lamella which were subsequently transmitted to cellulose in the cell wall proper. A subcellular version of this experiment that eliminated the pectin signal from the middle lamella would be informative for the question of pectins as load-bearing elements of the wall proper, particularly if time constants for decays in wall stress could be assessed for the different structural components of the primary cell wall. From dynamic mechanical measurements of (living) potato explants in combination with a mathematic model, Ulvskov et al. (2005) inferred that pectins can transmit dynamic, but not static, stresses to cellulose.
Turning to purely computational approaches, Dyson et al. (2012) developed a noteworthy mathematical model of cell wall yielding based on continuum mechanics. Their model considers hemicellulose and pectin to be acting as co-load-bearing elements, modeled with a mathematical function giving strongly non-linear stress/strain rate behavior resembling a yield threshold. The yield threshold was simulated by cross-links that stretch up to a limit, then break suddenly, with a role for pectins in the post-breakage yielding. The model mimics a number of the behaviors of growing cell walls; by its nature it cannot offer insights into the structural or biochemical bases for wall stress relaxation, but it does present an attractive approach for further efforts to relate structural models of cell walls to their mechanical behaviors beyond simple elastic moduli.
Wall loosening
As a final point, it is instructive to compare the wall-loosening actions of α-expansin and Cel12A, a family-12 endoglucanase that cuts both cellulose and xyloglucan (Table 1). These are the only two classes of proteins known to induce stress relaxation and creep of plant cell walls, although it should be noted that Cel12A is a fungal enzyme and we do not know of plant enzymes with similar wall-loosening action. Characterizations of two family-9 endoglucanases from plants indicate that they have a wide substrate specificity (Ohmiya et al., 1995; Yoshida and Komae, 2006), suggesting a wall-loosening action similar to Cel12A, but such action has not actually been documented. In support of this notion, ectopic expression of one of these enzymes in Arabidopsis promoted plant growth (Park et al., 2003), and transcript profiling identified several family-9 genes that are expressed in growing tissues of grasses (Buchanan et al., 2012).
Comparison of the characteristics of cell wall loosening by α-expansins and Cel12A, based on their activity with cucumber hypocotyl walls, based on published results (Yuan et al., 2001; Park and Cosgrove, 2012b)
| Property | α-Expansin | Cel12A |
|---|---|---|
| Hydrolytic action | No | Yes |
| Increased plasticity | No | Yes |
| Increase elasticity | No | Yes |
| Induces creep | Yes | Yes |
| Lag time for induction of creep | Seconds | 6min to >60 min |
| Property | α-Expansin | Cel12A |
|---|---|---|
| Hydrolytic action | No | Yes |
| Increased plasticity | No | Yes |
| Increase elasticity | No | Yes |
| Induces creep | Yes | Yes |
| Lag time for induction of creep | Seconds | 6min to >60 min |
Comparison of the characteristics of cell wall loosening by α-expansins and Cel12A, based on their activity with cucumber hypocotyl walls, based on published results (Yuan et al., 2001; Park and Cosgrove, 2012b)
| Property | α-Expansin | Cel12A |
|---|---|---|
| Hydrolytic action | No | Yes |
| Increased plasticity | No | Yes |
| Increase elasticity | No | Yes |
| Induces creep | Yes | Yes |
| Lag time for induction of creep | Seconds | 6min to >60 min |
| Property | α-Expansin | Cel12A |
|---|---|---|
| Hydrolytic action | No | Yes |
| Increased plasticity | No | Yes |
| Increase elasticity | No | Yes |
| Induces creep | Yes | Yes |
| Lag time for induction of creep | Seconds | 6min to >60 min |
Cel12A loosens plant cell walls by its hydrolytic activity, whereas no enzymatic activity has been found for α-expansin. Both proteins induce creep of isolated, heat-inactivated cell walls from cucumber hypocotyls and other plant materials, but with very different kinetics: α-expansin induces creep within seconds after application whereas Cel12A induced creep only after a substantial lag time that depended on enzyme concentration: a minimum of 6min at saturating enzyme concentration, >60min for low enzyme concentrations (Yuan et al., 2001). Cel12A reduced the elastic and plastic moduli of cucumber walls in uniaxial tensile tests, whereas α-expansin did not change these moduli. Note that the plastic modulus in these assays measured the irreversible component of uniaxial extension, which Nolte and Schopfer (1997) have re-interpreted to be the result of delayed viscoelasticity rather than true plasticity. Under this interpretation, Cel12A increases the viscoelastic hysteresis in the stress–strain curves, whereas α-expansin does not. By either interpretation, the results suggest that α-expansin can cause stress relaxation and creep by a mechanism (sliding at cellulose junctions?) that does not perceptibly alter cell wall structure, whereas the cutting action of Cel12A substantially reduces the number or strength of cellulose–cellulose contacts, resulting in a mechanically weaker cell wall. Attempts to simulate cell wall creep by cutting cross-links (e.g. Dyson et al., 2012) may not properly mimic expansin's loosening activity, so a refined model may be needed.
The loosening action of α-expansin bears some resemblance to the ‘stick–slip’ or ‘molecular Velcro’ model of plastic deformation of wet wood (Keckes et al., 2003; Cosgrove and Jarvis, 2012; Adler and Buehler, 2013) in which hydrogen-bonded contact regions between cellulose layers slip and then stick again, restoring mechanical strength until local stresses exceed the yield threshold and another local slip event occurs. In woody tissues this is a purely physical process, whereas in growing cells the selective slippage, relaxation, and creep of primary walls are mediated by expansins (and potentially other agents). The fact that α-expansin can loosen cell walls without reducing the plastic or elastic moduli is a likely explanation for the fact that wall extensibility and wall viscoelasticity are often uncorrelated, as discussed above. Another example of such a discrepancy is a micro-indentation study of plasticity along the growing root of Arabidopsis thaliana (Fernandes et al., 2012). Plasticity was measured as hysteresis in the approach and retraction phases of the force–deflection curves. This is a micro-scale version of the hysteresis seen in uniaxial tests of cell walls, which is the basis for estimating elastic and plastic moduli (Cleland, 1967, 1971). In the study by Fernandes et al. (2012), no correlation was found between the ‘index of plasticity’ on the living root surface and the growth rate along the root axis, despite the fact that cell growth and wall extensibility vary greatly along the axis (Pritchard, 1994). It might be useful to try this micro method with wall samples known to vary in wall plasticity (or delayed viscoelastic hysteresis), for example after Cel12A treatment, to validate that the method can indeed detect useful plasticity values for cell walls.
Concluding remarks
There is a growing recognition that stresses and strains in the growing cell wall feed back to cellular systems to regulate the cytoskeleton, vesicular trafficking of auxin transporters, and the cellular machinery involved in cellulose deposition. This provides the impetus for insightful dynamical models connecting wall mechanics with microtubule organization and studies to identify the molecular connections of these elements of the model. Careful thought needs to be given to the meaning of wall moduli based on surface indentation assays. They differ from the in-plane tensile elastic modulus, which is the parameter of greatest value for assessing the stress/strain properties of the cell wall. A deeper understanding of wall deformations—both reversible and irreversible—will require more extensive experimental testing in combination with quantitative models of how the structural components of the cell wall are linked to one another and what kinds of polymer motions occur during rapid, reversible deformations versus the slow irreversible enlargement of the growing wall.
Acknowledgements
Preparation of this review was supported by the US Department of Energy, Office of Science, Basic Energy Sciences as part of The Center for LignoCellulose Structure and Formation, an Energy Frontier Research Center under Award # DE-SC0001090.
References
Author notes
Editor: Nadav Sorek, University of California Berkeley
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