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Trends in Plant Science - Feature Review, to be published in September 2009 Mechanics and modeling of plant cell growth Anja Geitmann1, Joseph K. E. Ortega2 1 Institut de recherche en biologie végétale, Département de sciences biologiques, Université de Montréal, Québec H1X 2B2, Canada 2 Bioengineering Laboratory, Department of Mechanical Engineering, University of Colorado Denver, Denver, Colorado 80217-3364, USA Corresponding author: Geitmann, A. ([email protected]) Cellular expansive growth is one of the fundamental underpinnings of morphogenesis. In plant and fungal cells, expansive growth is ultimately determined by manipulating the mechanics of the cell wall. Therefore, theoretical and biophysical descriptions of cellular growth processes focus on mathematical models of cell wall biomechanical responses to tensile stresses, produced by the turgor pressure. To capture and explain the biological processes they describe, mathematical models need to be supplied with quantitative information on relevant biophysical parameters, geometry, and cellular structure. The increased use of mechanical modeling approaches in plant and fungal cell biology emphasizes the need for the concerted development of both disciplines and it underlines the obligation of biologists to understand basic biophysical principles. Mechanical aspects of plant and fungal cell growth Plant development is the result of three essential processes: cell expansive growth, cell division, and cellular differentiation. All three processes have key mechanical aspects that have prompted numerous attempts to generate theoretical, mechanical, and biophysical models. In the present review we will focus on cellular expansive growth in walled cells typical for plants, algae and fungi. Given that walled cells rarely migrate, cell expansive growth contributes in dramatic manner to the generation of a particular phenotype. Expansion is involved in the generation of both, increase in cell size and change in cell shape. Increase of cell volume during the differentiation of a meristematic cell into its destination cell type is typically between 10 and 1000-fold [1], but can reach up to 30 000-fold, for example in the case of xylem vessels [2]. While the increase in cellular surface (L2) that is necessary to accommodate the increase in volume (L3) generally is smaller by approximately one dimension, the amount of additional cell wall that has to be generated is nevertheless impressive. Implicit in cell expansive growth is a mechanical process that balances internal and external stresses with the compliance to allow expansion [3,4]. The balanced counterforce of primary wall stress to turgor pressure has prompted the comparison of plant cells with "hydraulic machines" [5]. Cell mechanics is therefore crucial for understanding plant cell functioning. The relevance of cell mechanical principles is particularly obvious in the light of recent studies that illustrate how mechanical changes influence and trigger cell biological processes. The local microinduction of expansin expression, and thus cell wall softening, is sufficient to induce morphogenetic processes leading to the initiation of leaf structures from the shoot apical meristem [6]. Microtubule orientation in the shoot apical meristem was found to follow the orientation of stress patterns in this organ and the experimental removal of individual cells caused microtubules to realign along the newly created stress patterns [7]. Enlisting engineering methods such as finite element modeling for plant biological applications [7-9] requires an increased and critical understanding of the mechanical and physical concepts and the challenges associated with their adaptation to plant and fungal biology. Given that cellular expansive growth in plant and fungal cells occurs almost exclusively prior to the deposition of the secondary cell wall, we will focus here on the mechanics of the primary cell wall. Both conceptual and mathematical models can contribute to our understanding of three critical aspects defining cell expansive growth: (i) the mechanical functioning of the structural network composing the plant and fungal cell wall, (ii) the mechanism of the cell wall modifying agents, and (iii) the behavior of the cell wall under the influence of the tensile stress resulting from the presence of the internal hydrostatic pressure, the turgor pressure. The objective of the present review is to provide an overview of the principles, concepts, and biophysical equations associated with the mechanical aspects of plant and fungal cell growth. Biophysical and biomechanical terms are explained and the most important advances in the field are summarized. Biophysical equations describing cell wall mechanics and expansive growth To model and predict the inherently mechanical process of cell expansive growth, equations have been derived for the underlying physical processes and coupled to the relevant biological processes with biophysical variables, thus forming biophysical equations [10]. In the case of expansive growth of cells with walls, two relevant physical processes are the net rate of water uptake and the rate of cell wall deformation which occurs in response to the cell wall stresses produced by the turgor pressure. These two simultaneous and interrelated physical processes provide the foundations for the derivation of biophysical equations with inclusive biophysical variables, whose magnitude and behavior are regulated by interrelated biological processes. Overall, the biophysical equations couple physical processes with interrelated biological processes, and provide a quantitative system model for the interrelated biological processes [10-13]. The increase in size of a growing algal, fungal, or plant cell can be described by (dV/dt)w/Vw = Lpr (Δπ - P) - (dT/dt)w /Vw (Eq. 1) (dV/dt)cwc/Vcwc = φ (P - Pc) + (1/ε ) dP/dt (Eq. 2) (rate of increase of water volume) = (net rate of water uptake) – (transpiration rate) (rate of increase of cell wall chamber volume) = (irreversible expansion rate) + (elastic expansion rate) Where V is the volume, t is the time, Lpr is the relative hydraulic conductance, Δπ is the osmotic pressure difference inside and outside the cell membrane, P is the turgor pressure, T is the water volume lost through transpiration, φ is the irreversible wall extensibility, Pc is the critical turgor pressure, and ε is the volumetric elastic modulus. Because the relative rate of increase of water volume, (dV/dt)w/Vw, is 1 essentially equal to the relative rate of increase in the volume of the cell contents, and essentially equal to the relative rate of increase of the cell wall chamber, (dV/dt)cwc/Vcwc, another equation can be obtained for the rate of change of turgor pressure [10,13]. dP/dt = ε { Lpr (Δπ - P) - (dT/dt)w /Vw - φ (P - Pc)} (Eq. 3) (rate of change of P) ∝ (net rate of water uptake) – (transpiration rate) – (irreversible expansion rate) It should be noted that the Lockhart Equations [14] are recovered from Equations 1 to 3 for the limiting cases when transpiration is zero or neglected, and elastic wall deformations are neglected (they are never zero). Importantly, the Lockhart Equations cannot model wall stress relaxation [11], pressure relaxation [15-18], and elastic deformations [18-20] exhibited by living and growing cell walls. Equation 2 has been modified for the case of elongation growth [10,19]. dL/dt = m (P – Pc) + (Lo/εL) dP/dt (Eq. 4) (elongation rate) = (irreversible extension rate) + (elastic extension rate) Where L is the length of the cell, m is the longitudinal irreversible wall extensibility, Lo is the initial length, and εL is the longitudinal volumetric elastic modulus. Equations 1 to 4 have been termed the augmented growth equations [10-13] (a detailed explanation can be found in the Supplementary material). Molecular models and the molecular interpretation of the biophysical variables Conceptually, growing (primary) cell walls of algal, fungal and plant cells can be viewed as being composed of a network of microfibrils cross-linked by tethers and embedded in a somewhat amorphous matrix of cell wall materials. Generally, microfibrils are synthesized in the plasma membrane, while the matrix materials are transported from cellular organelles to the plasma membrane and released to the inner cell wall via exocytosis. The molecular mechanisms by which cell wall materials are assembled and incorporated into the deforming cell wall to prevent thinning and eventual rupture, is highly debated. Furthermore, the molecular mechanisms responsible for the cell wall loosening which produces the irreversible cell wall deformation of expansive growth are poorly understood despite the increasing knowledge of the molecular players involved. The magnitude and behavior of the biophysical variables within the augmented growth equations have been determined for a variety of cells with walls [10], and can be used to guide and evaluate conceptual molecular models for cell wall assembly and loosening. Behavior of biophysical variables during expansive growth Because the relationship between cell wall stresses and turgor pressure is generally very complicated in tissues and organs, it is useful to focus on single cells where the relationship is more direct and generally only depends on the geometry of the cell wall. The magnitudes of cell-wall biophysical variables for fungal single-celled sporangiophores of Phycomyces blakesleeanus and algal single-celled internodes of Chara corallina have been determined with in vivo creep experiments employing the pressure probe [18,19,21,22]. These studies revealed that the magnitude of the irreversible wall extensibility, m, increases with the elongation rate, but by a disproportionate amount [10]. The disproportionate increases in m are accompanied by a decrease in the magnitude of (P - Pc). For different developmental stages of the sporangiophores [10], the magnitudes of both P and Pc decrease as the elongation rate increases. Also, the magnitudes of the volumetric elastic modulus, εL, are larger for nongrowing cells compared to growing cells, and for growing cells, the magnitudes of εL decrease as the elongation rates increase [10,19]. Previous models The most useful molecular models are those that predict behavior which can be compared to experimental results. For example, a model that predicts strain-hardening and loosening of the cell wall material can explain a number of experimental observations [23]. Conceptually, the basic molecular model consists of cellulose microfibrils oriented perpendicular to the long axis and tethered to each other by threads of hemicellulose molecules. It is assumed that the hemicellulose molecules are attached to the microfibrils by hydrogen bonds. An increase in longitudinal wall deformation (strain) resulting from an increase in turgor pressure would cause more hemicellulose threads to become load-bearing, thus resulting in strain-hardening. Wall loosening on the other hand is proposed to occur when the load-bearing hemicellulose threads are cut or become detached from the microfibril because large longitudinal strains break relevant hydrogen bonds. From this model it can be deduced how the yield threshold, Y (and Pc), changes when the wall is subjected to a sudden increase in longitudinal strain, thus explaining experimental observations [24]. Equations for m and Y are derived which describe their dependency on the relative magnitudes of strain-hardening and wall loosening. These equations for m and Y make three predictions: (i) that transient changes in elongation rate will result from stepchanges in P. This prediction is consistent with experimental results obtained from algal [24], plant [25], and fungal cells [18,21]; (ii) that m will be larger for faster growing cells compared to slower growing cells. This prediction draws support from both algal and fungal cells [10]; (iii) that Y (and Pc) will be smaller for faster growing cells. This prediction draws support from fungal cells [21]. A limitation of the strain-hardening model [23] is, however, that it ignores elastic deformations. Thus, it does not make predictions as to the behavior of εL as a function of elongation rate that is observed experimentally. A thermodynamic analysis was conducted for the conceptual molecular cell wall model consisting of cellulose microfibrils cross-linked by matrix hemicellulose (glucan) threads attached to the microfibrils by hydrogen bonds [1]. These analyses predict that additional secretion of long glucan threads, that can form multiple links (strands) between adjacent microfibrils, increases the magnitude of the yield threshold, Y (and Pc). Furthermore, cutting the glucan threads into smaller lengths, which will reduce the number of strands it can form between microfibrils, is expected to lower the magnitude of Y (and Pc). Support for these predictions awaits relevant experimental research. New models Recently, new models have been introduced to explain expansive growth. The LOS model suggests that the initiation of expansive growth can be explained and modeled by loss of stability (LOS) theory [26,27]. LOS theory is commonly used in engineering to study the buckling-stability of elastic structures in compression, and has been applied to elastic thin-walled spherical and cylindrical pressure vessels in an attempt to model plant cell growth [26,27]. The basic tenet is that the turgor pressure is gradually increased to a critical value, PCR, at which a loss of stability of the cell wall occurs, resulting in stress relaxation in the wall. The authors claim that the “current viscoelastic model of cell wall relaxation, which dates from the work of Preston, Cleland, Lockhart, and others in the 1960s, has serious shortcomings” and “that LOS also provides a more appropriate and versatile model of stress relaxation in growing plant cells” [26]. It is debated whether the LOS process is inconsistent with the generally accepted pressure relaxation and water uptake process for expansive growth [28]. However, the critical question is whether the LOS theory and its governing equation can model the stress relaxation that supposedly occurs after PCR. It is important to note, that the theory and its governing equation are only valid for the pressures before and when it reaches PCR. In fact, nothing can be said about the pressure behavior after PCR is reached, because it is undefined in the LOS theory. The LOS theory cannot even predict that stress relaxation occurs after PCR. By their own admission, the authors of the LOS theory state that “the dashed line in Fig. 1B” (showing pressure relaxation) “is a mathematical completion of the 2 Figure 1. Geometry dependence of surface stress patterns in thin-shelled pressure vessels. (a) A spherical shell experiences isotropic stress patterns. (b) In a cylindrical body the stress in circumferential direction is twice as large as the longitudinal stress. (c) The largest stress in a body with local outgrowth is located at concave bend between the main body and the outgrowth. Arrows in dark blue represent relatively larger stress than arrows in light blue. curve, but it has no physical meaning in that it does not represent any definable stress-strain relationship” [26]. Importantly, critical experiments demonstrating that stress relaxation and expansive growth are initiated by increasing the turgor pressure to a critical value, PCR, have not been reported. In addition, the relationship between PCR in the LOS theory and Pc in the Lockhart equations and augmented growth equations is unknown. Another new model focuses on pectin chemistry and calcium ion movements [29-33]. It suggests that the rate of irreversible wall extension is regulated by the rate of unsubstituted polygalacturonic acid (pectate) supplied to the cell wall (via the cytoplasm) and the calciumpectate chemistry that occurs within the cell wall. The experimental data supporting the model provide insight into the incorporation of cell wall materials into a stress-bearing cell wall, wall assembly, and the role of turgor pressure at a molecular level. Furthermore, the authors propose calcium-pectate chemistry that describes molecular mechanisms for cell wall loosening and cell wall assembly. Importantly, this model explicitly describes the relationship between the delivery of cell wall materials, its incorporation into the growing cell wall, and cell wall loosening. Future models The mechanical behavior of the growing cell wall demonstrates a relationship and interplay between irreversible and reversible (elastic) deformation. The relationship between irreversible and elastic cell wall deformation is explicit in the augmented growth equations (equations 1 to 4) and the interplay can be recovered from their solutions for different cases, e.g. pressure relaxations [11,16] and step changes in turgor pressure [19,20]. Current molecular models of the growing cell wall focus, almost exclusively, on molecular mechanisms that produce irreversible wall deformation. Generally, the molecular models neglect the elastic deformation and the interplay between the irreversible and elastic deformation that occurs during experiments and expansive growth. For instance, experimental results indicate that εL decreases as m increases [10]. This behavior might reflect the percentage change of the stress-bearing bonds in the cell walls growing at different rates, so as more stress-bearing bonds are broken to loosen the wall, fewer stress-bearing bonds remain to store energy for reversible (elastic) deformation. This relationship between m and εL can be used to guide and evaluate new more complete molecular models for cell wall assembly and loosening. Quantification of physical parameters at cellular level To supply mathematical models with relevant and accurate input, quantitative values for a number of physical parameters need to be provided. While educated guesses are often the only recourse, valuable quantitative information is becoming increasingly available through the application of various biomechanical and cytomechanical techniques [34-36]. The two central and measurable quantities involved in algal, fungal and plant cell growth are cell wall deformability and hydrostatic pressure. In terms of the Lockhart equations and the augmented growth equations (Supplementary material), their experimental quantification allows for the determination or estimation of ε (volumetric elastic modulus), φ (irreversible wall extensibility), and m (longitudinal irreversible wall extensibility), whereas measurement of the cellular hydrostatic pressure allows for the determination or estimation of P (turgor pressure). Numerous micromechanical testing methods have been developed to assess these parameters at tissue, cellular and molecular level [34,35]. The challenges of experimentation at cellular level lie primarily in the small size of single celled specimens and the mechanical and geometrical complexity of multicellular specimens that complicate the extraction of absolute values for biophysical parameters such as the Young's modulus of the cell wall. Mechanics of anisotropic plant cell growth: the microtubulecellulose connection Differentiated plant cells come in all shapes and sizes ranging from simple cylindrical cells (e.g. palisade mesophyll) to star-shaped complex structures (e.g. astro-sclereids)[37,38]. The fact that even these complex shapes are determined by the cell wall can easily be demonstrated by enzymatic digestion of the latter resulting in a perfectly spherical protoplast. Since hydrostatic pressure always produces a force perpendicular to the surface it acts on, the generation of complex geometries requires the mechanical properties of the cell wall to show non-uniform and (or) anisotropic distribution. Modeling the cell wall as fiber-reinforced composite material For the cell to globally elongate in a particular direction, cell wall deformability in this direction must be lower. Once a geometry different from a sphere has been established, this mechanical anisotropy may have to be even bigger in order to sustain elongation, since both surface geometry and local mechanical properties contribute to the spatial distribution of stresses on the cellular surface (Fig. 1) [39]. The orientation of cellulose microfibrils in the cell wall is generally recognized as a crucial parameter determining anisotropy in cell wall deformability under tensile stress [40-42]. Because of the cell wall's heterogeneous composition consisting of crystalline cellulose polymers and amorphous matrix components [43], the comparison with a fiber reinforced composite material has been helpful for modeling this material behavior [44,45]. Hence, in cells exhibiting elongation growth, microfibrils are typically oriented perpendicular to the long axis on the inner surface of the cell wall thus restricting deformation to the larger stresses in the circumferential direction [40,46-50]. The role of the fiber angle is being addressed in a recent model that combines the principles of glass blowing with that of a fiber reinforced material to model the mechanics of the cell wall (Dyson and Jensen, unpublished). Modeling the plant cell wall by defining a two-term strain energy function, one term each for the matrix and the microfibril phases, it was established that in addition to microfibril extensibility the matrix shear modulus is an important variable [51]. This aspect is occasionally neglected when discussing the effect of cellulose microfibrils on cell wall deformability. During elongation growth, the primary wall deforms predominately in the long axis, separating the space between the essentially transverse microfibrils and passively reorienting slightly tilted microfibrils toward the longitudinal axis. The latter process has been termed multi-net growth and is proposed to be responsible for the changing orientation of microfibrils through the thickness of the cell wall [48-50]. The 3 multi-net concept has been challenged since no significant reorientation of microfibrils was observed upon application of strain up to 30% during in vitro creep experiments [52]. However, simple geometrical calculations based on the influence of the initial microfibril angle and the applied strain on the final microfibril orientation demonstrated that the expected reorientation would have been too small to be recognized in this experimental setup [53]. Hence, these experiments are not necessarily inconsistent with the classical multi-net growth hypothesis. Interestingly, helical growth in fungal cells has been explained by a preferred direction of microfibril reorientation during irreversible cell wall deformation [54-56]. In anisotropic elongation growth perpendicular to microfibril orientation it is apparent that stretching and/or breaking of the (hemicellulose) connections between the microfibrils, and hence the mechanical properties of these linkages, are an important limiting factors regulating expansibility [2,57-59]. However, even in the case of tensile stress parallel to the direction of microfibril orientation, the cellulose fibers can only provide mechanical resistance if they are connected to each other - either directly, via hemicellulose tethers or by generating friction between each other, or indirectly, by generating shear within the amorphous matrix. The reason for this is that unless the microfibrils form complete hoops or spirals around the cell, tensile stress would simply cause them to slide against each other (Fig. 2). Since the typical length for microfibrils seems to be Figure 2. Mechanics of a composite material with parallel arrangement of the fiber component. (a) Hypothetical composite material in which the fiber elements form hoops around the cellular circumference. Wall extensibility in the direction parallel to the fibers would be largely determined by the mechanical properties of the fibers and their relative density. (b) Composite material consisting of fibers of limited length embedded and bonded to the matrix. Wall extensibility in direction parallel to the fibers is largely limited by the shear modulus of the matrix and the density of the fibers (c) Composite material in which the fibers (microfibrils) are connected by tethers (e.g. hemicellulose polymers) in addition to being embedded and bonded to the matrix. The mechanics of the tethers and their attachments largely influences both wall extensibility parallel to and perpendicular to the fiber orientation. The shade of red of the arrows indicates the overall wall extensibility of the material in the indicated direction (red - high extensibility). The terms between the arrows indicate the biophysical and structural parameters limiting extensibility in the indicated direction. The parameters represent the mechanical properties of fibers/microfibrils (red), tethers/hemicellulose links (green), and other matrix components (yellow). 4 below 10 µm [60], it is unlikely that individual microfibrils form complete hoops around the cellular perimeter. Hence, the length of the microfibrils, the mechanical properties of the connecting matrix tethers, their abundance relative to the density of microfibrils, and the deformability of the other matrix components should be important parameters determining mechanical behavior of the cell wall not only perpendicular, but also parallel to microfibril orientation (Fig. 2). This might explain how other cell wall components such as arabinogalactan proteins are able to mechanically influence cell wall anisotropy [61] or control elongation growth rates [62]. It has been proposed that matrix molecules do not actually act so much as a tether but rather as a spacer between the microfibril rods [63]. However, both functions are not mutually exclusive and the mechanical role of matrix molecules might depend on the particular situation and the angle of the microfibrils versus the direction of cell wall expansion [53]. The list of biophysical parameters determining expansibility will certainly have to be revised in the future, once the hierarchical organization and the interconnections between individual types of molecules are better understood (for a review of different conceptual models see [58]). Cytoskeletal control of microfibril orientation For developmental biologists the question arises how cells control the orientation of their microfibrils. The importance of the role of the microtubule cytoskeleton is undisputed in this context [46,64,65], but our understanding of the mechanism of the interaction and mutual control between microtubules and microfibrils is still rather poor. In many plant cells microtubule arrays are arranged parallel to the main microfibril orientation which led to the concept that movement of cellulose synthase enzyme complexes in the plasma membrane is constrained by interactions with the cortical microtubules [66]. However, this concept turned out to be inconsistent with observations of continued synthesis of organized cellulose microfibrils following the disruption of cortical microtubules by pharmacological agents [46,67] or mutation [68,69]. In mor1 (microtubule organization1) mutants microtubules are shortened and disorganized in a temperature-sensitive manner leading to a loss in growth anisotropy. However, cellulose microfibrils continue to be deposited transverse to the long axis of the plant organ even after prolonged disruption of cortical microtubule arrays and the onset of radial expansion. This is true for both mutation and drug induced interference with microtubules. Even more dramatically, change in growth anisotropy of root cells occurred in rsw4 (radially swollen 4) and rsw7 mutants despite the unaltered, horizontal orientation and abundance of both microtubules and microfibrils [70]. Several conceptual models have been proposed that could explain the role of microtubules in controlling cellular expansion other than by determining microfibril orientation. Microtubules might be required to concentrate and organize cellulose synthase complexes to ensure efficient cellulose assembly into higher order clusters of fibers [71]. Alternatively, microtubules might ensure the coordination between cellulose deposition and the delivery of proteins and other molecules that optimize cell wall structure [72]. Microtubule control of the mechanical cell wall properties might also be based on their influence on cellulose synthase turnover thus through the determination of microfibril length [73]. Shorter microfibrils might lead to an altered mechanical behavior of the cell wall thus allowing more expansion in the direction parallel to cellulose orientation. More than 30 mutants affecting the microtubule cytoskeleton are known to interfere with growth anisotropy [74] and hopefully the mechanics of the interaction between cytoskeleton and microtubules will be elucidated in the near future. Determining the actual length of microfibrils and their degree of clustering in the cells of these mutants will most certainly be necessary to validate any of the above models or allow the generation of better ones. Investigating the possible role of other cell wall components in the interaction between cell wall and cytoskeleton will be another important strategy to pursue [75]. There have been attempts to explain the orientation of microfibrils in cylindrical cells without the necessity of a guidance mechanism. These are essentially based on geometrical considerations, the hypothetical limitation of space into which microfibrils can be deposited, and the dynamics of cellulose synthases dispatching into the membrane [76-79]. These models have certainly provided food for thought and will help to develop experimental approaches that will be able to provide answers. Heterotropic growth behavior Cellulose-mediated anisotropy in the deformability of the cell wall is generally associated with overall anisotropic cellular expansion. Approximately spherical or polyhedric cells derived from the apical meristem elongate by anisotropic deformation of large portions of their surface. Typically, cells that have differentiated through this mechanism do not have any sharp concave bends in their surface. Numerous cell types, however, do have such concave bends or exhibit other types of complex geometries. Dramatic examples include star-shaped trichoblasts, astro-sclereids, stellate aerenchyma cells, lobed leaf epidermis cells, and the cylindrical protrusions typical for pollen tubes and root hairs. One of the mechanically most puzzling cases of concave bend formation is the elongation of Rhizobium induced infection threads in legume root hairs. These wall invaginations allow the nitrogen fixing bacteria to reach the root cortex by invading the root hair. The threads are able to maintain their shape against the turgor pressure of the surrounding cell [80]. Generating different geometries How are these more complex geometries generated? The common feature of many of the cell types with complex geometry is a relatively large cell body producing one or several roughly cylindrical or finger-like extensions that can be branched. These extensions are generated by spatially confined growth events that must rely on locally increased rates of cell wall deformation and deposition. To distinguish spatially confined growth events from global cell growth, we propose to name a local growth event heterotropic and global deformation homotropic (Fig. 3). The mechanics and geometry of heterotropic growth events varies largely between cell types and not in all cases have surface expansion rates been assessed quantitatively. Spatial gradients in the chemistry and distribution of the matrix polymers such as pectin play an essential role in the generation of these shapes [81]. In the stellate aerenchyma, the star-shaped cells are initiated by the detachment of adjacent parenchymatic cells at three or four way junctions. The resulting intercellular spaces increase in size to eventually become an anastomosing network. While developmental information is scarce and cell wall deformation patterns for this cell type have not been published to our knowledge, it is likely that the site of cell wall expansion is initially located at the concave bend and subsequently in the cylindrical walls of the branches (Fig. 3). By contrast, lobe formation in the jigsaw puzzle shaped epidermis cells seems to be generated by a cellulose-based reinforcement of non-growing regions and tip-growth like outgrowths of the lobes [82]. How "tip-focused" this growth really is remains to be seen. A high rate of cell wall expansion at the very pole of the lobe would generate friction with the adjacent concave bend of the neighboring cell. Therefore, it is more likely that the side walls of the lobes are the sites of highest expansion (Fig. 3). Branch formation in Arabidopsis trichomes is initiated by a highly spatially confined growth event on the surface of an existing trichome and subsequently branches elongate by large-surface expansion of the cylindrical side walls [38,83](Fig. 3). A similar mechanism is likely to act in cotton fibers - the single celled trichomes formed from the ovule epidermis of Gossypium hirsutum [84]. By contrast, cell wall expansion in pollen tubes, root hairs, moss protonemata, algal rhizoids, and fungal hyphae is confined to the apical tip of the cylindrical extension during the entire growth 5 Figure 3. Overview of cell wall expansion patterns resulting in various cellular geometries. Different types of cell wall expansion patterns can be combined in a single cell. They can occur simultaneously or at different times during differentiation. Areas on the cell surface undergoing expansion are marked in red. Question marks indicate lack of quantitative data on surface expansion patterns. For some categories alternative terms are provided. Typical cell types for each expansion pattern are listed on the right. period, the cylindrical walls of these cells do not expand [85-88]. To distinguish cells with a single site of increased growth activity from those with multiple sites, such as the lobed epidermal cells, we propose the terms monotropic for the former and pleiotropic for the latter (Fig.3). The tip-focused pattern of surface expansion is in part explained by the non-uniform distribution of cell wall components. In pollen tubes this is expressed by the changing pectin chemistry along the longitudinal axis [81] and the reinforcement of the cylindrical shank by callose [89] and cellulose. Recent evidence revealed, however, that the highest rate of surface expansion in root hairs and pollen tubes does not occur at the extreme tip or pole of the cell, but rather at an annular region around it [90]. The differences between "tip" growth and other growth patterns might therefore be quantitative rather than qualitative, with a growing region that is more spatially confined but not principally different from that in other cell types. This illustrates that despite being hailed for a long time as a very distinct growth mechanism, tip-growing cells might actually only represent the extreme end of a gradual spectrum defining the geometry of plant cell growth. Underlying mechanics The question is what defines the more or less spatially confined surface areas that are subject to expansion in heterotropic growth events and what prevents adjacent areas from being deformed? Intriguingly, these stable, adjacent areas often have to resist higher tensile stress than the growing regions due to cellular geometry resulting in differential distribution of stress patterns on the cellular surface [87] (Fig. 1). Clearly, spatially confined sites of cellular expansion require the establishment of local differences in cell wall mechanical properties [39] and theoretical models of the geometry and structure provide information on the exact degree of difference in mechanical properties that is necessary to generate a particular heterotropic shape. The simple radial symmetry of cylindrical, tip-growing protrusions such as those formed by pollen tubes, root hairs, and fungal hyphae, has made these cell types the subject of numerous theoretical models and, more recently, of cytomechanical studies. Ranging from relatively simple geometrical considerations [91,92] most treat the expanding cell using an elastic membrane model [9397]. Micro-indentation has revealed that the growing apical region in pollen has indeed lower cellular stiffness than the distal region of the same cell, which is at least in part due to the differences in cell wall mechanical properties [8,81,89,98]. The degree of apical stiffness also correlates with the dynamics of the growth rate, a relationship that is likely to be causal [99] own unpublished data). Controlling the shape The question arises how the cell generates and controls such spatially confined areas of higher cell wall deformability. Root trichoblasts about to form root hairs show localized cell wall regions of lowered pH [100], high xyloglucan endotransglycosylase activity [101] and increased expansin concentration [102,103]. Pollen grains about to germinate exhibit an increased accumulation of methyl-esterified pectin at the aperture [89]. Clearly, the biochemistry of the cell wall can change over distances as small as few micrometers. The mechanisms that are likely to be responsible for the establishment of 6 a local "hot spot" of mechanically different cell wall are localized activation of proton pumps and/or targeted secretion. Other than the secretion of proteins affecting the degree of cross-linking between cell wall polymers, secretion induced cell wall softening might also be due to a lack of alignment and loose coiling of newly delivered cell wall polymers or a lack of gel formation due to their methyl-esterification [81,104,105]. The targeting of vesicles containing cell wall precursor material or agents influencing cell wall properties is controlled by the cytoskeleton [38]. Contrary to the predominant role of microtubules in cellulose deposition, heterotropic growth is often accompanied by characteristic configurations of the actin cytoskeleton or a combination of the actin- and microtubule arrays at the future site of a polar outgrowth. This has been observed in pollen tubes, root hairs, algal zygotes and fungal hyphae [37,102,106-113]. This prominence of actin is consistent with the fact that in these cell types, contrary to animal cells, the actin microfilaments play an important and sometimes dominant role in cytoplasmic organelle transport. In pollen tubes both microtubule and actin arrays are involved in organelle transport, but the former is not critical for vesicle delivery [114]. Trichome branch initiation in Arabidopsis is dependent on actin- and microtubulemediated Golgi transport to the cell cortex [115]. Drug or mutation induced interference with microtubule functioning affects the initiation of a new branch whereas the actin cytoskeleton seems to be responsible for maintaining the polarity [109,116,117]. Root hair initiation is preceded by a reorientation of the microtubule cytoskeleton [118], but its disruption does not inhibit the process [119]. Drugs causing actin depolymerization or fragmentation on the other hand successfully interfere with root hair initiation [119-121] and germination in algal zygotes [122]. der1 plants, which possess a point mutation in the gene encoding actin2, have enlarged or misplaced root hair initiation sites [123,124]. While interference with the cytoskeleton and hence the delivery of cell wall material is sufficient to hamper or alter growth [109], it is important to note that the simple piling on of cell wall polymers through secretion cannot by itself produce the formation of a protuberance even when it is highly localized. This is evident from tip growing cells in which growth is arrested while secretion is still ongoing [125,126]. A force causing tensile stress in the cell wall is still a prerequisite to explain expansion. Models that base cellular expansion solely on material addition [127,128] therefore only reflect one aspect of cellular reality, just as do those that focus exclusively on cell wall deformation. Our increasingly precise knowledge about vesicle delivery [129,130] and surface expansion [93] during heterotropic growth events should therefore make their way into the theoretical models that attempt to have the ability to reflect cellular biology and predict its behavior. material neglecting their cellularity. This simplification is justified for macroscopic applications such as the use of wood in construction [131]. Other modeling approaches attempt to at least consider the porosity of plant tissues by modeling tissue mechanical properties based on theories developed from foams - manufactured materials comprising gas-filled cells formed from polymers [132,133]. The mention of "gas" indicates, however, that such a model is of limited use for living tissues, since the most important mechanical difference between a gas- and a liquid-filled foam pore is compressibility of the former. A more realistic model for turgid tissues would therefore be that of a liquid-filled cellular foam [134]. Other models have been developed that are more or less useful for particular applications [135], but here we will focus on the questions concerning the mechanics of tissues in which the mechanics of the individual cell actually makes a difference. A dramatic example for the role of individual cell growth behavior for overall organ structure is the phenomenon of twisted stem and root growth. This phenomenon receives a lot of attention at present and is thus an attractive target for modeling approaches. Most mutants exhibiting axial twisting in plant organs have been identified to be caused by alterations to microtubule functioning [74]. It is therefore essential to understand how the cytoskeleton acts at cellular level to influence morphology at organ level. It is known that in twisting roots cortical microtubule arrays have oblique orientations instead of normal transverse. The elongating cells of left-handed twisting roots have right-handed oblique microtubules and vice versa [136,137]. It is likely that helically arranged cortical microtubules act on cell shape by causing microfibrils to be oriented helically, but no proof has been provided hitherto. A helical microfibril arrangement in elongating cells is suggested to result in torsion along the longitudinal cell axis [138,139] but the question is how this translates into twisted growth of the entire organ given that the twisting cells are attached to each other. The arrangement of helical microfibrils in walls that are shared by neighboring cells results in a net orientation that is transverse. Internal walls therefore should not play a significant role in causing twisting. It is rather the helical pattern of the microfibrils in the outer wall of epidermis cells that should have the strongest influence on the handedness of helical growth [136]. This is not the only aspect to the mechanics of twisting, however. Due to geometrical reasons internal cell layers in twisting stems or roots should elongate less than outer cell layers [136,138]. Combining all these boundary conditions into a model for stem growth would be an enormous help for plant biologists and could allow predicting the effects of mutations and pharmacological agents on this behavior. However, such a model does not exist yet and will certainly require a major modeling effort. Single cell mechanics versus tissues While individually growing algal cells (internodes, rhizoids), fungal cells (hyphae, sporangiophores), and plant cells (pollen tubes, root hairs, trichomes, moss protonemata) do exist both in nature and in vitro (suspension culture cells) most growth activities in plant cells occur within the context of a tissue. In order to understand the mechanics of a plant tissue or organ and to be able to model their behavior, important additional parameters need to be taken into consideration: the connection between individual cells, the stress distribution within the cell walls throughout the tissue and (or) organ, and the different mechanical behavior of the cell walls throughout the tissue and (or) organ. On one side these parameters add significantly to the complexity of a model, on the other side they essentially beg for simplification. Depending on the objective of an individual theoretical model it may or may not make sense to simplify the cellularity of a tissue by neglecting the geometry and mechanics of the individual cell. There are different ways of coping with the complexity of a tissue in a mechanical model. Before engaging in a modeling attempt it needs to be clarified at what level of hierarchy the material can be considered as a continuum rather than a structure. Secondary plant tissues are often modeled as an elastic Conclusions Cells must obey the laws of physics, therefore, understanding cell mechanics is fundamental to understanding and explaining growth processes leading to morphogenesis. Theoretical modeling of the mechanical and physical underpinnings, using biophysical variables to couple the physical and biological processes, has provided a better understanding of growth processes and proven to be a useful approach. Future modeling approaches should focus on the relationships between the physical and biological processes, incorporating the increasing amount of biological and biochemical details that are known and introducing new experimental approaches and techniques to elucidate these relationships. This general approach can provide opportunities for exciting new collaborations between modelers (engineers, mathematicians, and physicists), cell biologists and biochemists that will answer relevant questions and validate proposed conceptual and mathematical models. Acknowledgements J.K.E. Ortega acknowledges funding by the National Science Foundation Grant MCB-0640542. A. Geitmann receives funding 7 from the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds Québécois de la Recherche sur la Nature et les Technologies (FQRNT), and the Human Frontier Science Program (HFSP). 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Journal of Plant Research 120: 61-70 139 Lloyd, C.W. and Chan, J. (2002) Helical microtubule arrays and spiral growth. Plant Cell 14: 2319-2324 Glossary Anisotropic material: A material whose properties differ in different directions. Creep test: The time-dependent strain or extension behavior of a material in response to constant stress. Elastic material: Material that immediately returns to its original shape after removal of the deforming load or stress. Isotropic material: A material whose properties are identical in all directions. Plastic material: Material that retains its deformed shape after the deforming load, or stress, is removed. Relaxation test: A constant strain is imposed on a material specimen and the stress decay is measured as a function of time. Shear stress: Force parallel to the surface of an object divided by the area of the surface. Strain hardening: Strengthening of a material by plastic deformation. Stress-strain curve: Graphical representation of the relationship between stress, derived from measuring the load applied on an object, and strain, derived from measuring the deformation of the object. Tensile stress: A tensile force perpendicular to the surface of an object divided by the area on which it is applied. Viscoelastic material: Material that exhibits both elastic and viscous flow properties when deformed. The stress-induced deformation is time dependent. 10 Supplementary data From Lockhart to the augmented growth equation Biophysical equations describing cell wall mechanics and expansive growth Expansive growth of algal, fungal and plant cells is the result of metabolic-mediated biochemical processes in conjunction with interrelated physical processes [1]. From a physical perspective, expansive growth of the cells of all of these evolutionary distant organisms can be viewed the same way. It is defined as a permanent increase in cell volume. Thus the cell wall must increase in surface area, Acw, and volume, Vcw (Vcw = Acw τcw, where τcw is the wall thickness), because the volume of the cell wall chamber, Vcwc, must equal the volume of cell contents, Vcc, which it encloses. While Vcw does not necessarily change at the same rate as Acw during the different developmental stages of a cell [2], the overall tendency is a linear relationship between both parameters, as long as primary growth occurs. During expansive growth, the rate of increase in the volume of cell contents, dVcc/dt, is predominately the result of water uptake, i.e. dVcc/dt ≈ dVw/dt. The production and accumulation of active solutes within the cell membrane generates the osmotic pressure that drives the water uptake from the cell exterior to its interior. The driving force for water uptake is typically measured as the difference in the osmotic pressure inside (πi) and outside (πo) the cell membrane, i.e., Δπ = πi πo. The water uptake increases the volume of both the cell contents and the cell wall chamber. When the volume of the cell contents increases, the internal hydrostatic pressure (Pi) increases because the cell wall resists being deformed. As Pi increases to magnitudes larger than the external pressure (Po) it drives water out of the cell. The net water uptake rate is the difference between the water flowing into the cell because of the osmotic pressure difference, Δπ, and the water flowing out of the cell because of the turgor pressure, P, which is defined as, P = Pi – Po, i.e. Δπ - P. Thus an increase in P reduces the net water uptake rate, and at one magnitude of P the net water uptake will stop (P = Δπ). The P produces stresses within the cell wall. The cell wall stretches (deforms) in response to these cell wall stresses and increases its surface area. The deformations are both irreversible (plastic) and reversible (elastic). The assimilation of new cell wall materials into the deforming wall controls cell wall thickness, τcw and increases the cell wall volume, Vcw. The characteristics of the cell wall deformation, or its mechanical behavior, depend on the mechanical properties of the cell wall and its “biological” state. Equations relating expansive growth of plant cells to the rate of water uptake and cell wall extension were first published by Lockhart [3]. Equation 1 describes the relationship between the rate of increase in water volume and the net rate of water uptake (in relative terms): (dV/dt)w/Vw = Lpr (Δπ - P) [1] (rate of increase of water volume) = (net rate of water uptake) The relative hydraulic conductance is defined as, Lpr = Lp Am/Vw, and it is assumed that the solute reflection coefficient of the cell membrane is unity. Equation 2 describes the relationship between the rate of increase in the volume of the cell wall chamber and the rate of irreversible cell wall expansion (in relative terms): [2] (dV/dt)cwc/Vcwc = φ (P - Pc) (rate of increase of cell wall chamber volume) = (irreversible expansion rate) The irreversible wall extensibility, φ, and the critical turgor pressure, Pc, are biophysical variables to be determined for each cell. Equation 2 can be derived from the constitutive equation (equation describing the stress-strain relationship) for a viscous dashpot with a Bingham fluid, i.e. de/dt = (σ - σc )/μ , where de/dt is the strain rate, σ is the stress, σc is the critical stress, and μ is the dynamic viscosity of the Bingham fluid [4]. Recent theoretical research indicates that Eq. 2 can be obtained from the diagonal component of a more general tensor equation [5] that can model growth anisotropies such as phototropism and gravitropism [6]. During expansive growth, because the rate of increase in volume of the water is approximately equal to rate of increase in volume of the cell wall chamber, a third equation can be derived for the steady-state turgor pressure: [3] P = (Lpr Δπ + φ Pc ) / (φ + Lpr ) Equations 1 – 3 describe and model expansive growth of cells with walls when the turgor pressure is constant. Over the years, these equations (Eqs 1-3) have been termed the Lockhart Equations. However, it is crucial to note that the Lockhart Equations (Eqs 2 and 3) and its underlying constitutive equations (viscous dashpot with a Bingham fluid) cannot model the behavior exhibited in Fig. Ia and Ib, Figure I. Schematic graphs of stress relaxation tests. (a) An initial extension (longitudinal strain) and longitudinal load (stress) are imposed on growing sporangiophores of Phycomyces blakesleeanus with a tension-compression machine. At constant strain, stress is monitored as a function of time. Green curve: Sporangiophore has normal turgor pressure. Cell wall stresses are imposed by the longitudinal load and the turgor. Stress eventually decays to zero. Red curve: Immediately prior to load application turgor pressure had been decreased to zero. Stress is only imposed by the load and decays to a constant value. Redrawn from [11]. (b) Pressure relaxation in growing tissues after removal from water source and in an environment of saturating humidity to suppress transpiration. Red curve: The presence of auxin (in the case of pea stem tissue; redrawn from [7] Copyright of the original figure: American Society of Plant Biologists. www.plantphysiol.org) or the administration of a light stimulus (sporangiophore; redrawn from [11]) accelerate the relaxation. (c) Elongation growth behavior of an internode of Chara corallina before and after a step-down and step-up in P (blue curve) produced with a pressure probe. At room temperature (red curve) cell length represents the sum of irreversible (growth) and elastic deformation, whereas at a colder temperature (green curve), the cell does not grow and the pressure induced deformations are purely elastic. Redrawn from [15] Copyright of the original figure: American Society of Plant Biologists. www.plantphysiol.org. 11 because the underlying constitutive equation cannot model the observed exponential decay in stress or the associated exponential decay in turgor pressure. Subsequently, a more detailed and explicit conceptual model of the expansive growth process emerged, e.g. [7,8]. Experimental evidence demonstrates that during expansive growth, cell wall stresses relax (decrease) because ongoing biochemical processes loosen the cell wall, resulting in a decrease in stress that is observed in Fig. Ia. The turgor pressure decreases (Fig. Ib) in response to the decrease in cell wall stresses and produces an increase in net water uptake. The increase in net water uptake increases the turgor pressure, which in turn increases the cell wall stresses. Then as before, the cell wall stresses begin to relax because of ongoing cell wall loosening. This process is repeated continuously during expansive growth. The main experimental evidence that supports this iterative pressure relaxation and water uptake concept of expansive growth, is that the turgor pressure decays in growing cells with walls when the water uptake is eliminated and transpiration is suppressed (e.g. [7-10]. The decay in turgor pressure (Fig. Ib) results from a decrease in cell wall stresses by ongoing biochemical processes (Fig. Ia). The effect of biochemical processes on cell wall stresses can be illustrated by a pre-treatment with IAA (auxin), which accelerates the decay of turgor pressure (Fig. Ib). This is consistent with the hormone's role in increasing expansive growth rate. Similarly, the stress decay rate increases in the cell wall after a light stimulus, a trigger that is known to increase the elongation growth rate [11]. Importantly, the Lockhart Equations (Eqs 2 and 3) cannot model the decrease in turgor pressure (pressure relaxation) exhibited in Fig Ib, or the iterative pressure relaxation and water uptake process for expansive growth, because Eqs 2 and 3 are only valid for constant turgor pressure, i.e. P = constant or dP/dt = 0. Specifically, Eq. 2 (and its underlying viscoelastic model) cannot model the stress relaxation and pressure relaxation results obtained from growing cells (Figs Ia, b). Some excellent papers and recent reviews correctly describe the iterative pressure relaxation and water uptake conceptual model of expansive growth, and then refer to the Lockhart Equations which cannot model the pressure relaxation, i.e. the decrease in turgor pressure. The Lockhart equations form the basis upon which most models are built, however, they have crucial shortcomings of being applicable only in situations when turgor is constant, and when water lost through transpiration is neglected or negligible. To model expansive growth of plant, algal and fungal cells in the more general cases, the Lockhart equations were augmented to account for changing turgor pressure and transpiration [4,12]. These augmented growth equations [13,14] have proven to be very useful in analyzing and interpreting the experimental results of growing algal cells (e.g. [15,16], fungal cells (e.g. [10,12,17-19], and plant organs (e.g. [7,8,20-23]. It is apparent that a slightly more complex viscoelastic model and corresponding biophysical equation is needed. This was achieved by the biophysical equation, Eq. 4 which was derived from the constitutive equation for a Bingham-Maxwell viscoelastic model, i.e. de/dt = (σ σc)/μ + (dσ /dt )/E, where E is the elastic modulus [4]. [4] (dV/dt)cwc/Vcwc = φ (P - Pc) + (1/ε ) dP/dt The volume of water lost via transpiration is T. Again, because (dV/dt)cwc/Vcwc = (dV/dt)w/Vw another equation can be obtained for the rate of change of turgor pressure [1,14]. dP/dt = ε { Lpr (Δπ - P) - (dT/dt)w /Vw - φ (P - Pc)} [6] (rate of change of P) ∝ (net rate of water uptake) – (transpiration rate) – (irreversible expansion rate) Equation 6 explicitly shows how the turgor pressure changes when there is a change in the net rate of water uptake, and/or the transpiration rate, and/or the expansive growth rate. As an example of its utility, Eq. 6 explicitly shows that when the water uptake is eliminated, Lpr (Δπ - P) = 0, and transpiration is suppressed, (dT/dt)w/Vw = 0, the governing biophysical equation for dP/dt and its solution, P(t), are obtained for a pressure relaxation test [4,7]: dP/dt = - ε φ (P - Pc) and P(t) = (Pi – Pc) exp (-ε φ t) + Pc [7] The turgor pressure behavior, P(t), describes the exponential pressure decay from Pi to Pc, and describes the pressure decay obtained experimentally for both plant and fungal cells (Fig. Ib). Equations 4, 5, and 6 have been termed the augmented growth equations [1,4,1214,24]. Some algal, fungal, and plant cells grow predominately in length, i.e. exhibit elongation growth. Equations 4 and 5 can be adapted for elongation growth [1,15]: dL/dt = m (P – Pc) + (Lo/εL) dP/dt [8] (elongation rate) = (irreversible extension rate) + (elastic extension rate) (dL/dt) w Ac = Lp Am (Δπ – P) – (dT/dt)w [9] (rate of change in water volume) = (net rate of water uptake) – (volumetric transpiration rate) Where L is the length of the cell, m is the longitudinal irreversible wall extensibility, Lo is the initial length of the cell, and εL is the longitudinal component of the volumetric elastic modulus (longitudinal volumetric elastic modulus). It is noted that during expansive growth the rate of increase of the volume of the cell contents is essentially the rate of increase in water volume. Equation 8 was shown to describe the elongation of the internode cells of C. corallina (Fig. Ic) and the sporangiophores of P. blakesleeanus before, during, and after a step-down and a step-up in turgor pressure. Again, it is important to note that the Lockhart Equations (Eqs 2 and 3) cannot model the elongation behavior exhibited in Fig. Ic. Equation 3 cannot model the step-down and step-up in turgor pressure (Fig. Ic, blue curve) and Eq. 2 cannot model the observed recovered elastic extension (nearly instantaneous decrease in length) and elastic extension (nearly instantaneous increase in length) that is produced by the respective step-down and step-up in turgor pressure (Fig. Ic, red and green curves). As it is shown in Fig. Ic, the second term in the augmented growth equation (Eq. 8) is needed to model the elastic responses. Generally, the augmented growth equations have been used in short-scale time periods (minutes to hours) to analyze and interpret the experimental results of growing algal cells (e.g. [15,16], fungal cells (e.g. [10,12,17-19]), and plant organs (e.g. [7,8,20-23]). More recently, Lewicka [25] has extended their application to large-scale time periods (days) and the underlying viscoelastic models have been reviewed [26]. (rate of increase of cell wall chamber volume) = (irreversible expansion rate) + (elastic expansion rate) Equation 4 is written in relative terms. The added term, (1/ε ) dP/dt, describes the elastic deformation of the cell wall during expansion. Importantly, this biophysical equation accommodates the condition when the turgor pressure is changing. Hence, Equation 4, and its corresponding constitutive equation for a Bingham-Maxwell viscoelastic model, can model the turgor pressure behavior and stress behavior exhibited by growing plant and fungal cell walls (Figs. Ia, b). Another biophysical equation (in relative terms) can be derived to account for the case when water is lost to transpiration [12]. (dV/dt)w/Vw = Lpr (Δπ - P) - (dT/dt)w /Vw [5] (rate of increase of water volume) = (net rate of water uptake) – (transpiration rate) 12 Parameters Acw – surface area of the cell wall Am – surface area of the cell membrane e – strain de/dt – strain rate E – longitudinal elastic modulus, or Young’s modulus L – length of the cell (a variable) Lo – initial length of the cell (a constant) dL/dt – elongation rate Lpr – relative hydraulic conductance of the cell membrane; L = Lp Am/Vw Lp – hydraulic conductivity of the cell membrane m – longitudinal irreversible wall extensibility P – turgor pressure, the difference in pressure inside and outside the cell; P = Pi – Po Pc – critical turgor pressure, related to the critical stress, σc, by geometry of the wall Pi – hydrostatic pressure inside the cell Po – hydrostatic pressure outside the cell dP/dt – rate of change of turgor pressure t – time T – volume of water lost by transpiration (dT/dt)w – volumetric transpiration rate (dT/dt)w/Vw – relative volumetric transpiration rate Vcc – volume of the cell contents Vcw – volume of the cell wall Vcwc – volume of the cell wall chamber Vw – volume of the water in the cell (dV/dt)cwc/Vcwc – rate of increase of relative cell wall chamber volume (dV/dt)w/Vw – rate of increase of relative water volume Y – yield threshold, related to the critical turgor pressure, Pc ε – volumetric elastic modulus εL – longitudinal component of the volumetric elastic modulus μ – dynamic viscosity Δπ – osmotic pressure difference inside and outside the cell; Δπ =πi - πo πi – osmotic pressure inside the cell πo – osmotic pressure outside the cell σ – longitudinal stress σc – critical longitudinal stress dσ /dt – longitudinal stress rate τcw – thickness of the cell wall φ – irreversible wall extensibility References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Glossary Bingham fluid: A fluid that flows like a Newtonian fluid after a critical stress has been exceeded. Constitutive equation: A mathematical equation relating the stress and the strain. Maxwell material: A material whose constitutive equation may be obtained from a Maxwell model which is viscous dashpot (damper) filled with a Newtonian fluid and an elastic spring connected in series. Newtonian fluid: Fluid whose stress versus rate of strain curve is linear and passes through the origin. The constant of proportionality is known as the dynamic viscosity. Young's modulus or elastic modulus: Measure of stiffness of an isotropic elastic material. The ratio of the stress and resulting strain, and can be determined experimentally from the slope of a stressstrain curve. 20 21 22 23 24 25 26 Ortega, J.K.E. (2004) A quantitative biophysical perspective of expansive growth for cells with walls. In Recent Research Developments in Biophysics (Pandalai, S.G., Editor. eds.). pp. 297-324, Derbyshire, P., Findlay, K., McCann, M.C., and Roberts, K. 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