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First published online September 28, 2007; 10.1104/pp.107.101964 Plant Physiology 145:763-772 (2007) © 2007 American Society of Plant Biologists OPEN ACCESS ARTICLE
Loss of Stability: A New Look at the Physics of Cell Wall Behavior during Plant Cell Growth[W],[OA]Department of Plant Biology, University of Vermont, Burlington, Vermont 05405 (C.W., P.M.L.); and Department of Physics, Guangxi National University, Nanning 530006, China (C.W.)
In this article we investigate aspects of turgor-driven plant cell growth within the framework of a model derived from the Eulerian concept of instability. In particular we explore the relationship between cell geometry and cell turgor pressure by extending loss of stability theory to encompass cylindrical cells. Beginning with an analysis of the three-dimensional stress and strain of a cylindrical pressure vessel, we demonstrate that loss of stability is the inevitable result of gradually increasing internal pressure in a cylindrical cell. The turgor pressure predictions based on this model differ from the more traditional viscoelastic or creep-based models in that they incorporate both cell geometry and wall mechanical properties in a single term. To confirm our predicted working turgor pressures, we obtained wall dimensions, elastic moduli, and turgor pressures of sequential internodal cells of intact Chara corallina plants by direct measurement. The results show that turgor pressure predictions based on loss of stability theory fall within the expected physiological range of turgor pressures for this plant. We also studied the effect of varying wall Poisson's ratio  on extension growth in living cells, showing that while increasing elastic modulus has an understandably negative effect on wall expansion, increasing Poisson's ratio would be expected to accelerate wall expansion.
It is generally accepted that the fundamental behavior underlying all rapid plant growth is an irreversible stretching of primary cell walls due to mechanical loads generated by cell turgor pressure. Turgor pressure, the osmotically maintained hydrostatic pressure in living plant cells, and the mechanics of the cell wall itself, are undoubtedly among the most fundamental physical factors that dictate both cell growth and cell morphologies in plants (Zonia and Munnik, 2007  ). During plant cell growth, the cell wall system performs seemingly contradictory roles. On the one hand, it must be rigid enough to allow turgor pressure to build up, while on the other, it must be loosened in some way to permit cell enlargement during growth.
The physics underlying wall extension growth appears to be distinct from the biochemical events that constitute the broader background for all cell growth. While the study of biochemical precursors and cross-linkages emphasizes the importance of enzymatic activities such as the synthesis of wall components and the cleavage of hemicellulose tethers, etc., the biophysical approach focuses on the purely physical aspects of extension growth (e.g. wall stresses and strains, mechanical properties of the walls, and cell geometry). Obviously, an understanding of the physical principles underlying wall stress relaxation is essential to understanding the full spectrum of wall behaviors during growth. The biochemical approach to cell wall loosening has made notable advances in recent years (Cosgrove, 2006
In actively growing cells, because the walls are thin compared to the size of the cell, tensile stresses in the walls can be very high. Typical plant cell turgor pressures in the range of 0.3 to 1.0 MPa translate into between 10 and 100 MPa of tensile stress in the walls. In this regard, early students of the problem including Probine and Preston (1962) Studies based on viscoelastic/creep models, using Nitella or Chara internodal cells, have broadened our understanding of cell wall mechanical properties. For instance, creep-based uniaxial experiments have detailed the yield characteristics of plant cell walls and have demonstrated that the cell wall can be described as a viscoelastic/plastic material. These experiments also reveal the most important material property relating to wall elasticity, namely the Young's modulus of the cell wall. With experimental determination of Young's modulus the wall's structural and mechanical anisotropy can then be described in meaningful terms. Viscoelastic/creep-based experiments carried out in different media have also confirmed that the extensibility of cell walls is generally promoted by acid pH, which in turn has provided strong support for the acid growth theory. In fact, these experiments have provided a means of probing the complex of events leading to cell expansion.
Although the viscoelastic/creep model has been remarkably fruitful, difficulties have also been noticed by some researchers. For instance, although a reduction in turgor pressure of only 0.02 MPa can result in the immediate cessation of growth in living cells (Taiz, 1984
We have proposed a new model of wall stress relaxation (Wei and Lintilhac, 2003
Turgor-driven growth of plant cells is a gradual process. In other words, in normal growing cells water transport and the resulting increases in turgor pressure are not subject to sudden, extreme excursions. This reality conforms to the fundamental prerequisite for loss of stability. It also requires that these loss of stability prerequisites can be met experimentally only under gradual ramp-loading conditions. In our uniaxial stretching experiments using isolated wall segments excised from growing Chara corallina cells we respected this requirement for a gradual ramp-loading regime. The results show that Chara cell wall materials respond to gradual ramp loading with a stress relaxation behavior that is entirely different from one based on a viscoelastic/creep mechanism (Wei et al., 2006
The work done in recent years in the lab of J.S. Boyer at the University of Delaware clearly shows the relationship between turgor pressure, cell wall materials, cytoplasmic input, and growth (Proseus and Boyer, 2005
The first goal of this study is to present what one can infer from the loss of stability model about the relationship between turgor pressure of a growing cell and cell geometry. It has long been supposed that cell geometry and cell turgor are linked in growing cells. For instance, noting the reported discrepancy between multiaxial and uniaxial yield stress values in Nitella walls, Taiz (1984)
To explain cell wall behavior during growth, several other models including Green's movable yield-point theory (Green et al., 1971
The above models adequately explain several features of extension growth but they do not encompass any direct relationship between cell geometry and cell turgor pressure. We believe that loss of stability theory can address the issue of cell geometry while respecting the underlying facts of turgor pressure regulation. As we have shown in a spherical cell model, turgor pressure prediction from loss of stability theory includes both the cell's geometry and the wall's mechanical properties (Wei and Lintilhac, 2003
The second goal of this study is to obtain a better understanding of the effects of basic material properties on wall extension growth. As we know, elastic modulus and Poisson's ratio are the two basic mechanical parameters that describe the elastic properties of materials satisfying Hooke's law. While the effect of wall elastic modulus on plant cell growth has been investigated in various ways, the effect of wall Poisson's ratio is still unstudied. Like any other elastic material, cell walls show typical two-dimensional stress-strain behavior, i.e. when cell wall material is under stress, strain is also induced in a direction normal to the direction of the stress. Poisson's ratio is the constant that reflects the proportionality between lateral contraction and longitudinal extension and directly reflects the compressibility of the material. We know from the familiar example of acid growth theory (Rayle and Cleland, 1992
From the point of view of biophysics, the key controlling variables determining the characteristics of plant cell growth are water uptake and cell wall stress relaxation. While water uptake has been adequately explained by the theory of cell water relations, the physical mechanism of wall stress relaxation is still problematic. As pointed out by Burgert and Fratzl (2007)
Results of cell pressure probe measurements are shown in Table I . Turgor value is presented as mean ± SD (n = 6). Sequential internodal cells in the intact plant are numbered 1, 2, 3, 4, and 5 (cell 1 is the oldest and the most basal, cell 5 is the youngest and most apical). Direct comparisons show that older cells have higher turgor values than younger cells. Moreover, cell length is not a factor in the regulation of turgor value (in many cases cell 2 is longer than cell 1, yet its turgor value is still lower than that of cell 1).
Figure 1 shows typical loading and unloading paths of a stretching experiment conducted on an excised wall ribbon. We excluded the first loading path for the calculation of the linear regression line (dashed line) because it differed significantly from all subsequent loading and unloading cycles. All wall ribbons were able to return to their original length when the peak force was  50 x 10–3 N, which demonstrates that the wall materials remained elastic throughout the experiment. The slope of the linear regression line, which allows us to calculate the elastic modulus, represents the elongation of the wall ribbon per 1 x 10–3 N change in load.
The calculated wall elastic moduli of each of the measured internodal cells are shown in Table I. These modulus values range from 213 to 361 MPa and agree with the results of previous studies (Wei and Lintilhac, 2003 ; Wei et al., 2006). In general, the walls of older cells had higher elastic moduli than those of younger cells. Results of cell wall thickness and cell radius measurements for each cell are also shown in Table I. The critical pressure values are determined by substituting the cell dimensions and the computed elastic moduli into Equation 7. The resulting critical pressure values are listed in Table I and correspond to turgor predictions from loss of stability theory. They are expressed as a range of pressures representing the full range of possible Poisson's ratios from 0.5 to 0 (see "Discussion").
Figures 2
and 3
are graphic representations of Equations 5 and 6, respectively, after substituting the measured values for wall thickness, cell radius, and elastic modulus. They coincide with the general graphical presentation of loss of stability theory (Panovko and Gubanova, 1965
The three images comprising Figure 4 show a typical number 3 cell at different stages during plasmolysis: Figure 4A, fully turgid (immersed in the original growth medium); Figure 4B, less turgid (immersed in approximately 100 mM mannitol solution); Figure 4C, plasmolysis (immersed in approximately 200 mM mannitol solution). All plasmolysis experiments on Chara internodal cells show the same lack of cell shrinkage, which is to say, during mannitol-induced plasmolysis the measured diameter of living Chara cells does not decrease at all. This confirms our theoretical assertion that the cell radius measured after plasmolysis does not represent R0, which should instead be calculated using the relevant equation.
Loss of Stability in a Thin-Walled Cylindrical Pressure Vessel
In turgid plant cells, as in any pressure vessel, it can be shown that as pressure builds within the cell, tensile stresses in the wall increase, leading inevitably to loss of stability, unless some other mechanism intervenes. Panovko's work on the loss of stability of a thin-walled spherical shell has been reviewed by Wei and Lintilhac (2003)
As shown in Figure 5
, a long thin-walled cylindrical vessel with radius R and wall thickness t is subjected to an internal pressure P. Because of the axial symmetry of the structure, it is conventional to specify the wall stress by its three components: stress in the circumferential direction (termed hoop stress)
Equations 1 and 2 illustrate the conventional truth that hoop stress must be twice the longitudinal stress; this is the basic feature of stress in any thin-walled cylindrical pressure vessel (Lockhart, 1965b ). Equation 3 is based on the fact that radial stress through the wall thickness is very much smaller than the other two components, and can be set equal to zero.
Strains in the wall can similarly be expressed with three components: hoop strain
In a thin-walled cylindrical pressure vessel, Hooke's law relating stresses and strains in three dimensions can be conveniently expressed in matrix form (see Supplemental Appendix S1). Substituting Equations 1, 2, 3, and 4 into this matrix the internal pressure can be expressed as a function of the relative deformations in the circumferential and longitudinal directions (see Supplemental Appendix S1),
 and  are Poisson's ratio related factors and t0, R0, and L0 are the initial wall thickness, the radius, and the length of the cell, respectively (see Supplemental Appendix S1).
We note that the exact numerical expressions of Equations 5 and 6 require the initial dimensions of the cell (t0, R0, and L0), all of which can be obtained using Equations A3 and A5 (see Supplemental Appendix S1). To explicitly show the functional relationships of P versus R and P versus L we plot the equations by assigning the parameters with their physiological values. For instance, Figures 2 and 3 are plotted by substituting the measured values for wall thickness, cell radius, and elastic modulus of cell number 3 into Equations 5 and 6, respectively. Moreover, to reveal the relative deformations the horizontal axes have been expressed as ratios R to R0 and L to L0 (note this is justified because both R0 and L0 are constant). It is apparent that after a monotonic increase in pressure, the slope of each curve becomes zero when the pressure reaches a certain critical value Pcritical. According to theory, this represents the instant when loss of stability occurs, initiating irreversible stress relaxation in the wall (Panovko and Gubanova, 1965
Clearly, the relative deformations R/R0 and L/L0 denote the change in cell size during growth. Loss of stability theory suggests that cell turgor pressure under normal conditions is constantly hovering just under the critical pressure determined by cell geometry and wall elastic properties (Wei and Lintilhac, 2003
The expression for the critical pressure (i.e. the working turgor pressure for a growing cell) can therefore be obtained by substituting the critical R/R0 back into Equation 6, giving:
Of course, the above procedure can also be applied to longitudinal terms, i.e. Equation 6. The critical pressure derived from Equation 6 turns out to be the same as that from Equation 5 (which is Eq. 7). This indicates that the mathematical outcomes of our model point to a rational conclusion—that the turgor pressure value that drives growth in girth is the same as that which drives growth in length, which is to say plant cells that are unconstrained by the pressures of surrounding tissues grow simultaneously in girth and in length (albeit not at the same rate). The graphical presentation of the equations, Figure 2, and Figure 3 also confirms that the Pcritical of corresponding curves (i.e. the peaks) have the same values.
As mentioned above, the Pcritical appearing in Equation 7 represents the working turgor pressure of growing cells. Therefore, the equation formalizes the relationship between the two elements: cell turgor pressure (Pcritical) and cylindrical cell geometry (as represented by t0 and R0). Despite the fact that the subscript 0 denotes the initial values for thickness and radius, t0 and R0 should not be mistaken for the wall thickness and cell radius of a very young cell. Rather, by the definition of logarithmic deformation (Eq. 4), t0 and R0 represent the hypothetical dimensions of a cell growing under conditions where the internal pressure is zero (clearly an impossibility). In the context of plant cell growth, therefore, the use of the terms t0 and R0 require some careful qualification. In traditional engineering usage they would simply represent the initial thickness and radius of a vessel wall before pressurization, but in unpressurized (plasmolyzed) plant cells we cannot assume that any measurement of thickness and/or radius represents an initial condition, because the cells have been continually pressurized since the last division and retain a permanent imprint of plastic deformation and continued wall relaxation, even after plasmolysis. The fact that cell diameter does not decrease with falling turgor values during plasmolysis (Fig. 4) confirms our assumption that the measured radius of the now flaccid cell cannot be taken as a valid determination of R0. The cylindrical solutions for loss of stability show that the true values of t0 and R0 are linked to the actual wall thickness t and cell radius R by Equations A3 and A6 (Supplemental Appendix S1). Thus Equation 7 does represent the relationship between cell geometry and cell turgor pressure. This relationship has been used to generate the predicted turgor pressures listed in Table I. Note that Equation 7 does not contain length terms (L, L0), reflecting the fact that in long cylindrical cells turgor pressure is not related to the length of the cell. This has also been confirmed by pressure probe measurement (see "Results"). It is significant that the relationship between cell geometry and cell turgor pressure as expressed in loss of stability theory is derived from physical first principles, and that the predicted turgor pressures fall within the expected physiological turgor range of the growing cells. This relationship shows that although the working turgor pressure of a growing plant cell can be shown to be a direct manifestation of the physical properties of its cell wall, it is also true that cell geometry plays a significant role in determining the effective value of Pcritical, the turgor pressure value at which local wall material relaxes through instability.
In general, Poisson's ratio is known to be between 0 and 0.5 for all materials (Poisson's ratio = 0.5 is associated with incompressible materials such as rubber). It has been reported that the Poisson's ratio of onion (Allium cepa) epidermal peel (a multicellular structure with more or less turgid cells) is in the range of 0.18 to approximately 0.30 (Wei et al., 2001 But while it follows that the effect of increasing elastic modulus on wall expansion must be negative (a stiffer wall stretches less), this study suggests that increasing Poisson's ratio has a positive effect on wall expansion growth. This can be seen in Figures 2 and 3, where an increase in the Poisson ratio of the wall lowers the value of critical pressure, thereby promoting cell wall expansion (low critical pressure means that wall stress relaxation happens at a lower turgor pressure, which implies that the cell needs to do less osmotic work to accomplish wall expansion).
This conclusion is confirmed by taking the total derivative of Equation 7:
This equation formalizes the combined effect of elastic modulus E and Poisson's ratio
This study assumes a structurally isotropic wall with the cellulose distributed randomly in three dimensions, but plant cell walls are highly anisotropic (Baskin, 2005
Furthermore, although the substitution of anisotropically specified wall parameters (such as Elastic Modulus and Poisson's ratio) into the stress and strain equations is possible, the analytical solutions to the equations are not available yet. As Panovko and Gubanova (1965)
The theory of loss of stability diverges sharply from previous thinking on the underlying mechanics of stress relaxation in growing plant cell walls. The physical principles in which the theory is embedded are the general concepts of stress and strain. Once the geometry of the pressure vessel and the mechanical properties of the wall materials are in place then these general concepts and equations become specific. In this study Equations 1, 2, and 3 are the stress characteristics of any thin-walled cylindrical vessel. Equation 4 is employed because loss of stability always deals with large deformations, and we choose to follow Panovko's logarithmic method in describing the large strains. Equation A1 (see Supplemental Appendix S1) represents the stress-strain relationship of an ideal thin-walled cylindrical vessel in that it assumes isotropic behavior.
All of the above equations are static equations because they describe the steady-state relationship of stress and strain. But what if the internal pressure changes over time? For instance, in growing plant cells one can reasonably assume that the turgor pressure changes to some extent as the cell imports water and as new wall material is laid down. To make every moment in the process subject to a static analysis, the change in pressure has to be slow enough so that the changes in stress and strain are small, and that all of the equations remain valid. Experimentally, this feature has been demonstrated in uniaxial loading experiments in which slow ramp-loading conditions were used to meet the requirements for loss of stability (Wei et al., 2006 In the case of cylindrical pressure vessels, just as in spherical ones, the occurrence of loss of stability is inextricably bound to the physics of the general stress-strain relationship, a fact that reaffirms our central tenet, which is that loss of stability is an inevitable result of the gradual loading process. After a monotonic increase in pressure, there comes a point where the slopes of the curves become zero. At this point, the stress cannot increase further and instability-induced stress relaxation follows.
Pcritical is the highest possible pressure allowed by wall mechanics and system geometry. It is defined by the differential equations dP/dR = 0 and/or dP/dL = 0. This is particularly clear in the graphical interpretation of the definition where Pcritical corresponds to the peak of the curve (Figs. 2 and 3). It is clear then that cell turgor pressure will never exceed Pcritical, regardless of the cell's ability to uptake water. In growing cells, the driving force for water uptake (i.e. water potential gradient across the cell membrane) can only raise turgor pressure to the level of Pcritical, at which point stress relaxation and cell wall expansion occur. For this reason we have suggested that the turgor pressure of actively growing cells should be constantly hovering near the critical value, Pcritical (Wei and Lintilhac, 2003 All of the cells used in this study were actively growing cells. By measuring the working turgor pressure of each cell with the cell pressure probe and then dissecting the same cell to determine its wall thickness and its wall elastic modulus (for the calculation of Pcritical), we can construct a direct comparison of observed turgor pressure and the turgor pressure predicted by loss of stability theory. Table I shows the measured turgor pressures (Pturgor) and the turgor predictions (Pcritical) expressed as a range of pressures corresponding to a range of Poisson's ratios from 0 to 0.5.
Although the agreement between predicted and measured turgor values is good, the comparison between Pcritical and measured turgor for each cell shows that Pcritical is always somewhat higher than the measured turgor pressure. This discrepancy probably results from the localized nature of loss of stability in heterogeneous cell wall materials from living cells. Panovko pointed out that in cases where the wall thickness is uneven or has regions of variable modulus, loss of stability will happen preferentially at the site where it has the lowest Pcritical value (Panovko and Gubanova, 1965 The link between turgor pressure and the critical pressure Pcritical is central to our suggestion that loss of stability represents the mechanism underlying plant cell wall stress relaxation during normal turgor-driven growth. We have shown that stress relaxation in the wall during normal growth can take place only when cell turgor pressure rises to the critical pressure value. We can therefore predict two short-term behaviors. The first of these predictions is that any reduction in turgor pressure will cause growth to cease immediately and the second is that any reduction of wall elastic modulus, with its attendant reduction in the critical pressure value, will increase the cell's volume growth rate.
In fact, this relationship has also been noted in elongating oat (Avena sativa) coleoptiles treated either with mannitol to depress growth or with auxin to stimulate growth (Burgess, 1985
Loss of stability emphasizes the physics of the wall relaxation process and as such it does not necessarily relate to our understanding of cell wall biochemistry. Although the terms wall loosening and stress relaxation have sometimes been used interchangeably to describe wall behavior during cell growth, they in fact refer to two different perspectives on the same phenomenon. Wall loosening refers to the unfastening of the wall matrix polymers, e.g. the scission of cross-links between cellulose microfibrils in the walls. This inevitably reflects a biochemical interpretation of wall behavior and implies changes in molecular size of wall matrix polysaccharides (Talbott and Ray, 1992a
The biochemical approach to cell wall loosening has made notable advances in the recent years. Cosgrove has recently reviewed the four wall loosening agents that directly or indirectly catalyze stress relaxation in plant cell walls (the primary wall and secondary wall loosening agents; Cosgrove, 2006
Cylindrical loss of stability solutions provide the link between cell geometry and cell turgor pressure. As in the spherical cell model (Wei and Lintilhac, 2003 ), the cylindrical cell's geometry and wall mechanical properties merge into a single expression. The necessary condition—a gradual increase in pressure so that every moment in the process can be represented as a static state—is entirely in accordance with our common understanding of plant cell growth. The critical pressure defined and predicted by the cylindrical loss of stability model basically agrees with the measured physiological turgor pressure of living Chara cells. Cell radius and wall thickness (but not cell length) are the key geometrical factors determining cell turgor pressure. Additionally, with regard to the elastic properties of wall materials, while an increase in elastic modulus inhibits cell wall expansion an increase in Poisson ratio promotes cell wall expansion. The loss of stability model only addresses the physical constraints that govern stress relaxation in growing cell walls. It says nothing about the molecular characteristics of cell wall materials or the nature of the biochemical controls affecting growth. Clearly, considering the variety of cell shapes that can be found in real living plants this study provides a limited view of wall stress relaxation in cylindrical plant cells only. The principal barriers to a more complete understanding arise first from the fact that real plant cells are not ideal cylinders, second from the fact that plant cell walls are not mechanically isotropic, and last from the fact that plant cell walls are continuously changing in their biochemistry and structure during growth. However, we believe that it is possible to gain meaningful insight into the nature of wall stress relaxation and plant cell growth by using loss of stability theory as a basis for constructing predictive models.
Plant Material and Growth Conditions Cultures of Chara corallina were grown in a medium of 5% soil mixture in distilled water. Fast growing intact plants each consisting of five to six internodal cells were used for measurements (very young cells less than 3 mm in length were considered unusable). The most mature internode (i.e. the most basal one), labeled cell number 1 in this study, grew at a rate of about 5% increase in length per day. The youngest internodes (cell no. 5) grew at about 20% increase in length per day.
The turgor pressures of internodal cells were measured directly using the cell pressure probe method (Steudle, 1993 To calculate turgor pressures predicted by loss of stability theory, cell radius was measured with an optical micrometer before probing the cell's turgor pressure.
After probing, the cells were opened with a razor blade and the cytoplasm was removed with a hair loop. The walls were then cut into longitudinal ribbons for elastic modulus measurements. Cell wall thickness was measured by means of image duplicating interference microscopy according to Mach-Zender (Aus Jena, Peraval, see Preston, 1974
To obtain the stress-strain relationships of the wall ribbons they were subjected to cyclic loading and unloading using a commercial version of a mechanical testing frame previously developed in this laboratory (Vitrodyne V-200, Liveco). This instrument incorporates a microprocessor-based feedback control system capable of operating either in strain-controlled or load-controlled mode. Details of this device have been described elsewhere (Wei et al., 2001
Elastic modulus was calculated from the results of the loading and unloading experiment, which we plotted as wall elongation versus tensile force (Fig. 1). The slope of the regression line,
To distinguish between R0 (the so-called initial radius) and R (cell radius at full turgor), dimensional changes were recorded as Chara internodes were bathed in a mannitol medium to reduce turgor pressure and induce plasmolysis. Cell radii were measured using an optical micrometer and repeatedly remeasured as 500 mM mannitol was slowly added until plasmolysis was observed (care was taken to ensure that the medium was well mixed).
The following materials are available in the online version of this article.
We would like to thank Dr. Jean-Guy Beliveau, College of Engineering and Mathematics, the University of Vermont, for his critical reading of this article. Received May 8, 2007; accepted September 16, 2007; published September 28, 2007.
The author responsible for distribution of materials integral to the findings presented in this article in accordance with the policy described in the Instructions for Authors (www.plantphysiol.org) is: Chunfang Wei (chunfang.wei{at}uvm.edu).
[W] The online version of this article contains Web-only data.
[OA] Open Access articles can be viewed online without a subscription. www.plantphysiol.org/cgi/doi/10.1104/pp.107.101964 * Corresponding author; e-mail chunfang.wei{at}uvm.edu.
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TOP
ABSTRACT
RESULTS
h, stress in the longitudinal direction 
h, longitudinal strain 
, was then used for the calculation of elastic modulus E: 
