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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

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Loss of Stability: A New Look at the Physics of Cell WallBehavior during Plant Cell Growth
[W][OA]
Chunfang Wei* and Philip M. Lintilhac
Department of Plant Biology, University of Vermont, Burlington, Vermont 05405 (C.W., P.M.L.); andDepartment 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 theEulerian concept of instability. In particular we explore the relationship between cell geometry and cell turgor pressure byextending loss of stability theory to encompass cylindrical cells. Beginning with an analysis of the three-dimensional stress andstrain of a cylindrical pressure vessel, we demonstrate that loss of stability is the inevitable result of gradually increasinginternal pressure in a cylindrical cell. The turgor pressure predictions based on this model differ from the more traditionalviscoelastic 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 pressurepredictions based on loss of stability theory fall within the expected physiological range of turgor pressures for this plant. Wealso studied the effect of varying wall Poisson’s ratio
n
on extension growth in living cells, showing that while increasing elasticmodulus has an understandably negative effect on wall expansion, increasing Poisson’s ratio would be expected to acceleratewall expansion.
It is generally accepted that the fundamental behav-ior underlying all rapid plant growth is an irreversiblestretching of primary cell walls due to mechanicalloads generated by cell turgor pressure. Turgor pres-sure, the osmotically maintained hydrostatic pressurein living plant cells, and the mechanics of the cell wallitself, are undoubtedly among the most fundamentalphysical factors that dictate both cell growth and cellmorphologies in plants (Zonia and Munnik, 2007).During plant cell growth, the cell wall system per-forms 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 insome way to permit cell enlargement during growth.The physics underlying wall extension growth ap-pears to be distinct from the biochemical events thatconstitute the broader background for all cell growth.While the study of biochemical precursors and cross-linkages emphasizes the importance of enzymatic ac-tivities such as the synthesis of wall components andthecleavageofhemicellulose tethers,etc., thebiophys-ical approach focuses on the purely physical aspectsof extension growth (e.g. wall stresses and strains,mechanical properties of the walls, and cell geometry).Obviously, an understanding of the physical princi-ples underlying wall stress relaxation is essential tounderstanding the full spectrum of wall behaviorsduring growth. The biochemical approach to cell wallloosening has made notable advances in recent years(Cosgrove, 2006). However, our appreciation of thephysical aspects of the process remains basically un-changed since the 1960s (Probine and Preston, 1962;Lockhart, 1965a, 1965b; Cleland, 1967; Ray et al., 1972).In actively growing cells, because the walls are thincompared to the size of the cell, tensile stresses in thewalls can be very high. Typical plant cell turgorpressures 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 includingProbine and Preston (1962), Lockhart (1965a), Cleland(1967),andalso Rayetal.(1972)all recognizedthatcellenlargement begins with stress relaxation in the walls.They interpreted stress relaxation as a viscoelastic/creep-based process. Accordingly, most of the relevantexperimental work on plant materials has been carriedout under this viscoelastic/creep paradigm.Studies based on viscoelastic/creep models, using
Nitella
or
Chara
internodal cells, have broadened ourunderstanding of cell wall mechanical properties. Forinstance, creep-based uniaxial experiments have de-tailed the yield characteristics of plant cell walls andhave demonstrated that the cell wall can be describedas a viscoelastic/plastic material. These experimentsalso reveal the most important material property re-lating to wall elasticity, namely the Young’s modulusof the cell wall. With experimental determination of Young’s modulus the wall’s structural and mechanicalanisotropy can then be described in meaningful terms.
* Corresponding author; e-mail chunfang.wei@uvm.edu.The author responsiblefordistribution of materialsintegral to theﬁndings presented in this article in accordance with the policydescribed in the Instructions for Authors (www.plantphysiol.org) is:Chunfang Wei (chunfang.wei@uvm.edu).
[W]
The online version of this article contains Web-only data.
[OA]
Open Access articles can be viewed online without a sub-scription.www.plantphysiol.org/cgi/doi/10.1104/pp.107.101964
Plant Physiology,
November 2007, Vol. 145, pp. 763–772, www.plantphysiol.org
Ó
2007 American Society of Plant Biologists 763
Viscoelastic/creep-based experiments carried out indifferent media have also conﬁrmed that the extensi- bility of cell walls is generally promoted by acid pH,which in turn has provided strong support for the acidgrowth theory. In fact, these experiments have pro-vided a means of probing the complex of events lead-ing to cell expansion.Although the viscoelastic/creep model has beenremarkably fruitful, difﬁculties have also been noticed by some researchers. For instance, although a reduc-tion in turgor pressure of only 0.02 MPa can result inthe immediate cessation of growth in living cells (Taiz,1984), viscoelastic/creep-based models cannot predicta reasonable working turgor value for growing cells because viscoelastic/creep behavior can be shown tooccur over a wide range of different load intensities inisolated cell wall materials (Preston, 1974; Dorrington,1980).We have proposed a new model of wall stress re-laxation (Wei and Lintilhac, 2003; Wei et al., 2006) based on the assumption that any changes in cell tur-gor pressure, and in the resulting wall stresses, mustoccur gradually. The model is based on the theory of loss of stability. The notion was srcinally developed by the great mathematician Leonhard Euler to modelthe behavior of columns in compression, and thenextended by physicists to treat tensile stress relaxation(Rzhanitsyn, 1955). The extension of the theory totensile systems led to the recognition that the samethinking could be applied to any thin-walled pressurevessel. With a gradual increase in internal pressure theresulting stresses in the wall will gradually increase toa critical value, at which time loss of stability mustoccur, leading to stress relaxation in the wall (Panovkoand Gubanova, 1965).Turgor-driven growth of plant cells is a gradualprocess. In other words, in normal growing cells watertransport and the resulting increases in turgor pres-sure are not subject to sudden, extreme excursions.This reality conforms to the fundamental prerequisitefor loss of stability. It also requires that these loss of stability prerequisites can be met experimentally onlyunder gradual ramp-loading conditions. In our uni-axial stretching experiments using isolated wall seg-ments excised from growing
Chara corallina
cells werespected this requirementfor a gradual ramp-loadingregime. The results show that
Chara
cell wall materialsrespond to gradual ramp loading with a stress relax-ation behavior that is entirely different from one basedonaviscoelastic/creepmechanism(Weietal.,2006).Inparticular, whereas a linear relationship between wallextension and log time is typical for creep-basedexperiments, it is not seen under ramp-loading condi-tions.The work done inrecentyearsin thelab ofJ.S. Boyerat the University of Delaware clearly shows the rela-tionship between turgor pressure, cell wall materials,cytoplasmic input, and growth (Proseus and Boyer,2005, 2006a, 2006b). In a quantitative comparison of iso-lated
Chara
wall tubes and living
Chara
cells (Proseusand Boyer, 2006a), Boyer’s group found that at allturgor pressures and temperatures, the isolated walltubes displayed turgor-driven growth indistinguish-able in every respect from that of living cells, exceptthat the rate decelerated over time in the isolated walltubes while the living cells continued to grow rapidly.Boiling the isolated walls gave the same results, indi-cating that enzyme activity does not directly controlwall extension in their experiments. These ﬁndingsstrongly suggest that wall extension growth is primar-ily a biophysical process, although ultimately depen-dant on enzymatic activity, and that under conditionswhere the enzymatic background can be subtractedthe biophysical process still proceeds normally. Theyalso found that in both living cells and in isolated walltubes, any reductionin turgorpressure belowa criticallevel terminated wall extension growth (Proseus andBoyer, 2006a). As mentioned earlier, this phenomenonis uniquely predicted by loss of stability theory (Weiand Lintilhac, 2003).The ﬁrst goal ofthisstudyis topresentwhatonecaninfer from the loss of stability model about the rela-tionshipbetweenturgorpressureofagrowingcellandcell geometry. It has long been supposed that cellgeometry and cell turgor are linked in growing cells.For instance, noting the reported discrepancy betweenmultiaxial and uniaxial yield stress values in
Nitella
walls, Taiz (1984) proposed that geometrical consider-ations may be as important as wall mechanical prop-ertiesinunderstandingthenatureofcellturgorpressure.It has also been suggested that the multiaxially deter-mined critical turgor pressure might be analogous tothepointofdimensionalinstabilityintheballoonmodelof Hettiaratchi and O’Callaghan (1974; Taiz, 1984). Thedimensional instability in the balloon model refers tothe sudden increase in expansion rate of a balloonwhen the balloon’s radius and the decreasing thick-ness of the membrane reach their critical values. Theanalogy breaks down, however, when one notes thatthe balloon is inﬂated by a compressible gas, so thatinstability can lead to aneurysm through the contin-ued expansion of the gas, but plant cells are ﬁlled withan incompressible liquid, which means that barringadditional water uptake the pressure immediatelydrops to zero once the cell wall expands. Also, unlikethe membrane of a balloon, plant cell walls are con-tinuously being thickened and strengthened so thatunder constant volume conditions the strength of thewall and the critical pressure value will continue torise (Wei and Lintilhac, 2003), further obviating thepossibility of aneurysm. This self-limiting controlcycle limits cell volume growth during any single re-laxation event and makes it unlikely that a suddenincrease in the expansion rate will lead to catastrophicfailure of the wall.To explain cell wall behavior during growth, severalother models including Green’s movable yield-pointtheory (Green et al., 1971) and Cleland’s two extensionprocesses notion (Cleland, 1971) have been suggested.In general, these models involve the occurrence of
Wei and Lintilhac764 Plant Physiol. Vol. 145, 2007
viscoelastic-plastic extension in the walls and someempirical rules that account for speciﬁc behaviors.Dumais et al. (2006) have recently presented ananisotropic-viscoplastic model of plant cell morpho-genesis by tip growth. To illustrate the role of turgor-induced wall stresses and the deposition of new wallmaterials in plant cell morphogenesis, they presentthree sets of equations whose solution demonstratesthe importance of cell geometry, and of wall stressesand wall strains in the study of plant cell morphogen-esis and growth. They also show that wall mechanicalanisotropy is critical in the patterning of wall expan-sion growth.The above models adequately explain several fea-tures of extension growth but they do not encompassany direct relationship between cell geometry and cellturgor pressure. We believe that loss of stability theorycanaddresstheissue ofcellgeometrywhilerespectingthe underlying facts of turgor pressure regulation. Aswe have shown in a spherical cell model, turgor pres-sure prediction from loss of stability theory includes both the cell’s geometry and the wall’s mechanicalproperties (Wei and Lintilhac, 2003). In this study wefurther develop the notion of loss of stability by rigor-ously extending the physics of loss of stability to freestanding cylindrical cells. We also test the predictivestrength of this approach by investigating the workingturgor pressures of living
Chara
internodal cells of different ages.The second goal of this study is to obtain a betterunderstanding of the effects of basic material proper-ties on wall extension growth. As we know, elasticmodulus and Poisson’s ratio are the two basic me-chanicalparametersthatdescribetheelasticpropertiesof materials satisfying Hooke’s law. While the effect of wall elastic modulus on plant cell growth has been in-vestigated in various ways, the effect of wall Poisson’sratio 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 thedirection of the stress. Poisson’s ratio is the constantthat reﬂects the proportionality between lateral con-traction and longitudinal extension and directly re-ﬂects the compressibility of the material. We knowfrom the familiar example of acid growth theory(Rayle and Cleland, 1992) that lowering wall elasticmodulus promotes cell wall expansion growth, but weknow little about the effect of wall compressibility andPoisson’s ratio.From the point of view of biophysics, the keycontrolling variables determining the characteristicsof plant cell growth are water uptake and cell wallstress relaxation. While water uptake has been ade-quately explained by the theory of cell water relations,the physical mechanismof wall stressrelaxation is stillproblematic. As pointed out by Burgert and Fratzl(2007), there has been very little theoretical coherenceinourunderstandingofcellwallextensibilityandwallstress relaxation. They note that ‘‘
.
cell wall extensi- bility has been often used in a rather imprecise way,describing elastic, viscoelastic, plastic and viscoplasticdeformation properties’’ (quote from Burgert andFratzl, 2007 [p. 198]). Our work is an attempt to codifywhat we now know in terms of strictly deﬁnable phys-ical ﬁrst principles.
RESULTS
Results of cell pressure probe measurements areshown in Table I. Turgor value is presented as mean
6
SD
(
n
5
6).Sequentialinternodalcellsintheintactplantare numbered 1, 2, 3, 4, and 5 (cell 1 is the oldest andthe most basal, cell 5 is the youngest and most apical).Direct comparisons show that older cells have higherturgor values than youngercells. Moreover, cell lengthis not a factor in the regulation of turgor value (inmany cases cell 2 is longer than cell 1, yet its turgorvalue is still lower than that of cell 1).Figure 1 shows typical loading and unloading pathsof a stretching experiment conducted on an excisedwall ribbon. We excluded the ﬁrst loading path for thecalculation of the linear regression line (dashed line) because it differed signiﬁcantly from all subsequentloading and unloading cycles. All wall ribbons wereable to return to their srcinal length when the peakforce was
#
50
3
10
2
3
N
, which demonstrates that thewall materials remained elastic throughout the exper-iment. The slope of the linear regression line, whichallows us to calculate the elastic modulus, representstheelongationofthewall ribbon per 1
3
10
2
3
N
changein load.
Table I.
Cell radius R, wall thickness t, wall elastic modulus E, and measured cell turgor pressures P
turgor
and the corresponding turgor predictions P
critical
of sequential internodal cells of intact Chara plants (cell no. 1 being the oldest and most basal cell
)Data from direct measurements are expressed as mean value
6
SD
(
n
5
6).
P
critical
is turgor prediction from loss of stability theory, expressed as arange of pressures corresponding to a range of Poisson’s ratios from 0.5 to 0 (see ‘‘Discussion’’).
Cell Parameter Cell No. 1 Cell No. 2 Cell No. 3 Cell No. 4 Cell No. 5
R
(
m
m) 424
6
33 435
6
32 338
6
26 291
6
23 218
6
18
t
(
m
m) 1.01
6
0.15 0.94
6
0.16 0.74
6
0.13 0.54
6
0.07 0.33
6
0.09
E
(MPa) 361 338 289 240 213
P
turgor
(Mpa;by probe)0.51
6
0.06 0.42
6
0.05 0.34
6
0.04 0.27
6
0.04 0.19
6
0.03
P
critical
(Mpa;prediction)0.57–0.86 0.49–0.73 0.42–0.63 0.30–0.45 0.22–0.32
Loss of Stability and Cell Wall Stress RelaxationPlant Physiol. Vol. 145, 2007 765
The calculated wall elastic moduli of each of themeasured internodal cells are shown in Table I. Thesemodulus values range from 213 to 361 MPa and agreewith the results of previousstudies (WeiandLintilhac,2003; Wei et al., 2006). In general, the walls of oldercells had higher elastic moduli than those of youngercells.Results of cell wall thickness and cell radius mea-surements for each cell are also shown in Table I. Thecritical pressure values are determined by substitutingthe cell dimensions and the computed elastic moduliinto Equation 7. The resulting critical pressure valuesare listed in Table I and correspond to turgor predic-tions from loss of stability theory. They are expressedas a range of pressures representing the full range of possible Poisson’s ratios from 0.5 to 0 (see ‘‘Discus-sion’’).Figures 2 and 3 are graphic representations of Equa-tions 5 and 6, respectively, after substituting the mea-sured values for wall thickness, cell radius, and elasticmodulus. They coincide with the general graphicalpresentation of loss of stability theory (Panovko andGubanova, 1965; Wei and Lintilhac, 2003), i.e. eachcurve has a peak where the slope changes from pos-itive to negative. Note that for any given Poisson’sratio value, the critical pressures for circumferentialand longitudinal loss of stability are identical. Forinstance, for
n
5
0.20, the peaks of the two curves are both positioned at
P
5
0.53 MPa.ThethreeimagescomprisingFigure4showatypicalnumber 3 cell at different stages during plasmolysis:Figure 4A, fully turgid (immersed in the originalgrowth medium); Figure 4B, less turgid (immersed inapproximately 100 m
M
mannitol solution); Figure 4C,plasmolysis (immersed in approximately 200 m
M
mannitol solution). All plasmolysis experiments on
Chara
internodal cells show the same lack of cellshrinkage, which is to say, during mannitol-inducedplasmolysis the measured diameter of living
Chara
cells does not decrease at all. This conﬁrms our theo-retical assertion that the cell radius measured afterplasmolysis does not represent
R
0
, which should in-stead be calculated using the relevant equation.
DISCUSSIONLoss of Stability in a Thin-Walled CylindricalPressure Vessel
In turgid plant cells, as in any pressure vessel, it can beshownthataspressurebuildswithinthecell,tensilestresses in the wall increase, leading inevitably to lossof stability, unless some other mechanism intervenes.Panovko’sworkonthelossofstabilityofathin-walledsphericalshell hasbeenreviewedbyWeiandLintilhac(2003). This early work outlined the material param-etersoflossofstability insphericalpressurevessels.Inextending this model from spherical to cylindricalgeometries we are exposing previously unexploredapplications of Panovko’s srcinal mathematical treat-ment and developing a predictive model that morenearly approximates conditions in axially extendingplant cells. To emphasize the basic problems and pin-point the onset of loss of stability we will present themathematical treatment in somewhat simpliﬁed form.
Figure 1.
Loading and unloading paths of a typical stretching exper-iment conducted on a wall ribbon obtained from a cell number 2. The
r
2
and the slopes of the linear regression (dashed line), b(1), werecalculated from data points in the loading and unloading cycles. Eachloading and unloading cycle lasted about 10 min.
Figure 2.
Relationship between turgor pressure and the nondimen-sional radius
R
/
R
0
, obtained by substituting measured values for wallthickness, cell radius, and elastic modulus of cell number 3 intoEquation 5. After a monotonic increase in pressure, loss of stability (atthe peak of the curve) occurswhen the pressure reaches a critical value
P
critical
. Allpoints onthe risingportionofthe curve(solid line)are stablestates, while points on the falling portion (dashed line) are unstablestates.
Wei and Lintilhac766 Plant Physiol. Vol. 145, 2007
As shown in Figure 5, a long thin-walled cylindricalvessel with radius
R
and wall thickness
t
is subjectedto an internal pressure
P
. Because of the axial symme-try of the structure, it is conventional to specify thewall stress by its three components: stress in thecircumferential direction (termed hoop stress)
s
h
,stress in the longitudinal direction
s
L
, and stress inthe radial direction (i.e. through the wall thickness)
s
t
.
s
h
5
PRt
ð
1
Þ
s
L
5
PR
2
t
ð
2
Þ
s
t
0
ð
3
Þ
Equations 1 and 2 illustrate the conventional truththat hoop stress must be twice the longitudinal stress;this is the basic feature of stress in any thin-walledcylindricalpressure vessel (Lockhart, 1965b). Equation3is based onthefactthat radial stressthrough thewallthickness is very much smaller than the other twocomponents, and can be set equal to zero.Strains in the wall can similarly be expressed withthree components: hoop strain
e
h
, longitudinal strain
e
L
, and strain through wall thickness
e
t
. Because thestrains in problems of loss of stability are large, theyare often calculated in accordance with logarithmicdeformation or true strain (Panovko and Gubanova,1965),
e
h
5
ln
RR
0
;
e
L
5
ln
LL
0
;
e
t
5
ln
tt
0
ð
4
Þ
where
R
0
,
L
0
, and
t
0
are the initial radius, length, andwall thickness, respectively.In a thin-walled cylindrical pressure vessel, Hooke’slaw relating stresses and strains in three dimensionscan be conveniently expressed in matrix form (seeSupplemental AppendixS1). Substituting Equations 1,2, 3, and 4 into this matrix the internal pressure can beexpressed as a function of the relative deformations inthe circumferential and longitudinal directions (seeSupplemental Appendix S1),
P
5
2
Et
0
R
0
ð
2
2
n
Þ
RR
0
a
ln
RR
0
ð
5
Þ
P
5
2
Et
0
R
0
ð
1
2
2
n
Þ
LL
0
b
ln
LL
0
ð
6
Þ
where
a
and
b
are Poisson’s ratio related factors and
t
0
,
R
0
, and
L
0
are the initial wall thickness, the radius,andthelengthofthecell,respectively(seeSupplemen-tal Appendix S1).We note that the exact numerical expressions of Equations 5 and 6 require the initial dimensions of thecell (
t
0
,
R
0
, and
L
0
), all of which can be obtained usingEquations A3 and A5 (see Supplemental AppendixS1). To explicitly show the functional relationships of
Figure 3.
Relationship between turgor pressure and the nondimen-sional length
L
/
L
0
, obtained by substituting measured values for wallthickness, cell radius, and elastic modulus of cell number 3 intoEquation 6.
Figure 4.
Photomicrographs of a cellnumber3duringmannitol-inducedplas-molysis. The dimensions of the cellremain constant as the cell turgor pres-suredecreasesfromthefullyturgidstate(A), to a less turgid (B), and ﬁnally toplasmolysis (zero turgor; C).
Loss of Stability and Cell Wall Stress RelaxationPlant Physiol. Vol. 145, 2007 767

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