Download Is the Loss of Stability Theory a Realistic Concept for Stress

Letter to the Editor
Is the Loss of Stability Theory a Realistic Concept for
Stress Relaxation-Mediated Cell Wall Expansion
during Plant Growth?
Plant cells grow by irreversible expansion of
their walls accompanied by a corresponding increase in water volume. There is general agreement
that irreversible (plastic) wall expansion in turgid
cells is brought about by stress relaxation of the
wall kept under tensional stress by turgor pressure
(Ray et al., 1972). Based on physical considerations
and supported by a large body of experimental
evidence, stress relaxation can be attributed to
chemorheological changes in the load-bearing polymer network, enabling a plastic deformation of
wall dimensions. In other words, growth of turgid
cells is initiated and maintained by chemical modifications of the wall (wall loosening) followed by
mechanical stress relaxation generating a driving
force for osmotic water uptake through the waterpermeable plasma membrane (Cosgrove, 1993;
Schopfer, 2006).
This traditional concept, designated as chemical
wall loosening (CWL) in the following, has been
challenged in a recent contribution to this journal
(Wei and Lintilhac, 2007). Extending a previous
publication (Wei and Lintilhac, 2003), these authors
presented an alternative concept for explaining
wall stress relaxation during cell growth that is
founded on the theory of loss of stability (LOS)
adopted from material sciences. The basic idea
of this theory is that solid bodies, or closed pressure vessels, placed under gradually increasing stress
will respond with abruptly facilitated deformation
once a critical level of stress is reached. Wei and
Lintilhac (2003) propose ‘‘that the walls of a growing
plant cell behave similarly; with turgor pressure rising
smoothly to a critical point determined by material
properties and cell geometry, followed by a loss of
stability that manifests itself as wall extension and
growth’’ (p. 306).
Applying the LOS theory to plant cells, Wei and
Lintilhac (2003, 2007) fail to appreciate that plant
cells behave as hydraulic systems modeled by
osmometers, the mechanical properties of which
are governed by osmotic water relations rather than
by physical mechanics of closed pressure vessels.
Growth of a turgid cell, defined as irreversible
volume increase by water uptake, can only be
brought about by lowering the water potential of
the cell content (Ci) below that of the outer medium
(Co), creating a driving force DC 5 Co 2 Ci for
www.plantphysiol.org/cgi/doi/10.1104/pp.108.121178
water influx. Dictated by the fundamental equation
C 5 P 2 Dp, DC can theoretically be produced either
by a decrease in turgor pressure (P) or an increase in
osmotic pressure (Dp) of the cell contents. Based on
extensive experimental evidence, cell growth can
take place in the absence of uptake or intracellular
liberation of solutes (Cosgrove, 1993; Schopfer, 2006).
Therefore, Dp can be ignored in the present context,
leaving P as the critical parameter governing the
water relations of growing cells. P contributes positively to Ci, and thus an increase in P will elevate Ci
over Co, producing an outwardly directed C gradient. If, in a turgid cell (DC 5 0), the turgor could be
raised by some mysterious water pump, this would
create a driving force for the extrusion of water, i.e.
cell shrinkage followed by a readjustment of turgor
to the previous level. The only condition for creating
a DC for driving water influx is a decrease in turgor
resulting from a decrease in wall stress due to wall
loosening. This is just the opposite of the prediction
of the LOS theory that stress relaxation leading to
growth occurs when the turgor increases to the
critical point. The basic problem resulting from
applying the LOS theory to plant cells becomes
apparent when we consider the causal chain implied
by this theory. Wei and Lintilhac (2003) correctly
state that, because water is an incompressible fluid,
turgor increase can be produced only by water influx
into the cell. However, they go on to conclude the
following: ‘‘As water enters a cell, turgor pressure
increases; once turgor pressure reaches its critical
value, the wall loses stability, with wall stress relaxation and cell enlargement resulting’’ (p. 309). This
obviously inverts cause and effect with respect to
stress relaxation and water uptake leading to growth.
It should be noted that the CWL and LOS concepts
differ with respect to the points of control offered for
affecting growth by the turgid cell. For accomplishing the transition from the nongrowing to the
growing state, CWL can be initiated by the activation of chemical loosening reactions in the cell
wall. In contrast, LOS suggests a lowering of the
critical wall stress from a nonpermissive level to the
level determined by the actual turgor. It can be
predicted that in this unstable situation even a small
reduction in turgor, for instance, experimentally
produced by applying an external osmoticum, will
cause growth to cease immediately (Wei and
Lintilhac, 2007). However, the experimental evidence obtained with numerous growing tissues
demonstrate that, generally, the growth rate is a
continuous function of turgor (in excess of a yield
Plant Physiology, July 2008, Vol. 147, pp. 935–938, www.plantphysiol.org Ó 2008 American Society of Plant Biologists
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935
Letter to the Editor
threshold), in agreement with the CWL concept
(Hohl and Schopfer, 1992).
In an attempt to support LOS, Wei and Lintilhac
(2007) cite a textbook (Burgess, 1985) referring
to elongating oat coleoptiles as follows: ‘‘Burgess
observed that if the bathing medium contained
mannitol at a concentration just sufficient to lower
cell turgor slightly without inducing plasmolysis,
growth stopped, whereas when auxin was added
to the medium resulting in a change in the elastic
properties of the wall, the growth rate increased
without any corresponding increase in turgor
pressure’’ (p. 770). Burgess (1985) wrote: ‘‘Growth
is dependent upon the turgor pressure within
cells; if the bathing medium contains mannitol at
a concentration just insufficient to cause plasmolysis, growth ceases. Applied auxin causes an
increase in growth rate after a lag period of a
few minutes. This increase is known to be due to a
change in the plasticity of the wall, and not, for
example, to an increase in the turgor pressure
within the cells. If auxin is added to the mannitolcontaining medium which inhibits growth, then
the change in the properties of the cell wall still
takes place, although no growth results, since it
cannot in the absence of turgor pressure’’ (pp. 75–76).
This represents a perfect description of the basic tenets
of CWL.
Peter Schopfer
Albert-Ludwigs Universität Freiburg
Institut für Biologie II, Botanik
Freiburg, Germany
[email protected]
LITERATURE CITED
Burgess J (1985) An Introduction to Plant Cell Development. Cambridge
University Press, Cambridge, UK
Cosgrove DJ (1993) Water uptake by growing cells: an assessment of the
controlling roles of wall relaxation, solute uptake, and hydraulic conductance. Int J Plant Sci 154: 10–21
Hohl M, Schopfer P (1992) Growth at reduced turgor: irreversible and
reversible cell-wall extension of maize coleoptiles and its implications for
the theory of cell growth. Planta 187: 209–217
Ray PM, Green PB, Cleland R (1972) Role of turgor in plant cell growth.
Nature 239: 163–164
Schopfer P (2006) Biomechanics of plant growth. Am J Bot 93: 1415–1425
Wei C, Lintilhac PM (2003) Loss of stability: a new model for stress relaxation
in plant cell walls. J Theor Biol 224: 305–312
Wei C, Lintilhac PM (2007) Loss of stability: a new look at the physics of cell
wall behavior during plant cell growth. Plant Physiol 145: 763–772
Response to Schopfer Letter
Schopfer’s comments on our recent articles begin
with the assertion that we are challenging the ‘‘traditional concept, designated as chemical wall loosening
(CWL)’’; and, furthermore, that by ‘‘Applying the LOS
theory to plant cells, Wei and Lintilhac (2003, 2007) fail
to appreciate that plant cells behave as hydraulic
systems modeled by osmometers, the mechanical
properties of which are governed by osmotic water
relations rather than by physical mechanics of closed
pressure vessels.’’
But the loss of stability (LOS) theory does not in any
way challenge either the principles of osmotic water
relations or the notion of biochemically mediated wall
loosening. On the contrary, in our series of articles on
the theory of LOS (Wei and Lintilhac, 2003, 2006, 2007),
we have repeatedly addressed the importance of these
factors in regulating plant cell growth. In the first of the
series, for instance, under the subheading ‘‘The Regulation of Turgor Pressure,’’ we state: ‘‘For a growing cell,
the water potential gradient across the cell membrane,
which is largely due to an osmotic potential gradient,
provides the driving force for water movement’’ (Wei
and Lintilhac, 2003, p. 309). Clearly, plant cells are both
osmometers and pressure vessels and any meaningful
www.plantphysiol.org/cgi/doi/10.1104/pp.104.900264
model has to be able to accommodate both perspectives.
Furthermore, the notion of ‘‘chemical wall loosening’’
(Schopfer’s CWL) is fundamental to the cell’s ability to
precisely manipulate PCR at the local level, thereby
enabling the fine-tuning of the underlying biophysical
control circuitry at the biochemical level. From the same
article, we note ‘‘it is likely that PCR values for growing
primary cell walls are continuously modifiable by
metabolic or biochemical means’’ (p. 309).
Regarding the water relations of growing cells, we do
not agree that water potential difference (DC 5 Co 2 Ci)
of a growing cell can be treated as zero. Schopfer
correctly states that DC is the driving force for water
influx into growing cells. But he then seems to assert that
the DC of a ‘‘turgid’’ cell must be zero, which requires
him to postulate a ‘‘mysterious pump’’ if pressures are to
rise even further to drive LOS behavior. This leads to the
conclusion that the ‘‘mysterious pump,’’ raising the
internal pressure even further, ‘‘would create a driving
force for the extrusion of water.’’ Simply put, if the DC of
a turgid cell equals zero, then how can cells continue to
uptake water and grow? But biophysically speaking, the
walls of a growing turgid cell constitute an elastic
system; therefore, Schopfer’s idea that any turgor
increase in an already turgid cell will lead to immediate
water extrusion without further wall extension is without merit.
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Letter to the Editor
Schopfer claims, ‘‘The only condition for creating a
DC for driving water influx is a decrease in turgor
resulting from a decrease in wall stress due to wall
loosening.’’ This implies a stepwise increase in the
turgor pressure of any growing cell, with wall
loosening preceding turgor loss and subsequent
water uptake. Again, his statement results from his
incorrect assumption that the DC of a turgid cell
must be zero and from his notion of singular
causality, whereby only biochemical wall loosening
can lead to stress relaxation and turgor loss. We note,
however, that because of structural inhomogeneities,
stress relaxation is not an event that will occur at the
same instant over the entire wall body (which would
indeed lead to a stepwise increase in turgor pressure). Instead, relaxation consists of a multitude of
local events that happen frequently at different sites
during growth, with each event finding the location
where PCR is at its lowest value at that moment (we
call this local LOS; Wei and Lintilhac, 2003). Thus, we
see LOS events ‘‘flickering’’ over the surface of the
cell during growth. New wall synthesis can thus
have two different results. If synthesis is homogeneous and rapid, it will thicken the cell wall
uniformly, thereby resulting in a higher value for
PCR and allowing turgor pressure to increase accordingly. If new synthesis is patchy, it will shift LOS
behavior to other parts of the wall that remain at a
lower PCR value. Of course, new synthesis can also
affect LOS behavior by modifying the modulus of
elasticity or geometry of the wall.
It is difficult to evaluate Schopfer’s comment citing
historical studies that relate growth rate to turgor
pressure without being able to examine them in detail.
Different experimental approaches and materials
highlight different aspects of growth dynamics and
make direct comparison difficult. For instance, he
asserts ‘‘the experimental evidence obtained with
numerous growing tissues demonstrate that, generally, the growth rate is a continuous function of
turgor .,’’ and offers Hohl and Schopfer (1992) as an
example. But this article refers to a multicellular
system, where the growth dynamics of the individual
cells are likely masked in the aggregate. We can offer
a counter-example, titled ‘‘Enlargement in Chara
studied with a turgor clamp: Growth rate is not
determined by turgor’’ (Zhu and Boyer, 1992). Clearly,
there are many difficulties to be overcome before
we will be able to generalize from our unicellular
model to a rigorous multicellular model.
Schopfer fails to understand what it is that LOS
theory is challenging, which is the thought process
that grew out of the traditional viscoelastic/creepbased mechanism for wall stress relaxation. Both LOS
and viscoelastic/creep theories were developed to
provide a context for modeling the physical (not
chemical!) mechanism for wall stress relaxation. The
use of pressure vessels as models for growing plant
cells is not new (Métraux and Taiz, 1978; Steudle et al,
1982; Sellen, 1983). Any article that has used the two
conventional equations sh 5 PR/t and sL 5 PR/2t
(where P, R, and t represent the internal pressure, cell
radius, and wall thickness, respectively) to describe
the hoop and longitudinal stresses in a cylindrical
cell is actually using a cylindrical pressure vessel
model. In reality, this has been a remarkably fruitful
model. Our own development of LOS theory invokes
both spherical and cylindrical pressure vessels as
models for cell growth, enabling us to extend Euler
and Panovko’s notion of LOS to the stress-strain
analysis of plant cell walls. Without biophysical
models, biochemical process would be hard pressed
to reveal the significance of stress-strain relationships,
geometrical inputs, and complex anisotropies, or
indeed to offer reasonably predictive details of any
sort.
Schopfer’s criticism that the perception that plant
cell growth involves turgor-induced physical stress
relaxation in the walls ‘‘obviously inverts cause and
effect with respect to stress relaxation and water
uptake leading to growth’’ is also not new and is
reminiscent of the now-discredited views expressed
by Burström, who noted in his 1971 article that ‘‘the
rigidity of the walls preventing the entry of water is
the cause of the turgor pressure, which arises as a
consequence of the resistance of the wall to an
expansion. How then can the turgor pressure cause
expansion?’’ Burström thus concluded: ‘‘The literature on plant cell growth would certainly improve if
the notion of turgor expanding the cell was abandoned and replaced by accepted equations for water balance or fluxes’’ (Burström, 1971, p. 488). In
response, Ray et al. (1972) observed: ‘‘Burström failed
to come to grips with the principle that irreversible
increase in plant cell volume involves simultaneous
water uptake (driven by a water potential difference)
and cell wall yielding that depends on turgor stress .’’
(p. 163).
The problem of plant cell volume growth lying as
it does at the intersection of pure biophysics,
cellular biochemistry, and the new realities of
nanostructural composite materials is inherently
difficult to circumscribe within a single theoretical
framework. The goal is to articulate a biophysical
framework that rests on accepted mechanical principles but that is robust enough to admit other
perspectives. We believe that the LOS theory has the
ability to do this, although its application to many of
the details of growth may be beyond our grasp at
present.
Chunfang Wei* and Philip M. Lintilhac
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.)
*Corresponding author;
e-mail [email protected].
Plant Physiol. Vol. 147, 2008
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Letter to the Editor
LITERATURE CITED
Burström (1971) Wishful thinking of turgor. Nature 234: 488
Hohl M, Schopfer P (1992) Growth at reduced turgor: irreversible and
reversible cell wall extension of maize coleoptiles and its implications for
the theory of cell growth. Planta 187: 209–217
Métraux JP, Taiz L (1978) Transverse viscoelastic extension in Nitella. Plant
Physiol 61: 135–138
Ray PM, Green PB, Cleland R (1972) Role of turgor in plant cell growth.
Nature 239: 163–164
Sellen DB (1983) The response of mechanically anisotropic cylindrical cells to
multiaxial stress. J Exp Bot 34: 681–687
Steudle E, Ferrier JM, Dainty J (1982) Measurements of the volumetric
and transverse elastic extensibilities of Chara corallina internodes by
combining the external force and pressure probe techniques. Can J Bot 60:
1503–1511
Wei C, Lintilhac PM (2003) Loss of stability: a new model for stress relaxation
in plant cell walls. J Theor Biol 224: 305–312
Wei C, Lintilhac PM (2006) Loss of stability, pH, and the anisotropic
extensibility of Chara cell walls. Planta 223: 1058–1067
We i C , L int ilh ac PM (2007) Loss of stability: a new look at the physics
of cell wall behavior during plant cell growth. Plant Physiol 145: 763–772
Zhu GL, Boyer JS (1992) Enlargement in Chara studied with a turgor clamp:
Growth rate is not determined by turgor. Plant Physiol 100: 2071–2080
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Copyright © 2008 American Society of Plant Biologists. All rights reserved.