Journal of Experimental Botany, Vol. 47, No. 297, pp. 519-528, April 1996
Journal of
Experimental
Botany
Tissue stresses in organs of herbaceous plants
III. Elastic properties of the tissues of sunflower
hypocotyl and origin of tissue stresses
Zygmunt Hejnowicz1 and Andreas Sievers2
Botanisches Institut, Universitat Bonn, Venusbergweg 22, D-53115 Bonn, Germany
Received 26 June 1995; Accepted 8 December 1995
Abstract
The finding that there are considerable tissue stresses
(TSs) in the hypocotyl of Helianthus annuus L. prompts
the question: how are the stresses generated? Here,
a one-dimensional model is formulated which, based
on (i) symplastic, turgor-induced extensions of tissues
which differ in moduli of elasticity, and (ii) static equilibrium, predicts the occurrence of longitudinal TSs in
stem-like organs, and gives their dependence on
turgor pressure. To calculate the longitudinal forces
which generate the TSs in a stem, the moduli of elasticity of the tissues need to be known. The moduli were
determined for uniaxial and multiaxial stresses for the
outer tissue (OT) and the inner tissue (IT) of the hypocotyl. In the OT, the moduli were strongly dependent
on applied uniaxial stress. The magnitudes of the calculated longitudinal forces (tensile and compressive)
in the hypocotyl, were comparable to those measured.
It follows that the TSs may arise without differential
growth of the tissues.
Key words: Epidermis, model, moduli of elasticity, turgor,
sunflower hypocotyl, tissue stresses.
Introduction
The definition of tissue stresses (TSs) has been given in a
previous paper in this series (Hejnowicz and Sievers,
1995a). In brief, the walls of all living cells in a plant
organ are under tension due to turgor pressure. The
parenchyma as a whole, however, is usually longitudinally
compressed while more-rigid tissues such as the epidermis
or the collenchyma are stretched. The stress which acts
1
on a tissue in an organ in excess of the turgor-induced
tensile stress is the TS for a particular tissue. Static
equilibrium requires that the sum of the forces generating
TSs acting in all tissues of an organ must be zero.
Therefore, a tensile TS is always accompanied by a
compressive TS in an organ. In another paper (Hejnowicz
and Sievers, 19956) the three-dimensional state of TSs
was determined for the hypocotyl of Helianthus on the
basis of measurements done on isolated tissues and
recalculated into the TSs using Poisson ratios. The TSs,
especially the tensile longitudinal stress in the epidermis,
attain quite high values. A question is: how are the TSs
generated?
It is commonly assumed by plant physiologists that the
TSs originate from a tendency of the various layers within
an organ to grow at different rates (differential growth)
(Sachs, 1865; Kraus, 1867; Thimann and Schneider, 1938;
Diehl et al., 1939; Burstrom et al., 1967; Kutschera and
Briggs, 1988), though it is generally accepted by plant
biomechanicians that TSs are due to turgor pressure, as
stated by Vincent and Jeronimides (1991). Recently,
Brown et al. (1995a, b) studied the role of the pith in the
growth and development of internodes in Liquidambar.
They emphasized differences in rates of cell multiplication
between inner and outer tissues, and the rapid production
and differentiation of vascular tissue as a histological
basis of compressive and tensile stresses in the internodes.
It is also commonly accepted that plant organs are
characterized by symplastic growth (Erickson, 1986).
This means that the growth of neighbouring cells is
integrated, i.e. adjacent cells do not alter their position
relative to each other and no new areas of contact are
formed, in contrast to animal organs where integrated
Present address: Department of Biophysics and Cell Biology, Silesian University, ul. Jagiellonska 28, PL-40-038 Katowice, Poland.
To whom correspondence should be addressed. Fax: +49 228 732677.
Abbreviations: IT, inner tissue; OT, outer tissue; TS, tissue stress.
2
© Oxford University Press 1996
520
Hejnowicz and Sievers
growth is accompanied by movements of cells which
thereby change their contacts. The differential growth, if
it occurs, must be within the range of the symplastic
growth regime and, therefore, may only be transient
unless there is a continuous development of curvature.
To illustrate this point, let us consider a cylindrical organ
growing longitudinally. Growth is defined as plastic (irreversible) extension; thus, differential elongation means
differential plastic extension in the longitudinal direction.
Symplastic growth requires the total extension of each
tissue to be the same. Thus, in the case of the occurrence
of differential growth, a difference in plastic extension of
different tissues in an organ must be compensated for by
elastic strains. The slower-growing tissue must be elastically stretched, i.e. a tension necessarily appears in this
tissue, while the tissue initially growing faster is brought
into compression. The tension (compression) enhances
(reduces) the further plastic extension of the tissue, so
that the difference in the elongation rates diminishes and,
finally, the rates of plastic extension of the tissues in the
longitudinal direction become equal (on the assumption
that the tissues differing in growth rate are distributed
radially and symmetrically in the organ; otherwise, even
such a transient differential growth will result in bending
of the organ). This would be the origin of TSs due to
differential growth.
Tissue stresses occur in both the elongating and nonelongating parts of the hypocotyl of sunflower (Hejnowicz
and Sievers, 19956). Our own unpublished observations
on longitudinal TSs indicate that the TSs disappear when
the organ is immersed in a solution which causes incipient
plasmolysis in cells, but recover fully after transferring
the organ into water. They also disappear reversibly when
the organ wilts. Hence turgor pressure is required for the
occurrence of TSs. There are two possibilities: (1) turgor
pressure brings about TSs without any differential growth
of various layers; or (2) turgor pressure is only necessary
to manifest TSs originating due to differential growth.
Although, as already mentioned, plant biomechanicians
consider that TSs are due to turgor pressure, the dependence of the TSs on the pressure and the moduli of
elasticity of the various layers has not yet been formulated.
This was undertaken in the present study.
In general, in an organ all stresses (in different directions) act on all strains, and a number of elastic moduli
and Poisson ratios would be required to describe its
elastic state. Moreover, experimental evidence indicates
that the elastic moduli of layers depend on turgor pressure
or applied stress (Murase et cil., 1980; Pitt, 1984; Niklas,
1992). Although the situation is complicated, it is possible
to obtain an insight into the problem of the origin of the
longitudinal TSs when considering a model built on
dependencies between the longitudinal elastic strains of
tissues and the stresses caused by turgor in a cylindrical
organ (Appendix A). If the organ is considered to be
represented by the hypocotyl of sunflower, the model
allows the turgor-derived TSs for the hypocotyl to be
calculated and compared with the experimentally determined TSs (Hejnowicz and Sievers, 19956). In this study
the moduli of elasticity are presented for the tissues in
the hypocotyl which is composed of two types—outer
(OT) and inner (IT) tissue—and they are used to calculate the magnitude of the forces (F) generating the TSs
in the hypocotyl according to equation 6A (Appendix A).
Materials and methods
Achenes of Helictnthus annuus L. cv. giganteum were germinated
in darkness at a temperature of approximately 20 °C. Only the
apical regions of the hypocotyls were used from plants having
60-80-mm-long hypocotyls. The OT was isolated by peeling
under a dissecting microscope and used as strips 1-1.5 mm
wide. The IT was obtained by peeling off the entire OT from
the hypocotyl segment. Immediately after peeling, the apical
end of the OT strip was either fixed mechanically in a light
plastic clamp, which had a hook at the fulcrum end, or glued
by means of cyanoacrylate to a narrow strip of plastic (20 mm
long) bent sharply at one end to make a hook. The basal end
of the strip was clamped at the bottom of the tensiometric
chamber, so that the free length of the strip was approximately
15 mm. A piece of lead (0.1-0.2 g) was added to the lower end
of the strip to facilitate the clamping. The hook above the upper
end of the strip was connected to the tensiometer described
previously (Hejnowicz and Sievers, 19956). Before the strip was
mounted, its width and length had been measured under a
dissecting microscope. The segments of IT were prepared and
mounted as described previously (Hejnowicz and Sievers,
1995*).
The specimens in the tensiometric chamber were immersed in
well-aerated distilled water. Mannitol was added to change the
osmolarity of the medium. In preliminary experiments, halfstrength Gamborg's B5 medium without hormones was used
(Sigma Chemie, Deisenhofen, Germany) to match the solution
used in long-term experiments with epidermal peels (Gorton
et al, 1989). However, specimens immersed in water or
Gamborg's medium did not differ in their elastic properties, so
that medium was not used again.
Measurements were performed in creep experiments: a specimen was rapidly loaded, or the load was rapidly increased, so
as to bear a new constant stress. The strain was monitored with
time. During the measurement of a uniaxial modulus the osmotic
pressure of the solution remained constant while the applied
load was changed. During measurements of a multiaxial modulus the applied load remained constant but the osmotic pressure
of the medium was changed. The osmotic pressure of the
solution was controlled by using mannitol solutions at molarities
of 50, 100. 150, and 200 mM. The change of tissue length,
determined on the tensiogram, was recalculated into conventional (Cauchy) strain (Niklas, 1992), i.e.
£= •
L
The cross-sectional dimensions of the specimens used in the
calculations were the original dimensions as measured under
nominal stresses.
The OT strips or the IT segments were recorded under tensile
or compressive forces, respectively.
Origin of tissue stresses
Determination of the osmotic pressure of the cell sap was
done by means of a Freezing Point Osmometer (OM 801; Vogel,
Giessen, Germany). Cell sap was expressed at full turgor from
apical and basal segments cut from 7-cm-long hypocotyls grown
in weak light at approximately 20 °C.
Results
Apparent uniaxial moduli
Figure 1 illustrates the changes in length of an OT strip
caused by changes in the stretching load (force). A small
initial load of 1.5 g (approximately 1.5xlO~ 2 N) was
constantly present. The first part of the tensiogram (A)
shows changes in length upon application and removal
of the load. In the second part (B) changes in length are
given as a function of increasing and decreasing the load
stepwise. It can be seen from Fig. 1 that the material
differed in two respects from that of an ideal elastic
material: (i) in addition to a rapid change of length on
loading, which represents the elastic change, there was
also a slow change with a decaying rate, and (ii) an excess
of strain remained at the end of a closed loop of force
change.
The former (i) slow change in length after changing
the force, is due to the viscoelastic and plastic behaviour
of the tissue. It represents the occurrence of creep, which
has two components: reversible, the retarded elastic component (transient creep) probably due to the osmo-elastic
properties of the tissue; and irreversible, the plastic component. The retarded elastic change is nearly completed
30 s after the force change. The reversible strain, se,
(elastic instantaneous and retarded elastic) was measured
30-40 s after the change and used to determine the
relationship between ee and force. The latter (ii) irrevers-
521
ible strain (indicated by the increasing length upon
removal of the load in successive cycles of stretching and
relaxing), obviously represents the plastic component in
the deformation of the tissue. This component was
observed in the studied tissues though they were derived
from behind the region of fast elongation in the hypocotyl.
The contribution of the elastic component can be
expressed by the 'degree of elasticity' which is the ratio
of elastic (reversible) deformation to total deformation
when a material is loaded to a certain stress and then
unloaded (Frey-Wyssling, 1952). The degree of elasticity
in the strip depends on the duration of stress. For the
intervals of loading in part A of Fig. 1, it was 0.94-0.97;
for part B it was approximately 0.90. If the strain is
measured upon loading, it should be corrected by multiplying the measured value of length increment by the
corresponding degree of elasticity. The strain obtained
after removing loads (Fig. 1A) does not need such a
correction, because the length change on reloading is
reversible by definition.
In Fig. 2 the data of Fig. 1 are replotted in the form of
a force/strain curve (analogous to a stress/strain curve).
Force per millimetre of strip width instead of stress was
used because we are interested in the product Edy.Ad
which is equal to dF/ds. Given a curve such as that in
Fig. 2A, in which the force per 1 mm of OT width is
plotted against the strain, the value of Ed x Ad can be
obtained by multiplying the slope of the curve at a
particular force by the length of the circumference.
The curves presented in Fig. 2 are typical of the OT.
Stiffening with the strain can be seen. Figure 2A refers to
part A of Fig. 1. It shows that force/strain values obtained
on loading are similar to those obtained on reloading.
Figure 2B refers to part B of Fig. 1. The force/strain
curve obtained from stepwise loading coincides well with
004
time
Fig. 1. A representative tensiogram for a strip of outer tissue (16 mm
long, 1 mm wide) from a sunflower hypocotyl. (A) Series of the loading
cycles with increasing load; (B) a cycle with stepwise increasing and
decreasing loads, showing the hysteresis. Arrows indicate instants of
load changing, with the value indicated above in grams (1 g * 10~2 N).
Numbers in parentheses give the actual full load.
Fig. 2. Stress/strain curves for the measurements shown in Fig. 1' (A)
for the series of loading (squares) and unloading (circles) cycles during
the interval A in Fig. 1, and (B) for the stepwise increase and decrease
of the load during the interval B in Fig. 1. Arrows indicate the direction
of load change. Hysteresis is seen in the cycle in B. Since the
measurements shown in Fig. 1 were done by addition of the load to the
original load of 1.5 x 10~2 N, the part of the stress/strain curve within
the range of the original load was taken from the other measurements
(inset in B and intermittent lines of the curves in A and B).
522
Hejnowicz and Sievers
the curves shown in Fig. 2A, indicating that the tensiogram obtained from stepwise loading can be used for the
determination of Ed x Ad. Stepwise loading was often used
for this purpose and Ed x Ad was calculated from AF/As
instead of from the tangent dF/de. Occasionally, values
of Ed x Ad obtained by the two methods were compared.
However, the force/strain curve obtained from the stepwise removal of the force (last part of B in Fig. 1 and
the curve with downward-pointing arrows in Fig. 2B) is
clearly different from that obtained by increasing the
force stepwise. The difference represents an elastic hysteresis of the tissue, a phenomenon repeatedly reported for
different tissues and cell walls (Cleland, 1967; Fujihara
et cil., 1978; Pitt, 1984; Hohl and Schopfer, 1992).
Hysteresis was observed in all experiments of stepwise
loading and reloading (but not in one-step loading and
reloading such as that shown in Figs 1A and 2A), both
in the case of the OT and the IT. The ratio of the area
under the unloading curve to the area under the loading
curve is known as the elastic efficiency, or simply the
resilience, of the tissue (Vogel, 1988), and indicates the
capacity of elastic material to store and subsequently
release strain energy. The resilience of the OT was
approximately 75-80%.
Figure 3 shows the product Ed x Ad, at full turgor
plotted against force: the dependence on force is evident.
The measurements of Ed x Ad were repeated on a broader
basis. The values were determined when the OT strips
were immersed in solutions of different molarities. Five
strips were measured for each molarity in successive steps
of the stretching force. The results (recalculated for the
whole circumference of the hypocotyl) are shown in
Fig. 4. The dependence on the force was high, but that
on the osmotic pressure turned out to be weak.
Figure 5 shows a representative example of the tensiometric record of the change in length of a peeled segment
(IT) when a compressive force was applied and removed.
20-1
0.1
02
03
Fig. 4. Dependence of the modulus for uniaxial tensile stress in the
outer tissue of sunflower hypocotyl (multiplied by the cross-sectional
area of the tissue) on the tensile force (for the whole circumference) at
different molarities of the medium. Standard errors of means are within
therange(±)0.3-l.l N.
Initially, the segment, mounted and immersed in water,
was prevented from expanding in water (Hejnowicz and
Sievers, 19956) by applying an appropriate load: 0.5 N
(«weight of 51 g) was necessary to achieve this in the
case of the particular peeled segment shown in Fig. 5.
Upon stepwise removing of the load the segment
expanded (phase A). To obtain the elastic component,
the released segment was loaded again successively with
varying forces and released from these forces (phase B).
The reversible length change was considered to represent
the elastic component. The value of El x As was calculated for each step. Then, a stepwise loading and reloading
was performed (phase C). Figure 6 shows the data from
the tensiogram in Fig. 5 in the form of force/strain curves:
A or B in Fig. 6 refers to the phase B or C in Fig. 5,
respectively. The values from loading and reloading
(Fig. 6A), as well as those from the stepwise loading
(Fig. 6B, upper curve), aligned along similar curves, while
the data from stepwise reloading (Fig. 6B, lower curve)
gave another curve, indicating some hysteresis. The resilience in the case shown in Fig. 6B was 72%; in general, it
was in the range 70-75% for the IT in water. It is
interesting that, unlike that for the OT (Fig. 2), the
loading curve for the IT was not convex with respect to
the strain axis, which means that the modulus did not
increase with strain. Within the range 0-0.4 N the
force/strain at full turgor was nearly linear, i.e. the
modulus El was nearly constant (approximately 0.4 N
was the force involved in the TSs and acting on the crosssectional area of the IT). Figure 7 shows the data summarizing the dependence of £jf x A8 on the force and
osmotic pressure. The value of El x Ae was only slightly
dependent on the osmotic pressure of the medium, except
at high pressure and low force.
force (Nl
Fig. 3. Dependence of the modulus for uniaxial tensile stress for the
outer tissue (multiplied by the cross-sectional area of the tissue) on the
tensile force (for the whole circumference) at full turgor. Standard
errors of means were within the range (±) 11 83 x 10" 2 N.
Apparent multiaxial moduli
The modulus Em was calculated from the ratio of the
change in osmotic pressure to the corresponding change
Origin of tissue stresses
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Fig. 5. A representative record for a segment of the inner tissue of
sunflower hypocotyl (10 mm long, cross-sectional area 2.5 mm 2 )
immersed in water and initially prevented from expanding by a load of
5:0.5 N (51 g). (A) The load was diminished in steps (10 g was removed
at each step); (B) a series of loading cycles with increasing load
(addition and subtraction of the load); (C) a cycle with increasing and
decreasing load showing the hysteresis. Arrows indicate instants at
which the load (with the value indicated) was changed; (—) load
subtraction, (+) load addition. Numbers in parentheses give the actual
full load in grams (1 g * 10~2 N). A small plastic component in length
change (increased length after returning to the minimal load) can be
seen. The decrease in length upon load subtraction served for the
estimation of the uniaxial modulus.
0 02
004
006 0
strain
002
004
006
Fig. 6. Stress/strain curves for the measurements presented in Fig. 5:
(A) for the series of loading (squares) and unloading (circles) during
the interval B in Fig. 5, and (B) for the stepwise loading (arrows up)
and unloading (arrows down) during the interval C in Fig. 5. Hysteresis
is seen in the cycle in B.
523
in the relative length, Em = l0P/Al. Figure 8 shows the
changes in length of the IT, caused by the changes in the
osmolarity of the solution in which the IT was immersed,
when the tissue was nearly free of uniaxial tension (a load
of only 1 g had to be applied to the cross-sectional surface
area of approximately 2.4 mm2 to make the tensiometric
measurement possible). It can be seen that reversible
changes in length occurred when the osmolarity changed.
The inset (Fig. 8) shows the relationship between the
changes in osmotic pressure of the external medium and
the strain, and indicates that the points relating the strain
changes to either the decreased or increased osmotic
pressure are aligned along a common curve. Both types
of data—from the increase and the decrease in the
pressure—were used in the calculation of £*. Figure 9
shows the modulus with reference to the osmotic pressure
of the medium and the uniaxial compression. In general,
£* was slightly higher at lower osmotic pressure of the
medium (higher turgor pressure in the tissue). The modulus appeared to be lower at 25 x 10" 2 N than at lower or
higher applied force except at very low turgor.
The measurements of the multiaxial modulus for the
OT were difficult. Since a layer of cortical parenchyma
was attached to the epidermis, the strips showed a tendency to bend at low load and low osmolarity of the
medium so that artifacts in this range are possible.
However, in our analysis the Edm for low load was needed
only at low turgor (high osmolarity of the medium) when
bending usually did not occur. This Em was 46.8 MPa
(Fig. 10). The uniaxial tension applied to the epidermal
strips reduced the bending, but also lowered the sensitivity
of the strips to turgor change, i.e. the apparent modulus
Em increased strongly with the tension (Fig. 10). Even at
the tensional force which was half of the force involved
in full TSs, the change from 200 to 0 mM mannitol
molarity ( mM)
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Fig. 7. Dependence of the modulus for uniaxial stress for the inner
tissue multiplied by the cross-sectional area of the tissue on the
compressive force (acting on the whole area) at different molarities of
the medium. Standard errors of means are within the range (+)0.35—1 N.
\
vi VJ
Fig. 8. A representative record of the change in length of a 10-mmlong segment of the inner tissue when the molarity of the medium was
changed. The segment was loaded with a weight of 1 g. Initially the
segment was equilibrated with distilled water (immersed for 30 min so
that the initial fast expansion ceased), then the water was replaced by
mannitol solution at 200 mM, at 150 mM, at 200 mM again, and so on
as indicated. The inset shows an analogue of the stress/strain curve
(relationship between strain and osmotic pressure of the medium) for
the measurements in this figure.
524
Hejnowicz and Sievers
Fig. 9. Dependence of the modulus for multiaxial stress for the inner
tissue on the molarity of the medium at different compressive forces.
Standard errors of means are in the range ( + )0 36-0.79 x 106 N m~ 2
the turgor pressure, dP, in the cells is equivalent to the
decrement, — dP, of the osmotic pressure of the medium.
The moduli determined as functions of the osmotic pressure in the integration of Eq. 6A from —0.5 MPa to 0
are used. Table 1 shows the values of the apparent moduli
(or their products with cross-sectional areas in the case
of uniaxial moduli) which were used in the numerical
integration of Eq. 6A. At a particular step of integration,
the values were chosen according to the already-generated
force on the basis of the data presented in Figs 4, 7, 9,
and 10. The result of the integration is shown in the
bottom row of Table 1. The calculated force at full turgor
amounts to approximately 0.39 N. It can be seen that the
force is strongly dependent on the turgor.
Discussion
v°^
Fig. 10. Dependence of the tissue composite modulus for multiaxial
stress for epidermal tissue on molarity at different tensile forces (for the
whole circumference). The modulus was higher than 1 0 8 N m ~ 2 at a
force >0.2 N. As accurate values could not be determined, the Edm axis
terminated at 10s N m" 2 . Standard errors of means were in the range
( + )0.8-1.4x I O 6 N m " 2 at the force 0.01 or 0 02 N.
resulted in a nearly imperceptible change in tissue length.
The corresponding apparent modulus was larger than
200 MPa. The reciprocal, (Edm)~l, which entered the
model, was negligible in comparison with (£•*)"'.
Determination of osmotic pressure of the cell sap
The osmotic pressure of the cell sap amounted to
0.51 MPa. Interestingly, this osmotic pressure is similar
to that of the solution which prevents the expansion of
the isolated IT (but the osmotic pressure in the cells of
the IT in such a solution is then higher than that measured
at full turgor, by an amount determined by the volumetric
elastic modulus).
Calculation of the forces involved in TSs
When the osmotic pressure of the external medium is
0.5 MPa there are no TSs in the hypocotyl (the osmotic
pressure of the mannitol solutions was calculated from
the relationship between water potential and the molarity
of mannitol (Michel et al., 1983)). With a decrease in
osmotic pressure the tissue stresses develop and attain the
full and normal value at zero pressure. The increment of
The one-dimensional model formulated on the basis of
the phenomenological relationships is simple and the
mathematics do not obscure the biological problem, but
it can give only approximate values of the forces involved
in the TSs. The state of isolated tissues differs from that
in the organ (Hejnowicz and Sievers, 1995a), and the
data from measurements on isolated tissues should be
corrected for lateral constraints which exist in the organ
but are lacking in the isolated state. It is also known that
the corrections due to lateral constraints are less than
30% in the case of the OT of sunflower hypocotyl
(Hejnowicz and Sievers, 1995a). Since the model could
give only approximate values, the correction could be
omitted. In spite of all the simplifying assumptions used
in the model, the magnitude calculated from the model
(0.39 N) fits quite well the value determined experimentally (0.37 N, Hejnowicz and Sievers, 19956).
One aspect of the biomechanical importance of the TSs
concerns the accumulation of a large tensile stress in the
epidermis (Hejnowicz and Sievers, 19956). This brings
about a prestressed state in the stems. Such a state is
mechanically favourable, as is known from prestressed
engineering constructions. In addition to the general
benefit of a prestressed state, the tensile TS in the hypocotyl allows the epidermis to 'work' in the hypocotyl with
considerably increased modulus of elasticity because the
modulus increases strongly with strain (Fig. 2). It can be
predicted that if the epidermis were only turgid but not
additionally stretched by the TS, the flexural stiffness of
the hypocotyl would be much lower. One may ask why
the TSs, if they are so favourable, are not additionally
enhanced by differential growth. Probably, enhancement
of the turgor-derived TSs by differential growth might
make the danger of buckling of the IT too high. One
may also ask why the magnitude of the TSs is just such
as that predicted by the model. After all, an organ does
not develop first without turgor, then take up water,
become turgid and the TSs appear. Such a series of events
Origin of tissue stresses
525
Table 1. Values which were used in numerical integration of equation 6A
The values are functions of molarity of the medium and of the already-attained force calculated from the integration up to the particular molarity.
Molarity of mannitol in the medium (mM)
200
Osmotic pressure of the medium (IT)
Apparent turgor pressure (0 15 — n)
Multiaxial composite modulus for the
IT (£«)
Multiaxial composite modulus for the
OT (EJJ
The product of uniaxial modulus and
cross-sectional area for the OT
(E{ x Ad)
As above but for the IT (£« x Ae)
Calculated force (F)
MPa
MPa
MPa
MPa
0.49
0.02
3.63
46.8
175
0.43
0.08
3.63
60.8
150
0.37
0.14
3.92
100
125
0.31
0.20
4.90
100
100
0.24
0.26
4.90
100
50
75
0.18
0.32
4.71
100
25
0.12
0.38
4.51
100
0
0.06
0.45
4.51
100
0
0.51
5.88
100
N
2.94
2.94
2.94
3.92
5.10
9.61
12.45
13.73
15.20
N
N
6.32
0.01
6.32
0.04
7.35
0.07
8.83
0.11
9.32
0.14
9.32
0.39
9.32
0.26
8.82
0.33
7.84
0.39
was assumed for the model because it was found that
there were no TSs at incipient plasmolysis and the TSs
recovered completely when the organ was immersed in
water. When answering this question reference is made
to the principle of minimum of strain energy: the turgorderived stresses predicted by the model fulfil this principle
(Appendix B). Every tissue, considered as an elastic
subject, must obey the basic physical principle of the
minimum of potential energy unless metabolic energy is
used to change the energy from its minimum. It turns out
that the state of a plant organ described by Eq. 6A is
that in which the internal energy of the elastic strains of
its tissues is at its minimum.
Meshcheryakov et al. (1992) have shown that in the
hypocotyl of intact plants there may be gradients of
turgor pressure across the cortex. In the hypocotyls of
Ricinus communis, there was a steep gradient of turgor
pressure increasing centripetally. It may well turn out
that such a gradient is common in hypocotyls. Our model
can easily be adapted for such a case by considering n>2.
This was done (not shown), with the conclusion that the
centripetal gradient of P generates tensile stress in the
peripheral layers even if they do not differ in elastic
moduli. Meshcheryakov et al. (1992) also showed that
the gradient of turgor pressure in the hypocotyl of Ricinus
was reduced or completely disappeared following certain
experimental treatments such as decapitation of cotyledons or excision of roots. If the same happens in the
hypocotyl of Heliantlms, the TSs previously measured
(Hejnowicz and Sievers, 19956) pertained to the state
when the turgor gradient had already disappeared from
the hypocotyl. Such TSs therefore correspond to those
calculated in this study where turgor was assumed to be
constant across the hypocotyl.
Apart from measurements done by Kutschera (1991),
which were discussed in previous papers of this series,
and those done by us, there have been only a few
quantitative determinations of longitudinal TSs in plant
organs. Miiller (1880, p. 191) had to apply 0.13 x 106
N m ~ 2 tensile stress to a sector of the peripheral tissue
of a young Ricinus stem to stretch it to its initial length.
This value is similar to that obtained by dividing the
tensile force acting on the OT of sunflower hypocotyl by
the cross-sectional area of this tissue (Hejnowicz and
Sievers, 1995ft). Muller (1880, p. 190) also determined
the force necessary to prevent the expansion of a block
of the 'pith' isolated from a segment of a Helianthus stem
250 mm long in water (the block probably represented
the pith crown because the stem of Helianthus has a
central cavity). This force recalculated into compressive
stress gave a value of 1.35 x 10 6 N m~ 2 , which is much
larger than that determined for the IT of Helianthus
hypocotyl (Hejnowicz and Sievers, 1995ft). One must
take into account, however, that a stem such as that
studied by Muller (1880) contains collenchyma which is
characterized by a higher modulus than the epidermis. In
the collenchyma, large tensile forces involved in TSs can
develop (Ambronn, 1881); thus a large compressive force
may also be concentrated in the pith crown. To restretch
the strand of collenchyma from a young internode of
Foeniculum officinale, Ambronn (1881) had to apply
approximately 380 g, which gives a stress of approximately 40 x 10 6 Nm~ 2 .
Recently Vincent and Jeronimidis (1991) calculated the
total tensile stress induced by turgor in stems of dandelion.
Psilotum and the herbaceous fossil plant Rhynia. Young's
modulus of tissue layers across the stem radius in these
stems was a function of the volume fraction of the cell
wall. In their method the TS was superimposed on the
turgor-induced stress which would exist in a layer if it
was isolated from the stem. The total tensile stress in the
outer layer of dandelion amounted to 6 x 106 N m~ 2 at a
turgor pressure 2 MPa, thus the stress in an isolated layer
would be equal to the turgor pressure in this layer and
526
Hejnowicz and Sievers
the tensile TS in the outer layer would be 4 x 106 N m~2
which is 1.7-fold that in the epidermis of sunflower
hypocotyl.
It should be mentioned that in the secondary xylem,
or even in the cork of woody stems, there occur 'growth
stresses' (Archer, 1987; Fortes and Rosa, 1992). The
source of growth stresses in wood is the addition of new
cells by the cambium to the wood, and longitudinal and
circumferential shrinkage developing in secondary walls
(Archer, 1989). It is reasonable to think that in young
organs, too, in the cells which function with unlignified
walls only, some developmental changes may occur in the
walls affecting the TSs. This would be another possible
mechanism of generation or modification of TSs in turgescent organs, in addition to differential growth and the
mechanism presented in this paper.
As far as the authors are aware, there are no data on
tissue composite moduli (£„) for the IT and/or the epidermis of the Helicmthus hypocotyl in the literature. To
obtain the moduli the measured values of Eu x A should
be divided by A. The average A8 amounts to 2.42 mm2
(Hejnowicz and Sievers, 19956). The average Ad is
0.42 mm2; however it has little value for the calculation
of Edu because the OT is a complex tissue. It contains the
epidermis and one or two layers of cortical parenchyma.
The epidermis is the layer which limits the changes in
strip length during the Edu measurements. However, it is
the cortical parenchyma which contributes mostly to the
cross-sectional area of the OT. The cross-sectional area
of the epidermis alone should be used for the calculation
of Eu for the epidermis; this area amounts to 0.13 mm2
for the whole circumference of the hypocotyl (Hejnowicz
and Sievers, 19956). It has already been calculated that
86% of the force applied to an OT strip is transmitted
through the epidermis (Hejnowicz and Sievers, 19956).
Multiplying the value of Edu x Ad by 0.86 and dividing by
0.13 mm2 gives the tissue composite modulus for the
epidermis. In intact, turgid hypocotyls, Edu for the epidermis would be 10 8 Nm~ 2 . In the literature, there are
data on the modulus for the unpeeled hypocotyls of
Helianthus obtained by the resonance-frequency method
(Uhrstrom, 1969). The mean value for the modulus was
37x 10 6 Nm" 2 (calculated for the cross-sectional area of
the tissue) when the hypocotyl segments were in water,
and 32xlO 6 Nm~ 2 in 0.1 M mannitol. The resonance-frequency method measures the tissue composite
modulus in the case of a homogeneous tissue; however,
in the case of an organ like a hypocotyl it measures an
effective modulus, Eef, resulting from the contribution of
the tissues in the organ. Indeed, E*j<Eef<Ed. The comparison of these data with those provided by Uhrstrom
(1969) indicates that the effective modulus is closer to
the modulus of the epidermis, it being the peripheral
tissue, which is in agreement with the equations governing
the resonance-frequency of free-bending oscillations
(Uhrstrom, 1969).
Acknowledgements
This work was supported within the project AGRA VIS by
Deutsche Agentur fur Raumfahrtangelegenheiten, DARA,
Bonn, and Ministerium fiir Wissenschaft und Forschung,
Diisseldorf. We are greatly obliged to Professor E Steudle
(Universitat Bayreuth, Germany) for the stimulating criticism
and numerous suggestions. We thank Dr Sandra K Hillman
(University of Glasgow, UK) for correcting the manuscript,
Dipl-Ing P Blasczyk and Dipl-Ing G Schnitzler for electrical
engineering, and Mrs H Geithmann for drawing the figures.
Symbols
A, area (m2); E, tissue composite modulus (Nm~ 2 ); F,
force (N); /, length (m); P, turgor pressure (Pa); U,
potential energy (J); A, difference; e, strain; v, Poisson
ratio. Subscripts and superscripts pertain to cl, outer
(dermal) tissue; g, inner (ground) tissue; / or j the tissue
/ or j ; m, multidirectional stress; o, initial state; w,
unidirectional stress.
Appendix A
Let us consider a cylindrical organ composed of several layers,
denoted by superscripts: / = 1,2,...,/», where n is the total
number of layers considered in the organ. It is assumed that
the layers have different moduli of elasticity calculated for their
cross-sectional areas (not for the cell walls only).
Turgor pressure (P) is a multiaxial stress in cells which results
in strains in all directions. It may give rise to longitudinal TSs
(uniaxial stresses) and corresponding strains in layers of an
intact organ. Here, only the longitudinal stresses and strains
are considered. The longitudinal strain caused directly by P is
denoted by em (m indicates the multiaxial stress). By contrast a
longitudinal uniaxial stress, caused by the longitudinal force, F.
distributed over the cross-sectional area, A, through the layer,
results in the longitudinal strain, eu. The two strains result in
an overall strain, e, which must be the same for all layers in an
organ because they do not glide past each other (symplastic
elastic extensions). Thus for each layer:
= s'm + s'u,
and
Ae =
+ Ae'u.
(1A)
It is assumed that small increments of both strain components
are proportional to the increments of relevant stresses.
Therefore:
= alxAP,
and
AF
—T
(2A)
where: a' and b' are coefficients empirically determined at
different values of P and F for an /-layer (tissue). The values of
a and b are small (of compliance type); however, they may be
represented by reciprocals which are large numbers of the type
Origin of tissue stresses
characteristic of elasticity moduli (E):a=\/Em, b=\/Eu. For
convenience Em and Eu are called the multiaxial and uniaxial
moduli (for longitudinal strains), respectively. The uniaxial
modulus is the same as the tissue composite modulus defined
by Niklas (1989). The relationships between strains and stresses
in an /'-layer can thus be written in the form:
AP'
and
~'~EZ'
Ae' =
AF
E,,xAr
(3A)
Equations 2A and 3A show a similarity to Hooke's principle
(linearity between strain and stress); however, they do not
represent this principle strictly, because Em and Eu may depend
on stresses or strains.
It is assumed that the forces F' do not cause movement in
the system, i.e. the system is in static equilibrium. This requires
that the sum of forces (and of their increments at any step of
the change) is zero in an organ:
F = 0,
AF =
and
(4A)
The force (and the resulting strain) is considered to be positive
when it causes an extension (as would be the case after
increasing turgor).
The turgor pressure in cells is balanced by the tensions in the
cell walls, so this balance does not need any further formulation
in our analysis. Also the second condition of static equilibrium—
the sum of torques in the organ is zero—does not need a
formulation because radial symmetry of the organ with respect
to all factors affecting the tissue stresses was assumed. In a
symmetrical organ, the sum of torques exerted by forces Ft
which fulfil Eq. 4A is zero, because the torques exerted by pairs
of forces acting in opposite segments of the organ cancel
mutually. However, as soon as the radial symmetry is lost,
which may occur during a tropic response, the resulting nonzero sum of the torques will tend to bend the organ. Here only
two kinds of tissue in the hypocotyl are considered: outer tissue
(OT) and inner tissue (IT) which are symbolized by superscripts
d (from dermal) and g (from ground), respectively. Since the
same extension is yielded in the two tissues, Eqs 1A and 3 A give:
d
AF
d
AP
~Ei
g
=
AF
EgxAg
AP
ITFS
Lj
(5A)
J__J_
when the tissue is transversely isotropic, i.e. the elastic properties
in the two transverse directions (in width, w, and in thickness,
t) are the same, or
when the properties are different (Hejnowicz and Sievers,
1995a), where vlw and v,, are the Poisson ratios in the w or /
directions for the longitudinal stress. This means that
(\-2VlJ{EX\
or
( l - v l l v - v , , ) (£„)-'
l
performs the role of (Em)~ i.e.
Ett = kxEm
(2B)
where k=\—2vlv) or k= 1 — vlw — vu for transverse isotropy or
anisotropy, respectively. We can choose to work with Eu in
Eq. IB. The increment of the strain energy per unit length of
the organ is the sum of increments for the tissues:
(AU)organ = -(Ae) 2 xEduxA"
+ -(As)2 xEgx
A*.
From Eqs 1A, 3A and 5A:
P
AF
AF
P
A
Ae = — ++
.
Eg
Eg Eg x Ag
Thus, the increment of strain energy per unit length of the
organ can be written in the form:
J
and
2
AF
1
JAP
AF )
+ ^gxAd[— +E*xAg)
The increment attains its minimum with respect to AF at
d(AU)/d(AF) = 0.
-_
pd
I
Fg I
•
1
EgxAg
A E-l
\ Fdx
Ad'
= 0.
Therefore
Ei E*r-
El x Ad x Eg x As
" ' ~ EUA*
1
1
E~i~~E{
which is identical to Eq. 6A.
(6A)
It may be seen that if the layers in an organ do not differ in
the moduli, the TSs do not appear (Fd = 0). If the moduli of
the OT are higher, this tissue is under tensile TS (Fd>0). To
calculate the magnitude of the forces involved in TSs, Eq. 6A
should be integrated after changing deltas into differentials.
Appendix B
It can be shown that Eq. 6A obeys the minimum of strain
energy in the organ. Namely, the increment of strain energy
(U) due to uniaxial stresses in each tissue per unit length is:
-(Ae)2xExA.
2,,J(£„)
AF(£U x / I T ' =
m
Introducing AFs=—AFd (from Equation 4A) and making a
simplifying assumption that APd = APg = AP, then
(AU)lissue =
In As, two types of modulus occur, Em and Eu, while Eq. IB
needs only one modulus. The two moduli are interrelated by
Poisson ratio because at the same strain increment
d(AF)
g
527
(IB)
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