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Elongation growth behavior of chara corallina

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Elongation growth behavior of chara corallina the effect of a longitudinal force on growth and an analysis of the resulting stress distribution
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Araneta, Eric Craig
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xi, 65 leaves : ; 28 cm

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Chara corallina -- Growth ( lcsh )
Stress concentration ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Includes bibliographical references (leaves 63-65).
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Department of Mechanical Engineering
Statement of Responsibility:
by Eric Craig Araneta.

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Full Text
ELONGATION GROWTH BEHAVIOR OF CHARA CORALLINA: THE EFFECT
OF A LONGITUDINAL FORCE ON GROWTH AND AN ANALYSIS OF THE
RESULTING STRESS DISTRIBUTION
by
Eric Craig Araneta
B.S., University of Central Florida, 1988
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Mechanical Engineering
2000


This thesis for the Master of Science
degree by
Eric Craig Araneta
has been approved
by

Date


Araneta, Eric (M.S., Mechanical Engineering)
Elongation Growth Behavior of Chara corallina: The Effect of a Longitudinal Force
on Growth and An Analysis of the Resulting Stress Distribution
Thesis directed by Professor Joseph K.E. Ortega
ABSTRACT
The elongation growth rates of single intemode cells of Chara corallina were
determined before and after they were subjected to uniaxial longitudinal forces of
different magnitudes. An increase in growth rates (accelerated growth) was
observed after the application of longitudinal forces. In general, the magnitude and
duration of the accelerated growth depended on the magnitude of the longitudinal
force. Following the accelerated growth, the growth rate returned to a steady value
that was typically greater than the basal growth rate (i.e., before the application of
the longitudinal force). Theoretical work was conducted that demonstrated how the
ratio of transverse stress and longitudinal stress in the cell wall was altered after the
application of longitudinal forces. Experiments were conducted in an attempt to
determine the longitudinal elastic modulus of in vivo cell intemodes. The results of
this study have implications on the common practice of hanging a weight on the end
of an intemode cell in order to measure the growth rate using a variety of electronic
displacement transducers.
m


This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Signed
Joseph K.E. Ortega
IV


DEDICATION
I would like to dedicate this thesis to my wife Erica, without whom this would not
have been possible.


ACKNOWLEDGEMENT
I wish to give special thanks to my advisor, Ken Ortega, for his guidance,
mentorship and patience in educating me in the fields of scientific investigation and
mechanical engineering.
I wish to thank Professor Samuel W. Welch for his input in the analysis
portion of this thesis, for serving on my thesis committee and for all his help and
guidance given to me during my education as a graduate student.
I wish to thank Elena Ortega and Jessica Olsen for their input, suggestions
and help with the experiments.
I wish to thank Ted Proseus and John Boyer of the University of Delaware
for supplying the initial Chara cuttings and for their instructions on the cultivation
and care of Chara corallina.
I wish to thank Professor Georgia Lesh-Laurie for her assistance with the
experiments and for serving on my thesis committee.
This work was supported by National Science Foundation grant IBN-
9603956 to J.K.E. Ortega.


CONTENTS
Figures.............................................................ix
Tables..............................................................xi
Chapter
1. General Introduction......................................... 1
1.1 Selection of Test Organism................................. 3
1.2 Statement of the Problem..................................... 5
1.3 Preview of Thesis Content.................................... 7
2. Materials and Methods........................................9
2.1 Plant Materials.............................................. 9
2.2 Longitudinal Force...........................................9
2.3 Longitudinal Elastic Modulus...................................11
3. Effect of a Longitudinal Force on the Longitudinal
Growth Rate.................................................. 14
3.1 Growth Rate Acceleration.......................................14
3.2 Duration of Acceleration Period as a Result of
Application of a Longitudinal Force.......................... 20
3.3 Resulting Steady Growth....................................... 23
3.4 Discussion.................................................... 26
4. Analysis of Cell Elongation................................... 29
vii


4.1 The Influence of Cell Wall Structure on Cell Elongation........... 29
4.2 Intemode Cell Wall Structure and Material Properties of Nitella.. 30
4.3 Theoretical Analysis of Cell Wall Stress Distribution..............32
4.4 Analysis of Cell Wall Stress Distribution for a Chara Intemode.... 42
4.5 Discussion........................................................ 49
5. Longitudinal Elastic Modulus...................................... 50
5.1 Discussion.........................................................58
Appendix
A. Development of Chara Cultures......................................59
B. Construction of Experimental Apparatus.............................61
References...........................................................63
vm


FIGURES
Figure
1.2.1 Schematic Illustration of the Apparatus from
Zhu and Boyer (1992) and Proseus et al. (1999),
for the Measurement of Cell Elongation.......................6
2.3.1 Schematic Illustration of the Apparatus
for Longitudinal Elastic Modulus Experiments................. 13
3.1.1 Length of a Chara Intemode as a Function
of Time for an Applied Force of 8.51 g....................... 16
3.1.2 Length of a Chara Intemode as a Function of
Time for an Applied Force of 3.56 g..........................18
3.1.3 Length of a Chara Intemode as a Function of
Time for an Applied Force of 2.4 g............................19
3.2.1 Duration of Acceleration Period as a Function
of Applied Force..............................................21
3.3.1 Ratio of Resulting Steady Growth Rate to B asal
Growth Rate as a Function of the Applied Force...............24
4.3.1 Cylindrical Thin-Walled Pressure Vessel........................34
4.3.2 Free Body Diagram of a Segment of a Cylindrical
Thin-Walled Pressure Vessel...................................36
4.3.3 Free Body Diagram of a Cylindrical Thin-Walled Pressure
Vessel........................................................38
4.3.4 Free Body Diagram of a Cylindrical Thin-Walled
Pressure Vessel with an Externally Applied Force..............41
IX


4.4.1 The Stress Ratio as a Function of Applied Force
for Various Turgor Pressures.................................48
5.1 Stress vs. Strain for a Typical Experiment....................52
5.2 Mean Stress vs. Mean Strain for Each of the Applied Forces....54
5.3 Mean Stress vs. Mean Strain For Trial One and Trial Two...... 56
B. 1. Apparatus for Longitudinal Force Experiments..................61
B .2. Apparatus for Longitudinal Elastic Modulus Experiments........62
x


TABLES
Table
3.2.1 Applied Longitudinal Forces with Their Associated
Acceleration Periods............................................22
3.3.1 Ratio of Resulting Steady Growth Rate to Basal Growth
Rate as a Function of the Applied Force.........................25
4.4.1 Membrane Stress Distribution for a Pressurized
Cylindrical Membrane with a Turgor Pressure of 0.5 MPa..........44
4.4.2 Membrane Stress Distribution for a Pressurized
Cylindrical Membrane with a Turgor Pressure of 0.7 MPa..........45
4.4.3 Membrane Stress Distribution for a Pressurized
Cylindrical Membrane with a Turgor Pressure of 1.0 MPa..........46
xi


1. General Introduction
The mechanics of plant cell wall extension are studied by plant physiologists
who want a better understanding of plant cell growth. Plant, algal and fungal cell
growth depend on two simultaneous and interdependent physical processes: the net
rate of water uptake and the net rate of cell wall extension (Ortega, 1985, 1990,
1994). The equation describing the net rate of water uptake was obtained by
applying the conservation of water mass to the plant cell (Ortega et al., 1988). The
conservation law states that the rate of change of water mass in the plant cell must be
equal to the difference between the flow rate of water into and out of the cell. The
application of this conservation law results in the mathematical expression known as
the first Augmented Growth Equation (Equation 1.1) (Ortega et al., 1988; Ortega
,1990, 1994):
v, =
fdV'
\ dt j
/Vw = L(Ax- P)-T
(1.1)
where vw is the relative rate of change in volume of the cell contents (mostly water),
Vw is the volume of the cell contents (mostly water), t is the time, L is relative
hydraulic conductance, A# is the difference in osmotic pressure between the cell sap
and the external medium, P is the turgor pressure (the pressure difference between
the cell interior and the external medium), and T is the relative transpiration rate.
1


The term, L(Ax P), represents the relative rate of water uptake and the term, T,
represents the relative rate at which water is lost from the cell by transpiration. For
cells that do not transpire (such as an alga in an aqueous growth medium) the term,
T, is zero.
The equation that describes the rate of cell wall extension is the second
Augmented Growth Equation (Ortega, 1985). The second Augmented Growth
Equation relates the relative rate of change in volume of the cell wall chamber to the
sum of the relative rate of irreversible (plastic) extension and the relative rate of
reversible (elastic) extension. The second Augmented Growth Equation (Equation
1.2) is:
f dV'
=
V dt j
/Vc= f dP}
\dt j
!e
(1.2)
where vc is the relative rate of change in volume of the cell wall chamber, Vc is the
volume of the cell wall chamber,

is the critical turgor pressure (a value of turgor pressure below which growth does
not occur and sometimes called the yield threshold, T) and £ is the volumetric
elastic modulus. In this equation, (p, the relative irreversible wall extensibility, is a
coefficient that reflects relevant enzymatic and metabolically mediated biological
processes that contribute to growth.
2


1.1 Selection of Test Organism
The alga Chara corallina is the organism that was used for the experiments
conducted for this thesis. Chara corallina is a photosynthetic green alga in the
division Chlorophyta that is in the kingdom Eukaryota. The alga Chara corallina is
closely related to the higher plants. The cell wall structure of Chara is composed of
alternating layers of transverse and longitudinal microfibrils (Wallace, 1967). The
cell wall structure of the higher plants is also composed of alternating layers of
transverse and longitudinal microfibrils (Taiz, 1984). This increases the possibility
of extrapolating experimental results to the higher plants. Also, Chara is very
closely related to Nitella. This is advantageous because there is a significant amount
of research conducted with Nitella, and some of the research conducted with Nitella
is relevant to Chara.
Chara has characteristics that make it well suited for experimentation. The
intemodes are large, hardy cells that are easy to cultivate and handle. The diameter
of a Chara intemode cell varies between 0.7mm and 0.9mm. In previous work (Zhu
and Boyer, 1992), the pressure probe was used to measure turgor pressure and the
diameter of the micro-pipette tip used to impale the intemode was approximately
40 pm (note that the cell intemode is 17 to 23 times larger in diameter than the
micro-pipette tip). Large tips on the micro-pipette reduces the possibility of
plugging the micro-pipette with the cell contents.
3


A most advantageous characteristic of Chara is that it lives in an aqueous
environment, and the hydraulic conductance of the cell is large enough that water
potential differences are negligible (Zhu and Boyer, 1992). Therefore, water is not a
limiting factor. Proseus, et al. (1999) showed that the main role of turgor pressure is
to stress the cell wall to facilitate cell wall extension and not to drive the water
transport of the cell. As indicated by equation 1.1, when the water supply is not a
limiting factor (i.e., the hydraulic conductance is very large), the turgor pressure, P,
is the same as the difference in osmotic pressure, An. Thus equation 1.1 may be
neglected. Proseus, et al. (1999) demonstrated that equation 1.2 is all that is needed
to model the growth of Chara. Because Chara cell growth may be modeled with
only the use of equation 1.2, the experimental work and analysis is significantly
simplified.
Also, because the intemode cell of Chara corallina grows primarily in
length, equation 1.2 may be revised to (Proseus, et al., 1999):
dL
dt
m(P-Pc) + ^
et dt
(1.3)
where (m s"1) is the elongation rate, m (m s"1 MPa'1) is the irreversible
dt
longitudinal extensibility of the cell wall, L0 (m) is the initial cell length and £,
(MPa) is the longitudinal elastic modulus.
4


1.2 Statement of the Problem
In previous work (Zhu and Boyer, 1992; Proseus et al., 1999), a wire with a
weight was attached to the basal end of the intemode cell of Chara to measure
elongation rate. The wire was then wrapped around a plastic disk that was part of a
rotational displacement transducer (see Figure 1.2.1)
Zhu and Boyer (1992) used a mass of 0.05 g with their apparatus. In the case
of the Proseus experiments the weight had a mass of 2.3 g. It is the application of
this weight that created an applied longitudinal force on the Chara intemode cell.
As the intemode changed length the wire turned the disk which in turn rotated the
rotational transducer.
It is not known what effect the applied weight (longitudinal force) has on the
growth rate of the Chara intemode. Furthermore, it is not known how the added
weight effected the stress distribution in the cell wall of the growing Chara intemode
cell.
The objective of this thesis is to investigate the effect of an applied
longitudinal force on the growth rate and stress distribution of an in vivo Chara
intemode cell.
5


Figure 1.2.1. Schematic Illustration of the Apparatus from Zhu and Boyer (1992)
and Proseus et al. (1999), for the Measurement of Cell Elongation. The intemode of
Chara was attached to the apparatus at its apical end. At its basal end, a tungsten
wire was attached. This wire was draped over a plastic disk which was attached to a
rotational transducer. A small weight (0.05 g in the experiments of Zhu and Boyer
(1992) and 2.3 g in the experiments of Proseus et al. (1999)) was attached to the free
end of the wire. The rotation of the disk was proportional to the change in
elongation of the cell intemode.
6


1.3 Preview of Thesis Content
This thesis is a combination of three investigations: 1) an investigation of the
longitudinal growth response to an applied longitudinal force (Chapter 3); 2) an
analytical investigation of the stress distribution within the cell wall (Chapter 4); and
3) a preliminary attempt to determine the elastic longitudinal modulus for the
growing cell wall of Chara.
Chapter 3 contains the results of experiments conducted to study the growth
rate of a single Chara intemode cell before and after a longitudinal force was
applied. The results demonstrate that the longitudinal growth rate increases
(accelerated growth) to a new steady growth rate after a longitudinal force was
applied to the intemode cell. It is shown that the duration of the accelerated growth
is a function of the magnitude of the applied longitudinal force. It is also shown that
after the acceleration period, the intemode cell resumes steady growth at a rate
greater than the original growth rate. The magnitude of the increase in steady growth
rate is also a function of the magnitude of the applied force.
In Chapter 4, the multinet theory of cell elongation is introduced. The effect
of cell wall structure on the cell wall mechanical properties is discussed and stress
distribution in the cell wall of a Chara intemode cell is determined using a
pressurized cylindrical membrane model. It is shown that the ratio of the transverse
stress to the longitudinal stress is two to one. Similar analysis shows that the
application of a longitudinal force changes the stress distribution within the cell wall
7


such that the ratio of the longitudinal stress to the transverse stress increases with an
increased applied longitudinal force.
In Chapter 5, a preliminary attempt is made to obtain the longitudinal elastic
modulus for the cell wall of Chara intemode cells. Instantaneous deformation
experiments were conducted to obtain the needed experimental data.
8


2. Materials and Methods
2.1 Plant Materials
Six cultures of Chara corallina Klein ex Willd., em R.D.W. {Chora australis
R. Br.) were grown in five gallon food grade plastic buckets. Each culture contained
at least 150 mm of soil, with the remaining volume of the bucket filled with nutrient
solution (for a detailed account of the development of the Chara cultures see
Appendix A). The nutrient solution consisted of ImM NaCl, 0.1 mM KC1, 0.1 mM
CaCl2, and O.lmM MgCl2. The nutrient solution was changed every four to six
weeks. The pH of the nutrient solution varied from 8.0 to 8.5 and was measured
with pH test paper. The cultures were maintained at a temperature of 21C to 22C,
with a continuous light source from fluorescent lamps. This light source consists of
one 4 ft. light fixture with two 40 watt fluorescent bulbs held 12 inches above the
surface of the water in the Chara cultures.
2.2 Longitudinal Force
To perform the longitudinal force tests, a thallus was chosen from one of the
Chara cultures. It has been observed (Proseus et al. 1999) that younger cells have a
greater growth rate than mature ones. Cells closer to the thallus apex are younger.
Therefore, the top four to five intemodes of the thallus were excised using a pair of
scissors. From this portion of the thallus the second node from the thallus apex was
9


chosen for excision. It was this single intemode that was excised for
experimentation. All excisions were made using a pair of scissors. To excise a
single intemode the following procedure was used. The branches at the apical end of
the selected intemode cell were excised. The intemode adjoining the selected
intemode at its apical end was then removed via excision with scissors. Following
this, the branches at the basal end of the intemode were excised. The basal end was
located at the base of the thallus, which anchored the entire plant to the growth
environments soil. The selected intemode was then removed from the remaining
intemodes via excision from the adjoining intemode at its basal end. The excised
intemode was hung by its apical end with a tungsten wire. The intemode was then
placed in a clear Plexiglas enclosure, which was filled with nutrient solution, from
the same culture from which the intemode was taken. For a schematic illustration of
the apparatus used for this experiment see Appendix B-l.
Throughout the experiment, air was bubbled into the nutrient solution with an
aquarium aerator (ELITE 799, from Rolf C. Hagen Corp. Mansfield, MA) that kept
the nutrient solution at a room temperature and well oxygenated. The temperature of
the nutrient solution was continuously monitored with a mercury thermometer placed
in the solution. The temperature was recorded at the beginning and end of the
experiment.
The growth rate of the intemode was measured optically with two 40X
microscopes (Lecia 40X wide field tube). This optical measurement technique was
10


similar to the technique used by Ortega et al. (1988, 1991). One microscope was
used to measure the displacement of the apical end of the intemode and the other
microscope was used to measure the displacement of the basal end of the intemode.
To illuminate the intemode a bilateral fiber-optic illuminator (Flexilux 90; HLU
Light Source 90AV from Schoelly Fiberoptic, Denzlingen, GmBH) with two swan-
neck light guides (from Schoelly Fiberoptic) was used. After an adaptation period of
150 min, during which the displacement of the apical and basal end of the intemode
was recorded every 10 min, a weight was hung from the basal end of the intemode.
The elongation growth rate was then measured for an additional 150 min.
Additionally, the diameter of the cell intemode was measured at three locations, one-
quarter, one-half and three-quarters of the length of the cell intemode from the basal
end. The cell diameter measurements were taken three times; 1) when the excised
intemode was first placed in the test apparatus, 2) immediately after the weight was
hung, and 3) every 30 min afterwards. To determine the effect of an applied
longitudinal form on the growth rate, several weights of different magnitudes were
used: 1.123 g, 2.4029 g, 3.56 g, and 8.51 g.
2.3 Longitudinal Elastic Modulus
To perform the longitudinal force tests, a new Plexiglas chamber was
constmcted to facilitate the application of the longitudinal forces and to optically
measure the growth (longitudinal and radial) of the cell (for a schematic illustration
of the apparatus used for this experiment see Appendix B-2). The Plexiglas housing
11


was filled with nutrient solution. The second node from the thallus apex was
excised. The excised node was clamped in a Plexiglas guillotine and the guillotine
and cell were placed onto the Plexiglas housing (see Figure 2.3.1), with the cell
hanging vertically in the nutrient solution. The intemode was guillotined with its
apical end up. The initial length and the diameter of the intemode were measured
and recorded. A tungsten wire was hung on the basal end of the intemode and
passed through the bottom of the Plexiglas enclosure. Nutrient solution from a
reservoir above the experimental apparatus was allowed to flow into the Plexiglas
enclosure via a clear tube. The flow rate into the Plexiglas enclosure was regulated
via an adjustable clamp to match the flow rate of the nutrient solution leaving the
Plexiglas enclosure through the hole in its base. After allowing 40 min for the
intemode to acclimate, a weight of 1.12 g was hung from the tungsten wire attached
to the intemode. The change in length was measured optically with a 40X
microscope (Lecia 40X wide field tube) and recorded, then the weight was removed.
After a period of fifteen minutes the next weight (2.4 g) was hung from the wire and
the change in length recorded. This procedure was repeated for each of the weights
of different masses. After the removal of the last weight, a rest period of fifteen
minutes was allowed to elapse. Then the process of adding the weights to the
intemode was repeated again. This resulted in two elongation measurements for
each applied force for each cell.
12


Figure 2.3.1. Schematic Illustration of the Apparatus for Longitudinal Elastic
Modulus Experiments. The Plexiglas guillotine consisted of two Plexiglas plates
(4L x 0.5W x 0.125D). Each plate had a semi-circle of diameter 0.85 mm cut out
of one of its long sides at the midpoint of the side. The intemode was guillotined
between the two plates at the site of the semi-circular cutouts.
13


3. Effect of a Longitudinal Force on the Longitudinal
Growth Rate
It was observed in these experiments and in the literature (Proseus et. al
1999) that there were several distinct periods of different growth behavior upon
excision of the Chara intemode and after application of a longitudinal force. An
initial period of accelerated growth occurred after the excision of a Chara intemode
from the thallus. There was then a period of steady growth, which for the purposes
of this analysis will be defined as the basal growth rate. A period of accelerated
growth follows after the application of the longitudinal force. The duration of this
period correlates with the magnitude of the force. At the end of this accelerated
growth period, a steady growth rate resumes. This final steady growth rate was
observed to be greater in magnitude than the basal steady growth rate.
3.1 Growth Rate Acceleration
When the Chara intemode was excised from the rest of the plant thallus and
placed into the experimental apparatus containing the growth medium, the intemode
experienced a period of accelerated growth. This can be seen as the curvilinear part
of the curve (0 to 90 minutes) in Figure 3.1.1, which is a graph of the elongation
growth vs. time. This accelerated growth was observed by Proseus et al. (1999). In
14


these experiments, the duration of accelerated growth after excision was typically
between 40 and 100 min.
The near linear portion of the graph from 90-150 minutes indicates a constant
growth rate of the excised intemode (a least squares method of linear regression on
this data yielded a correlation coefficient of 0.988). This rate is considered the basal
growth rate.
A longitudinal force of 8.51 g was hung from one end of the intemode after
60 min of steady growth was observed. An accelerated growth period was observed
for 110 min and was followed by a period of steady growth (90 min) after which
time the experiment was terminated.
15


Figure 3.1.1. Length of a Chara Intemode as a Function of Time for an Applied
Force of 8.51 g. This node demonstrated accelerated growth for a period of 90 min
after excision. Steady growth was observed for 60 min before the application of the
8.51 g force. After the application of the force accelerated growth was observed for
110 min and was followed by a period of steady growth (90 min) after which time
the experiment was terminated.
16


Similar results are obtained when a 3.26 g force is applied (Figure 3.1.2).
The correlation coefficient for the least squares linear regression analysis performed
for the period of 40 min to 150 min produced a correlation coefficient of 0.999,
which indicates that the growth rate, was essential constant, i.e. steady growth.
Figure 3.1.3 presents the results of an experiment conducted with a 2.4 g force. A
linear regression analysis for the period of 60 min to 150 min yielded a correlation
coefficient of 0.958, which indicates steady growth.
17


Figure 3.1.2. Length of a Chora Intemode as a Function of Time for an Applied
Force of 3.56 g. This node demonstrated accelerated growth for a period of 40 min
after excision. Steady growth was observed for 110 min before the application of the
3.56 g force. After the application of the force accelerated growth was observed for
60 min and was followed by a period of steady growth (140 min) after which time
the experiment was terminated.
18


Figure 3.1.3. Length of a Chara Intemode as a Function of Time for an Applied
Force of 2.4 g. This node demonstrated accelerated growth for a period of 60 min
after excision. Steady growth was observed for 90 min before the application of the
2.4 g force. After the application of the force accelerated growth was observed for
70 min and was followed by a period of steady growth (100 min) after which time
the experiment was terminated.
300.00
250.00
200.00
5150.00
100.00
50.00
0.00*
New steady growth rate ^
Acceleration period after application of force < > ^<1 ^ ^ ^ End of ex periment
1 ^ ^ End of e I Growth Rate| acceleration period
Acceleration period after intemode excision Period of steady growth w after application of force

a M
. + > Weight Added (2.4029 g) T ^ Begin steady growth
0.00
50.00
100.00
150.00 200.00
Time (min)
250.00
350.00
19


3.2 Duration of Acceleration Period as a Result of
Application of a Longitudinal Force
Accelerated growth was observed after a longitudinal force was applied, as
demonstrated in the curvilinear growth shown in Figures 3.1.1, 3.1.2, and 3.1.3. The
duration of the mean accelerated growth rate period as a function of the applied
longitudinal force's magnitude is demonstrated in Figure 3.2.1. Forces of 2.4 g, 3.56
g, and 8.51 g were applied, with resultant mean accelerated growth periods of 50.91
min, 57.27 min, and 112.5 min, as tabulated in Table 3.2.1.
As the magnitude of the force increased, the new steady growth rates tended
to have better linear fits, i.e. essential constant growth. Figure 3.1.3 illustrates
periods of acceleration and deceleration after the initial acceleration period incited by
the 2.5 g force. Figures 3.1.1 and 3.1.2 show more linear steady growth rates. The
correlation coefficients were 0.985, 0.996, and 0.994 for the applied forces of 2.4 g,
3.56 g, and 8.51 g, respectively.
20


Figure 3.2.1. Duration of Acceleration Period as a Function of Applied Force. This
figure shows a correlation between the applied force and the duration of the
acceleration period. As the magnitude of the applied force increased, the duration of
the acceleration period increased. The data displayed in the figure are the mean
values of the acceleration period SE.
Magnitude of Applied Force (grams)
21


Table 3.2.1. Applied Longitudinal Forces with their Associated Acceleration
Periods. This table illustrates the increased acceleration period as the magnitude of
the applied force increased, more notably between the 3.56 g and 8.51 g forces.
A pplied Force (grams)
2.4 3.56 8.51
Acce eration Period (minutes)
50 50 100
50 60 130
40 60 110
50 60 110
50 60 120
60 60 100
60 60 130
50 70 100
70 50
40 40
40 60
22


3.3 Resulting Steady Growth
It was observed that after the acceleration period, the ratio of the resulting
growth rate and basal growth rate as a function of the applied force was greater than
one (Figure 3.3.1). Therefore, the effect of the force was to increase the steady
growth rate of the cell intemode.
The magnitude of this increase in growth rate may also be a function of the
magnitude of the applied force, a trend noticed when comparing ratios from the
applied forces of 3.56 g and 8.51 g (Table 3.3.1). However, more data is necessary
to substantiate or refute this hypothesis.
A great deal of variability was observed in the data displayed in Table 3.3.1.
This may be associated with the maturity of the cell. The cause of this variability is
an area for further investigation.
23


Figure 3.3.1. Ratio of Resulting Steady Growth Rate to Basal Growth Rate as a
Function of the Applied Force. The ratio is greater than one, illustrating that the
effect of the applied force was to increase the steady growth rate of the cell
intemode.
i
(3
"5
<8
ffi
i
k.
(3
>.
g 2
CO
<0
o
o
co
oc 1 -
n = 8
n = 11
T
n = 11
0.0
0.5 1.0 1.5 2.0 2.5
3.0 3.5 4.0 4.5 5.0 5.5 6.0
Magnitude of Applied Force (grams)
6.5 7.0 7.5 8.0 8.5
9.0
24


Table 3.3.1. Ratio of Resulting Steady Growth Rate to Basal Growth Rate as a
Function of the Applied Force.
Q. < plied Force (grams)
2.4 3.56 8.51
Ratio of Resulting Steady Growth Rate to
Basal Growth Rate as a function of of Applied Force
5.022 0.917 1.732
1.247 2.704 2.965
3.600 0.328 2.181
0.826 0.983 3.636
2.940 1.353 1.683
1.823 1.148 2.712
0.870 1.285 2.667
1.667 1.400 5.256
1.661 1.309
1.538 0.731
1.924 5.567
25


3.4 Discussion
Similar cell behavior may be seen in the results presented in Figures 3.1.1,
3.1.2 and 3.1.3. In all cases, there was a period of accelerated growth after the node
was excised, followed by a period of steady growth (basal growth rate). A
longitudinal force was applied and the cell growth accelerated again and then
resumed a steady growth rate. This new steady growth rate was greater in magnitude
than the basal growth rate.
The accelerated growth after excision of the intemode cell was also observed
by Proseus et al. (1999). This acceleration may be caused by removal of the cell
intemode from its aqueous environment during excision. If so, then a water
regulatory mechanism may be responsible for this behavior. If this acceleration was
caused by the excision of the bounding nodes, this may imply the cell intemode
senses damage to the thallus or the intemode itself and regulates its metabolism in
response to this external stimuli. To determine the cause of the growth rate
acceleration after excision will require more investigation.
After the accelerated growth associated with excision, a constant growth rate
was observed. This may be the cell's metabolic response to a sensing mechanism
monitoring the cells turgor pressure or growth rate. The cell may also have altered
its turgor pressure in response to the increased strain in its cell wall. Bisson and
Bartholomew (1984) found that Chara regulates its vacuolar osmotic pressure in
response to changes in external osmotic pressure. This pressure is the mechanism
26


that Chara uses to regulate its turgor pressure. Therefore, Bisson and Bartholomew
(1984) concluded that Chara does not have a turgor regulatory mechanism. Since
the findings of Bisson and Bartholomew (1984) discount turgor regulation, it may be
that Chara has a mechanism for sensing the strain in its cell wall and regulates the
metabolic processes that govern the cell wall properties to maintain some cell wall
strain threshold.
To test the idea that the Chara intemode was not regulating its turgor in
response to the excision of its two end nodes, a simple experiment may be
conducted. A Chara thallus with at least three intact nodes would be excised. The
center intemode would be impaled to measure and monitor the turgor pressure.
Simultaneously, the growth rate of the intemode would be measured. When a
constant turgor pressure and growth rate were observed, the two nodes on each end
of the intemode would be excised. A detectable change in turgor pressure would
indicate that Chara regulates its turgor pressure in response to the excision of the
intemodes two bounding cells. This would be an area for future investigation.
When a longitudinal force was applied the cell behaved in a similar way after
excision. After the accelerated growth associated with the force, a constant growth
rate was observed. This growth rate was greater than the growth rate after excision.
The duration of the acceleration period increased as a function of the
magnitude of the applied force, as illustrated in Figures 3.1.1, 3.1.2, and 3.1.3. The
behavior of the cell length as a function of time after the application of the
27


longitudinal force resembles the creep response of a viscoelastic material. This
response was recorded by Kamiya et al. (1962) in their studies with Nitella.
A similar growth response was recorded by Proseus et al. (1999) when they
subjected the Chara intemode to a step-up in turgor pressure. In these experiments,
increased turgor pressure resulted in an accelerated growth rate, followed by a
constant growth rate greater than the basal growth rate. Similar results were seen
when a longitudinal force was applied to the existing turgor pressure force.
This behavior may indicate that the initial response of the Chara intemode
was to undergo viscoelastic creep superimposed on some new steady growth rate that
was greater than the basal growth rate. But in the case of Proseus et al. (1999), the
Chara cell intemode was subjected to a step down in turgor pressure and then a step-
up in turgor pressure to the original turgor pressure of the cell. The resulting steady
growth rate was greater than the basal growth rate at the same turgor pressure. This
suggests that this response is more complicated than viscoelastic creep superimposed
upon elastic and plastic longitudinal deformation. A metabolic regulatory
mechanism may be involved in these growth responses.
The theory that the application of a longitudinal force changes the steady
growth rate of the Chara intemode is supported by this thesis' data. It can also be
said that the effect of a longitudinal force on the growth rate of the intemode is
similar to the effect of a step up in turgor pressure.
28


4. Analysis of Cell Elongation
A brief introduction of the multinet theory of plant cell extension and the
mechanics of cell elongation is presented. This is followed by a review of the cell
wall structural and material properties of the alga Nitella. The argument is made that
the findings of these investigators may be applicable to the alga Chara corallina. A
theoretical analysis of the stress distribution in a cylindrical thin walled pressure
vessel with is undertaken. The results of this theoretical analysis show that for a
cylindrical thin walled pressure vessel the transverse stress is about twice the
longitudinal stress. Findings of previous investigators are presented which give the
ratio of the transverse elastic modulus to the longitudinal elastic modulus for Nitella.
4.1 The Influence of Cell Wall Structure on Cell
Elongation
Plant cell extension is governed by the interaction of cell wall structure,
cellular metabolism, and turgor pressure. The cell turgor pressure induces transverse
and longitudinal stresses in cell wall. Cellular metabolism controls the cell wall
material yielding in response to the turgor pressure (Taiz, 1984). This allows the
plant cell to enlarge.
For a cylindrical cell to grow predominately in length versus girth, the cell
enlargement must be directed in a preferential direction. This anisotropic expansion
29


in length versus width is a result of the structural anisotropy of the cell wall.
According to the multinet theory (Roelofsen and Howink, 1953), the cellulose
microfibrils of the cell wall are transversely deposited on the plant cell walls inner
surface. They are passively reoriented in a longitudinal direction by the predominant
longitudinal strain. In Nitella it was found that the cellulose microfibrils of the cell
wall are predominantly transversely aligned (Green, P.B. 1960). In the case of
Nitella, the multinet theory was shown to hold true (Gertel and Green, 1977).
4.2 Internode Cell Wall Structure and Material
Properties of Nitella
The cell wall structure of Nitella is a polymer composite of cellulose
microfibrils embedded in a matrix of polysaccharides (Probine and Preston, 1961;
Wallace, 1967). The arrangement of the cellulose microfibrils gives the cell wall its
anisotropic material properties.
The inner one-fourth of the cell wall is the main load-bearing portion of the
cell wall (Richmond et al., 1980). This portion directs growth in Nitella. Because
the microfibrils are transversely oriented in this area, the cell wall resists transverse
deformation and expands in the longitudinal direction (Gertel and Green, 1977).
This basic model of cell wall microfibril arrangement and resulting cell wall
mechanical anisotropy of Nitella is presumed to be true for Chara, since the two are
closely related and have similar wall structures.
Probine and Preston (1961) and Haughten and Sellen (1969) used excised
30


Nitella cell wall material to study algae cell wall properties. Creep tests, stress
relaxation and constant rate of strain tests were used to determine the cell walls
mechanical properties.
They found that cell wall material of Nitella has an elastic modulus in the
longitudinal direction that varied 50 x 108 to 200 x 108 . This modulus
cm cm
was a strong function of the growth rate and not reliant on initial cell length. Their
experiments also yielded results for the transverse elastic modulus that varied from a
value of 200 x 108 ^ynes t0 400 x 108 .
cm cm
Their results also indicated that the transverse elastic modulus is independent
E
of growth rate. Their ratio, , of the transverse elastic modulus, E to the
E,
longitudinal elastic modulus, E,, varied from a value of two (for mature non-
growing cells) to a value of five (for young fast growing cells).
Probine and Preston (1961) also found that the tensile modulus of the wall of
a mature, non-growing cell was much greater than that of the wall of a young,
rapidly-growing cell. Kamiya et al. (1962) experimented with mature cells and
observed longitudinal elastic modulus values that were of the same order of
magnitude of those found by Probine and Preston (1961). The conclusion reached
by Probine and Preston (1961) is that the mechanical properties of the Nitella cell
wall are anisotropic and this anisotropy arises from the anisotropy in structure.
31


4.3 Theoretical Analysis of Cell Wall Stress Distribution
Cell turgor pressure induces the stresses in the cell wall. It is these stresses
that result in the strain, observed as an increase in cell length and cell diameter. To
further understand the role of turgor pressure on the cell wall stresses, it would be
helpful to know the stress distribution of the cell wall induced by the turgor pressure.
To this extent, a theoretical analysis of the stress distribution of a thin walled
cylindrical shell with orthotropic properties is undertaken.
The Chara intemode may be modeled as a cylindrical cell with an internal
pressure greater than the external hydrostatic pressure. This pressure differential has
previously been defined as the turgor pressure, P. Since the thickness of the Chara
cell wall is much larger than its radius it is analogous to a cylindrical thin walled
pressure vessel. The typical cell wall thickness of Nitella (and presumably Chara)
varies from 6.0 pm to 8.8 |im (Kamiya et al., 1962). The typical radius of a Chara
cell intemode varies from 0.7 mm (700 pm) to 1.0 mm (1000 pm). The result is that
the average radius of a Chara cell intemode is 115 times greater than its wall
thickness, which meets the criteria of a thin walled cylindrical pressure vessel.
Because of the arrangement of the cellular microfibrils in the cell wall, the
elastic modulus in the transverse direction, Et, is not equivalent to the elastic
modulus in the longitudinal direction, E,. This results in a cell wall with orthotropic
32


properties. Therefore, the cylindrical Chara intemode shall be modeled as a
cylindrical thin walled pressure vessel with orthotropic properties.
As a first approximation and to simplify the model, the assumption that all
deformations are small shall be made. This allows the use of the theory of linear
elasticity. Also, time dependent phenomena (creep) shall be neglected. Two further
assumptions are held in this analysis: 1) plane sections remain plane; and 2) stress
distributions throughout the wall thickness will not vary. This results in a model of
the Chara intemode that is a cylindrical membrane with orthotropic material
properties. Using a membrane model to analyze plant cell extension has previously
been performed by Hettiaratchi and OCallaghan (1974). Therefore, the use of this
type of model has precedence.
The model used in this analysis consists of a cylindrical membrane of
thickness, t, with internal radius, r, and outer radius, r0, subject to an internal
pressure, which is greater than the external pressure, P0. If an element is selected,
sufficiently removed from the ends of the cylinder to avoid edge discontinuities, two
types of normal stresses are generated: transverse stress, a[, and longitudinal stress,
Figure 4.3.1.
33


Figure 4.3.1. Cylindrical Thin-Walled Pressure Vessel. Showing transverse stress,
element, dy.
34


For the transverse stress, also called the hoop stress, consider the section, dy,
in Figure 4.3.1, cut by planes in the z-x plane on each side and cut down the y-axis
by a plane in the z-y plane. This element is shown in Figure 4.3.2. Only loading in
the x-direction is shown and the internal reactions in the material are due to hoop
stress acting and incremental area, dy-t, produced by the pressure acting on
projected area, 2r dy.
A summation of forces in the x-direction leads to:
2X=0:
2[altdy]-P-2r-dy = 0 (4.1)
solving for the transverse stress,
P r
t
35


Figure 4.3.2. Free-Body Diagram of a Segment of a Cylindrical Thin-Walled
Pressure Vessel. Showing pressure, P, and transverse stress, 0\.
36


For the longitudinal stress (axial stress) consider a portion of the cylindrical
pressure vessel shown in Figure 4.3.3. The axial stress acts uniformly throughout the
wall and pressure acts on the endcap of the cylinder. For equilibrium of forces in y-
direction the summation of forces yields:
2X=:
(4.3)
solving for the longitudinal stress,
P-K-r2
x-(r2-r2)
substituting ra =r + t gives,
2 =
Pur2 P-K-r2 Pr2
k\r + tf -r2) 7t-{r2 + 2rt + t2-r2) (irt + t2)
(4.4)
(4.5)
since this is a thin wall with a small thickness, t, r is smaller and may be neglected,
such that,
Pr
21
(4.6)
37


Figure 4.3.3. Free Body Diagram of a Cylindrical Thin-Walled Pressure Vessel.
Showing pressure, P, and longitudinal stress, <72.
38


Therefore, for a thin-walled cylindrical pressure vessel subjected to an
internal pressure in excess of the external pressure we conclude that the ratio of
transverse stress to longitudinal stress is,
£l
a2
Pr
21
(4.7)
An external longitudinal force (m-g) applied to the cylindrical thin-walled
pressure vessel, such as is shown in Figure 4.3.4, where, m, is an arbitrary mass and,
g, is the acceleration due to gravity, will result in the following summation of forces
in the y-direction.
a2-x-[r* -r2)-P-a-r2 -m-g =0 (4.8)
such that the longitudinal stress becomes,
^2 =
_ P-7T r2 + {m g)
x-k2~r2)
(4.9)
substituting r0 =r + t and neglecting r results in,
Pr +
m-
(2,)
(4.10)
39


. The resultant ratio of the transverse stress to the longitudinal stress is,
Pr t o Pr
Pr + ' m-g' Vx-r ) Pr + l X-rJ
It
(4.11)
40


Figure 4.3.4. Free Body Diagram of a Cylindrical Thin-Walled Pressure Vessel with
an Externally Applied Force. Showing pressure, P, longitudinal stress, radius, r, and wall thickness, t.
41


4.4 Analysis of Cell Wall Stress Distribution for a
Chara Internode
In the preceding discussion it was found that the ratio of the transverse stress
to the longitudinal stress of cylindrical membrane may be represented by Equation
4.11:
Pr
G\ t Pr
<7, 2 Pr + m- [r-7l:) \r-jc)
It
(4.11)
If this equation is applied to a Chara cell intemode the result will be the stress
distribution, -, for the cell intemode. Although, to facilitate interpretation of the
stress ratio, this analysis shall use the ratio, , (the longitudinal stress to the
transverse stress). For this stress ratio Equation 4.11 shall be rearranged and the
result is Equation 4.23:
£2
<*i
Pr +
Kr-n)
2 (P-r)
(4.12)
To determine how the application of an applied longitudinal force effects the stress
distribution the magnitudes of the turgor pressure, the cell radius and the applied
forces were used with Equation 4.12. To make these calculations a turgor pressure
42


of 0.5 MPa, a cell radius of 0.4mm, and a cell wall thickness of 6.0 pm were used. A
turgor pressure of 0.5 MPa was used because the typical turgor pressure for an
excised Chara internode ranges from 0.5 to 0.7 MPa, (Zhu and Boyer ,1992 and
Proseus et al., 1999). The average diameter of the cell internodes in the experiments
conducted for this thesis was 0.8mm. A cell diameter of 0.8mm is also a typical cell
diameter found by Zhu and Boyer (1992). A cell wall thickness of 6.0 pm is typical
of young Nitella cells (Kamiya et al., 1962). Assuming the cell wall thickness of
Chara to be about the same as Nitella may not be an unreasonable assumption, as the
two algae are closely related. Since the results of these calculations are to contrast
relative stress ratios, not having exact values for turgor pressure and cell diameter
will not alter the findings. To show the effects of an applied force on cells with
different turgor pressures, the stress ratios for turgor pressures of 0.5 MPa, 0.7 MPa
and 1.0 MPa were calculated. The results of these stress ratio calculations for various
turgor pressures and various applied forces are shown in Tables 4.4.1,4.4.2,4.4.3.
43


Table 4.4.1. Membrane Stress Distribution for a Pressurized Cylindrical Membrane
with a Turgor Pressure of 0.5 MPa. The stress ratio , for a cylindrical membrane
O'!
as a function of applied longitudinal force, with a turgor pressure of 0.5 MPa and a
radius of 0.4 mm.
Transverse
Stress
(N/mA2)
Longitudinal
Stress
(N/mA2)
Stress Ratio
(longitudinal /
transverse)
Experimenter
Logitudinal
Force Applied
(grams)
0.00
3.33E+07
1.67E+07
0.5000
2.40
3.33E+07
1.82E+07
0.5468
Zhu & Boyer (1992)
Proseus et al. (1999)
3.56
8.51
0.05
2.30
3.33E+07
3.33E+07
3.33E+07
3.33E+07
1.90E+07
2.22E+07
1.67E+07
1.82E+07
0.5695
0.6661
0.5010
0.5449
44


Table 4.4.2. Membrane Stress Distribution for a Pressurized Cylindrical Membrane
with a Turgor Pressure of 0.7 MPa. The stress ratio, , for a cylindrical
membrane as a function of applied longitudinal force, with a turgor pressure of 0.7
MPa and a radius of 0.4 mm.
T ransverse
Stress per unit
thickness
Longitudinal
Stress Per unit
Thickness
Stress Ratio
(longitudinal /
transverse)
Experimenter
(N/mA2)
Longitudinal
Force Applied
(grams)
o-i
(N/mA2)
0.00
4.67E+07
2.33E+07
0.5000
2.40
4.67E+07
2.49E+07
0.5335
3.56
8.51
Zhu& Boyer (1992) 0.05
Proseus et al. (1999) 2.30
4.67E+07
4.67E+07
4.67E+07
4.67E+07
2.56E+07
2.89E+07
2.34E+07
2.48E+07
0.5496
0.6186
0.5007
0.5321
45


Table 4.4.3. Membrane Stress Distribution for a Pressurized Cylindrical Membrane
with a Turgor Pressure of 1.0 MPa. The stress ratio, , for a cylindrical membrane
i
as a function of applied longitudinal force, with a turgor pressure of 1.0 MPa and a
radius of 0.4 mm.
Experimenter
Transverse Longitudinal Stress Ratio
Stress Stress (longitudinal /
(N/mA2) (N/mA2) transverse)
Longitudinal
Force Applied
(grams)
cr,
0.00 6.67E+07 3.33E+07 0.5000
2.40 6.67E+07 3.49E+07 0.5234
3.56 6.67E+07 3.56E+07 0.5347
8.51 6.67E+07 3.89E+07 0.5830
Zhu& Boyer (1992) 0.05 6.67E+07 3.34E+07 0.5005
Proseus et al. (1999) 2.30 6.67E+07 3.48E+07 0.5224
46


The application of a longitudinal force to the cell intemode changes the stress
distribution and increases the ratio of longitudinal to transverse stress (i.e. the
longitudinal component of the stress distribution increases), as demonstrated in
Tables 4.4.1, 4.4.2, and 4.4.3.
These tables also illustrate that for a given cell radius, cell thickness, and
applied longitudinal force, the ratio of longitudinal to transverse stress increases as
the turgor pressure increases.
This indicates that for a cell with a smaller magnitude of turgor pressure the
effect of adding a longitudinal force is greater than that for a cell with a larger turgor
pressure at the same applied force.
Figure 4.4.1 displays the stress ratios as a function of applied force for three
different turgor pressures.
47


Figure 4.4.1. The Stress Ratio as a Function of Applied Force for Various Turgor
Pressures. Turgor pressures of 0.5 MPa, 0.7 MPa and 1.0 MPa, with a cell radius of
0.4 mm, and cell wall thickness of 6.0 pm are used for the calculations of the stress
ratios.
48


4.5 Discussion
From the previous theoretical analysis the conclusion is made that the
addition of a longitudinal force changes the stress distribution. It also increases the
longitudinal stress with no effect on the transverse stress.
Since the elastic strain of the cell wall is a function of the stress distribution
and cell wall material properties, a change of the former would change the strain of
the cell wall. This does not take into account any elongation due to metabolism
(growth).
Another finding of the analysis was that stress distribution on the cell wall
was dependent on the radius of the cell, the magnitude of the applied uniaxial
longitudinal force, and the magnitude of the cells turgor pressure (Equation 4.12).
Therefore, for cells with the same radius and magnitude of applied longitudinal force
but varying turgor pressures, the magnitude of the longitudinal stress will correlate
with the magnitude of the turgor pressure.
However, the stress distribution, , decreases as the cells turgor pressure
o-i
increases. As a result of this decrease, cells with a greater magnitude of turgor
pressure will have less of a change in the strain of the cell wall.
Therefore, in experiments where the same longitudinal force is applied to the
node to measure its elongation, the effect of this applied force varies with cell turgor
pressure and cell radius.
49


5. Longitudinal Elastic Modulus
An understanding of Choras cell wall material properties would aid in
understanding the effect of longitudinal force on cell elongation. Previous work on
Nitella was done using dead cell wall tissue (Probine and Preston, 1961).
Experimentation with in-vivo tissue may lead to more meaningful results.
Instantaneous deformation experiments were performed to quantify the longitudinal
elastic modulus of the Chara cell wall.
In these experiments a longitudinal force was applied to a cell intemode, the
change in elongation measured, and the force was removed (as described in section
2.3). This was repeated for each mass (trial 1). The cell internode was allowed to
rest for a period of fifteen minutes and the process was repeated (trial 2). This
protocol allows for two experimental trials for each cell intemode. For each
different value of applied force a stress and strain value were calculated. These
values were plotted on a graph representing stress as a function of strain (Figure 5.1).
To calculate the longitudinal stress in the cell wall of the intemode, the
following formula was used, Equation 4.10:
Pr +
2 ='
'm-g'
Klt-r j
(21)
(4.10)
50


For the thickness of the cell wall a value of 6.0 fJm was used. For Nitella a typical
cell wall thickness for young cells was estimated to be 6.0 jum and for old cells the
cell wall thickness is estimated to be 8.0 fJm (Kamiya et al., 1962). For a dry Nitella
cell wall specimen Probine and Preston (1962) found a cell wall thickness of 4.0
ftm. As in the preceding chapter a typical turgor pressure of 0.5 MPa was used in
the calculation of stress. Since young fast growing cells were used in the experiment
the value of cell wall thickness for young cells was used. The results of these
experiments are represented in the following figures.
Figure 5.1 is a plot of the stress vs. strain data from a typical experiment. It
can be seen in this figure that the stress-strain relationship is very near linear. A
linear regression (95% confidence level) for both trials was performed which yielded
an R-square value of 0.99 for trial one and 0.92 for trial two, indicating that this
stress strain relationship is close to linear. It can also be seen in this figure that the
strain values for the second trial are smaller than those for the first trial. This
behavior was observed in all of the longitudinal elastic modulus experiments.
51


Stress (Pa)
Figure 5.1. Stress vs. Strain for a Typical Experiment. Stress is calculated using a
turgor pressure of 0.5 MPa and a cell wall thickness of 6.0 fjm.
0.0000 0.0010 0.0020 0.0030
0.0040 0.0050 0.0060
Strain
0.0070 0.0080
52


The mean stress vs. the mean strain is plotted in Figure 5.2. It can be seen in
Figure 5.2 that the first three data points (the points corresponding to an applied
force of 1.12 g, 2.4 g and 3.56 g respectively) are very linear. A linear regression
(95% confidence level) performed on these data points yielded an R-square value of
0.96, indicating that these data are near linear. The last data point (corresponding to
an applied force of 8.51 g) shows curvilinear tendencies. The error bar for this last
point indicates a lot of scatter in the strain observed for the application of the 8.51 g
force.
53


Figure 5.2. Mean Stress vs. Mean Strain for Each of the Applied Forces. A turgor
pressure of 0.5 MPa and a cell wall thickness of 6.0 jum were used in the
calculations. Data points are the mean SE.
Strain
54


In each experiment the strain measured for each applied force in the first trial
was greater than the strain measured for the same applied force in the second trial.
To highlight this fact the mean stress vs. the mean strain for the first trials and the
mean stress vs. the mean strain for the second trials were plotted on the same graph
in Figure 5.3. Inspection of Figure 5.3 tends to indicate that the stress-strain
behavior for the second trials was better behaved than that for the first trials. The
mean strain for each applied force in the second trial was less than the mean strain
for the same applied force for the first trial. A visual inspection of all the data points
for the second trial shows a very linear relationship. A linear regression analysis
(95% confidence level) of the second trial data yields an R-square value of 0.98,
which indicates that this data has a good linear fit. For the first trial, the first three
data points show a linear relationship but the last data point (corresponding to an
applied force of 8.51 g) clearly displays nonlinear behavior. A linear regression
analysis (95% confidence level) performed on the first three data points of the first
trial yielded an R-square value of 0.86. A linear regression analysis (95%
confidence level) performed on all four data points of the first trial yielded an R-
square value of 0.64. This linear regression analysis supports the statement that the
first three data points of the first trial are more linear than are all four data points.
The standard error for each point in the first trial was less than the standard error for
its corresponding point in the second trial.
55


Figure 5.3. Mean Stress vs. Mean Strain for Trial One and Trial Two. Stress was
calculated using a turgor pressure of 0.5 MPa and a cell wall thickness of 6.0 /Jm.
Each data point is the mean SE, and is the mean of four experiments.
Strain
56


A linear regression performed on the first three data points of trial one means in
Figure 5.3 resulted in a slope of 333 MPa (= 33.3xlO8 ). This slope was the
cm
longitudinal elastic modulus. Since the last data point was obviously nonlinear it
was assumed to be out of the linear (elastic) range of the cell wall material and
neglected. A similar linear regression of the first three data points of the trial two
means yielded a longitudinal elastic modulus of 61.6xl08 ~- -ne? Since the trial
cm
two data was nearly linear for all four data points, a linear regression was performed
for all four data pints on the trial two means. This resulted in a longitudinal elastic
modulus of 58x10 ^mes. Proseus et al. (1999) found a longitudinal elastic
cm
modulus for Chara that ranged from 24x10
for non-growing cells to
cm
6.0x10
dviMzs
-z for fast growing cells. The range of longitudinal elastic modulus
cm
for Nitella obtained from Probine and Preston (1961) was 50x10
dynes
cm2
to 200 x 10 ^ with younger faster growing cells having a smaller longitudinal
cm
elastic modulus than older slower growing cells. The longitudinal elastic moduli
found in these experiments are of the same order of magnitude of those of Proseus et
al. (1999) and Probine and Preston (1961). For the trial two data the longitudinal
elastic modulus was within the range determined by Probine and Preston (1961), but
57


at the upper limit of the elastic moduli reported by Proseus et al. (1999). A linear
regression performed on the first three data points of the means in Figure 5.2 resulted
in a slope of 436 MPa or 43.6x 108 ,
cm
5.1 Discussion
The difference in strain behavior between the first and second trials for each
experiment may be attributed to strain hardening of the cell wall material. This
strain hardening effect has been observed in specimens of the fungi Phycomyces by
Ortega et al. (1975). In the first trial the increase in strain is very nearly linear to the
increase in stress, until the application of the 8.51 g force. It may be that the 8.51 g
force is large enough to strain harden the material. If this were the case, the strain
observed in the second trial would be less than the strain in the first trial for each
applied force, and indeed this is what was observed. The amount of scatter in the
data points for the 8.51 g force was caused by the data from one experiment. Further
experimentation (i.e., more data) should reduce the amount of standard error in the
strain.
58


APPENDIX
A. Development of Chara Cultures
The first step in culturing algae was to simulate the natural aquatic
environment in which they live. To create a pond environment required the
development of soil cultures. Anoxic soil from the bottom of an existing pond was
excavated and brought to the lab. A four to six inch layer of this anoxic soil was
placed into each of six five gallon food grade plastic buckets. Playground sand was
purchased at a hardware store and washed of any silt, fine sand and any debris. A
two-inch layer of washed playground sand was placed on top of the anoxic soil layer
in the six plastic buckets. The layer of playground sand acted as the base for the
Chara to anchor themselves. With the soil layers in place, the buckets were filled
with the nutrient solution that simulates the pond water of the algaes natural
environment. The recipe for the nutrient solution is presented in the methods and
materials section of this thesis. These soil cultures then sat undisturbed for at least
six to eight weeks to stabilize.
After the soil cultures stabilized, they were ready to receive Chara plantings.
Cuttings of Chara corallina were obtained from the laboratory of John Boyer,
Professor, College of Marine Studies, University of Delaware. Three cuttings each
of Chara were placed in each of the six soil cultures. The bottom of each cutting
59


was buried about a half inch deep into the layer of playground sand. After six weeks
the cuttings had firmly taken hold and signs of new growth were observed.
This process of starting six Chara cultures took one whole semester. After
another semester had passed the cultures had grown enough that excising parts of the
algae would not harm the health of the culture. It took two complete semesters to
raise a healthy crop of algae for experimentation.
60


B. Construction of Experimental Apparatus
B.l. Apparatus for Longitudinal Force Experiments
61


B.2. Apparatus for Longitudinal Elastic Modulus
Experiments
62


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