Download Mechanics and Modeling of Plant Cell Growth

Trends in Plant Science - Feature Review, to be published in September 2009
Mechanics and modeling of plant cell growth
Anja Geitmann1, Joseph K. E. Ortega2
1
Institut de recherche en biologie végétale, Département de sciences biologiques, Université de Montréal, Québec H1X
2B2, Canada
2
Bioengineering Laboratory, Department of Mechanical Engineering, University of Colorado Denver, Denver, Colorado
80217-3364, USA
Corresponding author: Geitmann, A. ([email protected])
Cellular expansive growth is one of the fundamental underpinnings of morphogenesis. In plant and fungal cells, expansive growth is ultimately
determined by manipulating the mechanics of the cell wall. Therefore, theoretical and biophysical descriptions of cellular growth processes focus on
mathematical models of cell wall biomechanical responses to tensile stresses, produced by the turgor pressure. To capture and explain the biological
processes they describe, mathematical models need to be supplied with quantitative information on relevant biophysical parameters, geometry, and
cellular structure. The increased use of mechanical modeling approaches in plant and fungal cell biology emphasizes the need for the concerted
development of both disciplines and it underlines the obligation of biologists to understand basic biophysical principles.
Mechanical aspects of plant and fungal cell growth
Plant development is the result of three essential processes: cell
expansive growth, cell division, and cellular differentiation. All three
processes have key mechanical aspects that have prompted numerous
attempts to generate theoretical, mechanical, and biophysical models. In
the present review we will focus on cellular expansive growth in walled
cells typical for plants, algae and fungi. Given that walled cells rarely
migrate, cell expansive growth contributes in dramatic manner to the
generation of a particular phenotype. Expansion is involved in the
generation of both, increase in cell size and change in cell shape.
Increase of cell volume during the differentiation of a meristematic cell
into its destination cell type is typically between 10 and 1000-fold [1],
but can reach up to 30 000-fold, for example in the case of xylem
vessels [2]. While the increase in cellular surface (L2) that is necessary
to accommodate the increase in volume (L3) generally is smaller by
approximately one dimension, the amount of additional cell wall that
has to be generated is nevertheless impressive.
Implicit in cell expansive growth is a mechanical process that
balances internal and external stresses with the compliance to allow
expansion [3,4]. The balanced counterforce of primary wall stress to
turgor pressure has prompted the comparison of plant cells with
"hydraulic machines" [5]. Cell mechanics is therefore crucial for
understanding plant cell functioning. The relevance of cell mechanical
principles is particularly obvious in the light of recent studies that
illustrate how mechanical changes influence and trigger cell biological
processes. The local microinduction of expansin expression, and thus
cell wall softening, is sufficient to induce morphogenetic processes
leading to the initiation of leaf structures from the shoot apical
meristem [6]. Microtubule orientation in the shoot apical meristem was
found to follow the orientation of stress patterns in this organ and the
experimental removal of individual cells caused microtubules to realign
along the newly created stress patterns [7]. Enlisting engineering
methods such as finite element modeling for plant biological
applications [7-9] requires an increased and critical understanding of
the mechanical and physical concepts and the challenges associated
with their adaptation to plant and fungal biology.
Given that cellular expansive growth in plant and fungal cells
occurs almost exclusively prior to the deposition of the secondary cell
wall, we will focus here on the mechanics of the primary cell wall. Both
conceptual and mathematical models can contribute to our
understanding of three critical aspects defining cell expansive growth:
(i) the mechanical functioning of the structural network composing the
plant and fungal cell wall, (ii) the mechanism of the cell wall
modifying agents, and (iii) the behavior of the cell wall under the
influence of the tensile stress resulting from the presence of the
internal hydrostatic pressure, the turgor pressure. The objective of the
present review is to provide an overview of the principles, concepts,
and biophysical equations associated with the mechanical aspects of
plant and fungal cell growth. Biophysical and biomechanical terms
are explained and the most important advances in the field are
summarized.
Biophysical equations describing cell wall mechanics and
expansive growth
To model and predict the inherently mechanical process of cell
expansive growth, equations have been derived for the underlying
physical processes and coupled to the relevant biological processes
with biophysical variables, thus forming biophysical equations [10].
In the case of expansive growth of cells with walls, two relevant
physical processes are the net rate of water uptake and the rate of cell
wall deformation which occurs in response to the cell wall stresses
produced by the turgor pressure. These two simultaneous and
interrelated physical processes provide the foundations for the
derivation of biophysical equations with inclusive biophysical
variables, whose magnitude and behavior are regulated by
interrelated biological processes. Overall, the biophysical equations
couple physical processes with interrelated biological processes, and
provide a quantitative system model for the interrelated biological
processes [10-13].
The increase in size of a growing algal, fungal, or plant cell
can be described by
(dV/dt)w/Vw = Lpr (Δπ - P) - (dT/dt)w /Vw
(Eq. 1)
(dV/dt)cwc/Vcwc = φ (P - Pc) + (1/ε ) dP/dt
(Eq. 2)
(rate of increase of water volume) = (net rate of water uptake) – (transpiration rate)
(rate of increase of cell wall chamber volume) = (irreversible expansion rate) + (elastic
expansion rate)
Where V is the volume, t is the time, Lpr is the relative hydraulic
conductance, Δπ is the osmotic pressure difference inside and outside
the cell membrane, P is the turgor pressure, T is the water volume
lost through transpiration, φ is the irreversible wall extensibility, Pc is
the critical turgor pressure, and ε is the volumetric elastic modulus.
Because the relative rate of increase of water volume, (dV/dt)w/Vw, is
1
essentially equal to the relative rate of increase in the volume of the cell
contents, and essentially equal to the relative rate of increase of the cell
wall chamber, (dV/dt)cwc/Vcwc, another equation can be obtained for the
rate of change of turgor pressure [10,13].
dP/dt = ε { Lpr (Δπ - P) - (dT/dt)w /Vw - φ (P - Pc)}
(Eq. 3)
(rate of change of P) ∝ (net rate of water uptake) – (transpiration rate) – (irreversible
expansion rate)
It should be noted that the Lockhart Equations [14] are recovered from
Equations 1 to 3 for the limiting cases when transpiration is zero or
neglected, and elastic wall deformations are neglected (they are never
zero). Importantly, the Lockhart Equations cannot model wall stress
relaxation [11], pressure relaxation [15-18], and elastic deformations
[18-20] exhibited by living and growing cell walls. Equation 2 has been
modified for the case of elongation growth [10,19].
dL/dt = m (P – Pc) + (Lo/εL) dP/dt
(Eq. 4)
(elongation rate) = (irreversible extension rate) + (elastic extension rate)
Where L is the length of the cell, m is the longitudinal irreversible wall
extensibility, Lo is the initial length, and εL is the longitudinal
volumetric elastic modulus. Equations 1 to 4 have been termed the
augmented growth equations [10-13] (a detailed explanation can be
found in the Supplementary material).
Molecular models and the molecular interpretation of the
biophysical variables
Conceptually, growing (primary) cell walls of algal, fungal and plant
cells can be viewed as being composed of a network of microfibrils
cross-linked by tethers and embedded in a somewhat amorphous matrix
of cell wall materials. Generally, microfibrils are synthesized in the
plasma membrane, while the matrix materials are transported from
cellular organelles to the plasma membrane and released to the inner
cell wall via exocytosis. The molecular mechanisms by which cell wall
materials are assembled and incorporated into the deforming cell wall
to prevent thinning and eventual rupture, is highly debated.
Furthermore, the molecular mechanisms responsible for the cell wall
loosening which produces the irreversible cell wall deformation of
expansive growth are poorly understood despite the increasing
knowledge of the molecular players involved. The magnitude and
behavior of the biophysical variables within the augmented growth
equations have been determined for a variety of cells with walls [10],
and can be used to guide and evaluate conceptual molecular models for
cell wall assembly and loosening.
Behavior of biophysical variables during expansive growth
Because the relationship between cell wall stresses and turgor pressure
is generally very complicated in tissues and organs, it is useful to focus
on single cells where the relationship is more direct and generally only
depends on the geometry of the cell wall. The magnitudes of cell-wall
biophysical variables for fungal single-celled sporangiophores of
Phycomyces blakesleeanus and algal single-celled internodes of Chara
corallina have been determined with in vivo creep experiments
employing the pressure probe [18,19,21,22]. These studies revealed that
the magnitude of the irreversible wall extensibility, m, increases with
the elongation rate, but by a disproportionate amount [10]. The
disproportionate increases in m are accompanied by a decrease in the
magnitude of (P - Pc). For different developmental stages of the
sporangiophores [10], the magnitudes of both P and Pc decrease as the
elongation rate increases. Also, the magnitudes of the volumetric elastic
modulus, εL, are larger for nongrowing cells compared to growing cells,
and for growing cells, the magnitudes of εL decrease as the elongation
rates increase [10,19].
Previous models
The most useful molecular models are those that predict
behavior which can be compared to experimental results. For example,
a model that predicts strain-hardening and loosening of the cell wall
material can explain a number of experimental observations [23].
Conceptually, the basic molecular model consists of cellulose
microfibrils oriented perpendicular to the long axis and tethered to
each other by threads of hemicellulose molecules. It is assumed that
the hemicellulose molecules are attached to the microfibrils by
hydrogen bonds. An increase in longitudinal wall deformation
(strain) resulting from an increase in turgor pressure would cause
more hemicellulose threads to become load-bearing, thus resulting in
strain-hardening. Wall loosening on the other hand is proposed to
occur when the load-bearing hemicellulose threads are cut or become
detached from the microfibril because large longitudinal strains break
relevant hydrogen bonds. From this model it can be deduced how the
yield threshold, Y (and Pc), changes when the wall is subjected to a
sudden increase in longitudinal strain, thus explaining experimental
observations [24]. Equations for m and Y are derived which describe
their dependency on the relative magnitudes of strain-hardening and
wall loosening. These equations for m and Y make three predictions:
(i) that transient changes in elongation rate will result from stepchanges in P. This prediction is consistent with experimental results
obtained from algal [24], plant [25], and fungal cells [18,21]; (ii) that
m will be larger for faster growing cells compared to slower growing
cells. This prediction draws support from both algal and fungal cells
[10]; (iii) that Y (and Pc) will be smaller for faster growing cells. This
prediction draws support from fungal cells [21]. A limitation of the
strain-hardening model [23] is, however, that it ignores elastic
deformations. Thus, it does not make predictions as to the behavior
of εL as a function of elongation rate that is observed experimentally.
A thermodynamic analysis was conducted for the
conceptual molecular cell wall model consisting of cellulose
microfibrils cross-linked by matrix hemicellulose (glucan) threads
attached to the microfibrils by hydrogen bonds [1]. These analyses
predict that additional secretion of long glucan threads, that can form
multiple links (strands) between adjacent microfibrils, increases the
magnitude of the yield threshold, Y (and Pc). Furthermore, cutting the
glucan threads into smaller lengths, which will reduce the number of
strands it can form between microfibrils, is expected to lower the
magnitude of Y (and Pc). Support for these predictions awaits
relevant experimental research.
New models
Recently, new models have been introduced to explain expansive
growth. The LOS model suggests that the initiation of expansive
growth can be explained and modeled by loss of stability (LOS)
theory [26,27]. LOS theory is commonly used in engineering to study
the buckling-stability of elastic structures in compression, and has
been applied to elastic thin-walled spherical and cylindrical pressure
vessels in an attempt to model plant cell growth [26,27]. The basic
tenet is that the turgor pressure is gradually increased to a critical
value, PCR, at which a loss of stability of the cell wall occurs,
resulting in stress relaxation in the wall. The authors claim that the
“current viscoelastic model of cell wall relaxation, which dates from
the work of Preston, Cleland, Lockhart, and others in the 1960s, has
serious shortcomings” and “that LOS also provides a more
appropriate and versatile model of stress relaxation in growing plant
cells” [26]. It is debated whether the LOS process is inconsistent with
the generally accepted pressure relaxation and water uptake process
for expansive growth [28]. However, the critical question is whether
the LOS theory and its governing equation can model the stress
relaxation that supposedly occurs after PCR. It is important to note,
that the theory and its governing equation are only valid for the
pressures before and when it reaches PCR. In fact, nothing can be said
about the pressure behavior after PCR is reached, because it is
undefined in the LOS theory. The LOS theory cannot even predict
that stress relaxation occurs after PCR. By their own admission, the
authors of the LOS theory state that “the dashed line in Fig. 1B”
(showing pressure relaxation) “is a mathematical completion of the
2
Figure 1. Geometry dependence of surface stress patterns in
thin-shelled pressure vessels. (a) A spherical shell experiences
isotropic stress patterns. (b) In a cylindrical body the stress in
circumferential direction is twice as large as the longitudinal
stress. (c) The largest stress in a body with local outgrowth is
located at concave bend between the main body and the
outgrowth. Arrows in dark blue represent relatively larger
stress than arrows in light blue.
curve, but it has no physical meaning in that it does not represent any
definable stress-strain relationship” [26]. Importantly, critical
experiments demonstrating that stress relaxation and expansive growth
are initiated by increasing the turgor pressure to a critical value, PCR,
have not been reported. In addition, the relationship between PCR in the
LOS theory and Pc in the Lockhart equations and augmented growth
equations is unknown.
Another new model focuses on pectin chemistry and calcium
ion movements [29-33]. It suggests that the rate of irreversible wall
extension is regulated by the rate of unsubstituted polygalacturonic acid
(pectate) supplied to the cell wall (via the cytoplasm) and the calciumpectate chemistry that occurs within the cell wall. The experimental
data supporting the model provide insight into the incorporation of cell
wall materials into a stress-bearing cell wall, wall assembly, and the
role of turgor pressure at a molecular level. Furthermore, the authors
propose calcium-pectate chemistry that describes molecular
mechanisms for cell wall loosening and cell wall assembly.
Importantly, this model explicitly describes the relationship between
the delivery of cell wall materials, its incorporation into the growing
cell wall, and cell wall loosening.
Future models
The mechanical behavior of the growing cell wall
demonstrates a relationship and interplay between irreversible and
reversible (elastic) deformation. The relationship between irreversible
and elastic cell wall deformation is explicit in the augmented growth
equations (equations 1 to 4) and the interplay can be recovered from
their solutions for different cases, e.g. pressure relaxations [11,16] and
step changes in turgor pressure [19,20]. Current molecular models of
the growing cell wall focus, almost exclusively, on molecular
mechanisms that produce irreversible wall deformation. Generally, the
molecular models neglect the elastic deformation and the interplay
between the irreversible and elastic deformation that occurs during
experiments and expansive growth. For instance, experimental results
indicate that εL decreases as m increases [10]. This behavior might
reflect the percentage change of the stress-bearing bonds in the cell
walls growing at different rates, so as more stress-bearing bonds are
broken to loosen the wall, fewer stress-bearing bonds remain to store
energy for reversible (elastic) deformation. This relationship between m
and εL can be used to guide and evaluate new more complete molecular
models for cell wall assembly and loosening.
Quantification of physical parameters at cellular level
To supply mathematical models with relevant and accurate input,
quantitative values for a number of physical parameters need to be
provided. While educated guesses are often the only recourse, valuable
quantitative information is becoming increasingly available through the
application of various biomechanical and cytomechanical techniques
[34-36]. The two central and measurable quantities involved in algal,
fungal and plant cell growth are cell wall deformability and hydrostatic
pressure. In terms of the Lockhart equations and the augmented growth
equations (Supplementary material), their experimental quantification
allows for the determination or estimation of ε (volumetric elastic
modulus), φ (irreversible wall extensibility), and m (longitudinal
irreversible wall extensibility), whereas measurement of the cellular
hydrostatic pressure allows for the determination or estimation of P
(turgor pressure). Numerous micromechanical testing methods have
been developed to assess these parameters at tissue, cellular and
molecular level [34,35]. The challenges of experimentation at
cellular level lie primarily in the small size of single celled
specimens and the mechanical and geometrical complexity of
multicellular specimens that complicate the extraction of absolute
values for biophysical parameters such as the Young's modulus of the
cell wall.
Mechanics of anisotropic plant cell growth: the microtubulecellulose connection
Differentiated plant cells come in all shapes and sizes ranging from
simple cylindrical cells (e.g. palisade mesophyll) to star-shaped
complex structures (e.g. astro-sclereids)[37,38]. The fact that even
these complex shapes are determined by the cell wall can easily be
demonstrated by enzymatic digestion of the latter resulting in a
perfectly spherical protoplast. Since hydrostatic pressure always
produces a force perpendicular to the surface it acts on, the
generation of complex geometries requires the mechanical properties
of the cell wall to show non-uniform and (or) anisotropic
distribution.
Modeling the cell wall as fiber-reinforced composite material
For the cell to globally elongate in a particular direction,
cell wall deformability in this direction must be lower. Once a
geometry different from a sphere has been established, this
mechanical anisotropy may have to be even bigger in order to sustain
elongation, since both surface geometry and local mechanical
properties contribute to the spatial distribution of stresses on the
cellular surface (Fig. 1) [39]. The orientation of cellulose microfibrils
in the cell wall is generally recognized as a crucial parameter
determining anisotropy in cell wall deformability under tensile stress
[40-42]. Because of the cell wall's heterogeneous composition
consisting of crystalline cellulose polymers and amorphous matrix
components [43], the comparison with a fiber reinforced composite
material has been helpful for modeling this material behavior
[44,45]. Hence, in cells exhibiting elongation growth, microfibrils are
typically oriented perpendicular to the long axis on the inner surface
of the cell wall thus restricting deformation to the larger stresses in
the circumferential direction [40,46-50]. The role of the fiber angle is
being addressed in a recent model that combines the principles of
glass blowing with that of a fiber reinforced material to model the
mechanics of the cell wall (Dyson and Jensen, unpublished).
Modeling the plant cell wall by defining a two-term strain
energy function, one term each for the matrix and the microfibril
phases, it was established that in addition to microfibril extensibility
the matrix shear modulus is an important variable [51]. This aspect is
occasionally neglected when discussing the effect of cellulose
microfibrils on cell wall deformability. During elongation growth,
the primary wall deforms predominately in the long axis, separating
the space between the essentially transverse microfibrils and
passively reorienting slightly tilted microfibrils toward the
longitudinal axis. The latter process has been termed multi-net
growth and is proposed to be responsible for the changing orientation
of microfibrils through the thickness of the cell wall [48-50]. The
3
multi-net concept has been challenged since no significant reorientation
of microfibrils was observed upon application of strain up to 30%
during in vitro creep experiments [52]. However, simple geometrical
calculations based on the influence of the initial microfibril angle and
the applied strain on the final microfibril orientation demonstrated that
the expected reorientation would have been too small to be recognized
in this experimental setup [53]. Hence, these experiments are not
necessarily inconsistent with the classical multi-net growth hypothesis.
Interestingly, helical growth in fungal cells has been explained by a
preferred direction of microfibril reorientation during irreversible cell
wall deformation [54-56].
In anisotropic elongation growth perpendicular to microfibril
orientation it is apparent that stretching and/or breaking of the
(hemicellulose) connections between the microfibrils, and hence the
mechanical properties of these linkages, are an important limiting
factors regulating expansibility [2,57-59]. However, even in the case
of tensile stress parallel to the direction of microfibril orientation, the
cellulose fibers can only provide mechanical resistance if they are
connected to each other - either directly, via hemicellulose tethers or
by generating friction between each other, or indirectly, by
generating shear within the amorphous matrix. The reason for this is
that unless the microfibrils form complete hoops or spirals around the
cell, tensile stress would simply cause them to slide against each
other (Fig. 2). Since the typical length for microfibrils seems to be
Figure 2. Mechanics of a composite material with parallel arrangement of the fiber component. (a) Hypothetical composite material in which the
fiber elements form hoops around the cellular circumference. Wall extensibility in the direction parallel to the fibers would be largely determined
by the mechanical properties of the fibers and their relative density. (b) Composite material consisting of fibers of limited length embedded and
bonded to the matrix. Wall extensibility in direction parallel to the fibers is largely limited by the shear modulus of the matrix and the density of
the fibers (c) Composite material in which the fibers (microfibrils) are connected by tethers (e.g. hemicellulose polymers) in addition to being
embedded and bonded to the matrix. The mechanics of the tethers and their attachments largely influences both wall extensibility parallel to and
perpendicular to the fiber orientation. The shade of red of the arrows indicates the overall wall extensibility of the material in the indicated
direction (red - high extensibility). The terms between the arrows indicate the biophysical and structural parameters limiting extensibility in the
indicated direction. The parameters represent the mechanical properties of fibers/microfibrils (red), tethers/hemicellulose links (green), and other
matrix components (yellow).
4
below 10 µm [60], it is unlikely that individual microfibrils form
complete hoops around the cellular perimeter. Hence, the length of the
microfibrils, the mechanical properties of the connecting matrix tethers,
their abundance relative to the density of microfibrils, and the
deformability of the other matrix components should be important
parameters determining mechanical behavior of the cell wall not only
perpendicular, but also parallel to microfibril orientation (Fig. 2). This
might explain how other cell wall components such as arabinogalactan
proteins are able to mechanically influence cell wall anisotropy [61] or
control elongation growth rates [62].
It has been proposed that matrix molecules do not actually act
so much as a tether but rather as a spacer between the microfibril rods
[63]. However, both functions are not mutually exclusive and the
mechanical role of matrix molecules might depend on the particular
situation and the angle of the microfibrils versus the direction of cell
wall expansion [53]. The list of biophysical parameters determining
expansibility will certainly have to be revised in the future, once the
hierarchical organization and the interconnections between individual
types of molecules are better understood (for a review of different
conceptual models see [58]).
Cytoskeletal control of microfibril orientation
For developmental biologists the question arises how cells
control the orientation of their microfibrils. The importance of the role
of the microtubule cytoskeleton is undisputed in this context [46,64,65],
but our understanding of the mechanism of the interaction and mutual
control between microtubules and microfibrils is still rather poor. In
many plant cells microtubule arrays are arranged parallel to the main
microfibril orientation which led to the concept that movement of
cellulose synthase enzyme complexes in the plasma membrane is
constrained by interactions with the cortical microtubules [66].
However, this concept turned out to be inconsistent with observations
of continued synthesis of organized cellulose microfibrils following the
disruption of cortical microtubules by pharmacological agents [46,67]
or mutation [68,69]. In mor1 (microtubule organization1) mutants
microtubules are shortened and disorganized in a temperature-sensitive
manner leading to a loss in growth anisotropy. However, cellulose
microfibrils continue to be deposited transverse to the long axis of the
plant organ even after prolonged disruption of cortical microtubule
arrays and the onset of radial expansion. This is true for both mutation
and drug induced interference with microtubules. Even more
dramatically, change in growth anisotropy of root cells occurred in
rsw4 (radially swollen 4) and rsw7 mutants despite the unaltered,
horizontal orientation and abundance of both microtubules and
microfibrils [70].
Several conceptual models have been proposed that could
explain the role of microtubules in controlling cellular expansion other
than by determining microfibril orientation. Microtubules might be
required to concentrate and organize cellulose synthase complexes to
ensure efficient cellulose assembly into higher order clusters of fibers
[71]. Alternatively, microtubules might ensure the coordination
between cellulose deposition and the delivery of proteins and other
molecules that optimize cell wall structure [72]. Microtubule control of
the mechanical cell wall properties might also be based on their
influence on cellulose synthase turnover thus through the determination
of microfibril length [73]. Shorter microfibrils might lead to an altered
mechanical behavior of the cell wall thus allowing more expansion in
the direction parallel to cellulose orientation. More than 30 mutants
affecting the microtubule cytoskeleton are known to interfere with
growth anisotropy [74] and hopefully the mechanics of the interaction
between cytoskeleton and microtubules will be elucidated in the near
future. Determining the actual length of microfibrils and their degree of
clustering in the cells of these mutants will most certainly be necessary
to validate any of the above models or allow the generation of better
ones. Investigating the possible role of other cell wall components in
the interaction between cell wall and cytoskeleton will be another
important strategy to pursue [75].
There have been attempts to explain the orientation of
microfibrils in cylindrical cells without the necessity of a guidance
mechanism. These are essentially based on geometrical
considerations, the hypothetical limitation of space into which
microfibrils can be deposited, and the dynamics of cellulose
synthases dispatching into the membrane [76-79]. These models have
certainly provided food for thought and will help to develop
experimental approaches that will be able to provide answers.
Heterotropic growth behavior
Cellulose-mediated anisotropy in the deformability of the cell wall is
generally associated with overall anisotropic cellular expansion.
Approximately spherical or polyhedric cells derived from the apical
meristem elongate by anisotropic deformation of large portions of
their surface. Typically, cells that have differentiated through this
mechanism do not have any sharp concave bends in their surface.
Numerous cell types, however, do have such concave bends or
exhibit other types of complex geometries. Dramatic examples
include star-shaped trichoblasts, astro-sclereids, stellate aerenchyma
cells, lobed leaf epidermis cells, and the cylindrical protrusions
typical for pollen tubes and root hairs. One of the mechanically most
puzzling cases of concave bend formation is the elongation of
Rhizobium induced infection threads in legume root hairs. These wall
invaginations allow the nitrogen fixing bacteria to reach the root
cortex by invading the root hair. The threads are able to maintain
their shape against the turgor pressure of the surrounding cell [80].
Generating different geometries
How are these more complex geometries generated? The common
feature of many of the cell types with complex geometry is a
relatively large cell body producing one or several roughly
cylindrical or finger-like extensions that can be branched. These
extensions are generated by spatially confined growth events that
must rely on locally increased rates of cell wall deformation and
deposition. To distinguish spatially confined growth events from
global cell growth, we propose to name a local growth event
heterotropic and global deformation homotropic (Fig. 3). The
mechanics and geometry of heterotropic growth events varies largely
between cell types and not in all cases have surface expansion rates
been assessed quantitatively. Spatial gradients in the chemistry and
distribution of the matrix polymers such as pectin play an essential
role in the generation of these shapes [81].
In the stellate aerenchyma, the star-shaped cells are
initiated by the detachment of adjacent parenchymatic cells at three
or four way junctions. The resulting intercellular spaces increase in
size to eventually become an anastomosing network. While
developmental information is scarce and cell wall deformation
patterns for this cell type have not been published to our knowledge,
it is likely that the site of cell wall expansion is initially located at the
concave bend and subsequently in the cylindrical walls of the
branches (Fig. 3). By contrast, lobe formation in the jigsaw puzzle
shaped epidermis cells seems to be generated by a cellulose-based
reinforcement of non-growing regions and tip-growth like
outgrowths of the lobes [82]. How "tip-focused" this growth really is
remains to be seen. A high rate of cell wall expansion at the very
pole of the lobe would generate friction with the adjacent concave
bend of the neighboring cell. Therefore, it is more likely that the side
walls of the lobes are the sites of highest expansion (Fig. 3).
Branch formation in Arabidopsis trichomes is initiated by a
highly spatially confined growth event on the surface of an existing
trichome and subsequently branches elongate by large-surface
expansion of the cylindrical side walls [38,83](Fig. 3). A similar
mechanism is likely to act in cotton fibers - the single celled
trichomes formed from the ovule epidermis of Gossypium hirsutum
[84]. By contrast, cell wall expansion in pollen tubes, root hairs,
moss protonemata, algal rhizoids, and fungal hyphae is confined to
the apical tip of the cylindrical extension during the entire growth
5
Figure 3. Overview of cell wall expansion patterns resulting in various cellular geometries. Different types of cell wall expansion patterns can
be combined in a single cell. They can occur simultaneously or at different times during differentiation. Areas on the cell surface undergoing
expansion are marked in red. Question marks indicate lack of quantitative data on surface expansion patterns. For some categories alternative
terms are provided. Typical cell types for each expansion pattern are listed on the right.
period, the cylindrical walls of these cells do not expand [85-88]. To
distinguish cells with a single site of increased growth activity from
those with multiple sites, such as the lobed epidermal cells, we propose
the terms monotropic for the former and pleiotropic for the latter
(Fig.3).
The tip-focused pattern of surface expansion is in part
explained by the non-uniform distribution of cell wall components. In
pollen tubes this is expressed by the changing pectin chemistry along
the longitudinal axis [81] and the reinforcement of the cylindrical shank
by callose [89] and cellulose. Recent evidence revealed, however, that
the highest rate of surface expansion in root hairs and pollen tubes does
not occur at the extreme tip or pole of the cell, but rather at an annular
region around it [90]. The differences between "tip" growth and other
growth patterns might therefore be quantitative rather than qualitative,
with a growing region that is more spatially confined but not principally
different from that in other cell types. This illustrates that despite being
hailed for a long time as a very distinct growth mechanism, tip-growing
cells might actually only represent the extreme end of a gradual
spectrum defining the geometry of plant cell growth.
Underlying mechanics
The question is what defines the more or less spatially confined surface
areas that are subject to expansion in heterotropic growth events and
what prevents adjacent areas from being deformed? Intriguingly, these
stable, adjacent areas often have to resist higher tensile stress than the
growing regions due to cellular geometry resulting in differential
distribution of stress patterns on the cellular surface [87] (Fig. 1).
Clearly, spatially confined sites of cellular expansion require the
establishment of local differences in cell wall mechanical properties
[39] and theoretical models of the geometry and structure provide
information on the exact degree of difference in mechanical
properties that is necessary to generate a particular heterotropic
shape.
The simple radial symmetry of cylindrical, tip-growing
protrusions such as those formed by pollen tubes, root hairs, and
fungal hyphae, has made these cell types the subject of numerous
theoretical models and, more recently, of cytomechanical studies.
Ranging from relatively simple geometrical considerations [91,92]
most treat the expanding cell using an elastic membrane model [9397]. Micro-indentation has revealed that the growing apical region in
pollen has indeed lower cellular stiffness than the distal region of the
same cell, which is at least in part due to the differences in cell wall
mechanical properties [8,81,89,98]. The degree of apical stiffness
also correlates with the dynamics of the growth rate, a relationship
that is likely to be causal [99] own unpublished data).
Controlling the shape
The question arises how the cell generates and controls such spatially
confined areas of higher cell wall deformability. Root trichoblasts
about to form root hairs show localized cell wall regions of lowered
pH [100], high xyloglucan endotransglycosylase activity [101] and
increased expansin concentration [102,103]. Pollen grains about to
germinate exhibit an increased accumulation of methyl-esterified
pectin at the aperture [89]. Clearly, the biochemistry of the cell wall
can change over distances as small as few micrometers. The
mechanisms that are likely to be responsible for the establishment of
6
a local "hot spot" of mechanically different cell wall are localized
activation of proton pumps and/or targeted secretion. Other than the
secretion of proteins affecting the degree of cross-linking between cell
wall polymers, secretion induced cell wall softening might also be due
to a lack of alignment and loose coiling of newly delivered cell wall
polymers or a lack of gel formation due to their methyl-esterification
[81,104,105].
The targeting of vesicles containing cell wall precursor
material or agents influencing cell wall properties is controlled by the
cytoskeleton [38]. Contrary to the predominant role of microtubules in
cellulose deposition, heterotropic growth is often accompanied by
characteristic configurations of the actin cytoskeleton or a combination
of the actin- and microtubule arrays at the future site of a polar
outgrowth. This has been observed in pollen tubes, root hairs, algal
zygotes and fungal hyphae [37,102,106-113]. This prominence of actin
is consistent with the fact that in these cell types, contrary to animal
cells, the actin microfilaments play an important and sometimes
dominant role in cytoplasmic organelle transport. In pollen tubes both
microtubule and actin arrays are involved in organelle transport, but the
former is not critical for vesicle delivery [114]. Trichome branch
initiation in Arabidopsis is dependent on actin- and microtubulemediated Golgi transport to the cell cortex [115]. Drug or mutation
induced interference with microtubule functioning affects the initiation
of a new branch whereas the actin cytoskeleton seems to be responsible
for maintaining the polarity [109,116,117]. Root hair initiation is
preceded by a reorientation of the microtubule cytoskeleton [118], but
its disruption does not inhibit the process [119]. Drugs causing actin
depolymerization or fragmentation on the other hand successfully
interfere with root hair initiation [119-121] and germination in algal
zygotes [122]. der1 plants, which possess a point mutation in the gene
encoding actin2, have enlarged or misplaced root hair initiation sites
[123,124].
While interference with the cytoskeleton and hence the
delivery of cell wall material is sufficient to hamper or alter growth
[109], it is important to note that the simple piling on of cell wall
polymers through secretion cannot by itself produce the formation of a
protuberance even when it is highly localized. This is evident from tip
growing cells in which growth is arrested while secretion is still
ongoing [125,126]. A force causing tensile stress in the cell wall is still
a prerequisite to explain expansion. Models that base cellular expansion
solely on material addition [127,128] therefore only reflect one aspect
of cellular reality, just as do those that focus exclusively on cell wall
deformation. Our increasingly precise knowledge about vesicle delivery
[129,130] and surface expansion [93] during heterotropic growth events
should therefore make their way into the theoretical models that attempt
to have the ability to reflect cellular biology and predict its behavior.
material neglecting their cellularity. This simplification is justified
for macroscopic applications such as the use of wood in construction
[131]. Other modeling approaches attempt to at least consider the
porosity of plant tissues by modeling tissue mechanical properties
based on theories developed from foams - manufactured materials
comprising gas-filled cells formed from polymers [132,133]. The
mention of "gas" indicates, however, that such a model is of limited
use for living tissues, since the most important mechanical difference
between a gas- and a liquid-filled foam pore is compressibility of the
former. A more realistic model for turgid tissues would therefore be
that of a liquid-filled cellular foam [134]. Other models have been
developed that are more or less useful for particular applications
[135], but here we will focus on the questions concerning the
mechanics of tissues in which the mechanics of the individual cell
actually makes a difference.
A dramatic example for the role of individual cell growth
behavior for overall organ structure is the phenomenon of twisted
stem and root growth. This phenomenon receives a lot of attention at
present and is thus an attractive target for modeling approaches. Most
mutants exhibiting axial twisting in plant organs have been identified
to be caused by alterations to microtubule functioning [74]. It is
therefore essential to understand how the cytoskeleton acts at cellular
level to influence morphology at organ level. It is known that in
twisting roots cortical microtubule arrays have oblique orientations
instead of normal transverse. The elongating cells of left-handed
twisting roots have right-handed oblique microtubules and vice versa
[136,137]. It is likely that helically arranged cortical microtubules act
on cell shape by causing microfibrils to be oriented helically, but no
proof has been provided hitherto. A helical microfibril arrangement
in elongating cells is suggested to result in torsion along the
longitudinal cell axis [138,139] but the question is how this translates
into twisted growth of the entire organ given that the twisting cells
are attached to each other. The arrangement of helical microfibrils in
walls that are shared by neighboring cells results in a net orientation
that is transverse. Internal walls therefore should not play a
significant role in causing twisting. It is rather the helical pattern of
the microfibrils in the outer wall of epidermis cells that should have
the strongest influence on the handedness of helical growth [136].
This is not the only aspect to the mechanics of twisting, however.
Due to geometrical reasons internal cell layers in twisting stems or
roots should elongate less than outer cell layers [136,138].
Combining all these boundary conditions into a model for stem
growth would be an enormous help for plant biologists and could
allow predicting the effects of mutations and pharmacological agents
on this behavior. However, such a model does not exist yet and will
certainly require a major modeling effort.
Single cell mechanics versus tissues
While individually growing algal cells (internodes, rhizoids), fungal
cells (hyphae, sporangiophores), and plant cells (pollen tubes, root
hairs, trichomes, moss protonemata) do exist both in nature and in vitro
(suspension culture cells) most growth activities in plant cells occur
within the context of a tissue. In order to understand the mechanics of a
plant tissue or organ and to be able to model their behavior, important
additional parameters need to be taken into consideration: the
connection between individual cells, the stress distribution within the
cell walls throughout the tissue and (or) organ, and the different
mechanical behavior of the cell walls throughout the tissue and (or)
organ. On one side these parameters add significantly to the complexity
of a model, on the other side they essentially beg for simplification.
Depending on the objective of an individual theoretical model it may or
may not make sense to simplify the cellularity of a tissue by neglecting
the geometry and mechanics of the individual cell. There are different
ways of coping with the complexity of a tissue in a mechanical model.
Before engaging in a modeling attempt it needs to be clarified at what
level of hierarchy the material can be considered as a continuum rather
than a structure. Secondary plant tissues are often modeled as an elastic
Conclusions
Cells must obey the laws of physics, therefore, understanding cell
mechanics is fundamental to understanding and explaining growth
processes leading to morphogenesis. Theoretical modeling of the
mechanical and physical underpinnings, using biophysical variables
to couple the physical and biological processes, has provided a better
understanding of growth processes and proven to be a useful
approach. Future modeling approaches should focus on the
relationships between the physical and biological processes,
incorporating the increasing amount of biological and biochemical
details that are known and introducing new experimental approaches
and techniques to elucidate these relationships. This general
approach can provide opportunities for exciting new collaborations
between modelers (engineers, mathematicians, and physicists), cell
biologists and biochemists that will answer relevant questions and
validate proposed conceptual and mathematical models.
Acknowledgements
J.K.E. Ortega acknowledges funding by the National Science
Foundation Grant MCB-0640542. A. Geitmann receives funding
7
from the Natural Sciences and Engineering Research Council of
Canada (NSERC), the Fonds Québécois de la Recherche sur la Nature
et les Technologies (FQRNT), and the Human Frontier Science Program
(HFSP). We are grateful to Firas Bou Daher for assistance in
nomenclature issues.
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Glossary
Anisotropic material: A material whose properties differ in
different directions.
Creep test: The time-dependent strain or extension behavior of a
material in response to constant stress.
Elastic material: Material that immediately returns to its original
shape after removal of the deforming load or stress.
Isotropic material: A material whose properties are identical in all
directions.
Plastic material: Material that retains its deformed shape after the
deforming load, or stress, is removed.
Relaxation test: A constant strain is imposed on a material specimen
and the stress decay is measured as a function of time.
Shear stress: Force parallel to the surface of an object divided by the
area of the surface.
Strain hardening: Strengthening of a material by plastic
deformation.
Stress-strain curve: Graphical representation of the relationship
between stress, derived from measuring the load applied on an
object, and strain, derived from measuring the deformation of the
object.
Tensile stress: A tensile force perpendicular to the surface of an
object divided by the area on which it is applied.
Viscoelastic material: Material that exhibits both elastic and viscous
flow properties when deformed. The stress-induced deformation
is time dependent.
10
Supplementary data
From Lockhart to the augmented growth equation
Biophysical equations describing cell wall mechanics and expansive
growth
Expansive growth of algal, fungal and plant cells is the result of
metabolic-mediated biochemical processes in conjunction with
interrelated physical processes [1]. From a physical perspective,
expansive growth of the cells of all of these evolutionary distant
organisms can be viewed the same way. It is defined as a permanent
increase in cell volume. Thus the cell wall must increase in surface
area, Acw, and volume, Vcw (Vcw = Acw τcw, where τcw is the wall
thickness), because the volume of the cell wall chamber, Vcwc, must
equal the volume of cell contents, Vcc, which it encloses. While Vcw
does not necessarily change at the same rate as Acw during the different
developmental stages of a cell [2], the overall tendency is a linear
relationship between both parameters, as long as primary growth
occurs.
During expansive growth, the rate of increase in the volume
of cell contents, dVcc/dt, is predominately the result of water uptake, i.e.
dVcc/dt ≈ dVw/dt. The production and accumulation of active solutes
within the cell membrane generates the osmotic pressure that drives the
water uptake from the cell exterior to its interior. The driving force for
water uptake is typically measured as the difference in the osmotic
pressure inside (πi) and outside (πo) the cell membrane, i.e., Δπ = πi πo. The water uptake increases the volume of both the cell contents and
the cell wall chamber. When the volume of the cell contents increases,
the internal hydrostatic pressure (Pi) increases because the cell wall
resists being deformed. As Pi increases to magnitudes larger than the
external pressure (Po) it drives water out of the cell. The net water
uptake rate is the difference between the water flowing into the cell
because of the osmotic pressure difference, Δπ, and the water flowing
out of the cell because of the turgor pressure, P, which is defined as, P
= Pi – Po, i.e. Δπ - P. Thus an increase in P reduces the net water
uptake rate, and at one magnitude of P the net water uptake will stop (P
= Δπ). The P produces stresses within the cell wall. The cell wall
stretches (deforms) in response to these cell wall stresses and increases
its surface area. The deformations are both irreversible (plastic) and
reversible (elastic). The assimilation of new cell wall materials into the
deforming wall controls cell wall thickness, τcw and increases the cell
wall volume, Vcw. The characteristics of the cell wall deformation, or its
mechanical behavior, depend on the mechanical properties of the cell
wall and its “biological” state.
Equations relating expansive growth of plant cells to the rate
of water uptake and cell wall extension were first published by
Lockhart [3]. Equation 1 describes the relationship between the rate of
increase in water volume and the net rate of water uptake (in relative
terms):
(dV/dt)w/Vw = Lpr (Δπ - P)
[1]
(rate of increase of water volume) = (net rate of water uptake)
The relative hydraulic conductance is defined as, Lpr = Lp
Am/Vw, and it is assumed that the solute reflection coefficient of the cell
membrane is unity. Equation 2 describes the relationship between the
rate of increase in the volume of the cell wall chamber and the rate of
irreversible cell wall expansion (in relative terms):
[2]
(dV/dt)cwc/Vcwc = φ (P - Pc)
(rate of increase of cell wall chamber volume) = (irreversible expansion rate)
The irreversible wall extensibility, φ, and the critical turgor
pressure, Pc, are biophysical variables to be determined for each cell.
Equation 2 can be derived from the constitutive equation (equation
describing the stress-strain relationship) for a viscous dashpot with a
Bingham fluid, i.e. de/dt = (σ - σc )/μ , where de/dt is the strain rate, σ
is the stress, σc is the critical stress, and μ is the dynamic viscosity of
the Bingham fluid [4]. Recent theoretical research indicates that Eq. 2
can be obtained from the diagonal component of a more general tensor
equation [5] that can model growth anisotropies such as phototropism
and gravitropism [6]. During expansive growth, because the rate of
increase in volume of the water is approximately equal to rate of
increase in volume of the cell wall chamber, a third equation can be
derived for the steady-state turgor pressure:
[3]
P = (Lpr Δπ + φ Pc ) / (φ + Lpr )
Equations 1 – 3 describe and model expansive growth of cells
with walls when the turgor pressure is constant. Over the years, these
equations (Eqs 1-3) have been termed the Lockhart Equations.
However, it is crucial to note that the Lockhart Equations (Eqs 2 and 3)
and its underlying constitutive equations (viscous dashpot with a
Bingham fluid) cannot model the behavior exhibited in Fig. Ia and Ib,
Figure I. Schematic graphs of stress relaxation tests. (a) An initial extension (longitudinal strain) and longitudinal load (stress) are imposed
on growing sporangiophores of Phycomyces blakesleeanus with a tension-compression machine. At constant strain, stress is monitored as a
function of time. Green curve: Sporangiophore has normal turgor pressure. Cell wall stresses are imposed by the longitudinal load and the
turgor. Stress eventually decays to zero. Red curve: Immediately prior to load application turgor pressure had been decreased to zero. Stress
is only imposed by the load and decays to a constant value. Redrawn from [11]. (b) Pressure relaxation in growing tissues after removal from
water source and in an environment of saturating humidity to suppress transpiration. Red curve: The presence of auxin (in the case of pea
stem tissue; redrawn from [7] Copyright of the original figure: American Society of Plant Biologists. www.plantphysiol.org) or the
administration of a light stimulus (sporangiophore; redrawn from [11]) accelerate the relaxation. (c) Elongation growth behavior of an
internode of Chara corallina before and after a step-down and step-up in P (blue curve) produced with a pressure probe. At room
temperature (red curve) cell length represents the sum of irreversible (growth) and elastic deformation, whereas at a colder temperature
(green curve), the cell does not grow and the pressure induced deformations are purely elastic. Redrawn from [15] Copyright of the original
figure: American Society of Plant Biologists. www.plantphysiol.org.
11
because the underlying constitutive equation cannot model the observed
exponential decay in stress or the associated exponential decay in turgor
pressure.
Subsequently, a more detailed and explicit conceptual model of the
expansive growth process emerged, e.g. [7,8]. Experimental evidence
demonstrates that during expansive growth, cell wall stresses relax
(decrease) because ongoing biochemical processes loosen the cell wall,
resulting in a decrease in stress that is observed in Fig. Ia. The turgor
pressure decreases (Fig. Ib) in response to the decrease in cell wall
stresses and produces an increase in net water uptake. The increase in
net water uptake increases the turgor pressure, which in turn increases
the cell wall stresses. Then as before, the cell wall stresses begin to
relax because of ongoing cell wall loosening. This process is repeated
continuously during expansive growth. The main experimental
evidence that supports this iterative pressure relaxation and water
uptake concept of expansive growth, is that the turgor pressure decays
in growing cells with walls when the water uptake is eliminated and
transpiration is suppressed (e.g. [7-10]. The decay in turgor pressure
(Fig. Ib) results from a decrease in cell wall stresses by ongoing
biochemical processes (Fig. Ia). The effect of biochemical processes on
cell wall stresses can be illustrated by a pre-treatment with IAA (auxin),
which accelerates the decay of turgor pressure (Fig. Ib). This is
consistent with the hormone's role in increasing expansive growth rate.
Similarly, the stress decay rate increases in the cell wall after a light
stimulus, a trigger that is known to increase the elongation growth rate
[11].
Importantly, the Lockhart Equations (Eqs 2 and 3) cannot
model the decrease in turgor pressure (pressure relaxation) exhibited in
Fig Ib, or the iterative pressure relaxation and water uptake process for
expansive growth, because Eqs 2 and 3 are only valid for constant
turgor pressure, i.e. P = constant or dP/dt = 0. Specifically, Eq. 2 (and
its underlying viscoelastic model) cannot model the stress relaxation
and pressure relaxation results obtained from growing cells (Figs Ia, b).
Some excellent papers and recent reviews correctly describe the
iterative pressure relaxation and water uptake conceptual model of
expansive growth, and then refer to the Lockhart Equations which
cannot model the pressure relaxation, i.e. the decrease in turgor
pressure. The Lockhart equations form the basis upon which most
models are built, however, they have crucial shortcomings of being
applicable only in situations when turgor is constant, and when water
lost through transpiration is neglected or negligible. To model
expansive growth of plant, algal and fungal cells in the more general
cases, the Lockhart equations were augmented to account for changing
turgor pressure and transpiration [4,12]. These augmented growth
equations [13,14] have proven to be very useful in analyzing and
interpreting the experimental results of growing algal cells (e.g. [15,16],
fungal cells (e.g. [10,12,17-19], and plant organs (e.g. [7,8,20-23]. It is
apparent that a slightly more complex viscoelastic model and
corresponding biophysical equation is needed. This was achieved by the
biophysical equation, Eq. 4 which was derived from the constitutive
equation for a Bingham-Maxwell viscoelastic model, i.e. de/dt = (σ σc)/μ + (dσ /dt )/E, where E is the elastic modulus [4].
[4]
(dV/dt)cwc/Vcwc = φ (P - Pc) + (1/ε ) dP/dt
The volume of water lost via transpiration is T. Again,
because (dV/dt)cwc/Vcwc = (dV/dt)w/Vw another equation can be obtained
for the rate of change of turgor pressure [1,14].
dP/dt = ε { Lpr (Δπ - P) - (dT/dt)w /Vw - φ (P - Pc)} [6]
(rate of change of P) ∝ (net rate of water uptake) – (transpiration rate) – (irreversible
expansion rate)
Equation 6 explicitly shows how the turgor pressure changes
when there is a change in the net rate of water uptake, and/or the
transpiration rate, and/or the expansive growth rate. As an example of
its utility, Eq. 6 explicitly shows that when the water uptake is
eliminated, Lpr (Δπ - P) = 0, and transpiration is suppressed, (dT/dt)w/Vw
= 0, the governing biophysical equation for dP/dt and its solution, P(t),
are obtained for a pressure relaxation test [4,7]:
dP/dt = - ε φ (P - Pc) and P(t) = (Pi – Pc) exp (-ε φ t) + Pc [7]
The turgor pressure behavior, P(t), describes the exponential
pressure decay from Pi to Pc, and describes the pressure decay obtained
experimentally for both plant and fungal cells (Fig. Ib). Equations 4, 5,
and 6 have been termed the augmented growth equations [1,4,1214,24].
Some algal, fungal, and plant cells grow predominately in length, i.e.
exhibit elongation growth. Equations 4 and 5 can be adapted for
elongation growth [1,15]:
dL/dt = m (P – Pc) + (Lo/εL) dP/dt
[8]
(elongation rate) = (irreversible extension rate) + (elastic extension rate)
(dL/dt) w Ac = Lp Am (Δπ – P) – (dT/dt)w
[9]
(rate of change in water volume) = (net rate of water uptake) – (volumetric transpiration
rate)
Where L is the length of the cell, m is the longitudinal
irreversible wall extensibility, Lo is the initial length of the cell, and εL
is the longitudinal component of the volumetric elastic modulus
(longitudinal volumetric elastic modulus). It is noted that during
expansive growth the rate of increase of the volume of the cell contents
is essentially the rate of increase in water volume. Equation 8 was
shown to describe the elongation of the internode cells of C. corallina
(Fig. Ic) and the sporangiophores of P. blakesleeanus before, during,
and after a step-down and a step-up in turgor pressure. Again, it is
important to note that the Lockhart Equations (Eqs 2 and 3) cannot
model the elongation behavior exhibited in Fig. Ic. Equation 3 cannot
model the step-down and step-up in turgor pressure (Fig. Ic, blue curve)
and Eq. 2 cannot model the observed recovered elastic extension
(nearly instantaneous decrease in length) and elastic extension (nearly
instantaneous increase in length) that is produced by the respective
step-down and step-up in turgor pressure (Fig. Ic, red and green
curves). As it is shown in Fig. Ic, the second term in the augmented
growth equation (Eq. 8) is needed to model the elastic responses.
Generally, the augmented growth equations have been used in
short-scale time periods (minutes to hours) to analyze and interpret the
experimental results of growing algal cells (e.g. [15,16], fungal cells
(e.g. [10,12,17-19]), and plant organs (e.g. [7,8,20-23]). More recently,
Lewicka [25] has extended their application to large-scale time periods
(days) and the underlying viscoelastic models have been reviewed [26].
(rate of increase of cell wall chamber volume) = (irreversible expansion rate) + (elastic
expansion rate)
Equation 4 is written in relative terms. The added term, (1/ε )
dP/dt, describes the elastic deformation of the cell wall during
expansion. Importantly, this biophysical equation accommodates the
condition when the turgor pressure is changing. Hence, Equation 4, and
its corresponding constitutive equation for a Bingham-Maxwell
viscoelastic model, can model the turgor pressure behavior and stress
behavior exhibited by growing plant and fungal cell walls (Figs. Ia, b).
Another biophysical equation (in relative terms) can be derived to
account for the case when water is lost to transpiration [12].
(dV/dt)w/Vw = Lpr (Δπ - P) - (dT/dt)w /Vw
[5]
(rate of increase of water volume) = (net rate of water uptake) – (transpiration rate)
12
Parameters
Acw – surface area of the cell wall
Am – surface area of the cell membrane
e – strain
de/dt – strain rate
E – longitudinal elastic modulus, or Young’s modulus
L – length of the cell (a variable)
Lo – initial length of the cell (a constant)
dL/dt – elongation rate
Lpr – relative hydraulic conductance of the cell membrane; L = Lp
Am/Vw
Lp – hydraulic conductivity of the cell membrane
m – longitudinal irreversible wall extensibility
P – turgor pressure, the difference in pressure inside and outside the
cell; P = Pi – Po
Pc – critical turgor pressure, related to the critical stress, σc, by
geometry of the wall
Pi – hydrostatic pressure inside the cell
Po – hydrostatic pressure outside the cell
dP/dt – rate of change of turgor pressure
t – time
T – volume of water lost by transpiration
(dT/dt)w – volumetric transpiration rate
(dT/dt)w/Vw – relative volumetric transpiration rate
Vcc – volume of the cell contents
Vcw – volume of the cell wall
Vcwc – volume of the cell wall chamber
Vw – volume of the water in the cell
(dV/dt)cwc/Vcwc – rate of increase of relative cell wall chamber volume
(dV/dt)w/Vw – rate of increase of relative water volume
Y – yield threshold, related to the critical turgor pressure, Pc
ε – volumetric elastic modulus
εL – longitudinal component of the volumetric elastic modulus
μ – dynamic viscosity
Δπ – osmotic pressure difference inside and outside the cell; Δπ =πi - πo
πi – osmotic pressure inside the cell
πo – osmotic pressure outside the cell
σ – longitudinal stress
σc – critical longitudinal stress
dσ /dt – longitudinal stress rate
τcw – thickness of the cell wall
φ – irreversible wall extensibility
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Glossary
Bingham fluid: A fluid that flows like a Newtonian fluid after a critical
stress has been exceeded.
Constitutive equation: A mathematical equation relating the stress and
the strain.
Maxwell material: A material whose constitutive equation may be
obtained from a Maxwell model which is viscous dashpot (damper)
filled with a Newtonian fluid and an elastic spring connected in
series.
Newtonian fluid: Fluid whose stress versus rate of strain curve is linear
and passes through the origin. The constant of proportionality is
known as the dynamic viscosity.
Young's modulus or elastic modulus: Measure of stiffness of an
isotropic elastic material. The ratio of the stress and resulting strain,
and can be determined experimentally from the slope of a stressstrain curve.
20
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