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Wood Grain Pattern Formation: A
Brief Review
Eric M. Kramer*
Physics Department, SimonÕs Rock College, 84 Alford Road, Great Barrington, MA, 01230, USA
A
BSTRACT
In trees, new wood develops from a layer of stem
cells called the vascular cambium. A subpopulation
of cambial cells—the fusiform initials—are elon-
gated and capable of coordinated reorientation in
response to internal and external stimuli. Changes
in the orientation of fusiform initials in turn leads to
changes in the grain pattern of developing wood.
This article reviews the phenomenon of cambial
orientation, with an emphasis on a recent computer
model that takes the plant hormone auxin as the
orienting signal. New model results are presented
that demonstrate the surprisingly complex grain
patterns that can emerge from simple initial condi-
tions, in qualitative agreement with similar patterns
found in wood. Lastly, an alternative theory of
wood grain pattern that takes mechanical stress as
the orienting signal is critically evaluated.
Key words: Auxin; Cambium; Fusiform initial;
Pattern formation; Populus; wood grain
I
NTRODUCTION
Because the topic of wood grain pattern formation
will be new to many readers, we begin with a re-
view of the relevant biology (for more thorough
reviews, see Iqbal [1990], Larson [1994], and Sav-
idge and others [2000]). ‘‘Wood’’ is the common
word for the xylem that constitutes most of a tree
stem or branch (see Figure 1). Wood is formed by
the vascular cambium, a thin layer of meristematic
tissue that lies under the bark. During the growing
season the cells of the cambium undergo a process
of repeated cell division and radial expansion. Cells
retaining the cambial identity move gradually out-
ward, while the daughter cells left behind differ-
entiate into mature xylem. The end-products of
xylem differentiation are called xylem elements rather
than cells, because maturation frequently termi-
nates with enucleation or cell death. Seasonal
variations in the activity of the cambium produce
the familiar annual growth rings visible in stem
cross sections.
The vascular cambium is composed of two cell
types: fusiform initials and ray initials (Figure 1)
(Larson 1994). The cuboidal ray initials differentiate
into radial strands of tissue in the xylem called rays.
The elongated fusiform initials (aspect ratio > 10:1)
differentiate into all other classes of xylem ele-
ments. It is likely that the layer of initials is only one
or two cells thick in the radial direction, with
adjacent layers of cells having a distinct genetic
identity (Schrader and others 2004). The term
cambial zone is sometimes used to describe the layer
of initials and adjacent layers of dividing cells, and
we adopt that convention here.
Received: 26 June 2006; accepted: 28 June 2006; Online publication: 24
November 2006
*Corresponding author; e-mail: ekramer@simons-rock.edu
J Plant Growth Regul (2006) 25:290–301
DOI: 10.1007/s00344-006-0065-y
290
Wood grain is the direction of material anisotropy
in wood. It is the chief direction of water movement
through intact xylem (Rudinsky and Vite 1959;
Kozlowski and Winget 1963; Zimmermann and
Brown 1971; Shigo 1985; Schulte and Brooks
2003), and the preferred direction of crack propa-
gation during mechanical failure (Mattheck and
Kubler 1995). Grain is often visible on the surface of
debarked branches and sawn lumber as the direc-
tion of slight striations and cracks (see Figure 5).
The anisotropy is due to the fact that xylem ele-
ments are highly elongated and aligned parallel to
one another. The orientation of the xylem elements
reiterates the orientation of the fusiform initials that
give rise to them, so a theory of wood grain pattern
is properly a theory of cell orientation in the cam-
bium. We will also use the word grain to describe
the arrangement of fusiform cells in the cambium.
Because xylem is deposited gradually at the cir-
cumference of the branch, and because the orien-
tation of the elements reflects the orientation of the
fusiform initials, the mature xylem contains a
nearly complete record of the orientation of cambial
cells. A series of tangential sections through the
wood thus reveals how cambial orientation changes
with time (reviewed by Harris [1989] and Larson
[1994]). Studies of this kind show that the orien-
tation of fusiform initials can change rapidly in
response to stimuli. Some of the large-scale grain-
reorientation processes that may be familiar to
readers are the following: (1) When a lateral branch
dies, the grain of the surviving stem will reorient to
avoid the dead branch. (Figure 2a and 2b) (Phillips
and others 1981; Kramer and Borkowski 2004). (2)
When a tree stem suffers a localized injury, the
grain of newly formed wood tends to circumvent
the wound (Figure 2c) (Mattheck and Kubler
1995). (3) When a tree stem or branch is constricted
by a climbing vine, the grain of the underlying
branch will rotate to run parallel to the path of the
vine (Figure 2d) (Harris 1969; Mattheck and Kubler
1995). Considering the role of grain direction in
water transport and the mechanical strength of the
stem, the coordinated reorientation of fusiform
initials in response to injuries and branch death is
critical to the survival of the tree. During these
responses, the orientation of fusiform initials in the
cambium can change by 90in just a few weeks.
THE ROLE OF AUXIN
The vascular cambium is specialized to accumulate
and transport the plant hormone auxin (indole-
cambium
fusiform initial cell
ray initial cell
xylem (wood)
Figure 1. Sketch of the tree stem anatomy discussed in the paper (not to scale). The bark has been removed from a
portion of the stem to show the vascular cambium. The two cell types in the cambium can be distinguished based on their
very different shapes as seen in tangential section. Ray initials are relatively small and square, and typically occur in
lenticular aggregates. Fusiform initials may be hundreds of microns long. The orientation of the fusiform initials deter-
mines the grain pattern in newly formed wood.
ABCD
Figure 2. Schematic illustra-
tion of grain reorientation in
response to the death of a
lateral branch [(a) and (b)],
an elliptical wound (c), and
constriction by a vine (d).
Wood Grain Pattern Formation 291
acetic acid; IAA). Studies using radiolabel or GC-MS
techniques have repeatedly found that the radial
distribution of IAA is limited to the cambial zone
and adjacent layers of differentiating cells, with a
maximum at or near the layer of initials (Nix and
Wodzicki 1974; Lachaud and Bonnemain 1982,
1984; Uggla and others 1996; Tuominen and others
1997; Uggla and others 1998). In addition, members
of the AUX/LAX and PIN gene families—encoding
auxin influx and efflux facilitator proteins, respec-
tively—show high levels of expression in cambial
zone cells (Schrader and others 2003). Because of
the proximity of the cambium to the conducting
phloem (often just a few cells apart), it is difficult to
rule out the presence of IAA in the phloem trans-
port stream. However, auxin transport inhibitors
that have no effect on phloem translocation still
block nearly all auxin movement (Johnson and
Morris 1989; Sundberg and others 1994). This
finding supports the view that the main route of
auxin transport is through the cambial zone cells.
Furthermore, the IAA in tree stems moves basipe-
tally (from the leaves to the roots) with a speed
between 0.5 and 1.5 cm/h (Gregory and Hancock
1955; Hollis and Tepper 1971; Zamski and Wareing
1974; Little 1981; Odani 1985; Lachaud 1989),
much slower than phloem-mediated transport and
consistent with the chemiosmotic model of polar
auxin transport (Mitchison 1980; Goldsmith and
others 1981).
There have been many suggestions for the role of
auxin in the cambium (Sundberg and others 2000;
Aloni 2001). Aloni and Zimmermann (1983) sug-
gested that the auxin concentration in the cambium
establishes the size and number of vessels (hollow
xylem elements specialized for water conduction) in
hardwood trees. Sundberg and coworkers suggested
that the radial auxin gradient provides positional
information to differentiating cambial cells (that is,
a lower IAA concentration would indicate a larger
radial distance from the initial layer) (Uggla and
others 1996; Sundberg and others 2000). Although
IAA may play a role in these phenomena, here we
are interested in the evidence that IAA is involved
in wood grain pattern formation, and thus in the
orientation of fusiform cells.
The best evidence that IAA plays a role in the
control of wood grain pattern comes from experi-
ments in which the bark and cambium are removed
from a portion of a woody stem (Figure 3). Fayle
and Farrar (1965) worked with woody root cuttings
of white elm (Ulmus americana), sugar maple (Acer
saccharum), and American linden (Tilia americana).
They removed rectangular patches of bark and ap-
plied to the wound a lanolin paste containing either
0.1% IAA or no IAA. After several weeks, new
vessels developed near the wound. The new vessels
formed a dense radial pattern around wounds sup-
plied with IAA, and, conversely, they tended to
circumvent the wounds that lacked IAA (Figure 3a
and 3b). Many subsequent wounding experiments
have found qualitatively similar results (Figure 3c
and 3d) (Harris 1969; Kirschner and others 1971;
Harris 1973; Savidge and Farrar 1984; Zagorska-
Marek and Little 1986).
The conclusion of many authors, based on the
above studies, is that the orientation of fusiform
initials tends to parallel the presumed direction of
AB C D
Figure 3. Schematic illustration of two experiments that indicate auxin plays a role in wood grain pattern formation.
Panels (a) and (b) follow Fayle and Farrar’s (1965) study of woody root cuttings in several hardwood species. A surface cut
(white rectangle, panel a) that removes the cambium induces new vessels (gray) that circumvent the wound. Application
of a lanolin paste containing 0.1% auxin to the cut (dotted rectangle, panel b) changes the orientation of new vessels so
that many lead to the auxin source. Panels (c) and (b) follow Kirschner and coworkersÕ(1971) study of black locust
(Robinia pseudoacacia) stems. They remove the bark from a portion of the stem, leaving only a zigzag bridge of cambium and
bark intact. After several weeks, new vessels differentiate that follow the path of the bridge (c). Application of lanolin
containing 1% auxin to the top of a severed bridge mimics the inductive role of the stem (d).
292 Kramer
auxin flux through the cambium (Harris 1973;
Zagorska-Marek and Little 1986; Sachs 1991).
However, as remarked by Sachs (2000), this idea
requires some care in its interpretation. Because the
long axis of the fusiform cells is the probable
direction of active auxin transport (Zagorska-Marek
and Little 1986), it would naively seem that the
orientation of the initials determines the direction of
auxin flux, rather than the converse. To resolve this
issue, Sachs suggested that cellular polarity is only
redirected when the passive diffusion of auxin
occurs at an angle to the existing cellular polarity
(Sachs 2000). Although the details of his proposal
differ significantly from the model discussed below,
the functional distinction between the direction of
active transport (the grain direction) and the
direction of auxin diffusion is an important clarifi-
cation.
THE MODEL
In a series of recent articles we presented a mathe-
matical model of auxin-mediated wood grain pat-
tern formation (Kramer 2002; Kramer and Groves
2003; Kramer and Borkowski 2004). The goals of
this work were (1) to find a closed set of partial
differential equations describing the time evolution
of grain patterns, (2) to solve the model under
biologically relevant conditions, and (3) to refine
the model by comparison with real wood grain
patterns. The main points of the model are sum-
marized in Figure 4.
The cambial zone and adjacent layers of differ-
entiating cells is approximated as a two-dimen-
sional surface of negligible thickness, henceforth
called the cambial surface. All quantities of interest
are treated as continuous fields on this surface. The
concentration of auxin (and any other substance of
interest) is integrated through the radial thickness
of the cambium and so has units of ng/cm
2
. The
model does not resolve cell-scale details of the
tissue, and all quantities are treated as homoge-
neous on length scales less than 0.5 mm. Thus, the
grain orientation field at any point (x,y) on the
cambial surface may be defined by taking an
appropriate average over the orientation of the
fusiform cell walls in a volume centered on that
point, spanning the thickness of the cambial zone
and covering a tangential area of roughly
(0.5 mm)
2
. (Forest and others, 2006)
We denote the auxin concentration and grain
orientation fields as m(x,y,t) and u(x,y,t), respec-
tively. To date, the model has only been imple-
mented on a flat simulation domain (but note that a
flat domain can be mapped without deformation
onto the surface of a cylinder). We thus limit the
following discussion to vector calculus in the plane.
Next, we need to specify the flux of auxin within
(that is, tangent to) the cambial surface. We denote
the flux by j(x,y,t), and note that it has units of
ng/h/cm. Following Zagorska-Marek and Little
(1986), and by analogy with recent observations of
the auxin efflux facilitator PIN1 in vascular tissues
of Arabidopsis (Galweiler and others 1998), we as-
sume the grain direction is the direction of active
auxin transport. Thus, there is a contribution to the
flux in the direction of uwith magnitude mv, where
vis the speed of active transport. We take the
representative value v= 1.0 cm/h (Gregory and
ABCD
Figure 4. Schematic of the model. Panel (a) shows a portion of the cambium with vertical grain (gray lines), and a
recent circular injury (disk). (b) Because auxin is transported basipetally (that is, downward), auxin accumulates above
the wound and is depleted below (grayscale shows auxin concentration). (c) The fusiform initials (black bars) tend to
rotate until their apical ends point towards the zone of locally high auxin concentration and their basal ends point away.
(d) Newly formed grain (gray lines) circumvents the injury.
Wood Grain Pattern Formation 293
Hancock 1955; Hollis and Tepper 1971; Zamski and
Wareing 1974; Little 1981; Odani 1985; Lachaud
1989).
There are also diffusive contributions to the flux,
characterized by the coefficients D
||
and D
^
, parallel
and perpendicular to the grain direction, respec-
tively. It should be clarified that, because of the
presence of cell membranes that the anionic form of
IAA cannot cross, auxin is not free to diffuse through
the tissue (Gutknecht and Walter 1980). Thus, con-
tributions to the diffusion coefficients actually derive
in part from the apoplastic pathway (that is, diffusion
within cell walls) and in part from the transcellular
pathway involving membrane-bound carriers.
However, so long as the carriers are not operating
near saturation, the result on scales greater than
0.5 mm is equivalent to Fick’s law for diffusion (see
Appendix A of Kramer [2002] for a derivation in the
1-dimensional case). We estimate D
||
= 0.05 cm
2
/h
and D
^
= 0.01 cm
2
/h, although the later value is
only an order of magnitude estimate (Kramer 2002).
The complete expression for the flux of auxin is
j¼Djjrjj mþvm
^
uþD?r?mðÞ
^
wð1Þ
where wis a vector field perpendicular to u, hats
denote unit vectors ð^
u¼u=uÞ;and the operators
rjj ¼ur and r?¼wr describe the gradient
parallel and perpendicular to the grain direction,
respectively.
We also use the fact that auxin moves through
transporting tissues with relatively little conversion
(Lachaud and Bonnemain 1984; Sundberg and
Uggla 1998). As a first approximation, we neglect
the effects of auxin biosynthesis, conjugation, and
metabolism in the cambium. This is expressed using
the continuity equation
@m=@t¼rjð2Þ
By contrast with auxin transport, the cell and
molecular biology of fusiform reorientation is poorly
understood. Our equation for the grain as a function
of time may be described as ‘‘minimal’’ or ‘‘phe-
nomenological.’’ This is to say, it is the simplest
equation consistent with the phenomena we are
trying to describe. The key observation of Sachs
(2000)—that the diffusive flux of auxin is the likely
orienting signal—is captured by relating the rotation
of the grain vector to the gradient of the auxin
concentration as measured across the grain. In
terms of the grain angle /¼tan1ðuy=uxÞ, this gives
a chemotropic contribution to the evolution equa-
tion with the form @/=@tr
?m.
Computer simulations using only the chemo-
tropic term give grain fields that develop sharp
bends and other failures of the continuity condi-
tion. We therefore find it necessary to add a
Laplacian term to the model, @/=@tr
2/;
to maintain a sufficiently smooth grain pattern.
Although it is introduced ad hoc, the Laplacian
term captures the fact that fusiform initials tend to
line up parallel to their neighbors. When we
originally introduced the Laplacian, we suggested
an analogy with the spontaneous alignment of
elongated cells and other objects under crowded
conditions, possibly mediated by mechanical or
paracrine signals (Edelstein-Keshet and Bard Er-
mentrout 1990; De Gennes 1995; Mogilner and
Edelstein-Keshet 1995). More recently it occurred
to us that the Laplacian may be explained if indi-
vidual fusiform cells have a shape homeostasis
mechanism (Chen and others 1997). For example,
serial tangential sections of xylem in Norway
spruce (Picea excelsa) show that bent fusiform ini-
tials have difficulty maintaining the cambial iden-
tity and are preferentially eliminated from the
cambium (Wloch 1976).
Combining the above terms gives an evolution
equation for the grain angle
@/
@t¼Kr2/lr?m;ð3Þ
where Kand lare model parameters that charac-
terize the strength of the Laplacian and chemotropic
terms, respectively (Kramer 2002). Equations
(1)–(3), together, provide a complete set of non-
linear partial differential equations describing the
relationship between grain angle and auxin con-
centration in the cambium.
It is interesting to note that Eq. (3) is related to an
empirical description of wood grain in use since the
1980s (Phillips and others 1981; Pellicane 1994).
The ‘‘flow–grain analogy’’ is used by wood product
engineers to model the influence of knots on the
surrounding grain pattern. It assumes that the grain
field near a knot is everywhere tangent to the flow
field of an inviscid fluid around an obstacle of the
same shape (Wilcox 2000). It is not hard to show by
direct substitution that the grain angle field given by
this method satisfies r2/¼0:In other words, the
l= 0, time-independent limit of our model is
equivalent to the flow–grain analogy.
Although Eq. (1)–(3) are adequate for many
applications, a discretization of these equations on
the square lattice has the drawback that topological
defects (for example, the core of a circular grain
pattern) are immobilized by pinning to the lattice.
Thus, attempts to simulate the formation and evo-
lution of disordered grain patterns (see the next
section) require a revision to the model.
294 Kramer
In an earlier work (Kramer and Groves 2003), we
overcame defect pinning by working with the grain
vector field and dropping the condition that u=1
@u
@t¼Kr2uþl1
23u2
rmþurjjm
þ2K
L2u1u2
;
ð4Þ
where Lis a length. Eq. (4) reduces to Eq. (3) when
u= 1. The rightmost term in Eq. (4) is chosen to give
a grain field whose magnitude is approximately 1
almost everywhere, but decreases to near 0 within a
distance Lof sites where the grain direction is
changing rapidly. Thus, Lis a defect core size. Eq. (1),
(2), and (4), together, constitute the revised model.
Values for the parameters in Eq. (4) may be
determined as follows. Because the grain field is
defined to be an average over length scales less than
0.5 mm, we set L= 0.25 mm. In Kramer and
Groves (2003) a comparison between simulation
results and observed grain patterns in cotton-
wood (Populus deltoides) gave the estimates
K=5·10
)7
cm
2
/h and l=10
–4
cm/h/m, where m
is the average concentration in the cambial region
during the growing season, of order 10 ng/cm
2
(Tuominen and others 1997). As expected, the time
scale for changes in the grain is several orders of
magnitude slower than the auxin dynamics.
WHIRLED GRAIN
Simulations of the model described above reproduce
the qualitative features of wood grain dynamics as
illustrated in Figures 2 and 3 (Kramer 2002; Kramer
and Borkowski 2004). In model branch junctions,
the auxin flux out of a lateral branch determines the
amount of grain it ‘‘captures’’ from the subjacent
stem. Lateral branches with a higher level of auxin
export capture more grain. When a branch dies,
auxin export stops and the grain reorients to cir-
cumvent the dead branch. Similar conclusions apply
to injuries with or without an exogenous supply of
auxin.
If wood grain were limited to the fairly simple
patterns like those shown in Figures 2 and 3, then
there might not be enough variety in nature to
refine the quantitative aspects of the model, or
even to distinguish it from competing models.
However, there is a class of grain patterns that do
permit these refinements (Kramer 1999; Kramer
and Groves 2003). Whirled grain is the name given
to patterns with a complicated topology like those
shown in Figures 5, 6, and 8. For definiteness, we
define whirled grain to be any grain pattern that
has at least one circular arrangement of initials
greater than 0.5 mm in diameter. A lower limit on
the size is necessary because the model is only
intended to describe the collective behavior of
large numbers of initials. Contorted xylem ele-
ments that occur in isolation are excluded from
consideration (Sachs and Cohen 1982; Kurczynska
and Hejnowicz 1991).
Whirled grain is known to occur in more than a
dozen tree species, including both hardwoods and
softwoods (Figure 5 and Table 1). Its occurrence is
always limited to specific locations in the tree,
commonly at branch junctions and in places where
Figure 5. Whirled grain patterns in
(a)white pine (Pinus strobus), bar = 8 mm,
(b) Eastern hemlock (Tsuga canadensis),
bar = 10 mm, (c) quaking aspen (Pop-
ulus tremuloides), bar = 1mm and (d)
red maple (Acer rubrum), bar=4 mm.
Digital photographs are contrast en-
hanced to show grain patterns.
Wood Grain Pattern Formation 295
newly forming tissue is advancing across an open
wound or over the stump of a dead lateral branch.
In some species the occurrence of whirled grain is a
predictable outcome of development, whereas in
others it is infrequent and sporadic.
Whirled grain patterns have been used by several
authors to dissect developmental cues in the cam-
bium (Hejnowicz and Kurczynska 1987; Lev-Yadun
and Aloni 1990; Andre 2000). The earliest example
we know of is Fritz Neeff, who in the early 1900s
published two remarkable monographs on dynamic
changes in grain pattern (Neeff 1914, 1922). In his
1922 paper, Neeff observed that whirled grain pat-
terns are consistent with a polar orienting signal in
the cambium. Many of the illustrations in that
article include arrows superimposed on sketches of
wood grain, implying that wood grain has vector
symmetry. Neeff also included many sketches of the
time-evolution of whirled grain, clearly showing the
creation and elimination of circular patterns.
Whirled grain has been a valuable test case dur-
ing the development of our model. We used whirled
grain patterns to argue that wood grain may be
treated as a continuous vector field, and that the
continuity condition only breaks down at isolated
points and lines—called topological defects (Kramer
1999). We subsequently showed that each kind of
topological defect observed in cottonwood also oc-
curs in model simulations, including circle patterns,
patterns where three or four grain lines meet
(Y- and X-patterns), and line discontinuities (Kra-
mer and Groves 2003). We also used a digital
analysis of whirled grain images to estimate the
ratio K/lin cottonwood (Kramer and Groves 2003).
The use of topological defects to constrain math-
ematical models has had great success in the physics
Figure 6. Wood grain on the debarked surface of a cottonwood (Populus deltoides) branch 3 cm in diameter. Three healed
branch abscission zones are shown, arranged from youngest (left) to oldest (right). The grain pattern of the new wood is
initially disorganized, but gradually returns to a straight condition. These changes in grain pattern take 1–3 years. All three
abscission zones are approximately 8 mm in diameter.
Table 1. Species in which Whirled Grain Has Been Reported
Species Common name Reference
Abies alba Silver fir Neeff 1914
Acer rubrum Red maple See Figure 5
Eucalyptus sp. Eucalyptus Andre 2000
Fagus sylvatica European beech Neeff 1922
Fraxinus excelsior Common ash Hejnowicz and Kurczynska 1987
Melia azedarach Chinaberry Lev-Yadun and Aloni 1990
Pinus strobus White pine See Figure 5
Populus deltoides Cottonwood Kramer 1999
Populus tremuloides Quaking aspen See Figure 5
Quercus calliprinos Kermes oak Lev-Yadun 2000
Quercus ithaburensis Valonia oak Lev-Yadun and Aloni 1990
Tilia cordata Littleleaf linden Hejnowicz and Kurczynska 1987
Tsuga canadensis Eastern hemlock See Figure 5
Tsuga mertensiana Mountain hemlock Sutton and Sutton 1981
Whirled grain is defined to be at least one circular grain pattern greater than 0.5 mm in size (isolated circular vessels not counted).
296 Kramer
of liquid crystals and other partially ordered inor-
ganic systems (Davis and Brandenberger 1994; De
Gennes 1995). In biological pattern formation,
topological defects have received less attention.
Examples of topological defects in biology are fin-
gerprint whorls (Galton 2005), ‘‘pinwheels’’ in the
orientation map of the mammalian visual cortex
(Bonhoeffer and Grinvald 1991), and whorls in the
patterns of hair and fur (for example, on the crown
of the head) (Kidd 1903; Wunderlich and Heerema
1975). In the first two cases, the creation of topo-
logical defects and their spatial organization have
been used as benchmarks for computer models of
development (Lee and others 2003; Kucken and
Newell 2005).
In Kramer and Groves (2003) we modeled a
whirled grain pattern by starting with a random
grain field and evolving it over time. This is obvi-
ously an artificial initial condition, because most of
the grain in a real tree is relatively straight. Here we
return to the topic of whirled grain generation, with
a focus on the physiological events in the tree that
give rise to these patterns.
Trees in several genera have evolved a specialized
mechanism for casting off dead branches, analogous
to the abscission of leaves and fruit (Eames and
MacDaniels 1925). We have made a study of this
process in Populus. Branches with a diameter as
large as 5 cm maintain a zone of relatively soft tis-
sue at their base. If the branch looses vitality, the
tissue in this zone weakens and the branch de-
taches. The exposed abscission zone fills with a layer
of parenchyma—soft tissue composed of small cells
that lack a cambial identity. Within a few weeks of
branch abscission, a thin layer of cells within the
parenchyma converts to a cambial identity. New
fusiform initial cells begin to elongate, but initially
they have no preferred direction (Novitskaya 1998).
The result is a thin, circular disk of cambial tissue
where the grain field is randomized. Only gradually
do localized regions of ordered initials develop. Over
the next few growing seasons, the grain eventually
returns to a defect-free condition. Figure 6 illus-
trates three abscission zones in cottonwood at dif-
ferent stages of the healing process. Figure 7 shows
model results for an analogous situation.
Whirled grain also tends to form in the region
immediately above (more accurately, distal to) a
recently dead lateral branch. Figure 8 shows an
example of whirled grain in quaking aspen (Populus
tremuloides) that arose when the subjacent lateral
died. The development of whirled grain under these
conditions is unpredictable; the main stem from
which we excised the sample shown in Figure 8 had
ABC
E
DF
Figure 7. Model results showing the development of whirled grain in a circular abscission zone 10 mm in diameter.
Panels (a)–(c) show the grain pattern after 25, 50, and 100 days of active growth. Only enough grain lines are drawn to
give a sense for the curvature of the grain pattern. The simulation domain is (24 mm)
2
, but for clarity only the central
(15 mm)
2
square region is shown. Panels (d)–(f) magnify a portion of panels (a)–(c), respectively, (6 mm)
2
in size, to
illustrate the detailed topology of the grain pattern. Parameter values are as described in text. Initial conditions are straight
grain outside the abscission zone and randomly assigned grain directions inside. Initial auxin concentration is uniformly
10 ng/cm
2
, corresponding to an initial flux of 10 ng/cm/h per hour. Boundary conditions are periodic in both xand y.
Wood Grain Pattern Formation 297
several other dead laterals with no whirled grain.
Figure 9 shows analogous model results. We see
that whirled grain is ‘‘nucleated’’ in zones where the
grain direction has a large gradient, then expands to
fill the zone above the dead branch. Notably, the
occurrence of whirled grain in the model is also
sporadic, showing a subtle dependence on the initial
conditions (that is, the grain pattern near the live
lateral branch). The observation that whirled grain
can be nucleated in regions where the grain direc-
tion is changing rapidly appears in Neeff (1922) and
Lev-Yadun and Aloni (1990). The model provides
the first clear indication that auxin chemotropism is
an adequate explanation for this phenomenon.
T
HE
M
ECHANICAL
M
ODEL
There is a competing model of wood grain pattern-
ing that should be discussed in a review of this sort.
Claus Mattheck and coworkers have published
extensively on the idea that evolution has steered
various aspects of tree growth toward a mechanical
optimum (Mattheck 1991; Mattheck and Kubler
ABC
E
DF
Figure 9. Model results showing the development of whirled grain in the zone above a lateral branch 6 mm in diameter
(circle). At time 0 the lateral stops exporting auxin. Panels (a)–(f) show the grain pattern after 0, 2, 4, 5, 10, and 40 years
(approximating one growing season as 100 days of active growth). The total simulation domain is 60 mm tall ·34 mm
wide. For clarity, only the (10 mm)
2
square region immediately above the branch is shown. Parameter values are as
described in text. Boundary conditions: There is a homogeneous auxin source along the top edge of the simulation domain
with a flux of 10 ng/cm/h and, before time 0, a second auxin source around the circumference of the lateral branch with a
flux of 15 ng/cm/h. Auxin that passes out the bottom or side edges of the simulation domain is removed.
Figure 8. Wood grain on the de-
barked surface of a quaking aspen
(Populus tremuloides) stem 23 cm in
diameter. Whirled grain patterns are
visible in the region above a dead
lateral branch. Although the pattern
changes with time, the presence of
whirled grain above the branch can
persist for decades. Bar = 2 mm.
298 Kramer
1995; Mattheck 1998). Regarding wood grain pat-
terning, they suggest that wood grain tends to rotate
until it is parallel with the direction of largest tensile
stress in the tree stem. There is some precedent for
this idea in the hypothesis that some trees develop
spiral grain as a response to wind-imposed torques
on the tree crown (Wentworth 1931; Harris 1989;
Eklund and Sall 2000), and in the observation that
stresses imposed on a growing plant tissue can
influence the orientation of newly forming cell
walls (Lintilhac and Vesecky 1981, 1984). The the-
ory of Mattheck and coworkers makes little explicit
mention of the vascular cambium, but presumably
they are suggesting that the fusiform initials inter-
pret mechanical stresses within the cambium as an
orientating signal.
The mechanical theory of Mattheck and cowork-
ers has several drawbacks. (1) To our knowledge the
theory has not yet been presented as a complete set
of differential equations, so it is difficult to evaluate
its merits. (2) No attempt is made to account for the
fact that the cambium is a soft tissue sandwiched
between the much stiffer wood and bark. Instead
they approximate the cambium as a flat plate subject
to stresses only along its edges (Mattheck and Kubler
1995, pg 533). This is not likely to be a good model
for the distribution of stresses in the cambium. (3) It
is not clear how the mechanical theory can account
for the experiments discussed above (see Fig. 3), in
which auxin application causes significant changes
in the grain pattern. (4) As we remarked in the
previous section, whirled grain patterns exhibit a
vector symmetry that is more restrictive than a
strictly mechanical theory can account for (Neeff
1922; Kramer 1999). (5) The preponderance of evi-
dence suggests that spiral grain in tree stems is
determined by factors other than wind-imposed
torques (reviewed in Harris 1989). Thus, although
we do not rule out a role for mechanical stress in
grain pattern formation, it is unlikely to be the pri-
mary orienting signal.
P
ROSPECTS
In this article we have briefly reviewed a model of
wood grain pattern formation as developed by
Kramer and coworkers (Kramer 2002; Kramer and
Groves 2003; Kramer and Borkowski 2004). The
model combines two dynamic processes, character-
ized by widely separate time scales: auxin transport
through the vascular cambium (time scale minutes
to hours) and fusiform initial reorientation (time
scale days to weeks). We developed a set of partial
differential equations that captures the qualitative
features of this system in the simplest reasonable
way. Despite these simplifications, the model man-
ages to reproduce much of the phenomenology of
wood grain pattern. This includes straight and
helical grain on a cylindrical branch, the dynamics
of changing grain patterns at branch junctions, and
the generation of specific classes of topological
defects in whirled grain. The model is thus a proof-
of-principle that auxin chemotropism may be the
primary orientating signal in the cambium.
The model is intended to be a starting point for
more realistic approaches, including, for example,
(1) more accurate hormonal dynamics including
auxin biosynthesis and metabolism (Lachaud and
Bonnemain 1984; Sundberg and Uggla 1998); (2)
gene signaling networks, for example possible feed-
back between auxin concentration and cell carrier
expression (Davies 2004; Vieten and others 2005);
(3) subcellular resolution of the cell wall geometry,
auxin concentration, and auxin carrier localization
(Kramer 2004; Swarup and others 2005); (4) oscil-
lations of cambial orientation in some species,
possibly mediated by the chemotropic term, that
lead to wavy grain patterns (Harris 1989); and (5)
variation in model parameter values with develop-
mental stage, season, location in the tree, and
between species (Hollis and Tepper 1971; Little
1981; Lachaud and Bonnemain 1984). These
improvements await the results of additional
experiments.
ACKNOWLEDGMENTS
We thank Tobias Baskin, Mike Bergman, and Jen-
nifer Normanly for helpful conversations, and
Henrik Jonsson for a critical reading of the mansu-
cript. Research assistants Mike Borkowski and Joe
Groves made valuable contributions to the devel-
opment of the model. This work was supported
in part by Simon’s Rock College, and by the
National Research Initiative of the U.S. Department
of Agriculture Cooperative State Research, Educa-
tion and Extension Service, grant number 2003-
35103-13793.
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