Journal of Theoretical Biology 336 (2013) 44–51
Contents lists available at ScienceDirect
Journal of Theoretical Biology
journal homepage: www.elsevier.com/locate/yjtbi
Basic rules for polarised cell growth
M.Z.A.M. Jaffar 1, F.A. Davidson n
Division of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK
H I G H L I G H T S
Model provides a framework for exploring basic rules of polarised growth.
Links wall material to deposition self-similar shape.
Accurately predicts self-similar geometry in a range of organisms.
Predicts the location of the region of maximal wall deposition in a range of organisms.
art ic l e i nf o
a b s t r a c t
Article history:
Received 30 April 2013
Received in revised form
25 June 2013
Accepted 27 June 2013
Available online 10 July 2013
Growth by cell elongation is a morphological process that transcends taxonomic kingdoms. Examples of
this polarised growth form include hyphal tip growth in actinobacteria and filamentous fungi and pollen
tube development. The biological processes required to produce polarisation in each of these examples
are very different. However, commonality of the polarised growth habit suggests that certain “basic
physical rules” of development are being followed. In this paper we are concerned with trying to further
elucidate some of these basic rules. To this end, we focus on a simple and hence ubiquitous description of
the polarised cell, its geometry, and using a mathematical model investigate how geometry and the
deposition of new wall material could be related. We show that this simple model predicts both cell
geometry and the location of maximal wall-deposition in a range of examples.
& 2013 Elsevier Ltd. All rights reserved.
Keywords:
Curvature
Mathematical model
Tip growth
1. Introduction
Growth by cell elongation is a morphological process that transcends taxonomic kingdoms. Examples include hyphal tip growth in
actinobacteria and filamentous fungi, plant root-hair formation, pollen
tube development and the development of neurons in animals (see e.
g. Read and Steinberg, 2008; Flärdh, 2003; Rounds et al., 2011;
Cáceres, 2012 and the reference therein). In microorganisms, such as
fungi and bacteria, these structures have developed almost surely
because they afford an evolutionary advantage, producing a growth
habit well-suited to exploiting physically complex environments and
facilitating the (internal) redeployment of nutrients over long spatial
scales. For pollen tubes and neurons, again the polarised growth form
allows for the efficient transfer of materials (sperm cells) and
information (electrical impulses) over large spatial distances. The
biological processes required to produce this polarised growth form
are clearly very different when manifested in plant, bacterial, fungal or
mammalian cells. Even within different classes of the same organism,
n
Corresponding author. Tel.: +44 138 2384471.
E-mail address: [email protected] (F.A. Davidson).
1
Permanent address: Department of Mathematics, Universiti Putra Malaysia,
43 400 Serdang, Selangor, Malaysia.
0022-5193/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jtbi.2013.06.039
the transport of vesicles containing wall-building materials to the
growing tip can be achieved by very different mechanisms. For
example, in pollen tubes the circulation of vesicles is in entirely
opposite directions in angiosperm and gymnosperm (Kroeger and
Geitmann, 2012).
Despite their biological differences, the basic mechanics of tip
formation are similar in many cases including fungi, bacteria and
pollen tubes—a soft region in the cell wall is located at or near the
apex. This soft tip is stretched by internal forces and thus driven
forward. A combination of turgor pressure, the developing cytoskeleton and the structure of the cell wall itself make up the driving
forces (Read and Steinberg, 2008; Kroeger et al., 2011; Winship
et al., 2010). Sub-apically, the wall stiffens and thus a tube-shaped
cell is formed. The commonality of polarised growth structures
across these diverse organisms suggests that certain “basic physical rules” are being followed that are in some sense independent
of the precise mechanisms of delivery (Campàs et al., 2012).
Moreover, if we compare even fungi and actinobacteria, it is clear
that these rules are scaleable—tip growth is similar, despite the
orders of magnitude difference in cell size (Davidson, 2010).
At a fundamental level, modelling tip growth processes require
descriptions of (i) the cell wall and (ii) the delivery of materials to
maintain the cell wall and produce new growth (Dumais et al., 2006).
M.Z.A.M. Jaffar, F.A. Davidson / Journal of Theoretical Biology 336 (2013) 44–51
To mathematically describe the cell wall, both geometrical and
biomechanical models have been developed. The former allows for
the most basic, but qualitatively accurate description of possible
self-similar tip-like shapes (e.g. Bartnicki-Garcia et al., 1989). More
detailed biomechanical models take account of, for example, wallmaterial delivery and the balance of forces on the cell-wall, which
is assumed to be a thin, differentially elastic or elasto-plastic
membrane (e.g. Dumais et al., 2006; Goriely and Tabor, 2003;
Dumais et al., 2004; Rojas et al., 2011). In these latter models, the
tip shape is not predetermined, rather it evolves naturally through
the mathematical rules for the material properties. See Goriely and
Tabor (2008) for a comprehensive review of both approaches in
modelling fungi and actinomycetes and Kroeger and Geitmann
(2012), Geitmann (2010) and the references therein for pollen tube
growth. Bernal et al. (1997) provide an overview of the transfer of
modelling ideas from inert to biological materials in determining
possible tip morphologies. More detailed properties of wall development continue to be investigated, for example in Eggen et al.
(2011), the maturation of wall building material in fungal hyphae
is explicitly modelled. Moreover, a model for pollen tube growth
that encompasses both material deposition and wall deformation
has recently been presented in Rojas et al. (2011). In this paper we
are concerned with trying to elucidate some basic physical rules
for polarised growth. Given that the delivery mechanism of wallbuilding materials and the properties of the wall itself can be very
different in different organisms, we focus on a simple and hence
ubiquitous description of the polarised cell: its geometry. However, the model analysed here lies at the interface between the
approaches discussed above in that it relates the geometry of the
tip to the deposition of wall-building materials in a mechanistic
way. By developing a model first proposed in Goriely et al. (2005)
as a way of describing hyphal tips of filamentous fungi and
bacteria, we construct a basic description of how geometry and
the deposition of new wall material could be related in a wider
class of organisms.
The paper is organised as follows. In Section 2, we discuss the
formulation of the model. To assist the reader and to provide
sufficient clarity for the subsequent discussion, this includes a
brief review of the basic model formulation given in Goriely et al.
(2005). Then, in Section 3, we present an extension of this model
and construct and analyse general forms for expressions determining the relationship between tip geometry and wall-material
deposition. Finally, in Section 4, we present some examples of tip
geometries and discuss how they are described by the model. We
draw some brief conclusions and in particular note that the model
is not only capable of capturing tip geometry, but also accurately
predicts the location of maximum wall deposition in a wider class
of organisms.
2. Construction of the model
2.1. Basic description
It is a common feature of polarised growth in many organisms
that a self-similar shape is formed at the apex of the cell that
moves forward with constant average velocity. It is well-known
that many organisms including fungi and pollen tubes can exhibit
both random fluctuations in extension rate and pulsatile growth
(see e.g. Rojas et al., 2011; Sampson et al., 2003), but the average
velocity can be reasonably assumed constant. Thus, it is assumed
that the medial section of the polarised cell can be described by a
curve, C, that translates at a constant speed, U0, in a given spatial
direction. The curve, C, can be parameterized by arc length, s, and
time, t, see Fig. 1. A moving frame of reference, ðx; yÞ, is associated
with the tip, the vertex being located at the origin in this moving
Uo
45
y
Y
x
g
s
n
f
σ
t
θ
C
X
Fig. 1. The curve, C, and the associated variables.
frame. The tip is assumed to be axisymmetric about the y-axis.
Hence a three-dimensional representation of the tip can be
generated by rotating the curve C around the y-axis through 2π
radians. Further assumptions can be made as follows: (i) the arc
length, s, can itself be parameterised by the material coordinate, s,
and time, t, i.e. s ¼ sðs; tÞ and (ii) the tip shape does not explicitly
depend on t, i.e. the tip is self-similar and any change in the profile
is due to the movement of material points with respect to each
other and not through translation of the curve (other than in the
Y-direction). Thus, relative to the origin in the fixed frame of
reference, ðX; YÞ, the curve can be identified by
rC ðsðs; tÞ; tÞ≔ðXðs; tÞ; Yðs; tÞÞ ¼ ðX 0 þ f ðsðs; tÞÞ; Y 0 þ gðsðs; tÞÞ þ U 0 tÞ;
where ðX 0 ; Y 0 Þ is the location of the origin of the moving frame at
t¼0 (this can be set to be ð0; 0Þ without loss of generality), U0 is
the translation speed introduced above and ðf ; gÞ is the coordinate
of the material point s on C with respect to the moving frame of
reference. The angle θ marked in Fig. 1 is defined to be the angle
between the normal n to the curve at s and the y-axis. The
dynamics of the curve can therefore be expressed as
drC ðsðs; tÞ; tÞ
df ∂s dg ∂s
¼
;
þ U 0 ≕Wt þ Un;
ð1Þ
dt
ds ∂t ds ∂t
where W and U denote the magnitude of the tangential, t, and
normal, n, components of the velocity, respectively, where
df dg
;
¼ ð cos θ; sin θÞ and n ¼ ð sin θ; cos θÞ:
ð2Þ
t¼
ds ds
From (1), and on taking the scalar product with n and t in turn, it
follows that
U ¼ U 0 cos θ
and
W¼
∂s
U 0 sin θ:
∂t
ð3Þ
A key assumption of the model is that W can be set to zero. This is
in line with the long-standing normal growth hypothesis, first
proposed in Reinhardt (1892). This hypothesis states that any
material point embedded in the cell wall will move normal to the
wall. Reinhardt (1892) made this conclusion based on observations
and projection methods, but discounted turgor as being the main
cause of this phenomenon. It was not until 2000 that the
orthogonal movement of particles adhered to the outside (and
inside) surface of fungal hyphae was reported (Bartnicki-Garcia
et al., 2000). In that paper, the key conclusion was that this
orthogonal movement of wall material was a result of turgor
pressure (which is by definition orthogonal to the wall and equal
at all areas of the tip). Thus it is proposed that the forward motion
of the tip shape is a result of this internal pressure and the
anisotropic delivery of wall-building materials released from an
46
M.Z.A.M. Jaffar, F.A. Davidson / Journal of Theoretical Biology 336 (2013) 44–51
advancing vesicle supply centre (VSC) or Spitzenkörper. This orthogonal growth hypothesis is necessary for a unique tip profile to be
described in 3-D in the model presented in Gierz and Bartnicki-Garcia
(2001). The role of turgor pressure in shape and forward translation
on tip growth continues to be debated (see e.g. Kroeger et al., 2011;
Money, 1997). However, for the simple geometric description that
forms the basis of the analysis detailed here, the normal growth
hypothesis provides a sufficient and consistent description of the
movement of the material points, s. Hence, from (3),
∂s
¼ U 0 sin θ:
∂t
ð4Þ
To complete the geometric description, the curvature κ is defined by
κ≔
dθ
:
ds
ð5Þ
In response to wall-building material deposition and internal
pressure, it is assumed that the tip deforms. A key metric of this
deformation is the stretching ratio (the motion of particles on the
curve relative to each other):
∂sðs; tÞ
:
∂s
ð6Þ
Note that from a biomechanical point of view, in one dimension,
the strain is defined as
ϵ≔
∂
ðsðsÞsÞ ¼ λ1
∂s
ð7Þ
where the final equality is obtained by applying the chain rule
again, using the definition of κ given in (5) and recalling from (3)
that U ¼ U 0 cos θ, which is the normal component of velocity at
each point along the curve.
2.3. Tip growth
Growth rate of the tip can be described in terms of the
incremental area ΔA as follows. Consider the tip formed by
rotating the curve C around the y-axis. The circumference of this
tip at distance jgj behind the apex is then 2πf (cf. Fig. 1). Hence, a
thin band on the surface of the tip at this location has area, ΔA,
which is approximately given by
ΔA≈2πf Δs≈2πf
∂s
Δs ¼ 2πf λΔs:
∂s
ð8Þ
Differentiating both sides of (8) with respect to t, dividing by f and
simplifying leads to
1 d
1 df 1 ∂λ
ΔA ¼
þ
:
ΔA dt
f dt λ ∂t
ð10Þ
1 df 1 ∂λ
þ
¼ Dðs; t Þ:
f dt λ ∂t
ð11Þ
To derive the equation that will form the focus of the rest of the paper,
we make a final assumption. When a tip has large diameter relative to
its length and the growth process is relatively slow (see e.g. Esser and
Fischer, 2006 in the case of fungi), it may be reasonably assumed that
the stretching rate in the azimuthal direction is significantly smaller
than in the longitudinal (tip to distal) direction, i.e.
1 df 1 ∂λ
⪡
f dt λ ∂t
Hence, if λ 4 1 or λ o1 then this represents the wall being
stretched or compressed in the longitudinal direction. (If λ ¼ 1
then this component of strain on the wall is zero and the curve C is
not deformed.) It is generally assumed that λ≥1 and that walldeposition and internal driving forces cause local wall expansion.
The stretching rate ∂t λ (or equivalently the strain rate ∂t ϵ) can now
be computed by differentiating (6) with respect to t. On doing this,
using (4), (the chain rule and on dividing both sides by λ) yields a
related, relative measure that will be useful below:
1 ∂λ
d
¼ U 0 ð sin θÞ ¼ κU;
λ ∂t
ds
1 d
ΔA ¼ Dðs; t Þ;
ΔA dt
where Dðs; tÞ is a function of arc length and time, encapsulating the
embedding of new wall material. Hence, in its most general form
the equation governing the tip profile in response to wall-material
deposition is
2.2. Wall stretching
λ≔
The next key assumption in the model relates wall deposition
to the local increase in area. In the absence of deposition, it would
be reasonable to assume that the walls of any of the organisms
considered here would thin as they were stretched by turgor.
(From a biomechanical standpoint, wall material of this form is
often assumed to be divergence-free or incompressible in nature.)
In the model discussed here, an assumption is made that is
equivalent to new wall material being introduced at a rate that
exactly balances this thinning. Now, the geometric model takes no
direct account of the thickness of the wall or its specific material
properties. Hence, this balance law translates to the deposition
rate being set equal to the relative rate of increase of area, i.e.
ð9Þ
The growth of an area element per (unit) area element is denoted by
the left-hand side of (9), while the right-hand side can be thought as
the sum of local azimuthal and longitudinal stretching rates.
and hence from (11),
1 ∂λ
≈Dðs; t Þ;
λ ∂t
which, from (7) gives the approximate formula
κ cos θ ¼
1
dðs; t Þ;
α
ð12Þ
with α ¼ U 0 =D0 where Dðs; tÞ ¼ D0 dðs; tÞ, with d being the nondimensional and D0 setting the fundamental time-scale of the wall
building process. Eq. (12) will form the focus of the following sections.
3. Curvature-driven wall deposition
The choice of deposition function, d, clearly determines the
curve dynamics derived above. In Goriely et al. (2005) the reasonable argument put forward was that the biology in many examples
of tip growth including actinomycetes and fungi, determines that
wall-deposition is maximal at the apex, where also the curvature
is generally the greatest. Hence, it was assumed there that
dðs; tÞ∝κp and moreover that p≥1. Indeed, the analysis presented
in Goriely et al. (2005) was restricted to the special case p ¼2 ,
which, as will be discussed below, results in a closed form
expression for the tip shape. Although self-consistent, both of
these assumptions merit a little more discussion. That wall
deposition occurs where the curvature is maximal is in fact
consistent with recent results concerning the distribution of
wall-binding sites (Rojas et al., 2011; Geitmann, 2010). The
assumption that p≥1 seems to be unnecessary, as any value of
p 4 0 will result in the desired behaviour of d i.e. a monotonically
increasing function of κ. We now present an analysis of the general
case p 40 and derive generalized formulae for the tip shape by
obtaining expressions for the tip f, g, and the curvature, κ as
functions of θ. To this end, we set dðs; tÞ ¼ ðκ=κ0 Þp where κ 0 4 0 is a
scaling constant.
M.Z.A.M. Jaffar, F.A. Davidson / Journal of Theoretical Biology 336 (2013) 44–51
We first derive a formula for κðθÞ. Substituting dðs; tÞ ¼ ðκ=κ 0 Þp
in (12) yields
1 κ p
κ cos θ ¼
:
ð13Þ
α κ0
It follows directly from (13) that either κ ¼ 0 or a generalized
formulae for the longitudinal curvature is
κ ¼ ðακ p0 Þ1=ðp1Þ ð cos θÞ1=ðp1Þ ;
ð14Þ
if we assume p≠1. If p ¼1, then it follows directly from (13) that
κ ¼ 0 or cos ðθÞ ¼ 1=ðακ 0 Þ, a constant. No useful tip shape can be
generated in either case and so from now on we assume κ≠0
and p≠1.
Next we derive a corresponding result for f ðθÞ. Recall that
κ ¼ dθ=ds ¼ ðdθ=df Þdf =ds and cos θ ¼ df =ds and hence on substituting these in (14), we get
!1=ðp1Þ
df
1
p2
¼
ð cos θÞ
:
ð15Þ
dθ
ακ p0
Integrating both sides of (15) with respect to θ gives
Z θ
1
f ðθÞ ¼
ð cos ϕÞðp2Þ=ðp1Þ dϕ;
p 1=ðp1Þ
ðακ 0 Þ
0
ð16Þ
where we have used f ð0Þ ¼ 0.
Finally, we derive a general expression for gðθÞ. Noting that
dg=df ¼ tan θ and using (16) gives
dg
dg df
1
¼
¼ tan θ
ð cos θÞðp2Þ=ðp1Þ :
dθ
df dθ
ðακp0 Þ1=ðp1Þ
Integrating both sides of (17) with respect to θ yields
Z θ
1
sin ϕ
g ðθ Þ ¼ dϕ;
ðακp0 Þ1=ðp1Þ 0 ð cos ϕÞ1=ðp1Þ
ð17Þ
ð18Þ
Next, from (16), it follows that provided the limit exists
8 π
ΓðhÞ
>
>
; p∈ð0; 1Þ∪ð3=2; 1Þ;
>
<f 2 ¼K 1
Γ hþ
lim f ðθÞ ¼
2
>
θ-π=2
>
>
: 1;
otherwise;
merits a little explanation. It is straightforward to show that
h : ½0; 1Þ-½3=2; 1Þ;
ð20Þ
where Γ is the gamma function and h ¼ hðpÞ≔ðð1=2Þððp2Þ=
pffiffiffi
ðp1ÞÞ þ ð1=2ÞÞ. The constant K ¼ ð1=ðακp0 Þ1=ðp1Þ Þ π =2. This limit
h : ð1; 3=2-ð1; 0
and
h : ½3=2; 1Þ-½0; 1Þ:
Hence, it follows directly from the properties of the gamma
function that the limit detailed in (20) exists and is equal to
f ðπ=2Þ iff p∈ð0; 1Þ∪ð3=2; 1Þ. For p outwith this set, the integral in
(16) becomes unbounded as θ-π=2 and so limθ-π=2 f ðθÞ ¼ 1.
From (18), it follows that a limit exists for all p 4 0 in the
following sense:
8
1
>
p1
<g π ¼
; p∈ð0; 1Þ∪ð2; 1Þ
2
p 1=ðp1Þ p2
ðακ 0 Þ
ð21Þ
lim g ðθÞ ¼
>
θ-π=2
: 1;
p∈ð1; 2:
The limits derived above separate the curvature-driven process
into distinct cases that we now summarise.
3.1. Case 1: 0 o p o1
For p in this range, the asymptotic tip radius and length are defined
and finite. Thus, the model determines that the curved part of the tip
has finite length. A polarised cell structure could be modelled by
abutting this apical component onto a cylinder of radius f ðπ=2Þ for the
given value of p as determined by (20). Moreover, the asymptotic
curvature is infinite and hence the model predicts that the distal zone
of the cap (where it meets the cylindrical shaft) is the region of
maximum wall material deposition. This contradicts the known fungal
and bacterial physiology. However, in the formation of pollen tubes,
the wall-building material deposition occurs in an annular region
located behind (distal to) the apex (Kroeger et al., 2011; Geitmann,
2010; Rojas et al., 2011). Typical tip configurations are shown in Fig. 2
(a). Note that from (15),
lim
df
θ-π=2 dθ
where this time we have used gð0Þ ¼ 0.
Clearly, for each fixed 0 oθ o π=2, (16) and (18) result in positive
and negative values of f and g, respectively. It follows immediately
from (16) and (18), that for the special case p¼2, explicit (and as will
be shown, useful) closed form expressions can be gained by direct
integration. This is the special case detailed in Goriely et al. (2005) as
we discuss further below. For general values of p, and for each fixed
0 o θ o π=2, closed form expressions can be obtained for these
integrals, but these are unwieldy combinations of hypergeometric,
trigonometric and gamma functions that provide no obvious advantage over direct numerical integration of (16) and (18). However, it is
instructive to investigate the limits of κ, f and g and as θ-π=2 as these
limits in themselves provide important information regarding the
geometric properties of the tip and, consequently, the location of wallmaterial deposition. As we will now discuss, they determine the
radius of the sub-apical shaft, the length of the apical region and the
manner in which the tip connects with the shaft. That these limits
exist is not obvious a priori as, by inspection, the expression itself in
(14) and the integrands in (16) and (18) can become unbounded as
θ-π=2 for certain values of p.
First, we consider κ. From (14), it follows directly that
(
1; 0 op o1;
lim κ ¼
ð19Þ
0;
p 41:
θ-π=2
47
¼ 0;
and hence the cap can be joined to the cylinder in a C1 manner.
However, limθ-π=2 κ ¼ 1 for p in this range and hence the connection
to the cylinder cannot be made C2. Thus the entire tip shape (cap and
shaft) cannot be formed through a more usual (smooth) deformation
process. This forms an interesting point for further investigation, but is
not central to the application here as we focus only on the form of the
apical cap.
3.2. Case 2: 1 op≤3=2
In this case, as θ-π=2, both the radius and the length of the tip
tend to 1. This would seem to indicate that this does not
represent a biologically relevant case. However, the divergence
of the integrals in (16) and (18) is relatively slow and even for
values of θ within 103 of π=2, the integrals are only of Oð1Þ as
illustrated in Fig. 2(b). Of note, however, is that κ is monotonically
increasing in θ for p in this range and hence the model predicts
that maximum wall deposition occurs far behind the apex of the
cell: the least likely case.
3.3. Case 3: 3=2 o p≤2
For 3=2 o p o 2 the asymptotic tip radius is finite and given by
(20). The corresponding tip length given by (21) is infinite. Note
from (14), that in this case, the asymptotic curvature is zero and
the curvature has unique maximum at θ ¼ 0, i.e. at the apex. Hence
this case closely represents the observed geometry of fungal and
bacterial tips as well as correctly predicting that new wall synthesis is at a maximum at the apex. A typical profile is shown in
Fig. 2(c).
The special case discussed in Goriely et al. (2005) is p ¼2. As
alluded to the above, a neat, closed expression can be derived for
48
M.Z.A.M. Jaffar, F.A. Davidson / Journal of Theoretical Biology 336 (2013) 44–51
Fig. 2. Typical tip profiles. (a) Tip profiles generated using the expressions (16) and (18). (a) p ¼ 0.5 (red, solid); (b) p ¼ 1.25 (red, solid); (c) p ¼1.75 (red, solid); (d) p ¼2.5 (red,
solid). For comparison, the special case p ¼ 2 is also shown in each case (black, solid). In each case, α ¼ 1; κ 0 ¼ 1. In both (a) and (d), straight lines have been abutted to the
apical caps (red, dashed). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
the curve C in this case. Substituting p¼ 2 in (16) and (18) yields
f ðθ Þ ¼
1
ðακ20 Þ
Z
θ
0
dϕ ¼
1
θ;
ðακ 20 Þ
ð22Þ
and
g ðθ Þ ¼ 1
ðακ20 Þ
Z
θ
0
sin ϕ
1
dϕ ¼
lnð cos θÞ:
cos ϕ
ðακ 20 Þ
ð23Þ
By combining (22) and (23), an explicit description of the profile C
reads as
y¼
explicit representative profile that is useful for predicting the
effects of varying α and κ0 on this class of profiles.
1
ln cos ακ 20 x :
ðακ20 Þ
This was the tip profile derived in Goriely et al. (2005). As detailed
there, this profile has been derived previously to describe other
tip-formation processes. Finally, although not explicitly discussed
in Goriely et al. (2005), in this case it follows directly from (14)
that κ ¼ ακ20 cos ðθÞ and hence is a monotonically decreasing function of θ. This special case is compared to other profiles in Fig. 2.
Therefore, for all values of p in the range 3=2 o p≤2, the
corresponding tip profiles have qualitatively the same geometry
and the pattern of wall-building material deposition is also
predicted to be qualitatively similar. Therefore, the special case
p ¼2 does not lead to a unique geometry. Rather, it provides an
3.3.1. Case 4: p 4 2
In this case, as above, f and κ remain finite. The curvature κ is
monotonically decreasing and tends to zero as θ-π=2. However, g
now also remains finite in this limit. (Note however, that
gðπ=2Þ-1 as p↘2.) Hence in this case, the model predicts that
the apical component is of finite length, but can be joined
smoothly to a cylindrical shaft. Moreover, wall-deposition is again
predicted to be a maximum at the apex, tending to zero at some
now finite distance behind the apex. Of all configurations, this
perhaps provides the closest match to the observed physiology of
hyphal growth. Typical profiles are illustrated in Fig. 2(d).
3.3.2. Case 5: limiting cases: p-0 and p-1
First consider the extreme case p-0. This models the situation
where wall-material deposition is essentially independent of
curvature and occurs at all regions of the tip at a constant rate.
Considering θ to be fixed, 0 o θ o π=2, then from (14) it follows that
lim κ ðθÞ ¼
p-0
1
≕κ0 ðθÞ:
α cos θ
ð24Þ
M.Z.A.M. Jaffar, F.A. Davidson / Journal of Theoretical Biology 336 (2013) 44–51
Also, from (16)
1
lim f ðθÞ ¼ lim
p-0
ðακp0 Þ1=ðp1Þ
p-0
Z
¼α
Z
¼α
Z
θ
ð cos ϕÞðp2Þ=ðp1Þ dϕ
0
lim ð cos ϕÞðp2Þ=ðp1Þ dϕ
0 p-0
θ
0
ð cos ϕÞ2 dϕ ¼
α
0
ð cos θ sin θ þ θÞ≕f ðθÞ:
2
Similarly, from (18)
lim g ðθÞ ¼ lim p-0
θ-π=2 and we deduce that the limiting curvature is not uniform
in θ.
As above, from (16), it follows that
!
Z θ
1
ðp2Þ=ðp1Þ
lim f ðθÞ ¼ lim
ð
cos
ϕÞ
dϕ
p-1
p-1 ðακ p Þ1=ðp1Þ 0
0
!
θ
p-0
Z
¼ α
θ
0
1
ðακ p0 Þ1=ðp1Þ
Z
θ
0
sin ϕ
1=ðp1Þ
ð cos ϕÞ
sin ϕ cos ϕ dϕ ¼ ð25Þ
dϕ ;
g 0 ðt Þ ¼ α α
þ cos t;
4 4
¼
θ
cos ϕ dϕ ¼
0
1
sin θ≕f 1 ðθÞ:
κ0
ð28Þ
lim f
ð27Þ
where t ¼ 2θ. Hence, the limit profile is a cycloid (letting
π=2≤θ≤π=2 for the full apical cap) with “length” α=2 and
“diameter” α=2ðπÞ. Notice that α acts only as a scaling factor: it
does not alter the geometry of the tip profile. This is quite different
to the special case p ¼2 where α controls the geometry of the
profile.
Finally, we consider the other limit p-1, which models the
situation where the wall-building material deposition is very
focused to the areas of high curvature. In this case, for each fixed
0 o θ o π=2, it follows directly that κ-κ0 as p-1. However, for
any fixed p sufficiently large, from (19) it follows that κ-0 as
1
κ0
Z
0
θ
Z
1
ðακ p0 Þ1=ðp1Þ
sin ϕ dϕ ¼
θ
0
!
sin ϕ
ð cos ϕÞ1=ðp1Þ
dϕ ;
1
ð1 þ cos θÞ≕g 1 ðθÞ:
κ0
ð29Þ
Furthermore, taking p-1 in the expressions (20) and (21)
yields
p-1
Hence, the limits given in (25) and (26) are uniform in θ. From
(24), we see that the curvature is monotonically increasing in θ
and becomes unbounded as θ-π=2, similar to the general case.
The limiting profile is given by f0 and g0. Indeed, a little trigonometric manipulation reveals
α
α
t þ sin t;
4
4
Z
p-1
ð26Þ
where we have used h-3=2 as p-0 and
pffiffiffi
π
3
1
1
:
¼ Γ
¼
Γ ð1Þ ¼ Γ ð2Þ ¼ 1; Γ
2
2
2
2
0
1
κ0
lim g ðθÞ ¼ lim
p-1
Furthermore, taking p-0 in the expressions (20) and (21) yields
π απ
π π α
0 π
¼
¼f
and lim g
¼ ¼ g0
;
lim f
2
4
2
2
2
2
p-0
p-0
f ðt Þ ¼
¼
Similarly, from (18)
!
α
sin 2 θ≕g 0 ðθÞ:
2
49
π 2
¼
π 1
¼f1
κ0
2
and
lim g
p-1
π 2
¼
π 1
;
¼ g1
κ0
2
where we have used h-1 as p-1 and hence
pffiffiffi
ΓðhÞ
2
π
¼ pffiffiffi and noting lim K ¼
lim :
1
p-1 Γ h þ
p-1
π
2κ
0
2
Hence, the limits given in (28) and (29) are again uniform in θ and
an explicit representation of the tip profile is given by f 1 and g 1 .
Indeed, f 1 and g 1 satisfy the very simple relationship
1 2
1
2
¼ 2;
f 1 þ g1 þ
κ0
κ0
ð30Þ
i.e. the tip profile is a circular, with radius 1=κ 0 , consistent with the
limiting value of the curvature derived above. Notice that in this
case, this limiting profile is independent of α. Hence, e.g. the speed
of the tip does not have any direct bearing on either its scale or
geometry. As detailed above, this is in contrast to the special case
p¼ 2 discussed previously. The three closed from cases p ¼ 0; p ¼ 2
and p ¼ 1 are illustrated in Fig. 3. Straight lines have been abutted
to the apical caps in the p ¼ 0; p ¼ 1 cases to form a pseudo cross
section profile. Different choices of α are used in Fig. 3(a) and (b).
Fig. 3. Tip profiles generated by special cases. (a) Pseudo cross-sections of the apical parts generated using the expressions (25) and (26) (p ¼ 0, red); (28), (29) (p ¼ 1, blue)
and (22), (23) (p¼ 2, black). In each case, α ¼ 1; κ 0 ¼ 1. (b) Pseudo tip cross-sections computed as for (a) with the same colour coding but with α ¼ 2 and κ 0 ¼ 1. For p ¼0 and
p ¼ 1 in both (a) and (b), straight lines (dotted) have been abutted to the apical caps. (For interpretation of the references to color in this figure caption, the reader is referred
to the web version of this article.)
50
M.Z.A.M. Jaffar, F.A. Davidson / Journal of Theoretical Biology 336 (2013) 44–51
4. Discussion
In this paper we have developed a geometric model for
polarised growth of cells based on that first introduced in
Goriely et al. (2005). Despite this being a minimal method with
which to describe the complex processes involved, as shown in
Fig. 4, the model is capable of capturing the geometry of polarised
cells from different organisms. (The special cases p ¼ 0; 2; 1 were
used for ease of plotting and calculation of the scaling factor α.
Further fine tuning of the parameter p improves the fit but then
scaling factor α has to be computed numerically using (20).) A
number of geometrical models have previously been utilised to
describe the shape of polarised cells. The hyphoid introduced by
Bartnicki-Garcia et al. (1989) is one such model used to describe
fungal hyphae. Others include simple geometric shapes such as
ellipses used to describe pollen tubes (e.g. Fayant et al., 2010).
These simple geometric models are purely descriptive and do not
always capture the range of tip shapes even within a given species.
The model discussed here is capable of capturing the geometry as
least as well as these other models and in particular is significantly
better at predicting the different geometries associated with the
two different species of fungi shown. Moreover, because this
model links geometry to growth, it is capable of correctly predicting the location of the zone of maximal deposition of wall-building
materials (the green zones shown in Fig. 4). In pollen tubes,
maximal deposition is known to occur in a sub-apical band where
the curvature is highest (Kroeger and Geitmann, 2012). In fungi,
maximal deposition occurs at the hyphal apex and is associated
with the Spitzenkörper (the bright object at the apex in Fig. 4B and
C). This is again associated with the zone of highest curvature.
Note that from a purely visual inspection of the tips in Fig. 4A and
C, in particular, it is not obvious where this high curvature zone is
situated.
We have established that basic rules of curvature-dependent
growth coupled with the normal growth hypothesis appear to be
sufficient to provide a minimal description of tip growth in a
variety of biologically distinct cases. Of particular note is that our
extended analysis of this model has revealed that it is capable of
predicting apical and sub-apical wall deposition and therefore
captures key features of e.g. fungal hyphal extension and pollen
tube growth, respectively. This predictive ability is potentially
useful in e.g. studying the response of tip growth to the physical
environment. For example, contact guidance (thigmotropism) is
the ability of an organism to respond to topographical stimuli by
altering its axis of growth. It is known that certain species of fungi
exhibit thigmotropic behaviour, presumably to their advantage (e.
g. in finding infection sites Gow et al., 2012). Many hypotheses
have been formulated to describe the biochemical origins of this
behaviour and various channels (e.g. calcium) have been proposed
(see e.g. Brand and Gow, 2007; Kumamoto, 2008). However, these
biochemical channels do not appear to be ubiquitous (see
Stephenson et al., submitted for publication) and hence again, it
appears that this response mechanism can be driven in a variety of
ways. So again the simple geometric description offered here is
potentially useful in that it could link the disruption of the tip
geometry caused by interaction with a physical feature to the
realignment of the zone of maximal wall-deposition and hence to
the realignment of the growth axis (e.g. Stephenson et al.,
submitted for publication; Bowen et al., 2007) without the need
for a detailed (and species-specific) description of the underlying
biochemical response.
Undoubtedly, more complex bio-mechanical models that treat
the wall as an elastic-plastic shell are able to incorporate greater
detail regarding known wall-physiology. Moreover, models that
couple the cytoskeleton to the delivery of wall building materials,
to wall-deposition and thence to wall-deformation would provide
the most accurate description of the process. These are far more
complex models and would, for example, circumvent the potential
instability of simple curvature-dependent models of the type
considered here. However, the construction of such complex
models comes at the price of necessarily being more speciesspecific and mathematically intractability. Even the numerical
integration of such models offers a significant challenge. As such,
we believe that simple, geometric models of the form discussed
Fig. 4. Geometric model predicts tip shape and location of maximal wall-deposition. The figure shows microscope images of polarised cells overlaid with the growth model
discussed here (red dashed) and simpler geometric descriptions (yellow dashed). Regions of maximal wall-deposition predicted by the model shown in green. (A) Lilly pollen
tube with cap modelled by (25) and (26) (red dashed) and ellipse with major axis 1.5 times minor axis (yellow dashed). (B) Hypha of the fungus Nerospora crassa overlaid
with models (16) and (18) with p ¼2 (red dashed) and the hyphoid equation y ¼ xcotðx=αÞ (yellow dashed). (C) Hypha of the fungus Sclerotinia sclerotiorum with cap modelled
by (28) and (29). (D) Hypha of the actinobacterium Streptomyces coelicolor overlaid with model (16) and (18) with p ¼2. In all figures κ 0 ¼ 1 and (A) α ¼ 2 diam:=π ¼ 27:5=π;
(B) and (C) α ¼ π=diam: ¼ π=14:6; (D) α ¼ π=diam: ¼ π=15. Straight lines have been abutted to the apical caps in (A) and (C) (red, dashed). Scale bars in A–C are 10 μm and in
(D) 1 μm. Background micrograph images from Lovy-Wheeler et al. (2006) (A); Hickey et al. (2005) (B) and (C); and Flärdh (2003) (D) with kind permission. (For
interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
M.Z.A.M. Jaffar, F.A. Davidson / Journal of Theoretical Biology 336 (2013) 44–51
here do have value as they offer a formalism with which to better
understand basic rules that link wall deposition, speed and
geometry in polarised growth.
Acknowledgments
The author would like to thank Ministry of Higher Education,
Malaysia for supporting her throughout this work.
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