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Proc. R. Soc. B (2012) 279, 867–875
doi:10.1098/rspb.2011.1284
Published online 31 August 2011
Geometrical constraints in the scaling
relationships between genome size, cell size
and cell cycle length in herbaceous plants
Irena Šı́mová1,2,* and Tomáš Herben1,3
1
Department of Botany, Faculty of Science, Charles University, Benátská 2, Praha 2, Czech Republic
Center for Theoretical Study, Charles University in Prague and Academy of Sciences of the Czech Republic,
Jilská 1, 110 00 Praha 1, Czech Republic
3
Institute of Botany, Academy of Science of the Czech Republic, Průhonice, Czech Republic
2
Plant nuclear genome size (GS) varies over three orders of magnitude and is correlated with cell size and
growth rate. We explore whether these relationships can be owing to geometrical scaling constraints.
These would produce an isometric GS–cell volume relationship, with the GS–cell diameter relationship
with the exponent of 1/3. In the GS–cell division relationship, duration of processes limited by membrane
transport would scale at the 1/3 exponent, whereas those limited by metabolism would show no relationship.
We tested these predictions by estimating scaling exponents from 11 published datasets on differentiated and
meristematic cells in diploid herbaceous plants. We found scaling of GS–cell size to almost perfectly
match the prediction. The scaling exponent of the relationship between GS and cell cycle duration did
not match the prediction. However, this relationship consists of two components: (i) S phase duration,
which depends on GS, and has the predicted 1/3 exponent, and (ii) a GS-independent threshold reflecting
the duration of the G1 and G2 phases. The matches we found for the relationships between GS and both cell
size and S phase duration are signatures of geometrical scaling. We propose that a similar approach can be
used to examine GS effects at tissue and whole plant levels.
Keywords: scaling exponent; standard major axis regression regression; nuclear size; cell size;
cell cycle phases; root meristem
1. INTRODUCTION
Sizes of plant nuclear genomes vary over three orders of
magnitude from about 0.065 (Genlisea margaretae) to
152.23 pg/1C (Paris japonica) [1,2]. However, possible
ecological implications have begun to be explored only
recently [3,4], for a review see [5]. Although the full functional meaning of this variation is yet unknown, it is clear
that genome size (GS) must represent certain phenotypic
constraints. Cells must be larger to accommodate large
genomes, and larger genomes typically take more time to
multiply [6,7] (but see [8 –10]), thus seemingly constraining the cell growth rate and other plant functional traits.
These assumptions led to the formulation of the large
genome constraint hypothesis (LGCH) [11]. According
to the LGCH, species with large genomes are disadvantaged owing to the costs associated with accumulation
and replication of non-coding repetitive DNA. The strongest support for this hypothesis is the correlation between
GS and cell size [3,5,12,13] as well as between GS and
cell cycle duration [6,14].
In addition to these basic relationships, there are numerous reports of correlations between GS and various plant
traits, as well as environmental or evolutionary characteristics [5,11,12,14–20]. Angiosperms with larger genomes
often have lower specific leaf areas and, consequently,
lower maximum photosynthetic rate [5,11] and lower
growth rate [14,15], possibly owing to the competition for
phosphorus between DNA and RNA [16]. At the same
time, GS in herbs is often associated with larger seed mass
[12,17,18] and leaf lifespan [19]. Furthermore, herbaceous
species with large genomes are not adapted to extreme
environments, as they cannot have a large number of small
stomata [12]. The same constraint also holds for woody
species, which consequently have smaller genomes [12].
From an evolutionary viewpoint, plant lineages with large
genomes also show reduced capacity for fast evolutionary
differentiation [11], and are rare in species groups that
show adaptive radiation on oceanic islands [20]. Genome
size has thus been suggested to be one of the important
attributes that constrain plant fitness [21].
In spite of extensive research on correlations of GS with
various plant traits, we are still far from understanding
specific processes that underlie these correlations. One of
the possible yet little explored quantitative approaches is to
characterize scaling relationships between GS and proximal
variables such as cell size, tissue density and cell division
rate. Scaling relationships arise owing to universal (often
geometrical) processes or constraints that act over a large
range of scales and take the form of scale-invariant power
laws [22–26]. The signature of such a scaling relationship
(namely, the value of the scaling exponent, typically
measured as the slope in the log–log plot) can shed light
on the type of the underlying process or constraint [27–31].
Although scaling relationships are common in the biological world and have been extensively studied [32–38],
only scant attention has been paid to the scaling of GS
with other plant traits. In an early study, Baetcke et al. [39]
* Author for correspondence ([email protected]).
Electronic supplementary material is available at http://dx.doi.org/
10.1098/rspb.2011.1284 or via http://rspb.royalsocietypublishing.org.
Received 20 June 2011
Accepted 8 August 2011
867
This journal is q 2011 The Royal Society
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868 I. Šı́mová & T. Herben
Geometry in genome size relationships
found that the slope of the interspecific relationship between
GS and nuclear volume is approximately 1 (i.e. a linear
relationship). Similarly, Gregory [40] explored different
studies focusing on cell sizes of both plants and vertebrates.
He showed that, in plants, nuclear volume scales linearly
with cell volume; however, he did not use GS (dataset
from Price et al. [41]). The only attempt to capture the
scaling relationship between GS and cell cycle duration
comes from Cavalier-Smith [42] who predicted that cell
cycle duration should be proportional to the amount of
material that a cell needs to grow and develop, which in
turn is constrained by the rate of transport. Assuming that
the rate of transport is proportional to the cell surface
(which increases with the cell volume with the exponent
2/3), and the GS is an isometric function of cell volume,
he inferred that the scaling exponent between GS and cell
cycle duration should be 1/3. However, he did not test this
prediction using statistical techniques.
All of these pioneering studies used ordinary leastsquares (OLS) regression to estimate the scaling exponents.
While OLS regression is a standard tool for analysing data
where both predictors and dependent variables are clear,
its use should be restricted to cases when errors on the
x-axis are negligible relative to the errors on the y-axis,
with this constraint particularly important when the
relationship is not very strong (see [43]). Thus, this technique is not suited to analysing the relationship between
GS and other cell properties, where the independent variable is unclear [44] and relationships are not always very
strong [5,11]. In such cases, it is necessary to estimate the
scaling exponent by more appropriate techniques, such as
standard major axis regression (SMA, sometimes also
called reduced major axis, RMA [45,46]).
In the present paper, we attempt to identify some possible
relationships between GS and variables at the cell and tissue
level based on simple geometrical assumptions and test
them using empirical data on diploid herbaceous plant
species. We first formulate several predictions on scaling
relationships involving GS based on geometrical considerations. Next, we examine these predictions using datasets
from a large number of plant species. We take advantage
of the fact that the recent advent of fast cytometric techniques has made possible collection of extensive data on
many plant species [1,21]. This enabled us to assemble
empirical datasets on some of these relationships for quite
large numbers of species in order to examine to what
degree they conform to theoretical predictions. We have limited ourselves to considering diploid plants only. Although
polyploidy does increase total DNA content, it introduces
a number of different phenomena that do not occur when
the base (monoploid) DNA amount increases [1,21]. This
is supported by the evidence that the relationship between
intraspecific cell size and GS differs between diploids and
polyploids [47–50]. Moreover, it has been shown that
increase in ploidy level does not increase cell cycle duration
[8–10]. Analysing diploid and recent polyploid species
together would thus confound the effects of polyploidy
and monoploid DNA amount [21].
2. PREDICTIONS
We predict that GS, cell size and cell cycle length are
constrained by geometrical relationships and thus
should scale together. These geometrical relationships
Proc. R. Soc. B (2012)
act both directly (as in the GS –nuclear size relationship),
and indirectly, through limits for the transport of material
to the cell and within the cell or through the cell metabolism. As all these constraints are possibly bidirectional,
we explicitly do not assume any cause-and-effect chain
and limit ourselves to symmetrical scaling relationships.
The simplest geometrical constraint concerns the
relationship between nuclear volume and GS. As these
are measured in volume units and mass units, they
should correspond linearly to each other and their scaling
exponent should consequently be 1 (prediction 1).
Further, cell size should scale with nuclear volume
and hence with GS. The simplest possible relationship is
a geometrical constraint as larger cells are needed to accommodate larger nuclei either physically, owing to cytoskeletal
structure or owing to the necessary ratio between RNA synthesis and the rate of protein synthesis [51]. This scaling is
between volume–volume units (or volume–mass units),
meaning that the relationship should also be linear, i.e.
with the scaling exponent of 1 (prediction 2a). Consequently, the scaling exponent for the cell area–GS
relationship should correspond to 2/3 owing to the area :
volume ratio (prediction 2b). Also, cell length should
scale with GS according to the length : volume ratio, i.e.
with scaling exponent 1/3 (prediction 2c).
When analysing data on cell lengths, it is necessary to
take into account the fact that most empirical cell length
data come from measurement of stomata guard cells.
These cells are known to have rather constant sizes and
typically do not endoreduplicate [48] (which makes them
a favourite subject for comparative studies), but have a
strongly constrained (curved) shape. The scaling relationship with other cell variables may be affected by the
possible effect of this bend or curvature. We assume that
the shape of this bend could be expressed by a power function y ¼ ax b, where x is the measured length, and a and b
are positive numbers. This length of the bent shape (S) is
always greater than x and scales nonlinearly with the
measured length, with the scaling exponent depending on
the shape of the cell (i.e. parameters a and b of the
curve), but necessarily greater than 1 (derivation for the
simplest case of a quadratic parabola is in the electronic
supplementary material). If we denote this scaling exponent c, the measured length (x) will thus scale with GS
with the exponent (1/3) into (1/c), which is always lower
than 1/3 if c . 1 (prediction 2d).
Constraints on possible scaling relationships between
GS and cell growth and division rates should be due
either to matter transport to the cell [42] or to cell metabolism [52] as a function of GS and cell size. During cell
growth, DNA synthesis and cell division, the cell would
need amounts of material proportional to its volume,
with the rate of transport limited by cell surface area.
This area increases with a scaling of 2/3 in relationship
with cell volume. If transport across the cell surface is
the limiting factor, the cell cycle duration should thus be
proportional to the amount of material that needs to pass
into the cytoplasm, divided by the rate of transport [42]:
TðccÞ /
V1
/ V 1=3 / GS1=3 ;
V 2=3
ð2:1Þ
where T is duration of the cell cycle and V is the cell
volume. Alternatively, cell cycle duration can also be
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Geometry in genome size relationships
I. Šı́mová & T. Herben
869
Table 1. Published datasets used in the study.
reported variables
type of cells
species
families
reference
nuclear volume, GS
nuclear volume, GS
cell volume, nuclear volume
pollen width, GS
cell area, guard cell length, GS
pollen volume, GS
cell cycle duration, GS
G1, S, G2, M phase duration
G1, S, G2, M phase duration, GS
S phase duration, GS
root and shoot meristem
epidermisa of young leaves
shoot meristem
pollen grains
leaf epidermal and guard cells
pollen grains
root meristem
root meristem
root meristem
root meristem
16
5
14
178
20
10b
62
7
5
21
10
4
9
40
13
1c
16
5
4
11
[39]
[49]
[41]
[13]
[12]
[55]
[6]
[56–58]
[59]
[7]
a
We used nuclear volume only for data of 2C nuclei reported in this study.
We excluded one outlier.
Poaceae.
b
c
driven by cell metabolism. Both respiration and photosynthesis take place on organelle membranes (owing to the
location of ATP-synthesis complexes). Assuming that an
increase in cell size leads to a proportional increase in the
number of organelles [53], the rate of whole-cell metabolism should scale with GS with the scaling exponent 1.
Therefore, mass-specific metabolic rate should be independent of GS (GS1/GS1). Assuming the duration of the
cell cycle is inverse to the cell cycle rate, cell cycle duration should not depend on GS (scaling exponent 0).
When considering both role of cell metabolism and the
rate of transport as factors responsible for cell cycle duration, the scaling exponent between GS and cell cycle
duration should vary between 1/3 (limitation by the
surface : volume ratio only) and 0 (limitation by cell
metabolism only; prediction 3).
This prediction is however problematic as the cell cycle
consists of four phases and the duration of each phase
may depend upon GS differently owing to the differences
underlying or limiting each of these phases. We therefore
make specific predictions for the relationship between GS
and duration of each phase of the cycle. Because of the
demands of the growth phases (G1 and G2), we expect
that during these phases cell metabolic rate should be
especially important. Thus, the duration of these two
phases should not depend on GS; i.e. scaling exponents
of relationships between G phase duration and GS
should be 0. However, as metabolic rate might be constrained by the rate of transport through the cell
membrane, we expect that the scaling coefficient between
the duration of G phases and GS should vary between 1/3
and 0, as shown earlier (prediction 4a).
Limitation on the S phase is in principle owing to the
number of replication origins and replication rate per replicon. However, the dataset of Hof & Bjerknes [7] shows that
there is no significant relationship between GS and the fork
rate (replication rate per replicon). We thus suggest that the
main factor limiting duration of the S phase is the amount
of DNA polymerase and other factors (e.g. CDC6 and
CDT1, see [54]) important for the activation of replication
origins. Given that the amount of the DNA polymerase as
well as CDC6 and CDT1 needed for the replication scales
linearly with GS, it should be limited by the rate of transport through the nuclear surface. S phase duration will
then be constrained by the ratio between the total
amount of material needed for replication (which scales
Proc. R. Soc. B (2012)
at 1 with the GS) divided by the rate of transport (which
scales at 2/3 with the nuclear volume). The predicted exponent for the relationship between S phase duration and GS
should thus be 1/3 (prediction 4b). We hypothesize that
during cell division (M phase), GS (and cell size) limits
the development of nuclear membranes and cell walls.
Therefore, duration of the M phase should scale linearly
with the total surface area of nuclear membranes and cell
walls, which in turn scales at 2/3 with the cell volume
and thus at 2/3 with GS. We thus predict the scaling exponent between the duration of the M phase and GS to be 2/3
(prediction 4c).
3. METHODS
(a) Datasets
We re-analysed 11 published datasets reporting nuclear
volume, cell size, cell cycle duration and cell cycle phase duration (detailed description is given in table 1). Both
differentiated and undifferentiated cells were used to obtain
data on cell and nucleus sizes; for cell cycle duration, we
used only root meristem data where potential constraints
owing to GS would show up most clearly owing to their
high cell division rates. To reduce noise in the data, we
included data only on diploid herbaceous plants. We
excluded woody plants, which usually have much smaller
genomes because of strong selection on cambial and stomatal
cell size [12].
The data came from a total of 269 plant species from 46
families. Genome sizes (2C-value in pg) were taken from original publications and, if missing there, from the Angiosperm
DNA C-value database [1]. Ploidy levels were also taken
from original publications or from C-value database [1];
corrections of ploidy levels using other existing cytogenetic data were performed in cases of clear errors ( J. Suda
2011, unpublished data). We were able to compile cell cycle
phase duration data from different resources (table 1, rows 8
and 9) as they all were measured within the same temperature
range (20–238C) [7,56–59].
(b) Data analysis
We re-analysed each dataset using SMA regression
(R language, package lmodel2; [60]). The scaling exponent
for each relationship between GS and different cell parameters was calculated as the slope of the SMA regression
from the power law relationship (logarithmic transformation
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870 I. Šı́mová & T. Herben
Geometry in genome size relationships
Table 2. Standard major axis power law regression between GS and nuclear volume, cell size, whole cell cycle duration and
duration of cell cycle phases.
power law relationship
nuclear volume
nuclear volume –GS (shoot
meristem)
nuclear volume –GS (root
meristem)
nuclear volume (epidermal
cells) –GS
cell size
cell volume –nuclear
volume
pollen volume –GS
epidermal cell area–GS
pollen width –GS
guard cell length–GS
cell cycle duration
cell cycle duration–GS
G1 phase duration–GS
S phase duration –GS
G2 phase duration–GS
M phase duration –GS
S phase duration (synthesis
of all data available) – GS
prediction
no.
predicted
slope
estimated
SMA slope
95% confidence
intervals
1
1
0.96
1
1
1
r2
p
ref.
(0.77;1.20)
0.86 ,0.001
[39]
0.85
(0.68;1.05)
0.86 ,0.001
[39]
1
1.01
(0.86;1.19)
0.99 ,0.001
[49]
2a
1
0.99
(0.90;1.08)
0.98 ,0.001
[41]
2a
2b
2c
2d
1
2/3
1/3
,1/3
1.16
0.64
0.34
0.24
(0.96;1.41)
(0.47;0.87)
(0.29;0.39)
(0.19;0.30)
0.89 ,0.001
0.59 ,0.001
0.01
0.18a
0.75 ,0.001
[55]
[12]
[13]
[12]
3
4a
4b
4a
4c
4b
1/3–0
1/3–0
1/3
1/3–0
2/3
1/3
0.69b
n.s.
0.36
n.s.
0.40
0.36
—
—
(0.21;0.53)
—
(0.20;0.79)
(0.27;0.48)
—
—
0.06
0.48
0.50
0.02
,0.01
0.87
0.46
0.07
0.64 ,0.001
[6]
[56–59]
[56–59]
[56–59]
[56–59]
[56–59]
a
p ¼ 0.08 for one-sided test.
Scaling exponent is estimated from the nonlinear regression.
b
of all variables). We calculated confidence intervals for all scaling
exponents. The match of the estimated slopes with the
theoretical predictions was carried out by assessing (i) simple
numerical closeness of the theoretical and empirical values and
(ii) statistical fit of the theoretical and empirical values using
95% confidence intervals of the empirical values.
As numerically high values of intercepts in untransformed
data can seriously distort estimated values of scaling exponents
in log–log transformed data [61], we checked all relationships
for the intercept effect. We tested for nonlinearity of the
relationship in the log–log transformed data by testing significance of the quadratic term in quadratic polynomial
regression fitted by OLS. If the quadratic term was significant,
we estimated the scaling exponent from nonlinear regression
(SPSS, v. 10; SPSS Inc., Chicago, IL, USA). We used nonlinear
regression to fit the equation y ¼ Cx z þ a, using the loss function proposed by Ebert & Russell [61] for model II regression
(analogue to SMA) estimation of the y-axis intercept.
4. RESULTS
(a) Nuclear volume and cell size
There was a strong linear relationship between GS
and nuclear volume. The SMA scaling exponent between
nuclear volume and GS was 0.96 for shoot meristem cells,
0.85 for root meristem cells and 1.01 for epidermal cells
of young leaves (table 2 and figure 1a). Confidence intervals
of all these relationships covered the predicted value
of 1. Also, in accord with our prediction, GS (and nuclear
volume) scaled linearly with the cell volume (table 2
and figure 1b–d). The strongest relationship occurred
between the shoot meristem cells and nuclear volume
(scaling exponent 1.01, r 2 ¼ 0.98; table 2 and figure 1b).
A similar, but weaker relationship was found between
Proc. R. Soc. B (2012)
epidermal cell area and GS (scaling exponent 0.64, close
to the predicted value of 2/3; table 2 and figure 1c). Nevertheless, these results with epidermal cells may be partly
affected by endoreduplication [48]. There was a very weak
indication of a relationship between GS and pollen width
having a scaling exponent of 0.34 (table 2). Although
this was in agreement with our prediction (1/3), this
relationship was not significant, as the p-value was 0.18
(0.08 for one-tailed test). In the case of guard cell length
(table 2 and figure 1d), as the observed scaling exponent
was 0.24, it was less than 1/3, and thus in accord with our
prediction. The highest deviation from the predicted value
was found in the relationship between pollen volume and
GS (table 2), with the scaling exponent being 1.16 instead
of the predicted 1.
(b) Cell cycle duration
There was strong nonlinearity in the power-law function
of the relationship of cell cycle duration to GS, as
showed by the high significance of the GS as a quadratic
term (p ¼ 0.004). This was the only relationship with
a significant quadratic term. This relationship also
had a high value of the intercept in non-transformed
data (figure 2a,b). The value of the intercept (6.14) corresponded to the minimum value of cell cycle duration.
The scaling exponent estimated by nonlinear regression
(0.69) did not match the value predicted by the surface :
volume or metabolic limitation (table 2).
The relationships between GS and individual cell cycle
phases varied widely. Although there was no relationship
between G phases and GS (table 2 and figure 3a,c), the
relationship between duration of the S phase and GS was
significantly positive (table 2 and figure 3b) and the scaling
exponent (0.36) corresponded to the predicted 1/3 (when
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Geometry in genome size relationships
871
(b)
log (cell volume of shoot meristem)
log (nuclear volume of shoot meristem)
(a)
3.5
I. Šı́mová & T. Herben
3.0
2.5
2.0
0.5
1.0
1.5
log (2C DNA)
4.0
3.5
3.0
2.5
2.0
1.5
2.0
2.5
3.0
3.5
log (nuclear volume of shoot meristem)
(d)
(c)
1.8
log (guard cell length)
log (epidermal cell area)
3.8
3.6
3.4
3.2
3.0
1.7
1.6
1.5
1.4
1.3
2.8
0.0
0.5
1.0
log (2C DNA)
1.5
2.0
0.0
0.5
1.0
log (2C DNA)
1.5
2.0
Figure 1. Relationship between genome size and nuclear and cell sizes: (a) relationship between genome size (GS) and nuclear
volume of shoot meristem cells (slope ¼ 0.96); (b) relationship between nuclear volume and cell volume of shoot meristem cells
(slope ¼ 0.99); (c) relationship between GS and epidermal cell area (slope ¼ 0.64); and (d) relationship between GS and guard
cell length. Black line represents SMA regression line (slope ¼ 0.24); grey lines are SMA 95% confidence intervals. The
regression results are listed in table 2, rows 1, 4, 6 and 8.
(a)
30
(b)
1.4
log (cell cycle duration)
cell cycle duration (h)
25
20
15
10
5
1.2
1.0
0.8
0.6
0
0
10
20
30 40 50
2C DNA (pg)
60
70
–0.5
0.0
0.5
1.0
log(2C DNA)
1.5
2.0
Figure 2. The relationship between cell cycle duration and genome size: (a) untransformed, (b) with logarithmic transformation. The line shown in part (b) is the quadratic function. The model containing log(GS) both as linear and quadratic
term has a significantly better fit than the linear-only model (p-value ¼ 0.004, n ¼ 83).
the same dataset as for other phases was used). When we
analysed the larger set of all available S phase data, the scaling exponent remained the same and the relationship was
Proc. R. Soc. B (2012)
even tighter (table 2 and figure 4). Duration of the M
phase increased with GS marginally significantly (table 2
and figure 3d). The estimated scaling exponent 0.40 did
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872 I. Šı́mová & T. Herben
Geometry in genome size relationships
(b)
(a) 0.8
0.6
log (S phase duration)
log (G1 phase duration)
1.0
0.4
0.2
0.5
1.0
log (2C DNA)
0.7
0.6
1.5
0.80
(d) 0.5
0.75
0.4
log (M phase duration)
log (G2 phase duration)
0.8
0.5
0
(c)
0.9
0.70
0.65
0.60
0.55
0.5
1.0
log (2C DNA)
1.5
0.5
1.0
log (2C DNA)
1.5
0.3
0.2
0.1
0
–0.1
0.50
0.5
1.0
log (2C DNA)
1.5
Figure 3. Relationships between duration of individual cell cycle phases and genome size: (a) G1 phase, (b) S phase (slope ¼
0.36), (c) G2 phase and (d) M phase (slope ¼ 0.40). Black line is the slope of SMA regression; grey lines are SMA 95%
confidence intervals. Regression results are listed in table 2, rows 10–13.
log (S phase duration)
1.0
0.9
0.8
0.7
0.6
0.5
–0.5
0
0.5
1.0
log (2C DNA)
1.5
Figure 4. The relationship between S phase and genome size
(slope ¼ 0.36). Black line is the slope of SMA regression;
grey lines are SMA 95% confidence intervals. Regression
results are listed in table 2, row 14.
not match numerically the value we predict (0.67), although
the difference was not significant owing to high variation
and the low number of observations.
Proc. R. Soc. B (2012)
5. DISCUSSION
We found strong support for the role of geometry in the
relationship between GS, nuclear size and cell size. Scaling exponents of these relationships almost exactly
matched expectations based on geometrical considerations. In the case of pollen volume, however, the
scaling exponent (1.16) was above the predicted value
of 1, although the confidence interval (0.96 –1.41) did
include this value. This was probably owing to variation
in the shape of pollen grains and the fact that an individual pollen grain could contain several nuclei. In addition,
their sizes can be affected by other factors, namely potential selection for the small sizes necessary for successful
transport to female flowers. All these factors are likely
to confound the relationship by adding unexplained variation. However, the fact that the size of pollen grain still
scales with GS despite such a large amount of unexplained variability, indicates that GS still acts as an
important constraint for the pollen grain size. Indeed,
the difference from the predicted value of 1 was not
significant when using the 95% confidence interval.
Most importantly, the relationships between GS and
both cell size and nuclear size are of simple isometry
with no nonlinear component. This indicates that size
relationships within cells are not constrained by transport,
as this process typically involves a mass or volume over
plane relationship, which is not isometric. However, our
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Geometry in genome size relationships
data show that nucleus : genome in a cell is constrained by
aspects of its cellular background, such as cytoskeletal
structure or ratio between rates of RNA synthesis and of
protein synthesis, that are simply proportional to its
size. While it has been pointed out earlier by many
authors that small cells cannot contain large genomes
[40,55], the fact of simple isometry shows that the reverse
is also true, i.e. large cells do not contain small genomes,
and this relationship is likely owing to a geometrical
constraint.
Although there was a strong relationship between GS
and cell cycle duration, it did not match our prediction
that was based on a single geometrical constraint. This
is clearly owing to differences in correlations between
GS and different cell cycle phases. Whereas there was
almost no relationship between GS and the duration
of growth phases (G1 and G2), M phase and particularly S phase scaled with GS rather strongly. Still none
of the relationships involved simple isometry, indicating
the strong role of transport or similar processes in
determining cell cycle duration.
The fact that cell growth phases (G1 and G2) were
independent of GS, and consequently, of cell size, indicates that the duration of phases of cell growth is not
limited by any transport process involving the nucleus.
These phases are probably driven by the rate of cell
metabolism, which is possibly maximized by number of
organelles increasing with GS [53]. In contrast with the
G phases, duration of the S phase increased with GS relatively strongly, with the scaling exponent corresponding
to the 1/3 predicted by the limitation of the DNApolymerase transport rate into the nucleus. Duration of
this part of the cell cycle thus seems to be limited by a
geometrical constraint on DNA replication. The role of
geometry in the M phase is unclear owing to the weak
relationship, with the estimated value not only differing
from the predicted value of 2/3 (limitation by membrane
or cell wall construction), but also weakly differing
from zero.
Taking into account the different results that we found
for individual cell cycle phases, the relationship between
GS and total cell cycle duration seems to consist of two
components. The first is the duration of the G phases
(which are virtually independent of the GS) in determining the cell cycle duration threshold. The second is the
role of the S (and possibly M) phases in determining
the slope of this relationship. GS thus plays a significant
role in the whole plant growth rate by increasing the
time needed for DNA polymerization. However, owing
to the different proportions of the cell cycle constituted
by particular cell cycle phases among different plants
[59,62], the slope of the scaling relationship between
GS and cell cycle duration is not universal. Its numerical
value may, however, be indicative of the relative time that
cells spend in the individual cell division phases.
It is known that a decrease in cell cycle duration is
associated with an increase in relative growth rate
[47,63], and thus indirectly with photosynthetic rate
[64] despite the fact that the relationship need not be
straightforward in a population of proliferating cells.
These relationships scale up to other plant traits [5] and
are fundamental for other biological rates. Cell size scaling has been much less studied in spite of its key role
in controlling organ or tissue complexity and thus
Proc. R. Soc. B (2012)
I. Šı́mová & T. Herben
873
whole-organism metabolic rate [52]. The separate scaling
relationships for cell cycle duration and cell size indicate
that their actions are likely to be independent.
While the geometrical scaling strongly indicates intimate functional ties between GS, nuclear volume, cell
size and cell cycle duration, it does not say anything
about the causal direction of the relationships (see also
[44]). In particular, it is not clear whether larger genomes
result only from passive accumulation of junk DNA, as is
commonly believed. In this scenario, the passive accumulation of junk DNA then constrains plant traits and life
histories [11]. However, this is not the only possible
basis for evolution of large genomes. For example, small
cells may be economically disadvantaged because of
their higher metabolic rates and consequently higher
energetic demands [65,66]. Increase in the GS may
thus be a possible option to diminish the energetic costs
of the cell [67] as large cells cannot have small genomes.
Moreover, organisms have been demonstrated to both
increase and decrease the GS in their cells by accumulation and deletion mechanisms (see [68] for review)
and to change the proportion of heterochromatin in
response to GS (data from Nagl [69]), suggesting possible
mechanisms to cope with excess DNA amounts.
In conclusion, the numerical fit between the predicted
and empirical scaling exponents provides strong support
for the concept of GS as a phenotypic trait (nucleotypic
theory of Bennett [3], etc.). It has also made it possible
to identify specific components of this limitation. In particular, size relationships and rate relationships are likely
owing to different limitations. At a finer scale, absence
of any relationship between GS and both G phases is a
strong indication that the surface area : volume ratio is
not limiting during cell growth, but does limit the S
phase and mitosis. These findings, although tentative
and concerning only undifferentiated meristematic cells,
show the power of the scaling relationships at the cellular
level in spite of the low number of data points available.
We are convinced that even finer analysis of geometrical
and mechanical constraints in the GS relationships will
be possible as additional comparative data are collected.
A similar approach might also be possible for examining
GS effects at tissue and whole plant levels.
We thank Matyáš Fendrych, Lukáš Kratochvı́l, Jonathan
Rosenthal, Zuzana Starostová, David Storch, Jan Suda,
Arnošt Šizling, Martin Weiser, Viktor Žárský, Hirokazu
Tsukaya and three anonymous reviewers for discussions
and/or comments on an earlier draft of this paper. The
research reported here was supported by research grants
MSM0021620845, MSM 0021620828, LC06073 from the
Czech Ministry of Education and AV0Z60050516 from the
Academy of Sciences of the Czech Republic.
REFERENCES
1 Bennett, M. D. & Leitch, I. J. 2010 Plant DNA C-values
database (release 5.0, December 2010). See http://www.
kew.org/cvalues/.
2 Pellicer, J., Fay, M. & Leitch, I. J. 2010 The largest
eukaryotic genome of them all? Bot. J. Linn. Soc. 164,
10–15. (doi:10.1111/j.1095-8339.2010.01072.x)
3 Bennett, M. D. 1987 Variation in genomic form in
plants and its ecological implications. New Phytol. 106,
177 –200. (doi:10.1111/j.1469-8137.1987.tb04689.x)
Downloaded from http://rspb.royalsocietypublishing.org/ on June 16, 2017
874 I. Šı́mová & T. Herben
Geometry in genome size relationships
4 Thompson, K. 1990 Genome size, seed size and germination temperature in herbaceous angiosperms. Evol.
Trends Plants 4, 113– 116.
5 Knight, C. & Beaulieu, J. 2008 Genome size scaling
through phenotype space. Ann. Bot. (Lond.) 101,
759– 766. (doi:10.1093/aob/mcm321)
6 Francis, D., Davies, M. & Barlow, P. 2008 A strong
nucleotypic effect on the cell cycle regardless of ploidy
level. Ann. Bot. (Lond.) 101, 747–757. (doi:10.1093/
aob/mcn038)
7 Hof, J. V. & Bjerknes, C. 1981 Similar replicon properties
of higher-plant cells with different S periods and genome
sizes. Exp. Cell Res. 136, 461 –465. (doi:10.1016/00144827(81)90027-6)
8 Bayliss, M. W. 1976 Variation of cell cycle duration
within suspension cultures of Daucus carota and its consequence for the induction of ploidy changes with
colchicine. Protoplasma 88, 279 –285. (doi:10.1007/
BF01283252)
9 Gong, J., Traganos, F. & Darzynkiewicz, Z. 1993 Simultaneous analysis of cell cycle kinetics at two different
DNA ploidy levels based on DNA content and cyclin B
measurements. Cancer Res. 53, 5096–5099.
10 Storchová, Z., Breneman, A., Cande, J., Dunn, J.,
Burbank, K., O’Toole, E. & Pellman, D. 2006
Genome-wide genetic analysis of polyploidy in yeast.
Nature 443, 541 –547. (doi:10.1038/nature05178)
11 Knight, C., Molinari, N. & Petrov, D. 2005 The large
genome constraint hypothesis: evolution, ecology and
phenotype. Ann. Bot. (Lond.) 95, 177–190. (doi:10.
1093/aob/rnci011)
12 Beaulieu, J., Leitch, I., Patel, S., Pendharkar, A. &
Knight, C. 2008 Genome size is a strong predictor of
cell size and stomatal density in angiosperms. New
Phytol. 179, 975 –986. (doi:10.1111/j.1469-8137.2008.
02528.x)
13 Knight, C. A., Clancy, R. B., Götzenberger, L., Dann, L. &
Beaulieu, J. M. 2010 On the relationship between pollen
size and genome size. J. Bot. 2010, 7. (doi:10.1155/2010/
612017)
14 Gruner, A., Hoverter, N., Smith, T. & Knight, C. A. 2010
Genome size is a strong predictor of root meristem growth
rate. J. Bot. 2010, 4. (doi:10.1155/2010/390414)
15 Leishman, M. 1999 How well do plant traits correlate with
establishment ability? Evidence from a study of 16 calcareous grassland species. New Phytol. 141, 487–496.
(doi:10.1046/j.1469-8137.1999.00354.x)
16 Hessen, D., Jeyasingh, P., Neiman, M. & Weider, L. 2010
Genome streamlining and the elemental costs of growth.
Trends Ecol. Evol. 25, 75–80. (doi:10.1016/j.tree.2009.
08.004)
17 Maranon, T. & Grubb, P. J. 1993 Physiological basis and
ecological significance of the seed size and relative growth
rate relationship in Mediterranean annuals. Funct. Ecol.
7, 591– 599. (doi:10.2307/2390136)
18 Knight, C. & Ackerly, D. 2002 Variation in nuclear DNA
content across environmental gradients: a quantile
regression analysis. Ecol. Lett. 5, 66–76. (doi:10.1046/j.
1461-0248.2002.00283.x)
19 Morgan, H. & Westoby, M. 2005 The relationship
between nuclear DNA content and leaf strategy in seed
plants. Ann. Bot. (Lond.) 96, 1321–1330. (doi:10.1093/
aob/mci284)
20 Suda, J., Kyncl, T. & Freiova, R. 2003 Nuclear DNA
amounts in Macaronesian angiosperms. Ann. Bot.
(Lond.) 92, 153 –164. (doi:10.1093/aob/mcg104)
21 Leitch, I. J. & Bennett, M. D. 2007 Genome size and its
uses: the impact of flow cytometry. In Flow cytometry with
plant cells (eds J. Dolezel, J. Greilhuber & J. Suda),
pp. 153 –176. Weinheim, Germany: Wiley-VCH.
Proc. R. Soc. B (2012)
22 Peters, R. H. 1986 The ecological implications of body size.
Cambridge, UK: Cambridge University Press.
23 Connor, E. & McCoy, E. 1979 Statistics and biology of
the species-area relationship. Am. Nat. 113, 791 –833.
(doi:10.1086/283438)
24 Martinez, N. 1992 Constant connectance in community
food webs. Am. Nat. 139, 1208 –1218. (doi:10.1086/
285382)
25 Reich, P., Oleksyn, J., Wright, I., Niklas, K., Hedin, L. &
Elser, J. 2010 Evidence of a general 2/3-power law of
scaling leaf nitrogen to phosphorus among major plant
groups and biomes. Proc. R. Soc. B 277, 877 –883.
(doi:10.1098/rspb.2009.1818)
26 Tilman, D. 1999 The ecological consequences of
changes in biodiversity: a search for general principles.
Ecology 80, 1455–1474.
27 Niklas, K. 1994 The scaling of plant and animal bodymass, length and diameter. Evolution 48, 44–54.
(doi:10.2307/2410002)
28 West, G., Brown, J. & Enquist, B. 1997 A general model
for the origin of allometric scaling laws in biology. Science
276, 122 –126. (doi:10.1126/science.276.5309.122)
29 Moses, M. & Brown, J. 2003 Allometry of human fertility
and energy use. Ecol. Lett. 6, 295 –300. (doi:10.1046/j.
1461-0248.2003.00446.x)
30 Savage, V., Gillooly, J., Woodruff, W., West, G., Allen, A.,
Enquist, B. & Brown, J. 2004 The predominance of
quarter-power scaling in biology. Funct. Ecol. 18,
257 –282. (doi:10.1111/j.0269-8463.2004.00856.x)
31 Reich, P., Tjoelker, M., Machado, J. & Oleksyn, J. 2006
Universal scaling of respiratory metabolism, size and
nitrogen in plants. Nature 439, 457– 461. (doi:10.1038/
nature04282)
32 Brown, J. & West, G. 2001 Scaling in biology. Oxford,
UK: Oxford University Press.
33 Price, C. & Enquist, B. 2007 Scaling mass and morphology in leaves: an extension of the WBE model.
Ecology 88, 1132–1141. (doi:10.1890/06-1158)
34 Enquist, B., Kerkhoff, A., Stark, S., Swenson, N.,
McCarthy, M. & Price, C. 2007 A general integrative
model for scaling plant growth, carbon flux and functional trait spectra. Nature 449, 218–222. (doi:10.1038/
nature06061)
35 Milla, R. & Reich, P. 2007 The scaling of leaf area and
mass: the cost of light interception increases with leaf
size. Proc. R. Soc. B 274, 2109–2114. (doi:10.1098/
rspb.2007.0417)
36 Niklas, K., Cobb, E., Niinemets, U., Reich, P., Sellin, A.,
Shipley, B. & Wright, I. 2007 ‘Diminishing returns’ in
the scaling of functional leaf traits across and within
species groups. Proc. Natl Acad. Sci. USA 104, 8891 –
8896. (doi:10.1073/pnas.0701135104)
37 West, G., Enquist, B. & Brown, J. 2009 A general quantitative theory of forest structure and dynamics. Proc. Natl
Acad. Sci. USA 106, 7040–7045. (doi:10.1073/pnas.
0812294106)
38 Mori, S. et al. 2010 Mixed-power scaling of whole-plant
respiration from seedlings to giant trees. Proc. Natl Acad.
Sci. USA 107, 1447–1451. (doi:10.1073/pnas.09025
54107)
39 Baetcke, K., Sparrow, A., Nauman, C. & Schwemme,
S. S. 1967 Relationship of DNA content to nuclear and
chromosome volumes and to radiosensitivity (LD50).
Proc. Natl Acad. Sci. USA 58, 533– 540. (doi:10.1073/
pnas.58.2.533)
40 Gregory, T. 2001 Coincidence, coevolution, or causation?
DNA content, cell size, and the C-value enigma. Biol.
Rev. 76, 65–101. (doi:10.1017/S1464793100005595)
41 Price, H., Sparrow, A. & Nauman, A. 1973 Correlations
between nuclear volume, cell volume and DNA content
Downloaded from http://rspb.royalsocietypublishing.org/ on June 16, 2017
Geometry in genome size relationships
42
43
44
45
46
47
48
49
50
51
52
53
54
55
in meristematic cells of herbaceous angiosperms.
Experientia 29, 1028–1029. (doi:10.1007/BF01930444)
Cavalier-Smith, T. 1978 Nuclear volume control by
nucleoskeletal DNA, selection for cell volume and cell
growth rate, and the solution of the DNA C-value
paradox. J. Cell Sci. 34, 247.
Smith, R. 2009 Use and misuse of the reduced major axis
for line-fitting. Am. J. Phys. Anthropol. 140, 476–486.
(doi:10.1002/ajpa.21090)
Hodgson, J. et al. 2010 Stomatal vs. genome size in angiosperms: the somatic tail wagging the genomic dog? Ann. Bot.
(Lond.) 105, 573–584. (doi:10.1093/aob/mcq011)
Jolicoeur, P. 1975 Linear regressions in fishery research:
come comments. J. Fish. Res. Board Can. 32, 1491–
1494. (doi:10.1139/f75-171)
Legendre, P. & Legendre, L. 1998 Numerical ecology.
Amsterdam, The Netherlands: Elsevier Science.
Beemster, G., Vusser, K., De Tavernier, E., De Bock, K. &
De Inze, D. 2002 Variation in growth rate between
Arabidopsis ecotypes is correlated with cell division and
A-type cyclin-dependent kinase activity. Plant Physiol.
129, 854–864. (doi:10.1104/pp.002923)
Melaragno, J. E., Mehrotra, B. & Coleman, A. W. 1993
Relationship between endopolyploidy and cell size in epidermal tissue of Arabidopsis. Plant Cell 5, 1661 –1668.
(doi:10.1105/tpc.5.11.1661)
Jovtchev, G., Schubert, V., Meister, A., Barow, M. &
Schubert, I. 2006 Nuclear DNA content and nuclear
and cell volume are positively correlated in angiosperms.
Genome Res. 114, 77–82. (doi:10.1159/000091932)
Gray, J., Kolesik, P., Hoj, P. & Coombe, B. 1999 Confocal measurement of the three-dimensional size and shape
of plant parenchyma cells in a developing fruit tissue.
Plant J. 19, 229–236. (doi:10.1046/j.1365-313X.1999.
00512.x)
Cavalier-Smith, T. 2005 Economy, speed and size
matter: evolutionary forces driving nuclear genome
miniaturization and expansion. Ann. Bot. (Lond.) 95,
147 –175. (doi:10.1093/aob/mci010)
Kozlowski, J., Konarzewski, M. & Gawelczyk, A.
2003 Cell size as a link between noncoding DNA and
metabolic rate scaling. Proc. Natl Acad. Sci. USA 100,
14 080 –14 085. (doi:10.1073/pnas.2334605100)
DeLong, J., Okie, J., Moses, M., Sibly, R. & Brown, J.
2010 Shifts in metabolic scaling, production and efficiency across major evolutionary transitions of life. Proc.
Natl Acad. Sci. USA 107, 12 941 –12 945. (doi:10.1073/
pnas.1007783107)
Méchali, M. 2010 Eukaryotic DNA replication origins:
many choices for appropriate answers. Nat. Rev. Mol.
Cell. Biol. 11, 728 –738. (doi:10.1038/nrm2976)
Bennett, M. D. 1972 Nuclear DNA content and
minimum generation time in herbaceous plants.
Proc. R. Soc. B (2012)
56
57
58
59
60
61
62
63
64
65
66
67
68
69
I. Šı́mová & T. Herben
875
Proc. R. Soc. Lond. B 181, 109–135. (doi:10.1098/rspb.
1972.0042)
Kuroki, Y. & Tanaka, R. 1973 Duration of mitotic cell
cycle in root tip cells of male Rumex acetosa.
Jpn. J. Genet. 48, 19–26. (doi:10.1266/jjg.48.19)
Powell, M., Davies, M. & Francis, D. 1986 The influence
of zinc on the cell cycle in the root meristem of a zinc-tolerant and zoinc-nontolerant cultivar of Festuca rubra L.
New Phytol. 102, 419– 428. (doi:10.1111/j.1469-8137.
1986.tb00819.x)
King, P. 1980 Plant tissue culture and the cell cycle. Adv.
Biomed. Eng. 18, 1 –38.
Barlow, P. 1973 Mitotic cycles in root meristems. In The
cell cycle in development and differentiation (eds M. Balls &
F. S. Billet), pp. 133 –165. Cambridge, UK: British
Society for Developmental Biology Symposium.
R Development Core Team. 2010 R: a language and
environment for statistical computing. Vienna, Austria: R
Foundation for Statistical Computing. See http://www.
R-project.org/.
Ebert, T. & Russell, M. 1994 Allometry and model II
nonlinear regression. J. Theor. Biol. 168, 367 –372.
(doi:10.1006/jtbi.1994.1116)
Francis, D. & Barlow, P. 1988 Temperature and the cell
cycle. In Plants and temperature (eds S. P. Long & F. I.
Woodward), pp. 181–201. Cambridge, U: Company of
Biologists.
Ivanov, V. & Dubrovsky, J. 1997 Estimation of the cellcycle duration in the root apical meristem: a model of
linkage between cell-cycle duration, rate of cell production, and rate of root growth. Int. J. Plant Sci. 158,
757 –763. (doi:10.1086/297487)
Garnier, E. 1991 Resource capture, biomass allocation
and growth in herbaceous plants. Trends Ecol. Evol. 6,
126 –131. (doi:10.1016/0169-5347(91)90091-B)
Vinogradov, A. 1995 Nucleotypic effect in homeotherms:
body-mass-corrected basal metabolic rate of mammals is
related to genome size. Evolution 49, 1249– 1259.
(doi:10.2307/2410449)
Starostova, Z., Kratochvil, L. & Frynta, D. 2005 Dwarf
and giant geckos from the cellular perspective: the
bigger the animal, the bigger its erythrocytes? Funct.
Ecol. 19, 744– 749. (doi:10.1111/j.1365-2435.2005.
01020.x)
Vinogradov, A., Anatskaya, O. & Kudryavtsev, B. 2001
Relationship of hepatocyte ploidy levels with body size
and growth rate in mammals. Genome 44, 350–360.
(doi:10.1139/g01-015)
Gregory, T. 2005 The evolution of the genome. San Diego,
CA: Elsevier.
Nagl, W. 1974 Mitotic cycle time in perennial and annual
plants with various amounts of DNA and heterochromatin. Dev. Biol. 39, 342 –346.