RESEARCH ARTICLE
Cell length growth in fission yeast: an analysis of its bilinear
character and the nature of its rate change transition
th1, Anna Ra
cz-Mo
nus1, Peter Buchwald2 & Ákos Sveiczer1
Anna Horva
1
Department of Applied Biotechnology and Food Science, Budapest University of Technology and Economics, Budapest, Hungary; and
Department of Molecular and Cellular Pharmacology and Diabetes Research Institute, Miller School of Medicine, University of Miami, Miami, FL,
USA
2
Correspondence: Akos
Sveiczer,
Department of Applied Biotechnology and
Food Science, Budapest University of
Technology and Economics, Szent Gellert ter
4., Budapest, H-1111, Hungary.
Tel.: +36 1 463 2349;
fax: +36 1 463 3855;
e-mail: [email protected]
Received 29 March 2013; revised 20 June
2013; accepted 4 July 2013.
Final version published online 1 August 2013.
DOI: 10.1111/1567-1364.12064
Editor: Jens Nielsen
Keywords
fission yeast; time-lapse microscopy; cell
growth pattern; model fitting; bilinear
function.
Abstract
During their mitotic cycle, cylindrical fission yeast cells grow exclusively at
their tips. Length growth starts at birth and halts at mitotic onset when the
cells begin to prepare for division. While the growth pattern was initially considered to be exponential, during the last three decades an increasing amount
of evidence indicated that it is rather a bilinear function [two linear segments
separated by a rate change point (RCP)]. The main focus of this work was to
clarify this and to elucidate the further question of whether the rate change
occurs abruptly at the RCP or more smoothly during a transition period
around it. We have analyzed the individual growth patterns obtained by timelapse microscopy of 60 wild-type cells separately as well as that of the ‘average’
cell generated from their superposition. Linear, exponential, and bilinear functions were fitted to the data, and their suitability was compared using objective
model selection criteria. This analysis found the overwhelming majority of the
cells (70%) to have a bilinear growth pattern with close to half of them showing a smooth and not an abrupt transition. The growth pattern of the average
cell was also found to be bilinear with a smooth transition.
YEAST RESEARCH
Introduction
Mitotically proliferating unicellular eukaryotes generally
grow between two consecutive cell divisions to maintain a
size homeostasis in successive generations. At population
level, growth means a doubling in size; however, at cellular level, a critical size should be reached before cytokinesis. Therefore, growth and division must be strictly
connected to each other (Alberts et al., 2009; Sveiczer &
Racz-M
onus, 2013). The time profile of size increase is a
fundamental problem as linear growth is thought to support homeostasis, whereas exponential growth is rather
thought to operate against it. In the latter case, more
stringent control mechanisms are required to maintain
constancy of cell size (Fantes & Nurse, 1981). The theoretical justification used for the exponential growth model
is that larger cells have larger synthetic capacities, for
example, they contain more ribosomes, which can
produce more proteins (Mitchison, 2003). For this model,
it is very easy to formulate a corresponding ordinary
FEMS Yeast Res 13 (2013) 635–649
differential equation where the mass increase rate is proportional to the actual cell mass leading to an exponential
function as a solution. The problem with this hypothesis
is that it totally neglects all the other aspects of cellular
architecture and physiology. For example, growth in space
requires cytoskeletal polymerization, and it seems to be
unrealistic that either microtubules or actin filaments
could grow exponentially. In yeast cells, their cell wall
(together with the plasma membrane) should also expand
exponentially to support the exponential growth of the
total mass. These problems raise the possibility that overall growth may also follow a linear function in time.
Moreover, linear growth not necessarily means a constant
growth rate, because at specific cell cycle points the
growth rate may change resulting in bilinear (or more
generally, multilinear) growth patterns. Such profiles of
consecutive linear segments have (at least) one rate
change point (RCP), which is assumed to be regulated by
the cell cycle itself (Mitchison, 2003). As a consequence,
several cells of different model organisms have been
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636
intensively studied during the last three decades to elucidate the general rules of cellular growth; nevertheless, several questions open for debates still remain in this field.
Studying the growth of the very small (1–3 lm) rodshaped prokaryotic cells of Escherichia coli is rather problematic. Recent exact measurements by phase contrast
and fluorescent microscopy indicated that these Gramnegative rods grow in length bilinearly or trilinearly
rather than exponentially (Reshes et al., 2008). The
eukaryotic cells of the budding yeast (Saccharomyces cerevisiae) are much larger, but less symmetric. A sophisticated microfluidic technique and image analysis was
recently developed to characterize cell volume as a function of time (Goranov et al., 2009; Bryan et al., 2010).
The growth pattern of cell size was found to be tetralinear: four linear segments separated by three RCPs. Progression during the cycle was found to be the direct cause
of these rate changes as both bud formation and mitotic
onset seemed to decrease growth rate, while late mitotic
events accelerated it (Goranov et al., 2009). In mammalian cells, the first question to decide is which species and
which cell type(s) should be used as models. Usually individually growing cells, which are not affected by any tissue control, are studied. Cell volume is generally
determined by a Coulter Counter, as cell symmetry is
poor, not allowing the calculation of cell size by direct
microscopic measurements. Rat Schwann cells were found
to grow linearly, that is, independently of their size (Conlon & Raff, 2003). By contrast, mouse lymphoblastoid
cells’ size had neither a simple exponential nor a linear
pattern in time, but their growth rate was definitely size
dependent (Tzur et al., 2009).
The fission yeast Schizosaccharomyces pombe has been
an attractive model organism in research studies on cellular morphogenesis ever since the 1950s. Its cylindrical
cells have a constant diameter and grow exclusively at
their tips so that cell volume is essentially proportional to
cell length (Sveiczer & Horvath, 2013). Cell populations
can simply be grown on an agar surface making possible
the real-time measurement of cell length in a microscopic
field. From mitotic onset until cytokinesis, that is, during
the last c. 25% of the cycle, the cells practically cannot
grow in length, a period that is called the constant volume (or constant length) phase (Mitchison, 1957). Initially, it was thought that during the first c. 75% of the
cycle the growth pattern was exponential. However, in
the 1980s, new measurements made by Mitchison and
Nurse suggested a bilinear growth pattern with an RCP
(Mitchison & Nurse, 1985). This rate change was thought
to be connected to the so-called new end take-off
(NETO) event when the formerly unipolar growth
changes to a bipolar one. This point is generally considered as a specific and important one during the fission
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A. Horvath et al.
yeast cell cycle because, following division, newborn cells
grow only at their old ends (which already existed in the
previous cycle), and they only initiate growth at the new
ends (which were generated during the last division) at
NETO. It was argued that NETO occurred in interphase
after the cell had fulfilled two requirements: finished
DNA replication and reached a critical size of about 9.0–
9.5 lm (Mitchison & Nurse, 1985). However, it was also
challenged in different studies, whether RCP and NETO
always coincided or not (Mitchison & Nurse, 1985; Sveiczer et al., 1996; Baumgartner & Tolic-Norrelykke, 2009).
In the mid-1990s, we have confirmed that fission yeast
cells grew bilinearly and extended the former wild-type
(WT) analyses to several cell cycle mutants (Sveiczer
et al., 1996). It was established that in wee1 mutants, an
RCP was connected strictly to DNA replication (or at
least to its cycle position), and it was sharper and stronger than that of WT cells. In the meantime, Cooper
refused using bilinear functions for cellular growth (Cooper, 1998). He argued that an exponential model fitted
the data nearly as well as the bilinear one, and invoked
Occam’s razor as a principle to conclude that the simpler
model must be applied as the bilinear model has four
parameters, whereas the exponential one has only two.
Another 10 years later, we have developed a continuously
differentiable, linearized, biexponential (LinBiExp) model
to describe length growth in fission yeast (Buchwald &
Sveiczer, 2006). This is a general, five-parameter bilinear
model with two linear segments separated by a smooth
transition section. Before the introduction of this model,
bilinear models were assumed to have a sharp transition
at RCP, such models required four-parameter functions.
In the same work, we have also analyzed in detail model
selection criteria and clarified that when differently
parameterized models are applied to the same experimental data, the adequacy of models cannot be judged solely
on the basis of the sum of squared errors (SSE) or correlation coefficient (r2). Instead, quantitative model selection criteria [like the Akaike information criterion (AIC;
Akaike, 1974)] should be used to find the most adequate
model. This work established that, at least in the case of
the one analyzed WT cell, the smooth bilinear model was
somewhat more adequate than the exponential model
(Buchwald & Sveiczer, 2006). In the case of the one analyzed wee1 mutant cell, the bilinear model was much
more adequate by all criteria used, and it also showed a
more abrupt change.
To increase the spatial precision and temporal resolution of cell length measurements, recently, Baumgartner
& Tolic-Norrelykke (2009) labeled plasma membrane
proteins of WT fission yeast fluorescently and studied the
cells using confocal microscopy. They analyzed several
individual cells or rather their average for the most
FEMS Yeast Res 13 (2013) 635–649
Cell length growth in fission yeast
important part of their growing phase (10–100 min).
Using the same model selection criteria as Buchwald &
Sveiczer (2006), they concluded that a bilinear model
with an abrupt change was the most adequate one
(Baumgartner & Tolic-Norrelykke, 2009). Another novel
method of studying cell size increase is monitoring dry
mass by digital holographic microscopy (Rappaz et al.,
2009). Using this approach, Rappaz et al. found both linear and bilinear patterns in dry mass growth in different
fission yeast cells. Very recent microscopic analyses discriminated the two poles’ extensions and suggested that
in WT S. pombe cells, length growth at the old end is linear, whereas it follows a bilinear pattern at the new end
(Das et al., 2012).
Nevertheless, the debates on the exact nature of growth
profiles have not been closed yet, neither in fission yeast
nor in other cell types. As fluorescent intensities seem to
fluctuate considerably during consecutive measurements
of the same cell (Baumgartner & Tolic-Norrelykke, 2009),
we went back to our previously used technique. Here, we
report new measurements of WT fission yeast cells on
microscopic films made by late Prof. Murdoch Mitchison
(University of Edinburgh, UK) at a higher final magnification than previously (Sveiczer et al., 1996). We have fitted linear, exponential, and bilinear functions on these
growth profiles and used rigorous model selection criteria
to find the most adequate model as described by us previously (Buchwald & Sveiczer, 2006). Here, we analyze 60
individual cells’ growth patterns, meanwhile the method
was formerly developed for only one wild type and one
wee1 mutant cell (Buchwald & Sveiczer, 2006). In addition to the analyses of individual cells, an ‘average’ cell
was also generated, and its growth profile was analyzed.
Materials and methods
Film techniques and data analysis
The WT strain 972 h of S. pombe was originally
obtained from Prof. Urs Leupold (University of Bern,
Switzerland). The time-lapse films were made by Prof.
Murdoch Mitchison (University of Edinburgh, UK) as
described previously (Sveiczer et al., 1996). Before filming, the culture was grown overnight at 35 °C in EMM3
minimal medium, up to c. 2 9 106 cells mL1. Then, the
cells were growing at 35 °C in EMM3 minimal medium
between a coverslip and a pad of nutrient agar. Filming
started about 1 h after dropping the yeast suspension on
the pad. The photographs were taken with a Zeiss Photomicroscope with a Planapochromat objective 910 (NA
0.32) and a darkground condenser using an automatic
timer to take a frame every 5 min for up to 6 h. The
films were stored in a cool and dark place since 1996,
FEMS Yeast Res 13 (2013) 635–649
637
without any visible reduction in their quality. The negatives were later projected onto a screen. There were two
consecutive generations filmed on the screens, and the
distribution of both the cycle time (CT) and division
length (DL) proved that the culture was in the exponential phase of microbial proliferation (Sveiczer et al.,
1996). Thirty sister pairs of the first generation were
selected for the study, partly overlapping with those
examined in the previous work. The lengths of these 60
individual cells were measured in every frame from birth
to division to obtain growth patterns. Compared to our
former work (Sveiczer et al., 1996), we used a higher final
magnification of about 2150. This allowed the reduction
of the uncertainty of cell length measurements (i.e. optical resolution) to c. 0.23 lm. It is noteworthy that the
micrometer scale was also magnified c. 2150-fold to
obtain the exact cell length data, and the contrast of the
measured cells was still very good at this magnification.
To obtain more uniform profiles, the patterns were
smoothed using the resistant smooth (rsmooth) procedure
of MINITAB 14.13 (Minitab, State College, PA) using the
default 4235H, twice method, similar to the original publication (Sveiczer et al., 1996). The significance of the
smoothing process was formerly discussed in the study by
Buchwald & Sveiczer, 2006; see also the next section
(Model fitting). In these smoothed patterns, the time
points and the cell lengths corresponding to the onset of
the constant length period at the end of the cycle were
determined by eye. This constant length period was omitted from the pattern during model fitting, that is, only
the growing period of the cell cycle was studied.
The growth pattern of the ‘average cell’ was generated
from the raw measurement data, however, differently
from that of Baumgartner & Tolic-Norrelykke (2009).
Because we believe that superposition of normalized data
is a more realistic method to form the average cell than
using original untransformed data (as it allows the use of
the entire cycle for all of the cells), single cell patterns
were first normalized both in time and cell length; that is,
time data were divided by the actual cell’s CT and cell
length data were divided by the actual cell’s birth length
(BL). Consequently, the relative time scale was between 0
and 1, and the cell started its cycle with a relative length
of 1. Relative cell lengths were determined at a time resolution of 0.025 (relative CT) on every single cell pattern
by linear interpolation between the two nearest points,
and these relative cell lengths were averaged afterward.
Finally, this normalized average pattern was renormalized
by multiplying the relative time data by the mean CT of
142 min, and the relative length data were multiplied by
the mean BL of 7.7 lm, data that are the average values
of the 60 measured individual cells (see later in Results
and Table 2). In this case, smoothing was not applied.
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Published by John Wiley & Sons Ltd. All rights reserved
638
Model fitting
Model fitting was performed for the growth patterns with
three types of mathematical functions: linear, exponential,
and bilinear ones – all giving cell length (L) as a function
of time (t) during the growing phase of a cell. The linear
and exponential models have two parameters and are
given as L = ct + d, and L = jelt, respectively. The
bilinear model has five parameters and, therefore, a more
complex form: L ¼ gln½ea1 ðtsÞ=g þ ea2 ðtsÞ=g þ e. It is also
called a LinBiExp model, as it is a sum of two exponentials, linearized by the natural logarithm (ln) function. As
discussed before (Buchwald & Sveiczer, 2006; Buchwald,
2007), this function describes a model having two linear
segments with slopes a1 and a2 that are separated by a
RCP positioned at time s during the cell cycle. The transition between the linear segments at RCP can also be
smooth and not sharp, that is, a transition period is situated around the RCP with a width determined by the g
parameter. It is worth mentioning that this LinBiExp
model converges to a sharp bilinear function if g ? 0;
therefore, such a sharp bilinear expression was not fitted
separately to the data. However, if g is extremely small,
numerical problems might occur in calculating the exponential terms depending on the computer and the software used for these calculations. Depending on the actual
cell length data, in certain cases, this required imposing a
lower limit on the value of g; the value used here
(gmin = 0.01 lm) corresponds to a minimum transition
period of c. 1 min (also depending on the growth rates,
a1 and a2, see Appendix), far below the time spent
between the acquisition of two successive frames (5 min).
To avoid too slow transitions and to have actual linear
segments both before and after the RCP, an upper limit of
gmax = 0.5 lm was also imposed for g, this corresponds
to a maximum transition period of c. 50 min. The fifth
parameter of the model (e) is an additive constant, which
is not the intercept of the bilinear function, but represents
an approximate value of the fitted cell length at RCP (at
t = s, L = gln2 + e, and because g e, L e).
The linear, exponential, and bilinear (LinBiExp) fittings
were all performed using an Excel (Microsoft, Seattle,
WA) worksheet and its solver function to estimate the
parameter values of the models. Because LinBiExp uses a
smooth, continuously differentiable functional form, the
optimization process is relatively trouble-free; nevertheless, sufficient care is recommended to verify that a true
and not just a local optimization minimum for SSE is
reached (i.e. start with different initial parameter values
from both sides of the final values). It is important to
emphasize here that with unsmoothed data the algorithm
often reaches a local minimum, because raw growth patterns often contain several sharp shifts, as our preliminary
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A. Horvath et al.
analyses clearly showed it. The real growth of the cell
probably cannot be as rough as the raw data indicates. In
light of these, we concluded that smoothing is an important step in our study. We have also tried to ‘simulate’
our raw length growth patterns from a theoretical bilinear
pattern, considering the optical resolution of our measurements. The most adequate model fitted to such a
simulated raw pattern was wrongly an exponential one;
however, smoothing restored the original bilinearity of
the pattern with similar parameters (data not shown).
The reverse simulation, that is, starting from a theoretical
exponential function, however, was correct without any
changes during the process (data not shown). It is not
surprising that after these mathematical transformations,
a model with less parameter will be favored by the selection criteria, which is the exponential one. Therefore, if a
smoothed pattern of a cell is found to be bilinear, then it
is highly probable that the cell originally grew bilinearly
rather than exponentially. By contrast, if an exponential
model is favored for a cell’s length pattern, then we cannot rule out the possibility that it is an artifact generated
by a low resolution discrete measurement of a continuous
function. This means that we probably overestimate the
relative frequency of exponential patterns at the expense
of bilinear ones; however, this was not a serious problem
because of the low number of exponential patterns found
(see also Results).
Model selection criteria and statistical analyses
Because the various models discussed here use different
numbers of parameters (npar), it is not sufficient to simply rely on the squared correlation coefficient r2, which
can be simply calculated from SSE and is independent of
the number of parameters of the model. Instead, further
discriminations between rival models (model selection
criteria) are needed. Improvement in the residual standard deviation (s) is a first possibility, as it accounts at
least in part for the change in the degrees of freedom,
d.f. = nobs npar, as s = (SSE/d.f.)1/2. Here, nobs
represents the number of observations and npar the
number of parameters in the model. More accurate
indicators include, for example, the AIC, the Schwarz
Bayesian information criterion (SBIC), and others (Buchwald & Sveiczer, 2006). AIC can be calculated as
AIC = nobsln(SSE) + 2npar, meanwhile SBIC is given as
SBIC = nobsln(SSE) + nparln(nobs). They both attempt to
quantify the information content of a given set of parameter estimates by relating SSE to the number of parameters required to obtain the fit. The model associated with
smaller values of AIC and SBIC is more appropriate, and,
as shown by their definitions, SBIC is a more restrictive
criterion on increasing npar (Buchwald & Sveiczer, 2006).
FEMS Yeast Res 13 (2013) 635–649
639
Cell length growth in fission yeast
To decide which model is the most adequate to describe
the growth pattern of an individual fission yeast cell, we
discriminated first between linear and exponential functions using r2 (or SSE). The better model was than compared to LinBiExp using s, AIC, and SBIC. In some cases,
these three criteria did not favor uniformly the same
model; in such cases, the final decision was based on
AIC, because AIC correlated with a t-test much better
than either SBIC or s (for a detailed explanation, see
Results).
For individual cells whose growth pattern was found to
be bilinear by any of the three criteria used (s, AIC and
SBIC), further statistical analyses were performed. The
growing period was divided into three phases: the first linear phase, the (curved) transition phase, and the second
linear phase. A detailed mathematical description on how
this separation was done is included in the Appendix. The
transition phase was then omitted from the growing period, and the slopes of the first and second linear phases
were determined by linear regression and compared to
each other by a t-test. These statistical comparisons
(homogeneity of slopes) were performed at a significance
level of 0.05 using STATISTICA 9.0 (StatSoft, Tulsa, OK).
Results
Cell length growth in fission yeast is bilinear in
most cells
We have analyzed the growth patterns of 60 individual
WT fission yeast cells. For each cell, the experimentally
measured and smoothed cell length data were plotted
versus time and fitted with the growth models (linear,
exponential and bilinear) to estimate their parameters, and
the model selection criteria (SSE, s, AIC and SBIC) were
calculated. SSE was used first to discriminate between the
two models having two parameters (linear vs. exponential). The linear model gave better fit in 19 cases (32%),
while the remaining 41 cases (68%) were better fitted by
an exponential model. Next, the more sophisticated model
selection criteria were used to discriminate between the
differently parameterized models as described in the Methods section. Most of the cell length growth patterns were
found to be bilinear, namely 70% according to the AIC
(Table 1). The bilinear pattern was also favored in more
than half of the cases by the SBIC, which is more stringent
than AIC (i.e. it favors more strongly the least parameterized model). The highest ratio (80%) of bilinear pattern
was found according to the s criterion, which is the least
stringent in regard to model parameter number among
the three criteria used here. A bilinear pattern can be
divided into two growing periods of constant growth rates
separated by a RCP: a slower growth in the first part folFEMS Yeast Res 13 (2013) 635–649
Table 1. Distribution of length growth patterns in the analyzed cells
as judged by different model selection criteria
AIC
N
s
SBIC
%
N
%
N
%
Bilinear
Linear
Exponential
42
13
5
70.0
21.7
8.3
32
17
11
53.3
28.3
18.3
48
9
3
80.0
15.0
5.0
Total
60
100.0
60
100.0
60
100.0
Data represent number (N) and percent (%) of cells with the corresponding growth pattern (bilinear, linear or exponential) as judged on
the basis of the model selection criteria indicated in the header (AIC,
SBIC and s).
lowed by faster growth in the second part (sometimes after
a transition part of accelerating growth; Figs 1 and 2).
A considerable proportion of cells were found to have a
linear length growth pattern (about 22% according to
AIC); an illustrative example is shown in Fig. 3. In these
cases, the growth pattern can obviously be considered as
one single period with the same growth rate. Among the
observed cells, there were only five growing exponentially
(one of them is shown in Fig. 4) according to the AIC criterion used, which represents a very low fraction of the
total sample cells (c. 8%). As discussed in the Model fitting section (Methods), one cannot be certain that these
cells really grew exponentially as the original bilinear
growth function might easily (and mistakenly) be transferred into an exponential one via the low resolution measurements and the mathematical transformations. On the
basis of these data, we conclude that length growth in fission yeast is mainly upward curved (i.e. there is a tendency
for the growth rate to increase during the growth cycle).
Accordingly, an exponential model is favored over a linear
one (at least in c. 70% of the studied cells); however, in
the majority of the cases, a bilinear function seems to be
even more adequate than an exponential one indicating
that the growth rate does not increase continuously, but
there are two periods of relatively constant growth rate
with a slower first phase followed by a faster second one.
The BL, DL, and CT of the 60 studied individual cells
are summarized in Table 2, together with their mean values standard deviations. In the case of bilinear patterns, the parameter values of the best-fitted bilinear
functions are also shown. The strength of the RCP in
these cases can be characterized by the ratio a2/a1. Since
the classic paper of Mitchison & Nurse (1985), different
studies have found this ratio to be about 1.3. However,
this rule seems to be valid for an average cell, but not
necessarily for individual ones. In our 42 bilinearly growing cells, this ratio is between 1.11 and 2.29 with a mean
value of 1.63 (Table 2). The discrepancy is due to the fact
that a fraction of cells actually grow linearly reducing the
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A. Horvath et al.
640
14
α1 = 0.0429 μm min–1
α2 = 0.0902 μm min–1
ε = 9.84 μm
τ = 57.6 min
η = 0.01 μm
13
0.2
R (μm)
Cell length (µm)
12
11
10
0.0
–0.2
0
20
40
60
80
100 120 140
Time (min)
9
RCP
Growth period I
Growth period II
Constant length period
Bilinear
Exponential
Linear
8
7
6
0
20
40
60
80
100
120
140
Time (min)
Fig. 1. Illustrative growth profile of an
individual fission yeast cell corresponding to a
bilinear model with abrupt (sharp) transition
(cell no. 52.1 in Table 2). Experimental cell
length is shown as a function of time
(symbols) together with the three different
fitted models as indicated (differently colored
lines). The optical resolution of cell length was
about 0.23 lm during the measurements, but
the patterns were smoothed before model
fitting (see text). Parameters for the most
adequate model (bilinear, green line) are given
in the inset above the graph. Note that
practically there is no transition period
between the two linear growing periods of
constant rate. Residuals for the most adequate
model (bilinear) are shown in the right side
inset.
16
α1 = 0.0413 μm min–1
15
α2 = 0.0782 μm min–1
ε = 10.2 μm
τ = 59.7 min
η = 0.500 μm
14
0.2
R (μm)
Cell length (μm)
13
12
11
0.0
–0.2
0
20
40
60
80 100 120 140 160
Time (min)
10
Growth period I
Transition period
Growth period II
Constant length period
Bilinear
Exponential
Linear
RCP
9
8
7
6
0
20
40
60
80
100
120
Time (min)
strength of the average RCP observed (see also the discussion of the average cell’s pattern, later).
Bilinear patterns often have a smooth and not
an abrupt, sharp transition
For cells whose growth pattern was found to be bilinear, a
further question arises regarding the nature of the transition between the two linear segments: is it sharp (abrupt)
or smooth? In other words, does it have a clear abrupt
RCP or a slower curved transition, respectively? This is a
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140
160
Fig. 2. Illustrative growth profile of an
individual fission yeast cell corresponding to a
bilinear model with smooth transition (cell no.
54.1 in Table 2). Experimental cell length is
shown as a function of time (symbols)
together with the three different fitted models
as indicated (differently colored lines). The
optical resolution of cell length was about
0.23 lm during the measurements, but the
patterns were smoothed before model fitting
(see text). Parameters for the most adequate
model (bilinear, green line) are given in the
inset above the graph. Note that there is a
transition period of 23 min between the two
linear growing periods of constant rate.
Residuals for the most adequate model
(bilinear) are shown in the right side inset.
somewhat arbitrary decision (i.e. exactly when does the
transition period start and end, plus how narrow does it
have to be to be considered an ‘abrupt’ transition), but to
make this as objective as possible, in the present work, we
set up clear quantitative rules as described in details in the
Appendix. The transition periods (ttr) of the 42 cells
judged to have a bilinear pattern were in the 0.4–46.7 min
range with an average of 8.3 min (Table 2). It would be
rather difficult to estimate the error of the ttr in the case
of any of these individual cells, which could be calculated
from errors of bilinear model fitting. The formerly used
FEMS Yeast Res 13 (2013) 635–649
641
Cell length growth in fission yeast
14
γ = 0.0494 μm min–1
δ = 7.86 μm
13
0.2
R (μm)
Fig. 3. Illustrative growth profile of an
individual fission yeast cell corresponding to a
linear model (cell no. 50.2 in Table 2).
Experimental cell length is shown as a function
of time (symbols) together with the three
different fitted models as indicated (differently
colored lines). The optical resolution of cell
length was about 0.23 lm during the
measurements, but the patterns were
smoothed before model fitting (see text).
Parameters for the most adequate model
(linear, blue line) are given in the inset above
the graph. Residuals for the most adequate
model (linear) are shown in the right side
inset.
Cell length (μm)
12
11
10
0.0
–0.2
0
20
40
60
80 100 120 140 160
Time (min)
9
Growth period
Constant length period
Linear
Bilinear
Exponential
8
7
0
20
40
60
80
100
120
140
160
Time (min)
14
κ = 7.66 μm
μ = 0.00543 min–1
13
0.2
R (μm)
Cell length (μm)
12
Fig. 4. Illustrative growth profile of an
individual fission yeast cell corresponding to an
exponential model (cell no. 101.1 in Table 2).
Experimental cell length is shown as a function
of time (symbols) together with the three
different fitted models as indicated (differently
colored lines). The optical resolution of cell
length was about 0.23 lm during the
measurements, but the patterns were
smoothed before model fitting (see text).
Parameters for the most adequate model
(exponential, orange line) are given in the
inset above the graph. Residuals for the most
adequate model (exponential) are shown in
the right side inset.
11
–0.2
10
0
20
40
60
80
100 120 140
Time (min)
9
Growth period
Constant length period
Exponential
Bilinear
Linear
8
7
0
software (WINNONLIN) was able to calculate the standard
deviations for the fitted bilinear model’s parameters. For
the WT cell analyzed and published previously (Buchwald
& Sveiczer, 2006), the relative standard deviations (or
coefficients of variation) were 8.34%, 4.06%, and 88.49%
for a1, a2, and g, respectively. The high error of g is
unfortunately a consequence of the small number of data
points being situated in the transition period. As ttr is
proportional to g=ða2 a1 Þ (see Appendix), its error is
determined by the error of these three parameters, however, it would be very difficult to express it explicitly as
the formula contains both the operations division and
FEMS Yeast Res 13 (2013) 635–649
0.0
20
40
60
80
100
120
140
Time (min)
subtraction (Banfai & Kemeny, 2012). Nevertheless, the
error of g must predominate in the error of ttr. As a consequence, we may conclude that the coefficient of variation for ttr might be about 100% or so, for the WT cell
analyzed formerly by us (Buchwald & Sveiczer, 2006). We
propose that a similar size error for ttr might also be
considered in the present study.
Considering the mean CT of the cells (142 min) and
the time resolution of the measurements (5 min), we
defined a bilinear pattern to be sharp (abrupt) if the
transition period was < 2.5 min (first criterion; note that
the midpoint between two successive measurements is
ª 2013 Federation of European Microbiological Societies.
Published by John Wiley & Sons Ltd. All rights reserved
Best model
Linear
Linear
Bilinear
Bilinear
Bilinear
Bilinear
Exponential
Bilinear
Bilinear
Bilinear
Bilinear
Linear
Bilinear
Bilinear
Bilinear
Bilinear
Linear
Bilinear
Bilinear
Bilinear
Linear
Linear
Bilinear
Bilinear
Linear
Linear
Linear
Linear
Bilinear
Bilinear
Bilinear
Bilinear
Bilinear
Bilinear
Bilinear
Linear
Bilinear
Exponential
Bilinear
Linear
Bilinear
Cell number
17.1
17.2
18.1
18.2
24.1
24.2
25.1
25.2
35.1
35.2
39.1
39.2
42.1
42.2
43.1
43.2
44.1
44.2
45.1
45.2
48.1
48.2
49.1
49.2
50.1
50.2
51.1
51.2
52.1
52.2
53.1
53.2
54.1
54.2
60.1
60.2
61.1
61.2
62.1
62.2
63.1
8.25
7.25
7.70
7.82
8.17
6.77
9.10
8.73
6.77
7.70
7.24
7.70
7.67
7.71
7.70
8.24
7.22
7.70
6.81
7.70
7.70
7.25
6.85
8.17
8.71
7.70
7.24
6.77
7.24
7.70
7.70
7.24
7.70
7.72
8.18
8.18
8.26
8.17
8.24
6.77
8.26
BL (lm)
13.76
12.82
16.07
15.61
14.68
13.75
13.75
13.75
14.68
14.21
13.28
13.75
14.68
15.61
13.75
13.28
12.35
13.75
13.75
14.68
14.21
13.28
13.52
13.28
14.68
13.75
12.35
14.21
13.28
13.28
13.75
13.28
15.14
14.21
13.28
14.21
15.61
13.75
15.61
14.68
13.28
DL (lm)
120
130
160
155
165
175
120
130
175
145
140
150
145
155
140
135
125
140
145
140
135
160
175
140
130
140
140
170
130
135
140
145
145
150
120
125
145
115
150
155
125
CT (min)
a2 (lm min1)
–
–
0.0868
0.0762
0.0748
0.0514
–
0.0795
0.0559
0.0716
0.0842
–
0.0727
0.0694
0.0847
0.0700
–
0.0597
0.0734
0.0775
–
–
0.0585
0.0845
–
–
–
–
0.0902
0.0687
0.0852
0.0731
0.0782
0.0622
0.0601
–
0.0866
–
0.0594
–
0.0643
a1 (lm min1)
–
–
0.0414
0.0364
0.0329
0.0410
–
0.0408
0.0439
0.0394
0.0452
–
0.0551
0.0445
0.0425
0.0394
–
0.0426
0.0486
0.0474
–
–
0.0348
0.0369
–
–
–
–
0.0429
0.0454
0.0427
0.0469
0.0413
0.0440
0.0422
–
0.0531
–
0.0433
–
0.0435
–
–
2.10
2.10
2.27
1.25
–
1.95
1.27
1.82
1.86
–
1.32
1.56
1.99
1.78
–
1.40
1.51
1.63
–
–
1.68
2.29
–
–
–
–
2.11
1.51
1.99
1.56
1.89
1.41
1.42
–
1.63
–
1.37
–
1.48
a2/a1
–
–
75.0
50.0
87.1
92.9
–
74.0
69.4
63.0
83.1
–
71.7
75.5
72.3
74.2
–
51.5
82.6
73.2
–
–
73.6
71.9
–
–
–
–
57.6
59.2
85.5
86.5
59.7
74.6
44.5
–
76.0
–
30.0
–
45.0
s (min)
Table 2. Data for the individual fission yeast cells used for model fitting, and the model parameters in case of the bilinearly growing cells
–
–
0.258
0.272
0.500
0.010
–
0.500
0.219
0.010
0.500
–
0.010
0.010
0.231
0.088
–
0.126
0.010
0.010
–
–
0.350
0.010
–
–
–
–
0.010
0.010
0.500
0.010
0.500
0.500
0.010
–
0.010
–
0.042
–
0.036
g (lm)
–
–
9.6
11.5
20.2
1.6
–
21.9
31.1
0.5
21.8
–
1.0
0.7
9.3
4.9
–
12.5
0.7
0.6
–
–
25.0
0.4
–
–
–
–
0.4
0.7
19.9
0.6
23.0
46.7
1.0
–
0.5
–
4.5
–
2.9
ttr (min)
–
–
Smooth
Smooth
Smooth
Sharp
–
Smooth
Smooth
Sharp
Smooth
–
Sharp
Sharp
Smooth
Smooth
–
Smooth
Sharp
Sharp
–
–
Smooth
Sharp
–
–
–
–
Sharp
Sharp
Smooth
Sharp
Smooth
Smooth
Sharp
–
Sharp
–
Smooth
–
Smooth
Transition
642
A. Horvath et al.
ª 2013 Federation of European Microbiological Societies.
Published by John Wiley & Sons Ltd. All rights reserved
FEMS Yeast Res 13 (2013) 635–649
FEMS Yeast Res 13 (2013) 635–649
Bilinear
Exponential
Bilinear
Bilinear
Bilinear
Bilinear
Bilinear
Bilinear
Bilinear
Bilinear
Bilinear
Exponential
Bilinear
Bilinear
Bilinear
Bilinear
Linear
Exponential
Bilinear
63.2
64.1
64.2
65.1
65.2
69.1
69.2
70.1
70.2
74.1
74.2
89.1
89.2
94.1
94.2
99.1
99.2
101.1
101.2
7.69 0.56
7.70
6.87
7.24
8.26
8.62
6.78
7.71
7.24
7.70
7.48
7.70
6.77
7.70
8.17
7.72
8.13
7.72
7.67
8.65
BL (lm)
14.04 0.90
14.21
15.14
15.14
13.28
12.35
14.21
15.14
13.75
15.61
15.14
15.14
12.82
13.75
13.28
14.21
13.52
13.28
13.28
14.68
DL (lm)
142 16
140
180
175
130
110
165
150
145
140
165
145
140
145
125
125
130
130
130
125
CT (min)
0.0438 0.0063
0.0516
–
0.0421
0.0393
0.0465
0.0416
0.0402
0.0513
0.0567
0.0399
0.0536
–
0.0413
0.0461
0.0623
0.0315
–
–
0.0389
a1 (lm min1)
0.0704 0.0103
0.0572
–
0.0603
0.0678
0.0737
0.0593
0.0627
0.0574
0.0744
0.0716
0.0668
–
0.0693
0.0845
0.0743
0.0554
–
–
0.0650
a2 (lm min1)
1.63 0.31
1.11
–
1.43
1.73
1.59
1.43
1.56
1.12
1.31
1.80
1.25
–
1.68
1.83
1.19
1.76
–
–
1.67
a2/a1
66.3 18.3
40.0
–
93.2
64.8
58.9
70.0
32.1
75.0
63.0
90.0
67.4
–
86.0
63.8
75.3
17.0
–
–
26.8
s (min)
0.13 0.19
0.010
–
0.500
0.027
0.010
0.010
0.010
0.010
0.010
0.225
0.010
–
0.010
0.010
0.010
0.010
–
–
0.010
g (lm)
8.3 12.1
3.0
–
46.5
1.6
0.6
1.0
0.8
2.8
1.0
12.0
1.3
–
0.6
0.4
1.4
0.7
–
–
0.6
ttr (min)
–
Smooth
–
Smooth
Sharp
Sharp
Sharp
Sharp
Smooth
Sharp
Smooth
Sharp
–
Sharp
Sharp
Sharp
Sharp
–
–
Sharp
Transition
Cell number identifies the 30 measured sister pairs of cells. The best model was chosen by AIC, SBIC, and s, the first criterion being the dominant. In columns 6–12, data are given only in cases,
when the most adequate model was the bilinear one. Considering the transition smooth or sharp was based on the numerical values of both g and ttr, see text.
Mean standard
deviation
Best model
Cell number
Table 2. Continued
Cell length growth in fission yeast
643
ª 2013 Federation of European Microbiological Societies.
Published by John Wiley & Sons Ltd. All rights reserved
A. Horvath et al.
644
2.5 min far away from any of them). However, because a
lower limit had to be applied to the g parameter of the
bilinear (LinBiExp) model in certain cases to avoid
numerical problems (see Methods), we also considered a
bilinear pattern sharp (second criterion), if g had to be
restricted to this lower limit (0.01 lm, corresponding to
a transition period of c. 1 min with some dependence on
a1 and a2; see Appendix). By contrast, a bilinear pattern
was considered smooth if g was larger than 0.01 lm and
the transition was longer than 2.5 min. As examples,
Figs 1 and 2 represent individual cells having sharp and
smooth bilinear patterns, respectively. According to these
two criteria, bilinear growth patterns were sharp in 23
cases (55%) and smooth in 19 cases (45%). Hence, a
large proportion (nearly half) of individual cells changes
growth rate during a transition period, but not abruptly;
therefore, a sharp bilinear pattern cannot be a general
rule for cell length growth in fission yeast.
Bilinear patterns result from an increased rate
of growth
In all the 42 cases where bilinear growth was indicated by
AIC, the two linear segments were analyzed by a t-test to
compare the corresponding growth rates. The differences
between the two slopes were significant (P < 0.05) in 39
cases, meaning that the t-test confirmed the adequacy of
the bilinear model in 93% of these cells. SBIC is a more
stringent criterion than AIC; therefore, it favored the
bilinear pattern in only 32 cells (Table 1). In a separate
analysis of those 10 cells where AIC suggested bilinearity
and SBIC suggested some simpler (linear or exponential)
model, the t-test confirmed a significant difference
between the slopes before and after the RCP in eight
cases. On the other hand, a separate analysis of those six
cells that were judged as bilinear by the residual standard
deviation (s) as criterion, but not by the more stringent
AIC, the slopes were significantly different in only one
case. Although the number of analyzed cells is not sufficient to draw a definitive conclusion, these results indicate that using AIC to discriminate among models is
much more consistent with the results of the t-test than
either SBIC or s (the former one being too stringent,
while the latter one not stringent enough). Therefore,
AIC was considered the dominant criterion to decide the
cell’s growth pattern in cases where the three criteria were
not uniform (see Methods).
have found four different growth patterns (linear, exponential, smooth bilinear and sharp bilinear) in different
individual cells, moreover, we could not be absolutely
sure in that our model selection was correct in every case,
we wondered how this somewhat artificially generated
average cell behaves. Moreover, studying an average cell is
a general method in growth studies, and it was also
applied for fission yeast by Baumgartner & Tolic-Norrelykke (2009). Analyses were done using the same procedure as for the individual cells: cell length (however,
without any smoothing) was plotted versus time, and the
three different models were fitted and compared (Fig. 5a).
The growth profile of this average cell was found to be
definitely bilinear by any selection criteria used. Between
the two-parameter models, SSE favored the exponential
function over the linear one. The average bilinear pattern
has a smooth transition period, which might be partly
caused by merging the growth profiles of the 60 individual cells. The transition period of this average cell is
c. 10 min, whereas the average transition time of the
individual bilinear patterns was 8.3 min (see above). The
growth rates of the segments before and after the RCP
(a1 = 0.0478 lm min1, a2 = 0.0628 lm min1; Fig. 5a)
were also found to be significantly different by the t-test
comparing the homogeneity of the slopes. The strength of
growth rate at RCP (a2/a1) is 1.31, in nice agreement
with former results (Mitchison & Nurse, 1985; Sveiczer
et al., 1996; for a comparison with individual cells and
discussion, see above).
Finally, for an additional perspective, the bilinearity of
growth was also visualized for the average cell using a different method (Fig. 5b; Buchwald & Sveiczer, 2006). During the growth period, at every time point, the current
growth rate was calculated from the two consecutive cell
length data as DL/Dt, and then it was plotted as a function of time. The first order derivatives (see also Appendix) of the best fitting exponential and bilinear models
(dL/dt) were also calculated and compared to the growth
rate (Fig. 5b). For the exponential function, the derivative
is also exponential, whereas for the bilinear function, the
derivative is a sigmoid-like stepwise function. By calculating length growth rate (the difference quotient of measured data), the noise is significantly increased, and there
is considerable scatter around the fitted models; nevertheless, the data is fitted much better by the sigmoid step-up
than the exponential function (Fig. 5b).
Discussion
Growth of the average cell
To analyze the average growth profile, a hypothetic ‘average cell’ was created by superposition of the individual
cell data as described in the Methods section. Because we
ª 2013 Federation of European Microbiological Societies.
Published by John Wiley & Sons Ltd. All rights reserved
Fission yeast is one of the most frequently investigated
model organisms in studies of cellular growth and also of
the coupling between growth and division. Nongrowing
fission yeast cells also stop division (Rupes et al., 2001);
FEMS Yeast Res 13 (2013) 635–649
645
Cell length growth in fission yeast
(a)
15
14
α2 = 0.0628 μm min–1
13
ε = 10.8 μm
τ = 64.3 min
ttr
0.2
η = 0.087 μm
R (μm)
Cell length (μm)
α1 = 0.0478 μm min–1
12
0.0
–0.2
11
0
20 40 60 80 100 120 140 160
Time (min)
10
Growth period I
Transition period
Growth period II
Constant length period
Bilinear
Exponential
Linear
RCP
9
8
7
0
40
60
80
100
120
140
160
(b) 0.075
ttr
0.070
0.065
α2
0.060
70% of (α2 − α1)
(α2 − α1)
0.055
30% of (α2 − α1)
0.050
α1
0.045
0.040
0.035
0
although it is not clear, whether this is a consequence of
a specific morphogenetic checkpoint (Sveiczer et al.,
2002) or simply of the size control (Rupes & Young,
2002). It has been generally accepted that these rodshaped yeast cells have a constant diameter during their
mitotic cycle (Mitchison, 1957; Kelly & Nurse, 2011);
however, even this view has been challenged in a few
cases (Kubitschek & Clay, 1986). Some recent observations concluded that a spatial (polar) gradient of the
Pom1 kinase at the cell cortex ensures the connection
between cell size and mitotic onset (Martin & BerthelotGrosjean, 2009; Moseley et al., 2009). The molecular players (together with their interactions and the developed
mechanisms) involved either in cell growth or in the size
control in fission yeast has long been very extensively
FEMS Yeast Res 13 (2013) 635–649
20
Time (min)
Growth rate (μm min–1)
Fig. 5. Growth profile of the ‘average’ fission
yeast cell obtained from data of 60 individual
cells. (a) Experimental cell length is shown as a
function of time (symbols) together with the
three different fitted models as indicated
(differently colored lines). This pattern was not
smoothed before model fitting (see text).
Parameters for the most adequate model
(smooth bilinear, green line) are given in the
inset above the graph. Note that there is a
transition period of c. 10 min between the
two linear growing periods of constant rates.
Residuals for the most adequate model
(smooth bilinear) are shown in the right side
inset. (b) Time profile of the rate of length
growth [DL/Dt for the experimental data, with
the same symbols as in (a)] together with the
first-order derivative (dL/dt) of the best-fitting
bilinear (green line) and exponential (orange
line) model functions of the average cell. The
determination of the transition period for the
bilinear model is also indicated (see Appendix
for details).
20
40
60
80
100
120
140
160
Time (min)
studied (Moreno & Nurse, 1994; Sveiczer et al., 1996;
Hachet et al., 2011; Grallert et al., 2012; Valbuena et al.,
2012), but many questions has still not been answered in
the area. A very recent observation proposes a novel
mechanism, namely that the NETO event (see Introduction) is triggered in G2 by the appearance of M-phase
promoting factor at the spindle poles (Grallert et al.,
2013), which is highly significant from our perspective,
since once NETO is generally thought to be connected to
the rate change in growth (RCP), moreover, it is a sizecontrolled event (Mitchison & Nurse, 1985). In a series of
reviews on cell growth and cell cycle published in late
2012 in Current Opinion in Cell Biology, three papers
highlighted the indisputable role of S. pombe as a model
organism in all these fields in the past, present, and
ª 2013 Federation of European Microbiological Societies.
Published by John Wiley & Sons Ltd. All rights reserved
646
probably in the future as well (Davie & Petersen, 2012;
Hachet et al., 2012; Navarro et al., 2012).
The growth pattern of length in individual fission yeast
cells has usually been considered bilinear since the seminal work by Mitchison & Nurse (1985), although somewhat later the possibility of pseudo-exponential growth
was also introduced (Miyata et al., 1988). Later, the bilinearity of the growth profile was verified in several cell
cycle mutants of S. pombe, and the RCPs were even connected to some cell cycle events (Sveiczer et al., 1996,
1999). Around the same time, a dispute also emerged
because a bilinear pattern is difficult to distinguish from
an exponential one by the applied mathematical methods
considering the range of the growth data (Cooper, 1998;
Mitchison et al., 1998). A few years later, improved mathematical tools were introduced to these studies: on one
hand, the bilinear pattern was formulated as a continuously differentiable LinBiExp function that allows a
smooth curved transition between the linear segments
(Buchwald, 2005), and on the other hand, more rigorous
model selection criteria such as AIC were applied to discriminate among rival models having different parameter
numbers (Buchwald & Sveiczer, 2006). When this method
was developed for cell length growth pattern studies in
fission yeast, however, only one WT and one wee1
mutant cell was used, therefore, a general conclusion
could not be drawn for the growth profile of this species
(Buchwald & Sveiczer, 2006).
In the present work, we have analyzed the growth pattern of 60 individual WT fission yeast cells together with
that of the ‘average’ cell calculated from the superposition
of all the individual cells. Different cells in the same culture may grow differently, and to our knowledge, this is
the first report on the distribution of growth pattern of
cells in S. pombe. We have established that the majority
of the cells (70%) grow following a bilinear pattern, and
that in nearly half of these cases, there is a smooth and
not a sharp, abrupt transition between the two linear segments. For bilinearly growing cells, a significant difference
between the slopes of regression lines before and after the
RCP has been confirmed by t-tests, comparing the homogeneity of the slopes as well. We are convinced that
studying individual cells is more important than their
superposition, as the latter one is a bit artificial, however,
we have modified the former superposing algorithm
(Baumgartner & Tolic-Norrelykke, 2009) by normalizing
the raw measurements first, which enables considering
the whole cycle of every cell. Growth of this average cell
is also best described by a bilinear function with a
smooth transition period (c. 10 min) of accelerating
growth rate between the two linear segments of constant
growth and different slopes. Therefore, the growth pattern
of fission yeast is best described by a fully general,
ª 2013 Federation of European Microbiological Societies.
Published by John Wiley & Sons Ltd. All rights reserved
A. Horvath et al.
five-parameter bilinear model such as the LinBiExp model
as the following minimum set of parameters is needed:
two for the slopes of the two different linear segments,
one for the position of the RCP, one for the width of the
transition period, and one for the starting length (or an
equivalent in this special function here). A bilinear model
with an abrupt transition can be considered as a special
case of the general model with smooth transition, as in
the former one the width of the transition collapses to
zero. Therefore, the only advantage of applying a model
with two linear segments intersecting at RCP is that it
uses only four and not five parameters.
Cell length growth in fission yeast was recently studied
in detail by Baumgartner & Tolic-Norrelykke (2009)
using cells with green fluorescent protein labeled plasma
membrane and time-lapse confocal microscopy that
allowed increased spatial precision and temporal resolution. They fitted several different mathematical functions
on the obtained growth patterns (among them the general five-parameter LinBiExp and a four-parameter abrupt
bilinear model, L = a1t + b1, t < RCP and L = a2t + b2,
t > RCP, as well), and used the same model selection criteria as we did to find the most adequate model. They
concluded that the sharp bilinear model without any
transient period was the best to describe the growth pattern of fission yeast. This is somewhat unusual even considering their own data, as a smooth transition is quite
evident in some of the published individual and average
growth curves. Unfortunately, fitting of the smooth bilinear model (LinBiExp) was inadequately optimized, as
with a sufficiently narrow transition period (i.e. a sufficiently small g), it should reproduce exactly the results of
the four-parameter bilinear model, which was not the
case here (Baumgartner & Tolic-Norrelykke, 2009).
Adjusting the smoothness of the transition should only
improve the fit (SSE, r2), and then model selection criteria such as AIC or SBIC can be used to decide whether
the improvement was sufficient enough to justify the
addition of the corresponding new parameter (here, g).
In their work, an observably bilinear character was
not maintained over the investigated time range, resulting
in not the best possible fit and in unrealistic slope
values (Baumgartner & Tolic-Norrelykke, 2009). Hence,
probably most of these growth patterns are also better
fitted by smooth bilinear functions.
In conclusion, use of statistically rigorous model selection criteria and clear quantitative rules on judging the
width and smoothness of the transition period indicates
that the growth of fission yeast cells overwhelmingly follows a bilinear pattern and often tends to have a smooth,
gradual transition between two segments of constant
growth with different growth rates. Rate changes may
probably be connected to some cell cycle events, where the
FEMS Yeast Res 13 (2013) 635–649
Cell length growth in fission yeast
cytoskeleton of the cell might be remodeled. These events
may require some time to be executed, and the length of
this interval could have some variation among different
individual cells. As the error of estimating the transition
time is unfortunately quite large, it is a challenge for the
future to increase its precision. It remains to be clarified
why are there any linearly or exponentially growing individual cells in the fission yeast culture and what (if any)
cell cycle events are connected to the RCP in bilinearly
growing cells as formerly suggested (Sveiczer et al., 1996).
Analyzing growth patterns in cell cycle mutants and in
induction synchronous cultures might help to solve these
problems, such studies are under way in our laboratory.
A further question for the future whether there is any
correlation between the cell’s growth pattern and its physiological state or not. In a steady state fission yeast culture, BL and CT may be the most important parameters,
which may perhaps affect the growth model followed by
the cell. Our preliminary analyses suggest the lack of such
effects (data not shown). Because the BL (and also the
CT) range is rather narrow in a WT culture, these problems might also be studied in different cell size mutants
and in induction synchrony. Finally, it is noteworthy that
we have developed a novel method here to distinguish a
sharp bilinear pattern from a smooth one. Although it is
somewhat arbitrary, its basic features might easily be
adapted to growth pattern studies on any other cell types,
and it is also independent of the applied technique
measuring either cell length, or volume or mass.
Acknowledgements
This paper is a tribute to the memory of late Prof. Mur
doch Mitchison (1922–2011), who introduced A.S.
into
research on fission yeast and also provided us with his
time-lapse microscopic films used in this study. We are
also grateful for several deep scientific discussions with
Profs. Sandor Kemeny and Ern}
o Keszei. Several rightful
comments and suggestions of the anonymous reviewers
are highly appreciated. This project is supported by the
Hungarian Scientific Research Fund (OTKA K-76229)
and by the New Szechenyi Plan (Project ID: TAMOP4.2.1/B-09/1/KMR-2010-0002).
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Appendix
A fully general bilinear pattern, such as those considered
here, consists of two different linear segments separated by
a curved transition period around a RCP. To be able to
analyze the slopes of the two linear segments before and
after the RCP as well as the nature of the transition
between them, one needs to clearly delineate the borders of
both segments. Generally, we considered the first linear
growing period to start at cell birth; however, in a few rare
cases, one or two starting points at the beginning of the
growth have been ignored. The end of the second growing
period coincides with the onset of the constant length
phase (see Methods). Between these two linear segments of
different slopes (i.e. different growth rates), there can be a
curved transition period of accelerating growth whose
points should be considered separately from those of the
two linear segments. Some objective criteria are needed
to define the borders of this transition segment; the
mathematical procedure we used is described below.
A quantitative criterion is needed to define the width
of the curved (nonlinear) transitional region of a general
bilinear function, which here is represented by LinBiExp
(Buchwald & Sveiczer, 2006; Buchwald, 2007). The most
reasonable approach is to start from the sigmoid character of the derivative. LinBiExp can be written as a
function of time (t) as L ¼ gln½ea1 ðtsÞ=g þ ea2 ðtsÞ=g þ e.
The derivative of this bilinear function dL=dt ¼
ða1 þ a2 eða2 a1 ÞðtsÞ=g Þ=ð1 þ eða2 a1 ÞðtsÞ=g Þ indeed satisfies
the condition of having a constant value of a1 at t s
and another constant value of a2 at t ≫ s with a smooth
transition between them. In other words, this is a sigmoid
function producing a characteristic S-shaped smooth
stepup curve. The slope changes from a1 to a2 during the
transition around s; the width of the transition being
adjusted through g (but it also depends on a1 and a2).
The transitional section can be defined as the middle segment corresponding to some arbitrary percent of the total
(a2 a1) change. For example, the middle 40% is a reasonable choice (see Fig. 5b) even based on simple visual
criteria. This also makes sense because for a normal distribution, about 40% of the area under the curve is
included in its middle segment that has a width of one
standard deviation: The corresponding cumulative distribution function, which has a shape very similar to the
sigmoidal shape of the LinBiExp derivative, has values of
31% and 69% at 0.5r. The corresponding condition of
having a transition section in which the slope change is
between 30% and 70% of the total slope change can be
written for the derivative of LinBiExp function as
a1 þ 0:3ða2 a1 Þ a1 þ a2 eða2 a1 ÞðtsÞ=g
a1 þ 0:7ða2 a1 Þ:
1 þ eða2 a1 ÞðtsÞ=g
FEMS Yeast Res 13 (2013) 635–649
Cell length growth in fission yeast
This can be solved for the corresponding t times to get
the borders of the transitional section (ttr) as
g
0:7
ln
:
s
a2 a1 0:3
In other words, the width of the transitional section
along the independent variable axis (t) is proportional to
g=ða2 a1 Þ and with the current suggestion, it will be
considered as the segment of width of ð2g=ða2 a1 ÞÞ
ln ð0:7=0:3Þ around the RCP s (which also is the time
corresponding to the point of intersection of the two
linear asymptotes of the bilinear function).
FEMS Yeast Res 13 (2013) 635–649
649
In the present study, this definition of ttr was used to
decide whether a bilinear length growth pattern is smooth
or sharp. The first derivative of the general bilinear (LinBiExp) function describing the growth of the average cell
is shown in Fig. 5b to compare the model calculated
growth rate with the experimentally obtained values. The
derivative of the exponential growth function, which fitted better the growth data than the linear function, was
also included here for comparison. The exponential
model is represented by L = jelt, and its first derivative
is obviously dL/dt = jlelt.
ª 2013 Federation of European Microbiological Societies.
Published by John Wiley & Sons Ltd. All rights reserved