TY - JOUR
AU1 - Horváth, Anna
AU2 - Rácz-Mónus, Anna
AU3 - Buchwald, Peter
AU4 - Sveiczer, Ákos
AB - 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 time-lapse 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. fission yeast, time-lapse microscopy, cell growth pattern, model fitting, bilinear function 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 & Rácz-Mónus, 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 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 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 μm) rod-shaped prokaryotic cells of Escherichia coli is rather problematic. Recent exact measurements by phase contrast and fluorescent microscopy indicated that these Gram-negative 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 & Horváth, 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 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 μm (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 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 × 106 cells mL−1. 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 ×10 (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, 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 μm. 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 μm, data that are the average values of the 60 measured individual cells (see later in Result and Table 2 ). In this case, smoothing was not applied. 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 = γ·t + δ, and L= κ·eμt, respectively. The bilinear model has five parameters and, therefore, a more complex form: L=η·ln[eα1(t−τ)/η+eα2(t−τ)/η]+ε. 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 α1 and α2 that are separated by a RCP positioned at time τ 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 η parameter. It is worth mentioning that this LinBiExp model converges to a sharp bilinear function if η → 0; therefore, such a sharp bilinear expression was not fitted separately to the data. However, if η 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 η; the value used here (ηmin = 0.01 μm) corresponds to a minimum transition period of c.1 min (also depending on the growth rates, α1 and α2, 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 ηmax = 0.5 μm was also imposed for η, this corresponds to a maximum transition period of c.50 min. The fifth parameter of the model (ε) 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 = τ, L = η·ln2 + ε, and because η ≪ ε, L ≈ ε). 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 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 = nobs·ln(SSE) + 2npar, meanwhile SBIC is given as SBIC = nobs·ln(SSE) + npar·ln(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). 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 Result). 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 followed by faster growth in the second part (sometimes after a transition part of accelerating growth; Figs 1 and 2). 1 Distribution of length growth patterns in the analyzed cells as judged by different model selection criteria   AIC  SBIC  s  N  %  N  %  N  %  Bilinear  42  70.0  32  53.3  48  80.0  Linear  13  21.7  17  28.3  9  15.0  Exponential  5  8.3  11  18.3  3  5.0  Total  60  100.0  60  100.0  60  100.0    AIC  SBIC  s  N  %  N  %  N  %  Bilinear  42  70.0  32  53.3  48  80.0  Linear  13  21.7  17  28.3  9  15.0  Exponential  5  8.3  11  18.3  3  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). View Large 1 Distribution of length growth patterns in the analyzed cells as judged by different model selection criteria   AIC  SBIC  s  N  %  N  %  N  %  Bilinear  42  70.0  32  53.3  48  80.0  Linear  13  21.7  17  28.3  9  15.0  Exponential  5  8.3  11  18.3  3  5.0  Total  60  100.0  60  100.0  60  100.0    AIC  SBIC  s  N  %  N  %  N  %  Bilinear  42  70.0  32  53.3  48  80.0  Linear  13  21.7  17  28.3  9  15.0  Exponential  5  8.3  11  18.3  3  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). View Large 1 View largeDownload slide 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 μm 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. 1 View largeDownload slide 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 μm 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. 2 View largeDownload slide 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 μm 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. 2 View largeDownload slide 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 μm 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. 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. 3 View largeDownload slide 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 μm 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. 3 View largeDownload slide 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 μm 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. 4 View largeDownload slide 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 μm 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. 4 View largeDownload slide 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 μm 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. 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 α2/α1. 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 strength of the average RCP observed (see also the discussion of the average cell's pattern, later). 2 Data for the individual fission yeast cells used for model fitting, and the model parameters in case of the bilinearly growing cells Cell number  Best model  BL (μm)  DL (μm)  CT (min)  α1 (μm min−1)  α2 (μm min−1)  α2/α1  τ (min)  η (μm)  ttr (min)  Transition  17.1  Linear  8.25  13.76  120  –  –  –  –  –  –  –  17.2  Linear  7.25  12.82  130  –  –  –  –  –  –  –  18.1  Bilinear  7.70  16.07  160  0.0414  0.0868  2.10  75.0  0.258  9.6  Smooth  18.2  Bilinear  7.82  15.61  155  0.0364  0.0762  2.10  50.0  0.272  11.5  Smooth  24.1  Bilinear  8.17  14.68  165  0.0329  0.0748  2.27  87.1  0.500  20.2  Smooth  24.2  Bilinear  6.77  13.75  175  0.0410  0.0514  1.25  92.9  0.010  1.6  Sharp  25.1  Exponential  9.10  13.75  120  –  –  –  –  –  –  –  25.2  Bilinear  8.73  13.75  130  0.0408  0.0795  1.95  74.0  0.500  21.9  Smooth  35.1  Bilinear  6.77  14.68  175  0.0439  0.0559  1.27  69.4  0.219  31.1  Smooth  35.2  Bilinear  7.70  14.21  145  0.0394  0.0716  1.82  63.0  0.010  0.5  Sharp  39.1  Bilinear  7.24  13.28  140  0.0452  0.0842  1.86  83.1  0.500  21.8  Smooth  39.2  Linear  7.70  13.75  150  –  –  –  –  –  –  –  42.1  Bilinear  7.67  14.68  145  0.0551  0.0727  1.32  71.7  0.010  1.0  Sharp  42.2  Bilinear  7.71  15.61  155  0.0445  0.0694  1.56  75.5  0.010  0.7  Sharp  43.1  Bilinear  7.70  13.75  140  0.0425  0.0847  1.99  72.3  0.231  9.3  Smooth  43.2  Bilinear  8.24  13.28  135  0.0394  0.0700  1.78  74.2  0.088  4.9  Smooth  44.1  Linear  7.22  12.35  125  –  –  –  –  –  –  –  44.2  Bilinear  7.70  13.75  140  0.0426  0.0597  1.40  51.5  0.126  12.5  Smooth  45.1  Bilinear  6.81  13.75  145  0.0486  0.0734  1.51  82.6  0.010  0.7  Sharp  45.2  Bilinear  7.70  14.68  140  0.0474  0.0775  1.63  73.2  0.010  0.6  Sharp  48.1  Linear  7.70  14.21  135  –  –  –  –  –  –  –  48.2  Linear  7.25  13.28  160  –  –  –  –  –  –  –  49.1  Bilinear  6.85  13.52  175  0.0348  0.0585  1.68  73.6  0.350  25.0  Smooth  49.2  Bilinear  8.17  13.28  140  0.0369  0.0845  2.29  71.9  0.010  0.4  Sharp  50.1  Linear  8.71  14.68  130  –  –  –  –  –  –  –  50.2  Linear  7.70  13.75  140  –  –  –  –  –  –  –  51.1  Linear  7.24  12.35  140  –  –  –  –  –  –  –  51.2  Linear  6.77  14.21  170  –  –  –  –  –  –  –  52.1  Bilinear  7.24  13.28  130  0.0429  0.0902  2.11  57.6  0.010  0.4  Sharp  52.2  Bilinear  7.70  13.28  135  0.0454  0.0687  1.51  59.2  0.010  0.7  Sharp  53.1  Bilinear  7.70  13.75  140  0.0427  0.0852  1.99  85.5  0.500  19.9  Smooth  53.2  Bilinear  7.24  13.28  145  0.0469  0.0731  1.56  86.5  0.010  0.6  Sharp  54.1  Bilinear  7.70  15.14  145  0.0413  0.0782  1.89  59.7  0.500  23.0  Smooth  54.2  Bilinear  7.72  14.21  150  0.0440  0.0622  1.41  74.6  0.500  46.7  Smooth  60.1  Bilinear  8.18  13.28  120  0.0422  0.0601  1.42  44.5  0.010  1.0  Sharp  60.2  Linear  8.18  14.21  125  –  –  –  –  –  –  –  61.1  Bilinear  8.26  15.61  145  0.0531  0.0866  1.63  76.0  0.010  0.5  Sharp  61.2  Exponential  8.17  13.75  115  –  –  –  –  –  –  –  62.1  Bilinear  8.24  15.61  150  0.0433  0.0594  1.37  30.0  0.042  4.5  Smooth  62.2  Linear  6.77  14.68  155  –  –  –  –  –  –  –  63.1  Bilinear  8.26  13.28  125  0.0435  0.0643  1.48  45.0  0.036  2.9  Smooth  63.2  Bilinear  7.70  14.21  140  0.0516  0.0572  1.11  40.0  0.010  3.0  Smooth  64.1  Exponential  6.87  15.14  180  –  –  –  –  –  –  –  64.2  Bilinear  7.24  15.14  175  0.0421  0.0603  1.43  93.2  0.500  46.5  Smooth  65.1  Bilinear  8.26  13.28  130  0.0393  0.0678  1.73  64.8  0.027  1.6  Sharp  65.2  Bilinear  8.62  12.35  110  0.0465  0.0737  1.59  58.9  0.010  0.6  Sharp  69.1  Bilinear  6.78  14.21  165  0.0416  0.0593  1.43  70.0  0.010  1.0  Sharp  69.2  Bilinear  7.71  15.14  150  0.0402  0.0627  1.56  32.1  0.010  0.8  Sharp  70.1  Bilinear  7.24  13.75  145  0.0513  0.0574  1.12  75.0  0.010  2.8  Smooth  70.2  Bilinear  7.70  15.61  140  0.0567  0.0744  1.31  63.0  0.010  1.0  Sharp  74.1  Bilinear  7.48  15.14  165  0.0399  0.0716  1.80  90.0  0.225  12.0  Smooth  74.2  Bilinear  7.70  15.14  145  0.0536  0.0668  1.25  67.4  0.010  1.3  Sharp  89.1  Exponential  6.77  12.82  140  –  –  –  –  –  –  –  89.2  Bilinear  7.70  13.75  145  0.0413  0.0693  1.68  86.0  0.010  0.6  Sharp  94.1  Bilinear  8.17  13.28  125  0.0461  0.0845  1.83  63.8  0.010  0.4  Sharp  94.2  Bilinear  7.72  14.21  125  0.0623  0.0743  1.19  75.3  0.010  1.4  Sharp  99.1  Bilinear  8.13  13.52  130  0.0315  0.0554  1.76  17.0  0.010  0.7  Sharp  99.2  Linear  7.72  13.28  130  –  –  –  –  –  –  –  101.1  Exponential  7.67  13.28  130  –  –  –  –  –  –  –  101.2  Bilinear  8.65  14.68  125  0.0389  0.0650  1.67  26.8  0.010  0.6  Sharp  Mean ± standard deviation  7.69 ± 0.56  14.04 ± 0.90  142 ± 16  0.0438 ± 0.0063  0.0704 ± 0.0103  1.63 ± 0.31  66.3 ± 18.3  0.13 ± 0.19  8.3 ± 12.1  –  Cell number  Best model  BL (μm)  DL (μm)  CT (min)  α1 (μm min−1)  α2 (μm min−1)  α2/α1  τ (min)  η (μm)  ttr (min)  Transition  17.1  Linear  8.25  13.76  120  –  –  –  –  –  –  –  17.2  Linear  7.25  12.82  130  –  –  –  –  –  –  –  18.1  Bilinear  7.70  16.07  160  0.0414  0.0868  2.10  75.0  0.258  9.6  Smooth  18.2  Bilinear  7.82  15.61  155  0.0364  0.0762  2.10  50.0  0.272  11.5  Smooth  24.1  Bilinear  8.17  14.68  165  0.0329  0.0748  2.27  87.1  0.500  20.2  Smooth  24.2  Bilinear  6.77  13.75  175  0.0410  0.0514  1.25  92.9  0.010  1.6  Sharp  25.1  Exponential  9.10  13.75  120  –  –  –  –  –  –  –  25.2  Bilinear  8.73  13.75  130  0.0408  0.0795  1.95  74.0  0.500  21.9  Smooth  35.1  Bilinear  6.77  14.68  175  0.0439  0.0559  1.27  69.4  0.219  31.1  Smooth  35.2  Bilinear  7.70  14.21  145  0.0394  0.0716  1.82  63.0  0.010  0.5  Sharp  39.1  Bilinear  7.24  13.28  140  0.0452  0.0842  1.86  83.1  0.500  21.8  Smooth  39.2  Linear  7.70  13.75  150  –  –  –  –  –  –  –  42.1  Bilinear  7.67  14.68  145  0.0551  0.0727  1.32  71.7  0.010  1.0  Sharp  42.2  Bilinear  7.71  15.61  155  0.0445  0.0694  1.56  75.5  0.010  0.7  Sharp  43.1  Bilinear  7.70  13.75  140  0.0425  0.0847  1.99  72.3  0.231  9.3  Smooth  43.2  Bilinear  8.24  13.28  135  0.0394  0.0700  1.78  74.2  0.088  4.9  Smooth  44.1  Linear  7.22  12.35  125  –  –  –  –  –  –  –  44.2  Bilinear  7.70  13.75  140  0.0426  0.0597  1.40  51.5  0.126  12.5  Smooth  45.1  Bilinear  6.81  13.75  145  0.0486  0.0734  1.51  82.6  0.010  0.7  Sharp  45.2  Bilinear  7.70  14.68  140  0.0474  0.0775  1.63  73.2  0.010  0.6  Sharp  48.1  Linear  7.70  14.21  135  –  –  –  –  –  –  –  48.2  Linear  7.25  13.28  160  –  –  –  –  –  –  –  49.1  Bilinear  6.85  13.52  175  0.0348  0.0585  1.68  73.6  0.350  25.0  Smooth  49.2  Bilinear  8.17  13.28  140  0.0369  0.0845  2.29  71.9  0.010  0.4  Sharp  50.1  Linear  8.71  14.68  130  –  –  –  –  –  –  –  50.2  Linear  7.70  13.75  140  –  –  –  –  –  –  –  51.1  Linear  7.24  12.35  140  –  –  –  –  –  –  –  51.2  Linear  6.77  14.21  170  –  –  –  –  –  –  –  52.1  Bilinear  7.24  13.28  130  0.0429  0.0902  2.11  57.6  0.010  0.4  Sharp  52.2  Bilinear  7.70  13.28  135  0.0454  0.0687  1.51  59.2  0.010  0.7  Sharp  53.1  Bilinear  7.70  13.75  140  0.0427  0.0852  1.99  85.5  0.500  19.9  Smooth  53.2  Bilinear  7.24  13.28  145  0.0469  0.0731  1.56  86.5  0.010  0.6  Sharp  54.1  Bilinear  7.70  15.14  145  0.0413  0.0782  1.89  59.7  0.500  23.0  Smooth  54.2  Bilinear  7.72  14.21  150  0.0440  0.0622  1.41  74.6  0.500  46.7  Smooth  60.1  Bilinear  8.18  13.28  120  0.0422  0.0601  1.42  44.5  0.010  1.0  Sharp  60.2  Linear  8.18  14.21  125  –  –  –  –  –  –  –  61.1  Bilinear  8.26  15.61  145  0.0531  0.0866  1.63  76.0  0.010  0.5  Sharp  61.2  Exponential  8.17  13.75  115  –  –  –  –  –  –  –  62.1  Bilinear  8.24  15.61  150  0.0433  0.0594  1.37  30.0  0.042  4.5  Smooth  62.2  Linear  6.77  14.68  155  –  –  –  –  –  –  –  63.1  Bilinear  8.26  13.28  125  0.0435  0.0643  1.48  45.0  0.036  2.9  Smooth  63.2  Bilinear  7.70  14.21  140  0.0516  0.0572  1.11  40.0  0.010  3.0  Smooth  64.1  Exponential  6.87  15.14  180  –  –  –  –  –  –  –  64.2  Bilinear  7.24  15.14  175  0.0421  0.0603  1.43  93.2  0.500  46.5  Smooth  65.1  Bilinear  8.26  13.28  130  0.0393  0.0678  1.73  64.8  0.027  1.6  Sharp  65.2  Bilinear  8.62  12.35  110  0.0465  0.0737  1.59  58.9  0.010  0.6  Sharp  69.1  Bilinear  6.78  14.21  165  0.0416  0.0593  1.43  70.0  0.010  1.0  Sharp  69.2  Bilinear  7.71  15.14  150  0.0402  0.0627  1.56  32.1  0.010  0.8  Sharp  70.1  Bilinear  7.24  13.75  145  0.0513  0.0574  1.12  75.0  0.010  2.8  Smooth  70.2  Bilinear  7.70  15.61  140  0.0567  0.0744  1.31  63.0  0.010  1.0  Sharp  74.1  Bilinear  7.48  15.14  165  0.0399  0.0716  1.80  90.0  0.225  12.0  Smooth  74.2  Bilinear  7.70  15.14  145  0.0536  0.0668  1.25  67.4  0.010  1.3  Sharp  89.1  Exponential  6.77  12.82  140  –  –  –  –  –  –  –  89.2  Bilinear  7.70  13.75  145  0.0413  0.0693  1.68  86.0  0.010  0.6  Sharp  94.1  Bilinear  8.17  13.28  125  0.0461  0.0845  1.83  63.8  0.010  0.4  Sharp  94.2  Bilinear  7.72  14.21  125  0.0623  0.0743  1.19  75.3  0.010  1.4  Sharp  99.1  Bilinear  8.13  13.52  130  0.0315  0.0554  1.76  17.0  0.010  0.7  Sharp  99.2  Linear  7.72  13.28  130  –  –  –  –  –  –  –  101.1  Exponential  7.67  13.28  130  –  –  –  –  –  –  –  101.2  Bilinear  8.65  14.68  125  0.0389  0.0650  1.67  26.8  0.010  0.6  Sharp  Mean ± standard deviation  7.69 ± 0.56  14.04 ± 0.90  142 ± 16  0.0438 ± 0.0063  0.0704 ± 0.0103  1.63 ± 0.31  66.3 ± 18.3  0.13 ± 0.19  8.3 ± 12.1  –  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 η and ttr, see text. View Large 2 Data for the individual fission yeast cells used for model fitting, and the model parameters in case of the bilinearly growing cells Cell number  Best model  BL (μm)  DL (μm)  CT (min)  α1 (μm min−1)  α2 (μm min−1)  α2/α1  τ (min)  η (μm)  ttr (min)  Transition  17.1  Linear  8.25  13.76  120  –  –  –  –  –  –  –  17.2  Linear  7.25  12.82  130  –  –  –  –  –  –  –  18.1  Bilinear  7.70  16.07  160  0.0414  0.0868  2.10  75.0  0.258  9.6  Smooth  18.2  Bilinear  7.82  15.61  155  0.0364  0.0762  2.10  50.0  0.272  11.5  Smooth  24.1  Bilinear  8.17  14.68  165  0.0329  0.0748  2.27  87.1  0.500  20.2  Smooth  24.2  Bilinear  6.77  13.75  175  0.0410  0.0514  1.25  92.9  0.010  1.6  Sharp  25.1  Exponential  9.10  13.75  120  –  –  –  –  –  –  –  25.2  Bilinear  8.73  13.75  130  0.0408  0.0795  1.95  74.0  0.500  21.9  Smooth  35.1  Bilinear  6.77  14.68  175  0.0439  0.0559  1.27  69.4  0.219  31.1  Smooth  35.2  Bilinear  7.70  14.21  145  0.0394  0.0716  1.82  63.0  0.010  0.5  Sharp  39.1  Bilinear  7.24  13.28  140  0.0452  0.0842  1.86  83.1  0.500  21.8  Smooth  39.2  Linear  7.70  13.75  150  –  –  –  –  –  –  –  42.1  Bilinear  7.67  14.68  145  0.0551  0.0727  1.32  71.7  0.010  1.0  Sharp  42.2  Bilinear  7.71  15.61  155  0.0445  0.0694  1.56  75.5  0.010  0.7  Sharp  43.1  Bilinear  7.70  13.75  140  0.0425  0.0847  1.99  72.3  0.231  9.3  Smooth  43.2  Bilinear  8.24  13.28  135  0.0394  0.0700  1.78  74.2  0.088  4.9  Smooth  44.1  Linear  7.22  12.35  125  –  –  –  –  –  –  –  44.2  Bilinear  7.70  13.75  140  0.0426  0.0597  1.40  51.5  0.126  12.5  Smooth  45.1  Bilinear  6.81  13.75  145  0.0486  0.0734  1.51  82.6  0.010  0.7  Sharp  45.2  Bilinear  7.70  14.68  140  0.0474  0.0775  1.63  73.2  0.010  0.6  Sharp  48.1  Linear  7.70  14.21  135  –  –  –  –  –  –  –  48.2  Linear  7.25  13.28  160  –  –  –  –  –  –  –  49.1  Bilinear  6.85  13.52  175  0.0348  0.0585  1.68  73.6  0.350  25.0  Smooth  49.2  Bilinear  8.17  13.28  140  0.0369  0.0845  2.29  71.9  0.010  0.4  Sharp  50.1  Linear  8.71  14.68  130  –  –  –  –  –  –  –  50.2  Linear  7.70  13.75  140  –  –  –  –  –  –  –  51.1  Linear  7.24  12.35  140  –  –  –  –  –  –  –  51.2  Linear  6.77  14.21  170  –  –  –  –  –  –  –  52.1  Bilinear  7.24  13.28  130  0.0429  0.0902  2.11  57.6  0.010  0.4  Sharp  52.2  Bilinear  7.70  13.28  135  0.0454  0.0687  1.51  59.2  0.010  0.7  Sharp  53.1  Bilinear  7.70  13.75  140  0.0427  0.0852  1.99  85.5  0.500  19.9  Smooth  53.2  Bilinear  7.24  13.28  145  0.0469  0.0731  1.56  86.5  0.010  0.6  Sharp  54.1  Bilinear  7.70  15.14  145  0.0413  0.0782  1.89  59.7  0.500  23.0  Smooth  54.2  Bilinear  7.72  14.21  150  0.0440  0.0622  1.41  74.6  0.500  46.7  Smooth  60.1  Bilinear  8.18  13.28  120  0.0422  0.0601  1.42  44.5  0.010  1.0  Sharp  60.2  Linear  8.18  14.21  125  –  –  –  –  –  –  –  61.1  Bilinear  8.26  15.61  145  0.0531  0.0866  1.63  76.0  0.010  0.5  Sharp  61.2  Exponential  8.17  13.75  115  –  –  –  –  –  –  –  62.1  Bilinear  8.24  15.61  150  0.0433  0.0594  1.37  30.0  0.042  4.5  Smooth  62.2  Linear  6.77  14.68  155  –  –  –  –  –  –  –  63.1  Bilinear  8.26  13.28  125  0.0435  0.0643  1.48  45.0  0.036  2.9  Smooth  63.2  Bilinear  7.70  14.21  140  0.0516  0.0572  1.11  40.0  0.010  3.0  Smooth  64.1  Exponential  6.87  15.14  180  –  –  –  –  –  –  –  64.2  Bilinear  7.24  15.14  175  0.0421  0.0603  1.43  93.2  0.500  46.5  Smooth  65.1  Bilinear  8.26  13.28  130  0.0393  0.0678  1.73  64.8  0.027  1.6  Sharp  65.2  Bilinear  8.62  12.35  110  0.0465  0.0737  1.59  58.9  0.010  0.6  Sharp  69.1  Bilinear  6.78  14.21  165  0.0416  0.0593  1.43  70.0  0.010  1.0  Sharp  69.2  Bilinear  7.71  15.14  150  0.0402  0.0627  1.56  32.1  0.010  0.8  Sharp  70.1  Bilinear  7.24  13.75  145  0.0513  0.0574  1.12  75.0  0.010  2.8  Smooth  70.2  Bilinear  7.70  15.61  140  0.0567  0.0744  1.31  63.0  0.010  1.0  Sharp  74.1  Bilinear  7.48  15.14  165  0.0399  0.0716  1.80  90.0  0.225  12.0  Smooth  74.2  Bilinear  7.70  15.14  145  0.0536  0.0668  1.25  67.4  0.010  1.3  Sharp  89.1  Exponential  6.77  12.82  140  –  –  –  –  –  –  –  89.2  Bilinear  7.70  13.75  145  0.0413  0.0693  1.68  86.0  0.010  0.6  Sharp  94.1  Bilinear  8.17  13.28  125  0.0461  0.0845  1.83  63.8  0.010  0.4  Sharp  94.2  Bilinear  7.72  14.21  125  0.0623  0.0743  1.19  75.3  0.010  1.4  Sharp  99.1  Bilinear  8.13  13.52  130  0.0315  0.0554  1.76  17.0  0.010  0.7  Sharp  99.2  Linear  7.72  13.28  130  –  –  –  –  –  –  –  101.1  Exponential  7.67  13.28  130  –  –  –  –  –  –  –  101.2  Bilinear  8.65  14.68  125  0.0389  0.0650  1.67  26.8  0.010  0.6  Sharp  Mean ± standard deviation  7.69 ± 0.56  14.04 ± 0.90  142 ± 16  0.0438 ± 0.0063  0.0704 ± 0.0103  1.63 ± 0.31  66.3 ± 18.3  0.13 ± 0.19  8.3 ± 12.1  –  Cell number  Best model  BL (μm)  DL (μm)  CT (min)  α1 (μm min−1)  α2 (μm min−1)  α2/α1  τ (min)  η (μm)  ttr (min)  Transition  17.1  Linear  8.25  13.76  120  –  –  –  –  –  –  –  17.2  Linear  7.25  12.82  130  –  –  –  –  –  –  –  18.1  Bilinear  7.70  16.07  160  0.0414  0.0868  2.10  75.0  0.258  9.6  Smooth  18.2  Bilinear  7.82  15.61  155  0.0364  0.0762  2.10  50.0  0.272  11.5  Smooth  24.1  Bilinear  8.17  14.68  165  0.0329  0.0748  2.27  87.1  0.500  20.2  Smooth  24.2  Bilinear  6.77  13.75  175  0.0410  0.0514  1.25  92.9  0.010  1.6  Sharp  25.1  Exponential  9.10  13.75  120  –  –  –  –  –  –  –  25.2  Bilinear  8.73  13.75  130  0.0408  0.0795  1.95  74.0  0.500  21.9  Smooth  35.1  Bilinear  6.77  14.68  175  0.0439  0.0559  1.27  69.4  0.219  31.1  Smooth  35.2  Bilinear  7.70  14.21  145  0.0394  0.0716  1.82  63.0  0.010  0.5  Sharp  39.1  Bilinear  7.24  13.28  140  0.0452  0.0842  1.86  83.1  0.500  21.8  Smooth  39.2  Linear  7.70  13.75  150  –  –  –  –  –  –  –  42.1  Bilinear  7.67  14.68  145  0.0551  0.0727  1.32  71.7  0.010  1.0  Sharp  42.2  Bilinear  7.71  15.61  155  0.0445  0.0694  1.56  75.5  0.010  0.7  Sharp  43.1  Bilinear  7.70  13.75  140  0.0425  0.0847  1.99  72.3  0.231  9.3  Smooth  43.2  Bilinear  8.24  13.28  135  0.0394  0.0700  1.78  74.2  0.088  4.9  Smooth  44.1  Linear  7.22  12.35  125  –  –  –  –  –  –  –  44.2  Bilinear  7.70  13.75  140  0.0426  0.0597  1.40  51.5  0.126  12.5  Smooth  45.1  Bilinear  6.81  13.75  145  0.0486  0.0734  1.51  82.6  0.010  0.7  Sharp  45.2  Bilinear  7.70  14.68  140  0.0474  0.0775  1.63  73.2  0.010  0.6  Sharp  48.1  Linear  7.70  14.21  135  –  –  –  –  –  –  –  48.2  Linear  7.25  13.28  160  –  –  –  –  –  –  –  49.1  Bilinear  6.85  13.52  175  0.0348  0.0585  1.68  73.6  0.350  25.0  Smooth  49.2  Bilinear  8.17  13.28  140  0.0369  0.0845  2.29  71.9  0.010  0.4  Sharp  50.1  Linear  8.71  14.68  130  –  –  –  –  –  –  –  50.2  Linear  7.70  13.75  140  –  –  –  –  –  –  –  51.1  Linear  7.24  12.35  140  –  –  –  –  –  –  –  51.2  Linear  6.77  14.21  170  –  –  –  –  –  –  –  52.1  Bilinear  7.24  13.28  130  0.0429  0.0902  2.11  57.6  0.010  0.4  Sharp  52.2  Bilinear  7.70  13.28  135  0.0454  0.0687  1.51  59.2  0.010  0.7  Sharp  53.1  Bilinear  7.70  13.75  140  0.0427  0.0852  1.99  85.5  0.500  19.9  Smooth  53.2  Bilinear  7.24  13.28  145  0.0469  0.0731  1.56  86.5  0.010  0.6  Sharp  54.1  Bilinear  7.70  15.14  145  0.0413  0.0782  1.89  59.7  0.500  23.0  Smooth  54.2  Bilinear  7.72  14.21  150  0.0440  0.0622  1.41  74.6  0.500  46.7  Smooth  60.1  Bilinear  8.18  13.28  120  0.0422  0.0601  1.42  44.5  0.010  1.0  Sharp  60.2  Linear  8.18  14.21  125  –  –  –  –  –  –  –  61.1  Bilinear  8.26  15.61  145  0.0531  0.0866  1.63  76.0  0.010  0.5  Sharp  61.2  Exponential  8.17  13.75  115  –  –  –  –  –  –  –  62.1  Bilinear  8.24  15.61  150  0.0433  0.0594  1.37  30.0  0.042  4.5  Smooth  62.2  Linear  6.77  14.68  155  –  –  –  –  –  –  –  63.1  Bilinear  8.26  13.28  125  0.0435  0.0643  1.48  45.0  0.036  2.9  Smooth  63.2  Bilinear  7.70  14.21  140  0.0516  0.0572  1.11  40.0  0.010  3.0  Smooth  64.1  Exponential  6.87  15.14  180  –  –  –  –  –  –  –  64.2  Bilinear  7.24  15.14  175  0.0421  0.0603  1.43  93.2  0.500  46.5  Smooth  65.1  Bilinear  8.26  13.28  130  0.0393  0.0678  1.73  64.8  0.027  1.6  Sharp  65.2  Bilinear  8.62  12.35  110  0.0465  0.0737  1.59  58.9  0.010  0.6  Sharp  69.1  Bilinear  6.78  14.21  165  0.0416  0.0593  1.43  70.0  0.010  1.0  Sharp  69.2  Bilinear  7.71  15.14  150  0.0402  0.0627  1.56  32.1  0.010  0.8  Sharp  70.1  Bilinear  7.24  13.75  145  0.0513  0.0574  1.12  75.0  0.010  2.8  Smooth  70.2  Bilinear  7.70  15.61  140  0.0567  0.0744  1.31  63.0  0.010  1.0  Sharp  74.1  Bilinear  7.48  15.14  165  0.0399  0.0716  1.80  90.0  0.225  12.0  Smooth  74.2  Bilinear  7.70  15.14  145  0.0536  0.0668  1.25  67.4  0.010  1.3  Sharp  89.1  Exponential  6.77  12.82  140  –  –  –  –  –  –  –  89.2  Bilinear  7.70  13.75  145  0.0413  0.0693  1.68  86.0  0.010  0.6  Sharp  94.1  Bilinear  8.17  13.28  125  0.0461  0.0845  1.83  63.8  0.010  0.4  Sharp  94.2  Bilinear  7.72  14.21  125  0.0623  0.0743  1.19  75.3  0.010  1.4  Sharp  99.1  Bilinear  8.13  13.52  130  0.0315  0.0554  1.76  17.0  0.010  0.7  Sharp  99.2  Linear  7.72  13.28  130  –  –  –  –  –  –  –  101.1  Exponential  7.67  13.28  130  –  –  –  –  –  –  –  101.2  Bilinear  8.65  14.68  125  0.0389  0.0650  1.67  26.8  0.010  0.6  Sharp  Mean ± standard deviation  7.69 ± 0.56  14.04 ± 0.90  142 ± 16  0.0438 ± 0.0063  0.0704 ± 0.0103  1.63 ± 0.31  66.3 ± 18.3  0.13 ± 0.19  8.3 ± 12.1  –  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 η and ttr, see text. View Large 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 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 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 α1, α2, and η, respectively. The high error of η is unfortunately a consequence of the small number of data points being situated in the transition period. As ttr is proportional to η/(α2−α1) (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 subtraction (Banfai & Kemeny, 2012). Nevertheless, the error of η 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 2.5 min far away from any of them). However, because a lower limit had to be applied to the η 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 η had to be restricted to this lower limit (0.01 μm, corresponding to a transition period of c.1 min with some dependence on α1 and α2; see  Appendix). By contrast, a bilinear pattern was considered smooth if η was larger than 0.01 μm 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). 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 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 (α1 = 0.0478 μm min−1, α2 = 0.0628 μm min−1; 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 (α2/α1) 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). 5 View largeDownload slide 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 [ΔL/Δt 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). 5 View largeDownload slide 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 [ΔL/Δt 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). 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 ΔL/Δt, 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 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); 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 rod-shaped 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 & Berthelot-Grosjean, 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 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 Result) 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 size-controlled 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 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, 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 = α1t + β1, t < RCP and L = α2t + β2, 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 η), 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, η). 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 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. Murdoch Mitchison (1922–2011), who introduced Á.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. Sándor Kemény and Ernő Keszei. Several rightful comments and suggestions of the anonymous reviewers are highly appreciated. 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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=η·ln[eα1(t−τ)/η+eα2(t−τ)/η]+ε. The derivative of this bilinear function dL/dt=(α1+α2e(α2−α1)(t−τ)/η)/(1+e(α2−α1)(t−τ)/η) indeed satisfies the condition of having a constant value of α1 at t ≪ τ and another constant value of α2 at t ≫ τ 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 α1 to α2 during the transition around τ; the width of the transition being adjusted through η (but it also depends on α1 and α2). The transitional section can be defined as the middle segment corresponding to some arbitrary percent of the total (α2 − α1) 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 ± 0.5σ. 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   This can be solved for the corresponding t times to get the borders of the transitional section (ttr) as   In other words, the width of the transitional section along the independent variable axis (t) is proportional to η/(α2−α1) and with the current suggestion, it will be considered as the segment of width of (2η/(α2−α1))·ln(0.7/0.3) around the RCP τ (which also is the time corresponding to the point of intersection of the two linear asymptotes of the bilinear function). 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= κ·eμt, and its first derivative is obviously dL/dt= κμ·eμt. © 2013 Federation of European Microbiological Societies. Published by John Wiley & Sons Ltd. All rights reserved
TI - Cell length growth in fission yeast: an analysis of its bilinear character and the nature of its rate change transition
JF - FEMS Yeast Research
DO - 10.1111/1567-1364.12064
DA - 2013-11-01
UR - https://www.deepdyve.com/lp/oxford-university-press/cell-length-growth-in-fission-yeast-an-analysis-of-its-bilinear-v3E9Av8En8
SP - 635
EP - 649
VL - 13
IS - 7
DP - DeepDyve
ER -