COMMENTARY
Is there a universal rule for cellular growth? – Problems in studying and interpreting this phenomenon
YEAST RESEARCH
DOI: 10.1111/1567-1364.12168
There is an enormous number of different cells in the biosphere, prokaryotes and eukaryotes, many of them living as unicellular organisms, while others forming complex multicellular beings. Some cells grow and divide generating offsprings; others are in a dormant state or generate gametes or spores. It is not too hard to conclude that it is unlikely that there is a general law describing the growth kinetics of every cell considering that several of them do not grow at all. However, when unicellular microorganisms (like bacteria or yeasts) reside in a stirred liquid medium containing appropriate nutrients both in qualitative and quantitative terms, the pure culture soon reaches an exponential phase of growth where the cell number (or cell mass) increases exponentially in time (Black, 2013). Notably, the known general growth kinetics of this asynchronous population tells us nothing about the growth kinetics of mass observed in the individual cells of the population. By the same logic, the ‘simplest’ way for the cells to grow is an exponential mode; however, there is no unequivocal proof that such a universal law exists. Theoretically, any continuous mathematical function might describe cellular growth, if it fulfils two requirements; namely, both cell mass must double during the cycle time, and also, the rate of cell mass increase must do the same – even if not necessarily for every individual cell – but at least for the hypothetical ‘average’ cell representing the population. This way, a symmetric cell division can generate two identical progenies whose physiological states are equivalent with each other as well as with that of their mother cell’s state at the beginning of the previous cycle (Mitchison, 2003). The exponential model has the advantage that cell mass is automatically proportional to the rate of cell mass increase; therefore, there is no need for any breakpoint(s) (or rate change point(s)) to be involved (Mitchison, 2003). Moreover, one need not bother either with different cellular structures (like cytoskeleton, cell membrane or cell wall) or discrete events during the cycle (like chromosome replication, segregation or cytokinesis). By contrast, a linear model is more problematic. As growth rate is constant along a linear growth pattern, it cannot follow cell mass increase during the division cycle. Therefore, one must involve breakpoints in the model where the
FEMS Yeast Res 14 (2014) 679–682
growth rate (abruptly or smoothly) changes to satisfy the general requirements, that is, that both cell mass and its growth rate must double within a cycle (Horvath et al., 2013). Were these breakpoints arbitrary without any sense, we should really throw this model out into a dustbin. However, the above mentioned cellular structures and discrete events give us several good reasons to suppose the existence of rate change points. For example, DNA replication is thought to accelerate growth rate, an effect also known as the gene dosage effect. By contrast, mitosis in eukaryotes decreases growth rate because mitotic chromosomes condense to such a compact state that makes RNA transcription nearly or totally impossible (Mitchison, 2003). In lower eukaryotes like fission yeast, mitotic chromosomes are less condensed than in higher eukaryotes, which makes possible some transcription even at this cell cycle stage, for example, the Cdc10 transcription factor is active and helps transcribe some genes, whose products are necessary for the next G1/S transition (Baum et al., 1998), however, this transcription activity is severely reduced compared with its level operating during interphase. A further point to be considered is that cytoskeleton rearranges before and during mitosis (or sometimes even in interphase); moreover, cells with a rigid wall must produce septa before cell division (Morgan, 2007) – these facts might also have negative effects on overall growth processes. Lacking a general law for the growth of cells, exploration of this field requires the study of adequate model organisms that are sufficiently representative, the use of sophisticated investigational techniques to increase accuracy, and the use of mathematical methods and adequate statistical criteria to explore different functions and select the best fitting one. During the last 5 years, two papers examined in detail cell length (proportional to cell volume) growth patterns for the unicellular fission yeast model organism, Schizosaccharomyces pombe (Baumgartner & Tolic-Norrelykke, 2009; Horvath et al., 2013). Both concluded that fission yeast cells grew bilinearly with the only discrepancy being on whether the observed rate change between the linear segments was sharp or smooth. The first observations on the bilinear growth mode of fission yeast cells were published nearly 30 years ago
ª 2014 Federation of European Microbiological Societies. Published by John Wiley & Sons Ltd. All rights reserved
680
(Mitchison & Nurse, 1985), and this result was verified and extended to several cell cycle mutants 10 years later (Sveiczer et al., 1996). A feasible explanation for this fact was also given at that time. Namely at the beginning of the cycle the rod shaped cell extended only at its old end (which had been a growing cell end in the previous cycle), while in mid G2 phase, at the so-called NETO (new end take off) event, the newly formed other cell end also started to extend (Mitchison & Nurse, 1985). This unipolar to bipolar growth switch was caused by cytoskeletal rearrangements forming a new growth zone (Martin & Chang, 2005), and it is easy to imagine that this event might accelerate the overall length growth rate. Very recently, Stephen Cooper (University of Michigan Medical School, Ann Arbor, MI) reanalysed the length growth patterns of many individual fission yeast cells using the data from the paper of Baumgartner & Tolic-Norrelykke (2009) and argued that contrary to the author’s original conclusion, their growth followed an exponential kinetics without any breakpoints (Cooper, 2013). Since in this publication the author vehemently criticizes all former studies on (bi)linear cell growth of fission yeast, but mainly our latest one (Horvath et al., 2013), we intend to consider here the most important theoretical and practical aspects of these results and also the differences between the two kinds of interpretation. As discussed in detail in his paper (Cooper, 2013), Cooper suggests that cell growth is something like inflating a balloon with air. In his model, the interior of a cell is a homogeneous and isotropic system where ribosomes are distributed equally; therefore, any elementary part of the cytosol generates new proteins (i.e. new cell mass) at the same rate, irrespectively of any space or time limitations. Moreover, the cell must extend in volume according to the internal protein synthesis rate, which must result in an exponential growth mode. By contrast, we assume the cells to be polarized and internally structured entities, that is, they are inhomogeneous and anisotropic. As a consequence, different parts of a cell may increase their mass at different rates, polarized growth zones may arise and cease via cytoskeletal rearrangements, and finally, the overall growth rate may also change at specific points during the cell cycle (e.g. gene dosage and mitotic effects, as discussed above). Whether a cell can increase its volume proportional to its protein content or not is also not necessarily obvious; it depends on whether its cytoskeletal network, its plasma membrane, and occasionally its cell wall may be synthesized at the required rate or not. Cooper (2013) also argues that we fundamentally misunderstood the connections between cell growth and size control, but this definitely is not the case. The old and important hypothesis of Brooks (1981) was mentioned in our paper (Horvath et al., 2013), but it was not ª 2014 Federation of European Microbiological Societies. Published by John Wiley & Sons Ltd. All rights reserved
Letter to the Editor
used as an argument against exponential growth at all – something that should be clear from a careful reading of our text. Our view on this point was published 10 years ago (Sveiczer et al., 2004), being very close to that of Cooper’s view. Turning to the technical points of studying fission yeast growth, one should monitor cell volume, mass, length or protein content during the division cycle of individual cells or of an ‘average’ cell generated from the individuals. The latest studies all dealt with cell length (Baumgartner & Tolic-Norrelykke, 2009; Cooper, 2013; Horvath et al., 2013). First, in sharp contrast to the view of Cooper, one always has to plot cell length versus time on a linear scale first and consider what functions to use to best fit the data. The corresponding functions can be chosen either on the basis of theoretical (physiological) consideration or of visual consideration, as the trend of the data might suggest a specific formula. Semi-logarithmic scale should be used only if an exponential formula is a convention to describe the phenomenon because of broadly accepted previous results or a natural law. Cooper (2013) believes that exponential growth during the cycle without any breakpoint is a universal rule; whereas we argue that it should be tested and used only when supported by experimental data. Moreover, small differences can be obscured on a logarithmic scale; therefore, rate change points are more noticeable on a linear scale. Functions have to be fitted to the data by statistically adequate methods such as nonlinear regression, which can now be performed conveniently with various softwares specifically designed for such purposes. These methods are suitable to find the appropriate parameter values of the model by optimizing the fit, typically by minimizing the sum of squared error (SSE). Then, the adequacy of different models (functions) can be judged based on the quality of the fit that they provide (i.e. the lowest optimized SSE). One also needs to remember that for differently parameterized functions, more sophisticated model selection criteria, such as the Akaike Information Criterion (AIC), have to be used (Buchwald & Sveiczer, 2006). SSE characterizes only the quality of the fit; the more complex model selection criteria, such as AIC (which also contains SSE in its formula), also account for the number of adjustable parameters used to achieve this fit tending to favour the simplest model (i.e. the one with the lowest number of parameters) that still gives an adequate fit. In other words, a model having a sufficiently low SSE (corresponding to a high correlation coefficient r2) does not necessarily mean that it is the best to describe the phenomenon; other aspects have to be considered as well. Distinguishing among different models is sometimes possible even simply visually; however, it is often less clear, and the above mentioned statistical methods are intended exactly for such purposes. Measurement errors might also cause FEMS Yeast Res 14 (2014) 679–682
Letter to the Editor
difficulties and cannot always be compensated by sophisticated mathematical methods. Nevertheless, it is unacceptable to conclude that observed (multi)linear growth patterns are nothing but artifacts generated by the applied methods as Cooper (2013) suggests. An additional analysis could be to focus on the differential (rate of change): calculate the change in cell length per unit time, DL/Dt and compare this to the corresponding first derivative of the fitted model, dL/dt, where L represents cell length measured as a function of time, t. For a multilinear model with sharp transitions, this first derivative should consist of steplike horizontal lines, whereas, for an exponential one, it should also be an exponential function. Unfortunately, this mathematical transformation extremely enlarges the scatter due to experimental error; therefore, it can be used only to visualize the data and not to distinguish between rival models. It has to be noted that in his paper (Cooper, 2013), Cooper argues that a first order function (line) can be drawn on the difference pattern of an average cell’s growth curve at two different temperatures [his fig. 6. – also generated from the data of Baumgartner & Tolic-Norrelykke (2009)]. Should he be right, the growth of the average cell would follow a quadratic (square) function rather than the exponential one favoured by the author. It is important to pinpoint here that growth mode (linear or exponential) and the presence (or absence) of RCPs are two different features of cellular extension; therefore, they should be studied separately. Cooper (2013) theoretically mixes these aspects as he cannot accept any other growth model than a pure exponential one without any RCP. Let us consider RCPs first. For the fission yeast cell cycle, there is such strong evidence for their presence (Mitchison, 2003), that not accepting them is essentially ignoring scientific reality. Just one example: as it was recently reviewed, it has been long known that the general transcription rate shows a step increase after DNA replication in S. pombe (Marguerat & Bahler, 2012), that is, gene dosage has a strong effect on RNA synthesis, therefore, also on protein synthesis and finally on overall cell growth as well. As the existence of RCPs has become obvious in several different model organisms besides fission yeast by now, we may raise the question: how the cells grow between two consecutive RCPs? Is there a default cellular growth mode if it is not perturbed by discrete events of the cell cycle? Unfortunately, this question cannot be answered at the moment. Were it once proved and verified to be the same function in very different cell types and in every individual cell studied, only in that case had we the right to conclude that there is a universal rule for the default growth mode. It is interesting that in our recent fission yeast study (Horvath et al., 2013), we
FEMS Yeast Res 14 (2014) 679–682
681
found that even individual cells of a steady-state population might follow different growth models; namely, of the 60 cells analysed, length growth pattern was found to be bilinear in 42 cases, linear in 13 cases and exponential in five cases. Whether this is a real heterogeneity or it comes from measurement error, is not clear at the moment (Horvath et al., 2013). It is also noteworthy to mention here that to date, models with RCPs were usually considered multilinear ones for simplicity, however, nothing rules out the existence of multi-exponential ones. The largest problem with such an approach is probably the resolution and the accuracy of the present techniques. As studying growth pattern and selecting the most adequate model even during a whole cycle is problematic, one can imagine how difficult (or nearly impossible) were examining this phenomenon restricted to, say, early G1 or early G2, parts that are no longer than 20–25% of the cycle. Although several cell cycle events can be selectively blocked and a specific cell cycle phase can be expanded this way, one cannot be sure that the applied method does not falsify the studied process. For example, when rat Schwann cells were arrested in S phase by aphidicolin, a linear volume growth pattern was observed (Conlon & Raff, 2003); nevertheless, this does not necessarily prove that these types of cells really grow linearly during an unperturbed cycle. Individual cells from a blocked cell cycle mutant culture of fission yeast definitely showed a decrease in length growth rate after some hours (Mitchison & Nurse, 1985). Some years ago, it was clearly demonstrated that in fission yeast cells larger than twice that of the normal wild-type size, some factors started to limit transcription and therefore cell growth (Zhurinsky et al., 2010). The total transcription rate plateaued, because the DNA-to-protein ratio became too low in these large cells. Rate limiting factors may be present in the cell, for example, the RNA polymerase complex, which limits general transcription in oversized cells, and the same factor might cause an RCP in growth in normal cell cycles via, for example, a gene dosage effect (Navarro et al., 2012). So, how to proceed with studying and understanding cell growth, this fundamental physiological phenomenon? First of all, more sophisticated techniques should be developed to measure more precisely the individual cell’s mass, volume, protein content or length. After generating more reliable growth patterns during the division cycle, statistical methods will help us to select the most adequate model to determine the positions of existing RCPs, whose connections to specific cell cycle events should also be clearly explored. Finally, we can try to examine how cells grow between these RCPs, that is, to find out if there is a general default growth mode (linear or exponential) or not. These are exciting challenges for the future.
ª 2014 Federation of European Microbiological Societies. Published by John Wiley & Sons Ltd. All rights reserved
682
Acknowledgement This project is supported by the Hungarian Scientific Research Fund (OTKA K-76229).
References Baum B, Nishitani H, Yanow S & Nurse P (1998) Cdc18 transcription and proteolysis couple S phase to passage through mitosis. EMBO J 17: 5689–5698. Baumgartner S & Tolic-Norrelykke IM (2009) Growth pattern of single fission yeast cells is bilinear and depends on temperature and DNA synthesis. Biophys J 96: 4336–4347. Black JG (2013) Microbiology, 8th edn. John Wiley & Sons Inc., Hoboken, NJ. Brooks RF (1981) Variability in the cell cycle and the control of proliferation. The Cell Cycle, (John PCL, Ed), pp. 35–61. Cambridge University Press, Cambridge. Buchwald P & Sveiczer A (2006) The time-profile of cell growth in fission yeast: model selection criteria favoring bilinear models over exponential ones. Theor Biol Med Model 3: 16. Conlon I & Raff M (2003) Differences in the way a mammalian cell and yeast cells coordinate cell growth and cell-cycle progression. J Biol 2: 7. Cooper S (2013) Schizosaccharomyces pombe grows exponentially during the division cycle with no rate change points. FEMS Yeast Res 13: 650–658. Horvath A, Racz-M onus A, Buchwald P & Sveiczer A (2013) Cell length growth in fission yeast: an analysis of its bilinear character and the nature of its rate change transition. FEMS Yeast Res 13: 635–649. Marguerat S & Bahler J (2012) Coordinating genome expression with cell size. Trends Genet 28: 560–565.
ª 2014 Federation of European Microbiological Societies. Published by John Wiley & Sons Ltd. All rights reserved
Letter to the Editor
Martin SG & Chang F (2005) New end take off: regulating cell polarity during the fission yeast cell cycle. Cell Cycle 4: 1046–1049. Mitchison JM (2003) Growth during the cell cycle. Int Rev Cytol 226: 165–258. Mitchison JM & Nurse P (1985) Growth in cell length in the fission yeast Schizosaccharomyces pombe. J Cell Sci 75: 357–376. Morgan DO (2007) The Cell Cycle. New Science Press Ltd, London. Navarro FJ, Weston L & Nurse P (2012) Global control of cell growth in fission yeast and its coordination with the cell cycle. Curr Opin Cell Biol 24: 833–837. Sveiczer A, Novak B & Mitchison JM (1996) The size control of fission yeast revisited. J Cell Sci 109: 2947–2957. Sveiczer A, Novak B & Mitchison JM (2004) Size control in growing yeast and mammalian cells. Theor Biol Med Model 1: 12. Zhurinsky J, Leonhard K, Watt S, Marguerat S, Bahler J & Nurse P (2010) A coordinated global control over cellular transcription. Curr Biol 20: 2010–2015.
Akos Sveiczer, Anna Horvath Department of Applied Biotechnology and Food Science Budapest University of Technology and Economics Budapest, Hungary E-mail: [email protected] Peter Buchwald Department of Molecular and Cellular Pharmacology and Diabetes Research Institute Miller School of Medicine University of Miami, Miami, FL, USA
FEMS Yeast Res 14 (2014) 679–682
Is there a universal rule for cellular growth? – Problems in studying and interpreting this phenomenon
YEAST RESEARCH
DOI: 10.1111/1567-1364.12168
There is an enormous number of different cells in the biosphere, prokaryotes and eukaryotes, many of them living as unicellular organisms, while others forming complex multicellular beings. Some cells grow and divide generating offsprings; others are in a dormant state or generate gametes or spores. It is not too hard to conclude that it is unlikely that there is a general law describing the growth kinetics of every cell considering that several of them do not grow at all. However, when unicellular microorganisms (like bacteria or yeasts) reside in a stirred liquid medium containing appropriate nutrients both in qualitative and quantitative terms, the pure culture soon reaches an exponential phase of growth where the cell number (or cell mass) increases exponentially in time (Black, 2013). Notably, the known general growth kinetics of this asynchronous population tells us nothing about the growth kinetics of mass observed in the individual cells of the population. By the same logic, the ‘simplest’ way for the cells to grow is an exponential mode; however, there is no unequivocal proof that such a universal law exists. Theoretically, any continuous mathematical function might describe cellular growth, if it fulfils two requirements; namely, both cell mass must double during the cycle time, and also, the rate of cell mass increase must do the same – even if not necessarily for every individual cell – but at least for the hypothetical ‘average’ cell representing the population. This way, a symmetric cell division can generate two identical progenies whose physiological states are equivalent with each other as well as with that of their mother cell’s state at the beginning of the previous cycle (Mitchison, 2003). The exponential model has the advantage that cell mass is automatically proportional to the rate of cell mass increase; therefore, there is no need for any breakpoint(s) (or rate change point(s)) to be involved (Mitchison, 2003). Moreover, one need not bother either with different cellular structures (like cytoskeleton, cell membrane or cell wall) or discrete events during the cycle (like chromosome replication, segregation or cytokinesis). By contrast, a linear model is more problematic. As growth rate is constant along a linear growth pattern, it cannot follow cell mass increase during the division cycle. Therefore, one must involve breakpoints in the model where the
FEMS Yeast Res 14 (2014) 679–682
growth rate (abruptly or smoothly) changes to satisfy the general requirements, that is, that both cell mass and its growth rate must double within a cycle (Horvath et al., 2013). Were these breakpoints arbitrary without any sense, we should really throw this model out into a dustbin. However, the above mentioned cellular structures and discrete events give us several good reasons to suppose the existence of rate change points. For example, DNA replication is thought to accelerate growth rate, an effect also known as the gene dosage effect. By contrast, mitosis in eukaryotes decreases growth rate because mitotic chromosomes condense to such a compact state that makes RNA transcription nearly or totally impossible (Mitchison, 2003). In lower eukaryotes like fission yeast, mitotic chromosomes are less condensed than in higher eukaryotes, which makes possible some transcription even at this cell cycle stage, for example, the Cdc10 transcription factor is active and helps transcribe some genes, whose products are necessary for the next G1/S transition (Baum et al., 1998), however, this transcription activity is severely reduced compared with its level operating during interphase. A further point to be considered is that cytoskeleton rearranges before and during mitosis (or sometimes even in interphase); moreover, cells with a rigid wall must produce septa before cell division (Morgan, 2007) – these facts might also have negative effects on overall growth processes. Lacking a general law for the growth of cells, exploration of this field requires the study of adequate model organisms that are sufficiently representative, the use of sophisticated investigational techniques to increase accuracy, and the use of mathematical methods and adequate statistical criteria to explore different functions and select the best fitting one. During the last 5 years, two papers examined in detail cell length (proportional to cell volume) growth patterns for the unicellular fission yeast model organism, Schizosaccharomyces pombe (Baumgartner & Tolic-Norrelykke, 2009; Horvath et al., 2013). Both concluded that fission yeast cells grew bilinearly with the only discrepancy being on whether the observed rate change between the linear segments was sharp or smooth. The first observations on the bilinear growth mode of fission yeast cells were published nearly 30 years ago
ª 2014 Federation of European Microbiological Societies. Published by John Wiley & Sons Ltd. All rights reserved
680
(Mitchison & Nurse, 1985), and this result was verified and extended to several cell cycle mutants 10 years later (Sveiczer et al., 1996). A feasible explanation for this fact was also given at that time. Namely at the beginning of the cycle the rod shaped cell extended only at its old end (which had been a growing cell end in the previous cycle), while in mid G2 phase, at the so-called NETO (new end take off) event, the newly formed other cell end also started to extend (Mitchison & Nurse, 1985). This unipolar to bipolar growth switch was caused by cytoskeletal rearrangements forming a new growth zone (Martin & Chang, 2005), and it is easy to imagine that this event might accelerate the overall length growth rate. Very recently, Stephen Cooper (University of Michigan Medical School, Ann Arbor, MI) reanalysed the length growth patterns of many individual fission yeast cells using the data from the paper of Baumgartner & Tolic-Norrelykke (2009) and argued that contrary to the author’s original conclusion, their growth followed an exponential kinetics without any breakpoints (Cooper, 2013). Since in this publication the author vehemently criticizes all former studies on (bi)linear cell growth of fission yeast, but mainly our latest one (Horvath et al., 2013), we intend to consider here the most important theoretical and practical aspects of these results and also the differences between the two kinds of interpretation. As discussed in detail in his paper (Cooper, 2013), Cooper suggests that cell growth is something like inflating a balloon with air. In his model, the interior of a cell is a homogeneous and isotropic system where ribosomes are distributed equally; therefore, any elementary part of the cytosol generates new proteins (i.e. new cell mass) at the same rate, irrespectively of any space or time limitations. Moreover, the cell must extend in volume according to the internal protein synthesis rate, which must result in an exponential growth mode. By contrast, we assume the cells to be polarized and internally structured entities, that is, they are inhomogeneous and anisotropic. As a consequence, different parts of a cell may increase their mass at different rates, polarized growth zones may arise and cease via cytoskeletal rearrangements, and finally, the overall growth rate may also change at specific points during the cell cycle (e.g. gene dosage and mitotic effects, as discussed above). Whether a cell can increase its volume proportional to its protein content or not is also not necessarily obvious; it depends on whether its cytoskeletal network, its plasma membrane, and occasionally its cell wall may be synthesized at the required rate or not. Cooper (2013) also argues that we fundamentally misunderstood the connections between cell growth and size control, but this definitely is not the case. The old and important hypothesis of Brooks (1981) was mentioned in our paper (Horvath et al., 2013), but it was not ª 2014 Federation of European Microbiological Societies. Published by John Wiley & Sons Ltd. All rights reserved
Letter to the Editor
used as an argument against exponential growth at all – something that should be clear from a careful reading of our text. Our view on this point was published 10 years ago (Sveiczer et al., 2004), being very close to that of Cooper’s view. Turning to the technical points of studying fission yeast growth, one should monitor cell volume, mass, length or protein content during the division cycle of individual cells or of an ‘average’ cell generated from the individuals. The latest studies all dealt with cell length (Baumgartner & Tolic-Norrelykke, 2009; Cooper, 2013; Horvath et al., 2013). First, in sharp contrast to the view of Cooper, one always has to plot cell length versus time on a linear scale first and consider what functions to use to best fit the data. The corresponding functions can be chosen either on the basis of theoretical (physiological) consideration or of visual consideration, as the trend of the data might suggest a specific formula. Semi-logarithmic scale should be used only if an exponential formula is a convention to describe the phenomenon because of broadly accepted previous results or a natural law. Cooper (2013) believes that exponential growth during the cycle without any breakpoint is a universal rule; whereas we argue that it should be tested and used only when supported by experimental data. Moreover, small differences can be obscured on a logarithmic scale; therefore, rate change points are more noticeable on a linear scale. Functions have to be fitted to the data by statistically adequate methods such as nonlinear regression, which can now be performed conveniently with various softwares specifically designed for such purposes. These methods are suitable to find the appropriate parameter values of the model by optimizing the fit, typically by minimizing the sum of squared error (SSE). Then, the adequacy of different models (functions) can be judged based on the quality of the fit that they provide (i.e. the lowest optimized SSE). One also needs to remember that for differently parameterized functions, more sophisticated model selection criteria, such as the Akaike Information Criterion (AIC), have to be used (Buchwald & Sveiczer, 2006). SSE characterizes only the quality of the fit; the more complex model selection criteria, such as AIC (which also contains SSE in its formula), also account for the number of adjustable parameters used to achieve this fit tending to favour the simplest model (i.e. the one with the lowest number of parameters) that still gives an adequate fit. In other words, a model having a sufficiently low SSE (corresponding to a high correlation coefficient r2) does not necessarily mean that it is the best to describe the phenomenon; other aspects have to be considered as well. Distinguishing among different models is sometimes possible even simply visually; however, it is often less clear, and the above mentioned statistical methods are intended exactly for such purposes. Measurement errors might also cause FEMS Yeast Res 14 (2014) 679–682
Letter to the Editor
difficulties and cannot always be compensated by sophisticated mathematical methods. Nevertheless, it is unacceptable to conclude that observed (multi)linear growth patterns are nothing but artifacts generated by the applied methods as Cooper (2013) suggests. An additional analysis could be to focus on the differential (rate of change): calculate the change in cell length per unit time, DL/Dt and compare this to the corresponding first derivative of the fitted model, dL/dt, where L represents cell length measured as a function of time, t. For a multilinear model with sharp transitions, this first derivative should consist of steplike horizontal lines, whereas, for an exponential one, it should also be an exponential function. Unfortunately, this mathematical transformation extremely enlarges the scatter due to experimental error; therefore, it can be used only to visualize the data and not to distinguish between rival models. It has to be noted that in his paper (Cooper, 2013), Cooper argues that a first order function (line) can be drawn on the difference pattern of an average cell’s growth curve at two different temperatures [his fig. 6. – also generated from the data of Baumgartner & Tolic-Norrelykke (2009)]. Should he be right, the growth of the average cell would follow a quadratic (square) function rather than the exponential one favoured by the author. It is important to pinpoint here that growth mode (linear or exponential) and the presence (or absence) of RCPs are two different features of cellular extension; therefore, they should be studied separately. Cooper (2013) theoretically mixes these aspects as he cannot accept any other growth model than a pure exponential one without any RCP. Let us consider RCPs first. For the fission yeast cell cycle, there is such strong evidence for their presence (Mitchison, 2003), that not accepting them is essentially ignoring scientific reality. Just one example: as it was recently reviewed, it has been long known that the general transcription rate shows a step increase after DNA replication in S. pombe (Marguerat & Bahler, 2012), that is, gene dosage has a strong effect on RNA synthesis, therefore, also on protein synthesis and finally on overall cell growth as well. As the existence of RCPs has become obvious in several different model organisms besides fission yeast by now, we may raise the question: how the cells grow between two consecutive RCPs? Is there a default cellular growth mode if it is not perturbed by discrete events of the cell cycle? Unfortunately, this question cannot be answered at the moment. Were it once proved and verified to be the same function in very different cell types and in every individual cell studied, only in that case had we the right to conclude that there is a universal rule for the default growth mode. It is interesting that in our recent fission yeast study (Horvath et al., 2013), we
FEMS Yeast Res 14 (2014) 679–682
681
found that even individual cells of a steady-state population might follow different growth models; namely, of the 60 cells analysed, length growth pattern was found to be bilinear in 42 cases, linear in 13 cases and exponential in five cases. Whether this is a real heterogeneity or it comes from measurement error, is not clear at the moment (Horvath et al., 2013). It is also noteworthy to mention here that to date, models with RCPs were usually considered multilinear ones for simplicity, however, nothing rules out the existence of multi-exponential ones. The largest problem with such an approach is probably the resolution and the accuracy of the present techniques. As studying growth pattern and selecting the most adequate model even during a whole cycle is problematic, one can imagine how difficult (or nearly impossible) were examining this phenomenon restricted to, say, early G1 or early G2, parts that are no longer than 20–25% of the cycle. Although several cell cycle events can be selectively blocked and a specific cell cycle phase can be expanded this way, one cannot be sure that the applied method does not falsify the studied process. For example, when rat Schwann cells were arrested in S phase by aphidicolin, a linear volume growth pattern was observed (Conlon & Raff, 2003); nevertheless, this does not necessarily prove that these types of cells really grow linearly during an unperturbed cycle. Individual cells from a blocked cell cycle mutant culture of fission yeast definitely showed a decrease in length growth rate after some hours (Mitchison & Nurse, 1985). Some years ago, it was clearly demonstrated that in fission yeast cells larger than twice that of the normal wild-type size, some factors started to limit transcription and therefore cell growth (Zhurinsky et al., 2010). The total transcription rate plateaued, because the DNA-to-protein ratio became too low in these large cells. Rate limiting factors may be present in the cell, for example, the RNA polymerase complex, which limits general transcription in oversized cells, and the same factor might cause an RCP in growth in normal cell cycles via, for example, a gene dosage effect (Navarro et al., 2012). So, how to proceed with studying and understanding cell growth, this fundamental physiological phenomenon? First of all, more sophisticated techniques should be developed to measure more precisely the individual cell’s mass, volume, protein content or length. After generating more reliable growth patterns during the division cycle, statistical methods will help us to select the most adequate model to determine the positions of existing RCPs, whose connections to specific cell cycle events should also be clearly explored. Finally, we can try to examine how cells grow between these RCPs, that is, to find out if there is a general default growth mode (linear or exponential) or not. These are exciting challenges for the future.
ª 2014 Federation of European Microbiological Societies. Published by John Wiley & Sons Ltd. All rights reserved
682
Acknowledgement This project is supported by the Hungarian Scientific Research Fund (OTKA K-76229).
References Baum B, Nishitani H, Yanow S & Nurse P (1998) Cdc18 transcription and proteolysis couple S phase to passage through mitosis. EMBO J 17: 5689–5698. Baumgartner S & Tolic-Norrelykke IM (2009) Growth pattern of single fission yeast cells is bilinear and depends on temperature and DNA synthesis. Biophys J 96: 4336–4347. Black JG (2013) Microbiology, 8th edn. John Wiley & Sons Inc., Hoboken, NJ. Brooks RF (1981) Variability in the cell cycle and the control of proliferation. The Cell Cycle, (John PCL, Ed), pp. 35–61. Cambridge University Press, Cambridge. Buchwald P & Sveiczer A (2006) The time-profile of cell growth in fission yeast: model selection criteria favoring bilinear models over exponential ones. Theor Biol Med Model 3: 16. Conlon I & Raff M (2003) Differences in the way a mammalian cell and yeast cells coordinate cell growth and cell-cycle progression. J Biol 2: 7. Cooper S (2013) Schizosaccharomyces pombe grows exponentially during the division cycle with no rate change points. FEMS Yeast Res 13: 650–658. Horvath A, Racz-M onus A, Buchwald P & Sveiczer A (2013) Cell length growth in fission yeast: an analysis of its bilinear character and the nature of its rate change transition. FEMS Yeast Res 13: 635–649. Marguerat S & Bahler J (2012) Coordinating genome expression with cell size. Trends Genet 28: 560–565.
ª 2014 Federation of European Microbiological Societies. Published by John Wiley & Sons Ltd. All rights reserved
Letter to the Editor
Martin SG & Chang F (2005) New end take off: regulating cell polarity during the fission yeast cell cycle. Cell Cycle 4: 1046–1049. Mitchison JM (2003) Growth during the cell cycle. Int Rev Cytol 226: 165–258. Mitchison JM & Nurse P (1985) Growth in cell length in the fission yeast Schizosaccharomyces pombe. J Cell Sci 75: 357–376. Morgan DO (2007) The Cell Cycle. New Science Press Ltd, London. Navarro FJ, Weston L & Nurse P (2012) Global control of cell growth in fission yeast and its coordination with the cell cycle. Curr Opin Cell Biol 24: 833–837. Sveiczer A, Novak B & Mitchison JM (1996) The size control of fission yeast revisited. J Cell Sci 109: 2947–2957. Sveiczer A, Novak B & Mitchison JM (2004) Size control in growing yeast and mammalian cells. Theor Biol Med Model 1: 12. Zhurinsky J, Leonhard K, Watt S, Marguerat S, Bahler J & Nurse P (2010) A coordinated global control over cellular transcription. Curr Biol 20: 2010–2015.
Akos Sveiczer, Anna Horvath Department of Applied Biotechnology and Food Science Budapest University of Technology and Economics Budapest, Hungary E-mail: [email protected] Peter Buchwald Department of Molecular and Cellular Pharmacology and Diabetes Research Institute Miller School of Medicine University of Miami, Miami, FL, USA
FEMS Yeast Res 14 (2014) 679–682
