On exponential or bilinear cell growth during division cycle
Stephen Cooper, University of Michigan Medical School
16 May 2006
An alternative view of the problem is presented in a previously published paper (Cooper, S. 2006 Distinguishing between linear and exponential cell growth during the division cycle: Single-cell studies, cell-culture studies, and the object of cell-cycle research. Theoretical Biology and Medical Modeling.http://www.tbiomed.com/content/3/1/10).
When comparing the two papers the following points should be noted:
First, the Buchwald-Sveiczer paper analyzes the data on only two cells. The previously published paper (Cooper, 2006) pointed out that it is not correct to use the data on only two cells to describe a general rule, particularly when the original data was selected from a larger number of cells. Until a complete presentation of all cells are given, it has been argued (Cooper, 2006) that artifacts in cell growth on a solid surface may give the published results of Buchwald and Sveiczer, To be more precise, it is not argued that the data of Buchwald-Sveiczer is incorrect, but that it is merely the result of two cells and these two cells are not necessarily representative of cell growth of all cells in the culture.
Second, Buchwald and Sveizer do not deal with the more important analysis of the biochemical basis for deciding between exponential and bilinear growth. A reading of both papers side-by-side will illuminate the problem.
Third, the postulation of gene dosage by Buchwald and Sveiczer is difficult to accept as merely increasing the rate of production of some cell product is not expected to lead to a doubling in the rate of cell growth. For example, if growth is depending on the activity of the protein synthesizing system (ribosomes, tRNAs, RNA polymerases, etc.) it is difficult to know how a doubling in the rate of production of some cell product could lead to a doubling in the rate of protein synthesis. Rather, one would expect, for any gene-dosage model, that there would only be a slow change in the rate of protein synthesis. A similar problem exists if one postulates cell-cycle dependent changes in protein breakdown or turnover.
Finally, there is ample experimental evidence supporting exponential growth during the division cycle, and this is summarized by Cooper (2006).
It is of interest to note that Buchwald and Sveiczer write that "...the statistical evidence suggesting a bilinear dependence rather than an exponential one is not strong enough to favor one model unequivocally over the other." I strongly agree with this statement. Given the limited set of data (two cells), and the theoretical problems as described in detail by Cooper (2006), I suggest, as I have written previously, that cells grow exponentially during the division cycle.
Competing interests
There are no competing interests.
Re: "On exponential or bilinear cell growth during division cycle"
Peter Buchwald, IVAX Research, Inc.
31 August 2006
In reply to Stephen Cooper's comments of May 16, 2006, we would like to emphasize again that the main goal of our article [1] was to provide a rigorous mathematical framework to compare the adequacy of bilinear vs. exponential models in fitting a given set of growth data regardless of the actual nature of the data. Among other things, this was intended to address the issues raised specifically by Cooper himself in an earlier Microbiology comment [2] since if different models are fitted to the same data and the numerical differences are relatively small, the correlation coefficient (r) alone is not adequate to decide which model is the more accurate. Occam's razor, a frequently quoted principle, is indeed a main modeling guideline, but is not sufficient: a good model should be as parsimonious as possible, but the simplest model is not necessarily the best one. The complexity of the model should be judged in the light of its goodness-of-fit as well as its generalizability and predictive power, and in addition to r, there are now more sophisticated quantitative model selection criteria that can and should be used when making such decisions.
While it is true that the current paper presents a detailed analysis of only two single cells and this cannot provide sufficient statistical evidence to settle the exponential vs. bilinear debate once and for all, it is also true that these cells are selected as representative of the growth process during the analysis of a large number of cell cycles (40-80 for each strain; see the Materials and Methods section and Table 2 of [3]) and analysis of single cell data cannot be simply dismissed as entirely useless. Certainly, using average ± SD type data obtained from a sufficiently large number of cells would be a statistically much more adequate approach, but we are certain that such data will be generated in the future, and, hopefully, the mathematical models and quantitative modeling criteria published here will be used to decide which model is more adequate for describing cell growth.
Furthermore, whereas the WT cell data are somewhat less clear and disputable, the wee1Δ cell data are so clearly bilinear and not exponential that it is difficult to argue against them. A p value of 0.000002 is quite strong statistical evidence, even if it comes from analysis of a single cell. It should also be mentioned here that (as clearly stated in [1]) the final magnification was improved in the case of wee1Δ to eliminate the problems caused by its smaller size. Studying the geometry of fission yeast cells on solid agar has been the most frequently used technique for many decades; therefore, Cooper's argument about the possible artifacts of this technique is a weak and probably incorrect one.
Growth data on single organisms might not be the perfect way to decide the adequacy of growth models, but they are neither futile nor worthless. As a relevant example, Cooper himself used two individual growth profiles from his own measurements of his two grandchildren in his recent paper [4] (Figure 3) to illustrate that such individual profiles are more scattered than the smooth profile corresponding to the 50th percentile stature-for-age chart developed by the US Department of Health and Human Service's Centers for Disease Control and Prevention (CDC) and National Center for Health Statistics using data from a large number of individuals (Figure 1 in the same paper [4]). However, even a cursory look at these single individual data makes it quite clear that these growth profiles follow the general human trendline and are, for example, not typical sigmoid growth profiles characteristic for many animal species (e.g., guinea pigs, rats, dogs). It is well-known that compared to other species, humans are unique even in their growth patterns because of their paedomorphic character, extended growth period, and unusually delayed and distinctive adolescent growth spurt [5-8]; they deviate unusually strongly [7] from the general West model for ontogenetic growth [9]. Therefore, even growth data from single, representative human individuals could be sufficient to make it clear that classic three- or four-parameter sigmoid models, such as the logistic, Weibull, Gompertz, or Richards models that work well for some animal species [9, 10], are inadequate for human growth, and other models such as the more complex seven-parameter JPSS model [11, 12] are needed. Similarly, single cell data may not provide the definitive answer in establishing the correct profile, but they can be the first steps in the right direction and cannot be dismissed in their entirety.
To return to the case of fission yeast, DNA replication is a relatively fast process at the scale of the cell division cycle; therefore, the effect of gene dosage on growth might be sharp. Moreover, any gene dosage effect, whether it is slow or sharp, must cause a discontinuous rate change (either an extended transient section or a sudden breakpoint, respectively) and this is inconsistent with a continuous exponential function. Passing through some cell cycle phase borderlines (and probably checkpoints as well) obviously causes cytoskeletal rearrangements and, as a consequence, abrupt changes in the length growth rate in fission yeast (called rate change points). Unfortunately, because the time interval between two consecutive rate change points is too small, it is difficult to establish whether growth is linear or exponential.
Theoretical biology searches for general laws that describe biological phenomena. When introduced for the first time, any model represents only a hypothesis, which could become a law only after being verified and validated in many different experiments. Furthermore, because of biological diversity, exceptions might always exist, even for laws that are usually valid. Therefore, one should never simply extend a law (or rather a hypothesis) that one favors and assume with certainty that it will be followed by any new, unstudied organisms; or worse yet, if any exception is seen, one should not immediately argue, as Stephen Cooper does here, that it must be some kind of an artifact.
Peter Buchwald
Akos Sveiczer
References
1. Buchwald P, Sveiczer A: The time-profile of cell growth in fission yeast: model selection criteria favoring bilinear models over exponential ones. Theor Biol Med Model 2006, 3:16.
2. Cooper S: Length extension in growing yeast: is growth exponential? - Yes. Microbiology 1998, 144:263-265.
3. Sveiczer A, Novak B, Mitchison JM: The size control of fission yeast revisited. J Cell Sci 1996, 109:2947-2957.
4. Cooper S: Distinguishing between linear and exponential cell growth during the division cycle: Single-cell studies, cell-culture studies, and the object of cell-cycle research. Theor Biol Med Model 2006, 3:10.
5. Bogin B: Evolutionary perspective on human growth. Annu Rev Anthropol 1999, 28:109-153.
6. Leigh SR: Evolution of human growth. Evol Anthropol 2001, 10:223-236.
7. Vinicius L: Human encephalization and developmental timing. J Hum Evol 2005, 49:762-776.
8. Walker R, Hill K, Burger O, Hurtado AM: Life in the slow lane revisited: Ontogenetic separation between chimpanzees and humans. Am J Phys Anthropol 2006, 129:577-583.
9. West GB, Brown JH, Enquist BJ: A general model for ontogenetic growth. Nature 2001, 413:628-631.
10. López S, France J, Gerrits WJJ, Dhanoa MS, Humphries DJ, Dijkstra J: A generalized Michaelis-Menten equation for the analysis of growth. J Anim Sci 2000, 78:1816-1828.
11. Jolicoeur P, Pontier J, Pernin MO, Sempe M: A lifetime asymptotic growth curve for human height. Biometrics 1988, 44:995-1003.
12. Jolicoeur P, Abidi H, Pontier J: Human stature: which growth model? Growth Dev Aging 1991, 55:129-122.
Theoretical Biology and Medical Modelling
On exponential or bilinear cell growth during division cycle
16 May 2006
An alternative view of the problem is presented in a previously published paper (Cooper, S. 2006 Distinguishing between linear and exponential cell growth during the division cycle: Single-cell studies, cell-culture studies, and the object of cell-cycle research. Theoretical Biology and Medical Modeling.http://www.tbiomed.com/content/3/1/10).
When comparing the two papers the following points should be noted:
First, the Buchwald-Sveiczer paper analyzes the data on only two cells. The previously published paper (Cooper, 2006) pointed out that it is not correct to use the data on only two cells to describe a general rule, particularly when the original data was selected from a larger number of cells. Until a complete presentation of all cells are given, it has been argued (Cooper, 2006) that artifacts in cell growth on a solid surface may give the published results of Buchwald and Sveiczer, To be more precise, it is not argued that the data of Buchwald-Sveiczer is incorrect, but that it is merely the result of two cells and these two cells are not necessarily representative of cell growth of all cells in the culture.
Second, Buchwald and Sveizer do not deal with the more important analysis of the biochemical basis for deciding between exponential and bilinear growth. A reading of both papers side-by-side will illuminate the problem.
Third, the postulation of gene dosage by Buchwald and Sveiczer is difficult to accept as merely increasing the rate of production of some cell product is not expected to lead to a doubling in the rate of cell growth. For example, if growth is depending on the activity of the protein synthesizing system (ribosomes, tRNAs, RNA polymerases, etc.) it is difficult to know how a doubling in the rate of production of some cell product could lead to a doubling in the rate of protein synthesis. Rather, one would expect, for any gene-dosage model, that there would only be a slow change in the rate of protein synthesis. A similar problem exists if one postulates cell-cycle dependent changes in protein breakdown or turnover.
Finally, there is ample experimental evidence supporting exponential growth during the division cycle, and this is summarized by Cooper (2006).
It is of interest to note that Buchwald and Sveiczer write that "...the statistical evidence suggesting a bilinear dependence rather than an exponential one is not strong enough to favor one model unequivocally over the other." I strongly agree with this statement. Given the limited set of data (two cells), and the theoretical problems as described in detail by Cooper (2006), I suggest, as I have written previously, that cells grow exponentially during the division cycle.
Competing interests
There are no competing interests.
Re: "On exponential or bilinear cell growth during division cycle"
31 August 2006
In reply to Stephen Cooper's comments of May 16, 2006, we would like to emphasize again that the main goal of our article [1] was to provide a rigorous mathematical framework to compare the adequacy of bilinear vs. exponential models in fitting a given set of growth data regardless of the actual nature of the data. Among other things, this was intended to address the issues raised specifically by Cooper himself in an earlier Microbiology comment [2] since if different models are fitted to the same data and the numerical differences are relatively small, the correlation coefficient (r) alone is not adequate to decide which model is the more accurate. Occam's razor, a frequently quoted principle, is indeed a main modeling guideline, but is not sufficient: a good model should be as parsimonious as possible, but the simplest model is not necessarily the best one. The complexity of the model should be judged in the light of its goodness-of-fit as well as its generalizability and predictive power, and in addition to r, there are now more sophisticated quantitative model selection criteria that can and should be used when making such decisions.
While it is true that the current paper presents a detailed analysis of only two single cells and this cannot provide sufficient statistical evidence to settle the exponential vs. bilinear debate once and for all, it is also true that these cells are selected as representative of the growth process during the analysis of a large number of cell cycles (40-80 for each strain; see the Materials and Methods section and Table 2 of [3]) and analysis of single cell data cannot be simply dismissed as entirely useless. Certainly, using average ± SD type data obtained from a sufficiently large number of cells would be a statistically much more adequate approach, but we are certain that such data will be generated in the future, and, hopefully, the mathematical models and quantitative modeling criteria published here will be used to decide which model is more adequate for describing cell growth.
Furthermore, whereas the WT cell data are somewhat less clear and disputable, the wee1Δ cell data are so clearly bilinear and not exponential that it is difficult to argue against them. A p value of 0.000002 is quite strong statistical evidence, even if it comes from analysis of a single cell. It should also be mentioned here that (as clearly stated in [1]) the final magnification was improved in the case of wee1Δ to eliminate the problems caused by its smaller size. Studying the geometry of fission yeast cells on solid agar has been the most frequently used technique for many decades; therefore, Cooper's argument about the possible artifacts of this technique is a weak and probably incorrect one.
Growth data on single organisms might not be the perfect way to decide the adequacy of growth models, but they are neither futile nor worthless. As a relevant example, Cooper himself used two individual growth profiles from his own measurements of his two grandchildren in his recent paper [4] (Figure 3) to illustrate that such individual profiles are more scattered than the smooth profile corresponding to the 50th percentile stature-for-age chart developed by the US Department of Health and Human Service's Centers for Disease Control and Prevention (CDC) and National Center for Health Statistics using data from a large number of individuals (Figure 1 in the same paper [4]). However, even a cursory look at these single individual data makes it quite clear that these growth profiles follow the general human trendline and are, for example, not typical sigmoid growth profiles characteristic for many animal species (e.g., guinea pigs, rats, dogs). It is well-known that compared to other species, humans are unique even in their growth patterns because of their paedomorphic character, extended growth period, and unusually delayed and distinctive adolescent growth spurt [5-8]; they deviate unusually strongly [7] from the general West model for ontogenetic growth [9]. Therefore, even growth data from single, representative human individuals could be sufficient to make it clear that classic three- or four-parameter sigmoid models, such as the logistic, Weibull, Gompertz, or Richards models that work well for some animal species [9, 10], are inadequate for human growth, and other models such as the more complex seven-parameter JPSS model [11, 12] are needed. Similarly, single cell data may not provide the definitive answer in establishing the correct profile, but they can be the first steps in the right direction and cannot be dismissed in their entirety.
To return to the case of fission yeast, DNA replication is a relatively fast process at the scale of the cell division cycle; therefore, the effect of gene dosage on growth might be sharp. Moreover, any gene dosage effect, whether it is slow or sharp, must cause a discontinuous rate change (either an extended transient section or a sudden breakpoint, respectively) and this is inconsistent with a continuous exponential function. Passing through some cell cycle phase borderlines (and probably checkpoints as well) obviously causes cytoskeletal rearrangements and, as a consequence, abrupt changes in the length growth rate in fission yeast (called rate change points). Unfortunately, because the time interval between two consecutive rate change points is too small, it is difficult to establish whether growth is linear or exponential.
Theoretical biology searches for general laws that describe biological phenomena. When introduced for the first time, any model represents only a hypothesis, which could become a law only after being verified and validated in many different experiments. Furthermore, because of biological diversity, exceptions might always exist, even for laws that are usually valid. Therefore, one should never simply extend a law (or rather a hypothesis) that one favors and assume with certainty that it will be followed by any new, unstudied organisms; or worse yet, if any exception is seen, one should not immediately argue, as Stephen Cooper does here, that it must be some kind of an artifact.
Peter Buchwald
Akos Sveiczer
References
1. Buchwald P, Sveiczer A: The time-profile of cell growth in fission yeast: model selection criteria favoring bilinear models over exponential ones. Theor Biol Med Model 2006, 3:16.
2. Cooper S: Length extension in growing yeast: is growth exponential? - Yes. Microbiology 1998, 144:263-265.
3. Sveiczer A, Novak B, Mitchison JM: The size control of fission yeast revisited. J Cell Sci 1996, 109:2947-2957.
4. Cooper S: Distinguishing between linear and exponential cell growth during the division cycle: Single-cell studies, cell-culture studies, and the object of cell-cycle research. Theor Biol Med Model 2006, 3:10.
5. Bogin B: Evolutionary perspective on human growth. Annu Rev Anthropol 1999, 28:109-153.
6. Leigh SR: Evolution of human growth. Evol Anthropol 2001, 10:223-236.
7. Vinicius L: Human encephalization and developmental timing. J Hum Evol 2005, 49:762-776.
8. Walker R, Hill K, Burger O, Hurtado AM: Life in the slow lane revisited: Ontogenetic separation between chimpanzees and humans. Am J Phys Anthropol 2006, 129:577-583.
9. West GB, Brown JH, Enquist BJ: A general model for ontogenetic growth. Nature 2001, 413:628-631.
10. López S, France J, Gerrits WJJ, Dhanoa MS, Humphries DJ, Dijkstra J: A generalized Michaelis-Menten equation for the analysis of growth. J Anim Sci 2000, 78:1816-1828.
11. Jolicoeur P, Pontier J, Pernin MO, Sempe M: A lifetime asymptotic growth curve for human height. Biometrics 1988, 44:995-1003.
12. Jolicoeur P, Abidi H, Pontier J: Human stature: which growth model? Growth Dev Aging 1991, 55:129-122.
Competing interests
No competing interests.