Cell size maintenance with exponential and linear mass increase
Can cells grow exponentially during the division cycle and maintain a constant cell size? Consider two possible cases of exponential growth for cells with variable cell sizes. For the first case (Fig. 1a
), three cells of the same newborn size have slightly different rates of mass increase. If all three of the cells in Fig. 1a
were to have the same interdivision time, the dividing cells would have disparate sizes. But if the interdivision times vary so that cells divide at the same cell size, then cell size is maintained even with exponential growth during the division cycle. A newborn cell that makes mass at a rate slightly faster than average will divide earlier than cells with an average or below average rate of mass increase (Fig. 1a
; arrows indicate division times). Conversely, a newborn cell producing mass at a rate slightly slower than average will divide later than cells with an average or above average rate of mass increase (Fig. 1a
). Variation in interdivision times allows maintenance of constant average cell size even with exponential mass increase. A second case (Fig. 1b
) starts with different sized newborn cells that synthesize mass at the same rate. As in Fig. 1a
, the earlier a cell reaches the division size, the earlier the cell will divide and the cell will have a shorter interdivision time. Size constancy is maintained even though mass increases at a constant rate for the three cells with different-sized newborn cells (Fig. 1b
). Mixtures of initial size variation and variation of rates of mass synthesis can be analyzed in the same manner; the analysis is strongly supported by a reanalysis of published experimental data on the variation of mammalian cell interdivision times as determined by time-lapse cinematography [3
Linear cell growth during the division cycle (Figs. 1c and 1d) can also produce size maintenance. Whether cells reach the division size earlier due to a larger initial cell size or due to a more rapid rate of mass increase, the cell size at division can be the same for all cells. Thus size maintenance is also consistent with linear growth.
The patterns shown in Figs. 1a,1b,1c,1d show that there is no impediment to size maintenance as long as interdivision times are not invariant. In all four panels in Fig. 1 the interdivision time varies depending on the time required for a newborn cell to reach a particular cell size.
To be precise, it is not proposed that cells always divide at "exactly" the same size. There is a statistical variation in mass increase and interdivision times that can lead to variations in cell size at division . The important point is that when cells deviate from the mean size there is a return to the mean size through compensating interdivision times during the next cell cycle. Large cells will have a relatively shorter interdivision time, leading to a return to the average cell size.
Further, it is not proposed that a large, newborn cell "controls" its mass increase to have a slower rate of mass increase (compared to smaller cells) thus compensating for the initial larger cell size. Nor do small cells increase their rate of mass synthesis to compensate for their initial mass deficit. Mass increase variation is postulated to have some inherent statistical variation  but with all cells, no matter what their extant size, having the same relative rate of mass increase.
There can be variability in cell mass increase with the rate of mass increase being independent of cell size. A large newborn cell could have a faster than average rate of mass increase. In this case, the interdivision time would be even shorter to compensate for both the larger initial newborn cell size and the greater than average rate of mass increase.
The size maintenance pattern is illustrated in Fig. 2, where the production of large and small cells can arise either by variation in interdivision times or deviations from equipartition. Newborn or baby cells (b) that have a relatively short interdivision time produce small (s) cells while newborn cells that have a relatively long interdivision time produce large (l) cells. Large and small cells may also be produced by deviation from equipartition at division so that an average-sized dividing cell produces one large (l) and one small (s) cell. The return of small and large cells to the average cell size occurs in the next generation by variation in interdivision times so that small cells (s) have a longer (on average) interdivision time than larger (l) cells (on average).
It may appear that this simple analysis is merely begging the question by not indicating how the cell "knows" to divide at a particular size. This question will be answered below. But first, two issues should be dealt with. An initial discussion will clarify the relationship of cell mass increase to cell size. This will be followed by a discussion of problems with the proposal of linear growth during the division cycle.
What is meant by the proposal that large cells grow faster than small cells?
What is meant by the Conlon and Raff proposal that, in yeast culture, large cells grow faster than small cells? And conversely, that in mammalian cells, large and small cells grow at the same rate? There are four different meanings that can be given the notion of the rate of mass increase and its relationship to cell size. These different meanings lead to some verbal confusion that requires clarification.
One meaning of the proposal that large cells make mass faster than small cells is that given two cells of disparate sizes, the absolute rate of increase in cell mass is greater in the larger cells. A cell of size 2.0 might add, in some time interval, 0.2 units of cell mass, while a cell of size 1.0, in that same time interval might add only 0.05 units of cell mass. This pattern is a clear and unambiguous difference in the rate of mass increase that is related to cell size.
A second meaning of cell size affecting the rate of mass increase considers that a cell of size 2.0 adds 0.2 unit of mass and a cell of size 1.0 adds 0.1 unit of cell mass over the same time interval. Of course, this case could arguably be said to be a constant rate of mass increase, as the rate of mass increase is proportional to the amount of extant mass. This second proposal is equivalent to mass increasing exponentially. This is because as extant mass changes during the cell cycle the absolute rate of mass increase also changes to reflect the newly added cell mass. After the cell of size 1.0 grows to size 1.1, in the next time interval, rather than 0.1 units of mass being added, there are 0.11 units of mass added to the cell mass. Just as interest is compounded in a bank account, and the funds grow exponentially, so mass in this second example increases exponentially.
A third meaning of the variation in mass increase with cell size is that the rate of mass increase is determined at birth and continues throughout the cell cycle, unaffected by continued cell size increase. A relatively small newborn cell could have a rate of addition of "X" units per time interval, and this rate would remain constant even as the cell increases its cell mass. The larger cell would add more than "X" units each time interval and not change this rate during the cell cycle. This pattern of increase would be called linear synthesis during the cell cycle.
It is interesting to think about these different meaning when considering the theoretical graph drawn by Conlon and Raff  to illustrate the return of cells of disparate sizes to the same cell size. As shown in Fig. 3 (redrawn from Fig. 1 of Conlon and Raff ), consider two cells, one of size 1.0 and one of size 10.0. During one generation of growth 5.5 units of mass are added by the smaller cell to produce a dividing cell of size 6.5, and 5.5 units of mass are also added to the larger cell to produce a dividing cell of size 15.5. As discussed by Conlon and Raff, upon cell division the daughter cells produced by this pattern of growth would be sizes 3.25 and 7.75. Repeating this each generation (5.5 units added to each cell independent of the extant newborn cell mass) leads, according to Conlon and Raff, to a convergence of cell size as shown in Fig 3.
But no indication of the length of the division cycles is given in Fig 3. If the interdivision times are the same for the large and small cells, which is implicit in, and not excluded by, the analysis in Fig 3 (Conlon and Raff's Fig 1), the relative rate of mass increase for the larger cell is 5.5/10.0 or 0.55 and the relative rate of mass increase for the smaller cell is 5.5/1.0 or 5.5. From this point of view, the ratio of the rates of mass increase is a factor of 10, with the smaller cell making mass from its mass at 10 times the rate (relative to extant mass) compared to the larger cell.
But if the absolute rates of mass increase were the same, then the smaller cell would have a much longer interdivision time than the larger cell. If, over a unit time, 1.0 unit of cell mass were added to the larger newborn cell, and that cell divided at size 11.0, then the interdivision time would be that unit time. The smaller cell, however, would require 10 time units for its interdivision time, because that is the time required to reach size 11.0 as 1.0 unit of material is added to each small cell during each unit of time. The smaller cell grows for a longer time before division. After this first division the new daughter cells produced by each of the initial cells would be the same size. By allowing interdivision time variation, cell size uniformity is restored in one generation.
A similar analysis can be made for exponential mass increase (i.e., mass added proportional to extant mass). If the cell of size 10.0 added 1.0 unit of mass in a unit of time, then the small cell would add 0.1 unit of mass in that same time interval. In this case, there would be even more time required for the small cell to reach the division size of 11.0. In any case, exponential growth coupled with interdivision time variation can allow size maintenance because both the large and the small newborn cells will divide at the same size, as described in Fig. 1.
Of course, the example given by Conlon and Raff (Fig 3) as discussed here is unrealistic. Cell sizes do not vary over a factor of 10 in exponential culture. But this re-analysis of Fig. 3 illustrates the power of considering different interdivision times as a factor in maintaining constant cell size.
Robert Brooks (personal communication) notes that in some of his experiments cell size is observed as very variable. He states that in experiments with Shields they found that size varied over a range of at least 6-fold. In response it can be pointed out that recent careful measurements of the size variation during the division cycle of cells grown under ideal conditions indicates that size variation is not broad [4-7]. Helmstetter (personal communication) points out that when cells are not grown under optimal conditions there are always some cells of odd or abnormal size. But these cells are cells that are dying or in some way impaired. These abnormally sized cells should not be considered as typical of the cells in a well-maintained, exponentially-growing cell culture.
The fourth part of our verbal analysis of mass increase as a function of cell size relates to bacterial cells. As will be seen below, one of the most important results in bacterial physiology is that as growth rate speeds up, cells get larger. As the growth rate of a cell is continuously varied by increasing the richness of the medium, there is a continuous variation in bacterial cell size with the faster growing cells being larger than slower growing cells . Bacterial cells with an interdivision time of 20 minutes are larger than cells with an interdivision time of 60 minutes. Although it could be said that larger cells make mass faster leading to the shorter interdivision time for larger cells, it is equally possible, and in fact preferred, to reformulate or verbalize this result by saying that faster growth produces larger cells. For a given medium the rate of mass increase is determined for the bacterial cells, and the cell size results from the growth rate. This idea, the fourth way of looking at the relationship of cell size and mass increase, will be illustrated below in the analysis of bacterial patterns of DNA replication and cell size maintenance.
As we shall see, in bacterial cells a constant period for DNA replication and a constant time between termination of replication and cell division explains the variation in bacterial cell size as a function of growth rate. This same explanation also applies to mammalian cells: the rate of growth determined by external conditions determines cell size. Rather than taking the results of Conlon and Raff and concluding that larger cells when placed in medium with more serum now grow faster, it is better, as with bacteria, to say that when cells are placed in a condition that provides faster growth (i.e., a shorter interdivision time), the cells grow larger. While this may appear, at first sight, to be a trivial and semantic difference, this distinction actually lies at the heart of the problem and is the key to the solution of size determination and size maintenance. Rather than thinking that cell size produces cells with a particular growth rate (e.g., large cells grow fast), it is preferable to think that a particular growth rate produces cells of a particular size (e.g, fast growing cells are made larger than slower growing cells).
What is wrong with linear growth?
There are problems inherent and unavoidable in any proposal of linear cell growth during the division cycle. Linear growth means that during the division cycle, as a cell proceeds from size 1.0 to size 2.0, cell mass is added at a constant amount per unit time. If a cell grows linearly, over tenths of a cell-cycle time, a cell increases its size from size 1.0, to 1.1, to 1.2, and so forth.
The main problem with linear growth (i.e., constant amounts of cell mass are added at constant time intervals) is that as the cell gets larger, the cytoplasm becomes inefficient. Inefficiency is defined here as producing less mass per extant mass compared to more efficient use of the extant mass. Efficient mass increase would exist when extant cell mass makes new mass as fast as possible. As a cell grows, more cytoplasm is present. With linear growth the extra cytoplasm does not increase the absolute rate of cell mass synthesis. In essence, the new cytoplasm does not work to make new mass. There is a decrease in the relative rate of mass increase (i.e., mass synthesis per extant mass) which means that the ribosomes, after some growth, are not working as efficiently as before there was growth.
One mechanistic model explaining the postulated absence of a change in the absolute rate of mass increase to produce linear growth is to propose that the new mass does not enter into active participation in mass synthesis until a cell division; new mass will not be "activated" to enter into mass synthesis until the next division. From this viewpoint, there is a constant rate of mass increase based on the original mass. As a cell approaches division, the efficiency of mass making new mass tends toward half that of the efficiency of the initial, newborn cell mass. An alternative mechanistic proposal to explain linear growth is that during a cell cycle the amount of material able to be taken up by a cell is constant, and only upon cell division is there an "activation" of the new cell surface so that there is an increase in the ability of the cell to take in material.
Even more important and troublesome is the result that if a cell grows linearly, at the instant of cell division there must be a sudden saltation or jump in the synthetic activity of the cytoplasm. Toward the end of the cell cycle, 1.9 units of cell mass make 0.1 unit of cell mass to achieve a cell mass of 2.0. Given linear growth, at the instant of division the 2.0 units of cell mass, now apportioned into two daughter cells, must now make, during the next time interval, 0.2 units of cell mass or twice as much as in the previous time interval. When the cell of size 2.0 divides, linear growth implies that the two new daughter cells now immediately activate the "quiescent" cytoplasmic material (or activate the previously inert cell surface uptake capabilities). Irrespective of mechanism, considering the two daughter cells together, linear growth during the division cycle inevitably implies that at division there is a sudden doubling in the rate of mass increase.
There is no known biochemical mechanism for these proposals to produce linear cell growth, or the sudden jump in the rate of mass increase. As currently understood, the new cytoplasm joins right in to make new mass. And there is no mechanism known to allow new cell surface to remain inert until a cell division. While the absence of any identification of these mechanisms does not mean that these mechanisms do not exist, there is no need to propose the existence of these mechanisms if cells grow exponentially.
The experimental evidence favors exponential mass increase during the cell cycle. In bacteria the evidence for exponential growth is extremely strong [9,10]. Analysis of data on eukaryotic cell size increase also supports exponential growth during the division cycle .
What of the experiments presented by Conlon and Raff  that cell mass increases linearly? Conlon and Raff studied cells cultured in 1% fetal calf serum, forskolin, and aphidicolin. Aphidicolin is an inhibitor of DNA synthesis. While mass increased, there was no concomitant increase in DNA. The cells were incubated for 216 hours (9 days). The cell volume was measured using a Coulter Counter, although in one experiment total protein content was measured.
Conlon and Raff realized that it is extremely difficult to distinguish linear from exponential growth over one doubling time. Therefore they measured mass increase over a longer period of time (approximately 3 or more normal interdivision times). The problem with this experiment is that the inhibited cells do not allow an exponential increase in cell number as DNA synthesis is inhibited. Therefore the experiment is subject to the critique that aphidicolin inhibition produced the observed results. The results may not, and very likely do not, reflect the situation in normal, uninhibited, and unperturbed cells. For example, there could have been exponential growth during the first "virtual cell cycle". Then the limitations of DNA content would lead to the observed linearity of growth as measured over the extended period of analysis. But this linearity should not be taken as an indication that during the normal cell cycle the cell mass increases linearly.
Even if cells grow linearly during the division cycle, if the rate of mass increase is measured over a number of cell cycles with uninhibited cells, then a priori there should be evidence of an approach to exponential mass increase. If the rate of mass increase during the first cycle is 1.0, during the second cycle it should be 2.0, during the third cycle 4.0, and so on. Thus, even on its own terms, with linear mass increase during the division cycle, the experiments of Conlon and Raff  on the pattern of mass increase are flawed by the presence of an inhibitor of DNA synthesis. An analysis of this idea is presented schematically in Fig. 4.
Raff (personal communication) disputes this interpretation of the aphidicolin experiments, proposing that "while the aphidicolin-arrest strategy is certainly artificial, it is not unrealistic...as many cells, including Schwann cells, grow a great deal after they have stopped dividing. Moreover...hepatocytes grow linearly, independent of their size, if a mouse is re-fed after it has been starved for a couple of days." As noted in Fig. 4, without inhibition, growing cells that grow and divide must, a priori, approach an exponential pattern (i.e., rate of 1, to 2, to 4, to 8 as cells multiply), and therefore the only meaningful discussion of the linear vs. exponential growth pattern relates to growth within the cell cycle. Regarding application of liver growth following starvation and refeeding, this complex situation seems particularly inapplicable to discussions of cell growth in cell culture as there are so many complicating factors. A detailed analysis of the proper systems for cell-cycle analysis has been presented .
The experiments of Conlon and Raff also show some internal inconsistencies that weaken the actual data. A comparison of cell volume increase and protein per cell increase in the same cells over a 96 hour period (Fig.3 of Conlon and Raff) shows that the volume increase was 4.75-fold (~2,000 μm3/cell increasing to ~9,500 μm3/cell) but the protein increase was only 2.93-fold (~0.16 ng/cell increasing to ~0.47 ng/cell). Until these differences are resolved, it is difficult to accept these experiments as supporting linear cell growth–or any other pattern of cell growth–during the normal division cycle. The discrepancies pointed out here suggest that the quantitative measures of cell size by Coulter Counter may not be able to distinguish different growth patterns.
Another problem arises in Conlon and Raff's  analysis of the pulse-chase experiments where cells starved for different times are pulsed and chased to measure protein turnover. They concluded: "...the rate of decrease in radiolabeled protein increased as the cells increased in size." That is, there was a greater release of labeled amino acids from cells that were inhibited with aphidicolin for longer periods of time and which were therefore larger . But the release data were plotted on rectangular coordinates. This led to the observation that the slope between the 0 hour and 2 hour points in their Fig.4 is steeper for the cells arrested for 72 hours compared to the cells arrested for 48 hours. The 72 hour cells were larger than the 48 hour cells. But considering the actual values, and reading the results from the published graph, the counts for the 72 hour arrested cells went from ~179 to ~121 in two hours, or a ratio of 0.67 for the two hour chase. The 48 hour arrested cells went from ~138 to ~94 for a ratio of 0.68 for the two hour chase. Thus, in contrast to the conclusion of Conlon and Raff  there is no apparent difference in the turnover of proteins as a function of cell size.
Robert Brooks (personal communication) has argued against this analysis, noting that the cells starved for 24 hours appeared to show "no turnover" as the line for this graph (Conlon and Raff's Fig.4) was flat. But in the text in the legend to their Fig.4(b) Conlon and Raff state, "The shallowness of the curve for the 24-hour-arrested cells is likely to be the result of the lower than expected value at 0 hours." This explanation comes from the initial counts in Fig.4(a) where it can be seen that there is some apparent error in the zero time value for the 24 hour starved cells in their Fig.4(b).
But an even more egregious error in analysis precedes even these technical problems. The cells studied by Conlon and Raff were not synchronized. The cells were not aligned and were in all phases of the cell cycle. Theoretically, it is impossible to determine the pattern of mass synthesis during the cell cycle on cells that are not synchronized. (For complete details see ). This is because of the age distribution of cells in a growing culture. The age distribution for growing cells in culture is given by 21-X, where X is the cell age during the cell cycle; X varies between 0.0 and 1.0 (newborn cells are age 0.0 and dividing cells are age 1.0). At age 0.0 the relative number of newborn cells is 2.0 (21-0 = 21 = 2) while the relative cell number of dividing cells is 1.0 at age 1.0 (21-1 = 20 = 1). This distribution of cell ages means that any incorporation measurement on asynchronous cells must, and will, yield an exponential pattern of uptake. This is illustrated in Fig. 5 for an idealized case where we imagine cells making all of their cytoplasm only at age 0.5. Because of the age distribution an exponential pattern of incorporation is observed when the entire culture is analyzed (Fig. 5a,5b). The details of the analysis are presented in the legend to Fig. 5. If the cells had been synchronized then one would have measured a peaked pattern as illustrated in Fig. 5c.
Robert Brooks (personal communication) argues that this critique is incorrect because "they  started with quiescent (G0/G1) cells." Quiescent cells with a G1-phase amount of DNA are not synchronized [13-15]. The reader is referred to these papers for a detailed analysis. Despite the widespread belief and acceptance that cells can be synchronized by growth arrest (i.e., by whole-culture synchronization methods), this idea is incorrect. Cells can only be synchronized by selective methods .
How can one determine whether mass increases exponentially or linearly during a normal, unperturbed, division cycle? To illustrate one approach to determining the pattern of mass increase during the division cycle, consider the following experiment. Grow cells for many generations in a radioactive amino acid (e.g., C-14 labeled amino acid) so that cell protein is totally labeled. Then add a pulse of a counter-labeled amino acid (e.g., H-3 labeled). As shown in Table 1, if cells grow linearly, the ratio of tritium (H-3) to C-14 should decrease as the cells become larger. With exponential growth the ratio of tritium to C-14 should be constant over the cell cycle. If one now one took such double-labeled cells, fixed them, and spread the cells out on a gradient such that the larger cells were preferentially at the bottom and the smaller cells at the top, if cells grew linearly there would be a decrease in the H-3/C-14 ratio as the larger and larger cells were assayed. If cells grew exponentially there would be a constant radioactivity ratio over the entire set of cell size fractions. The idealized results from Table 1 are illustrated in Fig. 6.
To summarize this critique of the aphidicolin-inhibition results, the experiments of Conlon and Raff do not measure the mass increase during the cell cycle. The experiments using inhibition of DNA replication merely measure the pattern of mass increase in a perturbed experimental situation on cells that are not synchronized. This experiment is not supportive of any particular pattern of mass increase during the normal division cycle. More important, as shown in Fig. 5, without synchronization of cells, it is impossible to determine the pattern of mass increase during the division cycle.
The bacterial cell cycle: Rules, patterns, and regulation
This analysis presented here explicitly deals with animal or eukaryotic cells. However, it is relevant to bring to bear on this problem the experience and results obtained regarding cell-size determination in bacteria. In 1968 the rules for the replication of DNA in a simple bacterium (Escherichia coli
) as well as the relationship of cell size to control of DNA replication were worked out [16
]. The pattern of DNA replication and cell size are determined by three rules:
1. A round of DNA replication is invariant (40 minutes) over a wide range of growth rates [16-19,21].
2. The time between termination of replication and cell division is invariant (20 minutes) over a wide range of growth rates [16-19,21,22].
3. At the time of initiation of replication, the cell mass per origin is a constant [16,20,23].
These rules are illustrated in Figs. 7 and 8. These three rules predict (Fig. 8c), that cell size should be a logarithmic function of growth rate. Cell size plotted on semi-logarithmic coordinates against the reciprocal of the interdivision time (i.e., the growth rate) gives a straight line. Faster growing cells are larger than slower growing cells. Ten years earlier, in 1958, before the rules predicting the size-growth rate relationship were determined, this experimental result  was clearly obtained in what has been called "the Fundamental Experiment of Bacterial Physiology" (Cooper, 1991). An analysis of the history, origins, and meaning of this experiment has been published (Cooper, 1993).
The important consequence of Figs. 7 and 8 is that we understand how cell size is controlled in bacteria. Cells initiate DNA replication at a certain cell size. This cell size (sometimes referred to as the "initiation mass") is a constant size within experimental limits (Cooper, 1997). The cell size at initiation is constant per origin present in the cell. A cell with two origins being initiated is twice as large as a cell with only one origin. The number of origins present at initiation and the cell age during the division cycle at which initiation occurs determines the average cell size of a cells growing in culture.
Analysis of size maintenance in animal cells
The ideas of the bacterial cycle can be directly applied to animal cells. Cells of different growth rates are shown in Fig. 9a
. The different lines, a-g, identify cells of different sizes because they pass through size 1.0 at different cell ages during a cell cycle span. Cell "g" is a faster growing cell than cell "a" with the others of intermediate growth rates. The earlier a cell reaches size 1.0, the larger the cells will be. Thus, in Fig. 9a
, the cell "g" is larger than the cell "a" because the cell "g" reaches size 1.0 earlier than the cell "a". As drawn in Fig. 9a
, the newborn "g" cell is size 1.0. The mother or dividing cell is size 2.0. We can imagine that the mean size of cells growing at this rate is approximately 1.5. In contrast, the "a" cell varies between newborn size of approximately 0.6 and dividing size of 1.2. The average size of the "a" cells is smaller than the "g" cell, approximately size 0.9. (The precise calculation of the average cell size requires consideration of the age distribution and the actual pattern of mass increase during the division cycle; for purposes of this analysis, these complications are omitted.) Other cells (b-f) may be similarly analyzed to see that faster growing cells are larger than slower growing cells.
As will now be seen, this variation in size is related to, and determined by, the growth rate.
It is proposed that mammalian cells initiate DNA replication at some relatively constant cell size. The time for S and G2 phases are relatively constant as the interdivision time varies , so the cell cycle age at initiation of S phase occurs earlier and earlier within the cell cycle as the growth rate increases (or as the interdivision time decreases). This is shown in Fig. 9b, where the interdivision time is varied but S- and G2-phase lengths are constant. In Fig. 9c the cell cycle patterns in Fig. 9b are normalized to a constant length. In Fig. 9c it is clear that the faster cells initiate S phase earlier in the cell cycle. This is because faster growing cells have a relatively short G1 phase. These faster growing cells achieve the initiation mass earlier in the cell cycle and thus these cells will be larger. As in bacteria, faster growth leads to larger average cell sizes. (For a discussion of the case of cells growing so fast as to not have a G1-phase as in cell "g" in Fig. 9b, see ).
The rate of cell growth is determined by medium composition. For example, as more and more nutrients are added to a minimal medium, bacterial cells grow at faster and faster rates. The interdivision time shortens as the medium becomes richer. For bacteria the mechanism for growth rate variation with medium composition is, in outline, well understood . The addition of nutrients to a medium represses the synthesis of enzymes that are not now needed (e.g., addition of leucine stops the synthesis of leucine synthesizing enzymes). This leads to a shift in the synthetic capacity of the cell to the protein synthesizing system (RNA polymerase, ribosomes, related materials, etc.) as these functions are not repressible by external components . This leads to a more rapid rate of mass increase and thus a shorter mass doubling time [24,26].
Although the details may vary, it is proposed here (and in fact supported by the experiments of Conlon and Raff) that the richer a medium is (e.g., more serum rather than less serum), the faster the cells will grow. The faster a cell grows, the larger it will be (Fig. 9). The variation of G1-phase length with interdivision time variation has been analyzed in detail [24,26].
Conlon and Raff  supply evidence for the relationship of cell size and growth rate in their Fig. 7. Cells that have become overcrowded by not being diluted back (their Fig. 7b) decrease their volume (their Fig. 7a).
The analysis presented above explains the variation of cell size as function of growth rate as observed by Conlon and Raff (slower growing cells are smaller than faster growing cells). Furthermore, the analysis can also explain the maintenance of cell size, even with exponential mass increase during the division cycle, as shown in Fig. 9. Larger than average cells will divide sooner as they reach the initiation mass earlier and smaller than average cells will delay initiation until the initiation mass is achieved. Cell division will follow after relatively constant S- and G2/M-phases. This is the underlying and fundamental explanation for the patterns described in Figs. 1 and 2.
Thus, we now have an answer to the question (raised in discussion of Fig. 2) "How does the cell 'know' when to divide so that size homeostasis is maintained?" The answer is that initiation of S phase is determined by the cell mass. A relatively large cell initiates S phase earlier than a relatively small cell. This earlier initiation is played out in an earlier cell division after a period equal to the S and G2/M phases. Since the S and G2/M phases are relatively invariant, an earlier initiation produces an earlier cell division. While the analysis in Fig. 1 discussed the size maintenance problem in terms of the cell dividing earlier if a newborn cell was larger and later if a newborn cell was smaller (or if the rate of mass increase was high or low), the deeper analysis presented now proposes that the decision to divide is determined not at the moment of cell division but earlier at the start of S phase. The initiation of S phase is determined by cell size and the faster a cell reaches the S-phase initiation size the earlier the cell will initiate S phase and the earlier it will divide. For reasons not yet understood, there is a relationship between initiation of S phase and cell division such that once S phase is initiated the cell will ineluctably proceed to division.
We now see the answer to the problem of cell size at division. Cell size at division is merely a surrogate indicator of cell size at initiation. Further, the time of cell division is a surrogate measure of the time of initiation of S phase. A cell that initiates S phase earlier in the cell cycle will have more time to increase its total mass prior to division. The larger newborn cell, having initiated S phase relatively early compared to its relatively smaller sister and cousin cells, will divide earlier as described in Fig. 1. Conversely, smaller cells will delay initiation of S phase; that delay will allow more mass increase before the actual cell division because S phase is somewhat delayed and thus division is postponed allowing mass to increase before the ultimate cell division. In this way, the cell size distribution is maintained.
Size variation during a shift from slow to fast growth
Immediately following the discovery of bacterial cell size variation with growth rate [8
] shift-up experiments of cells from slow growth (relatively small size) to faster growth (relatively large size) were performed [27
]. The phenomenon of "rate maintenance" was discovered in this shift-up experiment. Rate maintenance is the continuation of the rate of cell division for a constant period after the shift-up [19
]. The rate of mass increase changes immediately to the new rate at the instant of shift-up, while the rate of cell division continues for a period of time before abruptly
changing to the new rate. The rate maintenance phenomenon occurs over a wide range of shift-ups [19
]. The continuation of the original, slower rate of cell number increase, combined with an immediate transition to the new rate of mass increase, leads to an increase in cell size over the period of rate maintenance (Fig. 10a
). Rate maintenance is now understood to result from the constant S and G2 periods (C and D periods in bacteria) that do not allow new divisions to occur until the newly inserted replication forks pass through the S (i.e., C) period and the G2 (i.e., D) period. Without going into details here (see [12
] for a complete analysis and explanation), suffice it to say that the rate maintenance phenomenon leads to the observed variation in bacterial cell sizes as the rate of cell growth varies over a wide range.
Conlon and Raff  studied mammalian cells during a shift-up from slow to rapid growth and small to large cell size. Upon shifting slow cells to faster medium (e.g., shifting cells from low serum to high serum) there is a concomitant increase in cell size (Fig. 6e of Conlon and Raff ). One major difference from the bacterial shift-up result is that with animal cells the time for cell size to increase took a much longer time, between 6 and 9 days. To explain the difference between the bacterial shift-up result (Fig. 10a) and the mammalian cell shift-up result (Fig. 10b) one can postulate that for reasons unrelated to the cell cycle but merely related to cellular metabolism occurring continuously throughout the cell cycle, the change in external conditions does not immediately lead to the new rate of mass increase (Fig. 10b). The rate of mass increase is predicted to change relatively slowly as mammalian cells are shifted from serum-free (slow growth) medium to serum-containing (fast growth) medium. Of course, this is just the result reported by Conlon and Raff (their Fig. 6e).
This view of the change in cell size following a shift from slow to rapid growth is quite different from the description Conlon and Raff present for the case of yeast cells switched from a nutrient-poor to a nutrient-rich medium. They write , "When switched from a nutrient poor medium to a nutrient-rich medium, the cell cycle arrests and resumes only when the cells have reached the appropriate size for the new condition, which occurs within one cell cycle...Thus, the cells can adjust their size threshold rapidly in response to changing external conditions." The bacterial model of the shift-up allows a rapid change in cell size within one cell cycle without postulating any "arrest" of passage through the cell cycle.
Rather than postulate a mechanism that slows or actively shuts down the cell cycle, it is proposed that no change in cell division occurs until the increased initiations of S phases pass through the S and G2/M phases, as in the bacterial model. No additional mechanism need be proposed to "stop" some event of the cell cycle until cell size has increased.
The age-size distribution summarizes size control
One way to consider a growing culture is to see that every cell in a growing culture has an age and a size. The age/size structure of a population is a representation of each of the cells in the culture and its age and size. If there is no statistical variation, and cells move through the cell cycle with a perfectly precise exponential growth pattern, then the age/size distribution is seen in Fig. 11a
. The projections of the dots in this panel to the age axis (bottom, abscissa) and the size axis (left, ordinate) indicate that when cells are growing exponentially, there are a greater number of smaller cells relative to the population than there are younger cells. This reflects the age distribution discussed above.
There are, however, statistical variations in cell sizes at a given age and cells of a given age may have different sizes. The precise statistical distribution is not known, but one view of the possible result is shown in Fig. 11b. The shaded area indicates a cloud of points preferentially collected around the middle of the shaded area, with fewer cells at the outer edges. If this were a three-dimensional graph, there would be a peaked "ridge" up the center of the shaded area indicating that more cells reside with a particular age/size distribution than those at the edges of the age/size distribution.
It is possible to indicate the cells at particular times during the division cycle such as birth, division and initiation of DNA synthesis, as shown in Fig. 11c. There is no distribution in age at birth, since by definition age at birth is 0.0. The graphs at the upper and right sides of Fig. 11c are representations of the spread of the various distributions. There is variability in the age at initiation of DNA replication (I) and the age at division (D). The age at cell birth (B) is defined as constant (i.e., 0.0) but the size at birth (B) is not constant. It is expected that the size at division will be slightly less variable than the size at birth. This is attributable to the probability that any deviation from equipartition of dividing cells leads to a broadening of the size distribution in newborn cells. From Fig. 11c it can also be seen that larger newborn cells (distribution B) will reach the size at initiation (I) earlier than smaller newborn cells. This explains the size maintenance pattern drawn in Fig. 2. Note that in Fig. 11c the size at initiation is relatively narrow. It is this narrowing of size at initiation that leads to the slightly narrower size at division relative to the wider size distribution at birth.
An instructive way of looking at the age/size distribution is to replot cell ages using the age at initiation of DNA synthesis (I) as a starting point (Fig. 11d). By defining age at initiation as 0.0 one gets negative ages for cells before initiation and positive ages for cells after initiation. Fig. 11d shows that there is no distribution in the age at initiation (since by definition the age at initiation is 0.0 for all cells) but there is now a variation in the "age" of newborn cells. Smaller cells have a more negative age and take longer to reach initiation size than larger newborn cells; that is why there is a distribution of cell ages of newborn cells with respect to the time until initiation. The "bottleneck" at initiation of DNA synthesis enables cells born of different sizes to retain size homeostasis. Since all cells to the left of initiation must pass through the bottleneck of initiation on the way to division, all cells, of any newborn size, are realigned and assigned a new age and a new size as they pass through the act of initiation.