Department of Microbiology and Immunology, University of Michigan Medical School, Ann Arbor MI 48109-0620, USA

Abstract

Background

Conlon and Raff propose that mammalian cells grow linearly during the division cycle. According to Conlon and Raff, cells growing linearly do not need a size checkpoint to maintain a constant distribution of cell sizes. If there is no cell-size-control system, then exponential growth is not allowed, as exponential growth, according to Conlon and Raff, would require a cell-size-control system.

Discussion

A reexamination of the model and experiments of Conlon and Raff indicates that exponential growth is fully compatible with cell size maintenance, and that mammalian cells have a system to regulate and maintain cell size that is related to the process of S-phase initiation. Mammalian cell size control and its relationship to growth rate–faster growing cells are larger than slower growing cells–is explained by the initiation of S phase occurring at a relatively constant cell size coupled with relatively invariant S- and G2-phase times as interdivision time varies.

Summary

This view of the mammalian cell cycle, the continuum model, explains the mass growth pattern during the division cycle, size maintenance, size determination, and the kinetics of cell-size change following a shift-up from slow to rapid growth.

Background

Conlon and Raff have described experiments that they claim casts doubt on a basic assumption regarding the way mammalian cell size is maintained during proliferation

In an article accompanying the work by Conlon and Raff

The experiments of Conlon and Raff

"We show that proliferating rat Schwann cells do not require a cell-size checkpoint to maintain a constant cell size distribution, as, unlike yeasts, large and small Schwann cells grow at the same rate, which depends on the concentration of extracellular growth factors."

A reanalysis of the experiments and reasoning of Conlon and Raff, presented here, leads to a very different view of cell size control and cell size maintenance. It is first shown that there is no problem with either linear or exponential mass increase for size maintenance. Size maintenance does not depend on which pattern of cell mass increase occurs within a cell cycle. The preferred–and experimentally and theoretically supported–pattern of mass increase during the division cycle is exponential growth or exponential mass increase. An exponential growth pattern poses no problem for size maintenance. Constancy of cell size is fully compatible with an exponential pattern of mass increase as well as the hypothetical linear pattern of mass increase. No major difference between the size control systems of yeast, mammalian, or bacterial cells need be postulated to account for the constancy of cell size during the growth of cell cultures. In contrast to the proposed absence of a cell size control system in mammalian cells, it is shown that mammalian cells do have a very simple size control system. The formal elements of this system are similar to that found in the control of the bacterial cell cycle.

Discussion

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.

Exponential and linear growth patterns are both compatible with cell size maintenance

Exponential and linear growth patterns are both compatible with cell size maintenance. In panel (a) newborn cells of identical size increase mass at slightly different rates with an exponential pattern of mass increase. If cells divide at a constant cell size, here size 2.0, size will be maintained even though the rate of mass increase varies. This occurs as the cells divide at different times (division indicated by the downward arrows) as they reach the same size. In panel (b), exponential growth at identical rates from initial cells of different cell sizes also gives size maintenance as cells divide at the same size because there are different interdivision times for each cell; the larger initial cells have a shorter interdivision time and the smaller initial cells have a longer interdivision time. As shown in panels (c) and (d), linear cell increase (note the different ordinate scale compared to panels (a) and (b)) can also lead to cell size maintenance as cells divide at the same size, 2.0.

Linear cell growth during the division cycle (Figs.

The patterns shown in Figs.

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

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

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.

Interdivision time variation allows a return of slightly deviant sizes to a constant cell size

Interdivision time variation allows a return of slightly deviant sizes to a constant cell size. In panel (a) newborn "baby" cells (b) grow for slightly different times, producing either large (l) or small (s) newborn cells from large or small dividing cells. Deviation from equipartition for an average sized dividing cell can also produce large and small cells. The resolution of size differences is illustrated in (b) and (c) where the larger cell (l) has a shorter interdivision time dividing at average (a) cell size and the smaller cell (s) has a longer interdivision time also dividing at average cell size. The dividing average sized cells (a) produce newborn baby cells (b) of the original newborn size.

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

Hypothetical model of Conlon and Raff where constant size increase independent of cell size allows return of deviant cell sizes to a constant cell size over time (This figure is drawn directly from Conlon and Raff (Conlon and Raff, 2003))

Hypothetical model of Conlon and Raff where constant size increase independent of cell size allows return of deviant cell sizes to a constant cell size over time (This figure is drawn directly from Conlon and Raff (Conlon and Raff, 2003)). The assumption made for this figure is that both large and small cells increase their mass equally over time. Thus, a cell of size 10 and a cell of size 1 increase their mass over one doubling time by 11 units (the sum of the starting masses, 10 and 1). To the large and small cell an increase of 5.5 units of mass is proposed to occur as cells grow. Thus, the large cell grows to size 15.5 and the small cell grows to size 6.5, and at division the daughter cells now have sizes of 7.75 and 3.25 respectively. This continues for a number of generations as the founder cells, originally of disparate sizes, now converge to the same size.

But no indication of the length of the division cycles is given in Fig

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.

Of course, the example given by Conlon and Raff (Fig

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

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

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

What of the experiments presented by Conlon and Raff

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

Approach of cell mass to exponential even if cells had linear synthesis within cell cycle

Approach of cell mass to exponential even if cells had linear synthesis within cell cycle. Panel (a) illustrates cells dividing to produce two, four or eight times the original number of cells (thick line is cell number). The mass (thin line) increases linearly. It is clear that the cell size will not be maintained. In panel (b), even with linear mass growth within the cell cycle (thin line), as cells divide the rate of mass synthesis doubles and then quadruples as cell numbers increase. It is not proposed that mass increases linearly, but merely that even linear synthesis should exhibit, in an uninhibited situation, exponential mass growth.

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.

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 μm^{3}/cell increasing to ~9,500 μm^{3}/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

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 ^{1-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 (2^{1-0 }= 2^{1 }= 2) while the relative cell number of dividing cells is 1.0 at age 1.0 (2^{1-1 }= 2^{0 }= 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.

Unsynchronized cells cannot be used to determine cell-cycle pattern of synthesis

Unsynchronized cells cannot be used to determine cell-cycle pattern of synthesis. Panel (a) shows a series of age distributions starting with the initial age distribution reflecting the pattern Age Distribution = 2^{1-X}, where X is the cell age going from 0.0 to 1.0. In this

Robert Brooks (personal communication) argues that this critique is incorrect because "they

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

Analysis of linear and exponential growth by comparing long-term and short-term isotope incorporation.

LINEAR GROWTH

EXPONENTIAL GROWTH

Cell size

Size increase

Inc/size

**Cell age**

Cell size

Size increase

Inc/size

1

0.1

0.100

**0**

1.00

0.07

0.072

1.1

0.1

0.091

**0.1**

1.07

0.08

0.072

1.2

0.1

0.083

**0.2**

1.15

0.08

0.072

1.3

0.1

0.077

**0.3**

1.23

0.09

0.072

1.4

0.1

0.071

**0.4**

1.32

0.09

0.072

1.5

0.1

0.067

**0.5**

1.41

0.10

0.072

1.6

0.1

0.063

**0.6**

1.52

0.11

0.072

1.7

0.1

0.059

**0.7**

1.62

0.12

0.072

1.8

0.1

0.056

**0.8**

1.74

0.12

0.072

1.9

0.1

0.053

**0.9**

1.87

0.13

0.072

2

**1**

2.00

The center, bold-faced, column lists the cell ages from 0 to 1.0. At the left the linear increase of mass is related to the absolute increase in mass per interval (0.1 each interval for linear increase in mass during division cycle), and the ratio of incorporation per extant cell mass is given in the third column (0.1 to 0.053). Similar results for exponential growth except the mass increase per interval goes from 0.07 at the start of the division cycle to 0.13 at the end. The ratio of incorporation per extant mass in the right-most column is thus constant.

Comparison of the ratio of pulse label to total label for exponential and linear patterns of mass increase as described in Table 1

Comparison of the ratio of pulse label to total label for exponential and linear patterns of mass increase as described in Table 1.

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.

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 (

1. A round of DNA replication is invariant (40 minutes) over a wide range of growth rates

2. The time between termination of replication and cell division is invariant (20 minutes) over a wide range of growth rates

3. At the time of initiation of replication, the cell mass per origin is a constant

These rules are illustrated in Figs.

Diagram of patterns of DNA replication during the division cycle in bacteria

Diagram of patterns of DNA replication during the division cycle in bacteria. The different patterns go from an infinite interdivision time (i.e., essentially no or extremely slow growth) to cells with 90, 60, 50, 40, 35, 30, 25, and 20 minute interdivision times. In all cases, the rate of replication fork movement is 40 minutes for a round of replication or one-quarter of the genome every 10 minutes. All rounds of replication end 20 minutes before the end of the cell cycle. This is most clearly seen in the 60-minute cells where a newborn cell has one genome, which replicates for 40 minutes ending replication 20 minutes before cell division. The same rules are drawn here for a 90-minute and a very slow growing cell (infinite interdivision time). The large numbers in each pattern at the left indicate the number of origins to be initiated at each time of initiation of replication. Thus, in the 60-minute cells there is one origin in the newborn cell. Consider that the cell mass is given a unit value for each origin to be initiated. Thus, the newborn cell in the 60-minute case is given a size of 1.0 unit of mass. This means that the dividing cell in the 60-minute cells is size 2.0. Mass increases, in the 60-minute case, from 1 to 2. In the 90-minute cells the cell of size 1 is one third of the way through the cell cycle. Since mass increases continuously during the division cycle it is clear that the newborn cell in the 90-minute culture is less than 1.0 in size. Let us say it was something like size 0.7. In this case the newborn cell in the 90-minute cells would be size 0.7 and the dividing cell would be size 1.4. It is clear that the 90-minute cells are, on average, smaller than the 60 minute cells. Similarly, if we consider the very slow cells, the cell of size 1.0 is very near the end of the cell cycle, and the newborn cell is slightly above size 0.5. Since the very slow growing cells (top panel) go from sizes 0.5 to 1.0 and the 60 minutes cells go from size 1.0 to 2.0, the 60 minute cells are twice as large as the very slow growing cells. The 30-minute cells have two origins in the newborn cell and thus the newborn cells can be considered size 2.0 with the dividing cells 4.0. The 20-minute cells have a newborn cell of size 4.0 (four origins in the newborn cell) and a dividing size of 8.0. As one goes from extremely long interdivision time, to 60, to 30 to 20, the relative sizes go from 0.5, to 1, to 2 to 4, with the growth rates expressed as doublings per hour, or 0 (infinite interdivision time), 1 (60 minute interdivision time), 2 (30 minute interdivision time), and 3 (20 minute interdivision time). Cells that initiate DNA replication in the middle of the cycle may be considered as follows. The 40-minute cell has two origins in the middle of the cell cycle so the mid-aged cell is size 2.0. The newborn cell might be some size like 1.5 and the dividing cell something like 3.0. Thus, the 40-minute cell has an average size intermediate between the 60 and the 30-minute cell. Similarly, the 25-minute cell also initiates mid-cycle, but there are 4 origins at the time of initiation. Thus, the mid-aged cell in this case is size 4.0 and the newborn cell may be considered something like size 3.0. The cell sizes go from 3.0 to 6.0, and these cells are larger than the 30-minute cells and smaller than the 20 minute cells.

Size determination in bacteria

Size determination in bacteria. In panel (a) the rates of growth of cells from infinitely slow (very long interdivision time) minutes to 20 minutes (as illustrated in Fig. 6) are plotted with the relative sizes shown. Thus, the 60 minute cell goes from size 1.0 to 2.0 over 60 minutes. The 30-minute cell (third angled line from top) goes from size 2 to 4 over 30 minutes. And the 20-minute cell (top angled line) goes from size 4 to 8 over 20 minutes. Other rates of growth for 25, 35, 40, 50, 90 and "infinite" interdivision times are also shown. In panel (b) the same results are plotted over relative cell ages from age 0 (newborn) to 1.0 (dividing cell). The open circles indicate when initiation occurs, and corresponds to the numbers in the individual panels. Thus, in Fig. 6 the cells with a 60, 30, and 20 minute interdivision time initiation DNA replication in the newborn cell (age 0.0) at sizes 1, 2 and 4. Besides the cell age at initiation, the open circle also indicates the relative size of the cell at initiation (see numbers in Fig. 7). The cell sizes at age 0.0 for all cells is a measure of the average cell size in the culture. (Given an identical pattern of cell growth during the division cycle the relative cell size of the cells in a culture is precisely proportional to the newborn cell size). These size values are then plotted against the rate of cell growth (the inverse of the interdivision time or doublings per hour) as shown in panel (c). The log of the cell sizes are a straight line when plotted as a function of the rate of cell growth (the inverse of the interdivision time).

The important consequence of Figs.

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.

Mammalian cell size variation as growth rate varies

Mammalian cell size variation as growth rate varies. Panel (a) shows a given mammalian cell growing at different rates and with different sizes. The lines are parallel because the interdivision times are normalized to a relative cell age as cells are born at age 0.0 and divide at age 1.0. All lines are exponentially increasing cell sizes from smallest to largest. Where the lines cross the thick horizontal line indicates a cell of size 1.0. Since the fastest cell (cell g) has a size 1.0 at the start of the cell cycle these cells must go from a newborn sizes of 1.0 to a size at division of 2.0. The slowest cell (cell a) has size 1.0 toward the end of the cell cycle, so the newborn cell is slightly larger than size 0.5 at age 0.0. The size ranges of these cells goes over a factor of 2. In panel (b) the size patterns are re-interpreted in terms of initiation at a particular time during the cell cycle. In this figure the thick, short line on each pattern is the S phase, the thinner line to the right is the G2 phase and the thinner line to the left is the G1 phase. Given that S and G2 are relatively constant in length then the slower cells (e.g., cell "a") have a longer G1 phase than the faster growing cells (e.g., cell "g", which has no measurable G1 phase). This is because the interdivision time is the sum of S+G2+G1. If S and G2 are relatively constant as the interdivision time decreases (i.e., as cells grow at faster growth rates), the G1 phase gets smaller. When the interdivision time equals the sum of S and G2 as in cell "g", there is no G1 phase. Such a situation has been analyzed previously (Cooper, 1979). It is clear from panel (b) that as cells grow faster, the time during the division cycle at which initiation of S phase starts is earlier and earlier. This is illustrated even more directly in panel (c) where the phases are normalized to a unit length. The slowest cell (cell "a") has the shortest fraction of cells with an S or G2 phase and the fastest growing cell (cell "g") has the entire division cycle occupied by S and G2 phases. The topmost line in panel (c) is the fastest cell and it starts S phase early in the cell cycle. Thus we see that the faster a cell grows the earlier in the cell cycle the cell achieves a size of 1.0. This accounts for the result that the slower cell has a smaller cell size than the faster growing cell.

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

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

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.

Conlon and Raff

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.

Thus, we now have an answer to the question (raised in discussion of Fig.

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.

Size variation during a shift from slow to fast growth

Immediately following the discovery of bacterial cell size variation with growth rate

Comparison of shift-up of bacterial cells and mammalian cells

Comparison of shift-up of bacterial cells and mammalian cells. In the panel (a), after a shift of bacterial cells from slow-growth medium to fast growth medium there is an immediate change in the rate of mass synthesis to the new rate while the rate of cell division continues at the old rate for a fixed period of time (rate maintenance). At the end of this "rate maintenance" period, there is a sudden shift in the rate of cell number increase to the new rate. The thick line in panel (a) shows the change in cell size following the shift-up. In contrast, in panel (b) a slower and more gradual change in the rate of mass synthesis, concomitant with the cell number pattern also changing slowly over a period of time, will give a longer period of change in cell size. Conlon and Raff observed this slow pattern of mammalian cell size change.

Conlon and Raff

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

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.

Age-size structure of a growing culture

Age-size structure of a growing culture. Panel (a) is the age-size structure for a perfectly deterministic population growing exponentially in mass during the division cycle. The dots on the exponentially increasing line are placed at equal age intervals shown by their representation at the bottom of the panel. The representation of the dots at the left of panel (a) indicates that there is a greater concentration of smaller cells than younger cells. In panel (b) the age-size structure for a population with variation in size and interdivision times is illustrated. The cloud of points (indicated by a few points as representative of the population) is one possible age-size structure. In panel (c) the newborn cells are indicated by the filled circles, the dividing cells by open circles, and the cells in the act of initiation of DNA synthesis by + signs. It can be seen that the larger cells at birth will, on average, reach the size required for initiation of DNA replication more quickly than smaller cells. This is because the larger cells are closer to the initiation size (represented by I on the right side of panel (c)). The B and D distributions at the right of panel (c) indicate the size distributions of newborn (B) cells and dividing (D) cells. The B, D, and I distributions at the top of panel (c) illustrate the age distributions for newborn, dividing, and initiating cells. The size distribution of initiating cells is drawn with a narrower distribution. Variations in mass increase during the period after initiation lead to the widening of the size distribution at division. Panel (d) is a replotting of the pattern in panel (c) with the bottom time scale defined by setting the time of initiation of DNA synthesisas age 0.0. Cells before initiation have a negative age value, and cells after initiation have a positive age value. Initiation takes place, by definition in this panel, at age 0.0. There is some variation in the size of cells at initiation, but it is proposed that this variation is less than the variation at other events of the cell cycle. The narrowing of the age-size structure at the time of initiation is a graphic representation of the size-homeostasis mechanism. No matter what size cells are present at birth or division, these cells are returned to their proper age-size relationship at the instant of initiation of DNA synthesis. Larger cells at division produce larger newborn cells which then reach initiation size earlier than smaller cells which were produced by the division of smaller dividing cells. This is a restatement of the idea that larger cells get to initiation earlier because larger cells have less of a negative age value at cell birth. At the top and right panels of (c) and (d) are representation of the presumed variation of the sizes and ages of cells at particular events. The size at birth is always a little more widely distributed than the size at division due to a slight inequality of partition of mass at division. The size at initiation of DNA replication is drawn with a relatively small variability.

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.

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.

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.

Summary

Understanding mammalian cell size control

This analysis explains how cell size is maintained through a combination of interdivision time variation and cell growth rate variation. Exponential growth is possible and allowed during the division cycle, in contrast to the proposal of Conlon and Raff

It may be best to summarize these two contrasting views of size maintenance by looking at cell growth in a simple manner, and asking how the rate of mass increase is related to the passage of the cell through the cell cycle. The model of Conlon and Raff looks at the events of passage through the cell cycle as occurring independently of mass increase. The problem then remains as to how mass increase fits into, or coordinates with, the pattern or timing of passage through the cell cycle. It is as though the cell moves through the cell cycle without considering the mass problem, and then the mass of the cell looks at the cell cycle and says "I must grow at some rate so that I do not get too big or too small." In the Conlon/Raff model, a control exists that coordinates mass increase with the rate of cell division.

The model presented here–in contrast to the model of Conlon and Raff–situates mass as the driving force of the cell cycle. Mass increases at some rate that is determined by external conditions (medium, growth factors, pH, etc.). As the mass increases, the accumulation of mass starts or regulates passage through the cell cycle. A cell cannot grow to an abnormal larger size because at a certain cell size or cell mass the S phase is initiated and this event starts a sequence of events leading to mitosis and cytokinesis. A cell cannot get too small because if mass grows slowly (or even stops growing) then the later events of the cell cycle (S-, G2-, and M-phases) are delayed (or do not occur) until mass increases sufficiently to start S phase. A cell cannot get too large because at a certain size the cell initiates S phase leading to the relatively early cell division. A cell cannot get too small because if mass accumulation is inhibited then S phase initiations are also inhibited.

Thus, there is no problem relating mass increase and the cell cycle. Cell mass growth and cell cycle passage cannot be dissociated because one (mass increase) is the determinant of the other (S-phase initiation). For this reason one needs neither checkpoints nor control elements outside of mass increase.

The time for mass to double in a particular situation determines the doubling time of a culture. This is because initiations of S phase occur every mass doubling time, and cell divisions similarly occur every mass doubling time. Thus total mass increases at the same rate as total cell number.

The model presented here explains size determination, size maintenance, and the relationship of mass increase and cell number increase in a growing, exponential, unperturbed, mammalian cell cultures.

Acknowledgements

This paper is dedicated to Moselio Schaechter, who has supported and encouraged me over the years, who was a participant in what has been called "the Fundamental Experiment of Bacterial Physiology", the experiment that is the basis for the eukaryotic ideas presented here, and who has been a model scientist, person, and mentor.

Additional material on the ideas presented here may be viewed at

I thank Charles Helmstetter, Robert Brooks, Arthur Koch, Martin Raff, Ian Conlon, and Sanjav Grewal for their comments and suggestions. Further comments, thoughts, critiques, suggestions, or ideas from readers of this article are most welcome