Control and maintenance of mammalian cell size

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.

Figure 1

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.

Figure 2

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.

Figure 3

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.

Figure 4

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.

Figure 5

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 Gedanken analysis, it is assumed that cells of age 0.5 (i.e., cells in mid-cycle) are the only cells incorporating amino acid (cross-hatched bars). The asterisk (*) on a bar in each pattern indicates the newborn cells. One reads the cell ages by going from the asterisked bar to the right and then back to the left to finish off the age distribution. The number to the right of each pattern is the relative number of cells incorporating amino acid. Thus, in the uppermost pattern in Panel (a) the relative number is 1.46. After one-tenth of a generation we see that the oldest cells in the first pattern have divided to give double the number of cells and these cells are now the youngest cells in the culture. All of the other cells move up one-tenth of an age so that the cells that were age 0.4 are now age 0.5 (cross-hatched bar) and the rate of synthesis increases to 1.57. This is because there are more cells in the original culture of age 0.4 than there were of age 0.5. Continuing down the patterns in Panel (a) we see that as cells move to age 0.5 there is a continuous, and exponential, increase in the radioactivity. The cells above age 0.5 (in the original topmost diagram) divide and produce two cells each tenth of a cell cycle, so that over one total cell cycle there is an exponential increase in the rate of amino acid incorporation (a measure of cytoplasm increase). The total pattern of incorporation is plotted in panel (b) where the exponential incorporation during one cell cycle is indicated. Panels (a) and (b) thus show that even with a non-exponential pattern of incorporation, if a total culture is studied, the measured incorporation pattern will be exponential. If, however, cells are truly synchronized, as illustrated in Panel (c), a peaked incorporation pattern is observed, accurately reflecting the mid-cycle incorporation of amino acids into the cells at a particular cell-cycle age. Starting with newborn cells at age 0.0 and moving through the cell cycle at one-tenth of an age each pattern in (panel c) the incorporation (noted by the numbers to the right of the diagrams (panel c) shows a peaked pattern.

Figure 6

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

Figure 7

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.

Figure 8

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).

Figure 9

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.

Figure 10

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.

Figure 11

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.