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Surface-limited growth: A model for the synchronization of a growing bacterial culture through periodic starvation
J. theor. Biol. (1988) 134, 77-87
Surface-Limited Growth: A Model for the Synchronization of a Growing Bacterial Culture through Periodic Starvation N. B. GROVER
The Hubert H. Humphrey Center for Experimental Medicine and Cancer Research, Hebrew University-Hadassah Medical School, Jerusalem, Israel (Received 8 December 1987, and in revised form 25 March 1988) This article analyses the Surface-Limited Growth Model put forward to explain the very tight synchrony, over more than ten division cycles, obtained experimentally by subjecting a growing bacterial culture to alternating periods of starvation and dilution, using inorganic phosphate as the limiting substrate. The Model states that when an essential nutrient is in limited supply, the rate of growth of an individual cell will be proportional to its surface area (and the current concentration of the limiting substance) rather than to its volume. This decrease in dimensionality from volume to surface is expected to favor the smaller cells and so result ultimately in a narrower size distribution. The Surface-Limited Growth Model deals with cell growth under unusual nutritional conditions, and its predictions depend on how the cell replication cycle is assumed to behave under these same circumstances. Two alternatives are considered: the volume at which cells divide is the same during the starvation phase as during steady-state exponential growth, and the cells adjust immediately to the changing growth rate. In the latter case, we have tested both C + D constant with time and C + D variable (where C + D is the time between initiation of chromosome replication and the corresponding cell division), the incremental value at any instant being computed separately for each individual cell from its current effective growth rate. The simulation results are of two sorts depending on the auxiliary assumptions used. Either the dilution-starvation cycles have no effect whatsoever on the cell volume distribution, or the width of the distribution decreases gradually with time, approaching zero slowly and asymptotically, but the mean cell volume decreases as well--directly contradicting experimental observations. We conclude that the Surface-Limited Growth Model is incapable of explaining the synchronization of cells by periodic starvation of a growing bacterial culture.
Introduction H e a l t h y , w e l l - f e d b a c t e r i a l cells n o r m a l l y g r o w a s y n c h r o n o u s l y , a n d m a n y a t t e m p t s h a v e b e e n m a d e o v e r the y e a r s to p r o d u c e s y n c h r o n o u s c u l t u r e s b e c a u s e o f the o b v i o u s a d v a n t a g e s t h e y w o u l d offer for a n y b i o c h e m i c a l or p h y s i o l o g i c a l studies i n v o l v i n g s e q u e n t i a l c h a n g e s in i n d i v i d u a l cells d u r i n g t h e i r life cycle. M o s t m e t h o d s , w h e t h e r b a s e d on i n d u c t i o n or selection, give rise to a s y n c h r o n y t h a t d e t e r i o r a t e s r a p i d l y , u s u a l l y b e g i n n i n g in the s e c o n d o r t h i r d g e n e r a t i o n . T h e r e f o r e , t h e r e was c o n s i d e r a b l e t h e o r e t i c a l as well as p r a c t i c a l interest w h e n a t e c h n i q u e was d e s c r i b e d 77 0022-5193/88/170077+ 11 $03.00/0
© 1988 Academic Press Limited
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that purported to provide cells capable of maintaining very tight synchrony over more than ten cell division cycles (Kepes & Kepes, 1980, 1981). Briefly, ceils are grown in a fermenter with just sufficient inorganic phosphate to allow one mass doubling, after which the culture is diluted twofold, again with only enough phosphate for a single doubling. These dilutions and mass doublings in the phosphatelimited medium are repeated 10-16 times, after which the cells are found to exhibit synchronous behavior when transferred to regular unrestricted batch culture conditions; their mass at birth is then similar to what it had been prior to synchronization. The Surface-Limited Growth Model was first proposed publicly by R. H. Pritchard (EMBO Workshop on Regulation of the Cell Cycle in Prokaryotes, Sede Boqer, September 1984). The basic idea was elegant and deceptively simple: cells normally grow at a rate that is proportional to their mass or volume, but when an essential nutrient is in limited supply, the rate of growth of an individual cell will depend instead on how fast the cell can take up that nutrient from the surrounding medium and this rate, in turn, should be proportional to the cell's surface area (and, of course, to the current concentration of the limiting substance). According to this Model then, synchronization is merely a matter of geometry, the decrease in dimensionality from volume to surface being expected to favor the smaller cells and so result ultimately in a narrower cell size distribution. We have translated the Model into precise mathematical terms and studied the effect on a steady-state exponential Escherichia coli culture, of alternating periods of unrestricted and substrate-limited growth, by simulating the behavior of individual cells over many generations, and from this, constructing cell size distributions as a function of time and phosphate concentration. From the results, it is clear that the Surface-Limited Growth Model is not capable of reproducing the experimental observations. It should be borne in mind, however, that periodic phosphate starvation of a growing bacterial culture may disturb the normal physiology of a cell, since cells so treated do not display the usual pattern of D N A synthesis during the division cycle (Plateau et al., 1987); if this were the case, then the present analysis would be no more than an academic exercise.
The Model
Let M(t) represent the total cell mass t minutes after transfer to the limiting medium, and P(t) the amount of inorganic phosphate remaining. Then
P(O)-P(t)=kp[M(t)-M(O)],
(1)
where kp is a constant, since the decrease in phosphate is proportional to the increase in total mass. At the end of a starvation period, say t = T, all the phosphate has been consumed and the mass has doubled, so that P(T)=O=P(O)k p [ 2 M ( 0 ) - M(0)], whence kp = P(O)/M(O). Actually P(t)~O asymptotically, and in practice, one must terminate the simulation at a predetermined small value of P(t), say 10 -4 P(0), or after some suitable long time T has elapsed, say 5r, where ~"is the doubling time of the culture in the unrestricted growth medium. Substituting
CELL
for
SYNCHRONY
kp in
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eq (1) and rearranging, we get
P(t)
2
M(t)-2-y(t),
P(O)
(2)
M(O)
where y(t) represents the ratio M(t)/M(O). In the general case, where y(t) at the end of a dilution cycle approaches some value kr other than two, eqn (2) assumes the form
P(t) -1 P(O)
- -
y(t)-I
kr - 1
As long as the phosphate concentration is large enough, the cells will grow exponentially: dv(t) =/~v(t), dt
(3)
where v(t) is the volume (assumed proportional to mass) of a cell at time t and /z-= (In 2 ) / r is the steady-state specific growth rate. When P(t) becomes limiting, the Model states that growth will be proportional to cell surface area s(t) and to the amount of phosphate remaining. In other words dr(t)
dt
- k,,s(t)P(t)/P(O),
where k,, is the proportionality constant. If we adopt the usual geometry of circular cylinders with hemispherical polar caps, then dv dt
- - = 2k,,(2- y)(v+ 27rRa/3)/ R,
(4)
since s(t)=2(v+27rR3/3)/R, where R is the cell radius, assumed the same for all cells and constant with time. Here we have simplified the notation by using v for v(t) and y for y(t). Whether eqn (3) or eqn (4) obtains for a particular cell will depend on the cell's dimensions: the larger the cell, the less likely it is that the medium will be able to support its unrestricted growth. The precise level at which the transition occurs will depend on the value of the permeability coefficient of the cell membrane for inorganic phosphate. If we take this coefficient (uptake/phosphate gradient/unit surface area/unit time) to be the same for all cells, then the largest cell will be the first to enter the state of restricted growth. Let this occur when P(t) has reached the level ksP(0), where k~ is a constant less than one. Then it is easy to show that k., -
/zR 3 L o - 2 R 2k~ 3 L o - R '
(5)
where Lo is the length of newborn cells during steady-state exponential growth. These equations together govern the growth of the cells: when there is sufficient phosphate in the culture medium to support the exponential growth of a particular
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cell, eqn (3) obtains; as time passes and P(t) decreases (or y increases), the right-hand side of eqn (4) falls below that of eqn (3) and growth becomes limited, following eqn (4).
Assumptions The Surface-Limited Growth Model deals with cell growth under unusual nutritional conditions and its predictions, unfortunately but perhaps not expectedly, depend on how we assume the cell replication cycle to behave under these same circumstances. Two possibilities are treated. In the first, the volume at which the cells divide is assumed to be the same during the starvation phase as during unrestricted growth; in the second, the cells are considered capable of adjusting immediately to the slowly decreasing growth rate. In both instances, it is necessary to determine when a particular cell is going to initiate a round of c h r o m o s o m e replication and when it is ready to divide. These two major events in the cell cycle are most conveniently characterized by V~, the volume of a cell at c h r o m o s o m e initiation; Vd, the volume at the corresponding cell division; and C + D, the time it takes a cell to grow from the one to the other. In the first case, the cells are assumed to retain steady-state values of V~ and Vd, which therefore remain constant and the same for all cells, and the calculations are straightforward. In the second, we make use of an auxiliary variable r', the current effective doubling time of the culture (Grover et al., 1980), defined as r' -
vln2
dv/dt"
(6)
During steady-state exponential growth, z ' = r , as can be seen from eqn (3), but during the starvation phase, r'> r and, in fact, r ' . o o as P ( t ) ~ O . In order to initiate c h r o m o s o m e replication, a cell must accumulate a sufficient amount of what has been termed initiation potential (Helmstetter et al., 1968); under conditions of steady-state growth, this requires r min from the previous initiation. Here, because we are assuming the cells capable of adjusting immediately to the decreasing growth rate, the amount of potential accumulated during a short interval of time At is (z/r')At. When the total contribution of all such terms since the preceding initiation sums to r, initiation takes place again. Substitution of eqn (6) in the expression for the initiation potential and equating the sum to r, yields the somewhat unexpected result that V~ is again the same for all cells, fixed at its steady-state value, regardless of the rate at which phosphate is depleted or the amount of phosphate remaining. Cell division occurs C + D min after c h r o m o s o m e initiation. A question now arises as to the proper value to use for C + D. Here there are two schools of thought (Nanninga & Woldringh, 1985); the first maintains that under steady-state conditions, C + D is approximately 64 min and is independent of the growth rate, the
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second, that C + D is constant only at high growth rates but that it increases in proportion to r for doubling times in excess of 64 min or so. We have tested both possibilties: C + D fixed, independent of both the steady-state and the instantaneous growth rates, the same for all cells; C + D of each individual cell proportional to its current effective doubling time whenever r ' exceeds 64 min. The former implies cell division C + D rain after chromosome initiation; the latter requires the evaluation of the effective elapsed time since that initiation. This has two components. As long as r ' < 64 min, each time interval At contributes its full share to the elapsed time, but when r ' > 64 min, each At contributes only (64/T')At. When the total contribution from the two components reaches 64 min, the cell divides. The simplifying assumptions concerning cell geometry have been mentioned before: all cells are taken to be right circular cylinders of equal and constant radius terminated by hemispherical polar caps. In addition, all ceils are assumed to divide symmetrically and, while in steady-state exponential growth, at the same volume.
Parameter values Three growth rates were tested, corresponding to r = 45, 65, 100 min. The value of C + D was taken to be 64 min, independent of r and r', or equal to r or ~' for doubling times greater than 64 min. The constant kr was fixed at two, in accordance with the experimental design (Kepes & Kepes, 1980, 1981), and k~ set to 0.9; several simulations were also carried out for k~ = 0.1 (and ks > 1, which means that all cells enter surface-limited growth together at t = 0 and remain that way throughout the simulation), with essentially the same results. For the cell volume at initiation of c h r o m o s o m e replication, V~, we used 0"30 ixm 3 (Rosenberger et al, 1978a), and for cell radius R, 0.30 ~m at the highest growth rate and 0.25 I~m at the other two (Rosenberger et aL, 1978b). In all cases, Lo was computed analytically from V~, C + D, r, R and the simplifying assumptions listed in the preceding section.
Computer simulation We begin by constructing a volume distribution of cells in steady-state exponential growth at the particular doubling time being considered, based on the various assumptions described above and using the parameter values listed in the preceding section. This idealized population is divided into 100 groups of cells, according to cell age at t = 0 , and their growth and division are then monitored separately throughout the simulation, each group being treated as a homogeneous unit. At first, the cells in all groups are allowed to continue to grow exponentially, (eqn (3)), but at the end of each small time interval At, typically 0.1 min, they are tested to see whether their volume has increased from just below the initiation volume V~ to just above, in the preceding interval. When this has occurred, the precise time of initiation is determined by interpolation, and a clock is started, to record the time elapsed; where required by the auxiliary assumptions concerning the behavior of cells under conditions of substrate limitation, effective elapsed time is monitored instead. Each group is also tested to see whether cell division has taken place during the preceding interval, again interpolating where necessary. At t = 0,
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the phosphate in the culture begins to be consumed, and when its level P(t) drops below that specified by ks (usually 90% of its initial value), the largest cells in the culture are no longer able to grow exponentially and switch to surface-limited growth (eqn (4)). Thus at the end of each time interval At, the cells are also checked to determine whether any group has entered its surface-limited growth phase; following division, the newborn cells are examined to see whether their increased surface-tovolume ratio would be sufficient to enable them to revert to exponential growth with the amount of phosphate left in the medium. Since P(t)-> 0 asymptotically, the dilution-starvation cycle is terminated after a specified time, typically 57, or after a predetermined level of phosphate has been reached, usually 10 -4 P(0). A new cycle is begun by replenishing the phosphate and allowing all the cells to resume exponential growth, everything else remaining as is. Results We distinguish four cases, according to whether we assume that the chromosome replication cycle of a cell does or does not adjust to the slowly decreasing growth rate, and whether C + D is or is not taken to be independent o f the growth rate. Figure 1 is a plot o f the volume distribution o f cells at three different growth rates in steady-state exponential growth, at the end of the first starvation period, and after 15 dilution-starvation cycles, for the case in which the cells are assumed capable of retaining their steady-state properties and C + D is fixed. It is clear that the dilution-starvation cycles have practically no effect on the cell volume distribution under these assumptions; and this is true regardless o f the number of cycles simulated or the stopping criterion chosen. Figure 2 is a similar plot, but with C + D equal to the steady-state doubling time o f the culture for values o f 7/> 65 min. Thus, Fig. 2 ( r = 45 min) is merely a repeat o f Fig. 1 (7 = 45 min), and Fig. 2 (7 = 65 min) and 2 (7--100 min) are identical, since the volume at cell division is the same at the two growth rates (and, in fact, at all doubling times above 65 min). The results for the case in which the cells are assumed capable of adjusting immediately to their slowly decreasing growth rate and C + D is fixed at 64 min, the same for all cells, is presented in Fig. 3. Here the effect of the dilution-starvation cycles is dramatic: mean cell volume during the first cycle drops to a value that is
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FIG. 1. Volumedistribution of cells at three different growth rates in steady-state exponential growth, at end of first starvation period, after 15 dilution-starvation cycles.Cells are assumed capable of retaining steady-state properties; C + D = 64 rain. Arrows indicate value of mean cell volume.
CELL SYNCHRONY
BY S U R F A C E - L I M I T E D Steady-state
GROWTH--AN
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ANALYSIS
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., 0.2 0.4 0"6 0.8
FIG. 2. Volume distribution of cells at three different growth rates in steady-state exponential growth, at end of first starvation period, after 15 dilution-starvation cycles. Cells are assumed capable of retaining steady-state properties; C + D = 64 min for ~"= 45 rain, C + D = ~- for ~"= 65 and 100 min. Arrows indicate value of mean cell volume.
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Fig. 3. Volume distribution of cells at three different growth rates in steady-state exponential growth, at end of first starvation period, after 15 dilution-starvation cycles. Cells are assumed capable of adjusting immediately to growth rate; C + D = 64 min. Arrows indicate value of mean cell volume.
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FIG. 4. Coefficient o f variation o f cell volume distribution and (insert) mean cell volume v at end o f a dilution-starvation cycle as a function o f cycle number for three different growth rates. Cells are assumed capable o f adjusting immediately to growth rate and C + D = 64 min, as in Fig. 3.
84
N.B. Steady-state
i i
GROVER End tst dilution
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0.2 0-4 0.6 0-8 Cell volume (/.ira 3}
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FIG. 5. Volume distribution o f cells at three different growth rates in steady-state exponential growth, at end of first starvation period, after 15 dilution-starvation cycles. Cells are a s s u m e d capable o f adjusting immediately to growth rate; C + D = 6 4 m i n for current effective doubling time z' less than 64min, C + D = ~-' for r ' > 64 min. Arrows indicate value of m e a n cell volume.
independent of the growth rate (more detailed examination of the simulation results shows this to be V~In 2 or 0.21 ~m 3) and stays constant thereafter, whereas the width of the distribution decreases steadily and becomes narrower the shorter the doubling time of the steady-state culture. On continuing the simulations beyond cycle 15, we find that the coefficient of variation of the volume distribution does indeed approach zero, at all three growth rates, but extremely slowly, and that mean cell volume remains fixed at Viln2 (Fig. 4). Figure 5 displays the results for the case of immediate adjustment and the C + D in each cell equal to its current effective doubling time. As in Fig. 2, the simulations for z = 65 min and ~, = 100 min produce identical results; in fact, all four distributions are identical at any given time, since the two sets of assumptions give rise to the same volume at cell division when r~>65 min, as can easily be demonstrated analytically. Discussion
It is an unfortunate fact that the predictions of the Surface-Limited Growth Model depend on how we assume the cell replication cycle to behave under conditions of periodic starvation, assumptions that are extraneous to the Model itself. We have covered what can be considered the two extremes: cells retain their prestarvation, steady-state properties (essentially, the volume at which they divide) and they do not, but instead are capable of adjusting immediately to the slowly decreasing growth rate as the phosphate is depleted, and to its abrupt increase at the start of each new cycle. In both cases, it is still necessary to specify the time interval between the initiation of a round of chromosome replication and the corresponding cell division, C + D, but for different reasons. When the volume at cell division is fixed at its steady-state level, then we only need to know what that level is or, equivalently, the steady-state value of C + D. There seems to be some controversy in the literature on this point (Helmstetter et al., 1979; Nanninga & Woldringh, 1985). For cultures with doubling times less than about one hour, there is general agreement that C + D is constant, independent of growth rate. For slower growing cultures, some researchers report C + D to be constant whereas others find that it increases in proportion to the doubling time. We have tested both possibilities.
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When the cells are assumed capable of adjusting immediately to the changing nutritional conditions, the question of the C + D value is rather different--we are now concerned with the effective value of C + D during starvation. Either we take it to be constant, division occurring a fixed time after chromosome initiation whatever the growth in volume during that interval, or we say that the C + D in slow-growing cells is proportional to the doubling time, the steady-state doubling time when a cell is growing exponentially and the current effective doubling time when it is in surface-limited growth. All four cases were simulated at three growth rates: slow (steady-state doubling time, 100 min), intermediate (65 min), and fast (45 min). Not all the combinations are unique, of course, but the conclusions are clearly and unequivocally independent of growth rate. The results under the assumption of retention of prestarvation, steady-state properties are the same regardless of how one treats C + D - - p e r i o d i c starvation does not decrease the dispersion in the cell volume distribution whatsoever. The same can be said for the assumption that the cells are capable of adjusting immediately to the changing nutritional conditions, but only when C + D is taken to be proportional to the current effective doubling time of the cell (for r ' > 64 min). The other case, C + D constant, differs dramatically. Here the variance of the distribution decreases steadily, rapidly at first and more slowly during successive dilutionstarvation cycles, approaching zero asymptotically: perfect synchrony, just what the Model was designed to produce. Unfortunately, that is not all; mean cell volume also decreases, albeit only during the first cycle, reaching a level of V~ln2 (=0.21 ~m3), the same at all three growth rates. On this, the empirical observations are quite explicit: when transferred to fresh rich medium after the alternating periods of starvation and dilution, the cells not only exhibit synchronous growth, they do so with their original, steady-state doubling time and no lag period, implying that they retain their prestarvation volume. (Actually, what is required is that the limiting volume, Vi In 2, be equal to the volume of a newborn cell in steady-state exponential growth. For the case under consideration, this is simply ½Vi264/T, SO we could achieve synchronous growth without a decrease in cell volume, merely by starting with the right doubling time, around 135 min. This rather trivial exercise is presented here for the sake of completeness, more as a curiosity than for serious consideration.) Thus the Surface-Limited Growth Model, when it does predict cell growth synchrony, cannot in general predict the correct cell size under the same set of assumptions. We are forced, therefore, to reject it, despite its elegance and intuitive appeal. Throughout these simulations, the amount of inorganic phosphate added to the medium at the start of each dilution-starvation cycle was always just sufficient to allow the total cell mass to double exactly once. This is in accordance with the experimental procedures of Kepes & Kepes (1980, 1981), but these authors also stated that their cells, which grew with a doubling time of 65 min in the prestarvation culture, required 90 min for a single mass doubling in the limited medium, which presents somewhat of a paradox since when the rate of cell growth is proportional to the concentration of a limiting substrate, growth becomes slower and slower as
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the substrate is depleted, so the doubling of cell mass is asymptotic. Nevertheless, we have simulated a mass doubling after 90 min by increasing the initial phosphate concentration and terminating the dilution-starvation cycle prematurely (the actual values used were kr = 2.5 in place of 2 and T = 1.4~" instead of 5r). No reduction in the coefficient of variation of the volume distribution could be detected, for any of the assumptions discussed above, even after 50 dilution-starvation cycles. Perhaps it should be mentioned that an additional series of simulations was performed using the assumptions that predict synchrony and decreased cell volume, but with the growth rate during the substrate-limitation phase taken to be proportional to cell volume, rather than to cell surface area (in addition to the current phosphate concentration, of course). The results for mean cell volume are indistinguishable, and as far as dispersion is concerned, the coefficient of variation of the distribution remains perfectly constant throughout the simulation. The question of synchronization by repetitive dilution of a growing culture has been examined in more general terms by Koch (1986), who presented simulation results that define the degree of synchrony that can be obtained with this method, for bacteria of various morphologies utilizing limiting substrates taken up by different processes. The effect of a number of experimental design parameters is considered in detail, but the analysis is unreliable. The coefficient of variation of the canonical volume distribution is quoted as 23%, whereas, as can be demonstrated by elementary calculus, it is actually 20%. That the 23% is not an error of transcription or a misprint can be inferred from the fact that it also appears in the computer-generated graphic output. This is not a trivial mistake. The coefficient of variation in the case where "all daughter cells are born of equal size and the growth of cells at all times is proportional to their size" (Koch, 1986) does not depend on any additional assumptions and is the same regardless of cell geometry, namely x/~k-- 1, where k - I/In 2. This is an analytical result, rigorous and exact. A numerical error at that level suggests basic problems with the mathematics or the algorithms or the FORTRAN. (A personal communication from the author in his capacity as referee maintains that the figure of 23% refers to the case in which cell size at division is not fixed but normally distributed, with a coefficient of variation of 10%. This is certainly not how the published text reads, and some sort of clarification may be in order.) Despite the study by Koch we remain with our conclusion that the Surface-Limited Growth Model is incapable o f explaining the synchronization of cells by periodic starvation of a growing bacterial culture. This may be because it is inadequate, and additional elements are required or, less likely, because our assumptions concerning the effects of severe nutritional deprivation on the behavior of C + D are unrealistic. Alternatively, the problem could lie not with the model or our quantification of it but with the actual experimental t e c h n i q u e - - n o one has ever managed to make it work outside the laboratory where it originated and, according to one of the reviewers, there is also internal evidence suggesting that it may be flawed. -Indeed, a recent publication from that laboratory (Plateau et aL, 1987) would seem to support this supposition: cells conditioned by periodic phosphate starvation were found to exhibit abnormal DNA replication cycles after transfer to regular growth medium.
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If such cells indeed prove to have an altered p h y s i o l o g y that persists b e y o n d the entrainment phase, then our c o n c l u s i o n s may reflect a defect in the m e t h o d o l o g y o f cell preparation rather than in any m o d e l designed to account for the data it produces. The author wishes to express his thanks to Professor R. H. Pritchard for the time and thought he devoted to the material contained in this article. REFERENCES
GROVER, N. B., ZARITSKY, A., WOLDRINGH,C. L. & ROSENBERGER, R. F. (1980). J. theor. BioL 86, 421-439. HELMSTET'TER, C. E., COPPER, S., PIERUCCI, O. & REVELAS, E. (1968). Cold Spring Harbor Symp. quant. BioL 33, 809-822. HELMSTETTER, C. E., PIERUCCI, O., WEINBERGER, M., HOLMES, M. & TANG, M.-S. (1979). In: The Bacteria (Sokatch, J. R. & Ornston, L. N., eds), vol. VII, pp. 517-546. New York: Academic Press. KEPES, F. & KEPES, A. (1980). Ann. MicrobioL 131A, 3-16. KEPES, F. & KEPES, A. (1981). Biochem. Biophys. Res. Commun. 99, 761-767. KOCH, A. L. (1986)../. theor. Biol. 123, 333-346. NANNINGA, N. & WOLDRINGH, C. L. (1985). In: Molecular Cytology of Escherichia coli (Nanninga, N., ed.), pp. 259-318. New York: Academic Press. PLATEAU, P., FROMANT, M., KEPES, F. & BLANQUET, S. (1987). J. Bacterial. 169, 419-422. ROSENBERGER, R. F., GROVER, N. B., ZARITSKY, A. & WOLDRINGH, C. L. (1978a). J'. theor. Biol. 73, 711-721. ROSENBERGER, R. F., GROVER, N. B., ZARITSKY, A. & WOLDRINGH, C. L. (1978b). Nature (Lond.) 271,244-245.
Surface-Limited Growth: A Model for the Synchronization of a Growing Bacterial Culture through Periodic Starvation N. B. GROVER
The Hubert H. Humphrey Center for Experimental Medicine and Cancer Research, Hebrew University-Hadassah Medical School, Jerusalem, Israel (Received 8 December 1987, and in revised form 25 March 1988) This article analyses the Surface-Limited Growth Model put forward to explain the very tight synchrony, over more than ten division cycles, obtained experimentally by subjecting a growing bacterial culture to alternating periods of starvation and dilution, using inorganic phosphate as the limiting substrate. The Model states that when an essential nutrient is in limited supply, the rate of growth of an individual cell will be proportional to its surface area (and the current concentration of the limiting substance) rather than to its volume. This decrease in dimensionality from volume to surface is expected to favor the smaller cells and so result ultimately in a narrower size distribution. The Surface-Limited Growth Model deals with cell growth under unusual nutritional conditions, and its predictions depend on how the cell replication cycle is assumed to behave under these same circumstances. Two alternatives are considered: the volume at which cells divide is the same during the starvation phase as during steady-state exponential growth, and the cells adjust immediately to the changing growth rate. In the latter case, we have tested both C + D constant with time and C + D variable (where C + D is the time between initiation of chromosome replication and the corresponding cell division), the incremental value at any instant being computed separately for each individual cell from its current effective growth rate. The simulation results are of two sorts depending on the auxiliary assumptions used. Either the dilution-starvation cycles have no effect whatsoever on the cell volume distribution, or the width of the distribution decreases gradually with time, approaching zero slowly and asymptotically, but the mean cell volume decreases as well--directly contradicting experimental observations. We conclude that the Surface-Limited Growth Model is incapable of explaining the synchronization of cells by periodic starvation of a growing bacterial culture.
Introduction H e a l t h y , w e l l - f e d b a c t e r i a l cells n o r m a l l y g r o w a s y n c h r o n o u s l y , a n d m a n y a t t e m p t s h a v e b e e n m a d e o v e r the y e a r s to p r o d u c e s y n c h r o n o u s c u l t u r e s b e c a u s e o f the o b v i o u s a d v a n t a g e s t h e y w o u l d offer for a n y b i o c h e m i c a l or p h y s i o l o g i c a l studies i n v o l v i n g s e q u e n t i a l c h a n g e s in i n d i v i d u a l cells d u r i n g t h e i r life cycle. M o s t m e t h o d s , w h e t h e r b a s e d on i n d u c t i o n or selection, give rise to a s y n c h r o n y t h a t d e t e r i o r a t e s r a p i d l y , u s u a l l y b e g i n n i n g in the s e c o n d o r t h i r d g e n e r a t i o n . T h e r e f o r e , t h e r e was c o n s i d e r a b l e t h e o r e t i c a l as well as p r a c t i c a l interest w h e n a t e c h n i q u e was d e s c r i b e d 77 0022-5193/88/170077+ 11 $03.00/0
© 1988 Academic Press Limited
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that purported to provide cells capable of maintaining very tight synchrony over more than ten cell division cycles (Kepes & Kepes, 1980, 1981). Briefly, ceils are grown in a fermenter with just sufficient inorganic phosphate to allow one mass doubling, after which the culture is diluted twofold, again with only enough phosphate for a single doubling. These dilutions and mass doublings in the phosphatelimited medium are repeated 10-16 times, after which the cells are found to exhibit synchronous behavior when transferred to regular unrestricted batch culture conditions; their mass at birth is then similar to what it had been prior to synchronization. The Surface-Limited Growth Model was first proposed publicly by R. H. Pritchard (EMBO Workshop on Regulation of the Cell Cycle in Prokaryotes, Sede Boqer, September 1984). The basic idea was elegant and deceptively simple: cells normally grow at a rate that is proportional to their mass or volume, but when an essential nutrient is in limited supply, the rate of growth of an individual cell will depend instead on how fast the cell can take up that nutrient from the surrounding medium and this rate, in turn, should be proportional to the cell's surface area (and, of course, to the current concentration of the limiting substance). According to this Model then, synchronization is merely a matter of geometry, the decrease in dimensionality from volume to surface being expected to favor the smaller cells and so result ultimately in a narrower cell size distribution. We have translated the Model into precise mathematical terms and studied the effect on a steady-state exponential Escherichia coli culture, of alternating periods of unrestricted and substrate-limited growth, by simulating the behavior of individual cells over many generations, and from this, constructing cell size distributions as a function of time and phosphate concentration. From the results, it is clear that the Surface-Limited Growth Model is not capable of reproducing the experimental observations. It should be borne in mind, however, that periodic phosphate starvation of a growing bacterial culture may disturb the normal physiology of a cell, since cells so treated do not display the usual pattern of D N A synthesis during the division cycle (Plateau et al., 1987); if this were the case, then the present analysis would be no more than an academic exercise.
The Model
Let M(t) represent the total cell mass t minutes after transfer to the limiting medium, and P(t) the amount of inorganic phosphate remaining. Then
P(O)-P(t)=kp[M(t)-M(O)],
(1)
where kp is a constant, since the decrease in phosphate is proportional to the increase in total mass. At the end of a starvation period, say t = T, all the phosphate has been consumed and the mass has doubled, so that P(T)=O=P(O)k p [ 2 M ( 0 ) - M(0)], whence kp = P(O)/M(O). Actually P(t)~O asymptotically, and in practice, one must terminate the simulation at a predetermined small value of P(t), say 10 -4 P(0), or after some suitable long time T has elapsed, say 5r, where ~"is the doubling time of the culture in the unrestricted growth medium. Substituting
CELL
for
SYNCHRONY
kp in
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79
eq (1) and rearranging, we get
P(t)
2
M(t)-2-y(t),
P(O)
(2)
M(O)
where y(t) represents the ratio M(t)/M(O). In the general case, where y(t) at the end of a dilution cycle approaches some value kr other than two, eqn (2) assumes the form
P(t) -1 P(O)
- -
y(t)-I
kr - 1
As long as the phosphate concentration is large enough, the cells will grow exponentially: dv(t) =/~v(t), dt
(3)
where v(t) is the volume (assumed proportional to mass) of a cell at time t and /z-= (In 2 ) / r is the steady-state specific growth rate. When P(t) becomes limiting, the Model states that growth will be proportional to cell surface area s(t) and to the amount of phosphate remaining. In other words dr(t)
dt
- k,,s(t)P(t)/P(O),
where k,, is the proportionality constant. If we adopt the usual geometry of circular cylinders with hemispherical polar caps, then dv dt
- - = 2k,,(2- y)(v+ 27rRa/3)/ R,
(4)
since s(t)=2(v+27rR3/3)/R, where R is the cell radius, assumed the same for all cells and constant with time. Here we have simplified the notation by using v for v(t) and y for y(t). Whether eqn (3) or eqn (4) obtains for a particular cell will depend on the cell's dimensions: the larger the cell, the less likely it is that the medium will be able to support its unrestricted growth. The precise level at which the transition occurs will depend on the value of the permeability coefficient of the cell membrane for inorganic phosphate. If we take this coefficient (uptake/phosphate gradient/unit surface area/unit time) to be the same for all cells, then the largest cell will be the first to enter the state of restricted growth. Let this occur when P(t) has reached the level ksP(0), where k~ is a constant less than one. Then it is easy to show that k., -
/zR 3 L o - 2 R 2k~ 3 L o - R '
(5)
where Lo is the length of newborn cells during steady-state exponential growth. These equations together govern the growth of the cells: when there is sufficient phosphate in the culture medium to support the exponential growth of a particular
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cell, eqn (3) obtains; as time passes and P(t) decreases (or y increases), the right-hand side of eqn (4) falls below that of eqn (3) and growth becomes limited, following eqn (4).
Assumptions The Surface-Limited Growth Model deals with cell growth under unusual nutritional conditions and its predictions, unfortunately but perhaps not expectedly, depend on how we assume the cell replication cycle to behave under these same circumstances. Two possibilities are treated. In the first, the volume at which the cells divide is assumed to be the same during the starvation phase as during unrestricted growth; in the second, the cells are considered capable of adjusting immediately to the slowly decreasing growth rate. In both instances, it is necessary to determine when a particular cell is going to initiate a round of c h r o m o s o m e replication and when it is ready to divide. These two major events in the cell cycle are most conveniently characterized by V~, the volume of a cell at c h r o m o s o m e initiation; Vd, the volume at the corresponding cell division; and C + D, the time it takes a cell to grow from the one to the other. In the first case, the cells are assumed to retain steady-state values of V~ and Vd, which therefore remain constant and the same for all cells, and the calculations are straightforward. In the second, we make use of an auxiliary variable r', the current effective doubling time of the culture (Grover et al., 1980), defined as r' -
vln2
dv/dt"
(6)
During steady-state exponential growth, z ' = r , as can be seen from eqn (3), but during the starvation phase, r'> r and, in fact, r ' . o o as P ( t ) ~ O . In order to initiate c h r o m o s o m e replication, a cell must accumulate a sufficient amount of what has been termed initiation potential (Helmstetter et al., 1968); under conditions of steady-state growth, this requires r min from the previous initiation. Here, because we are assuming the cells capable of adjusting immediately to the decreasing growth rate, the amount of potential accumulated during a short interval of time At is (z/r')At. When the total contribution of all such terms since the preceding initiation sums to r, initiation takes place again. Substitution of eqn (6) in the expression for the initiation potential and equating the sum to r, yields the somewhat unexpected result that V~ is again the same for all cells, fixed at its steady-state value, regardless of the rate at which phosphate is depleted or the amount of phosphate remaining. Cell division occurs C + D min after c h r o m o s o m e initiation. A question now arises as to the proper value to use for C + D. Here there are two schools of thought (Nanninga & Woldringh, 1985); the first maintains that under steady-state conditions, C + D is approximately 64 min and is independent of the growth rate, the
CELL
SYNCHRONY
BY
SURFACE-LIMITED
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ANALYSIS
81
second, that C + D is constant only at high growth rates but that it increases in proportion to r for doubling times in excess of 64 min or so. We have tested both possibilties: C + D fixed, independent of both the steady-state and the instantaneous growth rates, the same for all cells; C + D of each individual cell proportional to its current effective doubling time whenever r ' exceeds 64 min. The former implies cell division C + D rain after chromosome initiation; the latter requires the evaluation of the effective elapsed time since that initiation. This has two components. As long as r ' < 64 min, each time interval At contributes its full share to the elapsed time, but when r ' > 64 min, each At contributes only (64/T')At. When the total contribution from the two components reaches 64 min, the cell divides. The simplifying assumptions concerning cell geometry have been mentioned before: all cells are taken to be right circular cylinders of equal and constant radius terminated by hemispherical polar caps. In addition, all ceils are assumed to divide symmetrically and, while in steady-state exponential growth, at the same volume.
Parameter values Three growth rates were tested, corresponding to r = 45, 65, 100 min. The value of C + D was taken to be 64 min, independent of r and r', or equal to r or ~' for doubling times greater than 64 min. The constant kr was fixed at two, in accordance with the experimental design (Kepes & Kepes, 1980, 1981), and k~ set to 0.9; several simulations were also carried out for k~ = 0.1 (and ks > 1, which means that all cells enter surface-limited growth together at t = 0 and remain that way throughout the simulation), with essentially the same results. For the cell volume at initiation of c h r o m o s o m e replication, V~, we used 0"30 ixm 3 (Rosenberger et al, 1978a), and for cell radius R, 0.30 ~m at the highest growth rate and 0.25 I~m at the other two (Rosenberger et aL, 1978b). In all cases, Lo was computed analytically from V~, C + D, r, R and the simplifying assumptions listed in the preceding section.
Computer simulation We begin by constructing a volume distribution of cells in steady-state exponential growth at the particular doubling time being considered, based on the various assumptions described above and using the parameter values listed in the preceding section. This idealized population is divided into 100 groups of cells, according to cell age at t = 0 , and their growth and division are then monitored separately throughout the simulation, each group being treated as a homogeneous unit. At first, the cells in all groups are allowed to continue to grow exponentially, (eqn (3)), but at the end of each small time interval At, typically 0.1 min, they are tested to see whether their volume has increased from just below the initiation volume V~ to just above, in the preceding interval. When this has occurred, the precise time of initiation is determined by interpolation, and a clock is started, to record the time elapsed; where required by the auxiliary assumptions concerning the behavior of cells under conditions of substrate limitation, effective elapsed time is monitored instead. Each group is also tested to see whether cell division has taken place during the preceding interval, again interpolating where necessary. At t = 0,
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the phosphate in the culture begins to be consumed, and when its level P(t) drops below that specified by ks (usually 90% of its initial value), the largest cells in the culture are no longer able to grow exponentially and switch to surface-limited growth (eqn (4)). Thus at the end of each time interval At, the cells are also checked to determine whether any group has entered its surface-limited growth phase; following division, the newborn cells are examined to see whether their increased surface-tovolume ratio would be sufficient to enable them to revert to exponential growth with the amount of phosphate left in the medium. Since P(t)-> 0 asymptotically, the dilution-starvation cycle is terminated after a specified time, typically 57, or after a predetermined level of phosphate has been reached, usually 10 -4 P(0). A new cycle is begun by replenishing the phosphate and allowing all the cells to resume exponential growth, everything else remaining as is. Results We distinguish four cases, according to whether we assume that the chromosome replication cycle of a cell does or does not adjust to the slowly decreasing growth rate, and whether C + D is or is not taken to be independent o f the growth rate. Figure 1 is a plot o f the volume distribution o f cells at three different growth rates in steady-state exponential growth, at the end of the first starvation period, and after 15 dilution-starvation cycles, for the case in which the cells are assumed capable of retaining their steady-state properties and C + D is fixed. It is clear that the dilution-starvation cycles have practically no effect on the cell volume distribution under these assumptions; and this is true regardless o f the number of cycles simulated or the stopping criterion chosen. Figure 2 is a similar plot, but with C + D equal to the steady-state doubling time o f the culture for values o f 7/> 65 min. Thus, Fig. 2 ( r = 45 min) is merely a repeat o f Fig. 1 (7 = 45 min), and Fig. 2 (7 = 65 min) and 2 (7--100 min) are identical, since the volume at cell division is the same at the two growth rates (and, in fact, at all doubling times above 65 min). The results for the case in which the cells are assumed capable of adjusting immediately to their slowly decreasing growth rate and C + D is fixed at 64 min, the same for all cells, is presented in Fig. 3. Here the effect of the dilution-starvation cycles is dramatic: mean cell volume during the first cycle drops to a value that is
Steady-state --
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FIG. 1. Volumedistribution of cells at three different growth rates in steady-state exponential growth, at end of first starvation period, after 15 dilution-starvation cycles.Cells are assumed capable of retaining steady-state properties; C + D = 64 rain. Arrows indicate value of mean cell volume.
CELL SYNCHRONY
BY S U R F A C E - L I M I T E D Steady-state
GROWTH--AN
End Ist dilution
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ANALYSIS
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., 0.2 0.4 0"6 0.8
FIG. 2. Volume distribution of cells at three different growth rates in steady-state exponential growth, at end of first starvation period, after 15 dilution-starvation cycles. Cells are assumed capable of retaining steady-state properties; C + D = 64 min for ~"= 45 rain, C + D = ~- for ~"= 65 and 100 min. Arrows indicate value of mean cell volume.
Steady-state
End Ist dilution
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i
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0
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Fig. 3. Volume distribution of cells at three different growth rates in steady-state exponential growth, at end of first starvation period, after 15 dilution-starvation cycles. Cells are assumed capable of adjusting immediately to growth rate; C + D = 64 min. Arrows indicate value of mean cell volume.
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FIG. 4. Coefficient o f variation o f cell volume distribution and (insert) mean cell volume v at end o f a dilution-starvation cycle as a function o f cycle number for three different growth rates. Cells are assumed capable o f adjusting immediately to growth rate and C + D = 64 min, as in Fig. 3.
84
N.B. Steady-state
i i
GROVER End tst dilution
"~r
0
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g ~5
After 15 cycles
lift,
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0
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0.2 0-4 0.6 0-8 Cell volume (/.ira 3}
0
:~:, ~,..
0.2 0.4 0'6 0.8
FIG. 5. Volume distribution o f cells at three different growth rates in steady-state exponential growth, at end of first starvation period, after 15 dilution-starvation cycles. Cells are a s s u m e d capable o f adjusting immediately to growth rate; C + D = 6 4 m i n for current effective doubling time z' less than 64min, C + D = ~-' for r ' > 64 min. Arrows indicate value of m e a n cell volume.
independent of the growth rate (more detailed examination of the simulation results shows this to be V~In 2 or 0.21 ~m 3) and stays constant thereafter, whereas the width of the distribution decreases steadily and becomes narrower the shorter the doubling time of the steady-state culture. On continuing the simulations beyond cycle 15, we find that the coefficient of variation of the volume distribution does indeed approach zero, at all three growth rates, but extremely slowly, and that mean cell volume remains fixed at Viln2 (Fig. 4). Figure 5 displays the results for the case of immediate adjustment and the C + D in each cell equal to its current effective doubling time. As in Fig. 2, the simulations for z = 65 min and ~, = 100 min produce identical results; in fact, all four distributions are identical at any given time, since the two sets of assumptions give rise to the same volume at cell division when r~>65 min, as can easily be demonstrated analytically. Discussion
It is an unfortunate fact that the predictions of the Surface-Limited Growth Model depend on how we assume the cell replication cycle to behave under conditions of periodic starvation, assumptions that are extraneous to the Model itself. We have covered what can be considered the two extremes: cells retain their prestarvation, steady-state properties (essentially, the volume at which they divide) and they do not, but instead are capable of adjusting immediately to the slowly decreasing growth rate as the phosphate is depleted, and to its abrupt increase at the start of each new cycle. In both cases, it is still necessary to specify the time interval between the initiation of a round of chromosome replication and the corresponding cell division, C + D, but for different reasons. When the volume at cell division is fixed at its steady-state level, then we only need to know what that level is or, equivalently, the steady-state value of C + D. There seems to be some controversy in the literature on this point (Helmstetter et al., 1979; Nanninga & Woldringh, 1985). For cultures with doubling times less than about one hour, there is general agreement that C + D is constant, independent of growth rate. For slower growing cultures, some researchers report C + D to be constant whereas others find that it increases in proportion to the doubling time. We have tested both possibilities.
CELL
SYNCHRONY
BY S U R F A C E - L I M I T E D
GROWTH--AN
ANALYSIS
85
When the cells are assumed capable of adjusting immediately to the changing nutritional conditions, the question of the C + D value is rather different--we are now concerned with the effective value of C + D during starvation. Either we take it to be constant, division occurring a fixed time after chromosome initiation whatever the growth in volume during that interval, or we say that the C + D in slow-growing cells is proportional to the doubling time, the steady-state doubling time when a cell is growing exponentially and the current effective doubling time when it is in surface-limited growth. All four cases were simulated at three growth rates: slow (steady-state doubling time, 100 min), intermediate (65 min), and fast (45 min). Not all the combinations are unique, of course, but the conclusions are clearly and unequivocally independent of growth rate. The results under the assumption of retention of prestarvation, steady-state properties are the same regardless of how one treats C + D - - p e r i o d i c starvation does not decrease the dispersion in the cell volume distribution whatsoever. The same can be said for the assumption that the cells are capable of adjusting immediately to the changing nutritional conditions, but only when C + D is taken to be proportional to the current effective doubling time of the cell (for r ' > 64 min). The other case, C + D constant, differs dramatically. Here the variance of the distribution decreases steadily, rapidly at first and more slowly during successive dilutionstarvation cycles, approaching zero asymptotically: perfect synchrony, just what the Model was designed to produce. Unfortunately, that is not all; mean cell volume also decreases, albeit only during the first cycle, reaching a level of V~ln2 (=0.21 ~m3), the same at all three growth rates. On this, the empirical observations are quite explicit: when transferred to fresh rich medium after the alternating periods of starvation and dilution, the cells not only exhibit synchronous growth, they do so with their original, steady-state doubling time and no lag period, implying that they retain their prestarvation volume. (Actually, what is required is that the limiting volume, Vi In 2, be equal to the volume of a newborn cell in steady-state exponential growth. For the case under consideration, this is simply ½Vi264/T, SO we could achieve synchronous growth without a decrease in cell volume, merely by starting with the right doubling time, around 135 min. This rather trivial exercise is presented here for the sake of completeness, more as a curiosity than for serious consideration.) Thus the Surface-Limited Growth Model, when it does predict cell growth synchrony, cannot in general predict the correct cell size under the same set of assumptions. We are forced, therefore, to reject it, despite its elegance and intuitive appeal. Throughout these simulations, the amount of inorganic phosphate added to the medium at the start of each dilution-starvation cycle was always just sufficient to allow the total cell mass to double exactly once. This is in accordance with the experimental procedures of Kepes & Kepes (1980, 1981), but these authors also stated that their cells, which grew with a doubling time of 65 min in the prestarvation culture, required 90 min for a single mass doubling in the limited medium, which presents somewhat of a paradox since when the rate of cell growth is proportional to the concentration of a limiting substrate, growth becomes slower and slower as
86
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the substrate is depleted, so the doubling of cell mass is asymptotic. Nevertheless, we have simulated a mass doubling after 90 min by increasing the initial phosphate concentration and terminating the dilution-starvation cycle prematurely (the actual values used were kr = 2.5 in place of 2 and T = 1.4~" instead of 5r). No reduction in the coefficient of variation of the volume distribution could be detected, for any of the assumptions discussed above, even after 50 dilution-starvation cycles. Perhaps it should be mentioned that an additional series of simulations was performed using the assumptions that predict synchrony and decreased cell volume, but with the growth rate during the substrate-limitation phase taken to be proportional to cell volume, rather than to cell surface area (in addition to the current phosphate concentration, of course). The results for mean cell volume are indistinguishable, and as far as dispersion is concerned, the coefficient of variation of the distribution remains perfectly constant throughout the simulation. The question of synchronization by repetitive dilution of a growing culture has been examined in more general terms by Koch (1986), who presented simulation results that define the degree of synchrony that can be obtained with this method, for bacteria of various morphologies utilizing limiting substrates taken up by different processes. The effect of a number of experimental design parameters is considered in detail, but the analysis is unreliable. The coefficient of variation of the canonical volume distribution is quoted as 23%, whereas, as can be demonstrated by elementary calculus, it is actually 20%. That the 23% is not an error of transcription or a misprint can be inferred from the fact that it also appears in the computer-generated graphic output. This is not a trivial mistake. The coefficient of variation in the case where "all daughter cells are born of equal size and the growth of cells at all times is proportional to their size" (Koch, 1986) does not depend on any additional assumptions and is the same regardless of cell geometry, namely x/~k-- 1, where k - I/In 2. This is an analytical result, rigorous and exact. A numerical error at that level suggests basic problems with the mathematics or the algorithms or the FORTRAN. (A personal communication from the author in his capacity as referee maintains that the figure of 23% refers to the case in which cell size at division is not fixed but normally distributed, with a coefficient of variation of 10%. This is certainly not how the published text reads, and some sort of clarification may be in order.) Despite the study by Koch we remain with our conclusion that the Surface-Limited Growth Model is incapable o f explaining the synchronization of cells by periodic starvation of a growing bacterial culture. This may be because it is inadequate, and additional elements are required or, less likely, because our assumptions concerning the effects of severe nutritional deprivation on the behavior of C + D are unrealistic. Alternatively, the problem could lie not with the model or our quantification of it but with the actual experimental t e c h n i q u e - - n o one has ever managed to make it work outside the laboratory where it originated and, according to one of the reviewers, there is also internal evidence suggesting that it may be flawed. -Indeed, a recent publication from that laboratory (Plateau et aL, 1987) would seem to support this supposition: cells conditioned by periodic phosphate starvation were found to exhibit abnormal DNA replication cycles after transfer to regular growth medium.
CELL SYNCHRONY
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ANALYSIS
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If such cells indeed prove to have an altered p h y s i o l o g y that persists b e y o n d the entrainment phase, then our c o n c l u s i o n s may reflect a defect in the m e t h o d o l o g y o f cell preparation rather than in any m o d e l designed to account for the data it produces. The author wishes to express his thanks to Professor R. H. Pritchard for the time and thought he devoted to the material contained in this article. REFERENCES
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