CELL DENSITY DEPENDENCE OF THE AGGREGATION CHARACTERISTICS OF THE CELLULAR SLIME MOULD DICTYOSTELIUM DISCOIDEUM

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1 J. Cell Sci. 19, (1975) 215 Printed in Great Britain CELL DENSITY DEPENDENCE OF THE AGGREGATION CHARACTERISTICS OF THE CELLULAR SLIME MOULD DICTYOSTELIUM DISCOIDEUM Y. HASHIMOTO,* M. H. COHEN AND A. ROBERTSONf Department of Biophysics and Theoretical Biology, University of Chicago, 920 East 58th Street, Chicago, Illinois 60637, U.S.A. SUMMARY We have measured fruiting body density and spore formation efficiency in Dictyostelium discoideum as functions of initial cell density. Experiments were performed on agar made up with distilled water and on buffered agar. Minor differences are seen; these are discussed. The functions show 4 regions of density dependence which can be accounted for by changes in aggregation characteristics with density and changes in the efficiency of spore differentiation. The results are discussed in terms of the relaying mechanism for signal propagation controlling cell aggregation. They extend earlier measurements by Bonner & Dodd and by Hohl & Raper, supply data for a quantitative model of the aggregation process, allow estimates of signal range, and show the importance of entrainment between neighbouring centres in defining aggregation territories. INTRODUCTION The cellular slime mould Dictyostelium discoideum was discovered by Raper in In its growth phase, it consists of free living, unicellular amoebae feeding on bacteria (Raper, 1940). The growth phase terminates in the absence of food bacteria, and, after a period of differentiation called interphase, the amoebae aggregate (Raper, 1940; Bonner, 1967). The aggregation is controlled by a co-operative signalling process (Shaffer, 1962). It is likely that c-amp is both the propagating signal and the chemotactic factor which induces the aggregative movements (Konijn, van de Meene, Bonner & Barkeley, 1967). After aggregation, morphogenesis and differentiation continue until multicellular fruiting bodies are formed, which consist mainly of spore and stalk cells (Raper, 1940; Bonner, 1967). The cellular slime moulds are organisms of particular interest for developmental biology (Bonner, 1958; Raper, i960). The unitary processes of development (Robertson & Cohen, 1972) are all exhibited simply and are well separated in time (Bonner, 1944, 1967; Raper, 1941). Among the cellular slime moulds, Dictyostelium discoideum is best known (Bonner, 1967). In particular, details of the way in which its aggregation is controlled are beginning to emerge in quantitative form (Robertson & Cohen, 1972). On leave from Tokyo Metropolitan Isotope Research Center. t Sloan Foundation Fellow, , and the author to whom correspondence should be sent.

2 2i6 Y. Hashimoto, M. H. Cohen and A. Robertson Because of their intrinsic interest and because of the possibility of insights afforded into larger questions in developmental biology, more extensive quantitative investigation into aggregation as morphogenesis and into its control is warranted. It seems intuitively obvious that the details of aggregation depend sensitively on the initial cell density. Measurement of this density dependence of the aggregation characteristics of Dictyostelium discoideum should then yield insight into the dynamics of aggregation. It can be inferred from experiments of Konijn & Raper (1961) that aggregation ceases at initial amoeba densities below about 5 x io 4 cm" 2. The existence of a critical density for aggregation can be explained by supposing that a signal generated by an aggregating amoeba has a finite range beyond which it cannot stimulate another amoeba to signal (Cohen & Robertson, 1971 a, b). By use of percolation theory (Shante & Kirkpatrick, 1971) in the description of signal propagation through random, low-density fields of amoebae (Cohen & Robertson, 1972) one can extract a value of 50 /tm for the range from the Konijn-Raper value of the critical density. The range is an important quantity, relating directly to the parameters of the c-amp signal, the extracellular phosphodiesterase which removes c-amp and the threshold concentration for relaying (Shante & Kirkpatrick, 1971; Robertson & Cohen, 1974). More accurate and complete data for the density dependence of the aggregation characteristics would yield a more accurate value for the range. Hohl & Raper (1964) have reported an approximate independence of the aggregate size frequency distribution on the initial cell density. In the same paper, they mention that fruiting body density increases at high densities. On the other hand, Bonner & Dodd (1962) find that territory radius is largely independent of the initial density of a bacterial food supply and suggest a centre-spacing substance to explain their results. Provided post-aggregative development is efficient, and each aggregate produces one fruiting body, fruiting body density is inversely proportional to territory area. Thus there may be a discrepancy between the Hohl & Raper results and the Bonner & Dodd results, unless the amoeba density at the onset of aggregation in the latter experiments was either relatively low or relatively constant. Further measurements of fruiting body density are clearly required. Accordingly, we have undertaken a series of experiments in which accurate measurements are taken of spore formation efficiency, fruiting body density, and mean number of spores per fruiting body for Dictyostelium discoideum on agar of various compositions. The results of these investigations are reported and discussed here. MATERIAL AND METHODS Spores of Dictyostelium discoideum, strain NC-4, descendants of a stock obtained from K. B. Raper, were plated on nutrient agar (Bonner, 1967) with the bacterium Aerobacter aerogenes and incubated at 22 C. At 36 h of incubation, while the amoebae were still in the growth phase, cultures to be used as sources of amoebae for experiment were suspended in cold PAD diluting fluid (Sussman, 1966). The bacteria were removed by centrifuging the suspension for 8 min at 4 C, resuspending and repeating twice. The volume density of the bacteria-free cell suspension was measured with a haemocytometer. The suspension was plated on agar (2 % agar in distilled water or phosphate buffer (K-K 2 ) solution (Bonner, 1967)) and spread as uniformly as possible with a glass rod. Subse-

3 Cell density dependence of aggregation characteristics 217 quent random movements of the amoebae early in interphase made the cell distribution still more uniform over the agar surface. The surface cell density was calculated from the agar area and the quantity of cell suspension plated out and checked by direct counting. Counts were made at least 2 h after plating, as we had found that no cell division was observed after this delay. This point was checked further by observation of films (see Bonner & Frascella, 1952). Cell densities obtained in this way ranged from 5 x io 8 to 2 x io 7 cm" 1. The density of a close packed monolayer of cells is 2 x io 9 cm"', using 7 /im as the mean cell diameter. The agar plates were placed in a 10-cm covered Petri dish containing wet filter paper and incubated at 22 C for 48, 72, 96 and 120 h. Ten plates were used for each point in the density measurements, except for low densities where up to 40 plates were used. After incubation, the number of fruiting bodies was counted. The spores were harvested into 1 ml of PAD diluting fluid, and the resulting volume density of spores measured with a haemocytometer. The total number of spores was then calculated from the known fluid volume. The spore formation efficiency (SFE), the total number of spores produced as a percentage of the total number of cells plated out, was then calculated, as was the average number of spores per fruiting body. In all number or density measurements, the statistical error (standard deviation) was kept below 10% when practicable. This was difficult or impossible at the lower densities. The standard deviation arising from purely statistical sources was recorded for each measurement. In general, systematic errors were unimportant. To establish the times at which specific developmental stages were reached at different densities, 16-mm time-lapse films were taken at various frame rates of plates at each density. Total magnification was 2'5- SYMBOLS USED IN THE TEXT N o Initial cell density. SFE Spore formation efficiency. DE Developmental efficiency. Np B Fruiting body density../f^p Mean number of spores per fruiting body. N,p Mean spore density. R, Range of relayed signal. N # Critical cell density for signal relaying. r. Mean aggregation territory size. RESULTS We define as a standard preparation, amoebae on a plate of distilled-water agar 0-5 cm thick, incubated for 48 h, and report our results first for such plates. Then we report the effects of increasing incubation time and of substituting K-K 2 solution for distilled water. Standard plates Spore formation efficiency. In Fig. 1, we present the results of our measurements of SFE on standard plates. There are 4 well defined regions of initial cell density, N o> within each of which the dependence of SFE on N o is quite characteristic: (1) Below 4X io 4 cm" 2 SFE is under 10% and drops with decreasing N o, forming a small tail on the rest of the curve; (2) from 4 x io 4 to i-6 x io 8 cm" 8 SFE rises rapidly to about 60%, (3) from i-6 x io B to i-8x io 6 cm" 2, it saturates in the range 60-65; an d (4) above i-8 x io 6 cm" 2, it falls continuously and is down to 13 % at i-6 x io 7 cm" 2.

4 2l8 Y. Hashimoto, M. H. Cohen and A. Robertson uj- 40 SO 20 1x10» 1X10 4 1X10 1 1x10* Cells cm" 1 I 1X10 7 Fig. i. Spore formation efficiency (SFE) as a function of initial cell density measured after 48-h incubation on plain agar. Developmental efficiency (DE) is plotted on the right-hand ordinate I I I I 1x10 1X10 4 ixio 5 Cells cm-' 1x10* 1x10' Fig. 2. Fruiting body density (N ra ) measured after 48-h incubation on plain agar.

5 Cell density dependence of aggregation characteristics 219 Fruiting body density. We present the results of our fruiting body counts for standard plates in Fig. 2. The same 4 regions show up in the fruiting body density. (1) Below 3 x io 4 cm~ 2 the fruiting-body density is of order 1 cm" 8 ; (2) between 3 x io 4 and 2 x io 6 cm" 2, it increases to about 10 cm~ 2 and saturates at this value; (3) from 2 x io 5 to 2 x io 6 cm" 2, it increases to about 20 cm" 8 ; and (4) from 2 x io 8 cm" 2 on it increases, saturating at about 50 at 2 x io 7 cm" 8. The boundaries of these regions are, of course, not sharply marked and differ somewhat in the SFE and in the fruiting body density. 1x10* - 1x10* 1X10 1 I I 1x10» 1X10 4 1x10* 1X10 4 1x10 7 Cells cm" 1 Fig. 3. Mean number of spores per fruiting body (^V, p ) measured after 48-h incubation on plain agar. The time-lapse films showed that, under our conditions, each aggregate formed one fruiting body. The fruiting-body density is therefore equal to the aggregate density, and no separate measurement of aggregate density is necessary. Mean fruiting body size. Our results for mean number of spores per fruiting body are shown in Fig. 3. There is no significant difference in the behaviour of this quantity in the first 2 density ranges, there being a monotonic increase with N o from 2 x io 4 spores at 5 x io 3 cm" 2 in both. The slope of the curve increases significantly in the third range. As the fourth region is approached, the curve bends over, saturating at 5 x io 4 spores per fruiting body in the fourth region. Subsidiary observations. Time-lapse films showed that in region (1) aggregation initially occurred without wave propagation. As aggregation proceeded and the density increased locally around centre cells, local signal propagation became observable. In region (2), signal propagation occurred over long distances but was confined to portions of the field. In region (3), signal propagation covered the entire field. In addition to the retardation of the course of development observed in region 4,

6 220 Y. Hashimoto, M. H. Cohen and A. Robertson 80 i X10 3 1x10 6 Cells cm" 2 Fig. 4. Spore formation efficiency (SFE) measured after 48 h {A), 72 (B), 96 (C) and 120 h (D) incubation on plain agar. Developmental efficiency (DE) is plotted on the right-hand ordinate. x x10* 1x10' Cells cm" 1 1x10 7 Fig. 5. Fruiting body density (N FB ) measured after 48 (^4), 72 (B), 96 (C) and 120 h (D) on plain agar.

7 Cell density dependence of aggregation characteristics 221 evidence for retardation of the course of differentiation was found. Incompletely differentiated spores were observed within region 4. Longer incubation times Incubations were carried out also for 72, 96 and 120 h. Mean sorocarp size was independent of incubation time throughout the entire density range. This was true for the SFE and fruiting body density only for densities of io 8 cm" 2 and lower. Above io 6 cm" 2 both SFE and fruiting body density increase monotonically with the duration w X10 3 1X10 4 1x10» 1X10 6 1x10 7 Cells cm" 1 Fig. 6. Spore formation efficiency (SFE) as a function of initial cell density measured after 48-h incubation on buffered agar. Developmental efficiency (DE) is plotted on the right-hand ordinate. of incubation. We have observed the slipping of sorocarps down the stalk after 96 h. Accordingly, both empty stalks and fruiting bodies were counted in arriving at final values of fruiting body density at 120 h. The results are displayed in Figs. 4 and 5. Note the tendency towards saturation at 120 h. Effects of buffer Figs are to be compared with Figs. 1-5 representing comparable data from buffered agar plates. Our general description of the results from plain agar hold, with the following detailed differences. (1) Buffered agar increases SFE at initial cell densities below 1 x io 6 cm" 2 and for the shortest incubation times (Fig. 6). The saturation values of SFE at longer incubation times are lower, and fall off more sharply at high cell densities (Fig. 9). (2) The number of fruiting bodies is enhanced at initial cell densities below 1 x io 6 cm" 2 on buffered agar (Fig. 7) but saturates at similar

8 222 Y. Hashimoto, M. H. Cohen and A. Robertson I 1X x10* 1x10* 1x10 Cells cm" 1 Fig. 7. Fruiting body density (N FB ) measured after 48-h incubation on buffered agar. 1x10 5 1x10 4 ixio 4 1X10 3 Cells cm" 1 1x10* 1x10 7 Fig. 8. Mean number of spores per fruiting body (./f^,) measured after 48-h incubation on buffered agar.

9 Cell density dependence of aggregation characteristics so o^ w 40 SO 20 00, 1x10* 1X10 4 1x10» Cells cm" 1 1x10* 1X10 7 Fig. 9. Spore formation efficiency (SFE) measured after 48 (^4), 72 (B), 96 (C) and 120 h (D) incubation on buffered agar. Developmental efficiency (DE) is plotted on the right-hand ordinate X10 3 1x10* 1x10' Cells cm" 2 j I 1X10 7 Fig. 10. Fruiting body density (N FB ) measured after 48 (A), 72 (B), 96 (C) and 120 h (D) on buffered agar.

10 224 ^- Hashimoto, M. H. Cohen and A. Robertson values on buffered and plain agar (Fig. 10). (3) The number of spores (Fig. 8) is lower at all initial cell densities below 1 x io 7 cm~ 2, suggesting a decrease in the efficiency of differentiation on buffered agar. DISCUSSION It is important that our measurements are reproducible and have internal consistency, which shows up as smooth variation with experimental parameters, e.g. density and incubation time. The 2 features of our experimental procedures responsible are care in the production of synchronized, bacteria-free amoebae, and care in the reduction of statistical errors. The data are of a quality to warrant detailed theoretical analysis and to make comparison with results of other experiments meaningful. The only earlier measurements which are directly related to ours are those of Bonner & Dodd (1962) on aggregation territory size and those of Hohl & Raper (1964) on fruiting body density and sorocarp size distribution. Bonner & Dodd provided a culture of amoebae with varying bacterial food densities. Their results for mean territory radius were essentially uncorrelated with initial food density, ranging from i-o to 1-4 mm with a mean at 1-2 mm. A mean territory radius of 1-2 mm corresponds to a mean aggregate density of 21 cm" 2 at the end of aggregation. Our finding that one aggregate gives rise to one fruiting body except in a statistically insignificant number of cases implies that the fruiting body density ranges from 16 to 32 cm" 2 with a mean at 21 cm" 2 in Bonner & Dodd's experiment. Our measurements of fruiting body density versus cell density in interphase show that such fruiting body densities occur in the range x io 6 cm~ 2 with the mean corresponding to a density of 2 x io 8 cm" 2, which is the density of a close packed monolayer. The conditions of the Bonner & Dodd experiment are such that the amoebae can be unsynchronized and the field non-homogeneous, with feeding fronts, etc., so it is dangerous to draw inferences from this comparison. Nevertheless, the comparison suggests that amoebae provided with a bacterial food supply of varying density tend to produce densities of amoebae in interphase in the areas of the culture cleared of bacteria which are near that of a close-packed monolayer. This accords with our direct observations of aggregation on growth plates; it would be interesting to establish how the density is so controlled. Hohl & Raper's results for fruiting-body densities are in essential agreement with ours. In particular, they find the strong increase in region 4 and weaker variation in region 3. There are 4 distinct regions of variation of the spore formation efficient (SFE) and the fruiting-body density N FB with initial cell density N o whereas there are only 2 distinct regions for the mean number of spores per fruiting body JV^. These 3 quantities are not independent. If N 8p is the mean spore density, then we have both and Combining (1) and (2) to eliminate N 8p gives N 8p = (SFE)N 0 (1) ^SP = -^pnf-n- (2) SFE = ^8p * FB /N 0, (3)

11 Cell density dependence of aggregation characteristics 225 the smooth variation of ^V ap with density in regions 1 and 2 indicates via (3) that the existence of regions 1 and 2 in the density dependence of SFE and N FB has the same origin for both. A recent theoretical analysis of aggregation in range 3 (Cohen & Robertson, 1972) makes clear that aggregation in general has 2 distinct dynamical aspects which ultimately give rise to the density dependence of aggregation characteristics: aggregative movements and new centre formation. Both are affected quite differently by signal propagation, which has its own characteristic density dependence. Our observation of 2 types of density dependence in the aggregation characteristics is consistent with the existence of 2 distinct dynamical processes in aggregation. We believe that the onset of signal propagation accounts for the peak in aggregative performance observed by Sussman & Noel (1952). This point is examined in detail by Gingle (1975); the steps in our density measurements reported in this paper are too large to reveal this behaviour. The fruiting-body density tends to saturate at the highest densities. As the sorocarp size has already saturated, equation (3) implies that SFE falls off there as N,," 1 (see Figs. 4, 9). Equation (2) implies, interestingly, that the spore density reaches a constant maximum value of 3 x io 6 cm~ 2 after 96 h independent of initial cell density, at high densities. The mean spore to stalk cell number ratio in a mature fruiting body is 2 to 1 (Bonner & Slifkin, 1949). It is not clear whether this includes base-plate cells. Assuming it does, then a limiting value for SFE of 66-7 % is reached when differentiation during interphase, aggregation, and subsequent development is perfectly efficient. Our observed values of SFE agree with the limiting value in region 3 within combined experimental errors. The accuracy of our measurement of SFE is greater than that of the spore to stalk cell-number ratio. We therefore prefer to take as the limiting value of SFE, the maximum value we have observed for SFE, 65 % for distilled-water agar. The difference is insignificant. We can define an overall developmental efficiency DE = SFE/max(SFE) (see Figs. 4, 9). In region 3, it remains above 90%, dropping dramatically on either side for distilled water agar. For buffered agar, the maximum DE is 71 %. If there were a significant amount of cell division during or after aggregation these figures would have to be reduced. However, Bonner & Frascella (1952) suggest that divisions are few, and we therefore ignore them. The causes of the loss of developmental efficiency on the low and high density sides of region 3 are quite distinct. On the high-density side, there is evidence from the time-lapse films that the time scale of aggregation is relatively unaffected but that the later stages are slowed. In particular, differentiation appears to be affected by crowding of the amoebae. On the other hand, below region 3, the time scale of aggregation initially decreases and subsequently increases at lower density. The remainder of development is normal, taking region 3 as the standard. The loss of efficiency is associated with amoebae left behind in low density regions of the field. All of the phenomena observed in regions 1-3 can be understood qualitatively in terms of the relative size of the mean interamoeba separation and the signal range for relaying (Shaffer, 1962) and for chemotaxis (Cohen & Robertson, 1971 a, b; Robertson & Cohen, 1974). First, as long as the separation of relaying-competent amoebae (Cohen 15 CEL 19

12 226 Y. Hashimoto, M. H. Cohen and A. Robertson & Robertson, 1972), signal propagation occurs as though the amoebae formed a continuous medium (Cohen & Robertson, 1972). At the beginning of aggregation, cells competent to signal autonomously emerge into a field of cells sensitive and competent to propagate signalling waves by relaying (Cohen & Robertson, 1972). Waves propagate outward from centres formed around these autonomous cells. The territories of these centres are defined by refractory boundaries (Gerisch, 1968; Robertson & Cohen, 1972). The territories shrink as new centres are formed around newly differentiated autonomous cells and as spirals and other complex centres emerge (Durston, 1974). Centres with shorter periods gradually entrain the territories of neighbouring centres with longer periods. The entrainment process reduces the number of centres below that expected from the rate of differentiation of new autonomous cells. At a certain stage of aggregation, the flow of cells away from the refractory boundary reduces the territory size more rapidly than does the differentiation of new centres. The field of cells then exhibits marked variation in its density, developing clear areas around the refractory boundaries. The entrainment process is subsequently reduced in efficiency. The high value of SFE and the slow, monotonic decrease of N^B with density in region 3 are the result of this interplay of new centre formation, entrainment, and amoeba flow, which leads to efficient aggregation. In region 2, both SFE and N FB fall dramatically. We attribute the fall to a change over from continuum wave propagation to percolation (Shante & Kirkpatrick, 1971; Cohen & Robertson, 1972; Robertson & Cohen, 1974) as the density is reduced. The interamoeba separation then becomes comparable to the range for relaying. Signal propagation cannot occur uninterrupted throughout the entire field. There are regions where the local density is too low, because of the random amoeba distribution, for the signal to propagate. The regions through which propagation can occur form a continuous, multiply-connected network bordering the non-propagating regions. The amoebae in the non-propagating regions flow into the propagating regions by chemotaxis because the range of the signal within which a chemotactic movement response occurs is substantially longer than the range for relaying (Shaffer, 1962; Konijn et al. 1967; Robertson & Cohen, 1974). Moreover, a signal wave propagating into the lower density regions is stronger than that associated with isolated centre cells within them because the former results from the collective action of many cells. Thus centres in the propagating regions can efficiently entrain, by chemotaxis, the chemotactic territories of centres in the non-propagating regions, even when the periods of the latter are shorter. Such an increase in entrainment would lead to a rapid decrease in fruiting body density, as is observed (Sussman & Noel, 1952; Shaffer, 1962). Moreover, some of the regions within which no signal propagation occurs can become sufficiently large that amoebae in their interior are out of range of the chemotactic signal and are not stimulated to aggregate. That this loss of aggregation efficiency is the cause of the fall of SFE in this region is confirmed by our time-lapse films. If the rapid fall of SFE is extrapolated, SFE would vanish at N o = 3-8 x io 4 cm" 2. We take this as the lower limit of region 2. This confinement of signal propagation by random fluctuations in the amoeba density is percolation. Accordingly, we call the regime from 4X10 4 to 4X io 8 cm" 8

13 Cell density dependence of aggregation characteristics 227 the percolation regime (region 2) and that from 4 x io 6 to 4 x io 6 cm" 2 the continuum regime (region 3). Since there is evidence that the developmental inefficiency in region 4 is probably not associated with inefficiency of aggregation, the continuum regime should probably be regarded as continuing to higher densities. The developmental inefficiency may simply be due to the accumulation of waste products or insufficient oxygen, etc., at densities of close packing and above. A better controlled environment would then increase region 3 at the expense of region 4. The density at which SFE extrapolates to zero is probably a fair estimate of the critical density for percolation, N* (Shante& Kirkpatrick, 1971), the density at which extended propagating regions disappear. The value of 3-8 x io 4 cm~ 2 obtained by extrapolation agrees quite well with the value of 5 x io 4 cm" 2 obtained by Konijn & Raper (1961) for the critical density, another indication of long term reproducibility of experiments with Dictyostelium discoideum. Using the relation (Shante & Kirkpatrick, 1971; Cohen & Robertson, 1972; Robertson & Cohen, 1974), nr^n* = 4-5 gives us a value of 61 fim for R», the relaying range for N* = 3-8 x io 4 cm" 2. Below N*, long-range signal propagation is impossible. Signal propagation is confined to small clusters of amoebae, within which aggregation occurs and into which additional amoebae are attracted by chemotaxis alone. As the density is reduced the clusters decrease rapidly in size (Robertson & Cohen, 1974) until, at least initially, signal propagation ceases to play an important role in aggregation. Chemotaxis towards the centres without relaying leads to a local density increase ultimately sufficient for the nearest neighbour to relay and enhance the chemotactic range of the centre. To build up a theory of aggregation in this chemotactic regime involves an intricate statistical analysis out of place here. The most significant feature is that even at these low densities fruiting bodies are still quite large, though few in number. The mean territory sizes r s involved are thus also large according to ^P = ^32 N 0, which gives values for r s of several mm in the density range 5 x IO 3 -I x io 4 cm" 2. This is quite difficult to understand without signal propagation, at least in the later stages of centre formation. In regions 3 and 4, the effect of buffering the agar is to reduce the efficiency of differentiation of some of the developmental competences. The fruiting body density is unchanged, and therefore centre density, territory size, autonomous cell differentiation, tip formation, and the dynamics of signal propagation and centre entrainment cannot be much affected. The rate of differentiation of relaying competent cells can still be affected, however, as long as the density of sensitive cells remains well above N*. The reduction in mean number of spores per fruiting body caused by buffering the agar implies a loss of developmental efficiency with regard to chemotaxis, contact formation, and/or later developmental competences. To resolve this point would require more detailed investigation. In regions 1 and 2, the effect of buffering is to increase SFE and N FU without much change in ^V gp or in the boundaries of the regions. 15-3

14 228 Y. Hashimoto, M. H. Cohen and A. Robertson The relaying range is therefore probably not much affected, and changes in the chemotactic range would not give the observed effects. Again, more detailed investigations are indicated. Finally, we should point out that our results do not require assuming the production of a centre-spacing substance controlling territory size. Territory size is a complicated function of initial amoeba density, and its control is adequately accounted for by refractory boundaries and by entrainment of neighbouring centres. We have explored entrainment further, concentrating on regions i and 2 of the density dependence curves, and will publish the results separately (A. Gingle, A. Robertson & M. H. Cohen, unpublished). This research was supported in part by NIH grant no. HD-04722, the Alfred P. Sloan Foundation, and the Otho S. A. Sprague Memorial Institute. REFERENCES BONNER, J. T. (1944). A descriptive study of the development of the slime mold/), discoidetim. Am.jf. Bot. 31, BONNER, J. T. (1958). The Evolution of Development, pp Cambridge: Cambridge University Press. BONNER, J. T. (1967). The Cellular Slime Moulds, 2nd edn. Princeton: Princeton University Press. BONNER, J. T. & DODD, M. R. (1962), Aggregation territories in the cellular slime molds. Biol. Bull. mar. biol. Lab., Woods Hole 122 (1), BONNER, J. T. & FRASCELLA, E. B. (1952). Mitotic activity in relation to differentiation in the slime mould Dictyostelium discoideum. J'. exp. Zool. 121, BONNER, J. T. & SLIFKIN, M. K. (1949). A study of the control of differentiation: the proportions of stalk and spore cells in the slime mold Dictyostelium discoideum. Am. J. Bot. 36, COHEN, M. H. & ROBERTSON, A. (1971a). Wave propagation in the early stages of aggregation of cellular slime molds. J. theor. Biol. 31, COHEN, M. H. & ROBERTSON, A. (19716). Chemotaxis and the early stages of aggregation in cellular slime molds. J. theor. Biol. 31, COHEN, M. H. & ROBERTSON, A. (1972). Differentiation for aggregation in the cellular slime molds. In Cell Differentiation (ed. R. Harris, P. Allin & D. Viza), pp Copenhagen: Munksgaard. DURSTON, A. J. (1974). Pacemaker activity during aggregation in Dictyostelium discoideum. Devi Biol. 37, GERISCH, G. (1968). Cell aggregation and differentiation in Dictyostelium. Ctirr. Top. devl Biol. 3, I57-I97- GINGLE, A. (1975). Ph.D. Thesis, University of Chicago. HOHL, H. R. & RAPER, K. B. (1964). Control of sorocarp size in the cellular slime mold Dictyostelium discoideum. Devi Biol. 9, KONIJN, T. M. & RAPER, K. B. (1961). Cell aggregation in Dictyostelium discoideum. Devi Biol. 3, 72S-7S6. KONIJN, T. M., VAN DE MEENE, J. G., BONNER, J. T. & BARKELEY, D. S. (1967). The acrasin activity of adenosine-3',5'-cyclic phosphate. Proc. natn. Acad. Sci. U.S.A. 58, RAPER, K. B. (1935). Dictyostelium discoideum, a new species of slime mold from decaying forest leaves. J. agric. Res. 50, RAPER, K. B. (1940). Pseudoplasmodium formation and organization in Dictyostelium discoideum. J. Elisha Mitchell scient. Soc. 56, RAPER, K. B. (1941). Developmental patterns in simple slime molds. Growth 5, RAPER, K. B. (i960). Levels of cellular interaction in amoeboid populations. Proc. Am. phil. Soc. 104,

15 Cell density dependence of aggregation characteristics 229 ROBERTSON, A. & COHEN, M. H. (1972). Control of developingfields.a. Rev. Biophys. Bioengnr ROBERTSON, A. & COHEN, M. H. (1974). Quantitative analysis of the development of the cellular slime molds: II. Lectures on Mathematics in the Life Sciences, vol. 6, pp Providence, R.I.: Am. Math. Soc. SHAFFER, B. M. (1962). The Acrasina. Adv. Morpiwgen. 2, SHANTE, V. K. S. & KIRKPATRICK, S. (1971). An introduction to percolation theory. Adv. Phys. 20, SUSSMAN, M. & NOEL, E. (1952). An analysis of the aggregation stage in the development of the slime moulds, Dictyosteliaceae. I. The populational distribution of the capacity to initiate aggregation. Biol. Bull. mar. biol. Lab., Woods Hole 103, SUSSMAN, M. (1966). Methods in Cell Physiology, vol. 2 (ed. D. Prescott), pp New York: Academic Press. {Received 18 February 1975)

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