J. Cell Sci. 19, 215-229 (1975) 215Printed in Great Britain

CELL DENSITY DEPENDENCE OF THE

AGGREGATION CHARACTERISTICS OF

THE CELLULAR SLIME MOULD

DICTYOSTELIUM DISCOIDEUM

Y. HASHIMOTO,* M. H. COHEN AND A. ROBERTSONfDepartment 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 Dictyosteliumdiscoideum as functions of initial cell density. Experiments were performed on agar made upwith 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 changesin 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 controllingcell 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 aggregationterritories.

INTRODUCTION

The cellular slime mould Dictyostelium discoideum was discovered by Raper in1935. In its growth phase, it consists of free living, unicellular amoebae feeding onbacteria (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 chemo-tactic factor which induces the aggregative movements (Konijn, van de Meene, Bonner& Barkeley, 1967). After aggregation, morphogenesis and differentiation continueuntil multicellular fruiting bodies are formed, which consist mainly of spore andstalk cells (Raper, 1940; Bonner, 1967).

The cellular slime moulds are organisms of particular interest for developmentalbiology (Bonner, 1958; Raper, i960). The unitary processes of development (Robert-son & Cohen, 1972) are all exhibited simply and are well separated in time (Bonner,1944, 1967; Raper, 1941). Among the cellular slime moulds, Dictyostelium discoideumis best known (Bonner, 1967). In particular, details of the way in which its aggregationis controlled are beginning to emerge in quantitative form (Robertson & Cohen, 1972).

• On leave from Tokyo Metropolitan Isotope Research Center.t Sloan Foundation Fellow, 1973-5, and the author to whom correspondence should be sent.

2i6 Y. Hashimoto, M. H. Cohen and A. Robertson

Because of their intrinsic interest and because of the possibility of insights affordedinto larger questions in developmental biology, more extensive quantitative investiga-tion into aggregation as morphogenesis and into its control is warranted.

It seems intuitively obvious that the details of aggregation depend sensitively onthe initial cell density. Measurement of this density dependence of the aggregationcharacteristics of Dictyostelium discoideum should then yield insight into the dynamicsof aggregation. It can be inferred from experiments of Konijn & Raper (1961) thataggregation ceases at initial amoeba densities below about 5 x io4 cm"2. The existenceof a critical density for aggregation can be explained by supposing that a signal gene-rated by an aggregating amoeba has a finite range beyond which it cannot stimulateanother 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 of50 /tm for the range from the Konijn-Raper value of the critical density. The rangeis an important quantity, relating directly to the parameters of the c-AMP signal, theextracellular phosphodiesterase which removes c-AMP and the threshold concentra-tion for relaying (Shante & Kirkpatrick, 1971; Robertson & Cohen, 1974). Moreaccurate and complete data for the density dependence of the aggregation characteris-tics would yield a more accurate value for the range.

Hohl & Raper (1964) have reported an approximate independence of the aggregatesize frequency distribution on the initial cell density. In the same paper, they mentionthat 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 ofa bacterial food supply and suggest a centre-spacing substance to explain their results.Provided post-aggregative development is efficient, and each aggregate produces onefruiting body, fruiting body density is inversely proportional to territory area. Thusthere 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 latterexperiments was either relatively low or relatively constant. Further measurementsof fruiting body density are clearly required.

Accordingly, we have undertaken a series of experiments in which accurate measure-ments are taken of spore formation efficiency, fruiting body density, and mean numberof spores per fruiting body for Dictyostelium discoideum on agar of various compositions.The results of these investigations are reported and discussed here.

MATERIAL AND METHODSSpores 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 aerogenesand 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 dilutingfluid (Sussman, 1966). The bacteria were removed by centrifuging the suspension for 8 minat 4 °C, resuspending and repeating twice.

The volume density of the bacteria-free cell suspension was measured with a haemocyto-meter. The suspension was plated on agar (2 % agar in distilled water or phosphate buffer(K-K2) solution (Bonner, 1967)) and spread as uniformly as possible with a glass rod. Subse-

Cell density dependence of aggregation characteristics 217

quent random movements of the amoebae early in interphase made the cell distribution stillmore uniform over the agar surface. The surface cell density was calculated from the agar areaand the quantity of cell suspension plated out and checked by direct counting. Counts weremade at least 2 h after plating, as we had found that no cell division was observed after thisdelay. This point was checked further by observation of films (see Bonner & Frascella, 1952).Cell densities obtained in this way ranged from 5 x io8 to 2 x io7 cm"1. The density of a closepacked monolayer of cells is 2 x io9 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 andincubated at 22 °C for 48, 72, 96 and 120 h. Ten 3-5-011 plates were used for each point in thedensity 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 into1 ml of PAD diluting fluid, and the resulting volume density of spores measured with a haemo-cytometer. 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 ofthe total number of cells plated out, was then calculated, as was the average number of sporesper fruiting body.

In all number or density measurements, the statistical error (standard deviation) was keptbelow 10% when practicable. This was difficult or impossible at the lower densities. Thestandard 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 differentdensities, 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

No Initial cell density.SFE Spore formation efficiency.DE Developmental efficiency.NpB 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 agar0-5 cm thick, incubated for 48 h, and report our results first for such plates. Then wereport the effects of increasing incubation time and of substituting K-K2 solution fordistilled 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, No>

within each of which the dependence of SFE on No is quite characteristic: (1) Below

4X io4 cm"2 SFE is under 10% and drops with decreasing No, forming a small tail

on the rest of the curve; (2) from 4 x io4 to i-6 x io8 cm"8 SFE rises rapidly to about

60%, (3) from i-6 x ioB to i-8x io6 cm"2, it saturates in the range 60-65; a n d (4)

above i-8 x io6 cm"2, it falls continuously and is down to 13 % at i-6 x io7 cm"2.

2 l 8 Y. Hashimoto, M. H. Cohen and A. Robertson

80 -

60

uj- 40

20

1x10» 1X104 1X101

Cells cm"11x10*

I

100

SO

1X107

Fig. i. Spore formation efficiency (SFE) as a function of initial cell density measuredafter 48-h incubation on plain agar. Developmental efficiency (DE) is plotted on theright-hand ordinate.

100 -

10

01

001 I I I I1x10° 1X104 i x i o 5

Cells cm-'1x10* 1x10'

Fig. 2. Fruiting body density (Nra) measured after 48-h incubation on plain agar.

Cell density dependence of aggregation characteristics 219

Fruiting body density. We present the results of our fruiting body counts for stan-dard plates in Fig. 2. The same 4 regions show up in the fruiting body density. (1)Below 3 x io4 cm~2 the fruiting-body density is of order 1 cm"8; (2) between 3 x io4

and 2 x io6 cm"2, it increases to about 10 cm~2 and saturates at this value; (3) from2 x io5 to 2 x io6 cm"2, it increases to about 20 cm"8; and (4) from 2 x io8 cm"2 on itincreases, saturating at about 50 at 2 x io7 cm"8. The boundaries of these regions are,of course, not sharply marked and differ somewhat in the SFE and in the fruitingbody density.

1x10* -

1x10*

1X101 I I1x10» 1X104 1x10*

Cells cm"11X104 1x107

Fig. 3. Mean number of spores per fruiting body (^V,p) measured after 48-h incubationon plain agar.

The time-lapse films showed that, under our conditions, each aggregate formedone fruiting body. The fruiting-body density is therefore equal to the aggregatedensity, and no separate measurement of aggregate density is necessary.

Mean fruiting body size. Our results for mean number of spores per fruiting body areshown in Fig. 3. There is no significant difference in the behaviour of this quantityin the first 2 density ranges, there being a monotonic increase with No from 2 x io4

spores at 5 x io3 cm"2 in both. The slope of the curve increases significantly in thethird range. As the fourth region is approached, the curve bends over, saturating at5 x io4 spores per fruiting body in the fourth region.

Subsidiary observations. Time-lapse films showed that in region (1) aggregationinitially occurred without wave propagation. As aggregation proceeded and the densityincreased locally around centre cells, local signal propagation became observable. Inregion (2), signal propagation occurred over long distances but was confined toportions 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,

220 Y. Hashimoto, M. H. Cohen and A. Robertson

80 i—

60

40

20

1X103 1x106

Cells cm"2

100

50

x107

Fig. 4. Spore formation efficiency (SFE) measured after 48 h {A), 72 (B), 96 (C) and120 h (D) incubation on plain agar. Developmental efficiency (DE) is plotted on theright-hand ordinate.

100

10

0-1

0011x10* 1x10 '

Cells cm"11x107

Fig. 5. Fruiting body density (NFB) measured after 48 (^4), 72 (B), 96 (C)and 120 h (D) on plain agar.

Cell density dependence of aggregation characteristics 221

evidence for retardation of the course of differentiation was found. Incompletelydifferentiated spores were observed within region 4.

Longer incubation times

Incubations were carried out also for 72, 96 and 120 h. Mean sorocarp size wasindependent of incubation time throughout the entire density range. This was truefor the SFE and fruiting body density only for densities of io8 cm"2 and lower. Aboveio6 cm"2 both SFE and fruiting body density increase monotonically with the duration

80

60

w 40

20

100

50

1X103 1X104 1x10»Cells cm"1

1X106 1x107

Fig. 6. Spore formation efficiency (SFE) as a function of initial cell density measuredafter 48-h incubation on buffered agar. Developmental efficiency (DE) is plotted onthe 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 finalvalues 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. 6-10 are to be compared with Figs. 1-5 representing comparable data frombuffered agar plates. Our general description of the results from plain agar hold, withthe following detailed differences. (1) Buffered agar increases SFE at initial celldensities below 1 x io6 cm"2 and for the shortest incubation times (Fig. 6). Thesaturation values of SFE at longer incubation times are lower, and fall off more sharplyat high cell densities (Fig. 9). (2) The number of fruiting bodies is enhanced at initialcell densities below 1 x io6 cm"2 on buffered agar (Fig. 7) but saturates at similar

222 Y. Hashimoto, M. H. Cohen and A. Robertson

100 —

10

0-1

0011X103

• I1x10*

Cells cm"11x10* 1x107

Fig. 7. Fruiting body density (NFB) measured after 48-h incubation on buffered agar.

1x105

1x104

ixiO4 1X103

Cells cm"11x10* 1x107

Fig. 8. Mean number of spores per fruiting body (./f ,̂) measured after 48-hincubation on buffered agar.

Cell density dependence of aggregation characteristics 223

80 —

60

soo^w 40

20

1x10* 1X1041x10»

Cells cm"1

1x10*

0 0 ,

100

SO

1X107

Fig. 9. Spore formation efficiency (SFE) measured after 48 (̂ 4), 72 (B), 96 (C) and120 h (D) incubation on buffered agar. Developmental efficiency (DE) is plotted onthe right-hand ordinate.

100

10

0-1

0011X103 1x10* 1x10 '

Cells cm"21X106

j I1X10 7

Fig. 10. Fruiting body density (NFB) measured after 48 (A), 72 (B), 96 (C) and 120 h(D) on buffered agar.

224 ^- Hashimoto, M. H. Cohen and A. Robertson

values on buffered and plain agar (Fig. 10). (3) The number of spores (Fig. 8) is lowerat all initial cell densities below 1 x io7 cm~2, suggesting a decrease in the efficiencyof differentiation on buffered agar.

DISCUSSION

It is important that our measurements are reproducible and have internal con-sistency, which shows up as smooth variation with experimental parameters, e.g.density and incubation time. The 2 features of our experimental procedures responsibleare care in the production of synchronized, bacteria-free amoebae, and care in thereduction of statistical errors. The data are of a quality to warrant detailed theoreticalanalysis 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) onfruiting body density and sorocarp size distribution. Bonner & Dodd provided aculture of amoebae with varying bacterial food densities. Their results for meanterritory radius were essentially uncorrelated with initial food density, ranging fromi-o to 1-4 mm with a mean at 1-2 mm. A mean territory radius of 1-2 mm correspondsto a mean aggregate density of 21 cm"2 at the end of aggregation. Our finding that oneaggregate gives rise to one fruiting body except in a statistically insignificant numberof cases implies that the fruiting body density ranges from 16 to 32 cm"2 with a meanat 21 cm"2 in Bonner & Dodd's experiment. Our measurements of fruiting bodydensity versus cell density in interphase show that such fruiting body densities occurin the range 1—3-5 x io6 cm~2 with the mean corresponding to a density of 2 x io8

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 fieldnon-homogeneous, with feeding fronts, etc., so it is dangerous to draw inferencesfrom this comparison. Nevertheless, the comparison suggests that amoebae providedwith a bacterial food supply of varying density tend to produce densities of amoebaein interphase in the areas of the culture cleared of bacteria which are near that of aclose-packed monolayer. This accords with our direct observations of aggregation ongrowth 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 withours. In particular, they find the strong increase in region 4 and weaker variation inregion 3.

There are 4 distinct regions of variation of the spore formation efficient (SFE) andthe fruiting-body density NF B with initial cell density No whereas there are only 2distinct regions for the mean number of spores per fruiting body JV^. These 3 quan-tities are not independent. If N8p is the mean spore density, then we have both

N8p = (SFE)N0 (1)and

^ S P = -^pNf-n- (2)

Combining (1) and (2) to eliminate N8p gives

SFE = ^8 p*F B/N0, (3)

Cell density dependence of aggregation characteristics 225

the smooth variation of ^Vap with density in regions 1 and 2 indicates via (3) that theexistence of regions 1 and 2 in the density dependence of SFE and NF B has the sameorigin for both. A recent theoretical analysis of aggregation in range 3 (Cohen &Robertson, 1972) makes clear that aggregation in general has 2 distinct dynamicalaspects which ultimately give rise to the density dependence of aggregation charac-teristics: aggregative movements and new centre formation. Both are affected quitedifferently by signal propagation, which has its own characteristic density dependence.Our observation of 2 types of density dependence in the aggregation characteristicsis consistent with the existence of 2 distinct dynamical processes in aggregation. Webelieve that the onset of signal propagation accounts for the peak in aggregative per-formance observed by Sussman & Noel (1952). This point is examined in detail byGingle (1975); the steps in our density measurements reported in this paper are toolarge to reveal this behaviour.

The fruiting-body density tends to saturate at the highest densities. As the sorocarpsize has already saturated, equation (3) implies that SFE falls off there as N,,"1 (seeFigs. 4, 9). Equation (2) implies, interestingly, that the spore density reaches a constantmaximum value of 3 x io6 cm~2 after 96 h independent of initial cell density, at highdensities.

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 itdoes, then a limiting value for SFE of 66-7 % is reached when differentiation duringinterphase, aggregation, and subsequent development is perfectly efficient. Ourobserved values of SFE agree with the limiting value in region 3 within combinedexperimental errors. The accuracy of our measurement of SFE is greater than thatof the spore to stalk cell-number ratio. We therefore prefer to take as the limiting valueof SFE, the maximum value we have observed for SFE, 65 % for distilled-water agar.The difference is insignificant. We can define an overall developmental efficiencyDE = SFE/max(SFE) (see Figs. 4, 9). In region 3, it remains above 90%, droppingdramatically on either side for distilled water agar. For buffered agar, the maximumDE is 71 %. If there were a significant amount of cell division during or after aggrega-tion 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 sidesof region 3 are quite distinct. On the high-density side, there is evidence from thetime-lapse films that the time scale of aggregation is relatively unaffected but that thelater stages are slowed. In particular, differentiation appears to be affected by crowdingof the amoebae. On the other hand, below region 3, the time scale of aggregation initiallydecreases and subsequently increases at lower density. The remainder of developmentis normal, taking region 3 as the standard. The loss of efficiency is associated withamoebae left behind in low density regions of the field.

All of the phenomena observed in regions 1-3 can be understood qualitatively interms of the relative size of the mean interamoeba separation and the signal range forrelaying (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 C E L 19

226 Y. Hashimoto, M. H. Cohen and A. Robertson

& Robertson, 1972), signal propagation occurs as though the amoebae formed a con-tinuous medium (Cohen & Robertson, 1972). At the beginning of aggregation, cellscompetent to signal autonomously emerge into a field of cells sensitive and competentto propagate signalling waves by relaying (Cohen & Robertson, 1972). Waves propa-gate outward from centres formed around these autonomous cells. The territories ofthese centres are defined by refractory boundaries (Gerisch, 1968; Robertson &Cohen, 1972). The territories shrink as new centres are formed around newly differ-entiated autonomous cells and as spirals and other complex centres emerge (Durston,1974). Centres with shorter periods gradually entrain the territories of neighbouringcentres with longer periods. The entrainment process reduces the number of centresbelow that expected from the rate of differentiation of new autonomous cells. At acertain stage of aggregation, the flow of cells away from the refractory boundaryreduces 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 areasaround the refractory boundaries. The entrainment process is subsequently reducedin efficiency. The high value of SFE and the slow, monotonic decrease of N^B withdensity 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 NF B fall dramatically. We attribute the fall to a changeover from continuum wave propagation to percolation (Shante & Kirkpatrick, 1971;Cohen & Robertson, 1972; Robertson & Cohen, 1974) as the density is reduced. Theinteramoeba separation then becomes comparable to the range for relaying. Signalpropagation cannot occur uninterrupted throughout the entire field. There are regionswhere the local density is too low, because of the random amoeba distribution, for thesignal to propagate. The regions through which propagation can occur form a con-tinuous, multiply-connected network bordering the non-propagating regions. Theamoebae in the non-propagating regions flow into the propagating regions by chemo-taxis because the range of the signal within which a chemotactic movement responseoccurs is substantially longer than the range for relaying (Shaffer, 1962; Konijn et al.1967; Robertson & Cohen, 1974). Moreover, a signal wave propagating into the lowerdensity regions is stronger than that associated with isolated centre cells within thembecause the former results from the collective action of many cells. Thus centres inthe propagating regions can efficiently entrain, by chemotaxis, the chemotactic terri-tories of centres in the non-propagating regions, even when the periods of the latterare shorter. Such an increase in entrainment would lead to a rapid decrease in fruitingbody density, as is observed (Sussman & Noel, 1952; Shaffer, 1962). Moreover, someof the regions within which no signal propagation occurs can become sufficiently largethat amoebae in their interior are out of range of the chemotactic signal and are notstimulated to aggregate. That this loss of aggregation efficiency is the cause of the fallof SFE in this region is confirmed by our time-lapse films. If the rapid fall of SFE isextrapolated, SFE would vanish at No = 3-8 x io4 cm"2. We take this as the lowerlimit of region 2.

This confinement of signal propagation by random fluctuations in the amoebadensity is percolation. Accordingly, we call the regime from 4X104 to 4X io8 cm"8

Cell density dependence of aggregation characteristics 227

the percolation regime (region 2) and that from 4 x io6 to 4 x io6 cm"2 the continuumregime (region 3). Since there is evidence that the developmental inefficiency in region4 is probably not associated with inefficiency of aggregation, the continuum regimeshould probably be regarded as continuing to higher densities. The developmentalinefficiency may simply be due to the accumulation of waste products or insufficientoxygen, etc., at densities of close packing and above. A better controlled environmentwould then increase region 3 at the expense of region 4.

The density at which SFE extrapolates to zero is probably a fair estimate of thecritical density for percolation, N* (Shante& Kirkpatrick, 1971), the density at whichextended propagating regions disappear. The value of 3-8 x io4 cm~2 obtained byextrapolation agrees quite well with the value of 5 x io4 cm"2 obtained by Konijn &Raper (1961) for the critical density, another indication of long term reproducibilityof experiments with Dictyostelium discoideum. Using the relation (Shante & Kirk-patrick, 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 io4 cm"2.Below N*, long-range signal propagation is impossible. Signal propagation is con-

fined to small clusters of amoebae, within which aggregation occurs and into whichadditional amoebae are attracted by chemotaxis alone. As the density is reduced theclusters decrease rapidly in size (Robertson & Cohen, 1974) until, at least initially,signal propagation ceases to play an important role in aggregation. Chemotaxis towardsthe centres without relaying leads to a local density increase ultimately sufficient forthe nearest neighbour to relay and enhance the chemotactic range of the centre. Tobuild up a theory of aggregation in this chemotactic regime involves an intricatestatistical analysis out of place here. The most significant feature is that even at theselow densities fruiting bodies are still quite large, though few in number. The meanterritory sizes rs involved are thus also large according to

• ^ P = 0-65 ^32N0,

which gives values for rs of several mm in the density range 5 x IO3-I x io4 cm"2. Thisis quite difficult to understand without signal propagation, at least in the later stagesof centre formation.

In regions 3 and 4, the effect of buffering the agar is to reduce the efficiency ofdifferentiation of some of the developmental competences. The fruiting body densityis unchanged, and therefore centre density, territory size, autonomous cell differentia-tion, tip formation, and the dynamics of signal propagation and centre entrainmentcannot be much affected. The rate of differentiation of relaying competent cells canstill be affected, however, as long as the density of sensitive cells remains well aboveN*. The reduction in mean number of spores per fruiting body caused by bufferingthe agar implies a loss of developmental efficiency with regard to chemotaxis, contactformation, and/or later developmental competences. To resolve this point wouldrequire more detailed investigation. In regions 1 and 2, the effect of buffering is toincrease SFE and NFU without much change in ̂ Vgp or in the boundaries of the regions.

15-3

228 Y. Hashimoto, M. H. Cohen and A. Robertson

The relaying range is therefore probably not much affected, and changes in thechemotactic range would not give the observed effects. Again, more detailed investiga-tions are indicated.

Finally, we should point out that our results do not require assuming the productionof a centre-spacing substance controlling territory size. Territory size is a complicatedfunction of initial amoeba density, and its control is adequately accounted for byrefractory boundaries and by entrainment of neighbouring centres. We have exploredentrainment further, concentrating on regions i and 2 of the density dependencecurves, 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. SloanFoundation, 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, 175-182.

BONNER, J. T. (1958). The Evolution of Development, pp. 76-95. Cambridge: CambridgeUniversity Press.

BONNER, J. T. (1967). The Cellular Slime Moulds, 2nd edn. Princeton: Princeton UniversityPress.

BONNER, J. T. & DODD, M. R. (1962), Aggregation territories in the cellular slime molds. Biol.Bull. mar. biol. Lab., Woods Hole 122 (1), 13-24.

BONNER, J. T. & FRASCELLA, E. B. (1952). Mitotic activity in relation to differentiation in theslime mould Dictyostelium discoideum. J'. exp. Zool. 121, 561-571.

BONNER, J. T. & SLIFKIN, M. K. (1949). A study of the control of differentiation: the propor-tions of stalk and spore cells in the slime mold Dictyostelium discoideum. Am. J. Bot. 36,727-734-

COHEN, M. H. & ROBERTSON, A. (1971a). Wave propagation in the early stages of aggregationof cellular slime molds. J. theor. Biol. 31, 101-118.

COHEN, M. H. & ROBERTSON, A. (19716). Chemotaxis and the early stages of aggregation incellular slime molds. J. theor. Biol. 31, 119-130.

COHEN, M. H. & ROBERTSON, A. (1972). Differentiation for aggregation in the cellular slimemolds. In Cell Differentiation (ed. R. Harris, P. Allin & D. Viza), pp. 35-45. Copenhagen:Munksgaard.

DURSTON, A. J. (1974). Pacemaker activity during aggregation in Dictyostelium discoideum.Devi Biol. 37, 225-235.

GERISCH, G. (1968). Cell aggregation and differentiation in Dictyostelium. Ctirr. Top. devlBiol. 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 Dictyo-

stelium discoideum. Devi Biol. 9, 137-153.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, 1152-1154.RAPER, K. B. (1935). Dictyostelium discoideum, a new species of slime mold from decaying

forest leaves. J. agric. Res. 50, 135-147.RAPER, K. B. (1940). Pseudoplasmodium formation and organization in Dictyostelium discoideum.

J. Elisha Mitchell scient. Soc. 56, 241-282.RAPER, K. B. (1941). Developmental patterns in simple slime molds. Growth 5, 41-76.RAPER, K. B. (i960). Levels of cellular interaction in amoeboid populations. Proc. Am. phil.

Soc. 104, 579-604.

Cell density dependence of aggregation characteristics 229

ROBERTSON, A. & COHEN, M. H. (1972). Control of developing fields. A. Rev. Biophys. Bioengnr.1. 409-464-

ROBERTSON, A. & COHEN, M. H. (1974). Quantitative analysis of the development of the cellularslime molds: II. Lectures on Mathematics in the Life Sciences, vol. 6, pp. 44-62. Providence,R.I.: Am. Math. Soc.

SHAFFER, B. M. (1962). The Acrasina. Adv. Morpiwgen. 2, 109-182.SHANTE, V. K. S. & KIRKPATRICK, S. (1971). An introduction to percolation theory. Adv.

Phys. 20, 325-357.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 initiateaggregation. Biol. Bull. mar. biol. Lab., Woods Hole 103, 259-268.

SUSSMAN, M. (1966). Methods in Cell Physiology, vol. 2 (ed. D. Prescott), pp. 397-412. NewYork: Academic Press.

{Received 18 February 1975)