The spatial pattern of aggregation centres of the slime mould Dictyostelium discoideum was analysed by using nearest-neighbour distributions. Both undisturbed cultures, and cultures that were initiated from cells dissociated from cultures that had already aggregated, formed non-randomly spaced patterns. However, the minimal distance between aggregates in undisturbed cultures was approximately ten times that observed in dissociated cultures.

In undisturbed cultures the aggregate size is regulated as a function of cell density (Bonner & Hoffman, 1963): As cell density increases aggregate density decreases and the aggregate size consequently increases. At the same cell density more and smaller aggregates were formed by dissociated cultures than by undisturbed cultures. Nevertheless, the same cell-densitydependent regulation of aggregate density existed in the dissociated cultures.

Here a model is developed to account for both the non-random spacing and the celldensity-dependant regulation of aggregate density. In this model, distance-dependent competition occurs between points in a random prepattern to generate patterns very similar to those observed in experiments. The cell-density-dependent regulation of the aggregate density can be explained by assuming that a constant fraction of the cell population has the capacity to initiate centres at the time of pattern determination regardless of the cell density. As the cell density is increased the fraction of potential centres that survive distance-dependent competition decreases and hence the aggregate size increases. These results suggest that distance-dependent competition may be a mechanism that evolved to control aggregate size at high cell densities.

Results obtained with this model indicate that the decrease in aggregate size in dissociated cultures is due primarily to an increase in the fraction of the cell population capable of initiating centres. This implies that as morphogenesis progresses a greater fraction of the cell population acquire a capacity which they will not normally express. Although this increase may have a function in later stages of morphogenesis, it may also provide a way to ensure aggregation in small populations of amoebae and at low cell densities.

Non-random spacing of structures occurs in a wide variety of organisms and contexts. The hair follicles of sheep (Claxton, 1964), hairs and bristles on insect cuticle (Maynard Smith & Sondi, 1961 ; Lawrence & Hayward, 1971), stomata on plant leaves (Sachs, 1978), pores on membranes (Markovics, Glass & Maul, 1974; Lacalli & Harrison, 1978) and heterocysts of filamentous blue-green algae (Wolk & Quine, 1975; Wilcox, Mitchison & Smith, 1973) are all arranged in patterns which exhibit little order beyond a minimal spacing between the structures. Aggregation centres of the cellular slime mould Dictyostelium discoideum (Bonner & Hoffman, 1963) as well as substructures of later stages of a related species of cellular slime mould, Polysphondylium pallidum (Speigel & Cox, 1980) are also arranged in non-randomly spaced patterns. Despite the relative simplicity of these patterns, the mechanisms which generate the patterns are in most instances poorly understood. Our relatively detailed understanding of at least some of the mechanisms involved in slime mould aggregation may allow us to understand the mechanisms involved in generating these patterns.

Aggregation centres in the cellular slime mould arise from homogeneous fields of starving amoebae. At the earliest stages, single cells and small groups of cells appear to initiate chemotactic signals which orient and attract nearby amoebae. The number of centres that are initiated determine the average territory size (the area from which cells migrate to a single centre) and, since no further growth occurs during morphogenesis, the minimal size of the multicellular structures formed.

Aggregate size (cells per aggregate) is regulated by cell density. At lower cell densities a maximal number of centres are formed by a population of cells and hence the aggregate size is small. As the cell density is increased, fewer centres are formed by the same number of cells and hence the aggregate size increases. Here a model is developed to account for this cell-density-dependent regulation of the number of centres initiated in large populations of amoebae.

In this model it is assumed that only a small fraction of the cell population is capable of initiating centres at the time of pattern determination. These potential centres are assumed to be arranged in a random prepattern. Nearest neighbours in this prepattern compete according to distance-dependent rules to generate a non-randomly spaced pattern. This simple model yields patterns that are very similar to those observed for aggregation centres and also predicts the dynamics of the cell-density-dependent regulation of the number of centres.

Growth and maintenance of cultures

Stock cultures of the axenic strain, Ax-3, of Dictyostelium discoideum were stored on silica gel according to the method of Reinhard (1966). The growth cultures were initiated monthly from clones initiated from this silica gel stock.

For experimental use, logarithmically growing cells were inoculated into 140 ml of axenic medium (Coccucci & Sussman, 1970) at an initial concentration of 2 ×105 cells per ml. The cultures were grown at 22 °C in one litre flasks on a rotating shaker at 70 –90 rev./min. The generation time under these conditions was 10 –14 h. The rate of cell division remained constant until the cell density reached 7 ×106—107 per ml and then decreased as the cultures approached stationary phase. For the experiments reported here, log-phase cells were obtained at cell densities of between 2 and 5 × 106 per ml.

Initiating morphogenesis

Morphogenesis was initiated following the method of Sussman (1966). For each developing culture, 5 × 107—108 amoebae were pelleted (200 g, 5 min), the nutrient medium was decanted and the cells were resuspended in buffered salts solution (0 ·02 M-KCI, 0 ·05 M-MgCl, 0 ·04 M phosphate buffer, pH 6 ·4, 34 mM streptomycin sulphate). The cells were washed twice with the salt solution and the cell density was determined after one of the resuspensions using a haemocytometer. After a final pelleting the cells were resuspended to a cell density of 4 × 107 cells per ml. Two dilutions of this suspension were then prepared: the first, D1, was a fourfold dilution (107 cells per ml) and the second, D2, was a tenfold dilution of D1 (106 cells per ml).

A set of black Millipore filters (BP00025) which were supported with two Millipore absorbent filters soaked with buffered salts solution was prepared. The cultures were initiated by evenly dispersing 0 ·25 –1 ·0 ml of the cell suspensions from one of the three tubes using a uniform procedure. Excess buffer was drawn from the lower filters using a Pasteur pipette. After the cultures had been initiated, the density of the three cell suspensions was carefully determined using a haemocytometer. Four samples of each tube were taken and 150 –200 cells were counted in each sample. These density determinations were used to calculate the number of cells that were dispersed on each filter.

In experiments employing cells dissociated during morphogenesis, the initial cultures were prepared by washing the cells free of nutrients and dispersing 5 × 107–108 cells on a 37 mm paper filter (Whatman No. 50, hardened) which was supported by two Millipore absorbent filters soaked with buffered salt solution. To dissociate cells from these cultures, the upper filter was removed and placed in a 30 ml conical tube. The cells were harvested from the filter by vortexing the filter with three successive washes of 5 ml buffered salts solution. To dissociate the cells, the cell suspension was forced through a small-bore pipette or a 16-gauge needle using moderate pressure. Dissociation was monitored until more than 90% of the cell population was singles, doubles or triples. After the cells were dissociated the cell suspension was counted, pelleted and resuspended at 4 ×107 cells per ml and treated as above.

Characterizing the spatial pattern of aggregates

After undisturbed or dissociated cultures had reached the tight aggregate stage, the cultures were photographed using a dissection microscope equipped with a 35 mm camera. The magnification was adjusted to include 30 –300 aggregates in a single frame. The photographs were enlarged to fit an 8 × 10 inch format and the distance between each aggregate and its nearest neighbour was determined using dividers and a ruler. If an aggregate was closer to the edge of a frame than its nearest neighbour, then it was not included. The distances were calibrated from photographs of a ruler taken at the same magnification. The aggregate density was determined from the photographs using the same calibration.

About 5 –6 h after cells were washed free of nutrients and dispersed on black Millipore filters, the first stages of aggregation became visible: small white aggregates and streams emanating from them appeared simultaneously over the filter surface. Since single cells are not visible on filters, the initial stages of centre formation could not be observed.

At the highest cell densities employed in this study (8 ×104 cells per mm2), the cells were arranged in a multicellular carpet about four to five cells thick before aggregation began. Under these conditions streaming was observed, which indicates that the cells were using relayed chemotactic signals to aggregate (Shaffer, 1957,1975).

At the lowest cell densities employed here (103 cells per mm2), the cell density was below that required to form a monolayer (9 ×103 per mm2). As a rough estimate of the spacing between cells at this density, we can calculate the spacing between cells in a rectangular lattice at this density. This yields a value of 32 μm. The diameter of a developing cell is about 9 –11 μm, so that the cells are on the average about 2 –3 cell diameters apart at this density.

At lower cell densities, numerous secondary centres occasionally appeared after the primary centres were established. Secondary centres have been reported before and several factors favour their formation : growth conditions (Garrod & Ashworth, 1972), light (Raper, 1940; Shaffer, 1961; Kahn, 1964), and high concentrations of cAMP (Bonner et al. 1969 ; Thadani, Pan & Bonner, 1977; Ryter, Klein & Brachet, 1979). Much of the variation due to secondary centres could be eliminated by incubating the cultures in the dark. Only experiments in which secondary centres did not form are presented here.

Cell-density-dependent regulation of aggregate density

Over the cell density range employed in these experiments, the maximum number of centres per cell was observed over the low cell density range (1 ×1039 ×103 cells per mm2). At higher cell densities the number of centres formed per cell declined and the aggregate size increased. In ten different experiments with undisturbed cultures, the maximum number of centres formed per cell varied between 28 ·7 and 280 centres per 106 cells (Waddell, 1980). Despite the variation in the maximum centres formed per cell, the number of centres formed per cell always declined as the cell density was increased. The results are consistent with results reported by Hashimoto, Cohen & Robertson (1975).

The cell-density-dependent regulation of aggregate density can be explained by two types of models which make different predictions about the final pattern of aggregation centres. The capacity of cells to initiate centres may be regulated by cell density. In this model the probability of a given cell initiating a centre may be regulated by the level of an extracellular molecule that varies as a function of the cell density. This model would predict a random pattern of aggregation centres. Alternatively, the decrease in the number of centres formed may be due to distance-dependent competition between centres. As the cell density increased, the number of centres within a competitive distance of each other would increase and fewer potential centres would survive. This model would predict a non-random pattern of aggregation centres.

Pattern of aggregation centres in undisturbed cultures

To characterize the pattern of aggregation centres, the aggregate density was determined and the cultures photographed. The distance between each aggregate and its nearest neighbour was measured and a frequency histogram was constructed. These histograms are referred to as nearest-neighbour distributions. The principal advantage of using this method to characterize the spatial pattern is the direct visualization they afford of the minimal spacing between centres.

Nearest-neighbour distributions for undisturbed cultures at six different cell densities are presented in Fig. 1. The maximum number of centres per cell in this experiment was 28 per 106 cells. The cell densities and aggregate densities associated with each nearest-neighbour distribution, and the results of a comparison of the observed nearest-neighbour distributions to the nearest-neighbour distributions expected for a random pattern are presented in Table 1. The non-parametric Kolmogorov-Smirov test (Rohlf & Sokal, 1969) was used to test for differences between observed and expected nearest-neighbour distributions. This test is sensitive to differences in both the means and the variances of distributions.

Table 1.

Fit of the nearest-neighbour distributions of Fig. 1 and Fig. 2 to nearest-neighbour distributions expected for random patterns at the same density

Fit of the nearest-neighbour distributions of Fig. 1 and Fig. 2 to nearest-neighbour distributions expected for random patterns at the same density
Fit of the nearest-neighbour distributions of Fig. 1 and Fig. 2 to nearest-neighbour distributions expected for random patterns at the same density
Fig. 1.

Nearest-neighbour distributions for undisturbed cultures. Cells of the Ax-3 strain were washed free of nutrient medium and dispersed on black Millipore filters. The cultures were incubated at 22 °C without light in humidified chambers. The dotted line depicts the nearest-neighbour distribution expected for a random pattern at the same aggregate density. Cell densities and aggregate densities are given in Table 1.

Fig. 1.

Nearest-neighbour distributions for undisturbed cultures. Cells of the Ax-3 strain were washed free of nutrient medium and dispersed on black Millipore filters. The cultures were incubated at 22 °C without light in humidified chambers. The dotted line depicts the nearest-neighbour distribution expected for a random pattern at the same aggregate density. Cell densities and aggregate densities are given in Table 1.

Very significant differences (P < 0 ·001) were obtained between the observed and expected nearest-neighbour distributions for all but the lowest cell density distribution. Most of the discrepancy from randomness was due to the lack of any aggregates within about 360 μm of each other (about 30 cell diameters). In a repeat experiment in which the maximum number of centres per cell was 139 per 106 cells the minimum distance between aggregates was 200 μm. Within a single experiment approximately the same minimum distance between centres was observed at all cell densities.

Pattern in dissociated cultures

When morphogenesis of D. discoideum is interrupted by washing the multicellular structures off the filters and dissociating them and replating on fresh filters, the cells rapidly recapitulate the stages of morphogenesis achieved before dissociation (Loomis & Sussman, 1970; Newell, Longlands & Sussman, 1971 ; Soli & Waddell, 1975). However, the structures formed after dissociation were much more numerous and smaller. The maximum centre-initiating capacity of the cultures dissociated at the tight-aggregate stage was well above those observed in undisturbed cultures. In three separate experiments with cultures dissociated at the tight-aggregate stage, 1859, 2164 and 2600 aggregates were formed per 106 cells at the lower cell densities. The maximum centre-initiating capacity of undisturbed cultures over the same cell density range ranged between 28 and 280 aggregates per 10° cells in 10 separate experiments.

Nearest-neighbour distributions for an experiment in which cells were replated at six different cell densities are presented in Fig. 2. The cell densities, aggregate densities, and the results of a comparison of the distributions to random nearest-neighbour distributions are presented in Table 1. Very significant differences from randomness were also obtained for the dissociated cultures. However, the discrepancy from randomness was due to the lack of any aggregates closer than 50 μm or about one-seventh the distance observed in the undisturbed cultures.

Fig. 2.

Nearest-neighbour distributions for cultures dissociated at the tight-aggregate stage. Initially, cells were dispersed on filters at a cell density of 4 × 104 cells per mm2. After 10 h, the cells were harvested, dissociated and redispersed on black Millipore filters at the cell densities given in Table 1. The nearest-neighbour distributions were determined from photographs taken 3 h after redispersal. The dotted line depicts the nearest-neighbour distributions expected for random patterns at the same density.

Fig. 2.

Nearest-neighbour distributions for cultures dissociated at the tight-aggregate stage. Initially, cells were dispersed on filters at a cell density of 4 × 104 cells per mm2. After 10 h, the cells were harvested, dissociated and redispersed on black Millipore filters at the cell densities given in Table 1. The nearest-neighbour distributions were determined from photographs taken 3 h after redispersal. The dotted line depicts the nearest-neighbour distributions expected for random patterns at the same density.

In both undisturbed and dissociated cultures the model hypothesizing celldensity-dependent regulation of potential centres can be eliminated. However, it is still possible that some intermediate model which involves both distancedependent competition and cell-density dependent regulation of centres is necessary to explain the results. If cell-density-dependent regulation is a significant factor, then we would predict that distance-dependent competition between centres would not be sufficient to explain the results. To determine if distancedependent competition is sufficient, a simple model for distance-dependent competition was developed.

A simple model for competition between centres

To develop a model for competition between potential centres, the following assumptions were made.

  1. Centres are initiated by a fraction of the cell population that is capable of initiating centres. These cells will be referred to as centre-initiating cells. However, the model does not require that centres are initiated by a special subpopulation. The model can also be interpreted in terms of a uniform cell population in which cells exhibit a low transition probability to centre-initiating capacity.

  2. Competition between centre-initiating cells is distance dependent. Centreinitiating cells which have many neighbouring cells that can also initiate centres will be less likely to survive than those with less.

  3. Only a certain fraction of the cell population is able to initiate aggregation centres at the time of pattern determination. During the period of time that potential centres are competing, this fraction does not increase significantly.

  4. The fraction of the cell population that is capable of initiating centres is not cell-density dependent.

  5. Centre-initiating capacity is randomly distributed amongst the cell population so that on a substratum the centre-initiating cells form a random prepattern for competition. The density of this prepattern is a function of the cell density and the fraction of the cell population able to initiate centres as a consequence of assumption 4.

Using these assumptions an algorithm for competition was developed and programmed in Fortran. A random-number generator was used to assign coordinates to points in an area. For the random-number generators used, the repeat period was 5 × 1018 (Lewis, Goodman & Miller, 1969; Learmouth & Lewis, 1973; Marasaglia, Ananthananayana & Paul, 1975). The suitability of the random-number generators was tested by comparing the nearest-neighbour distributions of the generated points with the nearest-neighbour distributions expected for a random pattern. The distributions were not significantly different from random distributions.

Since points at the edge of the area would not have an opportunity to compete with neighbours in all directions, the area was divided into two portions: a central area which contained approximately 1000 points and a buffer area which surrounded the central area. The buffer area was large enough to ensure that points on the edge of the central area could compete with their first ten nearest neighbours. Only the results of competition in the central area were used.

Competition was simulated in the following manner. First, a hypothetical probability of a centre-initiating cell being eliminated was assigned as a function of the distance between it and its nearest neighbour. In the simplest case competition was all-or-none: if two centres were within a critical distance, a, only one of the centres survived competition. If the two centres were not within this critical distance, then both of the centres survived. In other cases competition was all- or-none if two centres were within a critical distance of each other, but beyond this distance the probability of both centres surviving competition increased linearly with distance. These later models would be more reasonable for an inhibitory signal that decreased in effectiveness gradually with distance or if there is variability in the distance over which individual centres can influence the location of other centres.

During competition each of the points in the central area was considered in a random sequential order. When a point was considered, the distance to its nearest surviving neighbour was determined; then the point being considered was eliminated at the probability assigned by the model for that distance.

By considering points in a random sequential order, each point in a cluster of points has an equal probability of being the first or last point considered in one-to-one competition. Although points are considered one at a time, a given point has the opportunity to compete with all of its neighbours in succession.

In the simplest case, all-or-none competition, a point is eliminated only if there is a nearest neighbour within the critical distance, a at the time it is considered. If a point is within a cluster of n points and has n –1 nearest neighbours within a, then the point will be eliminated whenever it is considered before the other n – 1 points in the cluster. The point survives competition only when it is the last point in the cluster to be considered. Since there are n ! sequences in which to consider the points and the number of sequences in which a given point appears last in a sequence is (n-1) !, the probability of the point surviving is (n – l) !/n ! or simply 1/n.

The sequential competition algorithm assigns a probability of survival to a potential centre that depends upon its spatial relationships with nearby centres. Although competition between aggregation centres may actually occur sequentially, and the centres which are first to initiate competitive signals dominate, this is not the only situation which this algorithm models. For instance, if the centres compete by a diffusible inhibitory signal, then the likelihood of a given centre surviving would still depend upon its spatial relationship to nearby centres: centres which are surrounded by other centres would be less likely to survive than centres located at the edge of a group of centres.

Competition by this algorithm is probabilistic: each simulation of competition is one possible outcome. Therefore the global outcome of competition by this algorithm depends upon the particular prepattern used and the sequence in which the points are considered. However, when two different random-number generators were used to generate the prepattern and sequence of competition, the fraction of points surviving and the nearest-neighbour distributions of the surviving points were not significantly different.

For the simplest case in which competition ends abruptly at the critical distance, a, it is possible to derive an analytic expression for the overall probability of survival from Poisson statistics. The probability of survival of a given centre with n neighbours follows from consideration of the sequential order of competition (see above). Poisson statistics provides a way to determine the frequency with which points will have n neighbours within a given distance at a given density in a random pattern. By summing the products of the frequency of n neighbours and the probability of survival with n neighbours for each n from one to infinity we obtain the overall probability of a point surviving this type of competition in a random prepattern of a given density.

A detailed derivation is presented in the appendix. The formula for the overall probability of survival for all-or-none competition out to a distance a is:
The equation agrees with the results of simulations obtained using the algorithm (Fig. 6).

Analytic expressions for the overall probability of survival for the models in which competition declined linearly beyond the critical distance were not derived. Presently the simulations are the only means of determining the overall probability of survival for models in which competition declines linearly beyond a. No analytic expression for the nearest-neighbour distributions for either all-or-none competition or the other models has been derived.

Of course there are certainly several other ways of assigning a priori probabilities for the survival of centres that depend upon their spatial relationships with other centres. However, this algorithmic method, which assumes equal probability of survival for potential centres in a given situation, makes no assumption about competition beyond that it is distance dependent and occurs within a random prepattern of potential centres. Therefore, it is certainly one of the simplest possible ways to make the assignment of a priori probabilities. Most importantly, the outcome of competition by this method is explicitly realized so that direct comparisons can be made between predicted and observed patterns.

So far we have used the assumptions that competition between potential centres is distance dependent and that the prepattern for competition is random. The remaining assumptions, that the fraction of the cell population that can initiate centres at a given time is constant and this fraction is not celldensity dependent, were met by adjusting the density of points in the prepattern to that predicted by multiplying the cell density by the fraction of the cell population hypothesized to be capable of initiating centres.

Examples of competition by this algorithm are presented in Fig. 3. In these two examples the critical distance was kept constant and the density of the random prepattern varied. Potential centres are represented by + ; centres that survive competition have circles drawn about them. The radius of the circles is half the critical distance, a, within which competition was all-or-none. Consequently, none of the circles intersects. As the density of the prepattern increases fewer of the original points survive competition and the pattern of the surviving centres becomes more evenly spaced.

Fig. 3.

Examples of competition. In these examples of competition according to the algorithm for sequential competition, points in the random prepattem are represented by +. Circles are drawn about the points that survived competition. For further details see text.

In the two examples, only the average distance between the points in the prepattern (density) was different. The ratio of the critical distance to the average distance between points in the prepattem in the two examples was A, 0 ·59 and B, 1 ·78. Only a portion of the competitive area is shown for each example. The same prepattem was used in both cases, and consequently the identical arrangement of points is present in panel A and the upper right portion of panel B.

Fig. 3.

Examples of competition. In these examples of competition according to the algorithm for sequential competition, points in the random prepattem are represented by +. Circles are drawn about the points that survived competition. For further details see text.

In the two examples, only the average distance between the points in the prepattern (density) was different. The ratio of the critical distance to the average distance between points in the prepattem in the two examples was A, 0 ·59 and B, 1 ·78. Only a portion of the competitive area is shown for each example. The same prepattem was used in both cases, and consequently the identical arrangement of points is present in panel A and the upper right portion of panel B.

Comparison of simulated results to observed results

In simulations of competition three parameters were varied: the critical distance a, the distance over which competition declined linearly beyond a and the fraction of centre initiators. The values of the critical distance which give optimal fits are close to the observed minimal distance between aggregates. However, for the simplest model in which competition ended abruptly at a, the best fits to the observed results were obtained for values of a slightly greater than the observed minimal distance between aggregates. Since good fits were not obtained with models in which competition ended abruptly at a, models in which competition declined linearly beyond a were tried.

The results obtained with the models which yielded the best overall fits to the nearest-neighbour distributions and the dynamics of aggregate density as a function of cell density are presented in Fig. 4. The observed nearest-neighbour distributions of the experiment presented in Fig. 1 are redrawn beneath the distributions obtained from simulations. For all of the simulations the fraction of centre initiators was assumed to be 2 ·67 × 10 –5 and the critical distance was 364 μm. In one set of simulation (left panels) competition declined linearly for 182 μm beyond a and in the other (right panels) competition declined for 364 μm beyond a.

Fig. 4.

Comparison of undisturbed and simulated distributions. The nearest-neighbour distributions from Fig. 1 are drawn beneath simulated distributions for two models : in both models the fraction of centre initiators assumed was 2 ·67 × 10 –5, and the critical distance was 364 μm. In the left-hand panels, competition declined linearly to zero over an additional 182 μm. In the right-hand panels competition declined linearly to zero over an additional 364 μm.

Fig. 4.

Comparison of undisturbed and simulated distributions. The nearest-neighbour distributions from Fig. 1 are drawn beneath simulated distributions for two models : in both models the fraction of centre initiators assumed was 2 ·67 × 10 –5, and the critical distance was 364 μm. In the left-hand panels, competition declined linearly to zero over an additional 182 μm. In the right-hand panels competition declined linearly to zero over an additional 364 μm.

Good fits (differences between observed and simulated results not significant at the 0 ·1 level in the Smirov two-sample test (Conover, 1971)) were obtained at the lower cell densities with the model hypothesizing competition over a shorter distance. The model hypothesizing competition over a greater distance provided good fits at the highest cell densities (A and B). Therefore, although a single model was not sufficient to account for all of the nearest-neighbour distributions, they can be accounted for by two models which hypothesize competition over a narrow range of distances beyond a.

When one compares the aggregate densities predicted by the same two models with those observed in the experiment presented in Fig. 1, similar results are obtained (Fig. 5). The model hypothesizing competition over a greater distance predicted the results well at the highest cell densities and the model hypothesizing competition over the shorter distance predicted slightly better results at the lower cell densities. However, the differences in the predictions of the two models at the low cell densities were not significant.

Fig. 5.

Observed and predicted aggregate densities for undisturbed cultures. The aggregate predicted by the two models presented in Fig. 4 is plotted against the observed aggregate densities. For both models, a was 364 μm and the fraction of centre initiators assumed was 2 ·67 × 10 –5. The distance over which competition declined linearly to zero (r) was varied. (•), Observed ; ( ▵), simulated, r = 182 μm ; (▴), simulated, r = 364 μm.

Fig. 5.

Observed and predicted aggregate densities for undisturbed cultures. The aggregate predicted by the two models presented in Fig. 4 is plotted against the observed aggregate densities. For both models, a was 364 μm and the fraction of centre initiators assumed was 2 ·67 × 10 –5. The distance over which competition declined linearly to zero (r) was varied. (•), Observed ; ( ▵), simulated, r = 182 μm ; (▴), simulated, r = 364 μm.

Fig. 6.

The effect of varying the fraction of centre initiators on the fit of the models to the observed results. The fit to the nearest-neighbour distributions (A) was determined by dividing all the expected nearest-neighbour distributions predicted by a model into frequency intervals that contained at least 10 surviving centres. A grand x2 value was calculated from the differences between the observed and expected results over these intervals. This x2 statistic was then divided by the total number of frequency intervals into which the six predicted nearest-neighbour distributions were divided. The number of intervals varied between 63 and 82.

The overall fit of the predicted aggregate densities to the observed aggregate densities was determined by using the number of aggregates that appeared in the photographs from which the nearest-neighbour distributions were derived as the observed value in another x2 statistic. The predicted values for this x2 statistic were determined by multiplying the area in the photographs by the predicted aggregate densities. The differences between the observed and predicted numbers of aggregates at all six cell densities were summed to yield an overall measure of the fit between the predicted and observed results. (•) a = 364 μm, r = 182 μm; (○) a = 364 μm, r = 364 μm.

Fig. 6.

The effect of varying the fraction of centre initiators on the fit of the models to the observed results. The fit to the nearest-neighbour distributions (A) was determined by dividing all the expected nearest-neighbour distributions predicted by a model into frequency intervals that contained at least 10 surviving centres. A grand x2 value was calculated from the differences between the observed and expected results over these intervals. This x2 statistic was then divided by the total number of frequency intervals into which the six predicted nearest-neighbour distributions were divided. The number of intervals varied between 63 and 82.

The overall fit of the predicted aggregate densities to the observed aggregate densities was determined by using the number of aggregates that appeared in the photographs from which the nearest-neighbour distributions were derived as the observed value in another x2 statistic. The predicted values for this x2 statistic were determined by multiplying the area in the photographs by the predicted aggregate densities. The differences between the observed and predicted numbers of aggregates at all six cell densities were summed to yield an overall measure of the fit between the predicted and observed results. (•) a = 364 μm, r = 182 μm; (○) a = 364 μm, r = 364 μm.

These results can be interpreted in two ways: it is possible that a single intermediate model could explain the results at all cell densities. Since the method of extending competition beyond a is rather arbitrary, there are probably several other ways which may be more reasonable that would result in as good or bettter fits. On the other hand, these results may indicate that the distance over which centres compete increases with cell density. However, at this time it is sufficient to note that both the pattern and the dynamics of aggregate density as a function of cell density can be accounted for by distance-dependent competition over a narrow range of distances close to the minimal distance between centres.

The effect of varying the fraction of centre initiators

With a well-defined model of competition it is possible to ask what fractions of centre initiators can give rise to the observed spatial pattern and the dynamics of aggregate density as a function of cell density. More importantly, this allows us to eliminate a large subset of distance-dependent competition models that cannot predict the experimental results.

The best overall fit of the two models presented in Fig. 4 was obtained over a fairly narrow range of fractions of centre initiators extending from 2 ·5 ×10 –5 to 3 ·5 ×10–5 (Fig. 6). At lower fractions of centre initiators the differences rapidly became significant. This is due to the fact that if one assumes a lower fraction of centre initiators than the maximum number of aggregates per cell that was observed in an experiment, then it is impossible for the model to predict the results at the cell densities at which the number of aggregates per cell exceeds the hypothesized fraction of centre initiators in the cell population.

At higher fractions of centre initiators, the differences also became much greater. This implies that the fraction of the cell population able to initiate centres at the time of pattern determination is much less than the fraction of the cell population that is known to be capable of initiating centres under other conditions. For example, Konijn & Raper (1961) have demonstrated that at least 1 per cent of the cell population is able to initiate centres since efficient aggregation of populations of amoebae as small as 100 cells can be obtained. More recently, Glazer & Newell (1981) have demonstrated that at least 20 per cent of the cells of a wild-type strain are capable of initiating centres when mixed with a mutant strain that is defective in initiating centres.

The optimal fractions of centre initiators in this particular experiment indicate that only 1 in 33000-1 in 40000 cells had the capacity to initiate centres at the time of pattern determination. In several other experiments the optimal fraction of centre initiators to explain the cell-density-dependent regulation of aggregate density was close to the maximal centre-initiating capacity observed in the experiments. In a second experiment in which both the spatial pattern and the aggregate density was characterized, the optimal fraction of centre initiators was between 1 ·4 and 2 ·0 ×10−4. This implies that between 1 in 5000 and 1 in 7100 cells were capable of initiating centres at the time of pattern determination. Therefore, there appears to be considerable variation in the fraction of the cell population capable of initiating centres in separate experiments. Nevertheless, in each experiment distance-dependent competition appears to be sufficient to explain the cell-density-dependent regulation of the aggregate density.

Comparison of simulated and observed results for dissociated cultures

The aggregate densities observed in cultures dissociated at the tight-aggregate stage were 100-fold greater than those observed in the undisturbed cultures of the experiment presented in Fig. 1. The nearest-neighbour distributions for the experiment utilizing dissociated cells that was presented in Fig. 2 are compared with nearest neighbour distributions generated by simulation in Fig. 7. For these simulations a was 56 ·8 μm and the fraction of centre initiators assumed was 2 ·8 × 10−3 or 1 cell in 360 cells. In one set of simulations competition declined linearly beyond a for 24 μm and in the second set for 42 μm.

Fig. 7.

Comparison of nearest-neighbour distributions of dissociated cultures with simulated nearest-neighbour distributions of two models. The observed nearest-neighbour distributions of Fig. 2 are redrawn beneath the nearest-neighbour distributions predicted by the two models. For both models a was 56 ·8 μm and the fraction of centre initiators was 2 ·8 ×10 –3. The distance over which competition declined linearly beyond a (r) was varied. For the panels on the left r was 24 μm and for the panels on the right r was 42 μm.

Fig. 7.

Comparison of nearest-neighbour distributions of dissociated cultures with simulated nearest-neighbour distributions of two models. The observed nearest-neighbour distributions of Fig. 2 are redrawn beneath the nearest-neighbour distributions predicted by the two models. For both models a was 56 ·8 μm and the fraction of centre initiators was 2 ·8 ×10 –3. The distance over which competition declined linearly beyond a (r) was varied. For the panels on the left r was 24 μm and for the panels on the right r was 42 μm.

At the five highest cell densities (A –E), simulated distributions were obtained using one of these two models that were not significantly different from the observed distributions at the 0 ·01 level by the Smirov two-sample test (Conover, 1971). Good fits (not significantly different at the 0 ·1 level) were obtained for three of the distributions. At the lowest cell density (F) neither of the two models provided good fits. However, good fits were obtained to this distribution at slightly lower fractions of centre initiators (2 ·3 ×10−3). This may imply a violation of the fourth assumption, i.e. that the fraction of centre initiators is not cell-density dependent. However, the deviation is not great and occurs at the lowest cell density. At low cell densities small changes in cell density lead to rather large changes in aggregate density.

The two models also predict fairly well the cell-density-dependent regulation of the aggregate density (Fig. 8). As with the undisturbed cultures, the model hypothesizing competition over a greater distance was better at predicting the dynamics at the highest cell densities. At the lower cell densities both of the models closely fit the observed results.

Fig. 8.

Comparison of observed and predicted aggregate densities for dissociated cultures. The aggregate densities for the two models in Fig. 8 are plotted with the observed aggregate densities determined from the photographs used to derive the nearest-neighbour distributions presented in Fig. 2. For both models a was 56 ·8 μm and the fraction of centre initiators was 2 ·8 × 10–3. The distance over which competition declined linearly beyond a (r) was varied: (•) observed; (▵) simulated, r = 24 μm; (○) simulated, r = 42 μm.

Fig. 8.

Comparison of observed and predicted aggregate densities for dissociated cultures. The aggregate densities for the two models in Fig. 8 are plotted with the observed aggregate densities determined from the photographs used to derive the nearest-neighbour distributions presented in Fig. 2. For both models a was 56 ·8 μm and the fraction of centre initiators was 2 ·8 × 10–3. The distance over which competition declined linearly beyond a (r) was varied: (•) observed; (▵) simulated, r = 24 μm; (○) simulated, r = 42 μm.

The effect of varying the fraction of centre initiators

As with the undisturbed cultures, it is possible to determine the fractions of centre initiators that can account for the observed results by testing the overall fit of the simulated and observed results at several different fractions of centre initiators. The best overall fit was obtained over a fairly narrow range of fractions of centre initiators that extended from 2 ·5 × 10 –3 to 3 × 10 –3, or between 1 cell in 330 to 1 in 400 (Fig. 9). Above and below this range the differences rapidly became very significant. In this case, the position of the optimal range of fractions of centre initiators differs from the position of the range for the undisturbed cultures by almost two orders of magnitude. This result suggests that as cells progress through morphogenesis, a greater fraction of them acquire the capacity to initiate centres and compete for space with other centres. This additional capacity is not revealed unless cultures are dissociated and aggregation centres are established anew. However, this hidden capacity may be involved in other functions during later morphogenesis.

Fig. 9.

Sensitivity of the overall tit of simulated results to observe results for cultures dissociated at 10 h to changes in the fraction of centre initiators assumed. The overall fit of the nearest-neighbour distributions (A) and aggregate densities (B) were determined in the same way as that described in Fig. 8. (•) a = 56 ·8 μm, r = 42 μm.

Fig. 9.

Sensitivity of the overall tit of simulated results to observe results for cultures dissociated at 10 h to changes in the fraction of centre initiators assumed. The overall fit of the nearest-neighbour distributions (A) and aggregate densities (B) were determined in the same way as that described in Fig. 8. (•) a = 56 ·8 μm, r = 42 μm.

Bonner & Hoffman (1963) noted that the spatial pattern of aggregation centres was not random. A non-random pattern implies that centre initiation is not an independent process: centres influence the location of other centres. Two observations indicate that competition between potential centres exists: first, the number of centres formed per cell decreases as the cell density is increased above the cell density that elicits the maximum number of centres per cell ; secondly, centres are not observed within a critical distance of each other. Evidence has been presented for the existence of inhibitory substances which control aggregation territory size (Bonner & Hoffman, 1963) and of inhibitory substances which diffuse from centres to prevent centres in their vicinity from initiating centres. (Schaffer, 1963).

Here a model was developed to account for these two types of empirical evidence for competition and to show how the spatial pattern and densitydependent regulation of centres were related. Within the framework of this model an operational definition for potential aggregation centres arises which is based on the capacity to compete for space with other centres in a distancedependent fashion. This definition is more restrictive than other definitions (Konijn & Raper, 1961; Glazer & Newell, 1981) which were specifically designed to assay the maximal centre-initiating capacity of amoebae. Therefore, although all of the definitions may correspond to the same capacities, this is not necessary. The ability to compete for space in dense populations of amoebae may apply to ‘stronger’ centre initiators than the other definitions.

The author favours the interpretation that all of these different ways of defining centre-initiating capacity refer to a discrete class of cells which can both initiate centres and compete for space with other centres. The results presented here suggest that in large populations only a small fraction of the cell population has the capacity to initiate centres at the time of pattern determination. However, this is not inconsistent with the earlier studies of Konijn & Raper (1961), which are based on the efficiency of aggregation of small populations, since all of the cell population may acquire these capacities over sufficient periods of time. Indeed, the dramatic increase in the number of potential centres that must be hypothesized to account for the results observed in dissociated cultures demonstrates that this capacity to compete for space is not restricted to the small subpopulation that initiated the primary centres in undisturbed cultures.

This capacity to initiate centres may be acquired by cells in a stochastic manner without interactions with other cells. This is the simplest hypothesis and would predict a random prepattern of potential centres. However, it will be difficult to rule out a requirement for cell interactions to acquire the capacity to initiate centres, since any assay for this capacity requires the presence of other cells.

Distance-dependent competition between potential centres may occur in at least two ways: centres may compete by attracting and dominating other potential centres in their vicinity via the chemotactic mechanism. Alternatively, a mechanism distinct from the chemotactic mechanism may prevent the expression of centres by nearby cells. The former mechanism has been reported to be sufficient to explain the spacing of aggregation centres in Dictyostelium minutum (Gerisch, 1968). A similar mechanism appears to operate in Dictyostelium discoideum by means of the periodic signalling system (Gerisch, 1968). However, it is not clear whether this mechanism is sufficient to account for the spacing in D. discoideum.

The analysis presented here indicates that the territory area and hence the aggregate density may be regulated in two distinct ways: (1) by changing the distance over which potential centres compete; and (2) by changing the fraction of the cell population that is able to initiate centres. At low cell densities, at which distance-dependent competition between potential centres is less severe, the dominant factor is probably the fraction of the cell population able to initiate centres. As the cell density is increased, the distance over which centres compete becomes increasingly important. The mechanisms controlling how cells acquire the capacity to initiate centres and the distance over which centres compete have probably evolved in response to two types of selective pressure. In small populations and at very low cell densities there is probably a premium on ensuring the formation of centres. A stochastic acquisition of the capacity to initiate centres would meet this demand. In large populations and at high cell densities selection probably favours an optimal fruiting body size. The distancedependent competition between centres may meet this demand.

I want to thank Joseph Hegmann and Joseph Frankel for many helpful suggestions and discussions. I am indebted to Eric Six for a simplification of the analytic expression for the overall probability of survival in sequential competition. I also want to thank J. T. Bonner, D. Bozzone and K. Inouye for encouragement and reviewing drafts of this paper. The author was supported by NIH grants T01 HD00152-07 and 5T 32 CA09167.

APPENDIX

Derivation of the formula for the fraction of centres surviving competition in all-or-none competition

We can use the same logic that is used to derive the nearest-neighbour distribution for a random pattern of points (Pielou, 1969) to derive an analytic expression for the fraction of points surviving competition according to the algorithm given in Fig. 3. We choose a reference point and ask what the probability of survival of this point will be. This probability will depend upon the number of points within the critical distance, a, of our reference point. By Poisson statistics this is :
The probability that the reference point survives will depend upon the order in which the k +1 points are considered. The reference point survives if it is the last point considered for one-to-one competition by the sequential competition algorithm. It is eliminated in every other case. The total number of possible sequences will be as follows :
Let
total number of sequences = n!

number of sequences in which the reference point is considered last and hence survives competition = (n -1)!

To determine the probability of the reference point being eliminated, we multiply the probability of k points being within the distance, a, of the reference point by the probability of the point being eliminated for each case and add up all the possible cases :
The overall probability of survival will be:
Taking note that the above expression for P(survival)can be rewritten in the simplified from:
Bonner
,
J. T.
&
Hoffman
,
M.
(
1963
).
Evidence for a substance responsible for spacing of aggregation and fruiting in cellular slime moulds
.
J. Embryol. exp. Morph
.
11
,
571
589
.
Bonner
,
J. T.
,
Barkley
,
D. J.
,
Hall
,
E. M.
,
Konijn
,
T. M.
,
Mason
,
J. W.
,
O’Keefe
,
G.
&
Wolfe
,
P. B.
(
1969
).
Acrasin, Acrasinase and sensitivity to acrasin in Dictyostelium discoideum
.
Devl Biol
.
20
,
72
87
.
Claxton
,
J. H.
(
1964
).
The determination of pattern with special reference to that of the central primary skin follicles in sheep
.
J. theor. Biol
.
7
,
302
317
.
Coccucci
,
S.
&
Sussman
,
M.
(
1970
).
RNA in cytoplasmic and nuclear fractions of cellular slime mould amoebas
.
J. Cell Biol
.
45
,
399
407
.
Conover
,
W. J.
(
1971
).
Practical Nonparametric Statistics
.
New York
:
John Wiley and Sons
.
Garrod
,
D. R.
&
Ashworth
,
J. M.
(
1972
).
Effect of growth conditions on development of the cellular slime mould Dictyostelium discoideum
.
J. Embryol. exp. Morph
.
28
,
463
479
.
Gerisch
,
G.
(
1968
).
Cell aggregation and differentiation in Dictyostelium discoideum
.
Curr. Top. devl Biol
.
3
,
157
197
.
Glazer
,
P. M.
&
Newell
,
P. C.
(
1981
).
Initiation of aggregation by Dictyostelium discoideum in mutant populations lacking pulsatile signalling
.
J. gen. Microbiol. (In the press
.)
Hashimoto
,
Y.
,
Cohen
,
M.
&
Robertson
,
A.
(
1975
).
Cell density dependence of the aggregation characteristics of the cellular slime mould Dictyostelium discoideum
.
J. Cell Sci
.
19
,
219
229
.
Kahn
,
A. J.
(
1964
).
The influence of light on aggregation in Polysphondylium pallidum
.
Biol Bull
.
127
,
85
96
.
Konijn
,
T. M.
&
Raper
,
K. B.
(
1961
).
Cell aggregation in Dictyostelium discoideum
.
Devl Biol
.
3
,
725
756
.
Lacalli
,
T. C.
&
Harrison
,
L. G.
(
1978
).
Development of ordered arrays of cell wall pores in desmids: a nucleation model
.
J. theor. Biol
.
74
,
109
138
.
Lawrence
,
P. A.
&
Hayward
,
P.
(
1971
).
The development of simple patterns: spaced hairs in Oncopeltusfasciatus
.
J. CellSci
.
8
,
513
524
.
Learmouth
,
G.
&
Lewis
,
P.
(
1973
).
Statistical tests of widely used and recently proposed uniform random number generators. NPS55LW73111 A, Naval Postgraduate School, Monterey, Calif
.
Lewis
,
P.
,
Goodman
,
A.
&
Miller
,
J.
(
1969
).
Pseudorandom number generator for System/ 360
.
IBM Systems Journal No
.
2
.
Loomis
,
W. F.
&
Sussman
,
M.
(
1966
).
Commitment to the synthesis of a specific enzyme during cellular slime mould development
.
J. molec. Biol
.
22
,
401
404
.
Markovics
,
J.
,
Glass
,
L.
&
Maul
,
G. G.
(
1974
).
Pore pattern on nuclear membranes
.
Expl Cell Res
.
85
,
443
445
.
Marasaglia
,
G. R. Ananthananayanan
&
Paul
,
N.
(
1975
).
Super Duper Code LW/S 001 DL-1. University of Iowa Computing Center. Iowa City, Iowa
.
Maynard Smith
,
J
&
Sondi
,
K. C.
(
1961
).
The arrangement of bristles in Drosophilia
.
J. Embryol. exp. Morph
.
9
,
661
672
.
Newell
,
P.
,
Longlands
,
M.
&
Sussman
,
M.
(
1971
).
Control of enzyme synthesis and cellular interaction during development of the cellular slime mould Dictyostelium discoideum
.
J. molec. Biol
.
58
,
541
554
.
Pielou
,
E. C.
(
1969
).
An Introduction to Mathematical Ecology
.
New York
:
John Wiley and Sons
.
Raper
,
K. B.
(
1940
).
Pseudoplasmodium formation and organization in Dictyostelium discoideum
.
J. Elisha Mitchell Sci. Soc
.
56
,
241
282
.
Reinhard
,
D. T.
(
1966
).
Silica gel as a preserving agent for the cellular slime mould, Acrasis rosea
.
J. Protozool
.
13
,
225
226
.
Rohlf
,
F.
&
Sokol
,
R.
(
1969
).
Statistical Tables
.
San Francisco
:
W. H. Freeman and Co
.
Ryter
,
A.
,
Klein
,
C.
&
Brachet
,
P.
(
1979
).
Dictyostelium discoideum surface changes elicited by high concentrations of cAMP
.
Expl Cell Res
.
119
,
373
380
.
Sachs
,
T.
(
1978
).
Patterned differentiation in plants
.
Differentiation
11
,
65
73
.
Shaffer
,
B. M.
(
1957
).
Aspects of aggregation in the cellular slime moulds
.
Amer. Nat
.
91
,
19
35
.
Shaffer
,
B. M.
(
1961
).
The cells founding aggregation centres in the slime mould Poly-sphondlium violaceum
.
J. exp. Biol
.
38
,
833
847
.
Shaffer
,
B. M.
(
1963
).
Inhibition by existing aggregations of founder differentiation in the cellular slime mould Polysphondylium violaceum
.
Nature
255
,
549
552
.
Shaffer
,
B. M.
(
1975
).
Secretion of cyclic AMD induced by cyclic AMD in the cellular slime mould, Dictyostelium discoideum
.
Nature
255
,
549
552
.
Soll
,
D. R.
&
Waddell
,
D. R.
(
1975
).
Morphogenesis in the slime mould Dictyostelium discoideum
.
Devi Biol
.
47
,
292
302
.
Spiegel
,
F. W.
&
Cox
,
E. C.
(
1980
).
A one-dimensional pattern in the cellular slime mould Polysphondyliumpallidum
.
Nature
286
,
806
807
.
Sussman
,
M.
(
1966
).
In Methods in Cell Physiology
(ed.
D. M.
Prescott
).
New York
:
Academic Press
.
Thadani
,
V. P.
,
Pan
,
P.
&
Bonner
,
J. T.
(
1977
).
Complementary effects of ammonia and cAMP on aggregation territory size in cellular slime mould Dictyostelium mucuroides
.
Expl Cell Res
.
108
,
75
78
.
Waddell
,
D. R.
(
1980
).
Studies of dissociated cells in the slims mould, Dictyostelium discoideum
.
PhD Thesis, University of Iowa
.
Wilcox
,
M.
,
Mitchison
,
G.
&
Smith
,
R.
(
1973
).
Pattern formation in the blue-green alga Anabaena
.
J. Cell Sci
.
13
,
637
649
.
Wolk
,
C. P.
&
Quine
,
M. P.
(
1975
).
Formation of one-dimensional patterns by stochastic processes and by filamentous blue-green algae
.
Devi Biol
.
46
,
370
382
.