J. Embryol. exp. Morph. Vol. 70, pp. 15-98, 1982
Printed in Great Britain © Company of Biologists Limited 1982
75
The spatial pattern of aggregation centres
in the cellular slime mould
By DAVID R. WADDELL 1 2
From the Department of Biology, Princeton University
SUMMARY
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.
INTRODUCTION
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,
1
Author's address: Department of Biology, Princeton University, Princeton, New Jersey
08544, U.S.A.
2
Author's present address: Max-Planck-Institut fur Biochemie, D8O33 Martinsried be
Miinchen, Federal Republic of Germany.
76
D. R. WADDELL
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.
MATERIALS AND METHODS
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 x 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
Pattern formation in Dictyostelium discoideum
77
reached 7 x l 0 6 - 1 0 7 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 x 106 per ml.
Initiating morphogenesis
Morphogenesis was initiated following the method of Sussman (1966). For
each developing culture, 5 x 10MO 8 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, 005 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 x 107 cells per ml. Two dilutions of this suspension were then prepared: the
first, D 1 , was a fourfold dilution (107 cells per ml) and the second, D2, was a
tenfold dilution of D 1 (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 x 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 x 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 x 10 inch
format and the distance between each aggregate and its nearest neighbour was
78
D. R. WADDELL
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.
RESULTS
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 x 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 x 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 /Mm. The diameter of a developing cell is about 9-11 jum, 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 x 1039 x 103 cells per mm 2 ). 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
Pattern formation in Dictyostelium discoideum
79
30-f
10
1-5
20
Nearest-neighbour distance (mm)
2-5
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.
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.
80
D. R. WADDELL
Table 1. Fit of the nearest-neighbour distributions of Fig. 1 and Fig. 2 to nearestneighbour distributions expected for random patterns at the same density
Designation
Cell
Aggregate
in the
density
density
4
2
figures
(10 permm ) (mm- 2 )
Sample size of
distribution
Maximum
difference
Significance
between observed Kolmogorovand expected
Smirov test,
P< 0-0001)
distributions
A
B
C
D
E
F
7-47
3-73
2-99
1-87
1-40
0-933
Undisturbed cultures (Fig. 1)
0-801
202
0-672
207
172
0-619
0-520
152
0-402
114
0183
47
0-4201
0-4418
0-2968
0-2616
0-3374
01426
+
+
+
+
+
—
A
B
C
D
E
F
5-50
2-85
1-32
0-815
0-344
0199
Dissociated cultures (Fig. 2)
36-35
181
3204
158
2616
246
16-71
242
908
329
4-84
157
0-3673
0-3365
0-2697
0-2159
01304
0-2527
+
+
+
+
+
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 10G 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 nearestneighbour 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.
Very significant differences (P < 0001) 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 /tm of each other (about 30 cell diameters). In a
repeat experiment in which the maximum number of centres per cell was 139
Pattern formation in Diciyoslelium discoideum
81
6
per 10 cells the minimum distance between aggregates was 200 //m. 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;
Soil & 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 /im or about one-seventh the distance observed in the
undisturbed cultures.
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
82
D. R. WADDELL
20100
20100
20100
20100
2010-
20-
01
0-2
0-3
0-4
Nearest-neighbour distance (mm)
0-6
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 x 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.
Pattern formation in Dictyostelium discoideum
83
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 x 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 allor-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
84
D. R. WADDELL
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- \)\/n\ or simply \]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
85
Pattern formation in Dictyostelium discoideum
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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 prepattem (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.
86
D. R. WADDELL
overall probability of survival for all-or-none competition out to a distance a is:
P (survival) = ^ ( 1
-e~Xa2),
A = 77 (density of pre-pattern),
a = distance within which competition is all-or-none.
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.
Pattern formation in Dictyostelium discoideum
87
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 x 10~5 and the critical distance was
364 fim. In one set of simulation (left panels) competition declined linearly for
182 /im beyond a and in the other (right panels) competition declined for 364 /im
beyond a.
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.
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
D. R. WADDELL
a = 364/Ltm
ramp = 3 6 4 jum
Fig. 4. Comparison of undisturbed and simulated distributions. The nearestneighbour distributions from Fig. 1 are drawn beneath simulated distributions for
two models: in both models the fraction of centre initiators assumed was 2-67 x 10~5,
and the critical distance was 364 /tm. In the left-hand panels, competition declined
linearly to zero over an additional 182 /im. In the right-hand panels competition
declined linearly to zero over an additional 364 fim.
Pattern formation in Dictyostelium discoideum
89
140 i
120
100-
80-
60
40-
201
2
3
4
5
6
Cell density (10 4 per mm 2 )
7
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 /tm and the fraction of
centre initiators assumed was 2-67 x 10~5. The distance over which competition
declined linearly to zero (r) was varied. (#), Observed; (A), simulated, r = 182 [im;
(A), simulated, r = 364 /im.
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 x 10~5 to
3-5 x 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
90
D. R. WADDELL
10-
6T3
0)
t
4-
o
E
o
* 7001
o
Aggregate density
M6 0 0 -
° 500400300200100O—o
01 X 10"s
2X10"5
3X10'5
4X1O'S
5X 10 - 5
Fraction of centre initiators
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 /im, r = 182 /tm; (O) a = 364 /tin,
r = 364 /*m.
Pattern formation in Dictyostelium discoideum
91
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 x 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 ^m and the fraction of centre initiators assumed was
2-8 x 10~3 or 1 cell in 360 cells. In one set of simulations competition declined
linearly beyond a for 24 /im and in the second set for 42 /im.
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 (2-3xlO~ 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
92
D. R. WADDELL
a = 56-8 urn
ramp = 42 /im
Fig. 7. Comparison of nearest-neighbour distributions of dissociated cultures with
simulated nearest-neighbour distributions of two models. The observed nearestneighbour distributions of Fig. 2 are redrawn beneath the nearest-neighbour
distributions predicted by the two models. For both models a was 56-8 /tm and the
fraction of centre initiators was 2-8 x 10~3. The distance over which competition
declined linearly beyond a (r) was varied. For the panels on the left r was 24 fim
and for the panels on the right r was 42 fim.
93
Pattern formation in Dictyostelium discoideum
60-
50-
!40
30-
20-
10-
0
1
1
2
3
4
Cell density (10 4 per mm 2 )
5
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 /*m
and the fraction of centre initiators was 2-8 x 10~3. The distance over which competition declined linearly beyond a (r) was varied: ( # ) observed; (A) simulated,
r = 24 pim; (O) simulated, r = 42 pm.
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.
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 x 10~3 to 3 x 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
4
EMB 70
94
D. R. WADDELL
7 -
Pattern of centres
6-
5-
.2 800 -
600400 -
200 -
0J
0001 0002 0-003 0004 0005 0006 0007
Fraction on centre-initiators
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 pvn, r =
42 jim.
and aggregation centres are established anew. However, this hidden capacity
may be involved in other functions during later morphogenesis.
DISCUSSION
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
Pattern formation in Dictyostelium discoideum
95
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
4-2
96
D. R. WADDELL
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:
Pattern formation in Dictyostelium discoideum
97
Let
n = k +1 (reference + other points within a)
total number of sequences = n\
number of sequences in which the reference point is considered last and hence
survives competition = {n -1)!
P(survival) = (n-l)!/#t! = (n-\)\/(n-l)
In = \/n
P(elimination) = (1-1/n)
To determine the probability of the reference point being eliminated, we
multiply the probability of A: 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:
oo (\a2\n-l
I
\\
n—2 \fl— '•)• \
M/
P (elimination) = e~Xa2 2 7 — w l 1 —
The overall probability of survival will be:
P (survival) = 1 —P (elimination)
X2
Xn
Taking note that e* = 1 + X
+ — . . . —
the above expression for P (survival) can be rewritten in the simplified form:
P (survival) = ^ - 2 (1 - e~ Aa2 ).
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{Received 16 September 1981, revised 1 February 1982)