Whorls of the cellular slime mould Polysphondylium pallidum originate as spherical cell masses that during normal morphogenesis produce tips only at equidistant positions around their equator. We have observed a series of new patterns in whorls that differ from normal whorls only in that they are larger or more elongated. Among the novel patterns found were arrays of tips distributed fairly regularly over the whole whorl surface, as well as striped patterns detected at earlier stages with a tip-specific antigen. These altered patterns demonstrate that a whorl’s size and shape are by themselves important factors in pattern determination. We have compared the range of observed patterns to those predicted by a variety of different theories. We find that while no one theory can account in detail for all of our observations, predictions based on Turing’s scheme of pattern formation come the closest.

During morphogenesis, the cellular slime mould Polysphondylium pallidum periodically produces a ball of cells called a whorl mass. Within a half hour, protrusions known as tips appear at equidistant positions around the equator of the whorl mass (Fig. 1). Each tip ultimately gives rise to a secondary sorocarp.

Fig. 1.

Stereopair scanning-electron micrographs of normal and larger than normal spherically shaped whorls. (A) A whorl of normal size just after separation from the slug. (B) A whorl beginning to form four tips. The surface is slightly deformed by bulges. (C) Another whorl at a later stage also forming four tips. The tips are equally spaced around the equator. (D,E) Larger than normal whorls, still roughly spherical, displaying fairly regularly spaced tips over their entire surface. Bar, 10 μm

Fig. 1.

Stereopair scanning-electron micrographs of normal and larger than normal spherically shaped whorls. (A) A whorl of normal size just after separation from the slug. (B) A whorl beginning to form four tips. The surface is slightly deformed by bulges. (C) Another whorl at a later stage also forming four tips. The tips are equally spaced around the equator. (D,E) Larger than normal whorls, still roughly spherical, displaying fairly regularly spaced tips over their entire surface. Bar, 10 μm

Monoclonal antibodies specific for tip antigens reveal that the final tip pattern emerges by progression through a series of intermediate prepatterns (Byrne & Cox, 1986). First, tip antigen is expressed by all cells on the whorl’s surface. Then, the surface staining becomes localized to an equatorial band that soon fragments into patches. Initially, these patches are not equal in size or regularly spaced, but they become so just as tips themselves appear.

The process whereby a uniform surface expression of tip-specific antigen becomes progressively restricted to form the final equatorial tip pattern captures the essence of a central problem in development: how does a patterned state emerge from an initially unpatterned one?

A number of theories have been proposed to explain how patterns arise de novo. Among the simplest are those that model the formation of regularly spaced, identical pattern elements. To account for hair-follicle patterns in sheep, Claxton (1964) proposed that follicle primordia appear randomly in both space and time. Once born, each primordia becomes surrounded by a zone of inhibition that prohibits formation of new follicles in that zone. Such theories give rise to fairly regularly spaced arrays. Versions of Claxton’s model have been advanced to explain bristle spacing in insects (Wigglesworth, 1940; Lawrence, 1969), pore spacing in algae (Lacalli & Harrison, 1978) and aggregation-centre spacing in slime moulds (Waddell, 1982). The model, however, is not well suited to secondary tip formation in a P. pallidum whorl for two reasons. Patterns should arise sequentially because primordia appear at random times. In whorls, tips appear simultaneously. Further, the predicted final spacing of pattern elements is not nearly as regular as observed in whorls (Claxton, 1964; Byrne & Cox, 1987).

Goodwin & Cohen (1969) proposed a rather different model in which positional information is specified as a phase difference between two travelling waves. As applied to P. pallidum, such a pair of waves might propagate around the whorl equator, with integral multiples of a certain phase difference specifying the location of tips. This model also predicts incorrectly that tips should appear sequentially as the wave-pair propagates. An additional problem is that a wave source on the equator would not ordinarily generate an odd number of tips that are also equally spaced. (This is because as the wave-pair propagates bidirectionally from a point on the equator it will always specify pairs of tips equidistant from the point of initiation). Yet in whorls, odd and even numbers of tips do form symmetrically with equal frequency (Cox et al. 1988).

Failure of these simple models has led us to consider Turing’s (1952) reaction-diffusion theory and its relatives to explain the tip pattern. This class of theories can account for the rather precise spacing of tips, as well as for their simultaneous emergence from an initially unpatterned state (Byrne & Cox, 1987).

The present study was undertaken to test further the applicability of these theories, and to provide a more stringent set of criteria for evaluating all possible models. Turing-like theories predict that patterns depend critically and predictably on the size and shape of the field (Gmitro & Scriven, 1966). Accordingly, we examined tip patterns on whorls that were larger or more elongated than normal. In this way, an extensive catalogue of new patterns has been produced that any plausible theory must now explain. Among the novel patterns observed were lattice-like arrays of tips distributed over the entire whorl’s surface and multistriped patterns of a tip-specific antigen.

We have computed the patterns predicted by reaction-diffusion theories for spherical and spheroidal surfaces, and compared them to all of the patterns observed in whorls. We find good agreement. In particular, Turing theories predict that both striped and spotted patterns can form on a sphere or spheroid, and this is what we observe. In contrast, Claxton’s model does not predict striped patterns, and while stripes could be generated by a Goodwin-Cohen model a separate pair of propagating waves would have to be hypothesized. Thus our results to date are most economically explained by a Turing-like model.

Culture conditions

Wild-type PN500 was grown on a streak of Escherichia coli on GYP agar (glucose-yeast-peptone agar, Warren et al. 1976). Large whorls appeared spontaneously at low frequency under these conditions. Large whorls were also obtained by first growing amoebae on a lawn of E. coli on GYP agar. After two days, vegetative cells were harvested in phosphate buffer (Byrne & Cox, 1986) and washed three times by centrifugation. The amoebae were plated in drops of 1–5 ul at 108 –109 cells ml-1 on non-nutrient agar (2% agar dissolved in distilled water) and kept in the dark. Under these conditions large aggregates formed, which in turn yielded large culminating sorogens that sometimes deposited large whorls. The patterns observed on large whorls were independent of the growth conditions used to produce them.

Scanning electron microscopy

Whorls were prepared for scanning electron microscopy as previously described (Byrne & Cox, 1986).

Immunocytochemistry

Whorls fixed in methanol were incubated with a tip-specific mouse monoclonal antibody (PglOl), washed, incubated with a secondary anti-mouse antibody conjugated to biotin, washed and incubated with streptavidin conjugated to Texas red (details in Byrne & Cox, 1986). Stained specimens were sucked into a capillary, rotated by 22·5° increments under a fluorescent microscope and photographed at each angle. The resultant prints were cut and joined to form a montage of the full whorl’s circumference (Byrne & Cox, 1987). The montage was photographed with 4×5” sheet film, and the negatives were digitized using a Perkin-Elmer PDS two-dimensional scanning microdensitometer.

Computer analysis

Digitized images were transferred to a Silicon Graphics Iris computer. The images were edited in three steps. First, average background in each panel of the montage was determined by sampling pixels in that panel. The difference between this value and the average background for the whole image was subtracted from each pixel in the panel. Second, cut-marks produced by the montaging process were replaced with a slightly noisy ramp of intensity values based on interpolation between the average intensity to the left and right of the cut. Finally, mean background was subtracted from the whole image.

Formulas for spherical or spheroidal harmonic functions (Hobson, 1931; Flammer, 1957; Hunding, 1983) were used to plot harmonic patterns. Data for the spheroidal harmonic functions were kindly computed by Axel Hunding at the University of Copenhagen. Three-dimensional representations of harmonic functions were produced using a surfacegenerating program written by David Laur of the Interactive Computer Graphics Lab at Princeton University.

Final tip patterns on large whorls

On whorls of normal size, tips were equally spaced around the equator (Fig. 1B,C) (Cox et al. 1988). On larger than normal whorls, as determined by scanning electron microscopy, tips formed over the entire surface. These tips still appeared to be reasonably well spaced (Figs 1D,E; 2A,B). Sometimes, tips did not form at all on large, elongate whorls until the whorl segmented into a series of smaller whorls (Fig. 2C).

Fig. 2.

Stereopair scanning-electron micrographs of larger than normal, ellipsoidally shaped whorls. (A,B) Whorls with tips distributed in fairly regular arrays, comparable to those observed in more spherical whorls. (C) An especially elongated whorl in the process of segmenting into a chain of small whorls. Bar, 20 μm.

Fig. 2.

Stereopair scanning-electron micrographs of larger than normal, ellipsoidally shaped whorls. (A,B) Whorls with tips distributed in fairly regular arrays, comparable to those observed in more spherical whorls. (C) An especially elongated whorl in the process of segmenting into a chain of small whorls. Bar, 20 μm.

When large whorls were stained with a tip-specific monoclonal antibody and rotated under a fluorescent microscope, tips were found distributed in imperfect lattice-like patterns over the whorl surface (Fig. 3). The number of tips in the pattern was roughly proportional to whorl surface area. For example, on normal whorls, a single row of tips was present. On slightly larger than normal whorls, there were more tips, often arranged roughly into two to three rows (Fig. 3A,B). On very elongated whorls, numerous tips were present (Fig. 3C). An analysis of the spatial distribution of PglOl for the montages in Fig. 3 revealed that the average nearest-neighbour distances were similar (Fig. 3A: 45·9 μm, S.D. =2·8; Fig. 3B: 41·2 μm, S.D. = 8·3; Fig. 3C: 44·7 μm, S.D. = 8·8).

Fig. 3.

Tip patterns on large whorls as detected with a tip-specific monoclonal antibody. Each image is a montage of a series of 16 fluorescent micrographs of a whorl rotated through 360°. Arrows indicate the same tip at either end of the rotation. A bright-field image at the starting angle of the rotated whorl is on the left. To compare whorl size, the magnification is the same for each panel in this figure, and in Figs 5 and 6. As an estimate of ‘order’ in these distributions we have calculated R values (Clark & Evans, 1954). For any distribution, mean nearest-neighbour distances are R times as great as would be expected for a random distribution of the same density. (A) A whorl whose axis was at an angle to the axis of rotation. As a result, the whorl’s equator follows a concave downward path. Many tips are displaced to either side of the equator. R = 1·9. (B) A whorl with a two- to three-tiered array of tips covering its surface. R = 1·5. (C) An elongate whorl with a complex array of tips. Due to the whorl’s curvature, at certain angles during the rotation some tips at the top of the whorl were out of focus, and so these tips are not visible in the fluorescent montage. R = 1·5. Bar, 50 μm.

Fig. 3.

Tip patterns on large whorls as detected with a tip-specific monoclonal antibody. Each image is a montage of a series of 16 fluorescent micrographs of a whorl rotated through 360°. Arrows indicate the same tip at either end of the rotation. A bright-field image at the starting angle of the rotated whorl is on the left. To compare whorl size, the magnification is the same for each panel in this figure, and in Figs 5 and 6. As an estimate of ‘order’ in these distributions we have calculated R values (Clark & Evans, 1954). For any distribution, mean nearest-neighbour distances are R times as great as would be expected for a random distribution of the same density. (A) A whorl whose axis was at an angle to the axis of rotation. As a result, the whorl’s equator follows a concave downward path. Many tips are displaced to either side of the equator. R = 1·9. (B) A whorl with a two- to three-tiered array of tips covering its surface. R = 1·5. (C) An elongate whorl with a complex array of tips. Due to the whorl’s curvature, at certain angles during the rotation some tips at the top of the whorl were out of focus, and so these tips are not visible in the fluorescent montage. R = 1·5. Bar, 50 μm.

This intertip spacing is very close to the spacing (44· 8μm, S.D. = 7·5) reported for normal whorls (Byrne & Cox, 1987).

Emergent tip patterns on large whorls

When stained with anti-PglOl, large incipient whorl masses that had just separated from the slug revealed a relatively homogeneous distribution of Pg101 over their surface, with no evidence for a pattern (Fig. 4). In whorls at slightly later stages (as judged by their greater distance from the primary sorogen), two different patterns of Pg101 expression were detected: stripes and spots.

Fig. 4.

Expression of a tip-specific antigen (Pg101) in a large whorl at an early stage. (A) Bright-field image focused on the sides of the whorl. (B–E) Distribution of tip antigen at 90° angles around the whorl. (A and B are at 0°). The plane of focus was the whorl’s top surface. Antigen appears distributed over the whorl’s surface in no apparent pattern. Bar, 50 μm.

Fig. 4.

Expression of a tip-specific antigen (Pg101) in a large whorl at an early stage. (A) Bright-field image focused on the sides of the whorl. (B–E) Distribution of tip antigen at 90° angles around the whorl. (A and B are at 0°). The plane of focus was the whorl’s top surface. Antigen appears distributed over the whorl’s surface in no apparent pattern. Bar, 50 μm.

(i) Stripes

Striped patterns in large whorls were bands of relatively intense PglOl expression alternating with bands of less intense PglOl expression (Fig. 5). Staining within a band was not always uniform, but the average level of PglOl in bright versus dark bands was discernibly different. In this way, individual stripes on large whorls resembled the single equatorial band of staining observed as the first patterned expression of PglOl in whorls of normal size (Byrne & Cox, 1986). Multi - striped patterns were only observed in large elongated whorls, not large spherical ones.

Fig. 5.

Horizontal striped patterns of tip-specific antigen expression on large whorls. (The faint vertical stripes evident in some panels were artifactual remnants of the montaging process.) Specimens were fixed prior to definitive tip formation. The images were obtained and are presented as in Fig. 3, except that arrows were omitted due to the lack of clear-cut landmarks at either end of the rotation. Note that most stripes were correlated with bulges in the whorl, suggesting that some of these whorls may be precursors to segmented whorls (e.g. Fig. 2C). Other striped whorls may give rise to irregular lattice patterns at later stages, as suggested by the apparent fragmentation of certain bands (e.g. in C) into discrete patches. Bar, 50 μm.

Fig. 5.

Horizontal striped patterns of tip-specific antigen expression on large whorls. (The faint vertical stripes evident in some panels were artifactual remnants of the montaging process.) Specimens were fixed prior to definitive tip formation. The images were obtained and are presented as in Fig. 3, except that arrows were omitted due to the lack of clear-cut landmarks at either end of the rotation. Note that most stripes were correlated with bulges in the whorl, suggesting that some of these whorls may be precursors to segmented whorls (e.g. Fig. 2C). Other striped whorls may give rise to irregular lattice patterns at later stages, as suggested by the apparent fragmentation of certain bands (e.g. in C) into discrete patches. Bar, 50 μm.

The bands of antigen expression were always perpen-dicular to the central stalk, and were usually associated with bulges in the whorl. Since striped patterns were relatively rare, we have been unable to determine if stripe spacing is approximately constant and how it might be related to intertip spacing.

We also observed some large, elongate whorls in which the bright bands of the striped pattern were partially disrupted by distinct clusters of more intense antigen expression (Fig. 5C). In some of these whorls, stripes were apparent near one pole, while near the other pole rows of equally spaced antigen clusters were found (data not shown).

(ii) Spots

Most large whorls at early stages, before the emergence of visible tips, displayed somewhat regularly spaced spots or clusters of PglOl expression (Fig. 6). Individual clusters resembled those observed around the equator of whorls of normal size just prior to visible tip formation (Byrne & Cox, 1986, 1987). In large whorls, these clusters were distributed over the entire whorl surface in irregular lattice-like patterns apparently foreshadowing the types of final tip-distribution patterns observed at later stages.

Fig. 6.

Early spotted patterns of tip-specific antigen expression on large whorls. As in Fig. 5, specimens were fixed prior to definitive tip formation. The images were obtained and are presented as in Fig. 3. Antigen clusters are not confined to the equator. Such whorls are probably precursors to the tip patterns observed at later stages. Bar, 50 μm.

Fig. 6.

Early spotted patterns of tip-specific antigen expression on large whorls. As in Fig. 5, specimens were fixed prior to definitive tip formation. The images were obtained and are presented as in Fig. 3. Antigen clusters are not confined to the equator. Such whorls are probably precursors to the tip patterns observed at later stages. Bar, 50 μm.

We have found that whorls of different shapes and sizes express a broad spectrum of different patterns. The observed patterns vary from equally spaced equatorial tips on normal whorls, to fairly regular arrays of tips distributed over the surface of larger than normal whorls, to multistriped patterns on large, elongate whorls. The patterns on large whorls arise under normal growth conditions in wild-type strains, and appear on fruiting bodies in which most other whorls pattern normally. This demonstrates that whorl shape and size are themselves significant factors in pattern determination in P. pallidum.

By considering the repertoire of patterns expressed by whorls, we can ask which theories of pattern formation can account simply and economically for our observations. Any version of Claxton’s (1964) model for hair-follicle patterns in sheep is incapable of producing striped patterns in P. pallidum, since stripe formation would require the simultaneous, random appearance of an entire line of tip primordia. The probability of such an occurrence is negligible. Goodwin & Cohen’s (1969) model could produce stripes if a pair of travelling waves were emitted at one pole of a whorl. The phase difference between the waves would specify latitudinal position. As noted above, however, another wave-pair propagating around the equator must be hypothesized to account for regularly spaced tip patterns. As also discussed above, both of these models incorrectly predict that tips should appear sequentially.

A theory that at once generates both striped and spotted patterns, and predicts that a particular pattern emerges simultaneously over the domain has been proposed by Turing (1952). Turing’s scheme is based on the reaction and diffusion of two morphogens. One is autocatalytic and diffuses slowly, while the second morphogen diffuses more rapidly and inhibits production of the first. These properties lead to formation of periodic patterns of morphogen concentrations.

Turing’s mathematical analysis dealt with the properties of linear systems, and focused on reaction and diffusion in a ring of cells. In this one-dimensional problem, the morphogen pattern that emerges corresponds to a sine wave of a particular wavelength. It is the exponential growth in the amplitude of the entire sine-wave pattern that leads to a simultaneous emergence of evenly spaced morphogen peaks. Turing also showed that, because of the change in geometry, the characteristic patterns expected to form on a sphere’s surface are surface-spherical harmonics, which are quite different from sine waves (see the Appendix).

Since whorls are roughly spherical, we examined the spherical-harmonic patterns predicted by Turing’s theory for this geometry. The larger whorls observed in this study were sometimes elongated, and so we also examined patterns predicted by Turing’s theory for a prolate spheroid. Since previous studies (Byrne & Cox, 1986; McNally et al. 1987) have suggested that patterning in a whorl is restricted to its surface, we have examined the theoretical predictions for a spherical surface.

Figs 7 and 8 provide a representative sampling of the kinds of patterns that can be generated by Turing’s theory when applied to a spherical or spheroidal surface. These patterns should be compared to Figs 16. Several important points are illustrated in Figs 7 and 8.

Fig. 7.

Spherical and spheroidal harmonic patterns. The value of the function on the surface is indicated by grey level intensity, with black corresponding to the minimum and white the maximum*. The grey scale is nonlinear with a relatively sharp descent from white to black. Spherical and spheroidal harmonic functions are specified by two integer indices, m and n. (A) Spherical harmonic m = 0, n = 2. (B) Mercator projection of A. The abscissa varies from 0–360° and the ordinate varies from the south pole (180°) to the north pole (0°). (C) Spherical harmonic m = 6, n = 6. (D) Mercator projection of C. The patterns in A and C resemble the patterns of tip-specific antigen expression in whorls of normal size (Byrne & Cox, 1986). (E) Mercator projection of spherical harmonic m = 4, n = 5. The pattern is a zigzag about the equator. Compare to Fig. 3A. (F) Mercator projection of spherical harmonic m = 5, n = 7, forming a three-tiered checkerboard. Compare to Fig. 3B. (G) Mercator projection of spherical harmonic m = 0, n = 6, forming three stripes. Compare to Fig. 5B. (H) Spheroidal harmonic m = 6, n = 12. The ratio of the major to the minor axis of the ellipse is 5·0. Compare to Fig. 3C. (I) Spheroidal harmonic m = 0, n = 10. The ellipse axis ratio is 10·0. Compare to Fig. 5D.

*The opposite scheme is used for the striped patterns in A,B,G,I, which is equivalent to plotting the harmonic functions multiplied by -10. According to theory, either positive or negative harmonic patterns can emerge.

Fig. 7.

Spherical and spheroidal harmonic patterns. The value of the function on the surface is indicated by grey level intensity, with black corresponding to the minimum and white the maximum*. The grey scale is nonlinear with a relatively sharp descent from white to black. Spherical and spheroidal harmonic functions are specified by two integer indices, m and n. (A) Spherical harmonic m = 0, n = 2. (B) Mercator projection of A. The abscissa varies from 0–360° and the ordinate varies from the south pole (180°) to the north pole (0°). (C) Spherical harmonic m = 6, n = 6. (D) Mercator projection of C. The patterns in A and C resemble the patterns of tip-specific antigen expression in whorls of normal size (Byrne & Cox, 1986). (E) Mercator projection of spherical harmonic m = 4, n = 5. The pattern is a zigzag about the equator. Compare to Fig. 3A. (F) Mercator projection of spherical harmonic m = 5, n = 7, forming a three-tiered checkerboard. Compare to Fig. 3B. (G) Mercator projection of spherical harmonic m = 0, n = 6, forming three stripes. Compare to Fig. 5B. (H) Spheroidal harmonic m = 6, n = 12. The ratio of the major to the minor axis of the ellipse is 5·0. Compare to Fig. 3C. (I) Spheroidal harmonic m = 0, n = 10. The ellipse axis ratio is 10·0. Compare to Fig. 5D.

*The opposite scheme is used for the striped patterns in A,B,G,I, which is equivalent to plotting the harmonic functions multiplied by -10. According to theory, either positive or negative harmonic patterns can emerge.

Fig. 8.

Stereopair topographic-map representation of spherical and spheroidal harmonic functions. The surface has been deformed in proportion to the magnitude of the function there. The direction of deformation is along radial lines for the sphere (r in spherical coordinates), and along confocal hyperbolae for the spheroid ( η in prolate spheroidal coordinates). The amount of deformation is some fraction w of the sphere’s original radius, or the spheroid’s original ellipsoidal surface constant ξ = (τ2/(τ2 – 1))12, where τ is the ratio of the major to minor axis. The deformed shapes have been tilted and rotated to facilitate comparison to the scanning electron micrographs of Fig. 1. (A) Sphere slightly deformed (w = 0·025) by the m = 4, n = 4 spherical harmonic. Compare to Fig. 1B. (B) Greater deformation (w = 0·2) of a sphere by the same function as in A. Compare to Fig. 1C. (C) Deformation (w = 0·2) of a sphere by the spherical harmonic m = 6, n = 7 (a zigzag pattern). Compare to Fig. 1D. (D) Deformation (w = 0·025) of a spheroid by the spheroidal harmonic function m = 5, n = 6 (a zigzag pattern on a spheroid). The axis ratio is 2·0. Compare to Fig. 2A. (E) Deformation (tv = 0·0075) of a spheroid by the spheroidal harmonic m = 0, n = 10. The axis ratio is 10·0. This is the same function plotted in Fig. 6 with grey levels (The negative harmonic is also plotted here. See Fig. 7 footnote.) Compare to Fig. 2C.

Fig. 8.

Stereopair topographic-map representation of spherical and spheroidal harmonic functions. The surface has been deformed in proportion to the magnitude of the function there. The direction of deformation is along radial lines for the sphere (r in spherical coordinates), and along confocal hyperbolae for the spheroid ( η in prolate spheroidal coordinates). The amount of deformation is some fraction w of the sphere’s original radius, or the spheroid’s original ellipsoidal surface constant ξ = (τ2/(τ2 – 1))12, where τ is the ratio of the major to minor axis. The deformed shapes have been tilted and rotated to facilitate comparison to the scanning electron micrographs of Fig. 1. (A) Sphere slightly deformed (w = 0·025) by the m = 4, n = 4 spherical harmonic. Compare to Fig. 1B. (B) Greater deformation (w = 0·2) of a sphere by the same function as in A. Compare to Fig. 1C. (C) Deformation (w = 0·2) of a sphere by the spherical harmonic m = 6, n = 7 (a zigzag pattern). Compare to Fig. 1D. (D) Deformation (w = 0·025) of a spheroid by the spheroidal harmonic function m = 5, n = 6 (a zigzag pattern on a spheroid). The axis ratio is 2·0. Compare to Fig. 2A. (E) Deformation (tv = 0·0075) of a spheroid by the spheroidal harmonic m = 0, n = 10. The axis ratio is 10·0. This is the same function plotted in Fig. 6 with grey levels (The negative harmonic is also plotted here. See Fig. 7 footnote.) Compare to Fig. 2C.

First, the spheroidal patterns are essentially stretched versions of the spherical patterns. Second, both striped and spotted patterns are possible, and any number of stripes or spots can form. Third, for a particular number of stripes or spots, the possible patterns are limited. For example, there are only three spherical harmonic patterns consisting of six spots. One of these three patterns is composed of spots that are spaced equally around the equator (Fig. 7C,D). This is the pattern seen in whorls forming six tips. Thus, the similarities between our results and theory cannot simply be due to a chance correspondence between the observed patterns and an unlimited repertoire of harmonic functions. Fourth, besides the equatorial spotted pattern, spots can form in zigzag and checkerboard arrays (Fig. 7E,F,H). These patterns resemble the spotted patterns observed on larger whorls (Fig. 3), although the theoretically predicted patterns are clearly more regular. Fifth, the striped spherical harmonic patterns (Fig. 7B,G,I) are similar to the single-striped pattern observed during pattern genesis in normal whorls (Byrne & Cox, 1986) and the multistriped patterns observed in larger whorls in this study (Fig. 5). Note, however, that the striped patterns in whorls are always orthogonal to the stalk axis, while the harmonic patterns can form about any axis. It is possible that in whorls some influence of the stalk fixes the orientation of whorl patterns (Byrne & Cox, 1986; McNally & Cox, 1988).

While there appear to be many similarities between the patterns predicted by Turing’s theory and those observed in whorls, there are some discrepancies. The spotted arrays present on large whorls, although not random, were not as regular or ordered as the harmonic patterns. The striped patterns that we observed in whorls appeared to be transitional, eventually dispersing into spots. Turing’s theory cannot explain this transition. Neither can the theory fully explain why

certain patterns such as equatorial spots should ordinarily be favoured over some of the zigzag and checkerboard patterns of Figs 7 and 8.

How might one account for these deviations from the linear theory? As Gierer & Meinhardt (1972) have shown, nonlinear reaction-diffusion equations can give rise to imperfect spacing in spotted patterns, and under certain conditions striped patterns can form first and then disperse into spots (Meinhardt, 1988). Nonlinear effects can also be responsible for selecting certain patterns over others, as occurs in Rayleigh-Bénard convection (Golubitsky et al. 1984). Since we know little of the mechanics of the patterning apparatus in P. pallidum, we cannot at this time determine the exact form of these nonlinear effects, and whether they could account explicitly for the deviations from the linear theory that we have observed.

It should be emphasized that there is a close relationship between linear and nonlinear reaction-diffusion theories. The patterns generated by nonlinear theories are modifications of those predicted by Turing’s linear theory (see for instance Segel (1984)). It can be shown that the nonlinear terms do give rise to unequal spacing in spotted patterns, and that these same terms can favour the stability of one pattern over another.

Several other theories of patterning are Turing-like in that they too predict, when nonlinear effects are ignored, that harmonic functions underlie patterns (Keller & Segel, 1970; Goodwin & Trainor, 1983; Oster et al. 1983; Cummings, 1985). Some of these latter theories while mathematically similar to reaction-diffusion theories are based on rather different cellular properties such as adhesion and motility. In isolation, the results of the present study cannot distinguish among these various Turing-like theories. However, other data can. At this time, we favour a role for reaction and diffusion in whorl patterning based on two different lines of evidence. Turing-like theories which posit a role for cell motility (Keller & Segel, 1970; Oster et al. 1983) demand that morphogenesis accompany pattern genesis. In whorls, we see marked changes in tip-antigen distributions without obvious concomitant changes in whorl shape. This observation tends to rule out cell-motility models, although more detailed studies of cell motion in a whorl during pattern genesis are needed. A second line of evidence that supports a role for reaction and diffusion is that a number of diffusible molecules, including DIF, cAMP, adenosine and ammonia, are known to be involved in slime-mould differentiation (Williams, 1988). Further, in both P. pallidum (Byrne et al. 1982) and in the related cellular slime mould Dictyostelium discoideum (Kopachik, 1982) there is evidence for a diffusible tip inhibitor. Indeed, Gross et al. (1983) have hypothesized that DIF and ammonia are the activator and inhibitor in a reaction-diffusion mechanism that establishes the prestalk prespore pattern in D. discoideum. In P. pallidum, cAMP is known to induce stalk cell formation (Hohl et al. 1977), but the effects of the other putative D. discoideum morphogens are unknown.

Our approach of examining perturbed patterns to expose a range of ordinarily unexpressed patterns could find broader application in testing various theories of pattern formation. For example, a correlation between the harmonic patterns predicted by Turing’s theory and the normal patterns expressed by certain organisms has been noted before (Kauffman et al. 1978; Hunding, 1981; Murray, 1981; Goodwin & Trainor, 1983). Specifically, striped patterns in zebras (Murray, 1981), fruit flies (Kauffman et al. 1978; Russell, 1985) and feather stars (Lacalli & West, 1986) have been attributed to underlying harmonic functions. To a degree, an experiment comparable to that reported here has been done naturally in cheetah and jaguar tails where the striped coat pattern changes to spots as the tail broadens (Murray, 1981). Similarly, in Drosophila, mutants that produce checkerboard-like spot patterns at the cellular blastoderm stage might be expected if the normal striped pattern is generated by harmonic functions.

The results presented here plus those from previous studies (Byrne & Cox, 1986,1987) show that the origins of a simple spatial pattern in P. pallidum conform closely to the expectations of Turing’s (1952) proposal: the pattern arises first over the entire surface of the patterning domain, the kinetic intermediates on the way to the mature pattern are those predicted, and the intermediate and equilibrium patterns are sensitive to the geometry of the organism. Other studies in a variety of organisms have noted the close correspondence between the mature pattern and the predictions of Turing-like theories, but our results supply direct experimental evidence that the correspondence in P. pallidum is far more extensive and therefore much less likely to be fortuitous.

We are indebted to Axel Hunding for computing the data used to plot the spheroidal harmonic functions. We are grateful to David Laur and Kirk Alexander for much assistance with the computer graphics. We thank Gerry Byrne for advice on the capillary rotation technique and the digitization of the resultant montages. This work was supported by an American Cancer Society fellowship to JGM and research grants to ECC from the Whitehall Foundation and the National Science Foundation.

Turing’s (1952) linear reaction-diffusion equations are:

formula

where x and y are morphogen concentrations about equilibrium, a, b, c, and d are chemical rate constants, and Dx and Dy are the diffusion constants for x and y, respectively. ∇2 is the Laplacian operator and represents diffusion.

It can be shown that solutions to (1) take the form eλt H where H is a harmonic function. Such solutions will grow or decay depending on the sign of λ. The central concept underlying reaction-diffusion theory is that λ should be positive for (ideally) one harmonic function and negative for all others, so that just one pattern is amplified. This is how all Turing-like theories generate patterns.

Harmonic functions are solutions to the equation

formula

Each harmonic function Hk has associated with it a parameter k which is related to the spacing between peaks in Hk. The operator ∇2 contains all the spatial information in the problem, and so depends critically on geometry. By solving (2) for a spherical geometry, one obtains solutions known as spherical harmonics. Since whorl patterning is apparently restricted to the surface, we have ignored the radial dependence of these harmonics and concentrated on the surface dependence expressed in terms of the standard spherical angles (θ, ϕ). Surface-spherical harmonics are given by

formula

The functions are the associated Legendre poly nomials which can be calculated by using various recursion formulas. The formula used to generate the functions plotted here was

formula

The indices m and n are integers (with |m| ≤ n) which specify particular spherical-harmonic functions. When m = 0 striped patterns are generated; when m = n equatorial-spotted patterns are generated; for all other m,n the patterns are checkerboards with m columns and n − m + 1 rows.

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