Dislocation is a developmental mechanism in Dictyostelium and vertebrates Antony J. Durston1 Molecular Cell Biology, Sylvius Laboratory, Institute of Biology, University of Leiden, Leiden 2333 BE, The Netherlands Edited* by Morrel H. Cohen, Rutgers, The State University of New Jersey, Bridgewater Township, NJ, and approved August 29, 2013 (received for review February 28, 2013)

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he organism Dictyostelium discoideum has attracted eminent mathematicians, theoretical physicists, and computer scientists to biology. Pioneering observations by John Bonner, Theo Konijn, and their colleagues (1, 2), and by Brian Shaffer (3), respectively, in the 1960s and 1970s demonstrated that chemotaxis to cAMP and cAMP signal relay, respectively, are the key cell properties that mediate aggregation in this organism. These early insights have only recently fully borne fruit. The complex relayed waves of cAMP signaling and chemotaxis generated by these two simple properties mean that a mass of D. discoideum cells behaves as an isotropic excitable medium (4) and generates a range of complex and beautiful behaviors and structures: notably, structures based on scroll waves. It was thought initially that these waves mediated only the first (aggregation) stage of D. discoideum development, but it became evident that they organize morphogenesis of all multicellular stages (5–8). The waveforms are mathematically predictable and the structures they organize presumably are as well (Fig. 1). The prevalent waveform is the Archimedean spiral, a form that can arise when wavefronts are broken; its 3D manifestation is the scroll wave (6, 8) (Fig. 1 A–C). Investigation of these waves reveals how two simple cellular properties can generate very complex morphogenesis. This article explores unique inferences about an aspect of this process, reasoning that late D. discoideum aggregates manifest dislocation: The spontaneous functional separation of a wave field with smoothly varying properties into two or more connected domains. These may or may not be domains that support the propagation of distinct waveforms. A single homogeneous territory breaks up spontaneously into two or more, a topological bifurcation. The domains remain connected by function and sometimes by wave propagation. We can expect different behaviors and different morphogenetic consequences in the different domains. This article shows below that this is a developmental mechanism in D. discoideum. The concern herein is the evolution of an original D. discoideum scroll or concentric wave domain to connected scroll prestalk and concentric prespore subdomains, but dislocation may be relevant for other cases. It is interesting that exactly the same concept proposed herein from experimental observations in D. discoideum was deduced previously from theoretical considerations, indicating that a wave field containing a frequency gradient can www.pnas.org/cgi/doi/10.1073/pnas.1300236110

disintegrate, generating subdomains entrained to pacemakers with different frequencies (P 59 in ref. 9). This type of process is therefore likely to have general importance for morphogenesis. The article also identifies an example of dislocation in vertebrate somitogenesis, below. Results and Ideas from Weijer and Others Many observations point to the idea that 3D spiral (scroll in Fig. 1D) waves of cAMP signaling are important throughout later D. discoideum morphogenesis. All later structures are based on a cylindrical plan, as would be expected if scroll waves were the basic regulators. Scroll waves can be inferred directly during very late aggregation, where torus scroll rings can evidently organize ring-shaped aggregates containing clockwise or anticlockwise rotating cells (6, 7, 10) (Fig. 1E). These waves are also evident at the Mexican hat stage of culmination, where a vertical scroll wave is always detectable (7, 11) The most important questions, however, concern the cylindrical slug stage, where the slime mold’s regulative prestalk/prespore pattern of differentiation becomes evident. Siegert and Weijer (8) have shown that D. discoideum slug tips show rotational cell movements, indicating a scroll wave, in the prestalk region. Durston et al. (6, 7) and Siegert and Weijer (8) observed that a D. discoideum slug’s prespore zone shows synchronized forward pulsatile periodic movements, presumably reflecting backward plane waves. Weijer’s group (12) have also since been able to visualize posterior plane optical density waves directly in the prespore zone. The anterior scroll wave still defies direct visualization. Siegert and Weijer propose that the D. discoideum slug is organized by a twisted scroll wave (8) (Fig. 1F). The authors showed that Dictyostelium mucoroides slugs do actually show what appear to be helical cell movements (12), direct evidence possibly Significance One of the mechanisms generating structure during embryonic development is embryonic cells propagating periodic waves of activity, becoming an excitable medium. We examined this phenomenon in Dictyostelium discoidem, a simple organism with embryo-like development, and identified a unique phenomenon: a wave field that initially controls a whole population of Dictyostelium cell splits. The cell’s central portion comes to contain an Archimedean spiral/scroll wave and its peripheral portion contains concentric/target waves, because the central aggregate acts as an integral pacemaker. This mechanism (dislocation: the splitting of an embryonic wave field into subfields) was predicted previously; we also identified an example of it in a metazoan development (vertebrate somitogenesis). We have confidence we are dealing with a general mechanism. Author contributions: A.J.D. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper. The author declares no conflict of interest. *This Direct Submission article had a prearranged editor. 1

E-mail: [email protected].

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The excitable cells of Dictyostelium discoideum show traveling waves of signaling and generate a variety of complex wave forms during their morphogenesis. Important among these wave forms is the 3D spiral or scroll wave, which has been proposed previously to have a twisted variant: the “turbine wave.” Herein we argue that a D. discoideum scroll or concentric wave territory containing prespore and prestalk cell types can undergo “dislocation”: a wave field that initially controls aggregation of a whole developing population of Dictyostelium cells splits into two. This process leads to discontinuity between two connected domains of wave propagation and to specific phenomena, including high-frequency concentric pacemaker activity by the slime mold’s scroll-wave tip. The resulting morphogenetic events reveal a unique mechanism in morphogenesis.

Fig. 1. Waveforms in D. discoideum and in Beloussov–Zhabotinsky reagent, a chemical excitable medium, to illustrate the waveforms made in simple excitable media. (A) Concentric ring waves in Zhabotinsky reagent. The waves spread outward from a periodic point source pacemaker. New waves are initiated at regular intervals, leading to a target pattern of regularly spaced waves. (B) Zhabotinsky spirals. When a target ring is broken by shaking the medium, the wave ends propagate around to the back of their own refractory zone. They continue to curl around as the wave propagates out, generating a self-sustaining involute spiral that runs at the minimum period. (C) Spiral and concentric waves in an early D. discoideum aggregation field. The spirals can be high frequency, leading to closely packed waves. The concentrics are lower frequency. (Magnification: A and B, 1–10×; C, 10×.) (D) Spirals become scroll waves in three dimensions. Some of these scroll waves are bent and can also have their ends joined in torus rings. (E) A torus ring scroll wave in a late D. discoideum aggregate. The aggregating cells move from the inside outward over the top of the ring shaped aggregate. (F) Proposed scroll waves in slugs. (Lower) A D. mucoroides slug is proposed to contain a twisted scroll wave. (Upper) A D. discideum slug is known to contain a scroll wave in its tip and plane waves in its back part.

suggesting a twisted scroll wave in this species. In 1993, Steinbock et al. (13) made a mathematical model proposing that a D. discoideum slug contains a twisted scroll wave. This is a scroll in the prestalk zone and is twisted so that it generates plane waves in the prespore zone. The twisted scroll wave idea is very significant because it can account for how the slug migrates forward (enabled by the plane prespore waves) while being driven by a rotational high-frequency signal source in the tip (the scroll wave). Twisted scroll waves can translate rotational to linear movement. This principle (and the reverse) was discovered by Archimedes. The wave can pump water or move slug cells and has been massively exploited for many important engineering inventions, from jet turbine engines to hydroelectric power stations. We suspect, however, that this is not the whole story and that reality is more complicated. Herein, we address the question of how D. discoideum slug wave forms arise. When Are the Slug’s Wave Geometries Generated? Various observations indicate that the D. discoideum slug’s waveforms arise during late aggregation (the mound stage). The first D. discoideum aggregates generally have 2D concentric waves with low frequency (first 10 min and then 5 min), because of a 5-min pacemaker that is transiently gated by a decreasing refractory period for signal relay (14, 15) during early aggregation (Fig. 2A). There are other findings in the literature that, like this one, indicate the existence of different oscillators in aggregation stage D. discoideum cells (16). Other early aggregates are organized by spiral waves (that can be initiated by wave breakage because of inhomogeneities in the aggregation field, with an 2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1300236110

initial irregular period of around 10 min and decreasing to as little as 2.5 min.) (Fig. 2B). These waves become very shallow vertical scroll waves as the aggregates become several cell layers deep (mound stage). We reason that spirals will gravitate to the refractory period of the aggregating cells and it is known that the refractory period decreases with time (4, 15). In this study, all aggregates then made concentric waves during late aggregation (mound stage) (Figs. 2–4). These late concentric waves were initiated at the aggregate’s tip, which formed in the apex of the late aggregate and later became the tip of the slug. These waves had a shorter period (∼2–2.5 min.) than the previous spiral or concentric waves (15). We have reasoned that they run at the minimum (refractory) period and thought that this is because the tip acts as an integral continuous or very high-frequency source of signal (15). Our reason for thinking this was that a continuous or very high-frequency (1-min period) point source of cAMP (cAMP microelectrode) inserted into a field of late-aggregating cells initiated waves with the same geometry and frequency as the tip, although it did not initiate a tip (14). Originally spiral aggregates thus underwent a late spiral to concentric switch. These late concentric waves were initially generated from a small tight aggregate of cells that bobbed around in the looser, more fluid late-aggregation mound, and whirled/rotated (span) and clearly dislocated from the center of the mound. The existence of such a separate aggregate has been confirmed by later studies (see ref. 17 and below). This tight aggregate grew to become the morphologically distinct tip of the late aggregate and slug (Figs. 2–4). It is clear that this late aggregate was the precursor of the

Fig. 2. Dislocation: Observations that argue against twisted scroll waves and for dislocation during tip formation in D. discoideum. (A, i) Concentric ring waves of relaying initiated by a point source pacemaker in an early D. discoideum NC4 aggregation field . Other early NC4 aggregates are initially controlled by a spiral wave (see B). At this stage of development, we are dealing with very shallow 3D waves, which are to a first approximation, effectively 2D. For concentric centers: these are the relevant sections of expanding spherical waves. (A, ii–v) The same aggregate has later started to form a tip. A very small aggregate of cells (outlined by the blue disk) has sorted out to form an early stage tip at the late aggregation/mound stage. This aggregate rotates (black curved arrows); it thus presumably contains a spiral wave. The aggregate none-the-less initiates high-frequency concentric ring target waves of relaying in the main mound. There is thus a switch from low-frequency concentric to high-frequency concentric waves of relaying associated with tip formation in such aggregates. We suggest the tip aggregate acts as an integral continuous signal source. It is known that an artificial continuous point source of cAMP initiates high-frequency concentric waves of relaying in late aggregates. These observations argue for dislocation. They do not seem consistent with a twisted scroll wave: we never saw any sign of a scroll wave outside the tip in originally concentric aggregates during tip formation. (B, i) An early spiral aggregate. In three dimensions, this is a very shallow vertical 3D spiral (scroll) wave. (B, ii–v) Tip formation in this originally spiral aggregate. This process occurs exactly as in the concentric aggregate above. A small rotating mass of cells initiates very high-frequency concentric waves. This aggregate therefore undergoes a spiral to concentric switch.

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slug’s definitive tip/prestalk zone. In addition, the fact that the early tip span indicates that it already contained a spiral/3D scroll wave, as Siegert and Weijer (8) showed later in the slug tip. The prespore and prestalk zones are known to be initiated during late aggregation (17, 18) by chemotactic sorting of presumptive prestalk and prespore cells (19), which tend to differentiate from different cell cycle phases during previous starvation (20, 21). We believe we were witnessing this process and that the late aggregate tip initiates the prestalk zone. Because prestalk cells have a higher oscillation frequency than prespore cells (22), we are not surprised by the observation that the tip initiates maximum-frequency waves; it is a maximum-frequency signal source. The late aggregate’s concentric waves were clearly homologs of the later backward plane waves in the slug’s prespore region and our observations clearly also indicated that the aggregate tip is a maximum-frequency concentric pacemaker, even though it contains a scroll wave. The slug’s later prestalk/prespore pattern and scroll wave/plane wave geometry are possibly thus generated by cell sorting during aggregation. Our observations also indicated that cell sorting and spiral prestalk aggregation are reproduced during regulation of prespore pieces of slugs (6, 7). The mechanism of all this is unclear. A frequency difference between prestalk and prespore cells is likely to be important (22, 23), but our observations also suggested that cell contacts could be important. Our observations were confirmed by a study by Nicol et al. (17) (Fig. 3), which shows that sorting of prestalk and prespore cells begins in late aggregation, at the mound stage, and is finished by the slug stage. This study was of aggregation and cell sorting in a thin layer of cells, under agarose. The authors’ findings match some of ours above The whole aggregate becomes a whirling aggregate of cells, clearly organized by a spiral wave, and prestalk cells sort out from this to form a separate aggregate within the main aggregate The prestalk aggregate transiently stops whirling but later also transiently whirls again (becomes a spiral); it can have either the same or opposite (clockwise/anticlockwise) polarity as the main spiral (Fig. 3), and it is thus clearly separate. The prestalk aggregate then moves to the edge of the agarose and the prespore cells follow it (presumably this represents the concentric phase of wave propagation). The aggregate thus becomes a slug. Other studies have shown a rather complex picture with strain-dependent concentric Durston

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Fig. 3. Tip formation in 2D aggregates. (iii and iv) Nicol et al. (17) studied tip formation in originally spiral aggregates. The authors saw small prestalk aggregates transiently rotating within rotating main spiral aggregates. The rotation sense of the prestalk aggregates was sometimes the same as (ii) but equally often opposite (iii) to that of the main aggregate. This finding presumably indicates independent corotating or counterrotating spirals and demonstrates dislocation. The observation of counterrotating spirals rules out twisted scroll waves in these aggregates. After this phase, rotation of the aggregate stops and all cells follow the tip. Presumably this corresponds to the concentric wave propagation phase. i and ii show our observation of tip formation in an originally concentric aggregate for comparison.

and spiral waves at the mound stage, but do not yet give a clear picture of the tip’s role (24–26). At any rate, we can say (from our own work) that the late aggregate’s tip can act as a concentric pacemaker, even though it evidently contains a 3D spiral. The agarose technique has been criticized but what was seen by Nicol et al. (17) here is exactly in line with J. T. Bonner’s parallel studies (27). Bonner studied slug and fruiting body formation in small slugs that develop in thin layers of D. discoideum cells under mineral oil. His findings, summarized in his recent book (27), show that thin layers of D. discoideum cells can make motile slugs with or without scroll waves. The tip can transiently be represented by a whirlpool of cells (the 2D equivalent of a scroll) but can sometimes show disorganized active behavior. Bonner proposes that rotational movements in the tip simply accommodate the hyperactivity of the tip’s cells but are not required for slug movement. It thus appears there are various scenarios that can underly slug movement. It has also been observed that cells in slugs are often joined end to end, in strings, presumably by A contact sites. These contacts may play a synaptic role in signaling. It appears that when a tip acts as a continuous source, the contacts between the tip and

Fig. 4. What happens in aggregates that are making a tip? (i–iii) At certain points in the aggregate, waves from the original pacemaker and the new tip collide. When this happens, the colliding waves, which are signaling fronts, are extinguished because they are each followed by a refractory zone, which stops wave propagation. The panels show four (i–iv) steps in this process. Red ovals, signaling cells in the wavefront; black discs, refractory cells; blue discs, sensitive cells that can receive a signal. This process works on a cycle (vi). Sensitive cells are signaled to become signaling cells (signal relay) and they then become refractory for a defined time (refractory period) before becoming sensitive again. Because waves extinguish each other when they collide, a higher-frequency pacemaker (like the tip), which makes more waves, will compete successfully with a neighboring lower-frequency pacemaker (like the original center). It will take territory from the lowerfrequency pacemaker and eventually annihilate it (v). The high-frequency/continuous tip thus takes territory from and then destroys lowerfrequency early concentric and spiral centers. It is not at all surprising that the tip takes over from early low-frequency periodic concentric centers. It is more surprising that the tip takes over from early spirals, which are thought to soon run at the minimum refractory period of the aggregating cells. The answer presumably lies with two aspects: the cellular refractory period changes (decreases) with developmental time; and the cells that sort out into the tip are prestalk cells. These prestalk cells have a shorter refractory period than the remaining cells. The signal from the tip is thus higher frequency than the rest of the aggregate can produce. At the late aggregation stage, when these things happen, the aggregates are very shallow, of an order of one wavelength. This means that all of the phenomena above occur in very shallow sections of 3D concentric waves or very shallow vertical scroll waves; they are essentially 2D and we treat them as such.

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Fig. 5. A situation that never happens. We show a slug organized by an integral scroll wave, which—as we have argued above—never forms. We propose that even if such an originally spiral (scroll) aggregate could reach the slug stage in this form, it would then most likely dislocate. The higherfrequency prestalk cells (Left) have sorted out from and abut the lowerfrequency prespore cells (Right). The drawing shows two abutting cylinders with a narrow transitional zone between them (drawn in blue): these represent the prestalk (Left) and prespore zones (Right). The cylinders are shown as lying structures for convenience. The external excitation front of the scroll wave is represented by a red line in each cylinder. (i) The two scrolls are in phase. (ii) After one rotation of the prespore scroll, they are 30% out of phase because of a 30% frequency difference between the two zones. The two scrolls remain connected because of cross-signaling between neighboring cells. (iii) After three rotations of the prespore scroll, they are almost 100% phase-shifted: nearly 360° out of phase or actually back in phase. The two scrolls do not reconnect in phase, however, because their wave ends have stayed connected during the previous phase shift and this prevents them from reconnecting. This process will be repeated at least 10 times, leading to at least 10 closely packed coils of the wavefront. The consequence of this situation would most probably be dislocation: that the twisted scroll wave breaks and the higher-frequency prestalk scroll acts as an integral continuous signal source. This result is what is observed. The figure shows prestalk and prespore zones in a slug as two abutting cylinders: prestalk (Left), prespore (Right). Between them is a narrow transitional zone. Red line: the external signaling front of the scroll waves. (i) This is initially an integral scroll wave. (ii) The prestalk cylinder has advanced in phase (anticlockwise) by about 120°. (iii) The prestalk cylinder shows a further phase advance to about 360°. The signaling front stays connected so far.

the rest of the slug are broken so that the tip can act as an integral rather than a localized signal source. Could a D. discoideum Aggregate Develop a Twisted Scroll Wave? The observations above raise some questions. Are any of the situations above accounted for by Siegert and Weijer (8) twisted scroll waves? We never saw any experimental evidence for a twisted scroll in D. discoideum in aggregates transiting from early spiral waves to tip concentric waves, but could not absolutely rule out that we have missed them in these transitions from early spiral aggregates to tip concentric. In the transitions from early concentric aggregates to tip concentric, and in the tips that arose as counterrotating whirls of cells within a whirling aggregate in Nicol et al.’s (17) experiments, twisted scrolls are ruled out (Fig. 3). All of the observations point to dislocation and not to twisted scroll waves. There is also no experimental evidence for twisted scroll waves in D. discoideum. The waves have been proposed in models and there is an observation of helical cell movement that may support this notion in D. mucoroides. Is it, however, possible that late aggregates and slugs sometimes make an integrated (but twisted) prestalk/prespore spiral/scroll? Neither we nor anyone else has ever observed this, and there is an argument against it. Siegert and Weijer (8) argue that a slug’s scroll wave becomes twisted because of a difference in excitability between the prestalk and prespore regions, and that this twist leads to the plane waves in the prespore zone of the slug. The first excitability 4 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1300236110

difference that was predicted and published is that cell density classes corresponding to early prestalk and prespore cells have different oscillation frequencies during late aggregation, and thus presumably different refractory periods/adaptation periods (28). We have argued that this would already lead to prestalk/ prespore dislocation during aggregation (as above) (Figs. 2–4). The slug scroll’s external wavefront (the “red wire” in Fig. 5) will predictably become twisted as shown, because the prestalk block and prespore blocks rotate at different rates as a result of having different oscillator frequencies. The wavefront would bend but tend to stay intact, a result of cross-signaling between neighboring cells. Eventually, however, I think the wave front would break because a very big twist (approximately nine repeated rotations because of a 30% phase shift between the prestalk and prespore oscillators repeated every ∼2-min scroll rotation over at least 1 h) occurs in a small length of the slug scroll’s axis, and this would be too much. If the wave breaks, I think that the prestalk scroll (tip) would act as it does at the aggregation stage, and begin initiating high-frequency concentric waves in the remaining (prespore) population, dislocation. We think that this happens because the prestalk aggregate acts as an integral continuous source of signal. We know that a localized continuous source of cAMP acts as a high frequency concentric wave source at the aggregation stage and that the aggregate’s tip, which rotates, does the same. We also know that the prespore zone of the slug propagates plane (concentric) waves, whereas the tip contains a scroll wave. By the slug stage, the prespore cells lose their aggregation adenylyl cyclase (29). The difference between prestalk and prespore cells then becomes enormous and dislocation is even more likely. Importance of the Above Observations The above observations, indicating the importance of dislocation for prestalk/prespore scroll wave patterning, presumably have limited importance We made these observations in a certain D. discoideum strain (NC4) under certain aggregation conditions (low cell density). The observations need not necessarily be generally true and we could well have missed important facts and parameters in our considerations of dislocation. Perhaps tips are sometimes integrated scrolls of prestalk and prespore cells, as has been suggested (30, 31), and perhaps twisted scroll waves regulate prestalk/prespore patterning under appropriate conditions (8, 31, 32). The influence of many parameters remains to be investigated. Our calculations above concern essentially homogeneous populations of prestalk and prespore cells. Different assumptions made by Weijer and colleagues (below) can deliver stable twisted scroll waves, as possibly exist in D. mucoroides in appropriate models. Weijer’s group have modeled prestalk and prespore populations in a number of ways, and postulated different excitability properties, and they can generate robust and stable twisted scroll waves under appropriate conditions (31–33). One scenario is to assume that only a subpopulation of prespore zone cells relay, which can lead to a reduced wave velocity and can generate a frequency gradient within the prespore zone (34, 35). Twisted scroll waves possibly exist. Helical cell movements suggest their existence in D. mucoroides slugs (12), but there is no experimental evidence for them so far in D. discoideum. General Importance This simple morphogenetic system may yet provide general inspiration indicating general principles for embryonic development (Fig. 6). Recent findings strongly indicate that chemotactic movements organize vertebrate gastrulation (36), and that these timed gastrulation movements are the key to making the vertebrate bodyplan (37). The situation regarding oscillatory and relayed signaling is more complex but interesting. There is now much evidence that waves, coupled oscillators, and relaying are important in the development of metazoan animals. An early model proposed a progressive phase shift between two traveling Durston

waves originating at the organizer as a mechanism providing positional information in the vertebrate gastrula (9). Another early study actually detected synchronized oscillatory morphogenetic movement in the chicken gastrula (38), and another study detected scaling controlled by a regulative phase gradient, as predicted by the Goodwin–Cohen model (9, 39). There is much other evidence that signal relay, waves of signaling, and synchronized oscillations are important in embryonic development (40, 41). The most important case concerns an interaction between a complex traveling wave of signaling and synchronized periodic waves of coupled oscillations (somitogenesis clock) that regulates vertebrate somitogenesis (39, 42, 43). The signaling wave sequentially blocks the synchronous temporal oscillations, leading to genesis of spatially periodic segments (somites). The somitogenesis oscillations show a phase gradient, indicating wave propagation. There is also a second timer driven by this somitogenesis clock that generates the embryo’s coordinated axial Hox pattern of differentiation by time–space translation (37, 44, 45). The Hox and somitogenesis clocks both begin during gastrulation (37, 38, 44, 45). The findings above thus connect different timing and oscillatory/wave-based events during vertebrate gastrulation. As yet, there is no clear evidence for a role for 3D concentric or spiral excitation waves or twisted scroll waves in metazoan development: the observations are presently at too rudimentary a stage to permit conclusions about this, but this is likely to emerge in the future. There is however, at least one example of dislocation in metazoan development. Vertebrate somitogensis shows a progressive dislocation between a wave propagation domain (presomitic mesoderm) and an inert but functionally connected domain (somitic or somitomere mesoderm). It is clear that the same general principles will govern wave mechanisms wherever they are used in development. Conclusions The waveforms in a D. discoideum slug arise during late aggregation (mound stage) by cell sorting. An aggregate of prestalk Durston

cells sorts out and becomes distinct and separate from the aggregation mound and the aggregate of prestalk cells becomes the tip of the late aggregate. This aggregate contains a 3D spiral (scroll) wave. The prestalk aggregate can arise in a D. discoideum aggregation mound initially organized by concentric waves or by a spiral (vertical scroll) wave. Where the main aggregate is a spiral, the tip’s spiral may or may not have the same handedness (clockwise/anticlockwise) as that of the rest of the aggregate: it is therefore clearly separate and dislocated from the main aggregate. After its formation, the prestalk aggregate continues to grow, presumably by sorting out and recruitment of more prestalk cells into its scroll. It becomes the cylindrical morphologically distinct tip. The tip controls wave propagation in the whole aggregate and in the later slug because it acts as a highfrequency (continuous) pacemaker. The tip initiates concentric waves in the late aggregate and their homologs, plane waves, later in the posterior part of the slug. We inferred the continuous source property from the observation that a continuous or very high-frequency (∼1-min period) point source of cAMP inserted into a late D. discoideum aggregation field induces waves with exactly the same geometry and period as a tip, although it does not induce a tip. The analogy is strengthened by the fact that the tip and a continuous cAMP source share one more property: each can cause continuous attraction of the aggregating Dictyostelium cells instead of initiating obvious waves. The tip thus acts as a continuous source, even though it contains a 3D spiral scroll wave. Presumably this is because its wave propagation period is shorter than the wave period or mean refractory period of the remaining (mostly prespore) cells in the aggregate or slug. This is the tip’s only connection to the remaining cells. It dominates them by being a high-frequency signal source. Siegert and Weijer (8) propose that the D. discoideum slug’s wave geometry is generated because the slug contains a scroll wave that becomes twisted in the prespore region and therefore generates plane waves. This is an extremely beautiful and important proposition, but it is not supported by our observations (above). PNAS Early Edition | 5 of 6

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Fig. 6. Oscillators, wave propagation, and chemotaxis in embryogenesis of higher organisms. (A) Chemotaxis during gastrulation in the chicken At least four chemotactic systems mediate positive and negative chemotaxis during gastrulation in the chicken embryo (46). (B) The somitogenesis clock. A temporally oscillating system of gene expression (somitogenesis clock) interacts with a wave of activity to generate spatially periodic somites (47). (C and D) The somitogenesis clock also regulates the axial pattern of Hox gene expression. Hox patterning is driven by Hox temporal collinearity during gastrulation. This generates a pattern by time-space translation. The temporal collinearity is timed by the somitogenesis clock (D) (44, 48) but also requires a second signal to generate the correct spatial sequence of Hox expression in the axial pattern. We think that this second signal is delivered by interactions between the Hox genes (C) (44).

Our observations and others in D. discoideum development all point to dislocation rather than twisted scroll waves. We propose that the slug’s wave pattern and waveforms are set up during by dislocation during late aggregation (as above). Other observations indicate that twisted scroll waves could be important in some species, according to logical assumptions about cell heterogeneity. In small D. discoideum slugs formed in thin layers, investigations by J. T. Bonner indicate that slugs can form without any scroll waves. There are probably three roads to making a Dictyostelium slug and fruiting body: twisted scroll waves, dislocation, and Bonner’s nonspiral slugs, as described above. These observations may have general importance.

ACKNOWLEDGMENTS. I thank Stuart Newman, Steve Strogatz, Vidya Nanjundiah, Hans Meinhardt, Kees Weijer, John Bonner, Edward Cox, and Morrel Cohen and his team of reviewers for discussion and inspiration. Special thanks are due to K. W. for patiently explaining the basis of his thinking to me. I thank V. N. for mathematical instruction, J. B. for beautiful original ideas, and M. C. and his team for making sure I got everything straight. I thank the American Association for the Advancement of Science for permission to reproduce the images in Fig. 1 A and B from ref. 49 and the image in Fig. 1D from ref. 50. I thank Peter Newell and the Taylor and Francis Group for permission to reproduce the image in Fig. 1C and Kees Weijer for the image in Fig. 1F. I thank Springer for their kind permission to reproduce the image in Fig. 1E from ref. 6. I thank Bentham Press and Kees Weijer for permission to reproduce Fig. 6A from ref. 46. I thank Bentham Press for permission to reproduce Fig. 6 C and D from ref. 51. I thank Elsevier and Dr. Philip Maini for permission to reproduce Fig. 6B from ref. 47.

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Durston

Dislocation is a developmental mechanism in Dictyostelium and vertebrates.

The excitable cells of Dictyostelium discoideum show traveling waves of signaling and generate a variety of complex wave forms during their morphogene...
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