Polar Coordinate Model Research Articles

Regeneration in youngBunodactis verrucosa Pennant (actinaria, anthozoa).

YoungBunodactis verrucosa Pennant at the 12 tentacle stage are employed to test the applicability of the polar coordinate model to coelenterate regeneration. The animals are cut along every radius into fragments of 3 to 9 segments. Most fragments are patent 3-4 weeks later, but small fragments have a higher mortality rate than large fragments. Some fragments do not regenerate and occasionally tentacles fuse, thereby reducing the number of segments. Small fragments tend to regenerate more tentacies than large fragments, but large fragments may regenerate great numbers of supernumerary tentacles. Twenty-two percent of the fragments restore the missing number of tentacles, while 76% of all fragments produce an even number of tentacles.Fragments restoring the correct numbers of tentacles show a marked tendency to form the correct tentacles (regulative regeneration). Fragments regenerating two less than the number of tentacles already present show a marked tendency to reproduce tentacles of the types already present (miror image formation). Other fragments produce missing segments (forward regeneration), or those already present (reverse regeneration) at lower frequencies.No fragments beginning or ending with the number 1 directive tentacle fail to regenerate entirely, while first cycle segments maximally remote from segment 1 are associated with the absence of regeneration. No fragments beginning or ending with the number 4 directive tentacle fail to undergo forward regeneration, regulate or produce a mirror image when the appropriate number of segments are regenerated. In contrast, segment 4 is associated with a low frequency of reverse regeneration, and second cycle segments cut away from immediate contact with segment 4 show an increase in the frequency of reverse regeneration. Controls through morphogenic substances rather than polar coordinates seem to explain these results. Such substances would control the number and direction of tentacle regeneration.

Simpler rules for epimorphic regeneration: The polar-coordinate model without polar coordinates

Abstract The polar-coordinate model of French, Bryant & Bryant (1976) describes epimorphic regeneration in insects and amphibians, and correctly predicts some surprising phenomena. Their model rests on two main rules. It is shown here that the first of these, the Rule of Intercalation, expresses a requirement that the pattern of positional values of the cells shall be continuous, and that the other, the Rule of the Complete Circle, is not an independent hypothesis, but simply a consequence of the continuity requirement. The formulation of the rules in terms of continuity is coordinate-free and applicable in any number of dimensions: polar coordinates do not have the fundamental significance attached to them by French, Bryant & Bryant (1976) , nor are there any grounds to think that the system of positional values in an amphibian limb is two-dimensional. Some simple topology helps to clarify the concepts and prove these points. The phenomena codified by French et al. are to be expected in any epimorphically regenerative organ whose pattern of growth and pattern of positional values are defined by a maintained system of diffusible chemical signals.

Frequency of supernumerary limbs following blastemal rotations in the newt

AbstractAn examination has been made of the supernumerary limbs produced as a result of tissue interaction following ipsilateral rotation of blastemas at the stage of early digits. The frequency of supernumerary limb formation peaks sharply at angles of rotation near 180°, where tissues from widely separated circumferential locations are opposed. This behavior is compatible with predictions of the polar coordinate model with positional values clustered in the anteriorventral quadrant of the circumference (French et al., '76; Bryant and Iten, '76).

Structure of supernumerary limbs

The generation of supernumerary limbs by rotation of amphibian limb buds or regeneration blastemas is a phenomenon well known to embryologists1,2. The recent revival of interest in this aspect of limb morphogenesis has been largely due to the stimulus provided by the polar coordinate model for control of limb outgrowth and pattern3 and subsequent efforts to test its validity. The geometrical rules postulated in the polar coordinate model make precise predictions about the handedness of supernumerary limbs. Following antero-posterior (AP) or dorso-ventral (DV) inversions, they should be of the same handedness as the stump, whether single or in pairs, but after inversion of both axes (APDV) pairs should be of opposite handedness3. It is surprising, therefore, that the structure of supernumerary limbs has never been studied in sufficient detail to report whether they really are normal limbs, for only the skeletal elements are examined in cleared whole mounts (see Fig. 1). This only allows an assessment of the AP axis, and to determine the dorsal and ventral surfaces (and hence the handedness), the curvature of the digits is noted. I therefore decided to study the muscle patterns of all types of supernumeraries, and report here that, contrary to expectation, the majority of APDV supernumeraries are not normal limbs but are mirror-image duplications in the dorso-ventral axis, that is, either double-dorsal or double-ventral. This presents problems for contemporary models of pattern formation in the limb, and issues a warning to those who use supernumerary limbs as an experimental tool to investigate other areas of developmental biology.

Analytic Extensions of the Gaussian Plume Model

The Gaussian Plume Model is extended to provide analytic characterizations of (a) the dispersion parameters; (b) the Gaussian Plume Model in polar coordinates; (c) the wind rose in direction and velocity space; and (d) the impact of dispersion upon wind rose patterns. The final analytic result should facilitate the estimation of the impact of changes in source emissions upon ground level pollution concentrations and thus extend the calculational speed and usefulness of the Gaussian Plume Model.

Distal transformation from double-half forearms in the axolotl, Ambystoma mexicanum

The capacity for distal transformation of surgically produced symmetrical forearms of the axolotl, Ambystoma mexicanum, was tested. The polar coordinate model for pattern regulation in epimorphic fields states that positional information is a property of cells which reflects their position in the field, specifically, by their location on a radius and a circumference of the field. The model proposes that for complete distal transformation to occur, a complete circumference of positional values must be present at the plane of amputation. Previous experiments have shown that surgically constructed, symmetrical upper arms of axolotls and newts are not capable of regenerating a distally complete pattern. In the present experiment, double-posterior and double-anterior symmetrical forearms were produced by exchanging the anterior half of the right forearm with the posterior portion of the left forearm. The grafts were allowed to heal for 5–35 days, at which time the limb was amputated through the center of the grafted region. Ninety-five percent of the double-anterior (left) forearms and 77% of the double-posterior (right) forearms either did not regenerate or regenerated symmetrical patterns. The regenerated patterns always converged, i.e., lost structures along the line of symmetry. The length of the healing time did not affect the number of elements regenerated. The remaining limbs regenerated asymmetrical structures, either normal, or parts of normal, hands. These results are compared with those of similar experiments previously performed on the upper arms of axolots and newts and the hindlimbs of salamanders and are discussed in light of a model for distal transformation.

Intercalary regeneration around the circumference of the cockroach leg

Epidermal cells from different circumferential positions around the femur of Blabera craniifer can interact to form an intercalary regenerate. Removal of a longitudinal strip of integument (cuticle plus epidermis) from any position around the circumference leads to the cut edges healing, localized growth and intercalary regeneration of the missing section of the circumference, so that the resulting femur is approximately normal in size and pattern of cuticular structures. Grafting a longitudinal strip of femur integument into a different circumferential position on the host femur confronts epidermal cells from different positions along both the inner and outer longitudinal graft/host junctions. In numerous different situations this results in local growth and intercalary regeneration of that section of the circumference normally separating graft and host positions, by the shorter route around the circumference. Confrontation of opposite positions results in the intercalation of either of the intervening half circumferences. In one opposite confrontation, between mid-anterior and mid-posterior, there was also a third result where graft and host healed together, provoking no intercalary regeneration. Grafts made with reversed proximal/distal polarity show that a confrontation between different circumferential positions gives the same result, regardless of the proximal/distal levels involved, hence circumferential position is an independent aspect of position on the femur. These results strongly suggest that epidermal position is not specified with respect to two transverse axes running through the epidermis and internal tissue of the leg, but that there is a continuous circular sequence of positional values running around the circumference, in the epidermis. This is analogous to but independent of the sequence previously shown by Bohn (1967) and Bullière (1971) to run proximal/distal along a leg segment. Hence epidermal position on the femur is specified in two dimensions and can be represented in terms of the French, Bryant & Bryant (1976) polar co-ordinate model. Interactions along the edges of the strip-grafts conform to the Shortest Intercalation Rule (French et al. 1976). At the proximal and distal ends of strip-grafts intercalation restores normal sequences of positional values where possible. However, where the graft, together with the intercalary regenerates formed at the longitudinal graft/host junctions and the adjacent host tissue formed a complete sequence of circular values, then a supernumerary distal regenerate was formed, in agreement with the Complete Circle Rule of French et al. (1976). The problem of generating a continuous circular sequence of positional values by one or more circumferential gradients, is briefly discussed.

The failure of double-half forelimbs to undergo distal transformation following amputation in the axolotl, Ambystoma mexicanum.

Although capable of initiating early regenerative responses, axolotl forelimb stumps which are composed of double-half limb tissues fail to undergo the events that normally lead to the replacement of missing parts. In the present study, the posterior halves of right forelimbs were exchanged with the anterior halves of left forelimbs, or the dorsal halves of right forelimbs were exchanged with the ventral halves of left forelimbs. Forelimbs were amputated through the graft region 30 days after grafting. Limb stumps bearing double-dorsal, double-ventral or double-posterior tissues either produced hypomorphic regenerates or failed to form any externally visible outgrowth. When the limb stump bore double-anterior tissues, no externally visible structures were formed. Normal and multiple regenerates were never formed by double-half limbs. These results are discussed in terms of the polar coordinate model and suggest that the regeneration blastema requires a complete circumference of positional values in order to complete distal transformation.

Regeneration of symmetrical hindlimbs in larval salamanders.

The complete circle rule of the polar coordinate model of pattern regulation was tested for regenerating hindlimbs of Ambystoma larvae. The hindlimbs were made symmetrical in the circumference of either the thigh or the shank, and their ability to regenerate from both levels was observed. Thighs composed of two anterior halves failed to regenerate, whereas thighs composed of two posterior halves often regenerated distally complete, correspondingly symmetrical limbs; shanks composed of either two anterior or two posterior halves regenerated distally complete, correspondingly symmetrical limbs. These results are in contrast to what is predicted by the complete circle rule and suggest a modification of this rule.

Supernumerary limbs in the axolotl.

THE polar coordinate model of French et al.1, commonly called the clockface model, is a concept of great significance to developmental biology, as it presents a unified view of pattern regulation in Drosophila imaginal disks and insect and amphibian limbs. By postulating two rules, intercalation by the shortest route and the complete circle rule for distal transformation, the production of supernumerary limbs in insects2–5 and amphibians6,7 after axial reversal is elegantly explained. Following 180° rotation, the insect limb can produce two supernumeraries in various positions2–4, whereas in the newt the extra limbs are in constant circumferential positions8. Because a complete circle is needed for distal transformation, the model clearly predicts that at no other angle of ipsilateral rotation apart from 180° will supernumeraries be induced. However, Wallace (personal communication), the first to test such a prediction on amphibian limbs, has demonstrated that after 90° or 270° rotation, supernumerary limbs are produced. We report here that, unlike newts and more like insects, axolotls can produce supernumerary limbs anywhere on the circumference after 180° rotation of blastemas. In addition, supernumeraries can be produced at a variety of angles of rotation between 45° and 315°. This work calls for a reappraisal, at least in axolotls, of the clockface model.

Development of cuticular patterns in the legs of a cell lethal mutant ofDrosophila melanogaster.

The development of cuticular patterns in the legs ofDrosophila melanogaster was studied in the temperature-sensitive cell autonomous lethal mutant1 (1)ts726 by treating animals with heat pulses of two days' duration at different developmental stages, in order to find out whether or not models which account for regulation of imaginal discs in the late third instar also hold for earlier developmental periods. Eight kinds of phenotypes were found, each of which occurred only after heat pulses that started at particular time: (1) complete and incomplete mirror image duplications of mesothoracic legs: early second instar; (2) homoeotic transformation to wing hinge in mesothoracic legs: early second instar; (3) prothoracic leg fusions: early second instar; (4) hypertrophied sex combs: early third instar; (5) outgrowths: early third instar; (6) sex comb teeth on second tarsal segment: early third instar; (7) reversed bristle polarity in intersegmental membrane gaps: early third instar; (8) deleted individual bristles: middle of third instar. These phenotypes were compared with patterns predicted by two models that have been devised to account for regeneration data: the polar coordinate model, and the gradient-of-morphogenetic-potential model. Some of the data (especially the finding of circumferentially incomplete partial duplicates) are more readily predicted by the polar coordinate model, although neither model can be ruled out. Phenotypes (6) and (7) can be accounted for by postulating a tandemly repeated positional signal corresponding to tarsal segmentation. The homoeotic transformation may be due to a transdetermination event occurring in situ during regulative growth following cell death. Since deletion of individual sex comb teeth leads to altered sex comb rotation, it is suggested that adjacent sex comb tooth cells interact during rotation.

Wound healing in the imaginal discs of Drosophila: I. Scanning electron microscopy of normal and healing wing discs

Abstract Scanning electron microscopy was used to investigate the morphology of intact imaginal wing discs of third-instar larvae of Drosophila melanogaster. The disc stalk, nerve and tracheal entries and the surface ultrastructure of the columnar cells, the peripodial membrane cells, and the adepithelial cells are described. The behavior of various fragments of the wing disc during culture in vivo was also studied. After injuring a wing disc by cuts with a tungsten needle, during the first day of culture the epithelium curls and the wound surface contracts. Subsequent closure of the wound in 3 4 and 1 4 sectors, in fragments generated by straight cuts and in central squares, leads to the confrontations of cells from formerly separate positions, as was proposed in connection with the polar coordinate model of French, Bryant, and Bryant [(1976). Pattern regulation in epimorphic fields. Science 193, 969–981]. Wound healing comprises three steps: (1) Cell debris is removed; (2) occasional cell processes span the wound; (3) all cells at the wound edge contact cells on the opposite side. After 2–3 days, a continuous epithelium is re-established. The tissue distortion may lead to transient contacts of the columnar epithelium with the peripodial membrane and with itself. The latter can explain the occasional duplications of structures which, according to the fate map, arise from near the wound edge, and which have been previously reported from cultured imaginal disc fragments. The tissue movements appear to be due to the contractile properties of individual cells.

Pattern Formation in Asymmetrical and Symmetrical Imaginal Discs ofDrosophila melanogaster

The polar coordinate model for pattern regulation in epimorphic fields (French et al ., 1976) predicts that bilaterally symmetrical fields will show different kinds of regulative behavior depending on the direction of the cut. These predictions have been tested using the male genital disc of Drosophila melanogaster . First, a detailed fate map was established by examining the fate of disc fragments subjected to immediate metamorphosis in host larvae. Then the regulative abilities of various fragments were examined by culturing them for seven days in adult abdomens, before transfer to larvae for metamorphosis. When the disc was bisected by a vertical cut (parallel to the line of symmetry) then fragments smaller than half of the disc underwent duplication with some simultaneous regeneration, while fragments larger than half of the disc underwent regeneration. If the disc was bisected by a bilaterally symmetrical cut across the line of symmetry, wound healing resulted in the confrontation of cells from similar positions on the right and left sides of the fragment, and no regulation occurred. With the exception of regeneration occurring during duplication of small lateral fragments, these results are as predicted by the polar coordinate model.