Fig. 4 Symmetry of a doublet occurring in the dorso-anterior region.

Fig. 5 A mechanical device for showing the relations that extra legs in Secondary Symmetry bear to each other and to the normal leg from which they arise. The model R represents a normal right leg. SL and SR represent respectively the extra right and extra left legs of the supernumerary pair. A and P, the anterior and posterior spurs of the tibia. In each leg the morphologically anterior surface is shaded, the posterior being white. R is seen from the ventral aspect and SL and SR are in Position VP. From Bateson, W., Materials for the Study of Variation, London: Macmillan, 1894, p. 480.

Typically157 one leg (rarely more than one) of a beetle is abnormal in bearing a branch at some point in its length. This branch is regularly a doublet, consisting of

157 See Figures 1 and 2, pages 385 and 386.

two parts which may be fused at the point of branching off from the primary leg but which are commonly separate at their distal ends.

Distally from the point of branching there are thus three components — a primary leg and two supernumerary legs. These three lie in one plane and have the following symmetry: the two components of the supernumerary doublet are a complementary pair — one being a left and the other a right —as Bateson's Rule would suggest. Of these two, the leg nearest to the primary leg is complementary to it.

These relations are represented in Figure 3. (See page 387.) Each component is shown in diagrammatic cross section, and their dorsal, ventral, anterior, and posterior faces are indicated by the letters D, V, A, and P, respectively.

What is surprising about these abnormalities — in that it conflicts with the hypothesis offered above—is that there is no clear discontinuity by which the cases can be classified according to which sort of orienting information has been lost. The supernumerary doublet may be borne on any part of the circumference of the primary leg.

Figure 3 illustrates the symmetry of a doublet occurring in the dorsal region. Figure 4 (page 387) illustrates the symmetry of a doublet in the dorso-anterior region.

It appears, then, that the planes of symmetry are parallel to a tangent of the circumference of the primary leg at the point of branching but, since the points of branching may be anywhere on the circumference, a continuous series of possible bilateral symmetries is generated.

Figure 5 (page 388) is a machine invented by W. Bateson to demonstrate this continuous series of possible bilateral symmetries.

If the bilateral symmetry of the doublet is due to a loss of orienting information, we should expect the plane of that bilateral symmetry to be at right angles to the direction of the lost information; i.e., if dorso-ventral information were lost, the resulting limbs or doublet should contain a plane of symmetry which would be at right angles to the dorso-ventral line.

(The argument for this expectation may be spelled out as follows: a gradient in a lineal sequence creates a difference between the two ends of the sequence. If this gradient is not present, then the ends of the sequence will be similar, i.e., the sequence will be symmetrical about a plane of symmetry transverse to itself. Or, consider the case of the frog's egg. The two poles and the point of entry of the spermatozoon determine a plane of bilateral symmetry. To achieve asymmetry, the egg requires information at right angles to this plane, i.e., something which will make the right half different from the left. If this something is lost, then the egg will revert to the original bilateral symmetry, with the original plane of symmetry transverse to the direction of the lost information.)

As noted above, the supernumerary doublets may originate from any face of the primary leg, and therefore all intermediates occur between the expectedly discontinuous types of loss of information. It follows that if bilateral symmetry in these doublets is due to loss of information, then the information lost cannot be classified as antero-posterior, dorso-ventral, or proximo-distal.

The hypothesis must therefore be corrected.

Let us retain the general notion of lost information, and the corollary of this that the plane of bilateral symmetry must be at right angles to the direction of the information that was lost.

The next simplest hypothesis suggests that the lost information must have been centro-peripheral. (I here retain this bipolar term rather than use the simpler "radial.")

Let us imagine, then, some centro-peripheral difference —possibly a chemical or electrical gradient within the cross section of the primary leg; and suppose that the loss or blurring of this difference at some point along the length of the primary leg determines that any branch limb produced at this point shall fail to achieve asymmetry.

It will follow, naturally, that such a branch limb (if produced) will be bilaterally symmetrical and that its plane of bilateral symmetry will be at right angles to the direction of the lost gradient or difference.

But, clearly, a centro-peripheral difference or gradient is not a primary component of that information system which determined the asymmetry of the primary leg. Such a gradient might, however, inhibit branching, so that its loss or blurring would result in production of a supernumerary branch at the point of loss.

The matter becomes superficially paradoxical: the loss of a gradient which might inhibit branching results in branch formation, such that the branch cannot achieve asymmetry. It appears, then, that the hypothetical Centro-peripheral gradient or difference may have two sorts of command functions : (a) to inhibit branching; and (b) to determine an asymmetry in that branch which can only come into existence at all if the Centro-peripheral gradient is absent. If these two sorts of message functions can be shown to overlap or be in some sense synonymous, we shall have generated an economical hypothetical description of the phenomena.

We therefore address ourselves to the question: Is there an a priori case for expecting that the absence of a gradient which would prohibit branching in the primary leg will permit the formation of a branch which will lack the information necessary to determine asymmetry across a plane at right angles to the missing gradient?

The question must be inverted to fit the upside-downness of all cybernetic explanation. The concept "information necessary to determine asymmetry" then becomes "information necessary to prohibit bilateral symmetry."

But anything which "prohibits bilateral symmetry" will also "prohibit branching," since the two components of a branching structure constitute a symmetrical pair (even though the components may be radially symmetrical).

It therefore becomes reasonable to expect that loss or blurring of a Centro-peripheral gradient which prohibits branch formation will permit the formation of a branch which will, however, itself be bilaterally symmetrical about a plane parallel to the circumference of the primary limb.

Meanwhile, within the primary limb, it is possible that a Centro-peripheral gradient, by preventing branch formation, could have a function in preserving a previously deter-mined asymmetry.

The above hypotheses provide a possible framework of explanation of the formation of the supernumerary doublet and the bilateral symmetry within it. It remains to consider the orientation of the components of that doublet. According to Bateson's Rule, the component nearest to the primary leg is in bilateral symmetry with it. In other words, that face of the supernumerary which is toward the primary is the morphological counterpart of that face of the periphery of the primary from which the branch sprang.

The simplest, and perhaps obvious, explanation of this regularity is that in the process of branching there was a sharing of morphologically differentiated structures between branch and primary and that these shared structures are, in fact, the carriers of the necessary information. However, since information carried this way will clearly have proper-ties very different from those of information carried by gradients, it is appropriate to spell the matter out in some detail.

Consider a radially symmetrical cone with circular base. Such a figure is differentiated in the axial dimension, as between apex and base. All that is necessary to make the cone fully asymmetrical is to differentiate on the circumference of the base two points which shall be different from each other and shall not be in diametrically opposite positions, i.e., the base must contain such differentiation that to name its parts in clockwise order gives a result different from the result of naming the parts in anticlockwise order.

Assume now that the supernumerary branch, by its very origin as a unit growing out from a matrix, has proximo-distal differentiation, and that this differentiation is analogous to the differentiation in the axial dimension of the cone. To achieve complete asymmetry, it is then only necessary that the developing limb receive directional information in some arc of its circumference. Such information is clearly immediately available from the circumstance that, at the point of branching, the secondary limb must share some circumference with the primary. But the shared points which are in clockwise order on the periphery of the primary will be in anticlockwise order on the periphery of the branch. The information from the shared arc will therefore be such as to determine both that the resulting limb will be a mirror image of the primary and that the branch will face appropriately toward the primary.

It is now possible to construct a hypothetical sequence of events for the reduplications in the legs of beetles:

(1) A primary leg develops asymmetry, deriving the necessary information from surrounding tissues.

(2) This information, after it has had its effect, continues to exist, transformed into morphological differentiation.

The asymmetry of the normal primary leg is hence-forth maintained by a centro-peripheral gradient which normally prevents branching.

In the abnormal specimens, this centro-peripheral gradient is lost or blurred — possibly at some point of lesion or trauma.

Following the loss of the centro-peripheral gradient, branching occurs. The resulting branch is a doublet; lacking the gradient information which would have determined asymmetry, it must therefore be bilaterally symmetrical.

That component of the doublet which is next to the primary is oriented to be a mirror image of the primary by the sharing of differentiated peripheral structures.

(3) Similarly each component of the doublet is itself asymmetrical, deriving the necessary information from the morphology of shared peripheries in the plane of the doublet.

The above speculations are intended to illustrate how the explanatory principle of loss of information might be applied to some of the regularities subsumed under Bateson's Rule. But it will be noted that the data on symmetry in the legs of beetles have, in fact, been overexplained.

Two distinct but not mutually exclusive — types of ex-planation have been invoked: (a) the loss of information which should have been derived from a centro-peripheral gradient, and (b) information derived from shared peripheral morphology.

Neither of these types of explanation is sufficient by itself to explain the phenomena, but when combined the two principles overlap so that some details of the total picture can be referred simultaneously to both principles.

Such redundancy is, no doubt, the rule rather than the exception in biological systems, as it is in all other systems of organization, differentiation, and communication. In all such systems, redundancy is a major and necessary source of stability, predictability, and integration.

Redundancy within the system will inevitably appear as overlapping between our explanations of the system. Indeed, without overlapping, our explanations will commonly be insufficient, failing to explain the facts of biological integration.

We know little about how the pathways of evolutionary change are influenced by such morphogenetic and physiological redundancies. But certainly such internal redundancies must impose nonrandom characteristics upon the phenomena of variation. 158

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