A cycloid model for the chromosome

Ever since the acceptance of the chromosome theory of heredity, studies of mutation and recombination have required that any particular gene should be represented once only in a chromatid. With the realization, however, that a gene consists essentially of a segment of a DNA molecule containing a particular sequence of a few hundred nucleotide-pairs, the problem has arisen of how to reconcile this structure for the gene with that of the chromosome. Various lines of evidence have suggested that each gene might be represented many times in a chromosome, either transversely in duplicate DNA molecules, or longitudinally as duplicate genes within one DNA molecule. There now appears to be overwhelming evidence that germ-line chromosomes are not multistranded (see Callan, 1967), but the likely existence of duplicate copies of each gene within one DNA molecule (see later) poses an acute problem.

Callan & Lloyd (1960) offered a solution by postulating the existence of a master gene and, alongside it in the same DNA molecule, a set of slave genes composing a chromomere, with any changes due to mutation or recombination in the master gene copied in the slaves. Callan (1967) has now made an ingenious proposal for the copying mechanism by successive matching of the nucleotide sequence of the slaves against that of the master. This hypothesis has the added attraction of explaining the remarkable discovery which he and his associates have made of the polarized extension of lampbrush loops from chromomeres.

One of the difficulties, however, of this model of chromosome structure is that neighbouring master genes would be separated by a series of slave genes, and yet there is evidence from data on recombination suggesting that neighbouring genes may be contiguous. In particular, the genes ad-9 (adenine-9) and paia-1 (para-aminobenzoic acid-1) in Aspergillus midulans, which are thought to be neighbours, appear to be interrelated in recombination (see Whitehouse & Hastings, 1965). Moreover, the patterns of polarity in recombination within the gene can be accounted for by postulating that the pattern in any particular gene is partly determined by the behaviour of the neighbours on each side (Whitehouse, 1966). This would require a gene to be in contact with its neighbours on both sides. It thus appears that data on recombination provide no evidence for the existence of slave genes, and not only require each gene to be represented once only in the chromatid but also suggest that neighbouring genes may be immediately adjacent to one another. This conclusion is in direct conflict with the results of studies of lampbrush chromosomes (Callan, 1963) and of the nucleolar organizer (Beermann, 1960; Ritossa & Spiegelman, 1965; Callan, 1966) which point to serial repetition of each gene (see Callan, 1967).

This dilemma can be resolved, however, by postulating that the duplicate copies of each gene are removed from the chromatid at the time of crossing-over. Such removal could be brought about by a crossover between the first and last members of the linear series of identical genes. All except one would thereby be detached in the form of a single closed circle. After crossing-over had occurred with one of the homologous chromatids, the gene copies could be reinserted in the chromatid by the same mechanism as that which led to their removal from it, namely, a crossover between the gene in the chromatid and one of the copies of it in the ring. This mechanism for attaching and detaching the duplicate genes is the same as that proposed by Campbell (1962) (and now well established—compare Hayes, 1966) for the incorporation of a circular phage genome (including the sex factor) into that of its host, and for its removal again. The postulated gene-specific crossing-over between duplicate genes within one chromatid could be regulated in a similar way to normal crossing-over between homologous chromosomes, if this is also specific to structural genes, as evidence suggests (Whitehouse, 1966).

This intrachromatid crossing-over would imply that the chromosome can alternate between two states, with the duplicate genes detached as rings or integrated with the DNA axis. A convenient way of representing this situation is to consider the chromosome as having the form of a cycloid (Fig. 1). In studies of mutation or recombination, the loops in the chromosome would appear not to exist and the linear linkage map would match the linear chromosome axis. This would correspond to the line ADGJMPS in Fig. 1. From other points of view, however, the loops would form part of a thread which was continuous with the axis, corresponding to ABCDEF…NOPQRS in Fig. 1. Each loop of the cycloid would correspond to a chromomere (compare Callan, 1966 b). According to the present hypothesis, therefore, chromomeres are detached from the chromosome before crossing-over takes place with the homologue.

This cycloid model of the chromosome bears a superficial resemblance to the chain model proposed by Stahl (1961), in which the chromosome was supposed to consist of a linear series of circles. Stahl suggested that the phenomena associated with recombination between closely linked mutants, such as conversion and negative interference, might be due to events within a circle of DNA. On the present model, however, these phenomena are attributed to crossing-over by hybrid DNA formation in the linear part of the cycloid, and the loops contain duplicate copies of genes and do not take part in recombination. Thus, the cycloid model does not really resemble the chain model, and has been proposed for quite different reasons.

Keyl (1965 a, b) has proposed a chromosome model involving loops of DNA for the chromomeres, but differing from mine in having the loops joined by non-DNA material. Basic features of the cycloid model are the continuity of DNA throughout the chromosome and the ability to detach the loops by crossing-over.

The hypothetical steps in the process of detaching a chromomere are shown in Fig. 2 (a)–(d). A gene M, and 16 linearly arranged copies of it, are shown in diagram (i), and the succeeding diagrams illustrate stages in crossing-over between the first and last members of the series. It is assumed that the mechanism of crossing-over is the same as the normal process between homologous chromatids, and that this follows the operator model (Whitehouse, 1966). The detached chromomere resulting from the crossover and consisting of 16 copies of the gene serially arranged in a ring is shown in diagram (v). (The number of copies of a gene in a chromomere is likely to differ from one gene to another but to be a power of 2—see discussion of Keyl’s work below.) The master gene remaining in the chromatid would then be in a position to undergo normal crossing-over with the corresponding gene in a homologous chromatid. This gene would have shed its chromomere in the same way. One of the functions of the synaptinemal complex might be to hold detached chromomeres in place at pachytene while crossing-over occurred between homologous chromatids.

The reinsertion of the chromomere into the chromatid is shown in Fig. 2 (e)–(h) and, as indicated, could occur by precisely the same mechanism as its previous release. Any of the copies of the gene in the chromomere might be involved in the crossover which restores it to the chromatid or, alternatively, the crossover might regularly involve the same copy as took part in the releasing crossover.

Another open question, on the basis of the present hypothesis, is whether detachment of chromomeres occurs at meiosis for all the genes in both chromatids of each chromosome, or whether it is restricted to the genes which participate in crossing-over with the homologous chromosome. I have suggested that the pattern of polarity in recombination within a gene is partly due to the behaviour of the neighbouring genes (Whitehouse, 1966). If this is true, it would be necessary to suppose that when crossing-over took place at a particular locus, the chromomere not only of that gene but also of at least one neighbouring gene on each side had been detached from the chromatid. This suggests that release of all the chromomeres may occur at meiosis.

Callan (1967) has suggested how slave genes might be matched against a master gene and how in the process the DNA thread of the chromomere would be fed out as a loop which extended from one side in just the way he has observed lampbrush loops to develop. The chief features of this hypothesis are that, for each copy of the gene in turn, the two nucleotide chains of the copy would dissociate and pair with the complementary chains of the master gene, to be followed by correction of mispairing. This correction would always occur in the same direction, that is, from master to slave and not the converse. After correction, the slave chains would dissociate from the master chains and reanneal with one another.

A modified form of this model is given in Fig. 3 with matching of only one of the chains of the slaves against the master, and with breakage of the matching chains at one end of the gene. This breakage would facilitate the coiling and uncoiling of nucleotide chains. Matching of only one chain might suffice, since it is known that messenger RNA is the complement of one specific DNA chain. Since it is also known (see Guest & Yanofsky, 1966) that messenger synthesis begins at the operator and proceeds in the 5′ to 3′ direction (that is, grows from its 5′ end by adding nucleoside 5′-phosphates to the 3′-hydroxyl end), the complementary DNA chain would be polarized 3′ to 5′ with respect to the operator. It is this chain in the slaves which would need to agree with the master, and so the master chain which showed complementary pairing with successive slave chains would need to be that which was polarized 5′ to 3′ with respect to the operator, or in other words had the same nucleotide sequence (apart from thymine in place of uracil) as the messenger. This argument is based on the assumption that the messenger is synthesized with one DNA chain as the template. If the duplex DNA acts as template, modification might be needed, depending on how the nucleotide sequence in the DNA was recognized.

It is suggested that the initial step in the process of copy matching might be breakage of one of the nucleotide chains of the master copy of the gene at the operator. This breakage would be in the chain which had the 5′ to 3′ polarity, that is, the sequence -O-P-O-5′-4′-3′-O-P-O-in the atoms of its backbone reading from the operator end. Following this breakage, it is assumed that dissociation of the chains would occur over the length of the gene. A cycle of 6 steps is then postulated for the matching of each slave in turn against the master, namely: (1) breakage of the complementary chain of the slave at the terminus (non-operator) end of the gene ; (2) dissociation of the chains of the slave over the length of the gene; (3) annealing of complementary broken chains of master and slave ; (4) correction of mispairing in the direction from master to slave; (5) dissociation of master and slave chains; and (6) annealing of the two chains of the slave. These steps are shown diagrammatically in Fig. 3 (a)—(c) and again in Fig. 3 (c)–(e) for another cycle of matching. As in Callan’s model, the lampbrush loop would represent that part of the series of copies which had been corrected. The motive force for formation of the loop would be the attraction of complementary nucleotide chains, one from the master and one from each slave in succession.

In Fig. 3 the copy labelled M of the gene is taken to be the master, with breakage of the 5′ to 3′ chain at the operator. Copy number 1 is then the first to emerge as a lampbrush loop after correction, followed by number 2 and so on. Alternatively, if copy number 1 were the master and showed the 5′ to 3′ breakage at the operator, then copy M would be the first to be matched, followed by number 16. The first alternative (illustrated in Fig. 3) leads to the terminus (non-operator) end of each copy of the gene emerging first into the lampbrush loop. The direction of messenger synthesis would then coincide with that of loop movement. The second alternative leads to the operator end emerging first and to m-RNA being synthesized in the opposite direction to that of the movement of the loop. The alternative shown in Fig. 3 is considered to be the more likely, for reasons discussed later.

The mechanism of correction of mispairing would need to be able to distinguish a nucleotide chain from the master copy and one from a slave. This would require the correcting enzyme to recognize some distinguishing feature such as the operator region of the gene. In Fig. 3, at the times when correction would take place (at (iii), (v) and (vii)), the slave chain has the operator at the 3’ end (that is, the chain is attached to the operator and is polarized 3′ to 5′ with respect to it), while the master chain does not have the operator attached (or, if it were attached, it would necessarily be at the 5′ end). In addition to correction in a specific direction determined by a distinction of this kind, there must subsequently either be no correction between the chains of the slaves, or correction in a predetermined direction, namely, from the slave chain which had paired with the master to that which had not. The first alternative would mean that mispairing would persist in the slaves. The second would require a different recognition mechanism from that for correction from master to slave. This is because the slave chain which received corrections from the master would now have its role reversed and would have to pass the corrections on to the complementary slave chain. A model of the kind proposed by Callan (1967) in which both chains of each slave were corrected directly from the master would pose an acute recognition problem for the correcting enzymes, because correction from master to slave in one pair would correspond to correction from slave to master in the other. The question of whether both slave chains are corrected or not is discussed further later.

During the correction of the slaves it would be possible for the matching process to be broken off. This correcting of some but not all of the slaves might occur regularly, or only as an abnormality. These alternatives are related to the question of whether the process of matching slaves against master takes place once only per generation (after crossing-over has occurred with the homologous chromosome at meiosis), or whether it regularly precedes messenger synthesis. The number of lampbrush loops at diplotene far exceeds the number of chiasmata, so clearly the matching of copies is not directly related to the occurrence of crossing-over with the homologous chromosome. On the other hand, all the loops are associated with RNA and protein synthesis, which indicates that the genes concerned are functioning. It seems, however, most unlikely that all the genes are active in the oocytes. Those that are inactive at this stage might undergo matching of slaves to master in whatever tissue they first became active. The simplest hypothesis appears to be that the matching of slaves against master does not occur only once per life-cycle, but always precedes messenger synthesis at whatever stage of development the gene is functioning. This would explain the similarity of puffing in salivary gland chromosomes to lampbrush loop formation (compare Pelling, 1966). The peculiarity of the lampbrush phase of oocytes would then be merely that a large number of genes were active over a prolonged period. The hypothesis of matching whenever m-RNA was to be synthesized would also accommodate the data on somatic mutation and recombination, because the functional genes—the slaves—would always correspond to the master gene, at whatever stage they functioned.

If matching occurs regularly before m-RNA synthesis, there will be two categories of genes : those that function in the germ-line and those that do not. For genes which do not function in the germ-line, there is the possibility that, under special circumstances, selection might favour slight differences between the copies of a gene in the chromomere. Recombination between such non-identical copies might explain antibody variability (see Whitehouse, 1967).

There seem to be no exceptions to the rule that protein synthesis occurs throughout the length of lampbrush loops, although in giant granular loops RNA synthesis is confined to the region of the loop that has recently emerged from the chromomere (Callan, 1963). Presumably in these loops the messenger persists in contact with the loop and functions repeatedly for protein synthesis, whereas in the majority of loops the messenger breaks down or leaves the loop, and in either case is continually resynthesized. It might, therefore, be anticipated that the size of a loop was some indication of the scale of activity of the corresponding gene. It would then be expected (on the assumption that matching always occurs before m-RNA synthesis) that when the activity of a particular gene was low or of short duration, only a few of the copies would be corrected against the master, while for a high rate or prolonged activity, all the copies might be matched and act as templates for m-RNA synthesis. If an average gene codes for a polypeptide containing 500 amino-acid residues, it would occupy approximately 0·5 μ of DNA (1500 nucleotide pairs 0·34 mμ apart). If an average lampbrush loop in Trituras cristatus is 30 μ in circumference, it might contain 2⁶ copies of a gene. On the other hand, allowing for the loop being fed back into the chromo-mere, Callan (1963) has estimated that a giant granular loop and its chromomere might contain 1–2 mm of DNA, which would be equivalent to perhaps 2¹¹ or 2¹² copies of the gene. If a gene with so many copies were active at some stage of development for only a short period of time, it is unlikely that all these copies would be matched against the master. Interrupted slave matching thus seems probable, if it is accepted that correction of slaves always precedes m-RNA synthesis.

Callan (1963) has described the occurrence in T. cristatus of a small number of lampbrush loops which are symmetrical. In these loops, each half is the mirror image of the other and resembles a normal asymmetrical loop. The basal part of each half of the symmetrical loop resembles the end of a normal loop that appears to be re-entering the chromomere, and the part of each half which corresponds in appearance to the emerging end of a normal loop is that which is farthest from the chromosome axis. Callan has interpreted symmetrical loops as due to reversed repeats. This explanation is in keeping with the hypothesis of chromosome structure proposed here. A symmetrical loop would be accounted for if half the copies of the gene were reversed relative to the other half, and if the matching occurred separately in each half. The situation could be represented diagrammatically by including the master gene and the 16 copies of it twice in Fig. 3, so that the sequence in the DNA was L, 1–16, M, M, 16–1, N. According to this interpretation, the matching of copies against the master takes place at the tip of each half of the symmetrical loop. That the DNA thread emerges linearly from each half, and not as a loop, is in keeping with the model proposed here, in which the first slave to be matched is the one which is farthest from the master. According to Callan’s model (1967), the first slave to be matched adjoins the master, but the structure of symmetrical loops contradicts this.

Keyl (1965a, b) and Pelling (1966) have found evidence that the transverse band of dipteran salivary gland chromosomes, or its counterpart in normal chromosomes, the chromomere, is a unit of replication of DNA. By thymine autoradiography they had found (Keyl & Pelling, 1963) that DNA synthesis in the transverse bands of the salivary gland chromosomes of Chironomus thummi begins simultaneously in all the bands, but that those with larger DNA content take longer to replicate. Keyl (1964, 1965a, b, 1966) has found, by microspectrophotometry after Feulgen staining, that particular bands in two subspecies of C. thummi differ in DNA content by a power of 2 (either 1, 2, 4, 8 or 16 times as much DNA in a band in one subspecies as in the corresponding band in the other). Similar variation was found between different individuals within one of the subspecies. These findings have led Keyl (1965a, b) and Pelling (1966) to infer that each band is a unit of replication, and that the doubling in evolution of the DNA content of a particular band is due to doubling the length of the DNA molecules it contains. They have suggested that this duplication of particular chromomeres could occur most readily if the DNA of the chromomere were arranged in a ring, so that the end of the replicating segment lay close to its beginning. Keyl (1965a, b) has accordingly proposed a chromosome model with this feature, and has 2 Cell Sci. 2 suggested that the doubling arises through an error of replication. On the cycloid model, however, the doubling would be explained if the crossover between the first and last copies of the gene to detach the chromomere (see Fig. 2) occasionally took place between the first copy in one chromatid and the last copy in the sister chromatid.

Evidence for circularity in the DNA of the chromosome has been obtained by Hotta & Bassel (1965). By sedimentation analysis and electron microscopy, they have found that at least part of the DNA of the sperm of the boar (Sus domesticus’) is in the form of circles with circumferences ranging from 0·5 to 16·8 μ.

Direct support for the idea that the unit of replication of the DNA of the chromosome is circular and corresponds to the chromomere has been obtained from study of the hundreds of free nucleoli which occur in the nuclei of amphibian oocytes. Miller (1964) found from enzymic treatment and electron microscopy that each nucleolus of Triturus pyrrhogaster contains a circular DNA molecule surrounded by matrix. This construction closely resembles that of lampbrush loops and led him to suggest that a single chromosomal locus may replicate independently of the rest of the genome. Callan (1966) has made discoveries with Ambystoma mexicanum which lead to the same conclusion. He has found that the free nucleoli show the same range of form as those remaining attached to the chromosome in the region of the nucleolar organizer, and that whether free or attached they may be ring-shaped. Moreover, he has found that a typical lampbrush loop may occur at the nucleolar locus, and that this loop closely resembles the ring-shaped nucleoli. It is inferred, first, that the nucleolar locus is constructed in the same way as other gene loci, and secondly, that it is capable of generating several hundred DNA threads, each of which resembles its own lamp brush loop in appearance, but has the form of a closed circle. These conclusions are of outstanding importance because they imply that the lampbrush loop is the unit of replication of the chromosome, in precise agreement with Keyl and Felling’s comparable discovery for the salivary band.

Jacob & Brenner (1963) proposed the term ‘replicon’ for a unit of replication of DNA which was capable of controlling its own replication, as in bacteria and viruses. They suggested that this control would take place by means of a specific ‘initiator’, which would act on a recognition site, or ‘replicator’, and initiate DNA replication in a specific direction from that point. It is not yet known how the replication of chromomeres is controlled, but it is convenient to apply the term ‘replicon’ to these units of replication, even though control of their replication from elsewhere in the cell would imply a modification of the original definition of the term ‘replicon’. The circular configuration of the units of replication in bacteria and viruses on the one hand, and in the nucleoli of amphibian oocytes on the other, has been remarked upon by Miller (1964), and suggests a similarity in the control mechanism.

On the cycloid model, the unit of replication would be expected to correspond to all the copies of a particular gene, such as numbers 1–16 and M in Figs. 2, 3. One possibility is that the replicator is at the operator of the master copy (M) of the gene, and the replication terminus at the non-operator end of copy number 1 (see Figs. 2, 3). This would mean that each replicon consists of a chromomere and parts of the neighbouring interchromomeric segments, and that successive replicons along the chromosome would be contiguous. At the replicon junctions, one of the nucleotide chains of the DNA would need to be free to rotate about the other (compare Whitehouse, 1965, p. 201). If, as suggested above, the replicator coincided with the operator of the master gene, this would imply that the master gene always had the position shown in Figs. 2 and 3 relative to the copies, because if (as discussed earlier) the master were at the position of copy number 1 in the diagrams, its operator would not be at either end of the series of copies of the gene. The ring of DNA in each nucleolus of amphibian oocytes is evidently synthesized from the nucleolar lampbrush loop as template, and its synthesis is presumably controlled in the same way through the replicator. One possibility is that the rings are synthesized successively along the DNA of the chromomere and each in turn detached from the chromosome by the mechanism already proposed as a normal feature of meiosis, namely, crossing-over between the first and last copies of the gene as in Fig. 2 (i)—(v). Another possibility is that the crossover occurs first and successive replications of the ring follow.

Since the DNA of a lampbrush loop at the nucleolar locus can replicate many times to give rings of DNA in the nucleoli, and these rings (as well as the loop) can then function as RNA templates, both chains of the nucleolar slave genes presumably correspond with those of the master. For this locus, therefore, it would appear necessary for both chains of each slave to match against the master (as suggested by Callan, 1967), or for one to do so and then to correct the other slave chain. It is arguable, however, that matching may be abnormal at the nucleolar locus because it is not a normal structural gene specifying a polypeptide through m-RNA.

There are several aspects of the hypothesis of chromosome replication postulated here which appear to be relevant to the mechanism of crossing-over. I have suggested (Whitehouse, 1966) that crossing-over is organized on a gene-specific basis with a similar control mechanism to transcription. It now appears that the replication of the chromosome is also gene-specific. There is the possibility, therefore, of an interrelationship between the control of replication and of crossing-over. Thus, the model of crossing-over which I have proposed requires DNA synthesis to occur in both chromatids from the operator of the gene. If, as was suggested above, the replicator coincided with (or was adjacent to) the operator of the master copy of the gene, the synthesis necessary for crossing-over could be initiated in the same way as normal replication.

The operator model of crossing-over requires breakage at the operator of nucleotide chains of opposite polarity in the two recombining chromatids. The elaborate enzymic breakage organization that this would need can be dispensed with, if it is supposed that the breakage is a failure of joining of the phosphodiester backbones after the preceding replication. Such failure might be required in order to allow the chains of the next replicon to rotate about one another, if it was later in replicating. The failure to join would necessarily leave one chain of each polarity unjoined in sister chromatids, and similarly in the homologous chromosome. Moreover, if the primary breakage of nucleotide chains in crossing-over is really a failure to join after the previous replication, this would account for crossing-over, whether meiotic or mitotic, always occurring at the 4-strand stage. Evidence for a relationship between crossing-over and events at the preceding DNA replication has been obtained from studies of recombination in organisms as diverse as Lilium (Lawrence, 1961), Chlamydomonas (Hastings, 1964; Lawrence, 1965) and Drosophila (Grell & Chandley, 1965). These authors have shown that a change of temperature or treatment with chemicals or radiations at the time of the premeiotic DNA replication can affect recombination. This suggests, as Hastings (1964) has proposed, that the initial steps of the process of crossing-over may take place at this time, before the homologous chromosomes associate.

The chains remaining unjoined at the end of the replicon would necessarily be the newly synthesized ones. It follows that, in the intrachromatid crossover to detach a chromomere, the breakage of a nucleotide chain at the operator of copy number 1 of the gene (Fig. 2(a)) would always be in the old chain. This is on the assumption that the intrachromatid crossover follows the mechanism proposed for normal crossing-over.

The model of chromosome structure and organization proposed here has direct relevance to all the basic aspects of genetics—replication, recombination and transcription, and their regulation—and so could be tested by studies in all these fields.

Investigation of the mechanism of synthesis of nucleolar DNA, and its control, would be particularly instructive. Information about the correction of slave genes against a master gene might be obtained from electron microscopy of the tip of symmetrical lampbrush loops at different stages of development of amphibian oocytes, and perhaps also by isotope labelling, if mutation or recombination had occurred in the master gene. The idea that the replicator of a chromomere coincides with the operator of its master gene could be tested if it were possible to determine the direction of messenger synthesis in a lampbrush loop, since synthesis coinciding in direction with that of loop movement is predicted.

It is known that heterochromatic segments of chromosomes replicate late, and it would be interesting to know more about their detailed replication pattern. It is possible that the cycloid structure might apply only to the euchromatic regions, since chromomeres are largely restricted to euchromatin and may, therefore, be a specific feature of structural genes.

Apart from the circularity of the replicon, the organization of a chromosome appears to be quite different from that of its counterpart in bacteria and viruses, for which I have proposed the term ‘chromoneme’ (Whitehouse, 1965). The chromosomal organization, with its duplicate genes in chromomeres, is not known in bacteria. It appears significant that no example of an operon, in the sense of a group of structural genes with a common messenger, has yet been conclusively demonstrated in any chromosomal organism. It may be that the operon is not appropriate to the chromosomal (as distinct from the chromonemal) organization.

The term ‘operator’ has been used in this paper for the recognition site at one end of the gene. Jacob, Ullman & Monod (1964) have shown, however, that in the lactose operon of Escherichia coli, the operator, defined as a site of repressor action, is distinct from the site of initiation of transcription, for which they have proposed the term ‘promotor’. It is possible that for each chromomere in a chromosome there is only one operator, situated at one end of the master gene, and that for each slave what I have called an operator is a promotor.

I am most grateful to Professor H. G. Callan, F.R.S., for showing me the manuscript of his paper on the organization of genetic units in chromosomes. The ideas in the present paper are a development of his and owe much to discussion with him. I thank Mr G. J. Clark for preparing the diagrams.