Using Graph Theory to Describe and Model Chromosome Aberrations

Rainer K. Sachs^a1 Javier Arsuaga^a, Mariel Vázquez^a, Philip Hahnfeldt^b, and Lynn Hlatky^b

9 Figures, 1 Table

Running title: Chromosome aberration multigraphs

Submitted to Radiation Research, May 2002

Cells exposed to ionizing radiation undergo DNA breakage and misrejoinings that lead to large-scale rearrangements of the genome; such rearrangements have a number of important health-related implications. For acute irradiation during the G0/G1 phase of the cell cycle, the rearrangements typically found are chromosome aberrations (i.e. chromosome-type chromosomal aberrations, as defined in 1). Chromosome aberration spectra help characterize DNA repair/misrepair pathways and chromosome geometry during cell-cycle interphase (review: 2).

A colorful diversity of radiation-induced chromosome aberrations is now observed with chromosome painting techniques, revealing a higher degree of complexity than was previously assumed (3,4). Particularly informative are mFISH ² and SKY ² (5-9) where all pairs of autosome homologues, as well as the sex chromosomes X and Y, are simultaneously assigned their own (pseudo-)color, so that many aberrations become readily detectable. Still further technical developments (e.g. 10-14) include "bar coding" and other techniques where different parts of a single chromosome are assigned different colors, making intrachromosomal rearrangements more readily detectable.

Biophysical implications of observed aberration spectra for DNA repair/misrepair pathways and interphase chromosome geometry can be analyzed using "cycles" to describe the exchange process. A complete exchange involving just two DSBs (DNA double strand breaks), i.e. a 4-ends pairwise misrejoining as occurs for example in standard scenarios for the formation of a simple dicentric or translocation, is a "2-DSB cycle". More generally, an "n-DSB cycle" characterizes an exchange process where n different DSBs, presumably close in time and space, have participated in a single, irreducible reaction (7,15). In large-scale comparative genomics (review: 16) a very similar concept is used (e.g. 17), referring primarily to rearrangement configurations rather than rearrangement processes (review: 18, Ch. 10).

Graphs have been used in various applied fields and studied mathematically for more than two centuries (30). They have often been applied recently in computational biology (31), though not for studying radiogenic aberrations nor using the particular type of graph theory discussed below. Fig. 1 illustrates some basic definitions used throughout graph theory.

<Figure 1 about here>

Graph theory can be applied to aberration diagrams that have been considered by Savage³ and similar diagrams that have been presented by various authors (e.g. 7, Figs. 3 and 5; 26,32). By abstracting from such diagrams, one is led to aberration multigraphs, which unify three basic aberration elements: an initial configuration; an exchange process; and a final configuration. Each of these three basic elements is important in its own right (7,33).

An aberration multigraph refers in principle to the actual biophysical process of aberration formation, as it occurred in a cell, not just a final pattern observed cytogenetically. In any currently feasible experiment one does not directly observe entire aberration multigraphs, and many different aberration multigraphs can correspond to one observed pattern. Modeling the underlying process explicitly by using aberration multigraphs helps organize the observations. We now show how the unification of three basic aberration elements (initial configuration, exchange process and final configuration) into an aberration multigraph works, starting with some examples and then generalizing.

Examples of aberration multigraphs

Multigraphs consist of vertices and edges (Fig. 1). A vertex of an aberration multigraph (e.g. Fig. 2F) represents either a DSB free end ("free-end vertex") or a telomere ("telomeric vertex"). An edge represents a chromosome segment ("chromosome edge"), or initial partnership between the two free ends of a single DSB ("initial edge"), or final partnership between DSB free ends that have been misrejoined or restituted ("final edge"). Each of the three basic elements of a chromosome aberration (initial configuration, exchange process, and final configuration) can be considered as a submultigraph, obtained from the unified aberration multigraph by deleting some parts, and having some edges in common with each of the other two basic elements. Fig. 2 shows how an aberration multigraph for a simple dicentric is constructed; Fig. 3 gives the aberration multigraphs for a simple centric ring with accompanying acentric fragment, a simple translocation, and a simple inversion; Fig. 4 shows an aberration multigraph for a complex aberration.

<Figures 2, 3, and 4 about here>

For the time being we assume that all exchanges are complete and that all chromosome edges are long enough to be observable. If instead there are chromosome segments so short they are not seen in the assay, a different mathematical situation arises. In that case, if there are enough short segments, every observed aberration pattern is consistent with an exchange process consisting entirely of 2-DSB cycles, as can be shown by generalizing an argument given earlier (15); a quite elegant mathematical theory is available for determining how many 2-DSB cycles are needed to produce a given observed pattern 18). These results are not directly relevant when all chromosome edges involved are long enough to be observable, and will not play any further role in the analysis below.

Defining SECs ²

The aberration multigraph for the entire altered karyotype in one cell may have more than one connected component (connected components are defined in the caption to Fig. 1). In that case we are dealing with several different aberrations, presumably independent of each other. Formally, we can define an individual aberration as a connected aberration multigraph. Even for a connected aberration multigraph, the exchange process submultigraph may have more than one connected component (Fig. 5 shows an example). Having more than one component for the exchange process then identifies the aberration as a sequential exchange complex (SEC). This mathematical definition captures the current biology usage (review: 34) even in complicated situations.

<Fig. 5 about here>

Possible biophysical pathways for producing chromosome aberrations are breakage-and-reunion, based on non-homologous end-joining, recombinational misrepair, and the exchange theory pathway (35). The examples in Figs. 2-4 are most easily interpreted using a breakage-and-reunion scenario. However aberration multigraphs can also be used to analyze recombinational misrepair scenarios as follows: (1) restricting attention to those multigraphs where each connected component of the exchange process submultigraph is a 2-DSB cycle c2 (i.e. a 4-end pairwise exchange); (2) labeling each initial edge (e.g. by either "d" or "e") to identify if the corresponding break was caused directly by radiation or made by enzymes during the misrepair process (Fig. 4G). Likewise, one can consider exchange-theory scenarios simply by insisting that each connected component of the exchange process submultigraph be a 2-DSB cycle c2. In the rest of this article we sometimes assume the breakage-and-reunion pathway to illustrate the results. It is probably the dominant pathway for damage processing after acute irradiation during G0/G1 (23,36,37). Moreover, its analysis indicates the full range of questions that arise, since its possible final configurations include all possible final configurations of the other two pathways.

<Figs. 6 and 7 about here>

In aberration work it is important to have some standard way to describe observed aberration patterns. One would like a single approach applicable to different protocols such as solid staining, 2-color FISH, mFISH, SKY, bar coding, banding, length measurements for rearranged chromosomes, sequencing, etc. Aberration multigraphs suggested a generalization of PAINT (38) and mPAINT (7) descriptors, which gives a satisfying way to meet this requirement.

1. z = red ® blue ® green ® blue ® yellow ® red ® purple (in a bar coding experiment);
2. z = 23.5 Mb® 56.4 Mb on chromosome 3 (if breakpoint locations in Mb are measured);
3. z = AGCCA…TTATT (for a hypothetical "post-genome" aberration experiment, in which the exact base pair sequence of each chromosome segment is recorded; here there could in principle be up to several hundred million nucleotides given as part of the label z).

This generalization of mPAINT should suffice for the foreseeable future, no matter what advances are made in aberration detection techniques. It allows considerably more systematic comparisons than hitherto among different kinds of experiments or among different types of scoring within one experiment. It is compatible with the notation used for global comparisons of gene order in modern comparative genomics (16,18).

The cycle structure (7,15,17) of an aberration can be defined mathematically in terms of aberration multigraphs as follows. Consider the exchange process, a submultigraph of the aberration multigraph, obtained by deleting the chromosome edges and the telomeric vertices (Figs. 2- 6). In general this exchange process can have several different connected components, corresponding to several different reactions which could be separated in time and/or space. A theorem in graph theory (18,23,30) shows that for an aberration multigraph each connected component of the exchange process must have an even number of vertices in a cyclic arrangement. The decomposition of the exchange submultigraph into connected components thus defines the cycle structure in the way exemplified by Figs. 2-5. Specifically, the cycle structure for each aberration multigraph in Figs. 2 and 3 is c2, that in Fig. 4 is c3, and that in Fig. 5 is c2+c4. A cycle structure c5 would correspond to an exchange process that is a cyclic graph with 10 vertices (for 5 DSBs); cycle structure c1 would correspond to a single restitution (1 DSB); cycle structure c2+c3+c4 describes an aberration which starts from 9 DSBs and involves three separate reactions that have some chromosomes in common; etc.

There are so many different kinds of aberrations it has long been common to classify them into groups. Unlike primarily descriptive nomenclature such as mPAINT (or its generalization suggested above), and in parallel with such nomenclature, classifications have sometimes been based on "mechanistic" considerations. We next discuss classification methods that are mechanistic in the sense that they can use an entire aberration multigraph, whereas the descriptive nomenclature suggested above uses only the non-cryptic aspects of the final configuration submultigraph (Fig. 6).

Aberrations have often been classified according to the number of chromosomes involved (i.e. half the number of telomeric vertices in the aberration multigraph) and the number of DSBs involved (i.e. half the number of free-end vertices in the multigraph) (38). Recently (7,15) they have also been classified according to the cycle structure (given by the exchange process submultigraph of the aberration multigraph). Graph theory makes available additional ways to classify.

Simple examples of quantities that could be used to help classify, often used by graph theorists, are "diameter" and "girth" (30). Table I shows values of these particular quantities for some aberrations. The diameter of an aberration multigraph is the longest distance between any pair of vertices, where the distance between two given vertices is defined as the number of edges in a shortest connected path between the vertices. High diameter for a given number of DSBs corresponds to many different cycles in the exchange process, linked by the fact that they have some chromosomes in common. Thus, serendipitously, an aberration whose formation was spread out in space and/or time tends to have a higher diameter, in the graph-theory sense, for its multigraph than does an aberration involving the same number of DSBs in a tighter reaction. The girth of an aberration multigraph is the number of edges in a shortest cyclic submultigraph. Girth 2 usually corresponds to ring formation and girth 3 to inversions. Large girth can occur only if every reaction of the exchange process is complicated.

Diameter and girth are used throughout graph theory, but special properties of aberration multigraphs also make possible a somewhat deeper classification. All free-end vertices of an aberration multigraph have three incident edges (compare Figs. 2-5), so aberration multigraphs are closely related to what are called cubic multigraphs, defined as multigraphs where every vertex has three incident edges. An enormous amount is known about cubic multigraphs (30), mainly because they happen to be related to the famous four-color problem for maps. Results on cubic multigraphs suggest a mathematical classification of aberration multigraphs, as will be described elsewhere.

The connected components of the final configuration submultigraph represent the rearranged chromosomes (Figs. 2-6). One of the main problems in quantitative aberration analyses is that often the rearranged chromosomes have cryptic damage, where the experimental protocol does not uncover a misrejoining. An example is given by the paracentric inversion in Fig. 3F. The misrejoinings will be cryptic in an mFISH/mPAINT experiment, though a banding or bar coding experiment might uncover them. Typically, a more sophisticated experimental protocol decreases such ambiguities but does not eliminate them. Thus the observed pattern does not determine the final configuration of an aberration multigraph uniquely (given a specific protocol, the converse does hold, as represented in Fig. 6 by an arrow from the final configuration to the observed pattern).

Even when the final configuration is completely known, the initial configuration and/or the exchange process (i.e. the other two basic submultigraphs of the aberration multigraph) are not in general uniquely determined. Fig. 8 gives an example. In both panels the final configuration is the same. In this particular case the initial configurations are also the same in both panels. Thus the two aberrations involve the same number of chromosomes and DSBs, as well as having the same observed final pattern for any experimental protocol whatsoever. Nonetheless the two aberrations shown are different, as shown by the fact that the cycle structures of the exchange processes differ, being c2+c3 in Fig. 8A and c5 in Fig. 8B. The difference could, at least in principle, be uncovered by other kinds of experiments, e.g. premature chromosome condensation experiments.

<Figure 8 about here>

Since different aberrations can correspond to the same final configuration and different final configurations can correspond to the same observed final pattern, the number of different aberrations giving a particular observed pattern can be large. We next consider some methods of coping with this ambiguity, trying to determine, or at least give probabilistic estimates for, actual damage processing events from observable patterns.

Because aberration exchange processes are informative about enzymatic DNA repair/misrepair (23,28,35-37), attention has recently focussed on determining cycle structures. In cases where an observed final pattern defined by mPAINT descriptors is consistent with different possible cycle structures for the exchange process, Cornforth introduced the concept of obligate cycle structure, essentially as that consistent cycle structure which maximizes the number of cycles involved (7,23); the concept is closely related to the concept of maximal cycle decomposition (18) in comparative genomic analyses of gene order. For example the obligate cycle structure in Fig. 8 is the cycle structure c2+c3 of the aberration shown in panel A, since the alternative method (panel B) of making the given mPAINT pattern has cycle structure c5. The obligate cycle structure is considered the most conservative possibility for generating a given observed mPAINT pattern, being closest to the older view that all, or at least most, complete exchange processes are simply composed of one or more 2-DSB cycles, with higher-order cycles rare or altogether absent.

However, the obligate cycle structure is not always the one that occurs in aberration radiogenesis, so probabilistic analyses are needed (15). Specifically, suppose we are given an observed mPAINT pattern. We can first carry out the following four steps (7,23) to assign configurations:

1. count telomeres in the observed pattern, to find out how many chromosomes were involved;
2. by analyzing color junctions and cases of two contiguous centromeres involving a single color, infer the minimum number of DSBs that could have given rise to the observed pattern;
3. assuming the minimum break number, deduce the final configuration;
4. work out, for purposes of estimating the cycle structure, initial configurations which are consistent with that final configuration and have the minimum break number.

For example, in Figs. 2, 3A, 3E, 4, 5, and 7 the mPAINT descriptors and this procedure lead to the initial and final configurations given in the figures. Fig. 8 shows that even given the initial and final configurations, the cycle structure is not necessarily determined uniquely. However, at this point one can determine all the different ways the given final configuration can be generated from the given initial configuration by different exchange processes involving various possible cycle structures, and probabilities can thereby be assigned to the different cycle structures³. Here are some examples that we have tried:

1. the initial and final configurations of Fig. 8, involving 5 DSBs, gave c2+c3 (obligate cycle structure, 2 cases out of 8 for a probability of 1/4) versus c5 (6 cases out of 8, probability 3/4);
2. an example involving 4 DSBs gave c2+c2 (obligate cycle structure, 2 cases out of 4 for a probability of 1/2), versus c4 (2 cases out of 4 for a probability of 1/2);
3. a 6-DSB example gave c2+c2+c2 (obligate cycle structure, probability 1/4), versus c2+c4 (probability 1/2), versus c6 (probability 1/4).

Thus in each case there is a substantial probability of a cycle structure with cycles of order higher than given by the obligate cycle structure. This result was pointed out by Savage³, who analyzed a number of examples in detail.

In an actual experiment there will be not only a probabilistic mixture of cycle structures given the initial and final configurations, but also a probabilistic mixture of configurations given the observed mPAINT pattern. At low doses there will be a bias toward configurations having the minimum number of DSBs consistent with the observed pattern, as assumed in the simple probabilistic approach just described. For higher doses, aberrations with more DSBs than the minimum consistent number may play an important role, and a more extensive probabilistic calculation is required, using Monte Carlo simulations (e.g. 15,39). Then attention shifts from individual aberrations to probabilistic estimates for the entire observed aberration spectrum.

Aberration multigraph analysis has shown that this situation is more general. For example for aberrations involving 3 FISH colors (and no counterstain), we can unambiguously identify 3-DSB cycles but not 4-DSB cycles (this statement is contingent on the condition that no chromosome segments too short to be detected are involved; whether centromeres are scored or not is irrelevant in the present argument). Still more generally, a given observed pattern whose mPAINT descriptors involve n colors can never unambiguously identify the occurrence of an m-DSB cycle with m > n. If we assume enough DSBs, we can always produce any n-color observed pattern using at worst n-DSB cycles. Stated more formally, we have the following.

To get an example for part b, suppose n=2 and suppose the observed pattern has mPAINT descriptors

<Figure 9 about here>

The telomere removal trick

As we have discussed, it is in general hard to infer from an observed pattern the aberration cycle structure (which is the feature of an aberration most directly informative about enzymatic misrejoining pathways or interphase chromosome geometry) and the results can be ambiguous. If we somehow have some extra information on the initial configuration the situation is sometimes better. Here are some examples.

The proof of the theorem uses the fact that when telomeres are removed, one has a cubic multigraph (i.e. every vertex has three incident edges, as discussed in a different context above). This cubic multigraph structure leads to a remarkable symmetry: there is no longer, in an aberration multigraph with telomeres removed, any essential mathematical distinction between initial edges, misrejoining edges, and final edges. This unusual form of symmetry allows one to prove a number of unexpected results, including theorem 2. However, the argument is again lengthy and will be given elsewhere. In its place we add two comments.

1. Similar, but more complicated, results can be proved for cases where a few colors have two DSBs rather than just one.
2. For fairly low doses, the theorem's presupposition, that all colors have at most one DSB initially, tends to hold because if there are not many DSBs the chance that there are more than one on any one color is small. However, in general there could be a more complicated initial situation, so the telomere removal trick is only a partial answer to the problem of inferring exchange processes from observed final patterns, even assuming an mFISH/mPAINT protocol.

Two kinds of incompleteness can be incorporated into an aberration multigraph. When true incompleteness occurs some DSB free ends have no final partners, i.e. some of the final edges in the aberration multigraph are missing. Apparent incompleteness, caused by chromosome segments too short to observe, can be incorporated simply by giving the corresponding chromosome edges of the aberration multigraph an additional label (e.g. "s" for short). Incompleteness in general can then be defined as a situation where a free-end vertex has no detectable final edge incident on it. These possibilities give a substantially more natural way to describe incompleteness than has hitherto been available.

We introduced aberration multigraphs, which give the most systematic method to date of analyzing radiation-induced chromosome aberrations. Familiar aspects of aberrations, such as initial or final configurations, exchange processes, SECs, cycle structure, rearranged chromosomes, incompleteness, etc. re-emerge automatically as mathematical aspects of aberration multigraphs (Fig. 6). Using the multigraphs allows drawing some comparatively subtle but essential distinctions; for example there is a difference between the final configuration of rearranged chromosomes, as it actually occurs in a cell, and the observed pattern, which usually characterizes some aspects of the final configuration but not cryptic aspects (Figs. 3F and 6). We here emphasized graph-theoretical results relevant to the hard problem of trying to work out actual aberration exchange processes, which are the focus of interest biologically, from observed aberration patterns. Two theorems bearing on the problem, and also probabilistic estimates, were discussed.

Descriptors such as mPAINT descriptors or their generalizations, being determined by observed patterns, are not by themselves enough to classify aberrations fully. The present approach allows a sharp distinction between nomenclature, which depends only on an observed pattern, and "mechanistic" classifications, based on a multigraph as a whole. Mechanistic classifications involve various well-known mathematical concepts, for example the diameter of a multigraph.

Examples showed that using aberration multigraphs is more informative than studying separately the three basic aberration elements (initial configuration, exchange process, and final configuration; see Fig. 6). One main reason is that, serendipitously, aberration multigraphs are very closely related to cubic multigraphs, which happen to have received an unusually large amount of attention from mathematicians. In fact, by using the telomere removal trick discussed in the Results section, an aberration multigraph can be made into a cubic multigraph. Some normal genomes, and human mitochondrial DNA, consist of one or more circular chromosomes. Such a genome is especially amenable to aberration multigraph analysis because its aberration multigraphs are automatically cubic multigraphs, and the telomere removal trick, whose use can involve some loss of information, is never needed.

Graph theory has recently been applied to various aspects of computational biology (31), but the present paper's use of graph theory to analyze radiogenic chromosome aberrations and its use of cubic multigraphs are both new. As compared to the analyses used in modern large-scale comparative genomics (review: 16) a key difference is that aberration multigraphs specify the over-all process of aberration formation, not just the final configuration. A unique cycle structure is associated with the process, whereas in the comparative genomics case cycle decompositions are ambiguous (review: 18, Ch. 10). The radiation cytogenetics approach suggests that, in comparative genomics, searches in DNA sequences for details on breakpoints should be undertaken. Zooming in on breakpoints manifest in coarse-grained assays such as gene order or Zoo-FISH can in principle determine cycle structures, break point multiplets due to clustering or consecutive nearby breaks, and, for rearrangements made by ionizing radiation, characteristic features of particular kinds of radiation.

To summarize, we argued that in the long run a graph-theoretical approach will give the best way to describe aberrations, to analyze them mathematically, and to compare aberration data obtained using different protocols.

FIG. 1. Graphs and multigraphs. A graph consists of vertices and edges, with each edge connecting two different vertices and with at most one edge connecting any given vertex pair. For example, panel A shows a graph known as the wheel with six spokes and usually designated W₆. It has seven vertices (circles) and 12 edges (lines). Panel B shows a graph called the cyclic graph with six vertices. Graph theorists write this graph as C_{6 ,} but we here designate it c3 because our later treatment will emphasize DSBs rather than just vertices, with 2 vertices for 1 DSB and thus 3 DSBs for the 6-vertex graph shown. c3 is a subgraph of W₆, i.e. it is a graph obtained by deleting some of W₆, namely one vertex and 6 edges. When discussing graphs, only the pattern of vertices and edges matters, so that in panel B the hexagon and the lopsided figure are considered as the same graph, two ways to show c3. Panel C shows a subgraph of c3 which is not connected; rather it has three different connected components, each here consisting of two vertices and an edge. In general a connected component of a graph is obtained by starting at any vertex and tracing out as much of the graph as one can without lifting pencil from paper (repetitions allowed).

Multigraphs are like graphs except the restriction that at most one edge connect any given pair of vertices is dropped. Every graph is a multigraph, but not vice-versa. Panel D shows an example, which we designate c1, of a multigraph that is not a graph. Multigraphs that are not graphs are sometimes needed, e.g. when analyzing simple ring aberrations. To include all relevant cases, we will henceforth talk primarily of multigraphs, which automatically leaves open the possibility that any particular multigraph is also simply a graph. Submultigraphs, and connected components for multigraphs, are defined in essentially the same way as for graphs.

FIG. 2. A dicentric multigraph. Panel A shows the formation of a dicentric from two chromosomes, one painted red (color 1) and the other painted green (color 2). Panels B-F show relevant multigraphs (all of which are also graphs in this particular case).

Panel B gives the initial configuration, made up of telomeric vertices (a, d, e, and h), DSB free-end vertices (b and c, f and g), initial edges (shown dotted) specifying which free ends were originally partners, and chromosome edges (solid, colored lines). A prime indicates a centromere. The labels a-h are merely guides for the reader and such vertex labels will usually not be needed later.

Panel C shows the exchange process; the vertices are the DSB free-end vertices; the edges are the initial edges (dotted lines) from B plus two final edges (solid lines), specifying that b misrejoined with f, and c with g. A connected 4-vertex exchange process as shown in C characterizes a "2-DSB cycle", designated c2 and sometimes referred to as a 4-ends pairwise exchange or a complete reciprocal exchange. Here the "2" refers to the number of DSBs involved, half the number of free-end vertices in the exchange process multigraph.

Panel D shows the final configuration. Here the initial (dotted) edges of panel B have been removed and the final (solid) edges of panel C have been inserted instead. The two different connected components of the final configuration define the two rearranged chromosomes, the dicentric on the left and the acentric fragment on the right. The fact that there are corners here is irrelevant (compare the caption to Fig. 1B).

For any aberration, the number of DSBs involved is half the number of vertices in the exchange process submultigraph, which in turn equals the number of vertices in the aberration multigraph minus the number of telomeric vertices. These relations, and similar ones for the total number of edges, give for any aberration multigraph

# of vertices = 2(# of DSBs) + (# of telomeres) = (# of free-end vertices) + (# of telomeres).

# of edges = 3(# of DSBs) + (1/2)(# of telomeres)

An aberration multigraph will be called simple if two unrestituted DSBs are involved, complex if there are more than 2.

FIG. 3. Interpreting aberration multigraphs. Multigraphs give a way to see all relevant features

simultaneously. For example, the multigraph in panel A can be directly recognized as involving a centric ring. Panels B-D show explicitly the steps required, but after some practice with aberration multigraphs such extra diagrams are no longer needed. The detailed interpretation of panel A is obtained as follows.

Thus the aberration multigraph in panel A already contains all relevant information given in panels B-D. One can start with the intuitive picture of the aberration (panel D) and work upwards to construct the multigraph in A (see Figs. 2 and 4), but a preferred technique is to work directly with aberration multigraphs, which can also be constructed differently. For example, aberration multigraphs are often constructed by starting with an exchange process submultigraph and then adding chromatin edges.

E and F give two further examples, a reciprocal translocation and a paracentric inversion, on which the reader can practice. The crossing of two edges in F does not represent an extra vertex, just a convenient way to draw the multigraph. Panel G gives an example of a multigraph applicable to the recombinational misrepair model. The only difference from panel E is that one initial edge is labeled "d" to indicate a DSB produced directly by radiation, and the other initial edge is labeled "e" to indicate a DSB produced by enzymatic action during a cell's attempt to repair the first DSB.

The multigraph in panel A is not simply a graph because there is a pair of vertices (namely the free-end vertices on the right) connected by two different edges (compare Fig. 1D), but the multigraphs in panels E, F and G are graphs.

FIG. 5. A complex aberration involving 6 DSBs. The aberration multigraph gives the following properties for the three basic elements of the aberration:

As the flow chart indicates, most current cytogenetic experiments really only give information on one of the three basic aberration elements, the final configuration. Other kinds of experiments not considered here (for example pulsed-field gel-electrophoresis experiments for DNA fragment sizes prior to and during rejoining/misrejoining, or premature chromosome condensation experiments designed to catch the evolution of aberrations in time) do bear more directly on the other two basic elements (initial configuration and exchange process).

The flow chart indicates three major points of this paper:

FIG. 7. Examples of nomenclature. Panel A shows an intuitive picture of the formation of a simple dicentric with breakpoints at 1q22 (band 22 in the long arm of chromosome 1) and 5q13; pter and qter designate the ends of the short and long arms respectively. The mPAINT descriptors are given in A1. A2 gives the suggested descriptors if the protocol is banding instead. These suggested descriptors are similar to, but not identical with, "detailed" ISCN designations (21). Some of the differences between the suggested descriptors and the ISCN designations are designed to improve computer searchability in databases; others are suggested by the need for a general nomenclature applicable not just to banding but also to solid staining, mFISH/mPAINT, bar coding, etc. In general, for any protocol in which neither misrejoining is cryptic, the descriptors would have the form given in A3. For example, v = 1' in the mPAINT case; v = 1pter® 1q22 in the banding case; and for other protocols v could have other forms, in principle even an entire DNA sequence, as indicated by some further examples in the text. For some protocols the misrejoinings could be cryptic. For example, suppose chromosomes 1 and 5 are both counterstained in an experiment with some other chromosomes painted by FISH, and centromeres on counterstained chromosomes are not scored. Then the entire aberration is cryptic for this protocol and no descriptors apply.