We will now talk about the breakpoint graphs the workhorse of genome rearrangement studies. Let's take a look at two genomes, a red genome P, and a blue genome Q. And now let's arrange the black edges of Q in the same order they are arranged in genome P. So they will be arranged as +a -b -c +d. And how the genome Q will look like in the case of this new arrangement of its black edges? We do not change genome Q. We just show it differently. So after we look at genome Q: so, after a we should go to c. That's what we do. After c, we should go to b. After b, we should go to -d, and after -d, we should go to a. So that's a new representation of Q, where black edges are arranged exactly in the same way as they are arranged in P. As soon as we've done it, we can superimpose genome P with genome Q because black edges are arranged identically, and we will get something that is called the breakpoint graph. It's absolutely unclear yet what is the value breakpoint graph, so let's learn a few things about the breakpoint graph. So the first question, red and black edges in the breakpoint graph... what do they correspond to? Of course, they correspond to genome P, because breakpoint graph is simply obtained by superimposing of P and Q. What about blue and black edges? What do they correspond to? Of course, they correspond to genome Q. And now we ask a strange question, what about red and blue edges? What do they correspond to? And here, what happens when we limit our attention to red and blue edges, and they do not corresponding to anything. So a thing that you may notice, however, is that red and blue edges form alternating red-blue cycles in the breakpoint graph. Why? Because you may notice that at every node, there is a single red edge and a single blue edge meeting. And therefore the resulting graph must consist of alternating red-blue cycle, after every red edge, there is a blue edge, and after every blue edge, there is a red edge. We'll pay attention to one important parameter of the breakpoint graph, which is the cycle number. The cycle number is simply the number of red-blue alternating cycles in the breakpoint graph. Why do we care about the cycle number? But before I explain this, let's try an example, let's construct the breakpoint graph for these two genomes, P and Q. P consists of a single chromosome and Q consists of 2 chromosomes. The first thing we do, we arrange black edges of Q in the same order they are present in genome P. So it will be this arrangement of black edges in Q. And then we will have to represent genome Q in this order. How do we do this? Well, we start from a. After a, we know that we should go to -c. Afterwards we go to -f. Afterwards, we go to -e, and finally we return back to +a. And we continue the same way to show the smaller chromosomal with just two synteny blocks, right here. As soon as we represent it, genome Q, in this way, we can superimpose P and Q, and we get the breakpoint graph. And after we remove black edges from the breakpoint graph, we see that the cycle number of P and Q is equal to 3, because there are three cycles in this graph. Now, given genome P, I want to ask you, what is the genome Q that maximizes the cycle number? Well, the cycle number will be maximized if every cycle will be small. And the shortest cycle consist of just two edges, one red edge and one blue edge. Which means this is the breakpoint graph, with the maximum number of alternating cycles. And of course, this breakpoint graph corresponds to the case when genome Q and genome P are identical. And the number of cycles in this graph is of course simply the number of blocks in genome P. So in this case, the number is 4. So it's important to realize that genome rearrangements affect red-blue cycle. Every two breaks makes its mark on the breakpoint graph. And we know that starting from the breakpoint graph between genome P and Q, doesn't matter what series of rearrangement nature takes. That will be the breakpoint graph of two identical genomes, and we know how it looks like. It looks like the identity breakpoint graph with the maximum number of cycles. And we will explore this fact, despite the fact that the series of 2-break transforming P into Q is unknown, we know the number of cycles between P and Q, in this case 2, must change into the maximum number of cycles between P and Q, which in this case is 4. And this is just an example of one possible scenario for rearrangement of genome P into genome Q, but doesn't matter what this scenario is. It always will be the change from cycle(P,Q) to the maximum number of cycles in the graph. And armed with this observation, we are ready to prove the 2-break theorem that will prove to be very important for our analysis of the random breakage model.
