We'll now make a strange and counterintuitive step. You know that human chromosomes are linear. But we want to think about them as circular, cyclic chromosomes. Well, why would we do this? Why would we distort the biological reality? The only reason is that it will make our algorithmic problem much simpler while actually hardly affecting our biological conclusion. Remember, I told you that analyzing rearrangements in linear chromosomes is extremely difficult, even if we're dealing with a unichromosomal genome. Analyzing rearrangements in multichromosomal linear genomes becomes nearly impossible. And, surprisingly enough, analyzing rearrangements in multichromosomal genomes with cyclic chromosomes will turn out quite simple, we are about the discover it. So now we will look at the cyclic genomes. In this case, cyclic genome, a genome with a circle chromosome of consisting of two chromosomes, and we will arrange blocks in these chromosomes as directed black edges. For example, +a, -b, -d, +c will be arranged as four edges with corresponding orientation of these edges along the circle. It's just a start. We will now add to these black edges undirected red edges that connect adjacent synteny blocks. So the two chromosomal genome will be represented as a graph consisting of two cycles. These cycles, important, they have edges of two colors, red and black. Now what do we do next? First, let's notice that for every genome there are many different ways to visualize it. For example this is genome Q arranged in natural order along the cycle, but we can change arrangements of black edges of Q, and how in this case this cycle looks like. Well, after +a, we should go to -b. That's what we do. After -b, you should go to -d, that's what we do. And after -d, we should go to +c, and finally to +a. This is, this is the same genome and the same graph in fact, the same cycle, just visualized differently. Now let's try to figure out how a reversal looks like in terms of these graphs. So, we start with genome P: +a -b -c +d, and let's make a reversal on this genome, resulting with +a -b -d and +c. How this new genome Q will look like in the terms of our graph model. Well, after +a we should go to -b, after -b we should go to -d after -d we should go to +c and finally from +c we should go to +a So, therefore a transformation of genome P into genome Q corresponds also to transformations of the graph shown here. And how can we summarize what this transformation does. Certainly it removes these two red edges. And it adds two new red edges. So in summary, a reversal essentially now has geometric interpretation. It deletes two red edges and replaces them by two other red edges on the same four nodes. Now, let's now talk about fission, breaking a single chromosome into two. In this case, a unichromosomal genome P is broken into a genome with two chromosomes: the genome Q. So how does it look like in terms of our graph? So for, from +a, we should go to -b and from -b, we should go to +a, so it will look like this. And also from -c we should go to +d and from d we should go to c. So you see that geometrically also a single cycle is broken into two cycles. What does it mean geometrically again? Once again, we remove two red edges and add two new red edges. And similarly, in the case of fusion, the reverse operation, we would also remove two red edges and add two new red edges. And therefore, we define the notion of "2-break", a rearrangement that replaces two red edges with two new red edges on the same four nodes. What we have just accomplished, instead of four different operations, reversals, translocations, fusions and fissions, we have a single operation, a 2-break. I describe how to model reversals, fusions, and fissions as 2-breaks. What about translocations. It's the same story. Here are two linear chromosomes that after translocations transform into a blue chromosome and a green chromosome shown here. Once again we represent them as black-red paths. Let's circularize them by just gluing the ends using one of the red edges. And after we circularize them, let's perform this 2-break. We'll choose these two edges. We'll remove them and substitute them by other two red edges. And afterwards, we will simply remove the edges that we used for circularization, like here, which is essentially linearization of circular chromosomes. So if you look at this, then you see that we essentially mimic translocations using our graphs. And we will define 2-break distance as the minimum number of 2-breaks transforming one genome into another. And 2-break distance problem: input two genomes on the same set of synteny blocks, and output the 2-break distance between these genomes. It's now time to solve this problem. But before we do this, we need to introduce the key notion in studies of genome rearrangements which is called the breakpoint graphs.