We are almost ready to develop a fast algorithm for finding the 2-Break Distance. The only thing left for us to accomplish it is to prove the 2-Break Distance theorem. Let's consider an arbitrary transformation of genome P into genome Q. We don't know what 2-breaks accomplish this transformation. But what we know is that during this transformation, the breakpoint graph of P and Q changes into the breakpoint graph of Q with itself. Which means that the cycle number of P and Q is changing into the cycle number of Q with itself which in turn is simply the number of blocks in Q. Therefore, whatever transformation of P and Q accomplishes it, the number of red-blue cycles increases by the number of blocks in the genomes P and Q minus the number of cycles between P and Q. And therefore, the next question I will ask, how much each 2-break can contribute to this increase in the number of cycles? Let's look into this. Well, there are different scenarios on which two red edges from the breakpoint graph to break effect. Every 2-break removes two red edges. Consider for instance the case when these two red edges belong to the same cycle in the break point graph. In this case shown on this slide two red edges are removed, substituted by another two red edges, but the cycle number doesn't change. Here's another case where the same two red edges are removed from a single cycle in the break point graph. But the cycle number this time increases by one. And now lets consider the case when two removed red edges belong to two different cycles. You can see that in this case doesn't matter which two red edges replace these removed red edges. The cycle number doesn't change. Actually, the cycle number decreases by one in this case. And therefore, each 2-break increases the number of cycles by at most one. We almost proved this theorem. But actually, we haven't proved it yet. My previous slide was simply an invitation for you to look at the picture. Let's give a little bit more formal proof. So a 2-break adds two new red edges, and thus creates at most two new cycles in the breakpoint graph. On the other hand, it removes two red edges, and thus destroys at least one old cycle in the breakpoint graph. And therefore, the change in the number of cycle is at most 2 minus 1 which is at most 1, and on the other hand, if you watched carefully my proof, then you noticed, there is always a 2-break increasing the number of cycles by 1, and here it is, the central case here present. In example of such increase. And now is the 2-break distance theorem. We saw that each 2-break increases the number of cycles by at most 1. On the other hand, there exists a 2-break increasing the number of cycles by exactly 1. On the other hand it is so that every sorting by 2-break must increase the number of cycles by the number of blocks between P and Q, minus the number of cycles between P and Q, and therefore the 2-break distance between genomes P and Q is simply the number of blocks between P and Q minus the number of cycles within P and Q. And armed with the theorem, we actually can compute the 2-break distance between circularized human and mouse chromosomes. We know that human and mouse genomes can be decomposed into 280 synteny blocks. Each of these blocks will be larger than half a million nucleotides in length. The breakpoint graph on this block if we look at this actually contains 35 cycles. And therefore the 2-break distance between human and mouse equals to 245. Remember this number because it will be important in the next section. Now there are numerous 245 step scenarios, and we do not claim that we know which one of them is correct. Moreover, the true scenario may have more than 245 steps. Important thing for us is to remember that there are at least 245 steps to proceed to our next section, where we will analyse the random breakage model.