Well… I should use “proof” in quotes, but the simulations below should show that we will always have some “sticky” segments which survive many generations. Technically, I suppose, a “sticky segment” is one that is passed from one ancestor to another intact. However, “sticky” is usually used in the sense of segments which pass through many generations, intact.

Here are the ground-rules I used for this analysis:

1. Use a 200cM chromosome (about the size of chromosome 5, 6 or 7)
2. Assume 2 crossovers per generation. By definition, there will be an average of one crossover in 100cM per generation. So, on average, there will be two crossovers per generation in 200cM. Sometimes there are one or three crossovers; and infrequently there are none or four (or more) crossovers. Since it more or less evens out, I’m using the average of two crossovers per generation to illustrate what happens over 10-20 generations. This avoids any bias on my part.
3. Use a simplification. Each time there is a crossover, there is a switch from one ancestor’s DNA to another’s. However, the other ancestor’s DNA is subjected to the same one-crossover-per-100cM rule. So for the purposes of this discussion [about how segments are subdivided by crossovers, and not about which ancestor they come from], I will use a simplification: I’ll assume that the other ancestor’s DNA is exactly like the first ancestor’s DNA at that location, as far as crossovers are concerned. This means I’ll just continue to subdivide the segments in the initial 200cM segment, generation after generation. It’s a whole lot simpler and easier than starting with 2,048 chromosomes of different colors and keeping track of those. This simplification will give essentially the same result of segment subdivision, and is much easier to follow.
4. Assign crossovers to the middle of the largest segments. The DNA is very random, but to A) avoid any bias on my part, and B) show the worst case scenario, I will assign the two crossovers in each generation to the middle of each of the two longest segments in the 200cM chromosome. Of course, in real life some crossovers will subdivide a small segment (leaving a different, larger segment intact for another generation); or subdivide a large segment into very unequal parts (leaving one smaller segment, plus a somewhat larger segment – it all evens out in the average).

So apply these rules to the 200cM chromosome in the figure below.

Figure 1. Tracking Crossovers for 24 Generations

Note the first two crossovers subdivide the segment into roughly three equal segments. As outlined in Rule 3 above, the center segment will really be from a different ancestor, but for the purpose of following the subdivision of segments in general, we will continue with all the segments – on average this gives the same result as far as subdivided segment sizes go. Note: the segment which are subdivided are highlighted in yellow.

In the next generation two of the largest segments are subdivided, but the third fairly large segment remains intact. This is because we only have (on average) two crossovers per generation, so one of the three large segments in generation 2 will not be subdivided.

Moving to generation 3 we see the largest (66cM) segment is now subdivided by one of the two new crossovers for this generation, and then a 33cM segment is subdivided by the other crossover for this generation.  And we still have two 33cM segments and one 34cM segment passed intact.

Continue this process of subdividing the two largest segments in half in each generation, until we get to generation 11, where all the segments are now 8 or 9cM except one that is still 16cM. This 16cM segment has remained intact in 6 generations! In fact there are two other 16cM segments that remained intact for 6 generations.

Will it always happens this way? No – the DNA is very random, and a different result will happen every time. But if one of those “sticky” segments had been subdivided in an earlier generation, some other segment would not have been subdivided, and that segment would have remained “sticky” for another generation.

The takeaway here is that there are only two crossovers per generation in a chromosome about 200cM. Those two crossovers can only subdivide two of the many segments in the chromosome. And because of this, there are some segments that will pass down intact, generation after generation.

In actual practice, there are hot spots on each chromosome where crossovers are more likely to occur. The effect of these hot spots is that some smaller segments around the hotspots will be subdivided more frequently, and some other segments will be missed more frequently – leaving us with even more “sticky” segments elsewhere.

After generation 11, our process starts to subdivide some segments into segments so small that they would not show up as a shared segment – they are below the standard thresholds (about 7cM) for a match. But notice in generations 12 through 22, that there are still above-threshold segments. Even in generation 23 there is still an 8cM segment which has survived intact for 14 generations! Remember: if this particular segment had been subdivided, then some other segment would have not been subdivided.

The point is we should expect “sticky” ancestral segments. Particularly in the 7-10cM range. They are actually quite common. Even “sticky” segments in the 10-20cM range are usual, even after 7-10 generations.

Now, we have not studied the probability of matches at these great distances – that’s a different, somewhat harder, discussion. The point here is that there are many “sticky” segments in our DNA. They may come from generations that are generally way beyond our genealogies. Also we should not be surprised when we see a parent and child with essentially the same 7-10cM segment being shared with a Match. It happens pretty frequently with close relatives.

Here is another example based on a 100cM chromosome (think chromosomes 19-22). For this chromosome there is only an average of one crossover per generation – only one segment will be subdivided in each generation. I tried to place them more randomly this time [you can easily try your own pencil & paper sketch of this simulation]. I generally picked on the largest segment. After 11 generations over half of the segments are still over the (7cM) threshold. And several segments 10cM or over have survived, intact, for a number of generations.

Figure 2. Tracking Crossovers in a 100cM Chromosome.

Summary

1. Ancestral “sticky” segments in the 7-10cM range are normal. We will have Matches with these segments from time to time – and some of them may be fairly distant. But that’s another story.
2. Some segments over 10cM will survive from 9 or 10 or more generations back – it’s normal and expected. Again, matching is a different calculation.
3. The point is: we have many above-threshold segments from distant ancestors, back 10 generations and more!
4. Since we have many above-threshold ancestral segments from distant ancestors, on every chromosome, we should expect to have shared segments with distant cousins.

07C Segment-ology: Proof of Sticky Segments by Jim Bartlett 20151116