A Probability Problem In Sequencing

Question

[Tianyang Li]

Hi,

In sequencing, the asymptotic numbers of gaps and contigs have been give the Lander-Waterman model. But the results from the Lander-Waterman model is only asymptotic and doesn't work well when the sequence being sequenced is not long enough (e.g. RNA-seq).

Exact calculations when a circle is randomly covered with arcs have been give (Geometric Probability - Solomon, and several papers by Wendl)

I'd like to see a calculation for line segment that's similar to Chapter 4 of Solomon's Geometric Probability.

I'm wondering if there's any available work when a line segment is covered by shorter line segments?

Thanks!

[Michael]

I see no obvious connection between the first paragraph and the following two sentences, except that they refer to a definition of 'coverage'. Is it that what you are looking for. If so, it would help a lot if you can add a sentence stating what you are interested in.

[Tianyang Li]

@Michael I edited it a little so I hope it's more clear now.

Answer 1

[Zam Iqbal]

Can you clarify your question? Do you want exact solutions to the problem when the length of the shorter segment is comparable with that of the longer? Do you want to know, for example, what the probability of a given gene having coverage blah when the overall coverage of the transcriptome is blahblah? You could try Leeuwaarden, Lopker and Janssen "Connecting renewal age processes with M/D/1 and M/D/infinity queues through stick breaking"

[Aaronquinlan]

This should be a comment, not an answer.

[Tianyang Li]

Specifically, I'd like to see a similar solution to the one in Chapter 6 of Geometric Probability by Solomon. Thanks for you advice!

[Tianyang Li]

Specifically, I'd like to see a similar solution to the one in Chapter 4 of Geometric Probability by Solomon. Thanks for you advice!

Answer 2

[Tianyang Li]

This problem can actually be solved using the exact same method as Chapter 4 of Solomon's geometric probability.