# Negative binomial-distribution as background-model: How to calculate the p-value?

I would have a question to the statistics of the following paper:

Bioinformatics, Volume 28, Issue 23, 1 December 2012, Pages 3013–3020, https://doi.org/10.1093/bioinformatics/bts569

Simple summary:

This paper deals with the identification of interaction sites of a protein binding to various RNAs. Imagine the RNA as a string of positions where the protein can bind to (x-axis), with the number of bindings on the y-axis. You get random binding by chance: The more copies of this exact RNA you have in the cell, the more instances of random binding. This function of binding (random or specific) was found to be nicely fitted with zero-truncated binomial distributions. The idea is that only positions on the RNA that significantly exceed these negative binomial distributions-fit are considered "specific".

My question concerns the calculation of this significance of exceeding the background: They imply that this p-value depends on the amount of RNA and show how they account for different RNA concentrations in point "2.3 Peak finding".

Could you explain this concentration dependence to me? It is known that RNA concentration is proportional to the overall amount of observed bindings, so you basically just fill up your RNA-positions with more binding-events, BUT the proportions can be assumed to stay the same.

Isn't this model already robust to different RNA concentrations? What I mean is: Given you have sufficient RNA concentrations, would even higher RNA concentration lead to higher p-values? (without accounting for concentration) So would you expect to find more positions of specific binding, the higher the concentration of the RNA?