A Strategy for Searching a Rare Event 21: Probability Distribution of Bindings

Abstract

In molecular biology, the central dogma has been the most important framework to explain major processes to maintain life. The first step is transcription in which the genetic information of a very tiny section of DNA is transferred to a piece of mRNA. During the process, several proteins are involved, like RNA polymerase and transcription factors (TFs). Especially, the binding of a TF to the right segment of DNA is critical, and has been studied a lot since early seventies. For instance, a TF, called a lac repressor, makes gene expression regulate by site-specific binding to DNA, called a lac operon, in a living E. coli cell. The process is to search a rare event because a FT should find a right specific binding site among millions of sites on DNA. Earlier, it was reported that searching time in an experiment in vitro was around two orders of magnitude faster than the Smoluchowsky diffusion limit process. Since then, similar experimental results were interpreted by the facilitated diffusion(FD), which is a searching mechanism of 3D nonspecific binding with 1D sliding along DNA. Recently the FD mechanism has been supported by a few of single-molecule experiments, but also has been questioned due to the long sliding distance through crowding in the cytoplasm. In addition there are a variety of controversial issues going on about the detailed mechanisms like non-specific binding and unbinding, hopping or sliding for 1D searching, and so on. They were usually focused on binding steps between a searching molecule and target sites. Previously I reported a qualitative expectation that unbinding steps might be more important than binding steps in searching a rear event through a probability theory called Bayes’ theorem. This report shows that probability distributions of bindings may also play a key role for a rare event searching. The previous report showed that the probability of searching a rare event, like binding TFs to the repressor on chromosomal DNA, depends more on detaching process rather than binding process of a right targeting. As the previous report, ‘B’ denotes the events of a searcher binding with a right targeting site, ‘not B’ with the other wrong sites. ‘A’ denotes the events that a searcher bound with any targeting site keeps binding with the site, and ‘not A’ means falling apart immediately after binding. P(A), P(not A), P(B), and P(not B) are their probabilities respectively. According to Bayes’ theorem, the conditional probability of a right targeting site given the events kept binding is as follows;