A transcription factor is a molecule that binds to a specifi

A transcription factor is a molecule that binds to a specific DNA sequence in order to control the expression of the nearby genetic material. In this problem we will look at the process by which the factor finds the correct sequence to bind to. Assume that, factor can bind loosely to a DNA sequence that does not match the target sequence. After every 10 ns the molecule will move 1 base pair with a 50/50 chance that the move is to the left or to the right. After 100 ns. what is the probability that the transcription factor is 6 base pairs to the left of when it started? Where is the most likely place for the factor to be after 100 ns? What is the probability that it. ends up at the most likely place? Does your answer from part (b) mean that the factor will never find its target sequence? Why or why not?

Solution

a. Within 100ns, the number of movements = 100ns / 10ns = 10

So in these 10 movements, net position should be 6 positions left.

This is possible when it has travelled total 8 positions left and 2 positions right, irrespective of the order.

p for left = p for right = 0.5

P(left 6) = 10C8*(0.5)⁸ *(0.5)²   = 0.0439

b. Most likely place is the same place where it starts as at each and every movement, there is an equal chance of moving left or right, hence, on an avergae we can expect the position to remain the same as:

Expected change = -1*0.5 + 1*0.5 = 0

Probability = Moving 5 steps left and 5 steps right in 10 steps: 10C5 * 0.5⁵ * 0.5⁵ = 0.2461

c. It does not mean that it will never meet the target as the above event is just an expectation and reality may differ from expectation.