Biomod/2011/Caltech/DeoxyriboNucleicAwesome/Random Walk Formula
< Biomod | 2011 | Caltech/DeoxyriboNucleicAwesome
Random Walk Formula
General Modeling Idea
The random walk on DNA origami can be modeled as one dimensional random walk with a reflecting and an absorbing barrier (Figure 1). Tracks in the same column are grouped into rectangles, and each step is defined as walking from one rectangle to an adjacent one.
Consider a random walk on a line segment with N+1 sites denoted by integers (0,1,2, … , N) (Figure 2). The walker starts random walk at site i, 0 < i ≤ N. Let p be the probability for the walker to move one segment to the left, q be the probability for the walker to move one segment to the right. The probability for the walker to stay at a particular site for the next unit time is thus r = 1 – p – q. When the walker reaches site N, the partially reflecting barrier, it has a probability of β to be reflected back to site N – 1, and a probability of α = 1 – β to stay at site N in the next unit time. When reaching site 0, the absorbing barrier, it stays there for 100% probability and the random walk ends.
Random Walk Formula
Two assumptions are made in our case. 1) The DNA origami is immune to any free floating walkers in solution, meaning that free floating walkers cannot bind to an origami and starts random walking; 2) walkers are immediately absorbed when reaching the rectangles with WGs, despite the presence of two TR2 in the same rectangle.
Let be the probability that the walker reaches 0 for the first time after steps given its starting position being . obeys the following difference equation
for and . We define ; . Also, .
When we have
The generating function for can be expressed as
Following Netus (1963), the explicit expression of the generating function is
The explicit expression of can thus be deduced from using partial fraction expansion (Feller, 1971).
Assume that has distinct roots can then be decomposed into partial fractions
It follows that
can be similarly deduced from using the same method.
Possible positions of walkers on the origami can be grouped into two categories, either in the center of rectangles (Figure 2, Line 2) or at the sides (Figure 2, Lines 1&3), and the values of p, q and r are different in each category. Hence there is a need to find out the distribution of walkers in these two categories in order to estimate p, q and r. Due to the nature of our design, each walker has a probability of 50% to stay at its original track after one branch migration. Assuming no reflecting boundaries, when the walker is in the center (Figure 2, Line 2), the probability of staying in Line 2 after one branch migration is 1/2+1/2×1/2=3/4, while the probability of going to Lines 1 or 3 is 1- 3/4=1/4. Similarly, when it is in Lines 1 or 3, the probability of staying in the same line after one branch migration is 1/2+1/2×2/3=5/6 while the probability of going to Line 2 is1- 5/6=1/6. This process can be modeled using Markov chain as follows.