Biomod/2011/Caltech/DeoxyriboNucleicAwesome/Random Walk Formula
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 the adjacent one. Various formulae had been developed to model this process, but none of them can perfectly match our situation (Weesakul, 1961; Hardin & Sweet, 1969; Alessandro Blasi, 1976; Ahmed El-Shehawy, 1992).
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.