Biomod/2011/Caltech/DeoxyriboNucleicAwesome/Random Walk Formula
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The generating function for <math>h(t;i)\!</math> can be expressed as  The generating function for <math>h(t;i)\!</math> can be expressed as  
  <div class="center" style="width:auto; marginleft:auto; marginright:auto;"><math>H_i (s) = \sum_{t=  +  <div class="center" style="width:auto; marginleft:auto; marginright:auto;"><math>H_i (s) = \sum_{t=0}^\infty h(t;1)s^t</math>, <math> s <1 \!</math></div> 
Following Netus (1963), the explicit expression of the generating function is  Following Netus (1963), the explicit expression of the generating function is  
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Assume that <math>T_0 (s)\!</math> has <math>k\!</math> distinct roots <math>s_1,s_2,..., s_k , H_i (s)\!</math> can then be decomposed into partial fractions  Assume that <math>T_0 (s)\!</math> has <math>k\!</math> distinct roots <math>s_1,s_2,..., s_k , H_i (s)\!</math> can then be decomposed into partial fractions  
  <div class="center" style="width:auto; marginleft:auto; marginright:auto;"><math> H_i (s) \!</math></div>  +  <div class="center" style="width:auto; marginleft:auto; marginright:auto;"><math> H_i (s) = \frac {\rho_1} {ss_1} + \frac{\rho_2}{ss_2} +...+ \frac {\rho_k}{ss_k} \!</math></div> 
+  
+  where  
+  <div class="center" style="width:auto; marginleft:auto; marginright:auto;"><math> \rho_k = \frac {q^i {s_m}^i T_i (S_m)}{{T_0}^' (s_m)}, m\leqslant k \!</math></div>  
+  
+  It follows that  
+  <div class="center" style="width:auto; marginleft:auto; marginright:auto;"><math>h(t;i) = \sum_{m=1}^k \frac {\rho_m}{{s_m}^{t+1}} \!</math></div>  
+  
+  <math>h(t;N)\!</math> can be similarly deduced from <math>H_N (s)\!</math> using the same method.  
==References==  ==References== 
Revision as of 02:02, 3 October 2011
Thursday, August 28, 2014

Random Walk Formula
General Modeling IdeaThe 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 FormulaTwo 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 where and 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 where It follows that can be similarly deduced from using the same method. References
