Biomod/2011/Caltech/DeoxyriboNucleicAwesome/Simulation
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(Detailing random walk simulation) 
(Finishing up random walk section with some results of simulation) 

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  +  [[Image:RandomWalkTest22500x.jpg  thumb  300 px  right  A plot of the number of steps (on an average over 2500 iterations) it takes a walker to random walk from any point on the origami to the irreversible track at <16, 8>.]]  
+  ===Results===  
+  Results of the bulk data collection at right show that the average randomwalk duration, and thus the time for <math>(fluorescence_{initial}  fluorescence_{current})</math> to reach some standard level, increases with distance, though it changes less significantly the farther out one gets. Also important to note is that the "effective distance" (in terms of steps) along the short axis of our platform is a significantly less than the same physical distance along the long axis. This difference is due to our arrangement of track A and B: as can be seen in the left half of the diagram at the end of the [[#Overview]] section, alternating tracks A and B create a straight ''vertical'' highway for the walker to follow. ''Horizontal'' movement, in contrast, cannot be accomplished by purely straightline movement  it requires a backandforth weave that makes motion in that direction slower. The disparity in "effective distances" between the vertical and horizontal dimensions is something, in particular, that we should test for; however, a simple series of tests running random walks at a variety of points across the surface, and the comparison of the resulting fluorescence data to the control provided by this simulation should be sufficient to prove that our walker can, indeed, perform a 2D random walk.  
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Revision as of 14:50, 23 June 2011
Tuesday, July 22, 2014

Simulations
OverviewOur proposed sorting mechanism depends very heavily on a particular randomwalking mechanism that has not been demonstrated in literature before. The verification of this mechanism is thus a vital step in our research. Verification of the random walk in one dimension is fairly straightforward: as discussed in <LINK TO THE EXPERIMENTAL DESIGN SECTION>, a onedimensional track is easy to construct, and will behave like a standard 1D random walk, showing an average translation on the order of after n steps. Thus, we should expect the time it takes to get to some specific level of fluorescence to be proportional to the square of the number of steps we start the walker from the irreversible substrate. If we can, in an experiment, record the fluorescence over time when the walker is planted at different starting points and show that that fluorescence varies by this relationship, we'll have fairly certainly verified onedimensional random walking. Our particular case of 2D random walking, however, is not as easily understood, especially considering the mobility restrictions (ability to move to only 4 of 6 surrounding locations at any particular time) of our particular walker. As a control for the verification of 2D random walking, though, we still need to get an idea how long the random walk should take, and how that time will change as we start the walker at different points on the origami. We opt to do this by simulating the system with a set of movement rules derived from our design. We also use the same basic simulation (with a few alterations and extra features) to simulate our entire sorting system in a onecargo, onegoal scenario, to give us some rudimentary numbers on how long sorting should take, with one vs multiple walkers. Basic parameters and assumptions:
MATLAB CodeAt the core of the simulation is a function which runs runs one random walk on an origami of specified size. It can run in both a cargobearing (onecargo onegoal) and a purely randomwalk mode. The former has cargo positions corresponding to our particular origami preprogrammed and starting with multiple (specified by user) walkers at random locations on the origami, and terminates when all of the cargos have been "sorted" to the goal location (the x axis). The latter runs one walker starting at a specified location, and terminates when that walker reaches the specified irreversible track location. The function returns a log reporting when cargos were picked up and dropped off, and a count of the number of steps the simulation took. This function is utilized by separate cargobearing and randomwalk data collection programs that call the function many times over a range of parameters. The function code (saved as randomWalkFunction.m):
RandomWalk SimulationThe data we need from this simulator is a rough projection of the fluorescence response from our test of 2D random walking, which should change based on the starting location of the walker. Because this fluorescence is changed by a fluorophorequencher interaction upon a walker reaching its irreversible track, in the case where we plant all of the walkers on the same starting track, the time it takes (fluorescence_{initial} − fluorescence_{current}) in the sample to reach some standard value should be proportional to the average time it takes the walkers to reach the irreversible substrate. As this 'total steps elapsed' value is one of the outputs of our simulation function, we can generate a map of these average walk durations by running a large number of simulations at each point on the origami and averaging the results:
ResultsResults of the bulk data collection at right show that the average randomwalk duration, and thus the time for (fluorescence_{initial} − fluorescence_{current}) to reach some standard level, increases with distance, though it changes less significantly the farther out one gets. Also important to note is that the "effective distance" (in terms of steps) along the short axis of our platform is a significantly less than the same physical distance along the long axis. This difference is due to our arrangement of track A and B: as can be seen in the left half of the diagram at the end of the #Overview section, alternating tracks A and B create a straight vertical highway for the walker to follow. Horizontal movement, in contrast, cannot be accomplished by purely straightline movement  it requires a backandforth weave that makes motion in that direction slower. The disparity in "effective distances" between the vertical and horizontal dimensions is something, in particular, that we should test for; however, a simple series of tests running random walks at a variety of points across the surface, and the comparison of the resulting fluorescence data to the control provided by this simulation should be sufficient to prove that our walker can, indeed, perform a 2D random walk. 