Biomod/2011/Caltech/DeoxyriboNucleicAwesome/Simulation

=Simulations=

Overview
Our proposed sorting mechanism depends very heavily on a particular random-walking 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 , a one-dimensional track is easy to construct, and will behave like a standard 1D random walk, showing an average translation on the order of $$n^{\frac{1}{2}}$$ 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 one-dimensional 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 one-cargo, one-goal scenario, to give us some rudimentary numbers on how long sorting should take, with one vs multiple walkers.

Basic parameters and assumptions:
 * The unit of time is the step, which is the time it takes a walker to take a step given four good opposite track locations (good locations to step to) around it.
 * The walkable track are given coordinates like a grid (which shifts the even columns up by 0.5). The bottom-left is <1, 1>, the top-left <1, 8>, and the bottom-right <16, 1>.
 * Movement rules are based on column:
 * In even columns, a walker can move in directions <0, 1>, <0, -1>, <1, 0>, <-1, -1>.
 * In odd columns, a walker can move in directions <0, 1>, <0, -1>, <-1, 0>, <1, 1>.



MATLAB Code
At 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 cargo-bearing (one-cargo one-goal) and a purely random-walk mode. The former has cargo positions corresponding to our particular origami pre-programmed 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 of all walkers positions over time, 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 cargo-bearing and random-walk data collection programs that call the function many times over a range of parameters.

The function code (saved as randomWalkFunction.m): Toggle Code 

Examining Errors in Origami
This code can be used to generate diagrams like those below, visualizing the mobility of the walker. One immediate question thus far unanswered is the vulnerability of this layout to errors in the laying of track. We investigate this by, when generating the track layout in the beginning of randomWalkFunction, introducing a small (specified by input) percent chance that any single track will be omitted. Error rates at around 10% are bearable; error rates greater than that, however, are catastrophic, causing walkers to become permanently trapped in small sections of the track field.

Random-Walk Simulation
The 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 fluorophore-quencher 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: Toggle Code 

Results
Results of the bulk data collection at right show that the average random-walk 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 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 straight-line movement -- it requires a back-and-forth 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.

Cargo Sorting Simulation
This simulation investigates both the overall tractability of our sorting problem, and the degree to which it can be parallelized via the addition of multiple walkers onto a single origami. It runs by making repeated calls to randomWalkFunction in its cargo-bearing mode, testing the number of steps it takes to sort all five cargos to respective goals over a range of number of cooperating walkers: Toggle Code 

Results
While a single walker takes over a thousand steps to complete the sorting challenge, the addition of even a single walker vastly decreases the completion time, and additional walkers decrease it further, until a critical point is reached where the walkers are more getting in the way than helping with the sorting process. This is visible in the positive slope visible in the diagram at right that starts at around the 20 walker point.