Biomod/2011/Caltech/DeoxyriboNucleicAwesome/Simulation

Tuesday, April 16, 2024 Home Members Project Protocols Progress Discussion References

Simulations Simulation of Expected Results 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 <LINK TO THE EXPERIMENTAL DESIGN SECTION>, 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 [math]\displaystyle{ n^{\frac{1}{2}} }[/math] 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 substrate locations (good locations to step to) around it. The walkable substrates 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>. An illustration of the grid and motion rules used in the simulation. The bottom-left is the origin (<1,1> because MATLAB indexes by 1). The 2D platform that will be used for random walking, including substrate A (red), substrate B (blue), the marker (black), and the irreversible substrate (purple), is shown on the left. The grid on the right -- the grid corresponding to our numbering system -- is created by shifting even columns up by 0.5. This arrangement reveals through the vertical symmetry of the arrangement that movement rules are going to vary by column only. The valid moves in even and odd columns shown on the left are mapped onto the grid on the right to derive the moveset listed above.