Biomod/2011/Caltech/DeoxyriboNucleicAwesome/Simulation

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[[Image:RandomWalkTest2-2500x.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>.]]
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===Results===
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Results of the bulk data collection at right show that the average random-walk 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 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.
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Revision as of 14:50, 23 June 2011

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Saturday, August 23, 2014

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Contents

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 <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 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>.
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 track A (red), track B (blue), the marker (black), and the irreversible track (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.
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 track A (red), track B (blue), the marker (black), and the irreversible track (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.

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 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):

function [cargoLog, steps] = randomWalkFunction(x, y, length, numWalkers, ...
    startPos, endPos, cargoBearing, restricted)
%x: Width         y: Height
%length: max # of steps to run simulation
%numWalkers = number of walkers to simulate in cargoBearing state
%startPos = starting position for walker in randomwalk state
%endPos = irreversible track location in randomwalk state
%cargoBearing = running cargoBearing (1) vs randomWalking (0)
%restricted = whether we're paying attention to borders

%Random walking cargo pickup/dropoff simulation
%for origami tile, x (horizontal) by y (vertical) dim.
%Locations index by 1. x+ = right, y+ = up
% Gregory Izatt & Caltech BIOMOD 2011
% 20110615: Initial revision
% 20110615: Continuing development
%           Added simulation for cargo pickup/dropoff
%           Adding support for multiple walkers
% 20110616: Debugging motion rules, making display better
% 20110616: Modified to be a random walk function, to be
%           called in a data accumulator program

%Walkers:
% Set position randomly if we're doing cargo bearing simulation,
% or set to supplied startPos if not.
if cargoBearing
    currentPos = zeros(numWalkers, 2);
    for i=1:numWalkers
       currentPos(i, :) = [randi(x, 1), randi(y, 1)];
    end
     % If doing a cargobearing walk, set these to cargo positions too:
    cargoPos = [[3, 5]; [9, 5]; [15, 5]; [7, 2]; [11, 2]];
else
    currentPos = startPos;
    numWalkers = 1; %Want to make sure this is one for this case
end



%Initialize some things:
steps = 0; 
hasCargo = zeros(numWalkers);
sorted = 0;
cargoAPoss = [0, 1; 0, -1; 1, 0; -1, -1];  %Movement rules
cargoBPoss = [0, 1; 0, -1; -1, 0; 1, 1]; %''
log = zeros(length, 2*numWalkers + 1);
cargoLog = [];
collisionLog = [];

for i=1:length,
    for walker=1:numWalkers
        %Add current pos to log
        log(steps + 1, 2*walker-1:2*walker) = currentPos(walker, :);
        
        %Update pos to randomly
        %chosen neighbor -- remember,
        %these are the only valid neighbors:
        %   (0, +1), (0, -1)
        % IF (x/2)%1 = 0:
        %   (+1, 0), (-1, -1)
        % ELSE:
        %   (-1, 0), (+1, +1)

        temp = randi(4, 1);
        if (mod(currentPos(walker, 1),2) == 0)
            newPos = currentPos(walker, :) + cargoAPoss(temp, :); 
        else
            newPos = currentPos(walker, :) + cargoBPoss(temp, :); 
        end

        %If we just went out of bounds in the -y direction (toward
        % a goal) and had cargos, we drop off
        if cargoBearing && (hasCargo(walker) == 1 && (newPos(2) < 1))
            %Drop cargo, increment cargo-dropped-count
            hasCargo(walker) = 0;
            sorted = sorted + 1;
            %Don't move
            newPos = currentPos(walker, :);
            cargoLog = [cargoLog; steps, walker];
        end

        %General out-of-bounds case without cargo drop:
        if restricted && ((newPos(1) > x || newPos(1) < 1 || ...
                newPos(2) > y || newPos(2) < 1))
            %Don't go anywhere
            newPos = currentPos(walker, :);
        end

        %Hitting cargos case:
        if cargoBearing
            m = (find(cargoPos(:, 1) == newPos(1)));
            for n=m
                if cargoPos(m, 2) == newPos(2)
                    %Remove the cargo from the list of cargos and "pick up" if
                    %you don't already have a cargo
                    if hasCargo(walker) == 0
                        cargoPos(m, :) = [-50, -50];
                        hasCargo(walker) = 1;
                        cargoLog = [cargoLog; steps, walker];
                    %Anyway, don't move there
                    newPos = currentPos(walker, :);
                    end
                end
            end
        end
        
        % Already on irrev. cargo case:
        if (currentPos(walker, :) == endPos)
            return
        end
        
        %Hitting other walkers case:
        if numWalkers > 1
            m = (find(currentPos(:, 1) == newPos(1)));
            for n=m
                if (currentPos(n, 2) == newPos(2))
                    %Derp, don't go there
                    newPos = currentPos(walker, :);
                    collisionLog = [collisionLog; newPos, walker];
                end
            end
        end
        
        %Finally actually update the position
        currentPos(walker, :) = newPos;
        
    end
    
    % Step forward, update log
    steps = steps + 1;
    log(steps, 2*numWalkers + 1) = steps - 1;
    
    if (sorted == 5)
        log(steps+1:end, :) = [];
        break
    end
    
end

return

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 (fluorescenceinitialfluorescencecurrent) 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:

%%% Random walk bulk simulation that
%% runs a battery of tests and plots the results
%% to see how long random walks take on average to complete
%% based on distance from goal / platform size
% Gregory Izatt & Caltech BIOMOD 2011
% 20110616: Initial revision
% 20110623: Updating documentation a bit

%% Dependency: makes calls to randomWalkFunction.m

iterations = 2500; %Test each case of random walk this # times
xMax = 16;  %Scale of platform for test
yMax = 8; 
stopPos = [16, 8]; %Stop position
averages = zeros(xMax, yMax); %Init'ing this
trash = []; %Trash storing variable
%Cycle over whole area, starting the walker at each position
%and seeing how long it takes it to get to the stop position
matlabpool(4)
for x=1:xMax
    for y=1:yMax
        temp = zeros(iterations, 1);
        parfor i=1:iterations
            [trash, temp(i)] = randomWalkFunction(xMax, yMax, 1000000, ...
                1, [x, y], stopPos, 0, 1);
        end
        stdDev(x, y) = std(temp);
        averages(x, y) = mean(temp)
    end
end
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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>.
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 random-walk duration, and thus the time for (fluorescenceinitialfluorescencecurrent) 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 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.

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