Biomod/2011/Caltech/DeoxyriboNucleicAwesome/Simulation: Difference between revisions

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__TOC__
__TOC__
==Overview==
==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 [http://en.wikipedia.org/wiki/Random_walk#One-dimensional_random_walk standard 1D random walk], showing an average translation on the order of <math>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 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 [[Biomod/2011/Caltech/DeoxyriboNucleicAwesome/SPEX Experiments|SPEX experiments]], a one-dimensional track is easy to construct, and will behave like a [http://en.wikipedia.org/wiki/Random_walk#One-dimensional_random_walk standard 1D random walk], showing an average translation on the order of <math>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.
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:
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 unit of time is the '''step''', which is the time it takes a walker to attempt to interact with one of the surrounding six locations.
*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>.
*Every probe on the origami 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, n>, and the bottom-right <m, 1>, <m, n> being the number of probes on the origami (which can be anything).
**These layouts are inputted as a matrix in MATLAB, with the top-left being <1,1> and bottom-right being <m, n>; different objects on origami to be mounted on each probe are coded by number:
***0 = nothing
***1 = track 1
***10 = walker on track 1
***2 = track 2
***20 = walker on track 2
***3 = cargo
***4 = cargo goal
***40 = filled cargo goal
***5 = walker goal
***50 = filled walker goal
**To turn a hexagonal grid into the square one that the grid layout implies, even columns are shifted up by 0.5 in this representation. This leads to the restriction that the first column must be a "high" column, as described in the code's documentation (see below).
*Movement rules are based on column:
*Movement rules are based on column:
** In even columns, a walker can move in directions <0, 1>, <0, -1>, <1, 0>, <-1, -1>.
** In even columns, a walker can move in directions <0, 1>, <0, -1>, <1, 0>, <-1, -1>, <-1, 1>, <1, 0>.
** In odd 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>, <1, -1>, <-1, 0>.
*Every time step, each walker being simulated takes a step in a random direction, and attempts to interact with whatever it hits:
**If it tries to step off of the origami or onto something that isn't a track, it doesn't move.
**If it tries to step to a track of the same type or an occupied track of either type, it does nothing.
**If it tries to step to a track of the opposite type that's not occupied, it moves there.
**If it tries to step onto a cargo, it'll pick it up but not move.
**If it's carrying a cargo and tries to step onto a goal of the same type as the cargo, it'll drop the cargo but not move.


[[Image:MotionRulesJustification1.png|thumb|center|800px|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.]]
[[Image:MotionRulesDerivation.png|thumb|center|800px|An illustration of the grid and motion rules (for walking; directions of motion that won't result in a step aren't shown) used in the simulation. The bottom-left is the origin (<1,1> because MATLAB indexes by 1). The 2D platform, including track A (red), track B (blue), the marker (tan), cargo (gold), and goal (green), is shown on the left. The grid on the right -- the grid corresponding to our numbering system and representing viable track for a random walk -- is created by shifting even columns up by 0.5. This arrangement (which is, in essence, a visualization tool) 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==
==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.
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, a count of the number of steps the simulation took, and if desired, a move of the random walk. 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):
The function code (saved as randomWalkFunctionGeneric.m):
<code><pre>
<span class="_toggler-codeA">Toggle Code</span>
function [log, cargoLog, steps] = randomWalkFunction(x, y, length, ...
<div class="codeA" style="display:none;
     numWalkers, startPos, endPos, cargoBearing, restricted, error)
      border: 1px dashed rgb(0, 0, 0);
      padding-top: 10px; 
      padding-right: 10px; 
      padding-bottom: 10px; 
      padding-left: 10px;
      background-color: #FEF7EA;"><code><syntaxhighlight lang="matlab">
function [log, cargoLog, steps, M] = randomWalkFunctionGeneric(...
     length, layoutMode, startPos, numWalkers, cargoBearing, error,...
    record, spaceWalkOnly, departThreshold, arriveThreshold)


%x: Width        y: Height
%Random walking / cargo sorting  simulation for more general form
%length: max # of steps to run simulation
%track layouts. More flexible in terms of layout but ultimately
%numWalkers = number of walkers to simulate in cargoBearing state
%probably a touch slower.
%startPos = starting position for walker in randomwalk state
%Gregory Izatt & Caltech BIOMOD 2011
%endPos = irreversible track location in randomwalk state
%20110713: Init revision
%cargoBearing = running cargoBearing (1) vs randomWalking (0)
%20110714: Continuing init revision.
%restricted = whether we're paying attention to borders
%20110715: Continuing debugging of init revision.
%error = the chance of the failure of any single track
%20110718: Debugging issue where walkers n>=2 have occasional
%          unexplained jaints to 0,0 in the log
%20110719: Adding full movie capture capability, mostly working on
%          rendering origami during movie production
%20110816: Adding support for spacewalking and random
%           walker appearance / departure
%20110817: Adding support for ability to do /just/ a spacewalk
%           for diagnostic / control purposes
%Defines layouts:


%Random walking cargo pickup/dropoff simulation
%Layouts:
%for origami tile, x (horizontal) by y (vertical) dim.
%1 = Standard
%Locations index by 1. x+ = right, y+ = up
%2 = Mini-playground
% Gregory Izatt & Caltech BIOMOD 2011
%3 = 1D random walk / cargo sort
% 20110615: Initial revision
%4 = Wide & long random walk
% 20110615: Continuing development
 
%           Added simulation for cargo pickup/dropoff
%In layout specification:
%           Adding support for multiple walkers
%0 = nothing
% 20110616: Debugging motion rules, making display better
%1 = track 1
% 20110616: Modified to be a random walk function, to be
%10 = walker on track 1
%           called in a data accumulator program
%2 = track 2
% 20110628-30: Modified to take into account omitted positions
%20 = walker on track 2
%          , new probe layout, and automatic halting when
%3 = cargo
%          starting on impossible positions.
%4 = cargo goal
% 20110706: Fixed walker collision. It detects collisions properly
%40 = filled cargo goal
%          now.
%5 = walker goal
% 20110707: Adding support for errors -- cycles through and
%50 = filled walker goal
%          omits each track position at an input error rate
%Specification arrays are in origami coodinates as
%defined on the BIOMOD wiki's simulation page.
%Walking will assume that system, so make sure
%they're right!
%Also, make sure you start with an odd row (a high one)
% or the movement will be messed up.
 
if layoutMode == 1
    layout = ...
    [[1, 1, 2, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 2, 2]
    [2, 2, 1, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 1, 1]
    [1, 1, 0, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 0, 2]
    [2, 2, 3, 1, 2, 2, 1, 0, 3, 2, 1, 1, 2, 2, 3, 1]
    [1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2]
    [2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1]
    [1, 4, 2, 4, 1, 4, 2, 2, 1, 4, 2, 4, 1, 4, 2, 4]
    [2, 4, 1, 4, 2, 4, 1, 1, 2, 4, 1, 4, 2, 4, 1, 4]];
elseif layoutMode == 2
    layout = ...
    [[0, 1, 0, 2, 1, 1, 0, 2]
    [0, 2, 3, 1, 2, 2, 3, 1]
    [0, 1, 2, 2, 1, 1, 2, 2]
    [0, 2, 1, 0, 3, 2, 1, 1]
    [0, 1, 2, 2, 1, 1, 2, 2]
    [0, 2, 1, 1, 2, 2, 1, 1]
    [0, 4, 2, 2, 1, 4, 2, 4]
    [0, 4, 1, 1, 2, 4, 1, 4]];
elseif layoutMode == 3
    layout = ...
    [[0, 2, 0]
    [0, 1, 0]
    [3, 2, 0]
    [0, 1, 3]
    [0, 2, 0]
    [0, 1, 0]
    [3, 2, 0]
    [0, 1, 0]
    [0, 2, 0]
    [1, 4, 0]
    [2, 4, 0]];
elseif layoutMode == 4
    layout = ...
    [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 5]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 1]
    [0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 1, 0, 0]
    [0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 1, 0, 0, 0, 0]
    [0, 0, 0, 2, 1, 1, 2, 2, 1, 0, 0, 0, 0, 0, 0]
    [0, 2, 1, 1, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0]
    [1, 1, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
    [2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
elseif layoutMode == 410
    layout = ...
    [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 5]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 1]
    [0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 1, 0, 0]
    [0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 1, 0, 0, 0, 0]
    [0, 0, 0, 2, 1, 1, 2, 2, 1, 0, 0, 0, 0, 0, 0]
    [0, 0, 1, 1, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0]
    [0, 0, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
    [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
elseif layoutMode == 416
    layout = ...
    [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 5]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 1]
    [0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 1, 0, 0]
    [0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 1, 0, 0, 0, 0]
    [0, 0, 0, 0, 1, 1, 2, 2, 1, 0, 0, 0, 0, 0, 0]
    [0, 0, 0, 0, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0]
    [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
elseif layoutMode == 422
    layout = ...
    [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 5]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 1]
    [0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 1, 0, 0]
    [0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 1, 0, 0, 0, 0]
    [0, 0, 0, 0, 0, 0, 2, 2, 1, 0, 0, 0, 0, 0, 0]
    [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
elseif layoutMode == 457
    layout = ...
    [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 5]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 1]
    [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 1, 0, 0]
    [0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 0, 0, 0, 0]
    [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
elseif layoutMode == 434
    layout = ...
    [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 5]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 1]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 0, 0]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
elseif layoutMode == 5
    layout = ...
    [[1, 1, 2, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 5, 2]
    [2, 2, 1, 1, 2, 2, 0, 1, 2, 2, 0, 1, 2, 2, 1, 1]
    [1, 1, 0, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 0, 2]
    [2, 2, 0, 1, 2, 2, 1, 0, 0, 2, 1, 1, 2, 2, 0, 1]
    [1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2]
    [2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1]
    [1, 0, 2, 0, 1, 0, 2, 2, 1, 0, 2, 0, 1, 0, 2, 0]
    [2, 0, 1, 0, 2, 0, 1, 1, 2, 0, 1, 0, 2, 0, 1, 0]];
end
%Get the orientation right:
layoutSize = size(layout);
layout = flipud(layout);
startPos(1) = layoutSize(1) - startPos(1) + 1;
layout = transpose(layout);
startPos = [startPos(2), startPos(1)];
layoutSize = [layoutSize(2), layoutSize(1)];


%Initialize some things:
%Cargo positions:
cargoPos = [[3, 5]; [9, 5]; [15, 5]; [7, 7]; [11, 7]];
filledGoals = [];
omitPos = [[3, 6]; [7, 8]; [8, 5]; [11, 8]; [15, 6]];


steps = 0;
%Initialize some logging variables we'll need:
hasCargo = zeros(numWalkers);
steps = 0;
sorted = 0;
trackAPoss = [0, 1; 0, -1; 1, 0; -1, -1];  %Movement rules, even column
trackBPoss = [0, 1; 0, -1; -1, 0; 1, 1]; %'', odd column
log = zeros(length, 2*numWalkers + 1);
log = zeros(length, 2*numWalkers + 1);
cargoLog = [];
cargoLog = [];
collisionLog = [];


%Walkers:
%Movement rules:
evenPoss = [0, 1 ; 0, -1; 1, 0; -1, 0; 1, -1; -1, -1]; %For even columns
oddPoss = [0, 1 ; 0, -1; 1, 0; -1, 0; 1, 1; -1, 1]; %For odd columns
 
%Walkers positioning:
% Set position randomly if we're doing cargo bearing simulation,
% Set position randomly if we're doing cargo bearing simulation,
% or set to supplied startPos if not.
% or set to supplied startPos if not.
Line 75: Line 227:
       done = 0;
       done = 0;
       while done ~= 1
       while done ~= 1
             currentPos(i, :) = [randi(x, 1), randi(y, 1)];
             currentPos(i, :) = [randi(layoutSize(1), 1), ...
             done = checkPossible(numWalkers, currentPos, omitPos, ...
                                randi(layoutSize(2), 1)];
                                    cargoPos);
             done = (layout(currentPos(i, 1), currentPos(i, 2)) == 1 || ...
                    layout(currentPos(i, 1), currentPos(i, 2)) == 2);
       end
       end
      layout(currentPos(i, 1), currentPos(i, 2)) = ...
              layout(currentPos(i, 1), currentPos(i, 2)) * 10;
     end
     end
else
else
     numWalkers = 1; %Want to make sure this is one for this case
     numWalkers = 1; %Want to make sure this is one for this case
     currentPos = startPos;
     currentPos = startPos;
     if checkPossible(numWalkers, currentPos, omitPos, ...
     if currentPos(1) < 1 || currentPos(1) > layoutSize(1) || ...
                                    cargoPos) ~= 1
      currentPos(2) < 1 || currentPos(2) > layoutSize(2) || ...
        (layout(currentPos(1), currentPos(2)) ~= 1 && ...
        layout(currentPos(1), currentPos(2)) ~= 2)
           'Invalid start position.';
           'Invalid start position.';
          layout
          currentPos
           cargoLog = [];
           cargoLog = [];
           steps = -1;
           steps = -1;
          M = [];
           return
           return
     end
     end
    layout(currentPos(1), currentPos(2)) = ...
        layout(currentPos(1), currentPos(2)) * 10;
end
end
%Something to keep track of which walkers is carrying cargos:
hasCargo = zeros(numWalkers);


%Error: If there's a valid error rate, go omit some positions:
%Error: If there's a valid error rate, go omit some positions:
if error > 0
if error > 0
     for xPos=1:x
     for y=1:layoutSize(2)
         for yPos=1:y
         for x=1:layoutSize(1)
             %Only omit if it's not already blocked by something
             %Only omit if it's not already blocked by something
             if checkPossible(0, [xPos, yPos], omitPos, cargoPos)
             if rand <= error
                if rand <= error
                layout(x, y) = 0;
                    omitPos = [omitPos; [xPos, yPos]];
                end
             end
             end
         end
         end
Line 107: Line 270:


%Convenience:
%Convenience:
numOmitPos = size(omitPos, 1);
if cargoBearing == 1
numCargoPos = size(cargoPos, 1);
    numCargoPos = 0;
    for y=1:layoutSize(2)
        for x=1:layoutSize(1)
            if layout(x, y) == 3
                numCargoPos = numCargoPos + 1;
            end
        end
    end
end
 
if record == 0
    M = [];
else
    aviobj = avifile('RR2.avi');
end
 
%Indicator for premature completion
done = 0;


%Main loop:
%Main loop:
for i=1:length,
for i=1:length
     for walker=1:numWalkers
     for walker=1:numWalkers
       
         %Add current pos to log:
         %Add current pos to log
         log(steps + 1, 2*walker-1:2*walker) = currentPos(walker, :);
         log(steps + 1, 2*walker-1:2*walker) = currentPos(walker, :);
          
          
         %Update pos to randomly
         %Update pos to randomly chosen neighbor, based on motion rules
        %chosen neighbor -- remember,
         temp = randi(6, 1);
        %these are the only valid neighbors:
         if ~spaceWalkOnly
        %  (0, +1), (0, -1)
            if (mod(currentPos(walker, 1), 2) == 0)
        % IF x%2 = 0:
                newPos = currentPos(walker, :) + evenPoss(temp, :);
        %  (+1, 0), (-1, -1)
             else
        % ELSE:
                newPos = currentPos(walker, :) + oddPoss(temp, :);  
        %  (-1, 0), (+1, +1)
            end
 
         temp = randi(4, 1);
         if (mod(currentPos(walker, 1),2) == 0)
             newPos = currentPos(walker, :) + trackAPoss(temp, :);  
         else
         else
             newPos = currentPos(walker, :) + trackBPoss(temp, :);  
             newPos = currentPos(walker, :);
         end
         end
       
        %If this is out of defined boundaries, don't do anything:
        if ~(newPos(1) > layoutSize(1) || newPos(1) < 1 || ...
            newPos(2) > layoutSize(2) || newPos(2) < 1)       
            %Now react based on what kind of spot the new position is:
            oldPosIdent = layout(currentPos(walker, 1), ...
                                currentPos(walker, 2)) / 10;
            newPosIdent = layout(newPos(1), newPos(2));


        %If we tried to move onto the bottom two spots (in terms of y)
            %Can't move from one track to the same kind of track:
        %on an even column (i.e. a goal), we drop off cargo if we had it
            if newPosIdent == oldPosIdent
        %and there wasn't one there already.
                %'Cant step to same kind of track'
        %% Specific: 8th column has no goals! It has track instead.
            %Hitting cargos: pick up cargo if possible.
        if newPos(2) <= 2 && mod(newPos(1),2) == 0 && newPos(1) ~= 8
            elseif newPosIdent == 3 && cargoBearing
            if cargoBearing && hasCargo(walker) == 1
                if hasCargo(walker) == 0
                %Drop cargo, increment cargo-dropped-count, but
                    hasCargo(walker) = 1;
                %only if there isn't already a cargo here
                    layout(newPos(1), newPos(2)) = 0;
                temp = size(filledGoals);
                    cargoLog = [cargoLog; walker, newPos, steps];
                match = 0;
                for k=1:temp(1)
                    if filledGoals(k, :) == newPos
                        match = 1;
                        break
                    end
                 end
                 end
                 if match ~= 1
                 %'Hit cargo planter'
            %Hitting goals: drop a cargo if possible.
            elseif newPosIdent == 4 && cargoBearing
                if hasCargo(walker) == 1
                     hasCargo(walker) = 0;
                     hasCargo(walker) = 0;
                     cargoLog = [cargoLog; steps, walker];
                    layout(newPos(1), newPos(2)) = 40;
                     sorted = sorted + 1;
                     cargoLog = [cargoLog; walker, newPos, steps];
                    filledGoals = [filledGoals; newPos];
                     done = done + 1/numCargoPos;
                 end
                 end
                %'Hit goal'
            %Hitting walker goal: go there and trigger completion.
            elseif newPosIdent == 5 
                %'Hit walker goal'
                currentPos(walker, :) = newPos;
                layout(newPos(1), newPos(2)) = 50;
                done = 1;
            %Valid move, and we haven't been shot down yet?
            % Then actually move, and update layout to reflect that.
            elseif newPosIdent == 1 || newPosIdent == 2
                %'Moving'
                layout(currentPos(walker, 1), currentPos(walker, 2)) =  ...
                layout(currentPos(walker, 1), currentPos(walker, 2)) / 10;
                currentPos(walker, :) = newPos;
                layout(currentPos(walker, 1), currentPos(walker, 2)) =  ...
                layout(currentPos(walker, 1), currentPos(walker, 2)) * 10;
            end 
      end 
    end
   
    %Finish up bookkeeping log and step count for this step
    log(steps + 1, 2*numWalkers + 1) = steps;
    steps = steps + 1;
   
    if record
        hold on;
        xlim([0, layoutSize(1) + 1]);
        ylim([0, layoutSize(2) + 1]);
        %Plot walkers:
        for walker=1:numWalkers
            tempPos = currentPos(walker, :);
            if mod(tempPos(1), 2) == 0
                tempPos(2) = tempPos(2) - 0.5;
             end
             end
             %Don't move
             if hasCargo(walker)
            newPos = currentPos(walker, :);
                plot(tempPos(1), tempPos(2), ...
        end
                        'o', 'color', [0, 0.5, 0], 'MarkerSize', 15);
 
        %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:
        for k=1:numCargoPos
            if cargoPos(k, :) == newPos
                %Remove the cargo from the list of cargos and "pick up"
                % if you don't already have a cargo
                if hasCargo(walker) == 0 && cargoBearing
                    cargoPos(k, :) = [-50, -50];
                    hasCargo(walker) = 1;
                    cargoLog = [cargoLog; steps, walker];
                end
                %Anyway, don't move there
                newPos = currentPos(walker, :);
             end
             end
        end
            plot(tempPos(1), tempPos(2), 'o', ...
 
                        'color', [0, 0, 0], 'MarkerSize', 25);
        % Already on irrev. cargo case:
        if (currentPos(walker, :) == endPos)
            return
         end
         end
          
          
         %Hitting other walkers case:
         %Plot origami:
         if numWalkers > 1
         for x=1:layoutSize(1)
             for k = 1:numWalkers
             for y=1:layoutSize(2)
                 if all(newPos == currentPos(k, :)) && k ~= walker
                %Plot with coloration specific to probe identity
                     newPos = currentPos(walker, :);
                 if layout(x, y) == 0  || ...
                     collisionLog = [collisionLog; newPos, walker, k];
                  layout(x, y) == 10 || ...
                  layout(x, y) == 20
                    color = [1 1 1];
                elseif layout(x, y) == 1
                    color = [1 0 0];
                elseif layout(x, y) == 2
                    color = [0 0 1];
                elseif layout(x, y) == 3
                     color = [0 .5 0];
                elseif layout(x, y) == 4
                    color = [0 1 1];
                elseif layout(x, y) == 5
                     color = [1 1 0];
                elseif layout(x, y) == 40
                    color = [0 .5 0];
                elseif layout(x, y) == 50
                    color = [.5 .5 .5];
                 end
                 end
                tempY = y;
                if mod(x, 2) == 0
                    tempY = tempY - 0.5;
                end
                plot(x, tempY, 's', 'Color', color, 'MarkerSize', 10);
             end
             end
         end
         end
          
         M(i) = getframe;
         %Hitting the omitted positions case:
        aviobj = addframe(aviobj, M(i));
         %If we have any position matches with "omitted" list
         hold off;
        %, just don't go there.
        clf;
         match = 0;
    end
         for k=1:numOmitPos
   
             if omitPos(k, :) == newPos
    %If finished, done with everything
                 match = 1;
    if done >= 1
        if record
            aviobj = close(aviobj);
        end
        log((steps + 1):length, :) = [];
         return
    end 
   
    %If we're thinking about astronaut / orphaned walkers, do it here:
    %(astronaut walking isn't yet considered for cargobearing walks,
    %(but should be implemented in the future if it becomes an issue)
    if rand() > departThreshold && cargoBearing == 0
        numWalkers = numWalkers + 1;
        log = [log(:, 1:end-1) zeros(length, 2) log(:, end)];
        currentPos = [currentPos; [1, 1]];
         landingDone = 0;
         while landingDone ~= 1
            currentPos(end, :) = [randi(layoutSize(1), 1), ...
                                randi(layoutSize(2), 1)];
             if layout(currentPos(end, 1), currentPos(end, 2)) == 4
                 done = 1;
             end
             end
            landingDone = (layout(currentPos(end, 1), ...
                                  currentPos(end, 2)) == 1 || ...
                          layout(currentPos(end, 1), ...
                                  currentPos(end, 2)) == 2);
         end
         end
        if match == 1
            newPos = currentPos(walker, :);
        end
       
        %Finally actually update the position
        currentPos(walker, :) = newPos;
       
     end
     end
      
     if rand() > arriveThreshold && numWalkers > 0 && cargoBearing == 0
    % Step forward, update log
        walkerToRemove = randi(numWalkers);
    steps = steps + 1;
        numWalkers = numWalkers - 1;
    log(steps, 2*numWalkers + 1) = steps - 1;
        log(:, walkerToRemove*2-1:walkerToRemove*2) = [];
   
         currentPos(walkerToRemove, :) = [];
    if (sorted == 5)
         log(steps+1:end, :) = [];
        break
     end
     end
      
      
end
end
 
if record
return
     aviobj = close(aviobj);
 
end
 
%%Checks if a position is a possible place for a walker to be:
function [possible] = checkPossible(numWalkers, currentPos, ...
                                    omitPos, cargoPos)
    % If we're starting on an omitted position, or a goal, a cargo,
    % or another walker, just give up immediately, and return a -1:
    numOmitPos = size(omitPos, 1);
     numCargoPos = size(cargoPos, 1);
    possible = 1;
    for walker = 1:numWalkers
        thisWalkerPos = currentPos(walker, :);
        % Only run this for this walker if it's placed somewhere
        % valid (i.e. not waiting to be placed, x,y = 0,0)
        if all(thisWalkerPos)
            %Omitted positions:
            for k=1:numOmitPos
                if omitPos(k, :) == thisWalkerPos
                    possible = 0;
                    return
                end
            end
            %Cargo positions:
            for k=1:numCargoPos
                if cargoPos(k, :) == thisWalkerPos
                    possible = 0;
                    return
                end
            end
            %Other walkers:
            for k=1:numWalkers
                if (all(currentPos(k, :) == thisWalkerPos)) && ...
                        (k ~= walker)
                    possible = 0;
                    return
                end
            end
            %Goal positions:
            if mod(thisWalkerPos(1), 2)==0 && thisWalkerPos(1) ~= 8 ...
                    && thisWalkerPos(2) <= 2
                possible = 0;
                return
            end
        end
    end
return
return
</pre></code>
</syntaxhighlight></code></div>
===Examining Errors in Origami===
This code can be used to generate diagrams like those below, visualizing the mobility of the walker. One immediate question 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 probe 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.
[[Image:FullGridErrors.png | center | 800 px | thumb | Node graphs showing walker mobility of origami. Each junction represents a track, and each edge represents a step a walker can take. The left diagram shows no error, whereas the other two show increasing error rates. We observe that 10% error rates decrease walker mobility, but tend not to trap the walker in any particular location; 20% error rates or greater, over several tests, tend to cause catastrophic loss of mobility, making the sorting task impossible.]]


==Random-Walk Simulation==
==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 <math>(fluorescence_{initial} - fluorescence_{current})</math> 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:
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. [[Image:Caltech5000iter0ErrorRRWideLinearTrack.jpg | thumb | 300 px | right | A plot of the number of steps (on an average over 5000 iterations) it takes a walker to random walk from any point on the origami to the irreversible track at one end. This test was done assuming a 0% error rate, on the 3-track-wide linear random walking playground that we are using to investigate random walking (in a pseudolinear environment).]]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 <math>(fluorescence_{initial} - fluorescence_{current})</math> 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: <span class="_toggler-codeB">Toggle Code</span>
<code><pre>
<div class="codeB" style="display:none;
      border: 1px dashed rgb(0, 0, 0);
      padding-top: 10px; 
      padding-right: 10px; 
      padding-bottom: 10px; 
      padding-left: 10px;
      background-color: #FEF7EA;"><code><syntaxhighlight lang="matlab">
%%% Random walk bulk simulation that
%%% Random walk bulk simulation that
%% runs a battery of tests and plots the results
%% runs a battery of tests and plots the results
Line 287: Line 477:
% 20110701: Updating to use new, updated randomWalkFunction
% 20110701: Updating to use new, updated randomWalkFunction
% 20110707: Updated to use new error-allowing randomWalkFunction
% 20110707: Updated to use new error-allowing randomWalkFunction
% 20110719: Updating to use new and hopefully much faster
%              randomWalkFunctionGeneric
%% Dependency: makes calls to randomWalkFunctionGeneric.m
%Layout modes:
layoutMode = 4;
if layoutMode == 4
    yMax = 15;
    xMax = 9;
elseif layoutMode == 5
    yMax = 16;
    xMax = 8;
else
    'Layout not yet implemented.'
    return
end


%% Dependency: makes calls to randomWalkFunction.m
iterations = 500; %Test each case of random walk this # times
 
iterations = 2500; %Test each case of random walk this # times
xMax = 16;  %Scale of platform for test
yMax = 8;
stopPos = [15, 7]; %Stop position
averages = zeros(xMax, yMax); %Init'ing this
averages = zeros(xMax, yMax); %Init'ing this
trash = []; %Trash storing variable
trash = []; %Trash storing variable
%Cycle over whole area, starting the walker at each position
%Cycle over whole area, starting the walker at each position
%and seeing how long it takes it to get to the stop position
%and seeing how long it takes it to get to the stop position
matlabpool(4)
matlabpool(3)
for x=1:xMax
for x=1:xMax
     for y=1:yMax
     for y=1:yMax
         temp = zeros(iterations, 1);
         temp = zeros(iterations, 1);
         parfor i=1:iterations
         parfor i=1:iterations
             [trash, trash2, temp(i)] = randomWalkFunction(xMax, yMax, ...
             [trash, trash2, temp(i), trash3]=randomWalkFunctionGeneric(...
                 10000, 1, [x, y], stopPos, 0, 1, 0.0);
                 10000, layoutMode, [x, y], 1, 0, 0.1, 0, 0, 1, 1);
         end
         end
         stdDev(x, y) = std(temp);
         stdDev(x, y) = std(temp);
Line 311: Line 512:
end
end
matlabpool close
matlabpool close
</pre></code>
</syntaxhighlight></code></div>
[[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>.]]
===Results===
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. We can also use a similar simulation (run instead with tracks that don't continue past the start location of the walker, an arrangement which we have found to behave more like a linear track) data to generate approximate half-completion times, which we can compare with the SPEX results of the same random walk to both estimate the amount of time it takes the walker to perform a single branch migration on our origami, and to see if whatever our walker is doing on origami is looking like a random walk, as compared to a repeated jumping across or between origami platforms (whose half-completion times for this test would presumably not depend on the track length at all). That data is detailed on the SPEX results page.
 
==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. [[Image:250iters0ErrorCargoSort.jpg | thumb | 300px | right | A plot of the number of steps (on an average over 250 iterations) it takes n walkers to sort all five cargos to respective goals on a perfectly formed 16x8 track, as detailed above. The jaggedness in the curve is a result of the large spread of results for any given test.]] 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: <span class="_toggler-codeC">Toggle Code</span>
<div class="codeC" style="display:none;
      border: 1px dashed rgb(0, 0, 0);
      padding-top: 10px; 
      padding-right: 10px; 
      padding-bottom: 10px; 
      padding-left: 10px;
      background-color: #FEF7EA;"><code><syntaxhighlight lang="matlab">
%%% Cargo pickup/dropoff bulk simulation that
%% runs a battery of tests and plots the results
%% to see how long cargo sorts take on average
% Gregory Izatt & Caltech BIOMOD 2011
% 20110616: Initial revision
% 20110616: Fixing to fit cargoSort.m revision
% 20110623: Fixing documentation
% 20110701: Updating to work with new and improved randomWalkFunction
% 20110707: Updated to include error rate addition in randomWalkFunction
%% Dependency: Makes calls to randomWalkFunction.m
 
iterations = 50; %Test each #walker scenario this many times
maxNumWalkers = 30; %Test scenarios with up to this many walkers
averages = zeros(maxNumWalkers, 1);
medians = zeros(maxNumWalkers, 1);
 
'Initialized...'
 
matlabpool(3)  %Built-in parallel processing for speedup.
              %Change "4" to your # of cores.
             
for numWalkers=1:maxNumWalkers  %Iterate over possible #'s of walkers
    tempStorage = zeros(iterations, 1);
    parfor i=1:iterations %Iterate a bunch of times
        [trash, trash2, temp] = randomWalkFunction(16, 8, 10000, ...  
            numWalkers, [1, 1], [-100, -100], 1, 1, 0.0);
        tempStorage(i) = temp;
    end
    averages(numWalkers) = mean(tempStorage);  %Store the average
    medians(numWalkers) = median(tempStorage);
    numWalkers, averages
end
matlabpool close
 
averages
plot(averages)
</syntaxhighlight></code></div>
(Note: this code is now obsolete (as it relies on an obsolete version of the randomWalkFunction script), and may be rewritten in the future. This should not, however, impact the validity of this code's results.
===Results===
===Results===
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.
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.
 
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Latest revision as of 15:57, 31 October 2011

Friday, April 19, 2024

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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 SPEX experiments, 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 attempt to interact with one of the surrounding six locations.
  • Every probe on the origami 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, n>, and the bottom-right <m, 1>, <m, n> being the number of probes on the origami (which can be anything).
    • These layouts are inputted as a matrix in MATLAB, with the top-left being <1,1> and bottom-right being <m, n>; different objects on origami to be mounted on each probe are coded by number:
      • 0 = nothing
      • 1 = track 1
      • 10 = walker on track 1
      • 2 = track 2
      • 20 = walker on track 2
      • 3 = cargo
      • 4 = cargo goal
      • 40 = filled cargo goal
      • 5 = walker goal
      • 50 = filled walker goal
    • To turn a hexagonal grid into the square one that the grid layout implies, even columns are shifted up by 0.5 in this representation. This leads to the restriction that the first column must be a "high" column, as described in the code's documentation (see below).
  • Movement rules are based on column:
    • In even columns, a walker can move in directions <0, 1>, <0, -1>, <1, 0>, <-1, -1>, <-1, 1>, <1, 0>.
    • In odd columns, a walker can move in directions <0, 1>, <0, -1>, <-1, 0>, <1, 1>, <1, -1>, <-1, 0>.
  • Every time step, each walker being simulated takes a step in a random direction, and attempts to interact with whatever it hits:
    • If it tries to step off of the origami or onto something that isn't a track, it doesn't move.
    • If it tries to step to a track of the same type or an occupied track of either type, it does nothing.
    • If it tries to step to a track of the opposite type that's not occupied, it moves there.
    • If it tries to step onto a cargo, it'll pick it up but not move.
    • If it's carrying a cargo and tries to step onto a goal of the same type as the cargo, it'll drop the cargo but not move.
An illustration of the grid and motion rules (for walking; directions of motion that won't result in a step aren't shown) used in the simulation. The bottom-left is the origin (<1,1> because MATLAB indexes by 1). The 2D platform, including track A (red), track B (blue), the marker (tan), cargo (gold), and goal (green), is shown on the left. The grid on the right -- the grid corresponding to our numbering system and representing viable track for a random walk -- is created by shifting even columns up by 0.5. This arrangement (which is, in essence, a visualization tool) 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 of all walkers positions over time, a log reporting when cargos were picked up and dropped off, a count of the number of steps the simulation took, and if desired, a move of the random walk. 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 randomWalkFunctionGeneric.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 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 probe 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.

Node graphs showing walker mobility of origami. Each junction represents a track, and each edge represents a step a walker can take. The left diagram shows no error, whereas the other two show increasing error rates. We observe that 10% error rates decrease walker mobility, but tend not to trap the walker in any particular location; 20% error rates or greater, over several tests, tend to cause catastrophic loss of mobility, making the sorting task impossible.

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.
A plot of the number of steps (on an average over 5000 iterations) it takes a walker to random walk from any point on the origami to the irreversible track at one end. This test was done assuming a 0% error rate, on the 3-track-wide linear random walking playground that we are using to investigate random walking (in a pseudolinear environment).
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 [math]\displaystyle{ (fluorescence_{initial} - fluorescence_{current}) }[/math] 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 [math]\displaystyle{ (fluorescence_{initial} - fluorescence_{current}) }[/math] to reach some standard level, increases with distance, though it changes less significantly the farther out one gets. We can also use a similar simulation (run instead with tracks that don't continue past the start location of the walker, an arrangement which we have found to behave more like a linear track) data to generate approximate half-completion times, which we can compare with the SPEX results of the same random walk to both estimate the amount of time it takes the walker to perform a single branch migration on our origami, and to see if whatever our walker is doing on origami is looking like a random walk, as compared to a repeated jumping across or between origami platforms (whose half-completion times for this test would presumably not depend on the track length at all). That data is detailed on the SPEX results page.

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.
A plot of the number of steps (on an average over 250 iterations) it takes n walkers to sort all five cargos to respective goals on a perfectly formed 16x8 track, as detailed above. The jaggedness in the curve is a result of the large spread of results for any given test.
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

(Note: this code is now obsolete (as it relies on an obsolete version of the randomWalkFunction script), and may be rewritten in the future. This should not, however, impact the validity of this code's results.

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.


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