Qual

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Qual Project Source Code

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All source code used in my qualifying exam project can be found here in my github repo for the project. An online version of my qual with enhanced features like citation data popup graphics and animations can be found on this page


SI Fig 1

<html><iframe width="560" height="315" src="//www.youtube.com/embed/JH-hGjzhEFQ" frameborder="0" allowfullscreen></iframe></html>


SI Fig2 - Brownian Motion Matlab Code

Brownian Motion Matlab Code 


function [boom] = BradsBrownianMotionScript()
format compact;
format short;
close all;

% Note: This matlab script requires 'msdanalyzer' toolbox
%-------------############################------------------%
%                 STARTING PARAMETERS
%-------------############################------------------%
D = .3;                     % Diffusion Rate
Ds = .1;                    % Diffusion Scalar (Ds = Dn)
Dn = D/(D/Ds);              % new D after scaling L
d = 2;                      % dimensions
dT = 1;                     % time step
k = sqrt(d*D);	            % stdev of D's step size distribution
MSD = 2*d*D;                % mean squared displacement
L = sqrt(2*d*D);            % average diagonal (2D) step size
Lx = L/sqrt(2);             % average linear (1D) step size
Ls = 1/sqrt(D/Ds);          % scales Lx values for Dn


MSDtest = [1 0 0];			% test: D, Dn, or L
Scale = 1/10;				% scale of model
Ndots = 100;
Nsteps = Ndots;


xyl = ones(2,Ndots);
xyds = ones(2,Ndots);
lims = ((D+1)^2)*10;


%===========================================================%
%               LIVE PARTICLE DIFFUSION
%-----------------------------------------------------------%
for t = 1:Nsteps

    xyds = STEPxyds(Ndots, k);
	[xyl] = AMPARSTEP(Ndots, xyds, xyl);
	MAINPLOT(xyl, lims);

end
%===========================================================%


%===========================================================%
%               MSD RANDOM STEPS ANALYSIS
%-----------------------------------------------------------%
tracks = cell(Ndots, 1);

stepN = 1;
for t = 1:Nsteps 

	xyds = STEPxyds(Ndots, k);
	[xyl] = AMPARSTEP(Ndots, xyds, xyl);
    [tracks] = MSDfun(stepN, Nsteps, tracks, xyds);

stepN = stepN+1;
end
MSDfunction(tracks,Ndots,Nsteps,D,Dn,L,dT,k,Scale,MSDtest);





%===========================================================%
%               MSD UNIFORM STEPS ANALYSIS
%-----------------------------------------------------------%
stepN = 1;
for t = 1:Nsteps 

xyds = stepsize(Ndots, Lx);
		
	[xyl xyds] = MSDAMPARSTEP(Ndots, xyds, xyl, Ls);
    [tracks] = MSDfun(stepN, Nsteps, tracks, xyds);

stepN = stepN+1;
end
MSDfunction(tracks,Ndots,Nsteps,D,Dn,L,dT,k,Scale,MSDtest);


boom = 'done';
%-------------############################------------------%
end %         ##    END MAIN FUNCTION   ##
%-------------############################------------------% 
%%





%%
%-------------############################------------------%
%             ##      SUBFUNCTIONS      ##
%-------------############################------------------% 


%-------------------------------------------%
% STEP SIZE GENERATOR
%-------------------------------------------%
function xyds = STEPxyds(Ndots, k)

    xyds = (k * randn(2,Ndots));

end


%-------------------------------------------%
% MOVE PARTICLES MAIN FUNCTION
%-------------------------------------------%
function [xyl] = AMPARSTEP(Ndots, xyds, xyl)
	
	for j = 1:Ndots
        xyl(:,j) = xyl(:,j)+xyds(:,j);
	end
	
end


%-------------------------------------------%
% LIVE DIFFUSION PLOT
%-------------------------------------------%
function [] = MAINPLOT(xyl, lims)
%-------------------------------------------%
xlim = [-lims lims];
ylim = [-lims lims];
zlim = [-5 5];

%=================================%
%       MAIN 2D PLOT
%---------------------------------%
figure(1)
subplot(2,1,1), 
AMPARPlot = gscatter(xyl(1,:),xyl(2,:));
axis([xlim, ylim]);
set(AMPARPlot,'marker','.','markersize',[6],'color',[1 0 0])


%=================================%
%           3D PLOT
%---------------------------------%
figure(1);
subplot(2,1,2), 
gscatter(xyl(1,:),xyl(2,:)); view(20, 30);
axis normal;
grid off
axis([xlim, ylim, zlim]);
set(gca, 'Box', 'on');

end


%-------------------------------------------%
% MANUAL STEP SIZE FUNCTION
%-------------------------------------------%
function xyds = stepsize(Ndots, Lx)
%-------------------------------------------%

   Lx(1:2,1:Ndots) = Lx;
   xyd = randi([0 1],Ndots,2)';
   xyd(xyd == 0) = -1;
   xyds = (Lx.*xyd);
   
end


%-------------------------------------------%
% MSD SCALED STEPS FUNCTION
%-------------------------------------------%
function [xyl xyds] = MSDAMPARSTEP(Ndots, xyds, xyl, Ls)
	
	for j = 1:Ndots
        xyds(:,j) = xyds(:,j)*Ls;
        xyl(:,j) = xyl(:,j)+xyds(:,j);
	end	
	
end


%-------------------------------------------%
% MSD TRACKS GENERATOR
%-------------------------------------------%
function [tracks] = MSDfun(stepN, Nsteps, tracks, xyds)
    time = (0:Nsteps-1)';
    xymsd = xyds';
    xymsd = cumsum(xymsd,1);
    tracks{stepN} = [time xymsd];

end


%-------------------------------------------%
% MSD TRACKS ANALYSIS
%-------------------------------------------%
function [] = MSDfunction(tracks,Ndots,Nsteps,D,Dn,L,dT,k,Scale,MSDtest)

SPACE_UNITS = 'µm';
TIME_UNITS = 's';
N_PARTICLES = Ndots;
N_TIME_STEPS = Nsteps;
N_DIM = 2;

oD = D;				% raw		µm^2/s
D  = D*Scale;       % to-scale	µm^2/s

oDn = Dn;			% raw		µm^2/s
Dn = Dn*Scale;		% to-scale	µm^2/s

oL = L;				% raw		µm
L = L*Scale;		% to-scale	µm

dTbase = dT;		% raw time-step 
dT = dT*Scale;		% to-scale time-step
k = k;				% stdv of step distribution

ma = msdanalyzer(2, SPACE_UNITS, TIME_UNITS);
ma = ma.addAll(tracks);
disp(ma)

figure
ma.plotTracks;
ma.labelPlotTracks;

ma = ma.computeMSD;
ma.msd;

t = (0 : N_TIME_STEPS)' * dT;
[T1, T2] = meshgrid(t, t);
all_delays = unique( abs(T1 - T2) );

figure
ma.plotMSD;


cla
ma.plotMeanMSD(gca, true)

mmsd = ma.getMeanMSD;
t = mmsd(:,1);
x = mmsd(:,2);
dx = mmsd(:,3) ./ sqrt(mmsd(:,4));
errorbar(t, x, dx, 'k')

[fo, gof] = ma.fitMeanMSD;
plot(fo)
ma.labelPlotMSD;
legend off


ma = ma.fitMSD;

good_enough_fit = ma.lfit.r2fit > 0.8;
Dmean = mean( ma.lfit.a(good_enough_fit) ) / 2 / ma.n_dim;
Dstd  =  std( ma.lfit.a(good_enough_fit) ) / 2 / ma.n_dim;

Dheader1 = ['Raw Unscaled Values'];
Dhead1 = ['    D        Dn        L'];
Ddat1 = [oD oDn oL];
disp(' ')
disp(Dheader1)
disp(Dhead1)
disp(Ddat1)


yourtesthead = ['YOU ARE TESTING DIFFUSION FOR:'];
if MSDtest(1)
	yourtest = ['   D:   original diffusion rate'];
elseif MSDtest(2)
	yourtest = ['   Dn:  new diffusion rate'];
elseif MSDtest(3)
	yourtest = ['   L:  step length'];
else
	yourtest = ['   generic diffusion rate'];
end
disp(yourtesthead)
disp(yourtest)

disp(' ')
fprintf('Estimation of raw D coefficient from MSD:\n')
fprintf('D = %.3g ± %.3g (mean ± std, N = %d)\n', ...
    Dmean, Dstd, sum(good_enough_fit));




% Retrieve instantaneous velocities, per track
 trackV = ma.getVelocities;

 % Pool track data together
 TV = vertcat( trackV{:} );

 % Velocities are returned in a N x (nDim+1) array: [ T Vx Vy ...]. So the
 % velocity vector in 2D is:
 V = TV(:, 2:3);

 % Compute diffusion coefficient
varV = var(V);
mVarV = mean(varV); % Take the mean of the two estimates
Dest = mVarV / 2 * dT;



Dheader2 = ['Scaling to model...'];
Dhead2 = ['    D        Dn        L'];
Ddat2 = [D Dn L];

disp(' ')
disp(Dheader2)
disp(Dhead2)
disp(Ddat2)
fprintf('Estimation from velocities histogram:\n')
fprintf('Tested D = %.3g %s, compare to scaled Des value of %.3g %s\n', ...
    Dest, [SPACE_UNITS '²/' TIME_UNITS], D, [SPACE_UNITS '²/' TIME_UNITS]);

% printf('D.psd target value was %.3g %s\n', ...
%     Dest, msdDpsd, [SPACE_UNITS '²/' TIME_UNITS]);

end



SI Fig 3

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