Calculating MSD and Diffusion Coefficients
This page contains example MATLAB code for calculating the mean squared displacement (MSD) and diffusion coefficient from a movie of fluorescent microspheres.
Diffusion measurements from particle tracking data
The function CalculateDiffusionCoefficientFromTrajectories can be used to plot the MSD as a function of time for fluorescent particles and calculate their average diffusion coefficient. Remember that MSD = $ \left \langle {\left | \vec r(t+\tau)-\vec r(t) \right \vert}^2 \right \rangle=2Dd\tau $, where r(t) = position, d = number of dimensions, D = diffusion coefficient, and $ \tau $= time interval. The input argument Centroids can be generated by the function track, which creates a 4 column matrix listing the x- and y- coordinates, the frame number and the particle number. Additional input arguments to CalculateDiffusionCoefficientFromTrajectories include the pixel size, the movie frame rate, and an optional figure handle for plotting the MSD.
function out = CalculateDiffusionCoefficientFromTrajectories(Centroids, PixelSize, FrameRate, FigureHandle) Centroids( :, 1:2 ) = Centroids( :, 1:2 ) * PixelSize; numberOfParticles = max( Centroids(:,4) ); particles = cell( 1, numberOfParticles ); longestTrack = 0; particlesToSkip = []; if( nargin < 4 || isempty( FigureHandle )) FigureHandle = figure; end figure( FigureHandle ) for ii = 1:numberOfParticles % get the centroids for a single particle particle = Centroids(Centroids(:,4) == ii, :); numberOfSamples = size(particle,1); % number of samples in trajectory particles{ii} = struct(); numberOfMsds = round(numberOfSamples-1);% / 8); particles{ii}.msd = zeros(1, numberOfMsds); particles{ii}.msdUncertainty = zeros(1, numberOfMsds); particles{ii}.number = numberOfMsds; if numberOfMsds==0 particlesToSkip = [particlesToSkip ii]; else for tau = 1:numberOfMsds dx = particle(1:1:(end-tau), 1) - particle((tau+1):1:end, 1); dy = particle(1:1:(end-tau), 2) - particle((tau+1):1:end, 2); drSquared = dx.^2+dy.^2; msd = mean( drSquared ); particles{ii}.msd(tau) = msd; particles{ii}.msdUncertainty(tau) = std( drSquared ) / sqrt(length( drSquared )); end particles{ii}.diffusionCoefficient = particles{ii}.msd(1) / (2 * 2 * FrameRate^-1); longestTrack = max( longestTrack, numberOfSamples ); loglog( (1:numberOfMsds)/FrameRate, particles{ii}.msd(1:numberOfMsds), 'color', rand(1,3) ); hold on; end end % compute the average value of D and the ensemble average of msd d = zeros(1,numberOfParticles); averageMsd = zeros( 1, longestTrack ); averageMsdCount = zeros( 1, longestTrack); for ii = 1:numberOfParticles if sum(particlesToSkip==ii)==0 numberOfMsds = length(particles{ii}.msd); d(ii) = particles{ii}.diffusionCoefficient; averageMsd(1:numberOfMsds) = averageMsd(1:numberOfMsds) + particles{ii}.msd; averageMsdCount(1:numberOfMsds) = averageMsdCount(1:numberOfMsds) + 1; end end figure( FigureHandle ); averageMsd = averageMsd ./ averageMsdCount; loglog((1:longestTrack)/FrameRate, averageMsd, 'k', 'linewidth', 3); ylabel('MSD (nm^2)'); xlabel('Time (s)'); title('MSD versus Time Interval'); %hold off; out = struct(); out.diffusionCoefficient = mean(d); out.diffusionCoefficientUncertainty = std(d)/sqrt(length(d)); out.particles = particles;