Abstract

Table of Contents

Materials

Figures

Videos

Literature Cited

Supplementary Material

 

Matlab Code

% FRAPevolve2D.m

% A program to analyze a pair of images -- typically FRAP data -- and

% determine the molecular diffusion coefficient.

% Inputs: Two (observed) fluorescence images, taken at time “t1” and “t2”,

% as well as the scale of the images (microns/px), and various

% computational parameters. Also, the user inputs the background

% fluorescence intensity to be subtracted. This is important for

% accurate FRAP measurements -- knowing the background (measured e.g., by taking

% images of a blank slide) is important!

% Procedure: The initial (t1) image is 'evolved' for j = 1 to N time

% steps, where N is the maximum possible given the image scale and the

% final time (t2). At each step, the image is compared to the final (t2)

% image. The timestep j for which the evolved image best matches the

% final observed image (minimal chi^2) gives the Diffusion coefficient:

% D = dundefineddundefinedj/(2.undefined(t2-t1)), where dx is the pixel size.

% (See “Random Walks in Biology” by H. Berg for details, Chapter 1 and

% APPENDIX B )

% Averaging each point with its neighbors is done by convolution with a

% nearest-neighbor matrix.

% Each 'timestep' corresponds to time dundefineddx / (2D), even in two or three

% dimensions; this gives the correct macroscopic behavior.

% Re-scaling the image. A typical 20X image has a needlessly high pixel

% density, and 'N' (see above) will be quite large. The program allows

% the images to be re-scaled; I typically use a factor of 3.

% Noise: To account for non-ideal noise in the image, the program adds

% noise to the evolved image. This noise has the same standard deviation

% as that of a user-selected 'uniform' region of the initial image.

% Region of interest: 'Speckles' and other junk do not diffuse, and so

% disturb the analysis. The user can specify a “region of interest”, a

% binary map. Points not in these regions are not included in the

% calculation of chi^2. The region of interest can either be a

% user-input image file, or (better) can be created within the program

% by specifying polygonal regions.

% Raghuveer Parthasarathy

% 25 April 2004 -- first version

% 8 May, 2005: Simpler convolution.

% 22 Dec., 2005 Binary region of interest map

% cd 'C:\Documents and Settings\raghu\My Documents\Lipid Systems\Protein Segregation\Ab FRAP analysis'

clear all

close all

% turn off warning that image may be re-scaled to fit the screen

tw = iptgetpref( 'TruesizeWarning' );

iptsetpref( 'TruesizeWarning' , 'off' );

disp( ' ' ); disp( ' ' ); disp( ' ' ); disp( ' ' );

disp( ~undefined**_FRAPevolve2D.m_****~9~K~Hspan~M~1~Kspan~M~B~J~K~Hspan~M~1~Kspan~M~K~Hspan~M~K~Hp~M~Kp~M~K~Hp~M~1~1~Kp~M~K~Hp~M~1~1~Kp_class~L~4MsoNormal~4~M~Kspan~Mdisp~A~K~Hspan~M~1~Kspan~M~9_~9~K~Hspan~M~1~Kspan~M~B~J~K~Hspan~M~1~Kspan~M~K~Hspan~M~K~Hp~M~Kp~M~K~Hp~M~1~1~Kp~M~K~Hp~M~1~1~Kp_class~L~4MsoNormal~4~M~Kspan~Mdisp~A~K~Hspan~M~1~Kspan~M~undefined_Recommended~I_gray~Fscale_.tif_images.~9~K~Hspan~M~1~Kspan~M~B~J~K~Hspan~M~1~Kspan~M~K~Hspan~M~K~Hp~M~Kp~M~K~Hp~M~1~1~Kp~M~K~Hp~M~1~1~Kp_class~L~4MsoNormal~4~M~Kspan~Mdisp~A~K~Hspan~M~1~Kspan~M~undefined_Note_instructions_in_command_window_~Athis_window~B_as_well_as_the_dialog_boxes..~9~K~Hspan~M~1~Kspan~M~B~J~K~Hspan~M~1~Kspan~M~K~Hspan~M~K~Hp~M~Kp~M~K~Hp~M~1~1~Kp~M~K~Hp~M~1~1~Kp_class~L~4MsoNormal~4~M~Kspan~Mdisp~A~K~Hspan~M~1~Kspan~M~9_~9~K~Hspan~M~1~Kspan~M~B~J~K~Hspan~M~1~Kspan~M~K~Hspan~M~K~Hp~M~Kp~M~K~Hp~M~1~1~Kp~M~K~Hp~M~1~1~Kp_class~L~4MsoNormal~4~M~Kspan~M~7_images_should_be_a_grayscale_.tif~J_saved_as_array_A_of_type_uint8~J_BMP_also_works.~K~Hspan~M~1~Kspan~M~K~Hspan~M~K~Hp~M~Kp~M~K~Hp~M~1~1~Kp~M~K~Hp~M~1~1~Kp_class~L~4MsoNormal~4~M~Kspan~Mloadopt_~L_input~A~K~Hspan~M~1~Kspan~M~9Enter_1_to_choose_filenames_from_a_dialog_box~E_0_to_type_them_manually~I_~9~K~Hspan~M~1~Kspan~M~B~J~K~Hspan~M~1~Kspan~M~K~Hspan~M~K~Hp~M~Kp~M~K~Hp~M~1~1~Kp~M~K~Hp~M~1~1~Kp_class~L~4MsoNormal~4~M~Kspan~M~7_Load_first_FRAP_image~K~Hspan~M~1~Kspan~M~K~Hspan~M~K~Hp~M~Kp~M~K~Hp~M~1~1~Kp~M~K~Hp~M~1~1~Kp_class~L~4MsoNormal~4~M~Kspan~Mif~K~Hspan~M~1~Kspan~M_~Aloadopt~L~L1~B~K~Hspan~M~1~Kspan~M~K~Hspan~M~K~Hp~M~Kp~M~K~Hp~M~1~1~Kp~M~K~Hp~M~1~1~Kp_class~L~4MsoNormal~4~M~Kspan~M~7_dialog_box_for_filename~K~Hspan~M~1~Kspan~M~K~Hspan~M~K~Hp~M~Kp~M~K~Hp~M~1~1~Kp~M~K~Hp~M~1~1~Kp_class~L~4MsoNormal~4~M~Kspan~M~PpFileName~EpPathName~Q_~L_uigetfile~A~K~Hspan~M~1~Kspan~M~undefined.*' , 'Image to load...' );

fs = sprintf( 'Path Name: %s' , pPathName); disp(fs);

fs = sprintf( 'File Name: %s' , pFileName); disp(fs);

A1raw = imread(strcat(pPathName, pFileName));

% switch working directory to image directory

presentDir = pwd; cd(pPathName);

else

pFileName = input( 'Enter first image filename (assumes current directory) -- Don”t forget the extension!: ' , 's' );

A1raw = imread(pFileName);

end

fs = sprintf( 'Image file %s.' , pFileName);

% Load second FRAP image

if (loadopt==1)

% dialog box for filename

[pFileName,pPathName] = uigetfile( ~undefined.*' , 'Image to load...' );

fs = sprintf( 'Path Name: %s' , pPathName); disp(fs);

fs = sprintf( 'File Name: %s' , pFileName); disp(fs);

A2raw = imread(strcat(pPathName, pFileName));

% switch working directory to image directory

presentDir = pwd; cd(pPathName);

else

pFileName = input( 'Enter second image filename (assumes current directory) -- Don”t forget the extension!: ' , 's' );

A2raw = imread(pFileName);

end

fs = sprintf( 'Image file %s.' , pFileName);

% -------------------------------------

% Various calculation parameters

prompt = { 'Enter image scale (microns/pixel):' , ...

'Enter background to subtract from the first image (0.0 if already subtracted).' , ...

'Enter background to subtract from the second image (0.0 if already subtracted).' , ...

'Crop images? (1==yes):' , ...

'Scale mean intensity of image2, to match image 1? (1==yes):' , ...

'Re-scale image pixels? Enter factor (1==no change):' , ...

'Enter the time between the two images (s)' , ...

'Enter the resolution in D desired for the smoothed chi^2 curve (um^2/s):' };

dlg_title = 'Fluorophore parameters' ; num_lines= 1;

def = {'0.42' , '29.7' , '29.7' , '1' , '1' , '3' , '35.0' , '0.2' } ; % default values

answer = inputdlg(prompt,dlg_title,num_lines,def);

scale = str2double(answer(1));

backA1 = str2double(answer(2));

backA2 = str2double(answer(3));

crop = logical(str2double(answer(4)));

norm2 = logical(str2double(answer(5)));

resize = str2double(answer(6));

tmax = str2double(answer(7));

Dbox = str2double(answer(8));

% -----------------------------

% Subtract background, and display

A1sub = double(A1raw) - backA1;

A2sub = double(A2raw) - backA2;

figure(1);

subplot(1,3,1); imshow(uint8(A1suundefined255.0/max(max(A1sub)))); colormap( 'gray' ); title( 'Image 1' );

subplot(1,3,2); imshow(uint8(A2suundefined255.0/max(max(A2sub)))); colormap( 'gray' ); title( 'Image 2' );

% -----------------------------

% Crop, scale intensities

figure(2)

if (crop==1)

disp( ' ' ); disp( ~undefined Cropping image *' );

disp( ' Select the cropping rectangle from image 1 -- will apply to both images. Doubleclick when is done.' );

scaleA1sub = 255.undefined(A1sub - min(min(A1sub)))/(max(max(A1sub)) - min(min(A1sub))); % scale for display

[tempA1,rect] = imcrop(uint8(scaleA1sub)); % scale for display

[A1] = double(imcrop(uint8(A1sub), rect));

[A2] = double(imcrop(uint8(A2sub), rect));

else

A1 = A1sub;

A2 = A2sub;

end

if (norm2 == 1)

A2 = Aundefined(mean(mean(A1))/mean(mean(A2)));

end

close(2)

% -----------------------------

% measure noise

scaleA1 = 255.undefined(A1 - min(min(A1)))/(max(max(A1)) - min(min(A1sub))); % scale for display

figure(3); imshow(uint8(scaleA1)); colormap( 'gray' );

disp( ' ' ); disp( ~undefined Determine Noise *' );

disp( ' Select a region of fairly uniform intensity -- will define “noise” here. Doubleclick ends polygon definition.' );

noisereg = roipoly;

% for noise calculation:

Nnoisereg = sum(sum(noisereg)); % number of points in region of interest for noise calc.

meannoisereg = sum(sum(noisereg.*A1))/Nnoisereg; % mean intensity in region of interest for noise calc.

stdnoisereg = sqrt(sum(sum(noisereg.*(A1-meannoisereg).*(A1-meannoisereg)))/Nnoisereg);

close(3)

% -----------------------------

% Only use a 'region of interest' to compare images (optional)

% ROI should be a binary TIFF image, prepared separately, or defined by the

% user by selecting polygons to ignore

figure(3); imshow(uint8(Aundefined255.0/max(max(A1)))); colormap( 'gray' ); title( 'Image 1' );

prompt = { 'Use a binary region-of-interest map (defined here, or from a separate file)? (1==yes): ' , ...

'If yes, Region of interest: (1) Define by selecting polygons to ignore; (2) Load binary image that defines ROI' , ...

' If selecting polygons, number of polygonal regions to IGNORE' , ...

' If loading image, (1) to choose filename from a dialog box, (2) to type it manually' , ...

' If typing manually, ROI image filename (including extension)' };

dlg_title = 'Region of interest parameters' ; num_lines= 1;

def = {'1' , '1' , '1' , '1' , 'ROIfile.tif' } ; % default values

answer = inputdlg(prompt,dlg_title,num_lines,def);

useroi = logical(str2double(answer(1)));

defROIopt = uint8(str2double(answer(2)));

numROI = uint8(str2double(answer(3)));

loadROIopt = uint8(str2double(answer(4)));

rFileName = str2double(answer(5));

if (useroi==1)

switch defROIopt;

case 1

disp( ' ' ); disp( ~undefined Define Region of Interest *' );

reg = ones(size(A1));

for j=1:double(numROI),

% Select a polygon to IGNORE

fs = sprintf( ' Click corners of ignored polygon no. %d; double-click to end polygon definition' , j);

disp(fs);

partialreg = roipoly; % should use presently open Fig. 3

reg = reg.*not(partialreg); % multiply all selected regions to get their union

end

case 2

if (loadROIopt==1)

% dialog box for filename

[rFileName,rPathName] = uigetfile( ~undefined.*' , 'Image to load...' );

roiimage = imread(strcat(rPathName, rFileName));

else

roiimage = imread(rFileName);

end

fs = sprintf( 'Image file %s.' , rFileName); disp(fs);

if (size(roiimage,3) > 1)

roiimage = mean(roiimage,3); % make gray

end

if (max(double(roiimage(:))) > min(double(roiimage(:))))

% making sure that the image isn't all one value

mr = 0.undefined(max(double(roiimage(:))) + min(double(roiimage(:))));

else

mr = 0.undefinedmax(double(roiimage(:)));

end

reg = (roiimage > mr); % make ones and zeros;

otherwise

disp( 'Error! Bad region of interest paramters; using entire image!' );

msgbox( 'Error! Bad region of interest paramters; using entire image!' , 'Error' , 'error' ) ;

reg = ones(size(A1)); % Region of interest is the entire image

end

close(3);

else

reg = ones(size(A1)); % Region of interest is the entire image

end

Nreg = sum(sum(reg)); % number of points in region of interest;

% ----------------------------------------------------------

% re-sample image (re-scaling it)

% Also re-scale region of interest -- re-make it as binary

if (resize ~= 1)

A1 = imresize(A1, 1.0/resize);

A2 = imresize(A2, 1.0/resize);

regtemp = imresize(reg, 1.0/resize);

oldreg = reg;

reg = (regtemp > 0.5); % re-make binary, in case re-scaling led to values not 0 or 1

Nreg = sum(sum(reg)); % number of points in region of interest;

scale = scalundefinedresize;

end

% ----------------

% Display

dispscale = 225.0/max(mean(A1)); % value by which to scale image intensities for display

figure(2); clf

if (useroi==1)

subplot(2,2,1); imshow(uint8(Aundefineddispscale)); colormap( 'gray' ); title( 'Image 1' );

subplot(2,2,2); imshow(uint8(Aundefineddispscale)); colormap( 'gray' ); title( 'Image 2' );

else

subplot(1,3,1); imshow(uint8(Aundefineddispscale)); colormap( 'gray' ); title( 'Image 1' );

subplot(1,3,2); imshow(uint8(Aundefineddispscale)); colormap( 'gray' ); title( 'Image 2' );

end

% -----------------------------

% Evolve image (simulated diffusion)

dx = scale; % microns per pixel

Nmax = round(undefinedtmax / (dundefineddx/(undefined10.0))); % maximal number of steps to run

disp( ' ' );

fs = sprintf( ~undefined Calculation will take Nmax = %d steps' , Nmax); disp(fs);

if (Nmax > 100)

input( ' Are you sure you want to continue? Enter = yes. Control-C for no.' );

end

disp( ' ' );

fs = sprintf( ~undefined Range of D that can be probed is %.2e to %.2e um^2/s.' , dundefineddx/(2.undefinedtmax), dundefineddundefinedNmax/(2.undefinedtmax));

disp(fs);

minchi2=9e99;

c1evolve = A1;

N = size(A1);

progbar = waitbar(0, 'evolving image...' ); % will display progress

% in case no good fit is found, return minimal D and original image

bestN = 1;

bestevolvenoise = A1;

for j=1:Nmax,

nn = 0.2undefined[0 1 0; 1 0 1; 0 1 0];

c1evolveold = c1evolve;

c1evolve = conv2(c1evolve, nn, 'same' );

% conv2 zero-pads the edges of the output array, where the

% convolution is . Fill it in with the original array

% values -- pinning the edges.

c1evolve(1,:) = c1evolveold(1,:);

c1evolve(N(1),:) = c1evolveold(N(1),:);

c1evolve(:,1) = c1evolveold(:,1);

c1evolve(:,N(2)) = c1evolveold(:,N(2));

c1evnoise = c1evolve + stdnoisereundefinedrandn(N(1),N(2)); % adds noise of same std as the roi.

% compare the evolved image with the final image, ONLY in the region of interest!

chi2(j) = sum(sum(reg.*(A2-c1evnoise).*(A2-c1evnoise)))/Nreg;

Deff(j) = dundefineddundefinedj/(2.undefinedtmax); % Diffusion coefficient, if 'time' j were the best match to the data

if (chi2(j) < minchi2)

minchi2 = chi2(j);

bestN = j;

bestevolve = c1evolve;

bestevolvenoise = c1evnoise;

end

waitbar(j/Nmax, progbar, 'evolving image...' );

end

close(progbar);

disp( ' ' );

disp( ~undefined_RESULTS~I~9~K~Hspan~M~1~Kspan~M~B~J~K~Hspan~M~1~Kspan~M~K~Hspan~M~K~Hp~M~Kp~M~K~Hp~M~1~1~Kp~M~K~Hp~M~1~1~Kp_class~L~4MsoNormal~4~M~Kspan~MbestD_~L_dundefineddundefinedbestN/(2.undefinedtmax);

% smoothed chi^2 -- boxcar

Dsize = dundefineddx/(2.undefinedtmax); % D resolution per point in above scan

if (Dbox > Dsize)

nbox = floor(Dbox/Dsize); % no. points per boxcar = desired D resolution / res. in above scan

Npts = floor(Nmax / nbox); % no. boxcars

for j=1:(Npts-1),

smDeff(j) = mean(Deff((j-1~undefinednbox+1:undefinednbox));

smchi2(j) = mean(chi2((j-1~undefinednbox+1:undefinednbox));

end

[msmc, ci] = min(smchi2);

msmD = smDeff(ci);

fs = sprintf( ' Best-fit diffusion coefficient (smoothed to %.2f um^2/s resolution) = %.2f um^2/s.' , ...

Dbox, msmD);

disp(fs);

fs = sprintf( ' Minimal chi^2 (per pt.) = %.2e.' , msmc); disp(fs);

else

disp( ' Resolution in D too poor for desired smoothing.' );

smDeff = Deff;

smchi2 = chi2;

fs = sprintf( ' Best-fit diffusion coefficient (no smoothing) = %.2f um^2/s.' , bestD); disp(fs);

fs = sprinft( ' Minimal chi^2 (per pt.) = %.2e.' , minchi2); disp(fs);

end

% plot chi^2 and smoothed chi^2

figure(3);

plot(Deff, chi2, 'k-' , 'LineWidth' , 2.0);

hold on

plot([min(Deff) max(Deff)], [minchi2 minchi2], 'r:' );

plot([min(Deff) max(Deff)], undefined[minchi2 minchi2], 'b:' );

plot(smDeff, smchi2, 'g-' , 'Linewidth' , 2.0);

xlabel( 'Effective D, um^2/s' , 'FontWeight' , 'bold' );

ylabel( '\chi^2' , 'FontWeight' , 'bold' );

title( '\chi^2' , 'FontWeight' , 'bold' );

% Show figures

figure(2);

if (useroi==1)

subplot(2,2,3); imshow(uint8(bestevolvenoisundefineddispscale)); colormap( 'gray' );

title( 'Evolved Im.1, with noise' );

subplot(2,2,4); imshow(uint8(reundefined255)); colormap( 'gray' );

title( 'Region of interest map' );

else

subplot(1,3,3); imshow(uint8(bestevolvenoisundefineddispscale)); colormap( 'gray' );

title( 'Evolved Im.1, with noise' );

end

% return the warning that image may be re-scaled to fit the screen to its

% initial state

iptsetpref( 'TruesizeWarning' , tw);