Supported Membrane Formation, Characterization, Functionalization, and Patterning for Application in Biological Science and Technology

Wan‐Chen Lin1, Cheng‐Han Yu2, Sara Triffo1, Jay T. Groves3

1 Howard Hughes Medical Institute, Department of Chemistry, University of California, Berkeley, California, 2 Research Center of Excellence in Mechanobiology, National University of Singapore, Singapore, 3 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California
Publication Name:  Current Protocols in Chemical Biology
Unit Number:   
DOI:  10.1002/9780470559277.ch100131
Online Posting Date:  December, 2010
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Abstract

Supported membranes, formed as a single continuous lipid bilayer on a solid substrate, such as silica, have been used extensively as a model for protein‐protein and cell‐cell interaction, to study the molecular interactions at interfaces and the heterogeneities of plasma membranes. The advantages of a supported membrane system include the ability to control membrane composition and the compatibility it has with various surface‐sensitive microscopic and spectroscopic techniques. Recent advances in micro‐ and nanotechnology have greatly extended the use of supported membranes to address key questions in cell biology. Although supported membranes can be easily made by vesicle fusion, the samples need careful preparation for this process to be efficient. The protocols in this unit comprehensively describe procedures to prepare, functionalize, and characterize supported membranes. Curr. Protoc. Chem. Biol. 2:235‐269 © 2010 by John Wiley & Sons, Inc.

Keywords: supported membrane; supported lipid bilayer; small unilamellar vesicle (SUV); membrane functionalization; fluorescence recovery after photobleaching (FRAP); quantitative fluorescence measurement; photolithography

     
 
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Table of Contents

  • Introduction
  • Strategic Planning
  • Basic Protocol 1: Preparing Small Unilamellar Vesicles (SUVs) by Extrusion
  • Alternate Protocol 1: Preparing SUVs by Probe Sonication
  • Alternate Protocol 2: Preparing SUVs by Freeze‐Thawing
  • Basic Protocol 2: Preparing Membrane Supports by Piranha Etching
  • Alternate Protocol 3: Cleaning Substrates by Base Etching
  • Alternate Protocol 4: Cleaning Substrates with Air/Oxygen Plasma
  • Alternate Protocol 5: Cleaning Substrates with Ultraviolet Light/Ozone
  • Support Protocol 1: Preparing Substrates with Diffusion Barriers (Gridded Substrates)
  • Support Protocol 2: Preparing Substrates with Curvature Modulation
  • Basic Protocol 3: Formation and Functionalization of Supported Membranes
  • Support Protocol 3: Formation of Supported Membranes on Silica Beads
  • Support Protocol 4: Formation of Supported Intermembrane Junctions
  • Basic Protocol 4: Characterizing Supported Membranes
  • Support Protocol 5: Measuring the Scaling Factor
  • Reagents and Solutions
  • Commentary
  • Literature Cited
  • Figures
  • Tables
     
 
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Materials

Basic Protocol 1: Preparing Small Unilamellar Vesicles (SUVs) by Extrusion

  Materials
  • Stock lipid solutions in chloroform or a 2:1 (v/v) chloroform:methanol mixture, at a concentration of 0.1 to 10 mg/ml
  • Chloroform (ACS grade and above)
  • Nitrogen gas (industrial grade or better; if from a central supply, use a hydrophobic filter to remove oils from the gas)
  • 1 to 4 ml rehydration solution [deionized (DI) water or TBS or PBS; see reciperecipes; see for selection]
  • Deionized water (resistivity ≥18 MΩ and apparent pH 5.5)
  • Argon
  • 25‐ to 50‐ml glass round‐bottomed flask (cleaned by piranha or base etching; see protocol 4 and protocol 5)
  • Positive‐displacement pipet with capillary piston made of pure polypropylene (Gilson), or Hamilton syringes
  • Rotary evaporator attached to a vacuum pump
  • 40° to 50°C water bath
  • Benchtop vortex mixer
  • Extruder: Lipex extruder (Northern Lipids) or Avanti Mini‐Extruder (Avanti Polar Lipids)
NOTE: All stock lipid solutions should be stored in glass vials with Teflon caps or Teflon septa at −20°C or lower. Chloroform and methanol should be ACS grade or above.

Alternate Protocol 1: Preparing SUVs by Probe Sonication

  Materials
  • Isopropanol (ACS grade and above)
  • Cleaning solution: 1:1 (v/v) isopropanol/water
  • Deionized (DI) water (resistivity ≥18 MΩ and apparent pH 5.5)
  • 1 to 4 ml of lipid suspension (see protocol 1, steps 1 to 8)
  • Nitrogen gas (industrial grade or better; if from a central supply, use a hydrophobic filter to remove oils from the gas)
  • Argon
  • Emery sheet (3/0 grit or finer)
  • Ultrasonic processor equipped with a double‐stepped microtip (e.g., VCX750, Sonics & Materials) in a sound‐abating enclosure
  • Ice bath
  • Centrifuge that can reach 16,000 × g
  • Microcentrifuge tubes

Alternate Protocol 2: Preparing SUVs by Freeze‐Thawing

  Materials
  • 1 to 4 ml of lipid suspension (see protocol 1, steps 1 to 8)
  • Argon
  • Dry ice‐ethanol bath (or liquid nitrogen)
  • 50°C water bath
  • Microcentrifuge tubes

Basic Protocol 2: Preparing Membrane Supports by Piranha Etching

  Materials
  • 1:1 (v/v) isopropanol/water
  • Deionized water (resistivity ≥18 MΩ and apparent pH 5.5)
  • Sulfuric acid (H 2SO 4; ACS grade)
  • 30% hydrogen peroxide (H 2O 2; ACS grade)
  • Glass substrates
  • Teflon or glass rack/holder
  • Bath sonicator

Alternate Protocol 3: Cleaning Substrates by Base Etching

  Materials
  • 1 M sodium hydroxide (NaOH; ACS grade)
  • Deionized water (resistivity ≥18 MΩ and apparent pH 5.5)
  • Glass substrates

Alternate Protocol 4: Cleaning Substrates with Air/Oxygen Plasma

  Materials
  • Nitrogen gas (industrial grade or higher; if from a central supply, use a hydrophobic filter to remove oils from the gas)
  • Deionized water (resistivity ≥18 MΩ and apparent pH 5.5)
  • Plasma generator (SPI Plasma‐Prep II, SPI Supplies/Structure Probe)
  • Mechanical vacuum pump with oil filters (Leybold Vacuum Pumps, SPI Supplies/Structure Probe)
  • Oxygen gas with regulator, optional
  • Additional reagents and equipment for precleaning the glass substrates ( protocol 4, steps 1 to 4)

Alternate Protocol 5: Cleaning Substrates with Ultraviolet Light/Ozone

  Materials
  • Nitrogen gas (industrial grade or higher; if from a central supply, use a hydrophobic filter to remove oils from the gas)
  • Deionized water (resistivity ≥18 MΩ and apparent pH 5.5)
  • UV/ozone cleaner (UV/Ozone Procleaner Plus, BioForce Nanosciences)
  • UV protective safety glasses
  • Additional reagents and equipment for precleaning the glass substrates ( protocol 4, steps 1 to 4)

Support Protocol 1: Preparing Substrates with Diffusion Barriers (Gridded Substrates)

  Materials
  • Hexamethyldisilazane (HMDS; ACS grade or higher)
  • Photoresist (S1805 positive g‐line photoresist, Microchem)
  • Photoresist developer (MicroDev; Microchem)
  • Photoresist stripper (acetone, ACS grade or higher)
  • Deionized water (DI water; resistivity ≥18 MΩ and apparent pH 5.5)
  • Nitrogen gas
  • Isopropanol (ACS grade or higher)
  • General cleanroom (class 100 or above)
  • Spin coater (Laurell Technologies)
  • 90°C hot plate
  • Mask aligner (NXQ 4006, Neutronix‐Quintel)
  • Photomask (design of the mask, usually 1‐ to 3‐µm features, can be generated by L‐edit Pro; Tanner EDA; quartz masks with the designed feature, such as parallel‐line grid patterns, are then manufactured by a mask‐making vendor)
  • Metal target (99.99% chromium, Alfa Aesar)
  • Thin film evaporator (Edwards EB3 electron beam evaporator; Edwards)
  • Additional reagents and equipment for cleaning substrates using piranha etching ( protocol 4) and cleaning the patterned substrates using air/oxygen plasma cleaning or UV/ozone cleaning ( protocol 6 and 5)

Support Protocol 2: Preparing Substrates with Curvature Modulation

  Materials
  • Hexamethyldisilazane (HMDS; ACS grade or higher)
  • Photoresist (S1805 positive g‐line photoresist; Microchip)
  • Photoresist developer (MicroDev, Microchem)
  • Distilled water (DI water; resistivity ≥18 MΩ and apparent pH 5.5)
  • Nitrogen gas
  • Piranha solution (see protocol 4)
  • 5:1 buffered hydrofluoric acid (buffered oxide etch, 5:1 CMOS; Mallinckrodt Baker)
  • Photoresist stripper (Acetone, ACS grade or higher)
  • General cleanroom facility (class 100 or higher)
  • Spin coater
  • Hotplate
  • Mask aligner (NXQ 4006, Neutronix‐Quintel)
  • Photomask (design of the mask, usually 1 to 3 micron features, can be generated by L‐edit Pro; Tanner EDA; quartz masks with the designed feature, such as parallel lines, are then manufactured by a mask‐making vendor)
  • Plasma etcher for SiO 2 anisotropic etching (AutoEtch 590, Lam Research)
  • Additional reagents and equipment for cleaning the substrates using piranha etching and preparing piranha solution ( protocol 4)

Basic Protocol 3: Formation and Functionalization of Supported Membranes

  Materials
  • Nitrogen gas (industrial grade or higher; if from a central supply, use a hydrophobic filter to remove oils from the gas)
  • Clean substrates ( protocol 4, Alternate Protocols protocol 53, protocol 64, and protocol 75)
  • SUV suspension ( protocol 1, Alternate Protocols protocol 21 and protocol 32)
  • Spreading buffer (2× PBS or 2× TBS; see reciperecipes; see for selection)
  • Working buffer: deionized water or TBS (see recipe) or phosphate‐buffered saline (PBS; see recipe)
  • Blocking solution: 5 mg/ml casein in PBS (see recipe for PBS) or 0.01% (w/v) bovine serum albumin (BSA) in PBS (see recipe for PBS)
  • Aluminum foil, optional

Support Protocol 3: Formation of Supported Membranes on Silica Beads

  Materials
  • Spreading buffer: 2× TBS or 2× PBS (see reciperecipes)
  • SUV suspension ( protocol 1, Alternate Protocols protocol 21 and protocol 32)
  • Silica beads at 10 wt% solids in deionized water (Bangs Laboratories)
  • Working buffer: deionized water or TBS (see recipe) or phosphate‐buffered saline (PBS; see recipe)
  • 1.5‐ml microcentrifuge tubes
  • Benchtop vortex mixer
  • Mini centrifuge (max g force ∼2000 × g)

Support Protocol 4: Formation of Supported Intermembrane Junctions

  Materials
  • 0.5 M sucrose, warmed to 50°C
  • Dry lipid film in a round‐bottomed flask ( protocol 1, steps 1 to 6)
  • Supported membrane (see protocol 10)
  • Working buffer: DI water or TBS (see recipe) or PBS (see recipe)
  • Oven with temperature controls or a large water bath at 50°C
  • 10‐ml syringes
  • Microcentrifuge tubes

Basic Protocol 4: Characterizing Supported Membranes

  Materials
  • Fluorescent samples (i.e., supported membrane with or without protein functionalization; see protocol 10)
  • Standard supported membranes with known fluorophore concentrations (e.g., DOPC bilayers containing either BODIPY‐FL‐DHPE or Texas Red‐DHPE)
  • Fluorescence microscope equipped with:
    • Filter sets that match the excitation and emission spectrum of the fluorophore used
    • Light source (typically a mercury lamp or a xenon lamp)
    • Adjustable field diaphragm
    • 60× or higher objective lens
    • CCD camera

Support Protocol 5: Measuring the Scaling Factor

  Materials
  • Standard SUV suspensions with known fluorescent lipid molarity (see protocol 1 for SUV suspensions and protocol 13 for selection of standard fluorophore)
  • Fluorescent protein solutions with known protein concentrations
  • Buffer (same as that used to make the fluorescent protein solution)
  • Fluorescence microscope equipped with:
    • Filter sets that match the excitation and emission spectrum of the fluorophore used
    • Light source (typically a mercury lamp or a xenon lamp)
    • Adjustable field diaphragm
    • 20× objective lens
    • CCD camera
  • 96‐well plate (non‐glass bottom)
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Literature Cited

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   Wong, A.P. and Groves, J.T. 2002. Molecular topography imaging by intermembrane fluorescence resonance energy transfer. Proc. Natl. Acad. Sci. U.S.A. 99:14147‐14152.
   Yamazaki, V., Sirenko, O., Schafer, R.J., and Groves, J.T. 2005. Lipid mobility and molecular binding in fluid lipid membranes. J. Am. Chem. Soc. 127:2826‐2827.
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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 = dx*dx*j/(2.0*(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 dx*dx / (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('**** FRAPevolve2D.m ****');

disp(' ');

disp('** Recommended: gray-scale .tif images.');

disp('** Note instructions in command window (this window) as well as the dialog boxes..');

disp(' ');

% images should be a grayscale .tif; saved as array A of type uint8; BMP also works.

loadopt = input('Enter 1 to choose filenames from a dialog box, 0 to type them manually: ');

% Load first FRAP image

if (loadopt==1)

% dialog box for filename

[pFileName,pPathName] = uigetfile('*.*', '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('*.*', '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(A1sub*255.0/max(max(A1sub)))); colormap('gray'); title('Image 1');

subplot(1,3,2); imshow(uint8(A2sub*255.0/max(max(A2sub)))); colormap('gray'); title('Image 2');

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

% Crop, scale intensities

figure(2)

if (crop==1)

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

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

scaleA1sub = 255.0*(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 = A2*(mean(mean(A1))/mean(mean(A2)));

end

close(2)

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

% measure noise

scaleA1 = 255.0*(A1 - min(min(A1)))/(max(max(A1)) - min(min(A1sub))); % scale for display

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

disp(' '); disp('* 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(A1*255.0/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('* 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('*.*', '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 = 0.5*(max(double(roiimage(:))) + min(double(roiimage(:))));

else

mr = 0.5*max(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 = scale*resize;

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(A1*dispscale)); colormap('gray'); title('Image 1');

subplot(2,2,2); imshow(uint8(A2*dispscale)); colormap('gray'); title('Image 2');

else

subplot(1,3,1); imshow(uint8(A1*dispscale)); colormap('gray'); title('Image 1');

subplot(1,3,2); imshow(uint8(A2*dispscale)); colormap('gray'); title('Image 2');

end

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

% Evolve image (simulated diffusion)

dx = scale; % microns per pixel

Nmax = round(2*tmax / (dx*dx/(2*10.0))); % maximal number of steps to run

disp(' ');

fs = sprintf('* 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('* Range of D that can be probed is %.2e to %.2e um^2/s.', dx*dx/(2.0*tmax), dx*dx*Nmax/(2.0*tmax));

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 = 0.25*[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 undefined. 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 + stdnoisereg*randn(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) = dx*dx*j/(2.0*tmax); % 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('** RESULTS:');

bestD = dx*dx*bestN/(2.0*tmax);

% smoothed chi^2 -- boxcar

Dsize = dx*dx/(2.0*tmax); % 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)*nbox+1:j*nbox));

smchi2(j) = mean(chi2((j-1)*nbox+1:j*nbox));

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)], 2*[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(bestevolvenoise*dispscale)); colormap('gray');

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

subplot(2,2,4); imshow(uint8(reg*255)); colormap('gray');

title('Region of interest map');

else

subplot(1,3,3); imshow(uint8(bestevolvenoise*dispscale)); 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);