Simple Random Walk
The distribution approaches Gaussian after many steps
A random variable
FRAP - Fluorescence Recovery After
Photobleaching
Cartoon of FRAP
Bleach creates “hole” of fluorophores,
Diffusion is measured by “hole filling in”
Bleach high power
Monitor low power
Idealized photobleaching data
Y
= mobile fraction
X
D =w2/4D
A way of understanding diffusion:
Random Walk
| L/d |
2
N
L: End-to-end distance
d: Step size
N: number of steps
Lrms 
L2  d 2 N
 4(d 2 4 )t
 4Dt
Spread of molecules from one spot is proportional to
square root of time for random walk. Therefore, to go 2X
as far takes 4X as long.
A way of understanding diffusion:
Fick’s Law

J  D
x

 2
t
D
x 2
• J is flux
• D is diffusion constant
•  is concentration
Diffusion Constant
• Random thermal motions: (By Einstein)
D = kT
 = v/F
•  depends on size of particle and viscosity
of solution.
For spheres: scale as m1/3 (radius scaling)
 = 1 / 6r
(By Stokes)
B
A
intra
Both A and B will
have similar D in
Membrane
although
Very different
sizes
extracellular
Binding to immobilized
matrix will reduce
fraction of molecules
diffusing
Diffusion of membrane components can be
seen as a two dimensional diffusion problem
• Membrane is modeled as infinite plane
• Viscosity of the lipid bilayer is ~ 2 orders of magnitude
higher than water
• As shown by Saffman and Delbruck, the translational
diffusion coefficient for membrane components
depends only on the size of the membrane spanning
domain
Spot Photobleaching
•Bleach and monitor single diffraction limited spot
•Assumes infinite reservoir of fluorescent
molecules (hole can fill back in)
•Use D = w2/4D to obtain D
•Determine w = nominal width of Gaussian spot by
other optical method 1/e2 point
•Fit fluorescence recovery curve to obtain D



FK t   qP0C0 / A  K  / n! 1  n1  2t /  D 
n 0
n
1
Axelrod et al., 1976
Real FRAP data
More Diffusion types
Fully recovers
Important for
Large macromolecules:
Collisions, obstacles, binding
Different bleaching geometries yield different types
of information
1. Line photobleaching generates a one-dimensional diffusion
problem
Note that beam is still Gaussian
Line scan of single points
F
x
Allows collection of more fluorescence, averaging
Scanning over bleach spot improves ability to characterize
recovery curves
• Allows accurate characterization of the bleach
geometry and size for each individual experiment
• Simplifies fits of recovery curves to:


F x, t  / F ()  1   2 t  exp  x  t / w 2 t 
2
with
w 2 t   w 0 2 1  t /  D 
a2 t    0 / 1  t /  D 
a
where is a constant reflecting
extent of bleaching.
Koppel, 1979 Biophys. J. 281
• Allows compensation for photobleaching during
monitoring and sample drift.
Size dependence of dextrans (polysaccharides)
diffusion in solution
Not simple spheres:
Random coils
No simple m1/3 scaling
Verkman, J. Cell Biology 1999
Diffusion of FITC Dextrans, Ficolls in
MDCK Cell Cytoplasm
Verkman, J. Cell Biology 1999
Heavy dextrans very slow
Mobile fraction low: binding
More polarizable
FRAP in Cytoplasm
• Problem is much more complicated because of
three dimensional freely diffusing geometry.
Problems with FRAP of cytoplasmic
components (2 orders of magnitude faster
than membranes)
1. Diffusion is fast compared to bleaching and
monitoring rate D=ms : cannot truly scan
2 If use small bleach regions, redistribution may
occur during bleaching. In fact, often cannot
observe bleach of small region at all.
3. By enlarging the size of the bleach region, can
overcome this problem: but lose localization
Photobleaching of cytoplasmic components
One solution is to measure cytoplasmic diffusion by
comparing to characteristic times of known samples in
solutions of known viscosity.
e.g. Luby-Phelps et al., 1994.
SekSek et al. 1997.
D = kT/f
D =w2/4D
Not reliable, cytoplasm complicated
collection of fluid, cytoskeletal components,
endosome, etc: simple viscosity not
sufficient
Photobleaching of cytoplasmic components
Another solution is to use geometry such depth of
field is comparable to thickness of cell
High NA lens
Recovery is convolved
With depth of field
Low NA lens
Geometry approximates
cylinder bleached through Z
Diffusion becomes 2D
problem: easier
Compartmentalization and active transport
Eukaryotic cells tackle problem of organization by
compartmentalization
Nucleus: DNA replication & transcription
Mitochondria: energy production
Endoplasmic reticulum: protein synthesis
Golgi apparatus: protein sorting
Lysosymes: protein degradation and recycling
Plasma membrane: extracellular signalling
Move components between compartments
• Vesicle trafficking: endocytosis / exocytosis
• Cytoskeleton: filaments and motor proteins
Common applications often
bypass complicated analysis
Cells expressing
VSVG–GFP were
incubated at 40 °C to
retain VSVG–GFP in
the endoplasmic
reticulum (ER) under
control conditions (top
panel) or in the
presence of
tunicamycin (bottom
panel). Fluorescence
recovery after
photobleaching (FRAP)
revealed that VSVG–
GFP was highly mobile
in ER membranes at
40 °C but was
immobilized in the
presence of
tunicamycin(Nehls et al,
2000 Nature Cell Biology)
Fluorescence Loss in Photobleaching “FLIP”
continuous bleaching measure of mobility
Figure 3 | Fluorescence loss in photobleaching. Protein fluorescence in a
small area of the cell (box) is bleached repetitively. Loss of fluorescence in areas
outside the box indicates that the fluorescent protein diffuses between the
bleached and unbleached areas. Repetitive photobleaching of an endoplasmic
reticulum (ER) GFP-tagged membrane protein reveals the continuity of the ER in
a COS-7 cell. Image times are indicated in the lower right corners. The
postbleach image was obtained immediately after the first photobleach. The cell
was repeatedly photobleached in the same box every 40 s. After 18 min, the
entire ER fluorescence was depleted, indicating that all of the GFP-tagged
protein was highly mobile and that the entire ER was continuous with the region
in the bleach box. (Nehls et al, 2000 Nature Cell Biology)
Fluctuation (fluorescence) Correlation Spectroscopy (FCS)
Fluctuations in excitation volume
due to Diffusion, reactions
Compares probability of detecting photon
at time t with some latter time t + τ
Form for translational diffusion
N=concentration of molecules in focal volume
τD =diffusion time, R=ωz/ωxy of
observation volume
FCS of Rhodamine in Sucrose Solution
Higher concentrations
Shorter correlation times
webb
The slow component in living cells
Binding to mobile receptor
Concentration
Diffusion of receptor
Binding to immobile receptor
Concentration
Kd
On rate (M-1sec-1)
Off rate (sec-1)
Motility along microtubule
Mobile/immobile
Mean squared displacement
Mathematical model for autocorrelation
Two component autocorrelation curve
APPLICATIONS
– peptides bound to soluble receptors,
– ligands bound to membrane-anchored receptors,
– viruses bound to cells,
– antibodies bound to cells,
– primers bound to target nucleic acids,
– regulatory proteins /protein-complexes in interaction
with target DNA or RNA
– enzymatic products.
If the diffusion properties of the reactants are too similar, both
reactants have to be labeled with fluorescent dyes with different
excitation and emission spectra.
Cross-correlation spectroscopy
Dual channel fluctuation
Count rate
50000
Alexa488 RNA
Syto61
45000
40000
35000
3D Gaussian
confocal detection volume
~1 femtoliter
30000
25000
20000
15000
10000
0
1
2
3
4
5
6
7
8
9
10
seconds
fluorescent
molecules
diffusion
trajectories
Cross-correlation function Grg(t) = < Ig(t).Ir(t+t) >
1.1
Alexa488 RNA
Syto61
cross-correlation
1.08
1.06
Individual fluorescent molecules are detected as
single channel photon count fluctuations. Bound
molecules are detected as coincident dual channel
fluctuations.
Cross-correlation analysis provides a measure of the
number and rate of diffusion of bound molecules.
1.04
1.02
1
10
100
1000
microseconds
10000
Multiphoton bleaching
Need 3D treatment
Diffusion-related techniques: FRAP, FCS
and SPT
• Obtain diffusion coefficient
• Binding/mobile fraction
• Define active transport/directed flow mechanisms
• Define trafficking rates through intracellular
compartments (including cytoplasm, fast)
• Detect protein-protein interactions