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 powerMonitor low power

Idealized photobleaching data

YX

= mobile fraction

D =2/4D

A way of understanding diffusion: Random Walk

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.

NdL 2|/|L: End-to-end distanced: Step sizeN: number of steps

Dt

td

NdLLrms

4

)4(4 2

22

A way of understanding diffusion: Fick’s Law

xDJ

• J is flux • D is diffusion constant• is concentration2

2

xD

t

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 / 6r (By Stokes)

AB Both A and B will

have similar D inMembrane althoughVery different sizes

Binding to immobilized matrix will reduce fraction of molecules diffusing

intra

extracellular

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 = 2/4D to obtain D

•Determine = nominal width of Gaussian spot by other optical method 1/e2 point

•Fit fluorescence recovery curve to obtain D

0

100 /211!//

nD

nK tnnKACqPtF

Axelrod et al., 1976

Real FRAP data

More Diffusion types

Important for Large macromolecules:Collisions, obstacles, binding

Fully recovers

Different bleaching geometries yield different types of information

1. Line photobleaching generates a one-dimensional diffusion problem

Allows collection of more fluorescence, averaging

F

x

Note that beam is still GaussianLine scan of single points

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:

D

D

ttatt

with

ttxtFtxF

/1//1

/exp1)(/,

02

20

2

222

where a 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 coilsNo simple m1/3 scaling

Verkman, J. Cell Biology 1999

Diffusion of FITC Dextrans, Ficolls inMDCK Cell Cytoplasm

Heavy dextrans very slowMobile fraction low: bindingMore polarizable

Verkman, J. Cell Biology 1999

• Problem is much more complicated because of three dimensional freely diffusing geometry.

FRAP in Cytoplasm

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

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

Not reliable, cytoplasm complicated collection of fluid, cytoskeletal components, endosome, etc: simple viscosity not sufficient

Photobleaching of cytoplasmic components

D =2/4D

Another solution is to use geometry such depth of field is comparable to thickness of cell

High NA lens Low NA lens

Geometry approximates cylinder bleached through ZDiffusion becomes 2D problem: easier

Photobleaching of cytoplasmic components

Recovery is convolvedWith depth of field

Compartmentalization and active transport

Eukaryotic cells tackle problem of organization by compartmentalization

Nucleus: DNA replication & transcriptionMitochondria: energy productionEndoplasmic reticulum: protein synthesisGolgi apparatus: protein sortingLysosymes: protein degradation and recyclingPlasma 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 volumedue 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 concentrationsShorter correlation times

webb

Binding to mobile receptor Binding to immobile receptor Motility along microtubule

ConcentrationDiffusion of receptor

ConcentrationKd

On rate (M-1sec-1)Off rate (sec-1)

Mobile/immobileMean squared displacement

The slow component in living cells

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.

fluorescentmolecules

Cross-correlation spectroscopy

3D Gaussianconfocal detection volume

~1 femtoliter

diffusiontrajectories

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.

Cross-correlation function Grg(t) = < Ig(t).Ir(t+t) >

1

1.02

1.04

1.06

1.08

1.1

10 100 1000 10000microseconds

Alexa488 RNASyto61cross-correlation

Dual channel fluctuation

10000

15000

20000

25000

30000

35000

40000

45000

50000

0 1 2 3 4 5 6 7 8 9 10

seconds

Alexa488 RNASyto61

Count rate

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