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 YXYX = 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. L: End-to-end distance d: Step size N: number of steps

A way of understanding diffusion: Fick’s Law 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 m 1/3 (radius scaling)  = 1 / 6  r (By Stokes)

A B Both A and B will have similar D in Membrane although Very 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/e 2 point Fit fluorescence recovery curve to obtain  D 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 Gaussian Line 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: 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 coils No simple m 1/3 scaling Verkman, J. Cell Biology 1999

Diffusion of FITC Dextrans, Ficolls in MDCK Cell Cytoplasm Heavy dextrans very slow Mobile fraction low: binding More 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 2If 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., SekSek et al 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 Z Diffusion becomes 2D problem: easier Photobleaching of cytoplasmic components Recovery is convolved With depth of field

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 τDτD =diffusion time, R=ω z /ω xy of observation volume

FCS of Rhodamine in Sucrose Solution Higher concentrations Shorter correlation times webb

Binding to mobile receptorBinding to immobile receptorMotility along microtubule Concentration Diffusion of receptor Concentration Kd On rate (M -1 sec -1 ) Off rate (sec -1 ) Mobile/immobile Mean 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.

fluorescent molecules Cross-correlation spectroscopy 3D Gaussian confocal detection volume ~1 femtoliter diffusion trajectories 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 G rg (t) = microseconds Alexa488 RNA Syto61 cross-correlation Dual channel fluctuation seconds Alexa488 RNA Syto61 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