Theoretical Background:

Fluorescence correlation spectroscopy (FCS)

In contrast to conventional applications of fluorescence spectroscopy, FCS is a technique that derives thermodynamic and kinetic information not from an ensemble average but from small spontaneous deviations from equilibrium that give rise to fluctuations in fluorescence emission. In order to resolve such spontaneous fluctuations, the systems under observation have to be kept extremely small. Ideally, ensembles consisting of several or even single molecules at any time are probed. This condition is achieved by combining ultrasmall measurement volumes with extremely low sample concentrations. In confocal FCS geometries which are at present the most popular, the reduction of the measurement volume to approx. 1 fl is achieved by epiillumination of a high numerical aperture (NA > 0.9) objectives with parallel laser beam and imaging the focal spot onto a diaphragm or pinhole in the image plane.

Figure 1: Confocal setup for FCS detection. The laser light is focused to a diffraction-limited volume, which is imaged onto a pinhole or optical fiber of ~100 µm diameter. This results in an open volume element of approx. 10^-15 l .

The temporal fluctuations of the fluorescence signal F(t) recorded with a very sensitive detector (avalanche photodiode, APD) are analyzed by means of normalized fluorescence autocorrelation function G(τ) with lag time τ:

G(τ) ≡δF(t)δF(t +τ) F(t) ² (1)

where δF(t) ≡ F(t) -‚F(t)Ú, and ‚Ú is the temporal average over the duration of the experiment. The autocorrelation function reflects the decay of spontaneous fluctuations of F(t). The amplitude and the shape of G(τ) are determined by the underlying processes. The most common source of fluctuations in F(t) is the fluctuation of the number of molecules in the measurement volume due to the random 3D diffusion. In this case, the characteristic decay time of G(τ) reflects the average residence time of the molecules in the focal spot. Due to the Poissonian nature of the particle number fluctuations at low concentrations, the amplitude G(0) is inversely proportional to the average number of molecules present in the volume:

11

G(0) == (2) ^{N CV}eff

In case of a 3-dimensional Gaussian-shaped excitation intensity distribution, the autocorrelation function Gdiff(τ) for freely diffusing molecules of one species with concentration C is given by:

G_diff ()τ = ^{1 1 1} (3)CV_eff 1+

d 1+r₀²τ z₀²τ_d

where Veff is the size of the effective volume element:

3

V_eff =π²r₀²z₀ (4)

and τd the characteristic residence time of a molecule within Veff. The parameters r0 and z0 are the 1/e² half axes of the measurement profile W(r) in the sample space which is considered to be Gaussian in all three dimensions:

−2 ^{x2 + y 2} −2 ^{z 2}

W (r ^r) = e^r02 e ^z02 (5)

The relationship between τd and the diffusion coefficient D is given by

2

τ_d = ^r0 (6)

4D

The diffusion time τd which characterizes the decay of the autocorrelation function is then determined by the diffusion coefficient of the molecules D (how fast the molecules move) and the parameter r0 (the size of the volume over which the molecules move).

If the molecules additionally undergo intramolecular dynamics reflected as blinking of the fluorescence (on/off transitions) on a fast time scale during their residence time in the focal spot, G(τ) has to be modified as follows (in case where τf << τd):

(1− f + f ⋅ e ^{−τ τ f} )

G_{diff +fast} (τ ) = ^{D D} G_diff (τ ) (7)(1− f_D )

where fD is the average fraction of particles in the dark state, and τf is the inverse of the characteristic blinking rate: τf = 1/(kb+kd) with kb and kd being the respective transition rates from the dark to the bright state and vice versa.

The investigated system

The green fluorescent protein is in aqueous solution at neutral and higher pH present in an anionic state with absorption maximum at ~490 nm. It has been shown that at low pH the chromophore gets protonated by external protons leading to the shift of its absorption maximum towards a shorter wavelength of around 400 nm. The protein is no longer excitable at 488 nm (the Ar⁺ laser used in Praktikum) and appears dark (non-fluorescent) during the lifetime of the protonated form. Thus, if the protonation/deprotonation-induced blinking of fluorescence emission is measured on a single molecule scale as accomplished by FCS, protonation and deprotonation rate constants can be determined.

Fig. 2: Green fluorescent protein (GFP), a small protein (26 kDa, 1Da = 1g/mol), naturally occurring in jellyfish Aequorea victoria, has become a remarkable tool in fluorescence microscopy. Upon excitation in the visible spectral range (e.g. at 488 nm), the protein itself exhibits a strong native fluorescence which arises from a chromophoric unit inside the protein’s barrel-like tertiary structure (chromophore in green).

The protonation/deprotonation reaction: A^-+ H⁺ ↔ AH (8)

kd

R O + H⁺

R

OH

kb _A pK-5.7 _AH

bright state dark state abs.:~490 nm abs.:~400 nm

Fig. 3: The above figures show the protonation scheme with dark intervals given by the protonated form of the chromophore (left picture), as well as the correlation curves that are recorded from solutions of GFP in buffers of different pH. It can be seen that the dark fraction, given by the amplitude of the fast exponential process, increases with decreasing pH, while the fast time constant τf = 1/(kb+kd)= 1/(kb+kd^'[H⁺]) decreases.

Practical work:

1. Setup The FCS-measurements are carried out in sample droplets of an aqueous buffer solution with a custom-made microscope setup using excitation wavelength of 488 nm. The optics (mirrors, filters, objective, lenses, pinhole) are installed in the microscope module. The laser is coupled in externally. The excitation intensity needs to be adapted to the experimental conditions.

2. Materials The GFP is studied in buffer solutions of different pH values. The GFP mutant under investigation is eGFP (enhanced GFP, Clonetech), exhibiting stronger fluorescence and higher stability compared to the wild-type protein. The phosphate/citrate buffer at pH values ranging from 4.5 to 10.0 is used to prepare the solutions. The protein concentration should be less than 100 nM. At low pH values where the protein is less stable and partially denaturates already short time after mixing higher concentrations are used.

3. Data Evaluation The digital fluorescence signal (pulses corresponding to detected photons) is autocorrelated by a hardware correlator, a PC ALV-5000 multiple-τ correlator card (ALV, Langen, Germany). Evaluation of the curves is carried out by ORIGIN (MicroCal Software, Northampton, MA) by using the Marquardt-Levenberg fitting routine. The measured curved are fitted to the respective theoretical model (diffusion, blinking dynamics etc.) based on Eq. 7.

Questions and Tasks:

• In what sense can FCS be called a single-molecule method (even though the number of molecules probed at any time may be larger than one)? Compare, for example, with a standard absorption measurement of concentrated solution in a spectrophotometer.

• What conditions a system has to fulfill to be investigated by FCS?

• What are the main quantities measurable with FCS?

• Calculate the size of the experimentally defined measurement volume element Veff. Use the value of the diffusion coefficient of eGFP reported in literature: D = 8.7·10^-7 cm²/s (R. Swaminathan, Biophys. J., 72, 1900, 1997).

• Calculate the absolute concentration of your sample (in mol/l, exemplary for one measurement of choice)

• Calculate the power density of the excitation laser light within the focal spot (pay attention to the correct units). How would much larger or much smaller excitation power affect the results of the measurement (diffusion time, particle number, ...)?

• The autocorrelation curves at very low pH often vary between the different runs (10-20 s periods of measurement). Why?

• The equilibrium between the anionic and the protonated form of GFP at any pH is characterized by the dissociation constant Ka (the amino acid Tyrosine in the case of the here studied eGFP) or its negative decadic logarithm pKa ≡ - log10 Ka. The theoretical value of pKa is about 5.5. It can be determined experimentally from the measured dependence of the dark fraction (fraction of the molecules in the dark state) fD on pH. The theoretical dependence fD(pH) is described by the following equation:

f_D ( pH ) = ^c0₋⁺ _pH ^{10− pH} _{− pK} (9)

a

c₀ +10 +10

where c0 is a constant. In addition to protonation by free protons (see eq. 8) internal protonation is assumed. Plot the experimentally determined fD vs. pH, fit this dependence to eq. 9, and determine pKa.

• Derive the dependence fD(pH) for the case where the internal protonation is neglected by using the definition of fD, the definition of pH, and the definition of Ka: Ka = ([H⁺][A^-])/[HA]. Show that this function does not fit the experimental data.

• (Optional) Derive the equation 9. The possibility of internal protonation can be included in the derivation of eq. 9 in two ways. Either the total proton concentration is assumed to be that determined by pH (bulk) plus the protons

available for internal protonation c0, or the internal protonation can be described by the reaction scheme A^-↔ AH' with a dissociation constant Ka' = [A^-]/[HA'], and the total concentration of molecules in the protonated (dark) state is then [HA]+[HA']. Both ways lead to the same result with a fixed relationship between the constants c0, Ka' and Ka.

Literature

• Schwille P. and Haustein E. (2002) Fluorescence Correlation Spectroscopy: A Tutorial for the Biophysics Textbook Online

• Weiss S. 1999: Fluorescence spectroscopy of single biomolecules. Science 283:1676-1683

• Schwille P., and Kettling U. 2001. Analyzing Single Protein Molecules by Optical Methods, Curr. Op. Biotechnol. 12 (4): 382-386

• Tsien, RY. 1998. The green fluorescent protein. Annu. Rev. Biochem. 67: 509-544

• Haupts, U., S. Maiti, P. Schwille. and W.W. Webb. 1998. Dynamics of fluorescence fluctuations in green fluorescent protein observed by fluorescence correlation spectroscopy, Proc. Natl. Acad. Sci. USA 95:13573-13578

Who would like to read more...

• Magde, D., E.L. Elson, and W.W. Webb. 1972. Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy. Phys. Rev. Lett. 29:705-708

• Elson, E.L., and D. Magde. 1974. Fluorescence correlation spectroscopy. I. Conceptual basis and theory. Biopolymers. 13:1-27

• Rigler, R., Ü. Mets, J. Widengren, and P. Kask. 1993. Fluorescence correlation spectroscopy with high count rates and low background: analysis of translational diffusion. Eur. Biophys. J. 22:169-175

• Eigen, M., and R. Rigler. 1994. Sorting single molecules: Applications to diagnostics and evolutionary biotechnology. Proc Natl. Acad. Sci. USA. 91:57405747