## Fluorescence Correlation Spectroscopy – and what it can do for you

### June 15, 2012

Fluorescence Correlation Spectroscopy (FCS) in combination with a Laser-scanning confocal microscope (LSCM) is a commonly used tool to quantitatively determine properties of molecules diffusing through a defined volume and properties of the medium itself. When assuming no labelling-bias, the observed fluorescence intensity changes can yield information to determine physical properties such as the molecular diffusion coefficient and medium viscosity, but also hydrodynamic radii, average concentrations, kinetic data, and singlet-triplet dynamics. Here I want to show how you can use an LSCM-FCS setup to determine the diffusion coefficient of a fluorescent molecule (in this case Rhodamine B), the viscosity of a mixture containing Rhodamine B, and the hydrodynamic radius of fluorescent beads in water. For all measurements a 543 nm He-Ne laser was used. All calculation are based on only two values that can be determined during FCS analysis. How (A) the molecular diffusion time and (B) the number of molecules within the observed volume are derived from observed fluctuations in fluorescence intensity has briefly been described in my blog and can be found here. With just knowing (A) and (B) many interesting things can be calculated in a short amount of time. In the following for example the “exact” size of the confocal volume is determined, as well as the “diameter” of the molecule. Why the parentheses? Well, you are deriving certain values from the observation of other values and in a relatively complicated setup like this one (labeling, detecting, microscope…) there will always some kind of bias. I am still a beginner as well, so my intention for the future is to reduce these effects to a minimum by better understanding the theory behind LSCM-FCS and getting more practical experience. But now… let’s get started! All the (A) and (B) values were actually measured and give a better impression of what they actually mean in the context of the formulas. I also kindly want to thank Victor Krasnikov from the Department of Single Molecule Biophysics for supplying the microscope, the materials, and of course also the idea behind these trials!

**Diffusion coefficient of small dye molecules**

The Rhodamine B diffusion time through a confocal volume of unknown size and the number of particles within this volume were measured under non-diluted and twice diluted conditions. The diffusion time (τ) was 3.3 x 10^{-5} s under both conditions and the number of particles (N) within the confocal volume was determined to be 6.8 and 3.35 under both conditions, respectively. The diffusion coefficient (D) for Rhodamine B in water is D_{water} ≈ 300 µm^{2}s^{-1}. The observation volume (V), its radius (ω_{1}), and the concentration (C) were calculated as describes below.

_{ }(2)

By inserting τ and D into (1) the confocal radius ω_{1 }= 0.199 µm could be determined. Since the structural parameter S was set to 5 the half-length of the observation volume into the z-direction ω_{2} = 0.995 (2). In the following the volume (V) was calculated by inserting ω_{1 }and ω_{2} into (3) yielding V = 0.219 µm^{3}.

_{ }(3)

Since logically the diffusion time under both dilution conditions is the same, of course also the size of the determined confocal volume remains the same. The confocal concentration (C) for both concentrations could now be calculated by making use of the observed respective N values, and calculated V (4).

The concentration of the non-diluted solution was C_{non-diluted }≈ 31 molecules x µm^{-3} and C_{diluted }≈ 15 molecules/µm^{3}. Considering the 1:1 dilution a halved concentration in the confocal volume makes sense.

**Viscosity of binary mixtures**

Three different volume/volume mixtures of water containing Rhodamine B and ethylene glycol were made (1:2, 1:1, 2:1) and the diffusion coefficients were calculated based on the diffusion times. By inserting the measured diffusion times under the three respective conditions (8.6 x 10^{-5} s, 1.46 x 10^{-4} s, 2.65 x 10^{-4 }s) into (5) the respective diffusion coefficients were determined as D_{1:2} = 115, D_{1:1} = 68, and D_{2:1} = 37 µm^{2}s^{-1}.

A plot of the respective diffusion coefficients, including D_{water}, against the used ethylene glycol concentration results in the graph shown in **Figure 1**.

**Figure 1:** The Rhodamine B diffusion coefficients as a function of ethylene glycol concentration as determined by LSCM-FCS.

An exponentially fitted curve indicates that the measured diffusion coefficients non-linearly depend on the ethylene glycol and water ratio. This is counter-intuitive, but can probably be explained by the non-linearly behaving interactions of Rhodamine B with the increasing number of ethylene glycol molecules in the solution. The volume-volume mixture alone thus does not account for changing diffusion coefficients.

**Hydrodynamic radius of fluorescent beads **

For the fluorescent beads a diffusion time of τ = 1.8 x 10^{-5} s was observed. Inserting τ into (5) yields D_{bead} = 550 µm^{2}s^{-1}. Inserting D_{bead }and µ = 0.9 x 10^{-3} Pa s^{-1} into (6) leads to a fluorescent bead hydrodynamic radius of r = 0.43 nm.

(6)

The hydrodynamic radius indicates the hypothetical radius of the molecular shape based on the measured diffusion time. This radius is of course distinct from the actual molecular radius because many atomic properties influence the diffusion behaviour in a given solution.

Summing up, I hope these small experiments and the six formulas help you to understand what you can do with your FCS data. Rhodamine B was just because it is an easy-to-handle fluorescent molecule. In principle, however, the steps described above remain the same for labeled proteins or other diffusing particles. Of course, a modern software package (Zen from Zeiss) can do it all for you. But isn’t it handy to understand what just two parameters can mean for the determination of physical properties?