Fluorescence & FRET in Biophysics

Electronic States

The Stokes Shift and Mirror Image Rule

The Stokes shift $\Delta\tilde{\nu}$ is the difference (in wavenumber) between the absorption and emission maxima. It arises from vibrational relaxation in both the excited state (after absorption) and the ground state (after emission). Larger Stokes shifts indicate greater structural reorganisation upon excitation and are advantageous for microscopy because they allow clean separation of excitation and emission light.

The mirror image rule states that the emission spectrum is often an approximate mirror image of the absorption spectrum (plotted on a wavenumber scale). This arises when the vibrational mode structure is similar in $S_0$ and $S_1$. Deviations from the mirror image rule indicate significant excited-state structural changes (e.g., charge transfer, proton transfer, or excimer formation).

Common fluorophore Stokes shifts: Fluorescein (20 nm), Rhodamine B (25 nm), Cy3 (20 nm), Cy5 (20 nm), GFP (20 nm), DAPI (80 nm), Alexa Fluor 488 (20 nm). Environment-sensitive dyes like Laurdan and Prodan exhibit Stokes shifts that change dramatically ($\sim 50$–120 nm) with solvent polarity, making them valuable membrane probes.

Quantum Yield

The fluorescence quantum yield $\Phi$ is the fraction of absorbed photons that are re-emitted as fluorescence:

$$\boxed{\Phi = \frac{k_r}{k_r + k_{nr}} = \frac{\tau}{\tau_{\text{nat}}}}$$

where $\tau_{\text{nat}} = 1/k_r$ is the natural (or radiative) lifetime — the lifetime the fluorophore would have if there were no non-radiative decay. Since$k_{nr} \geq 0$, we always have $\tau \leq \tau_{\text{nat}}$ and$\Phi \leq 1$.

Measuring both $\tau$ (via time-correlated single photon counting, TCSPC) and$\Phi$ (via comparative or integrating sphere methods) allows separate determination of $k_r$ and $k_{nr}$.

FLIM: Fluorescence Lifetime Imaging

Fluorescence lifetime imaging microscopy (FLIM) maps the fluorescence lifetime at each pixel of an image. Since $\tau$ is sensitive to the local environment (pH, ion concentration, viscosity, FRET) but independent of fluorophore concentration and excitation intensity, FLIM provides contrast mechanisms that are inaccessible to intensity-based imaging.

FRET-FLIM: The most powerful application of FLIM is detecting FRET via changes in donor lifetime. In the presence of acceptor, the donor lifetime decreases from$\tau_D$ to $\tau_{DA} = \tau_D(1-E)$. FRET-FLIM is superior to intensity-based FRET because it is insensitive to donor/acceptor stoichiometry, spectral bleedthrough, and direct acceptor excitation.

Phasor analysis: An alternative to multi-exponential fitting, the phasor approach maps each pixel to a point on the phasor plot (a semicircle in the $g$–$s$plane). Single-exponential decays fall on the universal semicircle; multi-exponential or FRET populations appear inside the semicircle. This model-free approach is particularly useful for complex biological samples with heterogeneous lifetimes.

Stern-Volmer Quenching

When a quencher molecule Q is present at concentration $[Q]$, it provides an additional de-excitation pathway with rate $k_q[Q]$. The modified lifetime and quantum yield become:

$$\tau' = \frac{1}{k_r + k_{nr} + k_q[Q]}$$

Dividing the unquenched by the quenched quantities gives the Stern-Volmer equation:

$$\boxed{\frac{\tau_0}{\tau} = \frac{\Phi_0}{\Phi} = 1 + k_q \tau_0 [Q] = 1 + K_{SV}[Q]}$$

where $K_{SV} = k_q \tau_0$ is the Stern-Volmer constant. A linear plot of$\tau_0/\tau$ vs $[Q]$ confirms dynamic (collisional) quenching and yields $k_q$. For oxygen quenching of typical fluorophores,$k_q \approx 10^{10}$ M$^{-1}$s$^{-1}$, near the diffusion-controlled limit.

Deriving the Förster Radius $R_0$

The Förster radius $R_0$ is the distance at which the transfer rate equals the donor's intrinsic decay rate ($k_T = 1/\tau_D$), meaning transfer efficiency is 50%. Starting from Fermi's golden rule and integrating over the spectral overlap between the donor emission and acceptor absorption:

$$k_T = \frac{9000\, (\ln 10)\, \kappa^2 \Phi_D}{128\pi^5 N_A n^4 \tau_D r^6} \int_0^\infty F_D(\lambda)\, \epsilon_A(\lambda)\, \lambda^4\, d\lambda$$

where $F_D(\lambda)$ is the normalised donor emission spectrum (area = 1),$\epsilon_A(\lambda)$ is the acceptor molar extinction coefficient,$\kappa^2$ is the orientation factor, $n$ is the refractive index of the medium, $\Phi_D$ is the donor quantum yield, and$N_A$ is Avogadro's number. The spectral overlap integral is:

$$J(\lambda) = \int_0^\infty F_D(\lambda)\, \epsilon_A(\lambda)\, \lambda^4\, d\lambda$$

Setting $k_T = 1/\tau_D$ at $r = R_0$ and solving for $R_0$:

$$\boxed{R_0 = 0.211 \left[\kappa^2 n^{-4} \Phi_D J(\lambda)\right]^{1/6} \quad \text{(in nm, with } J \text{ in M}^{-1}\text{cm}^{-1}\text{nm}^4\text{)}}$$

Typical $R_0$ values for common FRET pairs: Cy3-Cy5 ($\sim 5.4$ nm), CFP-YFP ($\sim 4.9$ nm), Alexa488-Alexa594 ($\sim 5.4$ nm), ECFP-EYFP ($\sim 4.9$ nm).

Single-Molecule FRET (smFRET)

Ensemble FRET measurements report the average efficiency over all molecules, which can mask conformational heterogeneity. Single-molecule FRET (smFRET) resolves the efficiency distribution of individual molecules, revealing distinct conformational states and dynamic transitions between them.

In a typical smFRET experiment, a confocal microscope or total internal reflection fluorescence (TIRF) microscope excites individual doubly-labelled molecules. The donor and acceptor emission are spectrally separated and detected simultaneously. The apparent FRET efficiency for each burst or time bin is:

$$E^* = \frac{n_A}{n_A + n_D}$$

where $n_A$ and $n_D$ are the detected acceptor and donor photon counts. Corrections for background, crosstalk (donor emission leaking into the acceptor channel), direct excitation of the acceptor, and differences in detection efficiencies and quantum yields are required to obtain the true efficiency $E$.

Histograms of $E$ values from thousands of individual molecules reveal the number and population of distinct conformational states. Hidden Markov model (HMM) analysis of time traces from surface-immobilised molecules yields transition rates between states, providing a complete kinetic description of conformational dynamics.

Setting Up the Autocorrelation

The fluorescence intensity $F(t)$ fluctuates around its mean$\langle F \rangle$. We define the fluctuation$\delta F(t) = F(t) - \langle F \rangle$ and the normalised autocorrelation function:

$$G(\tau) = \frac{\langle \delta F(t)\, \delta F(t+\tau) \rangle}{\langle F(t) \rangle^2}$$

The fluorescence signal is $F(t) = \kappa \int I(\mathbf{r})\, C(\mathbf{r},t)\, d^3r$, where $I(\mathbf{r})$ is the molecule detection function (product of excitation intensity and collection efficiency), $C(\mathbf{r},t)$ is the local concentration, and $\kappa$ is a proportionality constant including the molecular brightness.

Extracting Physical Parameters from FCS

From a fit of $G(\tau)$ to the measured autocorrelation:

• Concentration: $C = N/V_{\text{eff}}$ from $G(0) = 1/N$
• Diffusion coefficient: $D = w_0^2/(4\tau_D)$ from the half-decay time of $G(\tau)$
• Hydrodynamic radius: $R_H = k_BT/(6\pi\eta D)$ via the Stokes-Einstein relation
• Molecular interactions: Changes in $\tau_D$ upon binding report on complex formation (larger complexes diffuse more slowly)

For two-component diffusion (e.g., free vs. bound ligand), the autocorrelation becomes a weighted sum of two diffusion terms with different $\tau_D$ values. Fluorescence cross-correlation spectroscopy (FCCS) uses two spectrally distinct labels to detect co-diffusion of interacting partners.

Practical Considerations for FCS Experiments

Calibration: The focal volume dimensions ($w_0$and $z_0$) must be calibrated using a dye with known diffusion coefficient (e.g., Alexa 488 in water, $D = 435\;\mu$m$^2$/s at 25°C). The structure parameter $s$ and $w_0$ are obtained from a fit to the autocorrelation.

Concentration range: FCS works best at nanomolar concentrations (1–100 nM), corresponding to $N \sim 0.1$–10 molecules in the focal volume. At lower concentrations, data acquisition takes prohibitively long; at higher concentrations, the relative fluctuations $\delta F/\langle F \rangle \propto 1/\sqrt{N}$ become too small to detect reliably.

Two-component analysis: For a mixture of free and bound species with diffusion times $\tau_{D,1}$ and $\tau_{D,2}$, the autocorrelation becomes a weighted sum: $G(\tau) = f_1 G_1(\tau) + f_2 G_2(\tau)$, where the weights$f_i$ depend on both the fraction and the molecular brightness of each species. Binding-induced changes in diffusion time are detectable when the mass ratio of free to bound species is at least $\sim 8$:1 (since $D \propto M^{-1/3}$ for globular proteins).

The Abbe Diffraction Limit

Ernst Abbe (1873) showed that the minimum resolvable distance for a conventional microscope is:

$$d_{\text{Abbe}} = \frac{\lambda}{2n\sin\theta} = \frac{\lambda}{2\,\text{NA}}$$

where $\lambda$ is the wavelength of light and NA is the numerical aperture of the objective. For visible light ($\lambda \approx 500$ nm) and a high-NA oil-immersion objective (NA = 1.4), this gives $d_{\text{Abbe}} \approx 180$ nm. This limit stood for over a century and was widely believed to be a fundamental barrier to optical microscopy.

The key insight that enabled super-resolution was recognising that the diffraction limit applies to the resolution of two simultaneously emitting fluorophores, but not to the precision with which a single isolated emitter can be localised. This distinction between "resolution" and "localisation precision" is the foundation of PALM/STORM.

Temporal Separation Strategy

The key insight of PALM (Betzig, 2006) and STORM (Zhuang, 2006) is to use photoswitchable or photoactivatable fluorophores to ensure that only a sparse subset of molecules is fluorescent in each frame. This allows individual PSFs to be isolated and fitted without overlap.

The process: (1) activate a random sparse subset of fluorophores; (2) image until they photobleach, fitting each PSF to determine its centre; (3) repeat thousands of times; (4) reconstruct a super-resolution image from all localised positions. The final resolution is determined by $\sigma_{\text{loc}}$, not the diffraction limit.

STED Microscopy Resolution

Stimulated emission depletion (STED) microscopy (Hell, 1994) uses a doughnut-shaped depletion beam to suppress fluorescence from the periphery of the excitation spot. The effective PSF is narrowed to:

$$\boxed{d_{\text{STED}} \approx \frac{\lambda}{2\text{NA}\sqrt{1 + I/I_{\text{sat}}}}}$$

where $I$ is the peak STED beam intensity and $I_{\text{sat}}$ is the saturation intensity of the fluorophore. In principle, resolution is unlimited as$I/I_{\text{sat}} \to \infty$; in practice, STED achieves 20–50 nm resolution with live-cell compatible intensities.

Comparison of Super-Resolution Techniques

Mobile Fraction

In practice, not all molecules are mobile. The recovery curve typically plateaus below the pre-bleach intensity:

$$F(t) = F_0 + (F_{\infty} - F_0)\, f(t)$$

The mobile fraction $M_f$ is:

$$\boxed{M_f = \frac{F_{\infty} - F_0}{F_{\text{pre}} - F_0}}$$

where $F_{\text{pre}}$ is the pre-bleach intensity, $F_0$ is the intensity immediately after bleaching, and $F_{\infty}$ is the plateau intensity. The immobile fraction ($1 - M_f$) may represent molecules bound to immobile structures (cytoskeleton, nuclear matrix), aggregated species, or molecules in membrane microdomains.

Typical FRAP Results in Biology

FRAP has yielded diffusion coefficients for a wide range of biological molecules:

• Lipids in membranes: $D \approx 1$–$5 \;\mu$m$^2$/s for free lipids, reduced in ordered domains
• Membrane proteins: $D \approx 0.01$–$0.1 \;\mu$m$^2$/s, much slower than lipids due to cytoskeletal interactions
• Cytoplasmic GFP: $D \approx 25$–$30 \;\mu$m$^2$/s (3–4x slower than in water due to macromolecular crowding)
• Nuclear proteins: $D \approx 0.5$–$5 \;\mu$m$^2$/s for freely diffusing; much slower ($< 0.01 \;\mu$m$^2$/s) for chromatin-bound factors
• Histones: Mobile fraction $\sim 3$% for H2B (most are stably bound to chromatin), with recovery times of minutes to hours

Live Cell Imaging

Genetically encoded fluorescent proteins (GFP and derivatives) revolutionised cell biology by enabling visualisation of protein localisation, trafficking, and dynamics in living cells. Multi-colour imaging with spectrally distinct fluorescent proteins allows simultaneous tracking of multiple cellular components. Light-sheet fluorescence microscopy (LSFM) reduces phototoxicity and enables long-term 3D imaging of developing embryos and organoids.

Key fluorescent protein families: The original Aequorea GFP ($\lambda_{\text{ex}} = 395/475$ nm, $\lambda_{\text{em}} = 509$ nm) was engineered into enhanced variants (EGFP, EYFP, ECFP, mCherry, mKate2) spanning the entire visible spectrum. Self-labelling tags (HaloTag, SNAP-tag) allow attachment of synthetic dyes with superior brightness and photostability to genetically encoded protein targets.

Photoconvertible and photoactivatable proteins: Proteins such as mEos, Dendra2, and PA-GFP can be irreversibly converted from one spectral form to another by UV illumination. Reversibly switchable fluorescent proteins (rsEGFP, Dreiklang) can be toggled between fluorescent and dark states, enabling RESOLFT super-resolution microscopy with low illumination intensities.

Protein-Protein Interactions

FRET-based assays are the gold standard for detecting protein-protein interactions in vivo. Bimolecular fluorescence complementation (BiFC), FRET-FLIM, and fluorescence cross-correlation spectroscopy (FCCS) each provide complementary information. FRET between labelled binding partners reports on the proximity ($< 10$ nm) and stoichiometry of complexes. Time-resolved FRET measurements distinguish static from dynamic heterogeneity in conformational ensembles.

Fluorescent Biosensors

Genetically encoded biosensors exploit FRET, circular permutation, or environment-sensitive fluorophores to report on intracellular signals in real time. Examples include: calcium sensors (GCaMP family), voltage indicators (ASAP, Voltron), cAMP sensors (Epac-based FRET), kinase activity reporters (AKAR), and redox sensors (roGFP). These tools enable spatiotemporal mapping of signalling dynamics with subcellular resolution.

Key Milestones