End-to-end diffusion coefficients and distance distributions from fluorescence energy transfer measurements: Enhanced resolution by using multiple donors with different lifetimes

Diffusion-Enhanced Förster Resonance Energy Transfer in Flexible Peptides: From the Haas-Steinberg Partial Differential Equation to a Closed Analytical Expression

In the huge field of polymer structure and dynamics, including intrinsically disordered peptides, protein folding, and enzyme activity, many questions remain that cannot be answered by methodology based on artificial intelligence, X-ray, or NMR spectroscopy but maybe by fluorescence spectroscopy. The theory of Förster resonance energy transfer (FRET) describes how an optically excited fluorophore transfers its excitation energy through space to an acceptor moiety-with a rate that depends on the distance between donor and acceptor. When the donor and acceptor moiety are conjugated to different sites of a flexible peptide chain or any other linear polymer, the pair could in principle report on chain structure and dynamics, on the site-to-site distance distribution, and on the diffusion coefficient of mutual site-to-site motion of the peptide chain. However, the dependence of FRET on distance distribution and diffusion is not defined by a closed analytical expression but by a partial differential equation (PDE), by the Haas-Steinberg equation (HSE), which can only be solved by time-consuming numerical methods. As a second complication, time-resolved FRET measurements have thus far been deemed necessary. As a third complication, the evaluation requires a computationally demanding but indispensable global analysis of an extended experimental data set. These requirements have made the method accessible to only a few experts. Here, we show how the Haas-Steinberg equation leads to a closed analytical expression (CAE), the Haas-Steinberg-Jacob equation (HSJE), which relates a diffusion-diagnosing parameter, the effective donor-acceptor distance, to the augmented diffusion coefficient, J, composed of the diffusion coefficient, D, and the photophysical parameters that characterize the used FRET method. The effective donor-acceptor distance is easily retrieved either through time-resolved or steady-state fluorescence measurements. Any global fit can now be performed in seconds and minimizes the sum-of-square difference between the experimental values of the effective distance and the values obtained from the HSJE. In summary, the HSJE can give a decisive advantage in applying the speed and sensitivity of FRET spectroscopy to standing questions of polymer structure and dynamics.

Effect of Diffusion on Resonance Energy Transfer Rate Distributions: Implications for Distance Measurements

Intrinsically disordered protein regions and many other biopolymers lack the three-dimensional structure that could be determined by X-ray crystallography or NMR, which encourages the application of alternative experimental methods. Time-resolved resonance energy transfer data are often used to measure distances between two fluorophores attached to a flexible biopolymer. This is complicated by the rotational and translational diffusion of the fluorophores and by nonmonoexponential donor decay in the absence of the acceptor. Equation I(DA)(t) = I(D)(t)·F(t) is derived here, which is applicable regardless of whether I(D)(t) is monoexponential. I(D)(t) and I(DA)(t) are the δ-excitation donor emission decays in the absence and in the presence of the acceptor; F(t) contains information about energy transfer, donor-acceptor distance distribution, and diffusion dynamics. It is shown that in the absence of rotational and translational diffusion, F(t) is a continuous distribution of exponentials, whereas in the presence of rotational and translational diffusion, F(t) is a sum of discrete exponentials. For each case it is shown how F(t) is related to the distance distribution. Experimental data obtained with a flexible tetradecapeptide in aqueous solution clearly demonstrate that F(t) is a sum of discrete exponential terms. A partial differential equation describing resonance energy transfer in the presence of both rotational and translational diffusion of the donor and acceptor tethered to the ends of a semiflexible chain is solved in this work using a combination of analytical and numerical methods; the solution is used to fit time-resolved emission of the donor, which makes it possible to determine the model parameters: contour length, persistence length, and the end-to-end translational diffusion coefficient.

Role of diffusion in excitation energy transfer and migration

Effect of diffusion on excitation energy transfer and migration in a dye pair sodium fluorescein (donor) and Rhodamine-6G (acceptor) has been studied for different viscosities by both steady state and time domain fluorescence spectroscopic measurements. The donor-donor interaction appears to be weaker as compared to donor-acceptor interaction and thus favors direct Forster-type energy transfer. Interestingly, at low viscosity (water in this case) transfer appears to be controlled by material diffusion/energy migration. Further, acceptor dynamics reveals the fact that direct Forster transfer dominates in viscous media.