An analytic solution to the Förster energy transfer problem in two dimensions.

An analytic solution of the F?rster energy transfer problem in two dimensions is presented for the case in which the orientation factor is independent of the donor-acceptor distance, and both the donors and acceptors are randomly distributed in a plane. A general solution based on the method of F?rster is possible since all distances are measured in units of R0. The analytic solution is extended to the cases of donors embedded in structures that exclude acceptors, and donors that bind acceptors. The validity of the analytic solutions is demonstrated by comparison with numerical simulation calculations. Numerical approximations to the exact solutions are given for ease of computation. Specific applications to the case of fluorescence quenching of a membrane-bound donor by membrane-bound acceptors are presented.     

Resolution of end-to-end distance distributions of flexible molecules using quenching-induced variations of the Forster distance for fluorescence energy transfer.

We describe a new method to recover the distribution of donor-to-acceptor (D-A) distances in flexible molecules using steady-state measurements of the efficiency of fluorescence energy transfer. The method depends upon changes in the Forster distance (Ro) induced by collisional quenching of the donor emission. The Ro-dependent transfer efficiencies are analyzed using nonlinear least squares to recover the mean D-A distance and the width of the distribution. The method was developed and tested using three synthetic D-A pairs, in which the chromophores were separated by alkyl chains of varying lengths. As an example application we also recovered the distribution of distances from the single tryptophan residue in troponin I (trp 158) to acceptor-labeled cysteine 133. The half-width of the distribution increases from 12 A in the native state to 53 A when unfolded by guanidine hydrochloride. For both TnI and the three model compounds the distance distributions recovered from the steady-state transfer efficiencies were in excellent agreement with the distributions recovered using the more sophisticated frequency-domain method (Lakowicz, J.R., M.L. Johnson, W. Wiczk, A. Bhat, and R.F. Steiner. 1987. Chem. Phys. Lett. 138:587-593). The method was found to be reliable and should be generally useful for studies of conformational distributions of macromolecules.     

Time-resolved fluorescence resonance energy transfer studies of DNA bending in double-stranded oligonucleotides and in DNA-protein complexes

Time-resolved F?rster resonance energy transfer (trFRET) has been used to obtain interdye distance distributions. These distributions give the most probable distance as well as a parameter, sigma, that characterize the width of the distribution. This latter parameter contains information not only on the flexibility of the dyes tethered to macromolecules, but on the flexibility of the macromolecules. Both the most probable interdye distance as well as sigma provide insight into DNA static bending and DNA flexibility. Time-resolved fluorescence anisotropy and static anisotropy measurements can be combined to provide a measure of the cone angle within which the tethered dyes appear to wobble. When this motion is an order of magnitude faster than the average lifetime that characterizes transfer, an average value of the dipolar orientational parameter kappa2 can be calculated for various mutual dye orientations. The resulting kappa2 distribution is very much narrower than the limiting values of 0 and 4, allowing more precise distances and distance changes to be determined. Static and time-resolved fluorescence data can be combined to constrain the analyses of DNA-protein kinetics to provide thermodynamic parameters for binding and for conformational changes along a reaction coordinate. The parameter sigma can be used to model multiple DNA-protein complexes with varying DNA bend angles in a global fitting of trFRET data. Such a global fitting approach has shown how the range of bends in single base DNA variants, when bound by the TATA binding protein (TBP), can be understood in terms of two limiting forms. Time-resolved FRET, combined with steady-state FRET, can be used to show not only how osmolytes affect the binding of DNA to proteins, but also how DNA bending depends on osmolyte concentration in the DNA-protein complexes.     

Improving lanthanide-based resonance energy transfer detection by increasing donor-acceptor distances

Lanthanide-based resonance energy transfer (LRET) is an established method for measuring or detecting proximity between a luminescent lanthanide (energy donor) and an organic fluorophore (energy acceptor). Because resonance energy transfer is a distance-dependent phenomenon that increases in efficiency to the 6th power of the distance between the donor and the acceptor, assay systems are often designed to minimize donor-acceptor distances. However, the authors show that because of the R(6) relationship between transfer efficiency and sensitized emission lifetime, energy transfer can be difficult to measure in a time-gated manner when the donor-acceptor distance is small relative to the F?rster radius. In such systems, the advantages inherent in time-resolved, ratiometric measurements are lost but can be regained by designing the system such that the average donor-acceptor distance is increased.     

Orientation of chlorophyll transition moments in the higher-plant light-harvesting complex CP29.

The Q(y) transition dipole moment vectors of all eight chlorophylls in the higher-plant antenna protein CP29 were calculated by an original method on the basis of linear dichroism and absorption spectroscopy. The contribution of individual chromophores was determined from difference spectra between wild type and mutant proteins in which a single chlorophyll has been removed by mutating pigment-binding residues. Recombinant proteins were constructed by overexpressing the apoprotein in bacteria and refolding of the pigment-protein complex in vitro [Bassi, R., Croce, R., Cugini, D., and Sandonà, D. (1999) Proc. Natl. Acad. Sci. U.S.A. (in press)]. The spectroscopic data are interpreted on the basis of a protein structural model obtained via the homology with the major antenna complex LHCII [Kuhlbrandt, W., Wang, D. N., and Fujiyoshi, Y. (1994) Nature 367, 614-621]. The results allow us to determine the orientation of six chromophores within the protein structure. The orientations of the two remaining chromophores are inferred by considering the symmetry properties of CP29 and fitting steady state absorption and linear dichroism spectra by independent chlorophyll spectral forms. As a consequence, four "mixed" sites with different chlorophyll a and b binding affinities are identified in CP29. Geometrical data and the F?rster mechanism for energy transfer suggest that excitation energy equilibrates rapidly among chlorophyll "pure" sites while energy preferentially flows outward from chlorophyll "mixed" sites. The orientation of the dipole moments of two chlorophyll molecules symmetrically located at the center of the protein and parallel to the carotenoid transition vectors suggests a role in energy transfer from xanthophyll to chlorophyll.     

Structural heterogeneity and quantitative FRET efficiency distributions of polyprolines through a hybrid atomistic simulation and Monte Carlo approach

Förster Resonance Energy Transfer (FRET) experiments probe molecular distances via distance dependent energy transfer from an excited donor dye to an acceptor dye. Single molecule experiments not only probe average distances, but also distance distributions or even fluctuations, and thus provide a powerful tool to study biomolecular structure and dynamics. However, the measured energy transfer efficiency depends not only on the distance between the dyes, but also on their mutual orientation, which is typically inaccessible to experiments. Thus, assumptions on the orientation distributions and averages are usually made, limiting the accuracy of the distance distributions extracted from FRET experiments. Here, we demonstrate that by combining single molecule FRET experiments with the mutual dye orientation statistics obtained from Molecular Dynamics (MD) simulations, improved estimates of distances and distributions are obtained. From the simulated time-dependent mutual orientations, FRET efficiencies are calculated and the full statistics of individual photon absorption, energy transfer, and photon emission events is obtained from subsequent Monte Carlo (MC) simulations of the FRET kinetics. All recorded emission events are collected to bursts from which efficiency distributions are calculated in close resemblance to the actual FRET experiment, taking shot noise fully into account. Using polyproline chains with attached Alexa 488 and Alexa 594 dyes as a test system, we demonstrate the feasibility of this approach by direct comparison to experimental data. We identified cis-isomers and different static local environments as sources of the experimentally observed heterogeneity. Reconstructions of distance distributions from experimental data at different levels of theory demonstrate how the respective underlying assumptions and approximations affect the obtained accuracy. Our results show that dye fluctuations obtained from MD simulations, combined with MC single photon kinetics, provide a versatile tool to improve the accuracy of distance distributions that can be extracted from measured single molecule FRET efficiencies.     

Energy transfer in lipid bilayers.

The quenching of fluorescence due to energy transfer between a dilute, random array of donor and acceptor chromophores in lipid bilayer was measured and compared to theoretical expressions developed to predict the decrease in emission intensity under these circumstances. The observed intensity was found to be the same function of quencher concentration in both planar, multilamellar dispersions and small, spherical vesicles. The degree of quenching was accurately predicted by a simple relation derived in this paper, as well as a more complex equation previously developed by Tweet, et al. The results suggest that significant quenching may be observed even when the average donor-acceptor separation exceeds the F?rster critical distance by severalfold. Application of these results to problems of current interest in membrane research are discussed.     

Energy transfer among CP29 chlorophylls: calculated Förster rates and experimental transient absorption at room temperature

The energy transfer rates between chlorophylls in the light harvesting complex CP29 of higher plants at room temperature were calculated ab initio according to the F?rster mechanism (F?rster T. 1948, Ann. Physik. 2:55-67). Recently, the transition moment orientation of CP29 chlorophylls was determined by differential linear dichroism and absorption spectroscopy of wild-type versus mutant proteins in which single chromophores were missing (Simonetto R., Crimi M., Sandonà D., Croce R., Cinque G., Breton J., and Bassi R. 1999. Biochemistry. 38:12974-12983). In this way the Q(y) transition energy and chlorophyll a/b affinity of each binding site was obtained and their characteristics supported by reconstruction of steady-state linear dichroism and absorption spectra at room temperature. In this study, the spectral form of individual chlorophyll a and b ligands within the protein environment was experimentally determined, and their extinction coefficients were also used to evaluate the absolute overlap integral between donors and acceptors employing the Stepanov relation for both the emission spectrum and the Stokes shift. This information was used to calculate the time-dependent excitation redistribution among CP29 chlorophylls on solving numerically the Pauli master equation of the complex: transient absorption measurements in the (sub)picosecond time scale were simulated and compared to pump-and-probe experimental data in the Q(y) region on the native CP29 at room temperature upon selective excitation of chlorophylls b at 640 or 650 nm. The kinetic model indicates a bidirectional excitation transfer over all CP29 chlorophylls a species, which is particularly rapid between the pure sites A1-A2 and A4-A5. Chlorophylls b in mixed sites act mostly as energy donors for chlorophylls a, whereas site B5 shows high and bidirectional coupling independent of the pigment hosted.     

Measurements of internal distance changes of the 30S ribosome using FRET with multiple donor-acceptor pairs: quantitative spectroscopic methods

We present analytical and experimental procedures for determining distance changes within the 30 S subunit of the Escherichia coli ribosome using F?rster resonance energy transfer (FRET). We discuss ways to contend with complexities when using FRET to measure distance changes within large multi-subunit macromolecular complexes, such as the ribosome. Complications can arise due to non-stoichiometric labeling of donor and acceptor probes, as well as environmental effects that are specific to each conjugation site. We show how to account for changes in extinction coefficients, quenching, labeling stoichiometry and other variations in the spectroscopic properties of the dye to enable more accurate calculation of distances from FRET data. We also discuss approximations that concern the orientation of the transition moments of the two dye molecules, as well as the impact of other errors in the measurement of absolute distances. Thirteen dye-pair locations with different distances using 18 independent FRET pairs conjugated to specific 30 S protein residues have been used to determine distance changes within the 30 S subunit upon association with the 50 S subunit, forming the 70 S ribosome. Here, we explain the spectroscopic methods we have used, which should be of general interest in studies that aim at obtaining quantitative distance information from FRET.     

Simulation vs. reality: a comparison of in silico distance predictions with DEER and FRET measurements

Site specific incorporation of molecular probes such as fluorescent- and nitroxide spin-labels into biomolecules, and subsequent analysis by F?rster resonance energy transfer (FRET) and double electron-electron resonance (DEER) can elucidate the distance and distance-changes between the probes. However, the probes have an intrinsic conformational flexibility due to the linker by which they are conjugated to the biomolecule. This property minimizes the influence of the label side chain on the structure of the target molecule, but complicates the direct correlation of the experimental inter-label distances with the macromolecular structure or changes thereof. Simulation methods that account for the conformational flexibility and orientation of the probe(s) can be helpful in overcoming this problem. We performed distance measurements using FRET and DEER and explored different simulation techniques to predict inter-label distances using the Rpo4/7 stalk module of the M. jannaschii RNA polymerase. This is a suitable model system because it is rigid and a high-resolution X-ray structure is available. The conformations of the fluorescent labels and nitroxide spin labels on Rpo4/7 were modeled using in vacuo molecular dynamics simulations (MD) and a stochastic Monte Carlo sampling approach. For the nitroxide probes we also performed MD simulations with explicit water and carried out a rotamer library analysis. Our results show that the Monte Carlo simulations are in better agreement with experiments than the MD simulations and the rotamer library approach results in plausible distance predictions. Because the latter is the least computationally demanding of the methods we have explored, and is readily available to many researchers, it prevails as the method of choice for the interpretation of DEER distance distributions.     

New donor-acceptor pair for fluorescent immunoassays by energy transfer

A novel F?rster donor-acceptor dye pair for an immunoassay based on resonance energy transfer (RET) is characterized with respect to its photophysical properties. As donor and acceptor, we chose the long-wavelength excitable cyanine dyes Cy5 and Cy5.5, respectively. Due to the perfect spectral overlap, an exceptionally high R(0) value of 68.7 A is obtained in solution. For biochemical applications, antibodies (IgG) are labeled with Cy5, while a tracer for competitive binding is synthesized by labeling bovine serum albumin (BSA) with an analyte derivative and Cy5.5. Binding the dyes to proteins at a low dye/protein ratio increases the fluorescence lifetimes and quantum yields, leading to an enhanced R(0) value of 85.2 A. At higher dye/protein ratios, the formation of nonfluorescent dimeric species causes a decrease in the fluorescence lifetime and quantum yield due to RET from monomeric dyes to dimers within one protein molecule. The F?rster distances could be calculated using the dimer absorption spectra to 83.9 and 83.6 A for Cy5 and Cy5.5, respectively. Upon binding of the Cy5-labeled IgG to the tracer, efficient quenching of Cy5 fluorescence is observed. Steady-state and time-resolved measurements reveal that approximately 50% of the quenching results in F?rster-type RET, while the residual quenching effect is caused by static quenching processes. The applicability of this dye pair is demonstrated in a homogeneous competitive immunoassay for the pesticide simazine.     

Impact of Emission Anisotropy on Fluorescence Spectroscopy and FRET Distance Measurements

The objective of this report is to provide a practical and improved method for estimating Förster resonance energy transfer distance measurement error due to unknown angles in the dipole orientation factor based on emission anisotropy measurements. We improve on the method of Dale et al. (1979), which has minor mistakes and is frequently interpreted in overly optimistic ways in the literature. To facilitate proper fluorescence intensity measurements, we also evaluated instrument parameters that could impact the measurement. The apparent fluorescence intensity of isotropic samples depends on the sample emission anisotropy, fluorometer geometry, and optical apertures. We separate parameters of the sample, and those of the cylindrically symmetric illumination source and detector in the equations describing results of unpolarized and polarized fluorescence intensity measurements. This approach greatly simplifies calculations compared with the more universal method of Axelrod (1989). We provide a full computational method for calculating the Förster resonance energy transfer distance error and present a graph describing distance error in the simplest case.     

Lateral energy transfer model for adjacent light-harvesting antennae rods of C-phycocyanins

Modeling of excitation transfer pathways have been carried out for the structure of Spirulina platensis C-phycocyanin. Calculations by F?rster mechanism using the crystal structure coordinates determined in our laboratory indicate ultra-fast lateral energy transfer rates between pairs of chromophores attached to two adjacent hexamer disks. The pairwise transfer times of the order of a few pico-seconds correspond to resonance transitions between peripheral beta155 chromophores. A quantitative lateral energy transfer model for C-phycocyanin light-harvesting antenna rods that is suggestive to its native structural organization emerges from this study.     

Conformational transitions of proteins engaged in DNA double-strand break repair, analysed by tryptophan fluorescence emission and FRET

We analysed protein-DNA and protein-protein interactions relevant to the repair of DNA DSBs (double-strand breaks) by NHEJ (non-homologous end-joining). Conformational transitions in mammalian DNA ligases III (LigIII) and IV (LigIV), as well as in PARP-1 [poly(ADP-ribose) polymerase-1], were analysed upon binding to double-stranded DNA by changes in tryptophan emission and FRET (F?rster resonance energy transfer) from tryptophan to DNA-conjugated Alexa Fluor? 532. For LigIII, two non-equivalent high- and low-affinity DNA-binding sites are detected interacting sequentially with DNA. PARP-1 displays a single high-affinity DNA-binding site and can displace bound DNA fragments from the low-affinity site of LigIII, consistent with its mediator role in LigIII-DNA interactions. For the LX [LigIV-XRCC4 (X-ray cross-complementation group 4)] complex, a single DNA-binding site is detected. Binding of Ku to DNA was accompanied by conformational changes in the protein and intermolecular FRET from dansyl chromophores of the labelled Ku to the Alexa Fluor? chromophores of Alexa Fluor? 532-conjugated DNA. The average distance of 5.7?nm calculated from FRET data is consistent with a location of Ku at the very end of the DNA molecule. Binding of LX to Ku-DNA complexes is associated with conformational changes in Ku, translocating the protein further towards the DNA ends. The protein-protein and protein-DNA interactions detected and analysed generate a framework for the characterization of molecular interactions fundamental to the function of NHEJ pathways in higher eukaryotes.     

Statistical interpretation of fluorescence energy transfer measurements in macromolecular systems.

A statistical method is presented for the interpretation of intramolecular distance measurements by the fluorescence energy transfer technique in systems for which the detailed geometries of the donor-acceptor pairs are unknown. This method enables calculation of the probability that a specified distance range corresponds to the actual distance to be measured. It makes use of the numerically calculated probability density function for the distance of interest. The two general systems considered are the single donor-acceptor pair and the multi-donor-single-acceptor transfer. In both systems, the statistical method incorporates the uncertainty in the orientation of the donor and acceptor dipoles. In addition, it can take into account the rotational mobility of the donor dipoles determined by time-dependent emission anisotropy measurements. When more than one donor is involved in the transfer process, the uncertainties associated with the number and location of individual donors and the size and shape of the donor distribution are also incorporated in calculating the distance ranges. Application of the method was demonstrated for a wide range of transfer efficiency and Ro values for the single donor-acceptor system. Specific examples are also presented for interpretation of both single donor-acceptor and multi-donor-single-acceptor energy transfer measurements performed in order to reveal the spatial relationship of the sigma subunit and the rifampicin binding site in the Escherichia coli RNA polymerase (see Wu, C.-W., Yarbrough, L. R., Wu, F. Y.-H., and Hillel, Z. (1976), Biochemistry, preceding paper in this issue). Analysis of these energy transfer data by methods which use average values of the unknown geometrical parameters of the system yielded results similar to those obtained by the statistical method. However, the statistical method represents a more realistic approach to the interpretation of energy transfer measurements since it provides information concerning the entire range of possible distances and their relative likelihood.     

Study of the interactions between tetracycline analogues and lysozyme

The interactions between tetracycline analogues (oxytetracycline, doxycycline, methacycline) and lysozyme (LYSO) were studied by using UV-visible absorption and spectrofluorimetric method. Tetracycline analogues can make LYSO's fluorescence quenching. The quenching mechanism is static quenching. Absorption spectrum and quenching constant all support this conclusion. The binding constants of tetracycline analogues with LYSO were obtained at various temperatures. According to the F?rster nonradiative energy transfer theory, we can get the distances of donators and acceptors. We also determined the main acting force between them is electrostatic gravitation according to the thermodynamic parameter.     

Mapping Multiple Distances in a Multidomain Protein for the Identification of Folding Intermediates

The investigation and understanding of the folding mechanism of multidomain proteins is still a challenge in structural biology. The use of single-molecule Förster resonance energy transfer offers a unique tool to map conformational changes within the protein structure. Here, we present a study following denaturant-induced unfolding transitions of yeast phosphoglycerate kinase by mapping several inter- and intradomain distances of this two-domain protein, exhibiting a quite heterogeneous behavior. On the one hand, the development of the interdomain distance during the unfolding transition suggests a classical two-state unfolding behavior. On the other hand, the behavior of some intradomain distances indicates the formation of a compact and transient molten globule intermediate state. Furthermore, different intradomain distances measured within the same domain show pronounced differences in their unfolding behavior, underlining the fact that the choice of dye attachment positions within the polypeptide chain has a substantial impact on which unfolding properties are observed by single-molecule Förster resonance energy transfer measurements. Our results suggest that, to fully characterize the complex folding and unfolding mechanism of multidomain proteins, it is necessary to monitor multiple intra- and interdomain distances because a single reporter can lead to a misleading, partial, or oversimplified interpretation.     

Effects of metallic silver particles on resonance energy transfer in labeled bovine serum albumin

Resonance energy transfer (RET) is widely used to detect proximity between biomolecules. In transparent solution the maximum donor-to-acceptor distance for RET is about 70 A. We measured the effects of metallic silver island films on RET from the intrinsic tryptophan of a protein to a bound probe as the acceptor. These preliminary experiments revealed a dramatic increase in the apparent F?rster distance increasing from 28.6 to 63 A. These results suggest the use of silver island films for detecting long range proximity between biomolecules and for biotechnology applications based on RET.     

Location of cyanine-3 on double-stranded DNA: importance for fluorescence resonance energy transfer studies

Fluorescence resonance energy transfer provides valuable long-range distance information about macromolecules in solution. Fluorescein and Cy3 are an important donor-acceptor pair of fluorophores; the characteristic F?rster length for this pair on DNA is 56 A, so the pair can be used to study relatively long distances. Measurement of FRET efficiency for a series of DNA duplexes terminally labeled with fluorescein and Cy3 suggests that the Cy3 is close to the helical axis of the DNA. An NMR analysis of a self-complementary DNA duplex 5'-labeled with Cy3 shows that the fluorophore is stacked onto the end of the helix, in a manner similar to that of an additional base pair. This provides a known point from which distances calculated from FRET measurements are measured. Using the FRET efficiencies for the series of DNA duplexes as restraints, we have determined an effective position for the fluorescein, which is maximally extended laterally from the helix. The knowledge of the fluorophore positions can now be used for more precise interpretation of FRET data from nucleic acids.     

The orientational freedom of molecular probes. The orientation factor in intramolecular energy transfer.

The measurement of the efficiency of F?rster long-range resonance energy transfer between donor (D) and acceptor (A) luminophores attached to the same macromolecular substrate can be used to estimate the D-A separation, R. If the D and A transition dipoles sample all orientations with respect to the substrate (the isotropic condition) in a time short compared with the transfer time (the dynamic averaging condition), the average orientation factor less than K2 greater than is 2/3. If the isotropic condition is not satisfied but the dynamic averaging condition is, upper and lower bounds for less than K2 greater than, and thus R, may be obtained from observed D and A depolarizations, and these limits may be further narrowed if the transfer depolarization is also known. This paper offers experimental protocols for obtaining this reorientational information and presents contour plots of less than K2 greater than min and less than K2 greater than max as functions of generally observable depolarizations. This permits an uncertainty to be assigned to the determined value of R. The details of the D and A reoreintational process need not be known, but the orientational distributions are assumed to have at least approximate axial symmetry with respect to a stationary substrate. Average depolarization factors are derived for various orientational distribution functions that demonstrate the effects of various mechanisms for reorientation of the luminophores. It is shown that in general the static averaging regime does not lend itself to determinations of R.