August 16, 2017 | Author: azzaass | Category: Spectroscopy, Emission Spectrum, Photon, Molecular Physics, Chemistry
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F¨ orster Resonance Energy Transfer (FRET) for Proteins Lambert K. Chao and Robert M. Clegg,

Physics Department,

University of Illinois-Urbana-Champaign, Urbana, Illinois

Advanced Article Article Contents • Examples Representing the Broad Applications of FRET and Proteins • What is the Basis for the FRET Phenomenon • Conditions for FRET and the Effect of the Transition Dipole Orientations

doi: 10.1002/9780470048672.wecb171

• What is Important: R6 or R0 , and When? 0 • Examples of FRET and Proteins

F¨orster Resonance Energy Transfer (FRET) is a spectroscopic technique applied throughout physics, chemistry, and biology to measure quantitatively the distance between selected locations on macromolecules and to determine the close association between interacting molecular components. Because FRET typically occurs over distances from 0.5 to 10 nm, it is especially useful for investigating many interesting biological molecular structures. It is also particularly valuable for following the dynamics and structural fluctuations of biological molecular systems. FRET can be applied in solution or under imaging conditions (such as in fluorescence microscopy, nanoscience, and even macroscopic imaging). In this article, we discuss the fundamentals of FRET. These principles apply to every FRET measurement. We present the basic rudiments and the relevant literature of FRET to provide the reader with the necessary background essential for understanding much of the past and modern literature. At the end of the article, we give a short discussion of several applications of FRET to proteins. The literature for FRET is vast, and many new applications are constantly being developed. We could not do justice to the many practitioners of FRET in such a short space, but armed with the background that is presented, we hope this basic information will help readers follow much of the literature and apply it in their own work.

The description of F¨orster Resonance Energy Transfer (FRET) in a form that is useful for quantitatively interpreting experimental results was first described in 1946 (1) and was later more quantitatively described by F¨orster in a series of publications (2–10). It has been popular and extensively used in biochemistry since the early 1950s. Many reviews have been published that cover not only the theory and analysis but also the application to protein structures. For the additional perusal of the reader, we list here some selected classic general overviews and discussions of the theory and analysis (11–52). These reviews contain many references to the literature that deal with specific topics, including proteins. In this article, we will concentrate on a discussion of the physical basis of the FRET mechanism and will present a few applications from the literature to determine macromolecular structures. The initial applications of FRET were by physicists and physical chemists. They dealt mainly with solution studies of freely diffusing molecular chromophores and with solid structures. But already in the early 1960s, the power of applying FRET to biological systems was realized: for instance, applications to proteins (16, 20, 23, 34) and to nucleic acids (53, 54). By this time, the theory had been

fully developed and tested; however, the applications were hindered by the limitations of a choice of suitable chromophores that could be attached covalently to specific sites of the structures. Thus, many original applications were carried out using either intrinsic chromophores (tryptophan or tyrosine) or dyes that were known to bind noncovalently to protein or nucleic acid structures. Quantitative interpretations of the early experimental results were thereby complex because the placement of the dyes on the biological macromolecules were usually not well known. However, many ingenious analysis methods were developed to extract structural information from the data. The limitation of available chromophores pairs that can be used to investigate structures of proteins has been removed in the last 20 years; a very large number of available chromophores that can be used as extrinsic labels of proteins can now be purchased commercially. All the research areas in this review are being actively and vigorously pursued, and despite the fact that FRET has been used extensively for over 50 years, new methods of measurement and analysis as well as new areas of application are continually being developed. The literature is extensive and sometimes daunting to the newcomer to the FRET field. The



F¨ orster Resonance Energy Transfer (FRET) for Proteins

following is an introduction to the basics of FRET, which enables the reader to read the vast, continually expanding FRET literature. We especially emphasize the aspects of FRET that are critical for determining structural and kinetic information about proteins and the biological structures that incorporate proteins.

Examples Representing the Broad Applications of FRET and Proteins The introduction of fluorescent proteins has been a great boon for use as FRET pairs that can be inserted into protein structures under genetic control. In vivo FRET studies have benefited greatly from the incorporation of the green fluorescent protein (GFP) gene into a host genome (55) to form protein hybrids. This method eliminates the external labeling of organic fluorescent dyes and allows labeling of specific proteins in vivo. Fluorescence lifetimes and photo-physical properties of fluorescent proteins have been characterized (56–65). It is easier to interpret time-resolved FRET studies quantitatively if the donor has only one fluorescence lifetime; although average lifetimes are often used. The original wild-type GFP and GFP variants exhibit complex (multiexponential) decays from their excited states (66, 67), which limit the reliability of lifetime measurements. Fluorescent proteins better suited for fluorescence lifetime imaging (FLI) and FLI-based FRET studies have been obtained by random and site-directed point mutations (63) (for a concise informative review of the development of monomeric fluorescent proteins, see Reference 68). Energy transfer is an integral part of photosynthetic systems [see review chapters in Govindjee et al . (69)]. Excitation energy transfer lies at the heart of the phenomenon and its mechanisms (70). The fluorophore of interest is chlorophyll and a few other intrinsic chromophores. The fluorescence intensity and lifetime of plants are tightly coupled to 1) the competition between the rapid shuttling of the excitation energy by FRET, 2) the dissipation of the excitation energy by quenching mechanisms, and 3) the eventual irreversible transfer of the energy into the reaction center, where it initiates the electron transfer chain of photosynthesis (71, 72). As a matter of fact, photosynthesis was one of the initial motivations for developing the dipole-dipole mechanism of FRET (1, 73), and the role of FRET and its mechanism is still a very active research topic in photosynthesis (70, 74–77). Single-molecule experiments in a microscope with the macromolecules of interest attached to a surface have made extensive use of FRET in the last several years (78–80). FRET allows one to directly observe conformational changes, and if the kinetics take place in the right time range (tens of microseconds to seconds), then the individual steps can be observed, and the kinetic rate constants can be determined. Another method that has single-molecule resolution is fluctuation correlation spectroscopy (81, 82). Thereby FRET can be used to measure kinetics of protein conformational changes and noncovalent binding reactions as the macromolecule passes through diffraction limited focused laser light in a fluorescence microscope (83). This technique has been further developed to achieve picosecond 2

time resolution, which allows fluorescence lifetime measurements to be made on the diffusing entities, and FRET to be determined from the lifetimes (60, 84, 85). Additional examples of the application of FRET to proteins are given at the end of the article, after discussing the physical basics of FRET.

What is the Basis for the FRET Phenomenon Hetero-and homo-FRET As the name implies, FRET involves the transfer of energy from a molecule in an electronically excited state (this molecule is called the donor) to another molecule (the acceptor) that is within a certain distance from the donor. The transfer of energy is brought about by a dipole–dipole interaction between the donor and acceptor. The acceptor is usually a different molecular species, but it can be the same as the donor. If the donor and acceptor are different molecular species, then the energy transfer is called hetero-FRET. If the donor and acceptor are the same, then it is called homo-FRET. For proteins, most applications in the literature use hetero-FRET to determine information about protein structures and conformational changes. In hetero-FRET, the fluorescence intensity of the donor decreases because a probability exists that the donor loses its excitation energy by transferring excitation energy to the acceptor, instead of fluorescing. If the acceptor can also fluoresce and energy transfer takes place, then emission from the acceptor can usually be observed. With true homo-FRET, the intensity of the fluorescence does not change; the energy is simply transferred from one identical molecule to the other, and the probability of emission remains the same. The only way to observe homo-FRET is to measure the decrease in fluorescence anisotropy (polarization). For measuring homo-FRET, the donor molecules are excited with polarized light, just as when measuring normal fluorescence anisotropy. Donor molecules oriented in a direction best suited for absorbing the polarized light are preferentially excited. Because the donor molecules are selectively excited according to their orientation relative to the excitation light polarization, their fluorescence emission will also be polarized preferentially in a direction related to the excitation light polarization. The extent of polarization depends on the rotational correlation time and time the molecules are in the excited state. If homo-FRET can take place, then those donor molecules excited by homo-FRET are not oriented solely relative to the excitation light, but they depend also on the relative orientation of the originally excited donor molecules and the molecules that accept the energy (as we will see in the general formula for energy transfer). This process will decrease the overall polarization of the sample. The measured anisotropy can then be used to interpret the extent of energy transfer.

The efficiency of FRET The number of energy quanta transferred from excited donors to acceptors divided by the number of quanta initially absorbed


F¨ orster Resonance Energy Transfer (FRET) for Proteins

by donors is called the efficiency of FRET. The maximum of this efficiency fraction is one. Whether energy transfer is more or less likely to occur in a particular situation will depend on what other paths are available for the excited donor molecule to give up its energy and how likely the donor will de-excite via these alternate paths. In other words, FRET is in direct kinetic competition with all other mechanisms of de-excitation of the donor. Therefore, for FRET to take place, the transfer must occur in the same time range, or faster, than all other de-excitation pathways (such as fluorescence emission, dynamic quenching, intersystem crossing to the triplet state, etc.). In practice, the actual transfer process is not measured directly, but is inferred by measuring its effect on other reaction pathways in kinetic competition with FRET. For example, the efficiency of FRET can be determined by comparing donor or acceptor fluorescence intensities in the presence and absence of FRET. This measurement can be done in steady-state or time-resolved experiments (25, 42, 45, 49, 86). The fluorescence quantum yield of the donor cannot be zero. On the other hand, the acceptor does not have to fluoresce. Because energy is conserved, the transfer must be resonant; that is, the energy lost by the acceptor must equal the energy gained by the acceptor. The probability that the energy can be transferred depends strongly on the distance between the molecules (see Eq. 1). This dependence makes FRET particularly suited for measuring molecular distances and determining spatial proximities.

The distance dependence of FRET and R0 For two isolated molecules (donor and acceptor), the rate of energy transfer k ET from the excited donor to the acceptor is proportional to the inverse sixth power of the distance between the two molecules R and is equal to

kET =

1   6 R0 R τD


τD is the lifetime of the donor excited state in the absence of the acceptor; it is the average time an excited donor remains in the excited state (typical values of τD are between 1–10 ns). The rate of de-excitation of the donor in the absence of the acceptor is 1/τD . The constant R 0 (see below) is defined as the distance R between the donor and acceptor molecules where the rate of transfer is equal to 1/τD . R 0 is often approximately 5 nm. The rate of transfer shown in Equation 1 has an exceptionally strong 1/R 6 distance dependence. As discussed above, the probability of energy transfer is in competition with all the other pathways of de-excitation. For instance, according to Equation 1, if the molecules are separated by less than about 0.5R 0 , then the rate of transfer is greater than 65 × (1/τD ). Therefore, on the average, essentially all the excitation energy will be transferred from the donor to the acceptor. If the molecules are separated by 2R 0 or greater, then the rate of transfer is less than 0.015 × (1/τD ). Essentially, on the average, no energy will be transferred. Thus, as a rule of thumb, because often R0 ≈ 5 nm, FRET can only be used to determine distances less than 10 nm. For reasons that will be evident later, the lower limit is approximately 0.5 nm.

The constant R 0 is dependent on several parameters: 1) the relative orientation of the transition dipole moments of the two molecules (these dipoles are the spectroscopic transition dipoles), 2) the extent that the fluorescence spectrum of the donor overlaps with the absorption spectrum of the acceptor, and 3) the surrounding index of refraction. We will deal with each of these below (see Equation 8). Because many proteins have diameters less than 10 nm, this distance dependence explains the usefulness of FRET for determining distances inside proteins as well as between interacting proteins, which is the reason that the name “spectroscopic ruler” was coined for FRET (20). FRET is a convenient method for determining the distance between two locations on proteins, or for determining whether two proteins interact intimately with each other. Fluorescence instrumentation is available in many laboratories, and a plethora of dyes and a wide variety of fluorescent proteins are now readily available. Therefore, FRET is a viable option for most researchers. With care, FRET can yield valuable information concerning protein–protein interactions, interactions of proteins with other molecules, and protein conformational changes.

Quantitative expressions for measuring the efficiency of FRET The efficiency of FRET is the fraction of times that an excited donor molecule will transfer its excitation energy to an acceptor. For instance, if the efficiency of transfer is 0.3, and if a molecule is excited 100 times, then on average it will transfer the excitation energy to the acceptor 30 times. Another way to express the efficiency is by means of the rate of energy transfer (Eq. 1). After the donor has been excited from the ground state (S0 ) into its first excited singlet state (S1 ), the donor can exit the S1 state by several pathways (see Fig. 1a). As indicated in Fig. 1a, all pathways that lead away from the excited state of a chromophore (either to the ground state—by some radiative or nonradiative process—or passage to the triplet state) are in direct kinetic competition with each other. FRET is one of these pathways. Each i th pathway exits the excited state with a certain rate constant (k i in seconds−1 ). Figure 1b is a schematic that depicts the kinetic pathways. Because they are in direct competition, the probability of going by any single pathway (the efficiency of that pathway) is the ratio of the rate of that pathway divided by the sum of the rates of all the pathways. The efficiency of energy transfer, Fig. 2a, can be expressed as

# times FRET pathway is chosen total# of times the molecule is excited  = kET ki



   1  1 6 6 = 1 + (R0 /R) (R0 /R) τD τD    =1 1 + (R/R0 )6


The k s are rate constants (with units of s−1 ), and the only ones that are included in Equation 2 are those that emanate



F¨ orster Resonance Energy Transfer (FRET) for Proteins



Figure 1 FRET and competing pathways. (a) A Perrin– Jablonski diagram shows common spectroscopic transitions for de-excitation of the donor fluorophore: Absorption of a photon excites the donor from the S0 to the S1 excited state, where it rapidly relaxes (typically in less than 1 ps) to the lowest vibrational level of the S1 state, which is known as internal conversion (IC). Internal conversion involves energy loss through vibrational interactions with the surroundings. The excited molecule in a higher vibrational state of S1 thereby releases heat to the surroundings, finally ending up in the lowest vibrational state of S1 . The donor can then undergo de-excitation from the lowest vibrational level of the excited state to the S0 ground state through several pathways. The typical fluorescence lifetime (independent of the other pathways) is on order of 1–10 ns. Through spin–spin or spin– orbital interactions, the S1 state can undergo intersystem crossing (ISC) into the excited triplet state where it spends some time (typically 10 µs– 10 s, depending on the concentration of oxygen in the solution) before phosphorescence or conversion to the ground state by internal conversion takes place. The ground state of oxygen is a triplet, and it can easily react with the triplet state of the chromophore, producing reactive oxygen species (radicals) that can collide with the chromophore, destroying it (photolysis). (b) This panel shows a schematic of the various competing pathways for leaving the excited donor state, including quenching, photolysis, fluorescence, and phosphorescence, which compete with FRET. Dexter transfer involves energy transfer to the acceptor molecule by exchange of electrons (see text) when the electronic orbitals of the donor and acceptor overlap. By measuring the effect of FRET on one of the competing processes (e.g., donor fluorescence), one can measure the FRET efficiency. It is not necessary to measure fluorescence to determine FRET efficiency, but this technique is the normal way. It is also possible to measure the fluorescence intensity of the acceptor (if the acceptor can fluoresce) to detect and quantify FRET (42, 48).

directly out of the excited state  of the donor (Fig. 1b). k ET is the rate of undergoing FRET, i ki is the sum of the rates over all the i th pathways (fluorescence emission, nonradiative de-excitation, quenching, photolysis, intersystem crossing to the triplet state, and including k ET ), and E FRET is the efficiency of energy transfer. The efficiency of fluorescence emission E fluor is defined analogously, and it is usually called the fluorescence quantum yield q em or φem . The efficiency can be defined this way for any of the pathways. It is also sometimes referred to as the quantum yield of the different pathways. The lifetime of an excited molecule (in seconds) is the inverse of the sum of the kinetic rates (in s−1 ) for all the pathways for exiting the excited state. Thus in the absence of an acceptor, the lifetime is

τ−A =

 i ,i =ET




And in the presence of an acceptor, the lifetime is

τ+A =

 all i




Therefore, the efficiency can also be given in terms of the measured fluorescence lifetimes in the presence and absence of 4

an acceptor


 all i

ki =

all i

ki − 

all i

1 τ+A − 1 τ−A τ+A = =1− τ−A 1 τ+A

i =ET


ki (5)

Thus one can determine the efficiency of FRET by measuring the nanosecond fluorescence lifetime of the donor in the presence (+A) and the absence (−A) of the acceptor. If one has the possibility of making this measurement, then it is a convenient, accurate, and robust way to measure the efficiency of FRET because the lifetimes can often be determined accurately, and not many corrections or experimental controls need to be made when calculating the efficiency. Several ways exist to determine the efficiency by measuring steady-state fluorescence. It is easily observed by inspecting the first equality in Equation 2. The best way to acquire steady-state values of the fluorescence intensity is photon counting. We will not go into the way it is done with hardware and electronics (it is usually done automatically with modern instrumentation), but simply note that by using photon counting one can easily


F¨ orster Resonance Energy Transfer (FRET) for Proteins

not there. The denominator is then simply the number of photons emitted by the donor in the absence of an acceptor. Thus, when the acceptor is far away, the efficiency is zero. When the acceptor is very close to the donor, the efficiency becomes one. Note the exact parallel of Equation 6 to Equation 5 (i.e., E FRET = 1 − τ+A /τ−A = 1 − #(photons)D,+A /#(photons)D,−A ). This parallel relationship occurs because the lifetime of the donor emission becomes shorter when another pathway (for instance, k ET ) opens up for an escape from the excited state. That is, the average time in the excited state becomes shorter. It reduces the number of photons emitted by the donor by transferring a certain number of energy quanta from the excited donor to the acceptor. Often one measures the intensity of fluorescence. The intensity is proportional to #(photons)D,±A /second. Then, provided the concentration of the donor is the same in both measurements, and the conditions of measurement remain identical, the efficiency is



EFRET = 1 −

#(photons)D,+A #(photons)D,−A


#(Photons)D,+A /second ID,+A =1− =1− #(photons)D,−A /second ID,−A Figure 2 FRET characteristics. (a) The FRET efficiency as a function of R/R0 is shown E = 1/(1 + (R/R0 )6 ). Proper selection of FRET pairs so that distances of interest lie near R0 where the FRET efficiency–distance slope is greatest will give the most sensitivity. (b) The spectral overlap requirement for FRET between the donor emission and the acceptor fluorescence spectrum is shown for the organic cyanine Cy3–Cy5 donor–acceptor pair.

measure a correct value of the intensity at any wavelength by counting the number of photons emitted. If one does not count photons, then the light detector must be corrected for the sensitivity of the photomultiplier at different wavelengths. We can rewrite Equation 2 as

# times FRET pathway is chosen total # of times the molecule is excited #(photons)D,−A − #(photons)D,+A = #(photons)D,−A (6) #(photons)D,+A =1− #(photons)D,−A


#(photons)D,±A means the number of photons emitted by the donor in the presence or absence of the acceptor, respectively. This “counting” quantification is very important when measuring the efficiency of FRET, because the quantities of interest in Equation 6 are the number of photons emitted and the number of photons absorbed, and not their energy. The experiment can be conducted either on an ensemble of molecules (where different donor molecules will be excited by each excitation event) or by repetitive experiments on a single molecule. Thus, the numerator of Equation 6 is just the number of photons that the donor emits in the absence of an acceptor minus the number of photons emitted by the identical number (or concentration) of donors in the presence of an acceptor. This difference is just the number of “quanta” that are transferred to the acceptor, which would have been emitted as fluorescence photons if the acceptor were

I D,±A is the measured fluorescence intensity of the donor in the presence and absence of the acceptor, which is measured under identical conditions and with the same donor concentration. If the concentration of the donor is not identical in both measurements, then it can usually be corrected simply by multiplying by the corresponding concentration ratio.

Conditions for FRET and the Effect of the Transition Dipole Orientations The value of R0 for a singular pair of D and A molecules is (1, 3, 6):

R06 =

9000(ln 10)κ2 φD NA 128π5 n 4  ∞  εA (v )F (v )v −4 d v 0


F (v ) d v 0

where, v is wave number (in cm−1 units), φD is the quantum yield of the donor, N A is Avogadro’s number, n is the index of refraction pertaining to the transfer, εA (v ) is the molar absorption coefficient of the acceptor (in units of cm−1 mol−1 ), F (v ) is the fluorescence intensity of the measured fluorescence spectrum of the donor, and κ2 is an orientation factor that results from the inner product between the unit vector of the electric near field of the donor dipole and the unit vector of the absorption transition dipole of the acceptor (see the Orientation Factor section below). The ratio of integrals in the bracket has units of cm6 /mol. Using the units given in the paragraph above, we have:

R06 = 8.79 × 10−25 n −4 φD κ2 J (v ) cm6


(9) 5

F¨ orster Resonance Energy Transfer (FRET) for Proteins



Figure 3 The electric field of the donor and the orientation factor. (a) The acceptor dipole p A in the electric field ED of the donor p D . r = |r | is the distance between the point dipoles. θD is the angle between pˆ D and ˆr . (b) This schematic shows the donor and acceptor dipoles and illustrates the angles and radial vectors used in the definition of the orientation factor κ2 . The coordinate system is chosen such that the p D and ED vectors are in the r–y plane; the p A vector can be in any direction, and is not supposed to be in either the r– y or r– x planes. The dipole field is symmetrical about the azimuthal angle of the p D vector.

J (v ) is the ratio of integrals given in square brackets in Equation 8. We describe the parameters of this important equation below.

Distance dependence For energy to be transferred from the donor to the acceptor, an interaction must occur between the two molecules, and the energy must be conserved (i.e., the energy lost by the donor equals that gained by the acceptor). Coulomb (charge–charge) interactions between the electron distributions in both molecules are responsible for the FRET interaction, and the energy transfer can be understood as an interaction between two classical oscillating dipoles. Actually, a classical derivation of the FRET equations (87, 26) results in the identical expression as a quantum mechanical derivation (3, 48). The electric field of an oscillating dipole is very large in the “near-field” region where the field can be described by a pure dipole field (Fig. 3a). A dipole field at some point in space a distance R from the center of the dipole is dependent on the angle relative to the dipole direction, and it varies with distance as 1/R 3 . The interaction energy between the two dipoles therefore varies as 1/R 6 . The interaction energy also depends on the relative orientations of the two dipoles and the orientation of each dipole to the separation vector between the two dipoles (see Figs. 3a and 3b). The dipole–dipole nature of the interaction explains the 1/R 6 dependence of the rate of energy transfer (Eq. 1).

Overlap integral It is important to realize that in this near-field zone of the dipoles (which extends maximally 10 nm for FRET) no photon exists; 6

the lack of photons in the near-field zone is essentially because of the Heisenberg uncertainty principle of quantum mechanics (88). That is, the energy is transferred without the emission or absorption of a photon. Of course, the energy transferred from the donor must equal exactly the energy transferred to the acceptor. However, it turns out that the interaction between the two molecules in the near-field zone can be described as if the acceptor is bathed in an oscillating electric field of the near-field zone of the donor, ED in Equation 10. This oscillating electric field (with a frequency equal to that of the fluorescence of the donor emission, F (v ) of Equation 8) is in the near-field zone of the donor where no photons exist and not in the far field zone where photons exist. The energy of such a Hertzian oscillating dipole is not emitted as photons until a distance of about one wavelength away from the donor. Energy is conserved during the transfer event; that is, the energy lost by the donor must equal the energy taken up by the acceptor. The change in the energy levels of the donor and the acceptor are the same energy levels that correspond to the spectroscopic transitions of the donor (the emission spectrum) and acceptor (the absorption spectrum). Therefore, to conserve energy, the emission (fluorescence) spectrum of the donor F (v ) must overlap with the excitation (absorption) spectrum εA (v ) of the acceptor (Fig. 2b). The more these spectra overlap, the stronger is the energy transfer; that is, a larger spectral overlap leads to a longer R 0 . The F (v ) and εA (v ) spectra (Fig. 2b) are those of the separate donor and acceptor components. These spectra must be the appropriate spectra that correspond to the identical conditions as where the FRET measurements are made. Often the spectra can simply be taken from the literature that refers to the separate dyes. But often, the dyes physically interact with the proteins, which changes the dyes’ absorption or emission spectra. The overlap integral involves the weighting function v −4 in addition to the emission spectrum of the donor and the absorption spectrum of the emitter (see Eq. 8). As mentioned above, although no photon is emitted or absorbed in the transfer process, the emission spectrum of the donor and the absorption spectrum of the acceptor are still involved in the overlap integral. This participation of the optical spectra is because FRET involves changes in the energy levels of D and A that are identical to those in normal absorption and emission events. An important condition for F¨orster transfer is that the interaction between the donor and acceptor is very weak, and the two molecular species retain their separate electronic and vibrational structures and energy levels. In FRET, the donor and acceptor are very weakly perturbed by dipole interactions, which is formally the same type and strength of interaction that describes the interaction of the chromophores with light. Therefore, the energy transitions, which can occur in the donor and acceptor molecules during FRET, are the same as their respective spectroscopic transitions when absorbing or emitting photons. This requirement for the conservation of energy, which is expressed by the overlap integral in the expression for R 0 (Eqs. 8 and 9), is also the reason for the word “resonance” in FRET. We emphasize once more: Although the emission and absorption spectra of the donor and acceptor are in the overlap integral, the FRET process does not involve the emission or the absorption of a “photon.” The quantum of energy transferred by FRET


F¨ orster Resonance Energy Transfer (FRET) for Proteins

takes place in the near field of the oscillating dipoles (quantum mechanically the transition dipoles), where the energy cannot yet be described in terms of real photons.

Orientation factor κ2 The parameter κ2 in Equation 8 is referred to as the orientation factor of FRET. This factor represents the angular dependence of the interaction energy between the acceptor transition dipole and the oscillating dipole electric field of the donor. κ2 is the source of much debate and many misunderstandings. It is the most difficult factor to control and usually the hardest to determine with confidence (28, 48). Therefore, we will spend more time discussing this factor. To understand this angular dependence of the acceptor transition dipole in the donor electric field, we first express the electric dipolar interaction between the donor and acceptor molecules classically. The energy of interaction between the donor and acceptor dipoles is directly proportional to the vector dot product of the electric field of the donor with the dipole moment of the acceptor. The equations for FRET have been derived classically (87, 26, 48) by assuming that the donor molecule is an oscillating point dipole [a Hertzian dipole (89)]. One assumes that the acceptor is located in the “near-field zone” (which extends much less than one wavelength of light of the corresponding frequency) of the oscillating dipole of the donor (Fig. 3). The acceptor absorbs the energy by interacting with the oscillating near field of the donor (the donor oscillates at the same optical frequency where the acceptor absorbs). The near field is not radiating (no transverse photon emission occurs in the near-field zone of a Hertzian dipole). And the electric field in the near field has both longitudinal and transverse components, in contrast to the far field zone, where the electric field has only transverse components. Nevertheless, the mechanism of absorption of energy by the acceptor from the oscillating electric field of the donor Hertzian dipole is identical to the mechanism of absorption of light of the same frequency. Therefore, the interaction of the “transition moment” of the acceptor with the electric field of the donor obeys the same rules as normal absorption of light by the acceptor (polarization dependence and all the spectral requirements for normal absorption). The rate of absorption is proportional to the square of the vector dot product of the acceptor transition dipole with the electric field of the donor, just as though this field were equivalent to the oscillating electric field of light impinging on the acceptor. However, the electric field in the near-field of the donor is not propagating, and the field direction has longitudinal as well as transverse vector components [propagating light (photons) has only vector components transverse to the direction of propagation]. This electric field is a reflection of the fact that in the near-field zone no photon could even exist according to the uncertainty principle. This simple classical description (which results in the correct theoretical description of FRET) is also perfectly consistent with a quantum derivation. The orientation factor can be best understood quantitatively by studying the interaction of the acceptor dipole in the near-field zone of the donor (Fig. 3a). The energy of interaction of the donor electric field ED , and the acceptor dipole

moment pA , is ED · pA . The rate of absorption is proportional to the square of this energy of interaction. Therefore, the rate 2  of energy transfer is proportional to ED · pA (Eq. 11). The field ED that surrounds an oscillating classical electric dipole pD is shown in Fig. 3a,

|pD | ED = 3 {3 cos θD rˆ − pˆ D } r  |pD | = 3 2 cos θD rˆ + sin θD θˆ D r


where |p D | is the time independent dipole strength, r = |r | is the distance from the point donor dipole (in FRET it is the distance from D to A), and rˆ is the unit vector pointing from the donor dipole to the position r, where the acceptor is located. θD is the polar angle between pˆ D and rˆ . θˆ D is a unit vector perpendicular to r that points in the direction of increasing θD (Fig. 3a). The caps designate unit vectors. Figure 3b shows the juxtaposition of two dipoles, and it defines the parameters used in the Equations 10, 11, and  12. As 2 we said above, k ET in Equation 1 is proportional to ED · pA , according to classic electrodynamics. We can write


2 kET ∝ UPD →PA = −ED · PA 2 |pD |2 |pA |2 2 cos θD rˆ · pˆ A + sin θD θˆ D · pˆ A 6 r (11) 2 2  |pD | |pA | 2 3 cos θD rˆ · pˆ A − pˆ D · pˆ A = r6 |pD |2 |pA |2 2 = κ r6 =

where κ2 has been defined as


κ2 = 2 cos θD rˆ · pˆ A + sin θD θˆ D · pˆ A


= {3 cos θD rˆ · pˆ A − pˆ D · pˆ A }2 cos θD = rˆ · pˆ D . The orientation factor κ2 can have values between 0 and 4. We have given κ2 in two different representations in Equation 12. Thus, the rate of energy transfer depends on the square of the dot product between the acceptor dipole (transition dipole) pA and the field ED of the donor dipole (transition dipole), pD (Eq. 11 and Fig. 3b). For any chosen locations and orientations of the donor and acceptor, the value of κ2 involves the cosine of the angle between the unit vectors pˆ D and pˆ A (i.e., pˆ D · pˆ A ) as well as the cosine of the angles between rˆ and pˆ A (i.e., rˆ · pˆ A ) and between rˆ and pˆ D (i.e., rˆ · pˆ D ). Therefore, for any constant selected angle between the donor and acceptor dipoles (that is, constant pˆ D · pˆ A ), the value of κ2 will depend on the position in space where the acceptor dipole is relative to the donor. The strength of the field of the donor molecule for any particular constant values of θD , θA and pˆ D · pˆ A changes with the distance |r | as 1/r 3 , that is, for any particular direction of rˆ relative to pD . As illustrated in Fig. 3a, for a particular angle between the orientations of the donor and acceptor dipoles (pˆ D · pˆ A ),



F¨ orster Resonance Energy Transfer (FRET) for Proteins

the angle between the acceptor dipole and the electric field of the donor depends on the position in space of the acceptor. Also, for constant relative orientations of the donor and acceptor dipoles (constant pˆ D · pˆ A ), and for constant cos θD rˆ · pˆ A , κ2 is constant (Eq. 11 and 12). In this case, the rate of FRET is solely a function of the distance between the donor and acceptor. However, if only pˆ D · pˆ A and θD are constant, then the distance dependence is more complicated than just the distance between the donor and acceptor. This intertwining relationship between the distance separating the donor and acceptor and the orientational dependence of κ2 illustrates the complexity of determining a value for κ2 . Because the orientations and spatial locations of the two chromophores may vary over the ensemble of molecules, and because they can also change during the time the donor is in the excited state, the measured effect of κ2 is usually an average over the appropriate spatial/temporal distributions. Whenever ED and the acceptor dipole moment pA have parallel orientations, the rate of FRET is maximum for that placement in space for the acceptor relative to the donor. For pA oriented parallel to ED , the possible maximum values of κ2 are between 1 and 4; that is, the actual maximum value depends on the value of θD (see below). The minimum value of κ2 is zero for every position of the acceptor relative to the donor whenever ED and the acceptor dipole moment pA are oriented perpendicular to each other. The possible introduction of error in the estimation of the FRET efficiency from the orientation factor is often of concern when measuring FRET in proteins or between proteins. This uncertainty can occur because the actual distribution of dye orientations is often not known, or because the dyes do not rotate freely and rapidly relative to the fluorescence lifetime of the donor. If the donor and acceptor molecules undergo rotational or translational movements during the time the donor is in the excited state, or if an ensemble of different donor and acceptor orientations exist, then the value of κ2 (and of course the rate) will be averaged over the corresponding ensemble of configurations. Because κ2 can, in principle, range from 0 (e.g., acceptor absorption dipole perpendicular to the electric field of the donor) to 4 (e.g., end-to-end stacked parallel dipoles), the variation in the rate of FRET can be extensive because of such movements. This possible variation in κ2 becomes especially apparent when one notes that the measurement of the rate of transfer (or the efficiency) varies directly as R06 , and therefore directly with κ2 (see below). If the condition of very rapid (compared with the lifetime of the donor) rotational movements of both donor and acceptor is met, then κ2 is rigorously 2/3, which arises from averaging over all possible orientations. If rapid re-orientation through all angles is not the case, then limitations on the degree of rotational freedom can have significant effects on the measured efficiency of energy transfer. However, as we discuss below, the assumption that κ2 = 2/3 is often a justified, and reasonable one to make (39, 48, 86). It is worthwhile to consider a few simple examples to get a feeling for the values of κ2 . If the two dipoles have orientations in space perpendicular to each other, and if the acceptor dipole is juxtaposed next to the donor, but in the direction perpendicular 8

to the direction of the donor dipole, then κ2 = 0 (that is, because, θD = π/2, so cos θD = 0, and θˆ D · pˆ A = 0). However, this example is only a special case where κ2 = 0. As was pointed out above, for any position in space of the acceptor molecule, κ2 will equal zero for all orientations of pA where ED · pA = 0. And for most of these positions where ED · pA = 0, the donor and acceptor dipoles are not perpendicular to each other (see Fig. 3a). Conversely, if the donor and acceptor dipoles are perpendicular, then most locations of the acceptor relative to the donor will have κ2 = 0. This relationship is easiest to observe by looking at the second equality in Equation 12 or by examining Fig. 3a. Another simple example is when the dipoles are parallel (pˆ A · pˆ D = 1). Then, if θD = 0 and rˆ · pˆ A = 1 (parallel dipoles, stacked on each other), then κ2 = 4. But when θD = π/2 and θˆ D · pˆ A = 1 (again parallel dipoles, but now next to each other), then κ2 = 1. In the latter case, pA is parallel to ED but cos θD = pˆ D · rˆ = 0, and the value of κ2 is four times smaller than when cos θD = pˆ D · rˆ = 1. These few examples demonstrate the complexity of the behavior of κ2 . In Fig. 3a, we have indicated both the orientation of pA and ED as well as the angle θD (where cos θD = pˆ D · rˆ ). More thorough discussions of kappa can be found in the literature (35, 48, 51). Because fluctuations always exist in positions and angles of the D and A molecules, the actual value of κ2 is an ensemble average or a time average. The most commonly used average value is κ2 = 2/3. As already mentioned, this result is rigorously true if during the excited state lifetime of the donor the orientations of the donor and acceptor can each individually reorient fully in an independent random fashion. However, even when this condition is not met (for instance when the anisotropy of the dyes is not close to zero), it has been found that the approximation κ2 = 2/3 is often satisfactory (39, 42, 45, 51, 90). It is often discussed in the literature as though κ2 = 2/3 pertains only to the case of very rapidly rotating D and A molecules. This statement is not true, because depending on the placement of the dyes and their relative orientations, it is possible for κ2 = 2/3 at every location of the acceptor relative to the donor, even when the dyes cannot rotate at all. Also, many dyes used for FRET have more than a single transition dipole, which can be excited at the same wavelengths. Because different transition dipoles of a fluorophore are usually not parallel to each other (they are often perpendicular to each other), the presence of multiple transition dipoles leads again to an averaging of κ2 (90). Finally, if one is interested in detecting a change in structure or extent of interaction (binding), then one may not be interested in exact estimates of κ2 . For instance, when fluorescent proteins are used in FRET experiments, κ2 can become a very important variable, and averages are often not applicable. This situation occurs because the chromophores are fairly rigidly held in the fluorescent protein structure, and the fluorescent proteins may have specific interactions either with each other or with other components of the complex under study (91, 92).

Index of refraction Note that in Equations 8 and 9, R06 ∝ 1/n 4 , which means that the distance over which FRET can take place is shortened for


F¨ orster Resonance Energy Transfer (FRET) for Proteins

higher indices of refraction in the molecular surroundings (the solvent). A high index of refraction—which is equal to the square root of the dielectric constant—means that the electrons in the molecules of the solvent are freer to respond to an electric field than for solvents with lower indices of refraction. The solvent is assumed not to absorb at the wavelengths in question, so the index of refraction is real (not a complex number). This screening from the response of the solvent molecules leads to a damping of the extent of the field, and therefore the two dipoles (donor and acceptor) must be closer together in a solvent with higher index of refraction to have the same strength of interaction as in a solvent with lower index of refraction. Note that the wavelengths are also shorter in a higher index of refraction; however, remember that FRET does not involve propagating light fields. The actual situation is somewhat more complex, and the reader is referred to an excellent discussion in the recent literature (93–95), which clears up common misconceptions as to the origin of the effect of the index of refraction for FRET. In addition, the index of refraction varies significantly over short distances in and on the surface of proteins (90); for very short distances, the concept of an index of refraction becomes suspect. However, usually an average value is chosen between 1.33 and 1.5, which correspond to values of water and crystals of polypeptides. The value chosen usually depends on whether the dyes are in direct contact with water or are inside the protein, where the dyes are removed from water. Detailed numerical methods (96, 76, 97, 70) can be used to take into account directly the interaction between charges if the protein structures are known and if the position and orientations of the chromophores are known. Such extensive numerical calculations, whenever the structures are well known, avoid the use of an “effective” index of refraction.

Quantum yield of the donor Equation 8 shows that R 0 is dependent on the quantum yield of the donor φD . This dependence is not strong; R 0 ∝ (φD )1/6 .  For instance, (R0 )φD = 0.1 (R0 )φD = 1 ≈ 0.7. Therefore, small changes in the quantum yield of the donor will not make large differences in the value of R 0 . Of course, if increasing φD by a factor allows one to make accurate measurements of distances or to observe conformational changes, then the change can be significant. However, the range of distances observable by FRET are not a strong function of the quantum yield of the donor. However, small displacements in the horizontal placement of the efficiency curve of Fig. 2a can significantly increase or decrease the sensitivity of a FRET measurement. In some circumstances, it is advisable to choose a donor–acceptor pair that have a smaller R 0 , preferably in the range of the distance one is interested in measuring (see section below on “What is important, R06 or R 0 ?”). One can sometimes shorten R 0 by simply adding a collisional dynamic quencher to the solution, which will lower the quantum yield of the donor in the absence of the acceptor, lowering τD , thereby adjusting R 0 to lower values (90).

Transfer at very short distances The F¨orster transfer mechanism (Eqs. 1, 8, and 10) is valid at all distances in the near-field region where the approximation of point dipoles is valid. At very small distances from the molecular dipoles, the finite extended distribution of the electrons make the point dipole approximation invalid (98, 99). In this case, the higher order terms such as quadrapoles and octapoles would have to be taken into account (14, 15). In addition, if no electric dipole interactions occur between the donor and acceptor (for symmetry conditions, making the transition dipole zero), then these electric quadrapole terms and interactions between the magnetic dipole and electric dipole terms have to be taken into account. The lack of electric dipole transitions is seldom the case, and the use of quadrapole terms has not yet been important for the interpretation of FRET in proteins. However, if the two participating dipoles are very close, then transfer by electron exchange [Dexter transfer (14, 15)] also presents a possible pathway for energy transfer. This type of energy transfer should not be referred to as FRET because it is a different mechanism than F¨orster Transfer and requires a partial overlap of the electronic orbitals of the donor and acceptor. The transfer probability drops off exponentially, following the exponential decrease in the overlap of the wave functions of the electronic orbitals. However, even in photosynthesis, where the chlorophylls are very close and oriented to maximize energy transfer, it has not been shown unequivocally whether Dexter transfer is an important pathway for energy transfer in the photosynthetic unit. Nevertheless, one should be aware of these other mechanisms of energy transfer.

Noncoherent energy transfer (FRET) or coherent transfer Lately there has been increased interest in coherent transfer. F¨orster transfer assumes that interactions with the molecular surroundings, or the internal vibrations of the molecules, are sufficiently rapid so no correlation exists between the act of excitation of the donor and the act of transfer. In other words (to give an anthropomorphic analogy), during the time when FRET can take place the donor shows no memory of how, or when, it was excited. This lack of memory is attributed to the chaotic, randomizing effects take place in subpicosecond times, rapidly relaxing the initially excited molecule to the lowest vibrational state of the excited electronic state (before fluorescence emission or FRET). A similar process takes place after the energy is transferred to the acceptor; the original acceptor state immediately following the transfer is rapidly randomized so that the transfer is irreversible. Whenever this randomization occurs, energy transfer can be analyzed from a probability viewpoint (F¨orster transfer), as is the case in the derivation of the Equations 2–7, 8, and 11. That is, in F¨orster transfer we consider the probability per unit time for each pathway where the probabilities are independent of time and independent of each other. This probabilistic treatment of all the rates of deexcitation from the excited state for F¨orster transfer holds true whether the donor and acceptor are identical (homo-FRET) or different molecules (hetero-FRET). If



F¨ orster Resonance Energy Transfer (FRET) for Proteins

molecules (or atoms, with which most of these coherent transfer experiments have taken place) are completely isolated from the environment (such as in a vacuum at very low temperatures, e.g. well below 1 K), or if one is observing the fluorescence emission in the femtosecond time range at low temperatures (
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