The formulation of the point dipole approximation for the resulting energy transfer rate constant was done by T. Forster in 1948 (Forster, 1948). The distance regime of the coupled molecules where this approximation holds (1-10 nm) needs to allow the approximation of the dipole of one pigment seen from the other pigment as point source (i.e. the distance has to be clearly bigger than the dipole length). Forster already suggested that the FRET mechanism could be the dominating process for energy transfer between coupled molecules in photosynthetic organisms. For Chl a he calculated the distance R_{0} (Forster Radius) to be 8 nm, when the probability for the energy transfer is the same as the sum of all other relaxation processes (Forster, 1948):

At the distance R_{0} 50% of the donor pigment’s energy is transferred to the acceptor pigment (i.e. the probability of the transfer is 50%). In eq. 49 t_{d} denotes the apparent average fluorescence lifetime of the excited donor state in absence of the acceptor (i.e. the average excited state lifetime denoted as f in eq. 35, but without the energy transfer rate constant k_{ET} (absence of the acceptor)). If the fluorescence decay does not occur monoexponentially, eq. 49 has to be treated carefully and t_{d }should be calculated from the experimental data according to eq. 43 only after a multiexponential fit of the donor system’s fluorescence decay (without the acceptor, see Lakowicz, 2006). For further fluorescence spectroscopic findings of Forster see (Forster, 1982).

Measurements of the depolarisation of fluorescence light in dependency of the fluorophore concentration were done in 1924, mainly by two groups: Gaviola and Pringsheim, and Weigert and Kappler. From these measurements, Perrin in 1925 deduced that a direct electromagnetic coupling between the dye molecules must occur. He realized that the transfer of energy between the pigments is not simply a pure emission and reabsorption effect of real photons that leads to an energy transfer of excited states. In Forster’s own words, this energy transfer mechanism

“is principally different from such a mechanism which would consist of the trivial one of reabsorption of fluorescence light, because emission and electronic transfers would [in such case] not occur in parallel but subsequently” (Forster, 1948).

Forster used Fermi’s Golden Rule, as given in eq. 44, to calculate k_{ET} = k_{FRET} . He assumed the dipole interaction as V_{12}. As seen in eq. 45, the wave functions of the two pigments can be expressed as a product state of the wave function of pigment one |ф_{1}^{е}’ ^ and pigment two

|^{ф}/’ *) in excited (e) or ground (g) state (Renger and Schlodder, 2005):

For a distance typically R_{12} > 2 nm between the center points of the coupled transition dipole moments of the two coupled pigments, the dipole potential is well described by the point-dipole approximation (see Figure 16):

In eq. 51 Д denotes the dipole moment of the i. pigment and R_{12} denotes the vector that points from the center of Д to the center of Д (see

Figure 16). The dipole strength, which is proportional to |д|^{2}, can be calculated from the absorption spectrum of the acceptor pigment and from the time-resolved emission of the donor pigment. Due to the vector products in eq. 51 the coupling strength V_{12} strongly depends on the orientation of the interacting dipoles (for the nomenclature see Figure 16):

Using the real part n of the complex refraction index of the medium between both molecules (surrounding medium) (Renger and Schlodder, 2005) and the dipole orientation factor к := (cos0_{12}-3cos0j cos0_{2}) , the energy transfer rate for Forster transfer k_{ET} = k_{FRET} as given in eq. 44 using the potential in eq. 52 can be written as

Figure 16. Scheme of the angels 0_{1} between fa and the center-center vector R_{12}, 0_{2} between fa_{2} and R_{12}, 0_{12} between fa and fa . fa and fa are the two Q_{y} dipole moments of two coupled chlorophyll molecules.

Eq. 44, i.e. Fermi’s Golden Rule, is a formula to analyse the transition probability of any coupled quantum states if the coupling is weak enough. Eq. 53 is a formula to calculate the probability for energy transfer according to the assumption of a point-dipole-approximation of these states (Forster coupling).

The point dipole approximation is good if the distance of the molecules is much bigger than the extension of the dipole moments (~1 nm). Typical distances of coupled pigments in photosynthetic pigment protein complexes can be in the order of some nm. Better approximations take into account higher orders of the multipole extension of the Coulomb potential or describe the dipole extension in the form of the so-called “extended dipole” approximation. This assumes monopole moments from partial charges found along the dipole extension in the molecule (Renger and Schlodder, 2005). When we discuss such approximation, we should keep in mind that Fermi’s Golden Rule, as given by eq. 44, is itself an approximation of first order perturbation theory for weak couplings.

Additionally, there are further approximations used in the Forster formula. One prominent simplification is the assumption of a dipole embedded in a homogeneous and isotropic refractive medium which can be described by a simple number of dielectricity. If we look at the two chlorophyll molecules exemplarily shown in Figure 16 and a typical protein environment of the chromophores as, for example, shown for the chlorophylls in the LHC (see Figure 20) it becomes clear that there is no homogeneous and isotropic medium surrounding the chromophores.

The partially classical description of the Forster transfer rate shown in eq. 53 deviates strongly when the wave functions of the electrons located in the excited states of pigments one and two start to overlap directly as mentioned above (i.e. in these short distances the full quantum mechanical Dexter transfer has to be evaluated). This also becomes clear from a quantum mechanical point of view in which the definition of several clearly defined distances and angels, as necessary for the formulation of eq. 53, fails.

As mentioned above, the dipole strength of Д and Д_{2} can be derived from the optical spectra. Forster showed that the term ^ |^{2}1//_{2}1^{2}p_{eff} (e)

oo

F_{D}(A)e(A )A^{4}dA

is proportional to the expression - , i.e. the energetic

D ^{F}^)^{d}

resonance between the fluorescence spectrum F_{d}(A} of the donor pigment and the spectral extinction e(A) of the acceptor divided by the (average) lifetime of the donor pigment in the absence of the acceptor t_{d }(see Lakowicz, 2006; Forster, 1948, 1982). The dipole strength of the acceptor pigment |Д_{2}|^{2} corresponds to the molar extinction coefficient s(A) while the dipole strength of the donor |д|^{2} is expressed by the isolated donor pigment’s average excited state lifetime f_{D} .

The transition probability per time unit k_{FRET} as derived by Forster finally denotes to

with the normalised overlap integral between the fluorescence of the

oo

j F_{D}(A)e(A)A^{4}dA

donor and the extinction of the acceptor — -:= /(Я),

FdW

9000 ln10 _{3 25 3}

the constant A cm=8,8-10 cm^{3}mol (dimension of the

128л-^{5}N_{a}

wavelength [A]= cm ) and the extinction coefficient [??(/!)] = (M-cm) ^{1}.

cm^{3}

In that case, the overlap integral has the dimension ^J .

A clear description how to evaluate the Forster formula, as given by eq. 54, numerically is found in (Lakowicz, 2006). Combining eq. 49 and eq. 54 one can rewrite the Forster radius to

'Ц

(measured in cm) with Ф_{? } — as found in eq. 37 for the donor mole-

0

cule. The rate constant for the Forster transfer can then be expressed as

In many cases there is a lack of information for the direct calculation of the distance (R_{12}) dependent rate constant -_{pret}(R_{12}) from eq. 56. For example, the value of the orientation factor к^{2} is not known for many

2

pigment-protein complexes. For molecules in solution is к^{2} but in

photosynthetic complexes к^{2} can be of much higher values. Certain pathways along the PBP antenna are strongly suppressed due to very small values of к^{2} near to zero, which causes the EET to select pathways along the PBS structures of cyanobacteria (Suter and Holzwarth, 1987). It is found that pairwise coupled pigments with к^{2} & 1 exist in all photosynthetic complexes and therefore FRET helps to explain the structural organisation of pigment-protein complexes.

In the expressions of the Forster formulae given in eq. 54 to eq. 56 there might occur definition problems if the fluorescence decay is multiphasic. For coupled systems of different states of equal or comparable energy (isoenergetic states) the eq. 35 to eq. 37 gathered from integration of the uncoupled system are not valid and the quantum yield for the fluorescence decay cannot be simply calculated by

^{k}mour+^{k}Annrn+^{k}ET^{+k}ic+^{k}isc =• Eq. 35 is only strictly valid for

т

a monoexponential decay, i.e. if there is not a backwards transfer k__{ET }in the coupled system (see Figure 8). That means that the inverse average lifetime of a fluorophore is not well described by the sum of all rate constants (see Figure 8) as there is a high probability for a backward transfer from the acceptor molecule to the donor if the donor and the acceptor pigment are isoenergetic or of comparable energy.

If we look at Figure 8 the primary quantum yield for energy transfer from an excited state Ф_{ЕГ} (Д_{12}) = ®_{FREr} (R_{12}) can be defined according to

focusing on Forster Resonance Energy Transfer k_{ET} - k_{FRET} as dominating EET mechanism (see eq. 54). It is assumed that all other rate constants that appear in the time development of the excited state are concurring mechanisms to the EET via FRET. This is not the case if rate constants exist which populate the state (instead of depleting it, i.e. if the energy is transferred from the acceptor to the donor pigment). In such cases, eq. 57 does not describe the final EET from the donor pigment to the acceptor pigment (i.e. the average amount of energy transferred after equilibration of the system); in these cases it only describes the relative probability that the excitation energy is transferred to the neighbouring molecule directly after excitation. Eq. 57 therefore does not describe the final efficiency of Forster Resonance Energy Transfer from the donor to the acceptor after the full equilibration of the excited state, but only the initial probability for EET.

This is of high importance to understand the difference of average times for energy transfer in comparison to rate constants for energy transfer. Looking at a typical photosynthetic system as shown in Figure 11 (imagine this figure without the quencher positioned at state N4) we find that if the energy is located at pigment N0 at zero time and if

kpnE^Rn)^ ( *d) * then ^ _{FRET} (Я_{12})«1. But due to the strong coupling of all 6 states shown in Figure 11, 1/6 (16,7%) of the excitation energy will still relax from state N0 after equilibration. Therefore this amount is not transferred effectively from N0 to other pigments of the chain shown in Figure 11 if no quencher exists.

In the following it is noted that the effective energy transfer (the amount of energy that is really transported to the acceptor after equilibration of the system) cannot be calculated by eq. 57 since this equation is only denoting the probability for a single transfer step. Therefore we will suggest an alternative calculation.

k_{FRET} (R_{12}) as generally defined is a rate constant describing the faith of a single exciton. We want to derive the typical formulas for the quantum yield ®_{Er}(R_{12}) = ®_{FREr}(R_{12}) as they are found in the literature where they are sometimes incorrectly derived from eq. 57 (see Lakowicz, 2006). For this reason we will not call ®_{fret}(R_{12}) as denoted by eq. 57 the “efficiency" of FRET (which would be an expression regarding the equilibrated system) but we will call it the “quantum yield” of FRET (which is an expression valid for a nonequilibrium situation).

Combining eq. 56 and eq. 57 delivers

For the simplified monoexponential chromophores with (r_{D}) ^{1} = 2>_{f}- k_{FRET} (R12 ) ^{and}(Tja)= (T_{D} )^{_1} + ^{k}FRET (^{R}!2 ) = X ^{k}i^{one obtains}

i i

Looking at eq. 58 and eq. 59 as they are found in the literature (see e.g., Lakowicz, 2006) one can immediately see the contradiction if one interprets a system as shown in Figure 11 without the quencher positioned at the state N_{4} if the energies of the electronic states of all coupled pigments are equal and the distance between the coupled pigments R_{12} = 0 . In that case the fluorescence decay lifetime of the coupled ensemble does not change if no additional dissipative channels are occurring (see the simulation of the DAS of the described system in Figure 12, upper panel). One would get Ф^г^_{12}) = 1 ^ = 0 from eq. 59 but ®_{FREr} (R_{12} = 0) =

D

R^{6}

j-^{0} - = 1 from eq. 58. Eq. 59 can be understood as a FRET efficiency

^{R}12 ^{R}0

even if the donor and acceptor pigment do not decay monoexponentially if t_{da} and T_{D} are calculated according to eq. 43, which is an appropriate calculation of the average decay times for multiexponentially decaying systems (see Lakowicz, 2006). These lifetimes t_{d} and T_{DA} cannot simply be gathered from the DAS (see Figure 12, upper panel); one would need molecular resolution to analyze the excited state lifetime of the donor pigment in presence of the acceptor T_{DA} as given in the lower panel of Figure 12.

Eq. 58 denotes the quantum yield for an initial FRET process but not an overall efficiency. Independently from the coupling strength, an isoenergetic ensemble of pigments is not necessarily quenched (situation without the quencher at position N4 in Figure 11) but all pigments exhibit the same apparent fluorescence decay time t_{d} . This case is found for example in the strongly coupled Chlorophyll complex of the solubilised LHCII trimer. In LHCII t_{d} = t_{da} as t_{d} of the monomeric LHC structure or even of single chlorophyll molecules is similar to t_{da} of the solubilized LHC trimers (see Lambrev et al., 2011, and references therein).

Eq. 59 becomes valid if the average lifetimes of the donor and the acceptor are not measured from the ensemble but with single molecule resolution. At the moment this is not possible in experiments for structures as shown in Figure 11 because it would require an optical resolution in the order of 1 nm (less than the distance of 2 pigments). The DAS in the wavelength domain of a structure as shown in Figure 11 (without the quencher located at state N4) exhibit a single decay constant for the whole spectral range as shown in Figure 12, upper panel. If one could perform a superresolution experiment delivering the possibility to get individual decay curves for the chromophores in the coupled structure shown in Figure 11, then it would be possible to construct spatial DAS indicating the amplitude spread of individual decay components along the space coordinate instead of the wavelength coordinate (see Figure 12, lower panel). In that case, a correct calculation of the average fluorescence lifetimes of the donor in absence or presence of the acceptor t_{d} and t_{da} , respectively, as given by eq. 43, would enable the principal possibility to correct formula 59 so that it is also valid for single isoenergetic molecules in a coupled chain as shown in Figure 11.

Eq. 59 combined with eq. 43 directly delivers the well known intensity formulation for t_{da}* f_{D}:

I

Eq. 60 calculates the spatially resolved DAS as shown in Figure 12, lower panel and is independent from the restrictions mentioned above if

n n

the spectra are normalised, i.e. A_{i}

^{i=1}D^{i=1}DA

case if the absorption of the donor and acceptor in the donor-acceptor pair is the same as the absorption of the isolated donor only.

Evaluating ^ in the spatial “pigment domain“ for single

D

chromophores (Figure 12, lower panel, with help of eq. 43) one obtains a value of about ^ » 0.16 for the initially excited donor pigment N1 in

D

the exemplary molecule chain shown in Figure 12 if we neglect the contributions А_{;}(Я)г_{;} АДД)4ns . In fact, ^ becomes

i 4ns DA D

exactly 0.16 for к ^00 as is expected from theory (see Figure 11). The value of 0.16 is also expected for the relation -^{DA} if one evaluates the

-D

intensity of the donor pigment N1 in comparison to the fluorescence of all five acceptor pigments N_{0} and N_{2} to N_{5}. Therefore the real FRET efficiency after equilibration calculates to 84% as is expected from the structure of the system employing eq. 60.

With the use of eq. 43, eq. 59 (right side) and therefore also eq. 60 are correct for the FRET efficiency after equilibration of the system. In such case the efficiency should be denoted with rj_{FRET} (R_{12}). Eq. 58 describes the quantum efficiency ®_{FRET} (R_{12}) of a single excited state that undergoes a single FRET transition.

The final formula is therefore suggested to denote to

where it would be necessary to determine all parameters from single molecules but not to measure them on the ensemble. That means, one necessarily needs to achieve a signal with superresolution from a single molecule which acts as donor in a strongly coupled chain and calculate the f_{D} and t_{da} , respectively, as given by eq. 43, and

for the real „quantum efficiency^{1} describing the transfer probability of a single quantum state.

However, there are more problems than the ones already mentioned. Eq. 61 contains additional problems. For example, the fluorescence intensity and average lifetime of the donor pigment in presence of the acceptor, I_{DA} and f_{DA} , respectively, might change due to quenching effects without an energy transfer from the donor to the acceptor pigment. In such cases one might get a kind of “donor depletion efficiency” instead of EET efficiency when one employs eq. 61.

To avoid problems that especially affect the evaluation of the donor fluorescence, it is an advantage to focus on the fluorescence rise kinetics of the acceptor pigment (Schmitt, 2011) and to compare the fluorescence dynamics of the acceptor in absence of the donor with the fluorescence dynamics in presence of the donor pigment. It was found that this method is very stable against uncertainties. Unfortunately, however, the rise kinetics at the acceptor pigment is hard to resolve (Schmitt, 2011).

One additional approximation found in eq. 54 is the fact that the calculation of k_{FRET} (R_{12}) is usually performed using the fluorescence and absorption spectra of isolated pigments. But the optical properties of donor and acceptor states might be slightly changed by the coupling. Therefore it is always necessary to analyse the stability of the obtained solutions for FRET efficiencies against variations of the parameters used for the calculation and compare the experimental results with complementary methods.