# Two Orders of Magnitude Variation of Diffusion-Enhanced Förster Resonance Energy Transfer in Polypeptide Chains

^{*}

^{†}

## Abstract

**:**

_{0}> 20 Å) lead to a fairly low contribution of diffusion. We introduced short-distance FRET (sdFRET) where Dbo, an asparagine residue conjugated to 2,3-diazabicyclo[2.2.2]octane, acts as acceptor paired with donors, such as naphtylalanine (NAla), tryptophan, 5-l-fluorotryptophan, or tyrosine. The Förster radii are always close to 10 Å, which makes sdFRET highly sensitive to diffusional motion. We recently found indications that the FRET enhancement caused by diffusion depends symmetrically on the product of the radiative fluorescence lifetime of the donor and the diffusion coefficient. In this study, we varied this product by two orders of magnitude, using both donors of different lifetime, NAla and FTrp, as well as a varying viscogen concentration, to corroborate this statement. We demonstrate the consequences of this relationship in evaluating the impact of viscogenic coadditives on peptide dimensions.

## 1. Introduction

_{6}, with two donors of largely different radiative lifetimes, NAla and FTrp. The peptides were sufficiently long to avoid steric-hindrance effects and sufficiently short to lead to sizable FRET (Figure 1). In addition, we varied the end-to-end diffusion coefficient by altering the ethylene glycol content of the solutions. As a viscogen, ethylene glycol allows practical experiments to be done at very high concentrations and guarantees, because of its small size, that the measured macroviscosity is closest to microviscosity relevant on a molecular scale. Because of these properties, ethylene glycol has previously proven instrumental in elementary questions on protein folding [16,17,18,19]. The theoretical framework of the analysis is found in the Materials and Methods because the foundations have been laid in previous studies [1,13,20].

## 2. Materials and Methods

^{3}M

^{−1}cm

^{−1}(FTrp) and 5.5 × 10

^{3}M

^{−1}cm

^{−1}(NAla) at 280 nm; they were adjusted to 10 µM in aerated solutions of varying ethylene glycol content, 25 °C, pH 5.0. The donor quantum yields were determined by comparison with N-acetyl-tryptophanamid (0.14, pH 7.0) [23,24,25]. The Förster radii of NAla/Dbo and FTrp/Dbo were determined by using the absorption and emission spectra of the single-labeled peptides as previously described [11,23].

_{6}-Dbo-NH

_{2}and 5-F-Trp-(Gly-Ser)

_{6}-Dbo-NH

_{2}, and, in the absence of FRET, from the donor-only peptides, NAla-(Gly-Ser)

_{6}-NH

_{2}and 5-F-Trp-(Gly-Ser)

_{6}-NH

_{2}. Upon excitation of NAla or FTrp at 280 nm and emission recording at 350 nm, the traces were analyzed by using the FLS920 instrument software (Edinburgh Instruments, Livingston, UK) as described [23] and by using the software ProFit (Quantumsoft, Zürich, Switzerland). The reproducibility of the reported fluorescence lifetimes for the monoexponential decays was ±3%.

## 3. Theoretical Analysis

_{rad}) by plotting it as a function of Dτ

_{rad}, or as a function of the product of the inverse viscosity times the inverse radiative lifetime or intrinsic decay rate (Dτ

_{rad}∝ η

^{−1}k

_{rad}). Towards this goal, it will not be sufficient to only change the ethylene glycol content. To be convincing, we had to switch donors to achieve dramatically different radiative lifetimes that, together with the variation in viscosity, would allow us to cover two orders of magnitude changes in Dτ

_{rad}. In the following, we outline the analysis that yielded the effective distances and the products of the inverse viscosities and radiative lifetimes.

_{DA}, and of the donor-only peptides, τ

_{D}, were converted into the corresponding decay rates, k

_{D}and k

_{DA}, according to Equation (1), which yielded the FRET rate constant, k

_{FRET}, according to Equation (2).

_{D}, is the ratio of the radiative decay rate, k

_{rad}, and the decay rate, k

_{D}(Equation (7)). The values of the quantum yield and radiative decay rate in the absence of ethylene glycol have been reported [13].

_{0}, equals 1.3328 under our conditions.

_{6}peptides in water, ${R}_{0}\left({\mathsf{\Phi}}_{\mathrm{D},{n}_{0}},{n}_{0}\right)$, have been reported (FTrp/Dbo: R

_{0}= 9.6 Å; NAla/Dbo: R

_{0}= 9.8 Å).

_{0}, is also proportional to the sixth root of the orientation factor, κ

^{2}. The factor, κ

^{2}, captures that the dipole moments of the donor and acceptor can adopt various orientations towards each other and can vary between 0 (perpendicular vectors) and 4 (collinear vectors). The consequences of possibly misjudging the value of κ

^{2}were recently discussed [27]. The orientation factor adopts a value of 0.67 when donor and acceptor dipoles sample all possible orientations randomly and rapidly in comparison to the time scale of the donor emission decay [28]; this is the usual assumption that we also employed. However, it adopts the slightly smaller value of 0.48, when dipole orientations are random, but remain virtually frozen during the radiative donor lifetime [29,30], which translates (due to the sixth root dependence) into 5% smaller Förster radii and effective distances. However, we recently applied sdFRET to short polyproline peptides and could show by MD simulations that probe orientations are randomized due to the flexible linkers that connect the optically active probes to the polyproline chain [10]. Furthermore, the probes we employ in sdFRET are small and the time of randomization, the reorientation time, is well within the picosecond time scale [31]. Thus, even if, in the presence of a viscogen, the reorientation time is increased by an order of magnitude, it is still small compared to the donor lifetime (NAla, 256 ns; FTrp, 19.8 ns), and the criteria discussed in reference [27] are met.

_{rad}, (Equation (13)), and the diffusion coefficient, D; that is, on the product, Dτ

_{rad}. We assumed that the diffusion coefficient is proportional to the reciprocal viscosity (Equation (14)). The radiative lifetimes have been determined as outlined above. If the effective distance depends symmetrically on the diffusion coefficient and the radiative lifetime (Equation (15)), it also depends symmetrically on the viscosity and radiative rate (Equation (16)), and vice versa. The relationship between the concentration of aqueous ethylene glycol solutions and their viscosity has been established in Ref. [32] (Equation (17)), where x

_{EG}is the mole fraction of ethylene glycol.

_{0}*(r) or N

_{0}*, instantly after short-pulse donor excitation, mirrors the distance distribution in all peptide chains in the experimental sample because the probability of donor excitation does not depend on the distance of the acceptor. Thus, N

_{0}*(r), when normalized to ∫N

_{0}*(r) dr = 1, is identical to the probability distribution in the ground-state equilibrium ensemble of chains, i.e., N

_{0}*(r) = p(r). In the widely used skewed Gaussian distribution (Equation (19)), the meaningful parameters, a and b, determine the shape of the distribution, whereas c is a normalization constant determined by the condition that the integral of the probability density over all possible distances has to equal unity, ∫p(r) dr = 1.

_{D}, the Förster radius, R

_{0}, as well as the parameters describing chain conformation and dynamics, a, b, and D, are fed into the HSE—a linear partial differential equation, which can only be solved numerically, the HSE returns the fluorescence decay kinetics and, through further calculation, previously outlined in reference [13], the effective distance, R

_{eff}. As we determined the k

_{D}and R

_{0}values for 48 donor/viscosity combinations, we could simultaneously analyze 48 equations, each with its own pair of k

_{D}and R

_{0}values, with the diffusion coefficient, D

_{η}, at viscosity, η, given by Equation (20)—where D and η

_{0}are the values in water—and with the parameters of the distance distribution, a and b, whose values were kept constant in all 48 equations.

_{eff}values could not be further minimized. The result is illustrated in Figure 7 in the Results and Discussion section. This kind of optimization does not directly provide standard deviations of the parameters of interest. Therefore, we further tested the reliability of the obtained values by performing the same global optimization analysis on subsets that contained only a limited number (15 to 35) of data points, randomly chosen from the set of 48 experimental data points (200 optimizations). The computations were carried out in MATLAB (MathWorks).

## 4. Results and Discussion

_{6}peptides occurs and how it is strongly influenced by ethylene glycol is illustrated in Figure 2. The peptide, NAla-(GS)

_{6}-Dbo-NH

_{2}, shows a fluorescence spectrum (red curve) with two regions of maximal intensity, one around 340 nm arising from NAla fluorescence and another at 440 nm due to Dbo fluorescence. Since Dbo in the absence of a donor cannot be optically excited at 280 nm [12], it receives its excitation from NAla through FRET. In the presence of increasing concentrations of ethylene glycol, the fluorescence from NAla increases and the fluorescence from Dbo decreases. This is a first indication of reduced diffusion, and of a reduced FDE in the presence of viscogen. A second cause is the well documented quenching of Dbo by protic solvents [33].

_{rad}, of the NAla-only peptide (256 ns, τ

_{rad}= τ

_{D}/Φ

_{D}) exceeds the radiative lifetime of the FTrp-only peptide (19.8 ns) by a factor of 13. While, in the presence of an acceptor, FRET shortens the donor lifetime (red traces in Figure 3a,b), ethylene glycol increases it by increasing the quantum yield of both donors, NAla and FTrp (Figure 3c,d).

_{rad}, or, eqivalently, ηk

_{rad}(Equations (13)–(16)). The viscosity of the experimental solutions increases exponentially with the ethylene glycol concentration (Figure 4a, Equation (17)) and the refractive index linearly (Figure 4b). The refractive index, in turn, affects the radiative lifetimes of the donors, NAla and FTrp (Figure 4d), according to Equation (8). Donor quantum yields were calculated according to Equation (7). The quantum yield of NAla increases with ethylene glycol content (Figure 4c, red). The quantum yield of FTrp (Figure 4c, blue) increases even more and almost coincides with that of NAla at the highest ethylene glycol concentration that we employed (92%). This stronger dependence of the FTrp quantum yield on the presence of the coadditive is also reflected in the case of the Förster radii (Equations (10)–(12)) of the donor/acceptor pairs, NAla/Dbo (red) and FTrp/Dbo (blue), shown in Figure 4e. While both graphs show an increase of R

_{0}, they do not run in parallel. At high viscogen concentrations, the FTrp/Dbo values even exceed the NAla/Dbo Förster radii. The effective donor/acceptor distances in NAla-(GS)

_{6}-Dbo-NH

_{2}(red) and in 5-F-Trp-(GS)

_{6}-Dbo-NH

_{2}(blue) could now be obtained from Equation (5) (Figure 4f).

_{rad}, of viscosity, η, and inverse radiative lifetime, k

_{rad}(Figure 5, top). The inverse product, (ηk

_{rad})

^{−1}, is directly proportional to D𝜏

_{rad}(Equations (13)–(16)). To ease the understanding of Figure 5, we depicted the courses of its components, the viscosity and the inverse lifetime, beneath it (Figure 5, middle and bottom panel). The critical step is going from solutions that contain high amounts of ethylene glycol and peptides labeled with NAla or NAla/Dbo to solutions that contain no or little amounts of ethylene glycol and peptides labeled with FTrp or FTrp/Dbo. This jump in viscosity when going from ethylene glycol to water is perfectly balanced when simultaneously going from NAla to FTrp. The continuity of the course of the effective distance proves that it is only the product, ηk

_{rad}, and, equivalently, D𝜏

_{rad}, which influences the FDE. With a decrasing FDE (Figure 5), the effective distance increases from ca. 8 Å to ca. 14 Å. Our choice of (GS)

_{6}model peptides seemed to be appropriate as their dimensions emerged to be a good match to the Förster radii (ca. 10 Å) of the sdFRET methods. The remarkable increase of the effective distance emphasizes that any FRET-based study and inferrence on peptide structure could be gravely incorrect if a possible FDE is simply ignored.

_{rad}or ηk

_{rad}symmetry has important implications. In Figure 6, the effective distance is again plotted against ηk

_{rad}. The dotted line marks the simultaneous switch from high to low viscosity and from a short to a long donor lifetime. If the coadditive has no impact on peptide dimensions, this course is continuous (case (2)). If the intrachain distances contract upon addition of the coadditive, they should expand again when the experimental series goes from highest to lowest ethylene glycol content (case (1)). If the intrachain distances expand upon addition of the coadditive, they should contract again when the experimental series starts again at 0% ethylene glycol (case (3)). In this study, we clearly observed a type-2 course. Even though ethylene glycol changes not only the viscosity and the refractive index of the solutions, but also the dielectric permittivity and other properties, it has, even at very high concentrations, no impact on the peptide dimensions.

^{2}/ns in the absence of ethylene glycol, and a skewed Gaussian distance distribution, p(r) = cr

^{2}∙exp(−a(r − b)

^{2}, with a = 1.27 × 10

^{−3}Å

^{−2}and b = −25.3 Å (Figure 7b) at all solution conditions. Because the optimization could not directly provide confidence intervalls of the parameters of interest and because we have repeatedly expressed our skepticism, whenever very sharp values of diffusion coefficients were reported in the literature [13,36], we tested the robustness of the analysis and the results by also analyzing random subsets, limited to 15–35 points, of the entire set of 48 data points. This procedure and the analysis of 200 optimizations yielded D = 53.4 ± 6.0 Å, a = 1.28 ± 0.06 × 10

^{−3}Å

^{−2}, and b = −25.7 ± 2.3 Å, which is very close to the results obtained from all 48 points.

^{2}/ns [40] to 58 Å

^{2}/ns [34]. Most often, ensemble measurements on peptides and proteins [34,36,39,40] were analyzed on the basis of the Haas-Steinberg equation [14]. To dissect the contributions of the equilibrium distance distribution and diffusion requires a global analysis [42]. Haas et al. simultaneously analyzed the donor fluorescence kinetics as well as the decay kinetics of the acceptor optically excited through FRET [39,40,42]. Other proposals to carry out a global analysis included the use of an external quencher to modulate the donor lifetime [15], or the simultaneous analysis of the donor decay kinetics of peptides labeled with two different donors [34]. We challenged both attempts in previous works [13,36]. However, the corrected end-to-end diffusion coefficient, 58 Å

^{2}/ns, reported in reference [34] and measured for labeled (GS)

_{16}peptides, is only slightly larger than our value of 55 Å

^{2}/ns for the (GS)

_{6}peptides that we studied here. At short chain lengths, the diffusion coefficient is expected to grow with an increasing length of the probe-intermittent chain [36,39], but we can also expect that this effect will level off for chains of sufficient length; naturally, the parts of the chain closest to the probes exert the largest effect on probe diffusion.

## 5. Conclusions

_{rad}symmetry can be employed to conclude on the impact of viscogenic coadditives on peptide chain dimensions. Measurements in the presence of a viscogen can even provide absolute parameters of chain dynamics and dimensions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The selected donor-acceptor and donor-only labeled peptides composed of Gly-Ser units. Donors are N-terminal naphthyl-1-l-alanine (NAla) and 5-fluoro-l-tryptophan (5-F-Trp); the acceptor is C-terminal Dbo, whose optically active group is the azo group of the bicyclic chromophore.

**Figure 2.**(

**a**) Steady-state fluorescence emission spectra and (

**b**) time-resolved fluorescence decays (λ

_{obs}= 350 nm) of NAla-(GS)

_{6}-Dbo-NH

_{2}upon excitation at 280 nm at ethylene glycol concentrations (vol/vol) ranging from 0% (red) to ca. 92% (blue). The donor lifetime increases from 10.3 ns (red curve) to 34.3 ns (blue curve).

**Figure 3.**(

**a**) NAla sdFRET measurements: Time-resolved fluorescence decays measured with NAla-(GS)

_{6}-NH

_{2}(black) and NAla-(GS)

_{6}-Dbo-NH

_{2}(red) at 350 nm after excitation at 280 nm in water. The lifetime of the donor-acceptor peptide, τ

_{DA}= 10.3 ns, and the lifetime of the donor-only peptide, τ

_{D}= 34.7 ns, yield an energy transfer efficiency of 70.4% and an effective distance of 8.5 Å. (

**b**) FTrp sdFRET measurements: Time-resolved fluorescence decays measured with 5-F-Trp-(GS)

_{6}-NH

_{2}(black) and 5-F-Trp-(GS)

_{6}-Dbo-NH

_{2}(red) at 350 nm after excitation at 280 nm in water. The lifetime of the donor-acceptor peptide, 2.0 ns, and the lifetime of the donor-only peptide, 1.5 ns, yield an energy transfer efficiency of 24.7% and an effective distance of 11.5 Å. (

**c**) Time-resolved fluorescence decays measured with NAla-(GS)

_{6}-Dbo-NH

_{2}and (

**d**) 5-F-Trp-(GS)

_{6}-Dbo-NH

_{2}at 350 nm after excitation at 280 nm, at ethylene glycol concentrations (vol/vol) ranging from 0% (red) to 92% (blue).

**Figure 4.**(

**a**) Viscosity and (

**b**) refractive index of aqueous ethylene glycol solutions plotted against the ethylene glycol concentration. (

**c**) Donor quantum yield of NAla in NAla-(GS)

_{6}-NH

_{2}(red) and of 5-F-Trp in 5-F-Trp-(GS)

_{6}-NH

_{2}(blue) plotted against ethylene glycol concentration. (

**d**) Radiative lifetimes of NAla in NAla-(GS)

_{6}-NH

_{2}(red) and of 5-F-Trp in 5-F-Trp-(GS)

_{6}-NH

_{2}(blue) plotted against the refractive indices that correspond to the ethylene glycol concentrations shown in (

**b**). (

**e**) Förster radii of the NAla/Dbo donor/acceptor pair in NAla-(GS)

_{6}-Dbo-NH

_{2}(red) and of the 5-F-Trp/Dbo donor/acceptor pair in 5-F-Trp-(GS)

_{6}-Dbo-NH

_{2}(blue) plotted against ethylene glycol concentration. (

**f**) The effective donor/acceptor distances in NAla-(GS)

_{6}-Dbo-NH

_{2}(red) and in 5-F-Trp-(GS)

_{6}-Dbo-NH

_{2}(blue) plotted against the ethylene glycol concentration.

**Figure 5.**(

**Top panel**) The effective donor/acceptor distance in NAla-(GS)

_{6}-Dbo-NH

_{2}(red) and 5-F-Trp-(GS)

_{6}-Dbo-NH

_{2}(blue) plotted against the product, ηk

_{rad}, of the viscosity of the aqueous ethylene glycol solutions and the intrinsic decay rates (reciprocal radiative lifetimes) of NAla in NAla-(GS)

_{6}-NH

_{2}and of 5-F-Trp in 5-F-Trp-(GS)

_{6}-NH

_{2}(blue). (

**Middle panel**) The corresponding viscosity values and (

**Bottom panel**) radiative decay rates of NAla in NAla-(GS)

_{6}-NH

_{2}(red) and of 5-F-Trp in 5-F-Trp-(GS)

_{6}-NH

_{2}(blue).

**Figure 6.**(

**a**) The effective donor-acceptor distance (R

_{eff}, relative scale) in donor/acceptor labeled peptides plotted against ηk

_{rad}. The dotted vertical line marks the point where the viscosity switches back from high to low ethylene glycol content and where a donor of high radiative lifetime is exchanged for one with low radiative lifetime. The three possible cases are (1), (2), and (3). (1) The end-to-end distances in the experimental peptide shorten upon addition of ethylene glycol, such that the switch back to low ethylene glycol content shows an up-jump of the effective distance (red and green line). (2) The distances are completely independent of the ethylene glycol concentration (red and blue line). (3) The peptide expands upon addition of ethylene glycol, such that the switch back to low ethylene glycol content shows a down-jump of the effective distance (red and black line). (

**b**) The distance distributions (probability density plotted against relative donor-acceptor distance) that correspond to the three cases in (

**a**) shown in corresponding colors. For instance, case (2) points to virtually identical distance distributions (red and blue) that do not change with the ethylene glycol concentration.

**Figure 7.**(

**a**) The effective donor/acceptor distance in NAla-(GS)

_{6}-Dbo-NH

_{2}(red) and 5-F-Trp-(GS)

_{6}-Dbo-NH

_{2}(blue) plotted against the product, ηk

_{rad}, as in the top panel of Figure 5. The black circles represent the best fit of the data points, as obtained from an optimization based on the numerical solution of the Haas-Steinberg equation (HSE, Equation (18)). It yielded a diffusion coefficient of 55.4 Å

^{2}/ns in the absence of ethylene glycol, and a skewed Gaussian distance distribution, p(r) = cr

^{2}∙exp(−a(r − b)

^{2}, with a = 1.27 × 10

^{−3}Å

^{−2}and b = −25.3 Å, as shown in (

**b**) for all employed ethylene glycol concentrations (0–92%). The normalization constant, c, is fixed by the condition that ∫p(r)dr = 1.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jacob, M.H.; Ghosh, I.; D’Souza, R.N.; Nau, W.M.
Two Orders of Magnitude Variation of Diffusion-Enhanced Förster Resonance Energy Transfer in Polypeptide Chains. *Polymers* **2018**, *10*, 1079.
https://doi.org/10.3390/polym10101079

**AMA Style**

Jacob MH, Ghosh I, D’Souza RN, Nau WM.
Two Orders of Magnitude Variation of Diffusion-Enhanced Förster Resonance Energy Transfer in Polypeptide Chains. *Polymers*. 2018; 10(10):1079.
https://doi.org/10.3390/polym10101079

**Chicago/Turabian Style**

Jacob, Maik H., Indrajit Ghosh, Roy N. D’Souza, and Werner M. Nau.
2018. "Two Orders of Magnitude Variation of Diffusion-Enhanced Förster Resonance Energy Transfer in Polypeptide Chains" *Polymers* 10, no. 10: 1079.
https://doi.org/10.3390/polym10101079