Förster energy transfer from nonexponentially decaying donors

Journal of Photochemistry and Photobiology B: Biology 87 (2007) 200–208 www.elsevier.com/locate/jphotobiol

Fo¨rster energy transfer from nonexponentially decaying donors Agnieszka Czuper a, Ignacy Gryczynski b, Jo´zef Kus´ba b

a,*

a Faculty of Applied Physics and Mathematics, Gdan´sk University of Technology, ul. Narutowicza 11/12, 80-952 Gdan´sk, Poland Department of Cell Biology and Genetics, University of North Texas Health Science Center at Fort Worth, 3500 Camp Bowie Boulevard, Fort Worth, TX 76107, USA

Received 14 November 2006; received in revised form 10 April 2007; accepted 16 April 2007 Available online 24 April 2007

Abstract Necessary modifications to the expression for the Fo¨rster energy transfer rate are discussed when fluorescence decay of the donor in the absence of acceptor is nonexponential. Discrete and continuous models of the nonexponentiality are taken into account. No general solution of the problem is found. It is, however, suggested that in many of the biochemical problems the most appropriate modification of the transfer rate can be that which is based on the assumption of the same constant value of the radiative decay rate for all donor molecules. The effect of the assumed form of the Fo¨rster energy transfer rate on the recovered values of the distance distribution and dynamics parameters of some exemplary bichromophoric systems is examined.  2007 Elsevier B.V. All rights reserved. Keywords: Resonance energy transfer; Transfer rate; Nonexponential fluorescence decay

1. Introduction

absence of acceptor, r – the donor–acceptor distance, and R0DA is so-called critical distance defined as

Fo¨rster energy transfer (FET) is a technique which relies on the distance-dependent transfer of energy from a donor (D) molecule to an acceptor (A) molecule. It has been intensively investigated for more than half a century and is still gaining importance [1–3]. In widefield fluorescence and confocal laser scanning microscopy, as well as in biophysics and molecular biology, FET is a useful tool to quantify molecular dynamics, such as protein–protein interactions, protein–DNA interactions, and protein conformational changes. It allows to measure the nanometer scale distances and changes in distances, both in vivo and in vitro. The rate constant for FET is given by [4–6]

R60DA ¼ 9000ðln 10Þj2DA QD J DA =ð128p5 N A n4r Þ

In Eq. (2) is a factor describing the relative orientation of the transition dipoles of the donor and acceptor in space, JDA – the overlap integral expressing the degree of spectral overlap between the donor emission and the acceptor absorption, NA – the Avogadro’s number, and nr is the refractive index of the medium. In a typical situation, when no static quenching component is observed and the donor fluorescence decay function ID(t) in the absence of acceptor is exponential, i.e. described by the equation

k DA ¼ ðCD =QD ÞðR0DA =rÞ6

I D ðtÞ ¼ I D0 expðt=sD Þ

ð1Þ

where CD is the rate of radiative deactivation of the donor excited state, QD – the quantum yield of the donor in the

Corresponding author. Tel.: +48 58347 29 66; fax: +48 58347 28 21. E-mail address: [email protected] (J. Kus´ba).

1011-1344/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jphotobiol.2007.04.003

ð3Þ

with ID0 being the donor fluorescence intensity at t = 0, and sD – the donor fluorescence lifetime, the radiative rate CD can be expressed as CD ¼ QD =sD

*

ð2Þ

j2DA

ð4Þ

Under such conditions Eq. (1) takes the commonly known form

A. Czuper et al. / Journal of Photochemistry and Photobiology B: Biology 87 (2007) 200–208

k DA ¼ ð1=sD ÞðR0DA =rÞ

6

ð5Þ

Here all parameters needed for evaluation of the rate kDA have a clear physical meaning. Values of the lifetime sD, the quantum yield QD, and the refractive index nr can be obtained from direct measurements, the overlap integral JDA can be easily calculated based on the measured donor emission and acceptor absorption spectra, and the coefficient j2DA can be evaluated based on the assumed angular distribution of the donor and acceptor dipoles in the medium. If static quenching component is present, the measured quantum yield has to be corrected to obtain the real quantum yield. This problem is well described in the literature, see e.g. [1,7] for detail. However, calculation of the rate kDA becomes ambiguous when the decay function ID(t) in the absence of acceptor appears to be nonexponential. In such a case, relation (4) is nonadequate because then more than one fluorescence lifetime or even a continuous lifetime distribution can be assigned to the donor. Unfortunately, there is no consensus in the literature how these multiple lifetimes should be utilized to calculate value of the rate kDA. The nonexponential decay is observed for many biomolecules, especially polypeptides and proteins. Decays of fluorescence intensity in proteins often exhibit a complex behavior, the origin of which can result from multiple (ground state) conformations, protein dynamics, spectral relaxation, or other interactions between fluorophores and their environment [1]. In the case of small molecules the heterogeneity of the decay may be caused by appearance of quantum beats [8], whereas for some dyes in fluid solutions two different time scales can characterize the population decay as a result of specific properties of the relaxation process (barrierless, or very small barrier, reaction) [9]. Heterogenous decays are also observed for ensemble-averaged fluorescence of quantum dots [10–12]. Such type of heterogeneity can be interpreted as a single molecule effect, i.e., the result of single quantum dot intermittency. In this article we will treat the kinetics for an ensemble of molecules with exponential rates, and the single molecule effects will not be addressed. The nonexponential fluorescence decay is generally described in terms of lifetime distribution, although two complementary approaches based either on discrete or on continuous lifetime distribution function can be here distinguished. According to the discrete lifetime distribution approach, also called the multiexponential model, the time dependence of the donor fluorescence decay is represented as an arithmetic sum of n exponents I D ðtÞ ¼ I D0

n X

aDi expðt=sDi Þ

ð6Þ

i¼1

where aDi are the preexponential  Pn  factors describing the fractional amounts a ¼ 1 of the initial intensity Di i¼1 ID0, decaying with the respective lifetimes sDi. Values of the number n, the lifetimes sDi, and the amplitudes aDi are usually obtained as these providing the best agreement of Eq. (6) with the decay observed experimentally.

201

According to the continuous lifetime distribution approach the nonexponential fluorescence decay of the donor is represented as an integral of exponential decays weighted by continuous lifetime distribution function aD(sD) Z 1 I D ðtÞ ¼ I D0 aD ðsD Þ expðt=sD ÞdsD ð7Þ 0

R 1  Here aD(sD)dsD denotes the fraction 0 aD ðsD ÞdsD ¼ 1 of the initial intensity ID0, decaying with lifetimes belonging to the interval (sD, sD + dsD). In Eq. (7) the specific forms of the distribution function aD(sD) are usually applied without a particular theoretical basis. Most often Gaussian- or Lorentzian-shaped distribution functions are chosen. However, in these cases the part of distribution exists below sD = 0, so the aD(sD) values need additional normalization. These disadvantages disappear when the fluorescence decay is analyzed using certain specific nonsymmetrical lifetime distribution functions, which lead, e.g., to the Kohlrausch (stretched exponential) decay function [13] I D ðtÞ ¼ I D0 exp½ðt=sD0 Þb 

ð8Þ

or to the Becquerel (compressed hyperbola) decay function [14]  I D ðtÞ ¼ I D0 1 þ ð1  bÞ

t sD0

1 1b

ð9Þ

where b (0 6 b 6 1) and sD0 are parameters which can be evaluated based on the best agreement of the decays (8) or (9) with those observed experimentally. The Kohlrausch and Becquerel laws become both a single exponential law in the limiting case where b = 1. The main advantage of these laws is that each of them having only two adjustable parameters is able to properly describe the time evolution of many luminescent systems [13,14]. The aim of this paper is to find out, how should Eq. (1) and/or (5) be interpreted or modified to properly describe the process of nonradiative energy transfer from nonexponentially decaying donors in frames of the discrete and continuous lifetime distribution approaches discussed above. From this point of view, in the next section we will discuss the radiative rate CD, quantum yield QD, and fluorescence lifetime sD, involved in these equations. 2. Models of FET from nonexponentially decaying donors The nonexponentiality of fluorescence decay of a single fluorophore system is usually explained by existence of a number of fractions of the fluorophore in different environments. Each fraction displays one of the decay times sDi. In general, each of the fractions may also display different radiative rate CDi, playing an essential role in the process of FET. In such a case precise evaluation of all actual values of CDi can easily become very difficult or even impossible. The problem becomes much more complicated when the continuous lifetime distribution is considered. Then

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one usually assumes existence of a continuum of different molecular environments or a continuum of partially relaxed excited states, which in general can generate a continuum of rates CD(sD). For this reason, in this paper we will restrict ourselves to the systems in which one can assume that the rates CD for all donor fractions remain approximately the same. Thus, we will assume that the nonexponentiality of the donor fluorescence decay is basically caused by fluctuations of the nonradiative rate kDnr, whereas the radiative rate CD remains approximately constant. In our further considerations such a model of FET will be referred to as ‘‘the constant CD model’’. The constant CD model finds its justification in the experimental data and is in accordance with generally accepted belief that the environmental sensitivity of the fluorescence intensity and quantum yield is due to nonradiative processes that compete with emission for deactivation of the excited state [1,15–17]. Most quenchers decrease the quantum yield and the excited-state lifetime by merely increasing the nonradiative rate. However, one has to admit existence of systems, which do not reveal such a behavior [18–20]. The constant CD model is also applicable in cases [8,9] when nonexponentiality of donor fluorescence decay is caused by dependence of the nonradiative rate kDnr on time. Then the donor fluorescence decay in the absence of acceptor can be described as [21] Z t k Dnr ðtÞdtÞ ð10Þ I D ðtÞ ¼ I D0 expðCD t  0

or alternatively, if the time dependence of the rate kDnr is not known, the observed experimental data can be fit using Eq. (6) or (7). It has been shown in [21] that if CD remains constant then regardless of the form of ID(t) the radiative rate can be calculated from the equation CD ¼ QD =hsD i

ð11Þ

where ÆsDæ is the average lifetime of the donor given by Z 1 hsD i ¼ S D ðtÞdt ð12Þ 0

and SD(t) is the survival probability of excited donor, related to ID(t) by S D ðtÞ ¼ I D ðtÞ=I D0

ð13Þ

To the best of our knowledge Eq. (11) has been first time proposed (without satisfactory justification) by Werner and Forster [22] for the case when donor decay is described by discrete lifetime distribution (6). The form of the average lifetime they proposed, n X hsD i ¼ aDi sDi ð14Þ i¼1

can be easily obtained from Eqs. (12), (13), and (6). Analogous calculations based on Eqs. (12), (13), and (7) lead to the following form for ÆsDæ in the case of continuous lifetime distribution

hsD i ¼

Z

1

aD ðsD ÞsD dsD

ð15Þ

0

While for Gaussian or Lorentzian-shaped distribution functions, aD(sD), calculation of ÆsDæ requires application of numerical methods, in the case of Kohlrausch and Becquerel decay functions analytical expressions for ÆsDæ can be used [13,14]. Introduction of Eq. (11) into Eq. (1) leads to the following expression for the donor–acceptor transfer rate valid within the constant CD model  6 1 R0DA ð16Þ k DA ¼ hsD i r which, we think is the best alternative to Eq. (5) in the case of nonexponentially decaying donor. Because in Eq. (16) the quantum yield used for calculation of the Fo¨rster radius is in fact an average value, so the R0DA itself can be here understood as an average value of different R0DAi characteristic for different fractions of the donor. One has to underline that the average ÆsDæ denotes here the average lifetime of the excited donor molecules and should not be confused with the average time of donor fluorescence decay Z 1 Z 1 sD ¼ tI D ðtÞdt= I D ðtÞdt ð17Þ 0

0

For exponential fluorescence decay sD ¼ hsD i ¼ sD , but when the decay is nonexponential sD and ÆsDæ are different [13]. Sometimes one observes dependence of the amplitude fractions aDi or a(sD) on the observation wavelength. In such a case to avoid wavelength dependence of the calculated value of the average lifetime ÆsDæ and the rate CD, one can use wavelength independent amplitude fractions proposed by Sillen and Engelborghs [17]. Eq. (16) differs from analogous equation  6 1 R0DA k DAi ¼ ð18Þ sDi r proposed for the case of discrete lifetime distribution by Lakowicz and coworkers [23,24] and referred to in a number of works [25–35]. From comparison of Eqs. (18) and (1) one can see that acceptance of Eq. (18) is tantamount to the assumption that the radiative rates for particular components of the donor are not equal to each other and are given by CDi ¼ QD =sDi

ð19Þ

Lakowicz et al. [23,24] suggest that the advantage of Eq. (18) is a constant value of the Fo¨rster radius, R0DA, for different components, but they also admit that they do not have experimental data to rigorously defend the assumed form of the equation. For convenience, in further considerations we will call the FET model expressed by Eq. (18) ‘‘the constant R0DA model’’. Eq. (16), as reflecting experimentally observed constancy of the rate CD (at least for selected cases), seems to be better justified then

A. Czuper et al. / Journal of Photochemistry and Photobiology B: Biology 87 (2007) 200–208

Eq. (18). An additional argument for validity of the constant CD model of FET is that it agrees with the results of considerations curried out by Steiner and coworkers [36,37]. Similarly as in [23,24], the authors address here only the case of discrete lifetime distribution. They suggest different values of the Fo¨rster radius for different fractions of the donor, so their expression for the transfer rate takes the form  6 1 R0DAi k DAi ¼ ð20Þ sDi r Finally they notice that especially when different decay times arose from purely dynamic interactions of the excited state, it is reasonably to assume that R0DAi is related to the average value, R0DA, by R60DAi R60DA ¼ sDi hsD i

ð21Þ

One can see from Eqs. (20) and (21) that in such a case the transfer rate takes the same constant value for all fractions of the donor which is in accordance with Eq. (16). Further valuable argument for the usefulness of Eq. (16) is that it remains in accordance with the concept of constant, independent on sDi, expression for the transfer rate kDA postulated in [38]. Apart from Eq. (4) or (11) the radiative rate CD needed for calculation of the transfer rate kDA can also be obtained from a modified Einstein relation between absorption and emission, the well known Strickler and Berg formula [39] Z 2:880  109 n2r 1 eD ðmÞ dm ð22Þ CD ¼ m hm3 i 0 where nr is the refractive index of the medium, m is the light frequency in cm1, eD(m) is the molar extinction coefficient, and Æm3æ is the mean value of 1/m3 evaluated over the emission spectrum fD(m) Z 1 Z 1 fD ðmÞ hm3 i ¼ dm fD ðmÞdm ð23Þ m3 0 0 Expression (22) assumes no interaction with a solvent, does not consider changes in refractive index between the absorption and emission wavelength, and assumes no changes in excited-state geometry. There are not many reports in the literature where the Fo¨rster transfer rate is calculated with use of Strickler and Berg formula. A good example here is the work written by Knox [40], showing good applicability of Eq. (22) for calculation of radiative rate and the transfer rate for chlorophyll a and bacteriochlorophyll a. In the case of nonexponentially decaying donors, the applicability of Eq. (22) seems to be limited to molecules for which the absorption and emission spectra of particular components remain essentially constant, which may be understood as an argument for the assumption that the radiative rate in these molecules is approximately the same for all components.

203

3. Importance of the kind of assumed model of FET to evaluation of the distance distribution and dynamics parameters in proteins One of the most important applications of FET is determination of distances or distribution of distances between donor and acceptor residues on macromolecules. The method relies on utilization of the inverse sixth power dependence of the transfer rate on the donor–acceptor distance. As it already was mentioned in Introduction, fluorescence decay of fluorophores attached to such macromolecules as e.g. proteins becomes mostly nonexponential. In this section we will try to estimate the consequences of the assumed form of the expression for donor–acceptor transfer rate [Eq. (16) vs. Eq. (18)] to the recovered values of parameters of intramolecular distance distribution and dynamics in a real experimental situation. In order to shorten our discussion we will restrict ourselves just to the case when only donor fluorescence is observed and the nonexponentiality of the donor florescence in the absence of acceptor is interpreted in terms of discrete lifetime distribution. Under such conditions the observed fluorescence intensity I(t) can be understood as being proportional to the sum of actual numbers ND*Ai(t) of the excited donors in the donor–acceptor pairs, multiplied by respective rates of radiative transition to the donors’ ground state. If one chooses the constant CD model of FET, and decides to describe the processes of energy transfer using Eq. (16), then I(t) is given by n 1 X IðtÞ ¼ C N DAi ðtÞ ð24Þ hsD i i¼1 where C is a constant. If instead of Eq. (16) one chooses Eq. (18) than n X 1 IðtÞ ¼ C N DAi ðtÞ ð25Þ s Di i¼1 In the presence of donor-to-acceptor diffusion the time evolution of the numbers ND*Ai(t) can be described by [41] Z rmax ð0Þ ð0Þ P ðrÞy i ðr; tÞdr ð26Þ N DAi ðtÞ ¼ N DA xDi expðt=sDi Þ rmin ð0Þ N DA

where is the total number of the excited donor–accepð0Þ tor pairs at t = 0, xDi is the initial fractional concentration of the excited molecules of the donor characterized by the lifetime sDi, P(r) is the distribution function of donor– acceptor distances, rmin and rmax is the minimum and maximum donor–acceptor distance, respectively, and the function yi(r, t) satisfies the equation  2  oy i ðr; tÞ o y i ðr; tÞ 1 dP ðrÞ oy i ðr; tÞ ¼D þ ot or2 P ðrÞ dr or  k DAi ðrÞy i ðr; tÞ

ð27Þ

with D being the mutual diffusion coefficient of donor and acceptor moieties. The initial and boundary conditions of Eq. (27) are yi(r, t = 0) = 1, ½oy i ðr; tÞ=orr¼rmin ¼ 0, and ½oy i ðr; tÞ=orr¼rmax ¼ 0.

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If one chooses the constant CD model of FET and assumes validity of Eq. (16) then according to Appendix ð0Þ A xDi ¼ aDi , and besides, the rates kDAi(r) in Eq. (27) become independent on the index i. Thus, in this case the functions yi(r, t) also become independent on the index i, which allows rewrite Eq. (24) in the form Z rmax n X IðtÞ ¼ I 0 P ðrÞyðr; tÞdr aDi expðt=sDi Þ ð28Þ rmin

i¼1 ð0Þ

where I 0 ¼ CN DA =hsD i is the total luminescence intensity at t = 0. On the other hand, if one assumes validity of ð0Þ Eq. (18) then according to Appendix B xDi ¼ aDi sDi = hsD i, and then Eq. (25) can be rewritten as n Z rmax X P ðrÞy Di ðr; tÞdraDi expðt=sDi Þ ð29Þ IðtÞ ¼ I 0 i¼1

rmin

Based on the condition of the best agreement of fluorescence decays predicted by Eq. (28) or (29) with respective data obtained experimentally one can estimate values of parameters of the distribution P(r) and of diffusion coefficient D. These parameters can also be evaluated from analysis of frequency-domain data. In frequency-domain fluorometry the measured quantities are the phase angle (/x) and the relative depth of modulation of the emitted light (mx) at various frequencies, x, of the excitation light. For the assumed model describing the fluorescence decay calculated (c) values of the phase angle and the modulation are determined from the expressions given below /cx ¼ arctan ðN x =Dx Þ ð30Þ mcx ¼ ðN 2x þ D2x Þ1=2

ð31Þ

where Nx and Dx are obtained from Z 1 Z 1 Nx ¼ IðtÞ sinðxtÞdt= IðtÞdt 0 0 Z 1 Z 1 Dx ¼ IðtÞ cosðxtÞdt= IðtÞdt

ð32Þ

0

ð33Þ

0

In Eqs. (32) and (33) I(t) is the fluorescence decay predicted using the considered model of FET. In order to illustrate the potential differences between this way obtained values of dynamics and distance distribution parameters, we will apply Eqs. (28) and (29) to analysis of the exemplary frequency-domain data selected from two articles written by Eis et al. [26,27]. These data concern solution structure of two forms of a single-domain CCHHtype zinc finger peptide. In both forms, the donor is a single tryptophan residue located at the midpoint of the sequence, and the acceptor is either a 5-(dimethylamino)-1-naphthalenesulfonyl (DNS) group attached at the amino terminus (DNS-ZF28), or a 7-amino-4-methyl-coumarin-3-acetyl (AMCA) group attached to the e-amino function of a carboxy-terminal lysine residue (AMCA-ZF29). Similarly as in [26,27], we will assume that the distribution P(r) is a Gaussian and is given by h i ( ðrRav Þ2 1 exp  for rmin 6 r 6 rmax 2 2r ð34Þ P ðrÞ ¼ Z 0; elsewhere

where Z is the normalization factor, Rav is the donor– acceptor most probable distance, and r is the standard deviation of the untruncated Gaussian function. In further considerations, instead of r we will use the half-width of the distribution (hw; full width at half-maximum probability), which is related to r by the equation hw = 2.355r. In order to shorten our discussion, from several cases investigated in [26,27], in this paper we will analyze fluorescence decay properties of DNS-ZF28 and AMCA-ZF29 only under the metal-free conditions. The critical distances for ˚ and DNS-ZF28 and AMCA-ZF29 were equal 19.5 A ˚ 25.2 A, respectively. Values of other basic parameters describing fluorescence decays of these systems are listed in Tables 1 and 2. Table 1 illustrates the nonexponentiality of the donor-only ZF28 and ZF29 peptides. For both of them the triple-exponential model is needed for an adequate fit of their intensity decays. We found it informative to calculate for each donor-only peptide the value of a parameter nD, which we have defined as   Z 1 I D ðtÞ 1 t dt ð35Þ nD ¼  exp  2hs i I hs i D

0

D0

D

and which we call the degree of nonexponentiality of the donor fluorescence decay. It is seen from Eq. (35) that nD describes the difference between the considered decay of the donor and the pure exponential decay of the same area under the decay curve. The normalization factor, 2ÆsDæ, provides that all possible values of nD are limited to the interval [0, 1). If the fluorescence decay of the donor is exponential, then its degree of nonexponentiality is equal to zero. Values of nD listed in Table 1 show that in the absence of acceptor the florescence decay of the tryptophan residue in ZF28 is significantly more nonexponential than that of ZF29. Distance distribution and dynamics parameters of DNS-ZF28 and AMCA-ZF29 found in [26,27] under no metal conditions are shown in Table 2 and are denoted by superscript ‘‘a’’. One has to stress that values of the Rav, hw, and D parameters have been obtained in [26,27] with the assumption that the donor–acceptor transfer rate is given by Eq. (18) and that the analyzed frequency-domain responses are described by Eqs. (29)–(33). Theoretical courses of the decay functions calculated by means of Eqs. (28) and (29) for parameter values obtained

Table 1 Donor multiexponential intensity decay parameters of ZF28 and ZF29 under no metal conditions Sample

sDi (ns)

aDi

nD

ZF28

0.040 1.60 3.90

0.662 0.260 0.078

0.364a

ZF29

0.108 1.02 2.78

0.282 0.474 0.244

0.147

a Values of the degree of nonexponentiality were calculated with use of Eqs. (35) and (6).

A. Czuper et al. / Journal of Photochemistry and Photobiology B: Biology 87 (2007) 200–208

205

Table 2 Recovered distance distribution and dynamics parameters of DNS-ZF28 and AMCA-ZF29 under no metal conditions ˚) ˚) Sample Rav (A hw (A D (106 cm2/s) v2R DNS-ZF28

AMCA-ZF29

20.1a (19.8–20.3)b 20.1c (20.0–20.2) 24.7d (24.6–24.8)

14.5 (12.5–16.3) 14.3 (13.5–15.1) 7.9 (7.4–8.4)

1.23 (0.81–1.78) 1.23 (1.03–1.48) Æ0æ e –

1.0

14.6a (11.4–16.9) 13.6c (10.6–15.8) 20.1d (19.4–20.9)

27.8 (24.3–32.7) 29.5 (26.2–33.8) 24.5 (22.6–26.3)

2.04 (1.62–2.82) 2.26 (1.90–2.84) 0.81 (0.58–1.13)

1.01

0.90 6.1

0.91 0.85

a Values obtained in [26,27] based on the FET model expressed by Eq. (18). b Values in parentheses are the 67% confidence limits. c Values obtained in this paper based on Eq. (18) and reconstructed experimental data files. d Values obtained in this paper based on Eq. (16) and reconstructed experimental data files. e Bracketed value indicates that this parameter was held fixed during the final analysis.

in [26,27] and reported in Table 2, together with respective frequency-domain responses calculated by means of Eqs. (30)–(33), are shown in Figs. 1 and 2. The thick solid lines correspond to the decays calculated with the assumption that the donor–acceptor transfer rate is given by Eq. (18), and the thick dashed lines – with the assumption that the rate is given by Eq. (16). The thin solid and dashed lines show the limits of bands associated with the respective thick lines when the parameters Rav, hw, and D are varying within their 67% confidence limits. In Fig. 1 one can see significant differences between the courses of thick solid and thick dashed lines, indicating that in the case of DNS-ZF28 the kind of assumed model of FET from nonexponentially decaying donor can play an essential role in the process of evaluation of the dynamics and distance distribution parameters of the investigated system. In Fig. 2 the differences between the thick solid and thick dashed lines are evidently smaller than in Fig. 1, and the confidence bands significantly overlap each other. The smaller differences between the lines can be explained by a fact that in the absence of acceptor the fluorescence decay of tryptophan residue in ZF29 is much less nonexponential than that in ZF28. In such a case, the differences between the constant R0DA model and constant CD model become less pronounced. It seems to be interesting to answer the question, what values of the dynamics and distance distribution parameters one would obtain if the frequency-domain responses in [26,27] were analyzed not by means of the constant R0DA FET model, but rather with use of the constant CD model expressed by Eq. (16). Unfortunately, we do not have the original experimental data files analyzed in [26,27]. However, based on the parameter values and on

Fig. 1. Effect of the assumed model of FET from nonexponentially decaying donors on the shape of theoretically predicted fluorescence decay functions (upper panel) and the respective frequency-domain responses (lower panel) of the DNS-ZF28. The thick solid lines are obtained using Eq. (18), and the thick dashed lines – using Eq. (16). Values of the parameters Rav, hw, and D are taken from [26,27]. The thin solid and dashed lines show the limits of bands associated with the respective thick lines when the parameters Rav, hw, and D are varying within their 67% confidence limits.

figures given in these papers we were able to reconstruct them. The results of analysis of these reconstructed files by means of Eq. (29) and (30)–(33) are listed in Table 2 below respective values rewritten from [26,27] and are denoted by superscript ‘‘c’’. One can see a very good agreement of values of the recovered parameters and confidence limits with those obtained from analysis of the original files. This allows us to state that for purposes of this work the reconstructed data files are sufficiently close to their originals. On the other hand, one obtains definitely different parameter values, as well as confidence intervals if the same reconstructed files are analyzed by means of Eq. (28) and (30)–(33). These results are denoted in Table 2 by superscript ‘‘d’’. Evidently, for both zinc finger peptides the two considered models of FET lead to essentially different recovered values of the distance distribution parameters Rav and hw and of the diffusion coefficient D. The recovered values of the distribution half-width and of the diffusion coefficient obtained with help of the constant CD model are significantly lower compared to those obtained with help of the constant R0DA model. In particular, in the case of DNS-ZF28 the best fit of the simulated data file to the constant CD model was achieved with value of the

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think that the value of the quantity nD can be used as the criterion of importance of the nonexponentiality of the donor fluorescence decay in a given particular case. In this article we have found that for bichromophoric systems with nD as low as 0.147 the influence of nonexponentiality of the donor fluorescence decay on the donor–acceptor transfer rate is still essential. In order to better determine the properties of FET in systems characterized by different values of nD and taking place in different geometries, further investigations are needed. 4. Conclusions

Fig. 2. Effect of the assumed model of FET from nonexponentially decaying donors on the shape of theoretically predicted fluorescence decay functions (upper panel) and the respective frequency-domain responses (lower panel) of the AMCA-ZF29. The thick solid lines are obtained using Eq. (18), and the thick dashed lines – using Eq. (16). Values of the parameters Rav, hw, and D are taken from [26,27]. The thin solid and dashed lines show the limits of bands associated with the respective thick lines when the parameters Rav, hw, and D are varying within their 67% confidence limits.

diffusion coefficient, D, set to zero, whereas the constant R0DA model yielded D = 1.23 · 106 cm2/s. On the other hand, the recovered values of the average donor–acceptor distance obtained using the constant CD model are significantly higher. It can seem to be strange that a good resolution between the two models of FET is also possible in the case of AMCA-ZF29, despite of significant overlap of the confidence bands associated with respective simulated data files (Fig. 2). This is probably a result of the fact that the information about the kind of model is not only contained in the position of investigated decay curve, but also in its shape and slope, which for the two considered models can be sufficiently different. Summarizing this section, we can conclude that in the case of the two forms of a single-domain CCHH-type zinc finger peptide the proper qualification of adequate model of FET is very important from the point of view of the estimated values of the distance distribution and dynamics parameters. One can generalize this observation and suppose that the proper qualification of adequate model of FET is important in quantitative description of all systems where the donor molecules in the absence of acceptor are characterized by nonexponential fluorescence decay. We

In the article we have addressed the problem of Fo¨rster energy transfer in the case when fluorescence decay of donor molecules in the absence of acceptor is nonexponential. Because of many different mechanisms potentially causing the nonexponentiality of the donor decay, no general solution of the problem is possible. In our final expressions we have limited ourselves to the systems that seem to be most frequent, where fluorescence of all donor molecules can be described with the same constant value of the radiative rate CD and varying values of the nonradiative rate kDnr. We have found that under such conditions the transfer rate kDA should be expressed by Eq. (16), which relates to the concentration averaged fluorescence lifetime of the donor instead to single, immediately measured lifetime present in the original formula derived by Fo¨rster. We also have shown that the discussed problem is crucial because for real systems in which the donor fluorescence decay appears to be nonexponential the different assumed forms of the rate kDA can introduce significant differences to the theoretically predicted courses of donor fluorescence decay in the presence of the acceptor. Appendix A The preexponential factors aDi can be understood as ð0Þ being proportional to the initial populations N Di of the excited molecules belonging to respective components of the donor, multiplied by respective rates, CDi, of radiative decay. If constant CD model is assumed, then ð0Þ

aDi ¼ CN Di CD

ðA:1Þ

where C is a constant. Eq. (A.1) implies that the initial fracð0Þ tional concentrations xDi of the excited molecules belonging to respective components are given by ð0Þ

ð0Þ

xDi  N Di =

n X

ð0Þ

N Di ¼ aDi

ðA:2Þ

i¼1

Analogously, for continuous lifetime distribution one obtains Z 1 ð0Þ ð0Þ ð0Þ N D ðsD Þ ¼ aD ðsD Þ ðA:3Þ xD ðsD Þ  N D ðsD Þ= 0

A. Czuper et al. / Journal of Photochemistry and Photobiology B: Biology 87 (2007) 200–208

Appendix B If the rates of radiative decay are taken in the form of Eq. (18) then ð0Þ

aDi ¼ CN Di

QD sDi

ðB:1Þ

In this case one obtains n X ð0Þ xDi ¼ aDi sDi = aDi sDi ¼ aDi sDi =hsD i ¼ fDi

ðB:2Þ

i¼1

where fDi are the fractions of quanta emitted by particular components, defined as Z 1 Z 1X n aDi expðt=sDi Þdt= aDi expðt=sDi Þdt fDi ¼ 0

0

i¼1

ðB:3Þ Analogously, for continuous lifetime distribution one obtains Z 1 ð0Þ xD ðsD Þ ¼ aðsD ÞsD = aðsD ÞsD ¼ fD ðsD Þ ðB:4Þ 0

where fD ðsD Þ ¼

Z

1

aD ðsD Þ expðt=sD Þdt Z 1Z 1 = aD ðsD Þ expðt=sD ÞdsD dt 0

0

ðB:5Þ

0

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