Polarized fluorescence decay surface for a mixture of non-interacting species in solution

13 September 1996

CHEMICAL PHYSICS LETTERS ELSEVIER

Chemical Physics Letters 259 (1996) 579-587

Polarized fluorescence decay surface for a mixture of non-interacting species in solution J.J. Fisz Institute of Physics, N. Copernicus University, ul. Grudziqdzka 5f7, PL 87-100 Toruh, Poland

Received 29 February 1996; in final form 10 July 1996

Abstract

A treatment for analyzing the excitation and emission multiwavelength polarized fluorescence decay surfaces for a mixture of non-interacting species in solution, is discussed. The polarized fluorescence decays are parameterized in terms of speciesassociated total fluorescence decay parameters, diffusion tensors and the polar angles defining the directions of absorption and emission dipole moments. The case of excitation of the species to higher excited states is also considered. To the memory o f my dear daughter Karolina

I. Introduction

The decay of polarized fluorescence components after a t%pulse excitation are considered traditionally in the literature in the following form (see e.g. Refs. [ 1,2] and works cited therein): ill(t) = ½/tot(t)[1 + 2 r ( t ) ]

,

i x ( t ) = ½/tot(t)[1 - r ( t ) ] .

(1)

/tot(t) represents the total fluorescence decay, m

/tot(t) = ill(t) + 2 i x ( t ) = Z a i e x p

(--t/~'F,i) .

(2)

i

itot(t) can also be replaced by the fluorescence decay collected at the so-called "magic" angle 54.7 °, i.e. /tot(t) = 3 imag(t), r ( t ) in (1) represents the emission anisotropy decay [3], n

r(t) = ill(t)-i±(t)

=Zbjexp(--t/rR,j). ill(t) + 2 i ± ( t ) J

(3)

/tot(t) and r ( t ) are multi-exponential functions, in general, ai and ZF,i are the total fluorescence decay parameters, and bj and 7R,j are the emission anisotropy decay parameters. The multi-exponential time-course of the 0009-2614/96/$12.00 Copyright Q 1996 Elsevier Science B.V. All rights reserved. PII S0009-261 4(96)00814-7

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emission anisotropy r ( t ) in (3) results from a particular model for the rotational motion of solutes. In the case of asymmetric diffusors, the sum in (3) contains five terms, for symmetric diffusors only three terms and in the case of spherical diffusors the evolution is mono-exponential. However, from an experimental point of view, all five terms in the time-evolution of r(t) are hard to resolve as discussed in Ref. [4]. Relations (1) were also employed in the problem of fluorescence depolarization for a mixture of independent fluorophores or the same fluorophore with different environments, i.e. to the species-associated emission anisotropy decays problem [5]. In this problem one assumes that different ZF,i are associated with different n rR,j. For this reason (3) is replaced by r(t) = y']~j Yij bjexp (--t/zR,j) and Yij = 1 if ~'R,j is associated with "rF,j and Yij = 0 otherwise. Finally, the polarized fluorescence decays corresponding to the situation considered here can be rewritten as ([5,6])

ai exp (--t/e'F,i) [1 + 2

ill (t) = ~1~

Yijbj exp (--t/rR,j)

i m

i±(t) = ~l ~ - ~ a i e x p ( - - t / T F . i ) [ 1 i

,

(4)

j t7

~-~/ijbjexp(--t/~'R..i)]

(5)

j

The above relations are used to analyze the polarized fluorescence decays of a mixture of non-interacting species. The polarized fluorescence decays collected at different excitation and emission wavelengths are analyzed globally with the fluorescence decay parameters, ai and rF,i, and the emission anisotropy decay parameters, bj and ~'R.j, as the fitted parameters [5-9]. In this work we wish to present an alternative approach to the above-mentioned problem. The treatment discussed here is based on a direct definition of polarized fluorescence decays. For this reason we do not need to introduce the y;j parameters to associate particular correlation times with particular fluorescence decay times. This will be accounted for directly in the scheme for the global analysis of the fluorescence decay surface by appropriate linking of the fitted parameters. The expressions for polarized components of fluorescence decays will be parameterized in terms of species-associated fluorescence decay parameters, diffusion tensors and the polar angles describing the directions of absorption and emission transition dipole moments. We wish to consider a more general case, relating to many real experimental cases, by assuming that at a certain excitation wavelength each species can be excited to the lowest two or more excited states and that the corresponding absorption transition dipole moments (for each species) can be directed differently. In our description of the problem the parameters ai and bj are not independent fitted parameters because different bj can be coupled among themselves by the angles describing the directions of absorption and emission dipole moments. Furthermore, they are also correlated with some of the a i parameters when a given species (or more of them) can be excited simultaneously to two or more excited states and the absorption transition dipole moments to these states have different directions. Also the decay times 7"R,j are not treated in our approach as independent parameters. They are coupled by the same components of the diffusion tensor. Note that for a symmetric rotor the decay of emission anisotropy contains three exponential terms with three different TR,j'S. In our case these three correlation times are expressed by the same two components of the diffusion tensor (we assume the symmetric diffusor model). It will be shown that our relations and the algorithm for linking particular fluorescence lifetimes with particular correlation times become equivalent to the method based on relations (4) and (5), for spherical diffusors (one correlation time for rotational motion for a single species) and when each species can be excited only to one excited state (a single absorption band or more absorption bands having the same directions of absorption transition dipole moments). It seems, therefore, that the approach discussed in this work can be useful for more precise analysis of the excitation and emission multiwavelength fluorescence decay surfaces for a mixture of the species non-interacting in the excited states.

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2. Excitation and emission multiwavelength polarized fluorescence decay surfaces We consider the situation shown in Scheme 1. The sample is composed of a greater number of the species non-interacting in the excited states (the species do not interconvert in the excited states and there is no excitation energy transfer in the system). It is assumed that each species can be excited at a given excitation wavelength to a greater number of higher excited states (the lowest two or more excited states) and that the absorption transition dipole moments are oriented differently in the molecule fixed frame. One assumes that the population density of the excited-state species (from the lowest excited state) decays with rate constant k( for a given sC'th species and that its rotational dynamics are described by the diffusion tensor D ((). It is also assumed that the direction of the emission dipole moment has a different direction with respect to the absorption dipoles.

,I

N** I

I

,'~ob:

' k'

P-ab

k

1"~' •

. .

~t I I I

ab

,hi* ~tIN~) ab N

Scheme 1

The intensity of the polarized fluorescence decay corresponding to the situation displayed in Scheme 1 can be written as N

l~i&(t'Jex'hem) =C(aex'Aem) Z

Z

Og('s('h'ex) K((/~em)

(=1 s=a,b....

x

~

Pexgs (S'20) p((S2o, 0Is'2, t) ,i°(ef) t')~em,(,.~, [ d~O0 dO.

(6)

noo C(flex, Aem) includes all instrumental excitation and emission wavelength-dependent factors, w(,s(aex) and K((aem) represent the absorption band of species sX'th to the excited state s at a~, and the emission band from (&) the lowest excited state for the same species at &m, respectively. --,.-P~(O°) and __P,~m,¢(/2)describe the angular dependence of the polarized excitation (absorption) and emission probabilities, respectively, and where 2 Pe(~]~(S2o)

3 COSZ(0i,/2ab,fs)

=

1+2

(&) Pem,((/2) "~3 cos2(ef,/.~em,() = 1 + 2

Z C2.p,(Oi'~°i)Z pl=--2 ql

P"q" (J~O) ~'2,ql k UA ' ~A ) ,

2

~ p2=-2

C2.p2(Of,@f) Z D

(2) ( / 2 ) t ~L'Z,q2 * [~(() P2,q2 ~ VE

~(~:)) • , WE

(7)

q2

( e i , [-~ab,(s) means the angle between the unit vector &i, representing the polarization direction of the exciting light, and the absorption dipole moment direction /2ab,¢s. (&f,/2emg) means the same for the direction of polarization of the detected light, 0f, and the direction of the emission dipole moment, /~ern,~:. The polar angles (0i, q~i) and (0f, q~f) define the orientations, in the laboratory frame, of Oi and Of, respectively, t"VA"(~:s), ~OA(~S))" and

582

J.J.

F i s z / C h e m i c a l P h y s i c s Letters 2 5 9 ( 1 9 9 6 ) 5 7 9 - 5 8 7

(0(E() , ~O(E ~) ) describe the orientations of the absorption and emission transition dipole moments in the moleculefixed frame, respectively. Cj,m(O,~o) are the modified spherical harmonics [10], Cj,m(O,~o) = [47r/(2j + 1 ) ] 1/2 Yj,m(O, ~p), where these latter are spherical harmonics. P((/20, 01/2, t) is the Green's function describing the conditional probability of finding an excited molecule at time t at angular orientation/2 (Euler angles transforming the laboratory-fixed frame to the molecule-fixed one), which was excited at t = 0 and its initial angular orientation was/20. The Green's function p((Oo, 01/2, t) fulfills the equation of motion 0

(8)

ot pe(/2o, 01/2, t) = - (:/(~) + k~) pe(/2o,0l/2, t ) ,

where /:/(()is the time-development operator and fI(() = Y'~=l D} ~) L]. The operators Li are identical to the quantum mechanical orbital momentum operators (i = x, y, z) and D} ~') are the components of the diffusion tensor. For the symmetric rotor Dx(~9 = D~~,) = D(_[') , D~ () = Dlt'), and for the spherical rotor Dx(~') = D},~) = D~) = D(~). The propagator p~(/20,0[/2, t) can be split into a purely dynamic part and a purely kinetic one, i.e. p¢(Oo,01/2, t) = p~(Oo,01/2, t) exp(-kCt). The solution to the equation of motion (8) for the Green's functions p}(/2o, 0]/2, t) (after neglecting k¢) in known in the literature [ 11 ] :

p~(/2o,Ol/2,t) = ~

2j87r +2 1 E

,

(9)

Cj,,,,j,m,n,(t )(¢) = e x p { - [j(j -I- 1 ) D ~ ) + n2(Dl') - D~))] t}~jmn,j'm'n' •

(lO)

jmn

r(()

~jmn,j'm'n t

(t) D mtn (j')' (12o) D(J) --ran * ( ~

~ - - :

j ' ml n ~

where the explicit form of the expansion coefficients is

The expansion coefficients for the Green's function p~ (Oo, 0]/2, t) can be easily obtained by multiplying the expansion coefficients for p~ ( 12o, 01/2, t) by exp ( - k(t), and the expansion of pt: (/2o, 01/2, t) in the basis set of Wigner rotation matrices is given by (9) with these new expansion coefficients. ( I"~ ~ (ef) After inserting into (6) the explicit forms for. tD(~i) ex,#s'"~:, Pem,~(/2) and the Green's function Pt,(/2o, 01/2, t), performing the integrations over 1"2oand /2 and after employing the orthogonality properties for the Wigner matrices [10], the polarized fluorescence decay surface (6) reads N I~i &(t, ~-'ex, '~-em) ----C(,'~ex, ~em) E O')~'E(~'ex) K~e(~'em) [ exp ( - k £ t) -t- 34 P2(0if) ~(~) (t) ] , (=1

(11)

where 0if is the angle between ei and ~f, and where m~,:~(Aex)= E

m(,s(Aex),

s=-a,b,.,

~b(~O(t)= E R(q¢)(t) E C~'s(Aex) T(q¢'s)" q--0,1,2 s--a,b....

(12)

ct',s(Aex) (s = a,b . . . . ) are the relative spectral contribution coefficients of the absorption bands to higher excited states for the ~'th species, where o)~,s(&x)

C~,s(Aex) = ~--]s=a,b,...m(,s(Aex) ' and where Y-]~-a,a,... C~.s(Aex) = I.

(13)

J.J.

583

F i s z / C h e m i c a l Physics Letters 259 (1996) 5 7 9 - 5 8 7

The correlation functions R(q¢)(t) and the functions describing the orientations of absorption and emission dipole moments T(q¢'~) are given by

R(o~)(t)

= exp[-(6D~)+

k() t],

T~o~,') = p2(o~ ~,~ ) P2(O~)),

R(,e)(t)

= exp[-(50~)+

Oll()+ k ¢ ) t ] ,

7"(1("s)

3 sin 20(A~'.s)sin20(~)COS(q~(A~,'s)

T(J ,s)

3 sin20~ ,s) sin20(E~') cos2(~o(A~,,s'

R(J ) (t) = e x p [ - ( 2 D ~ ) + 4DII~:) + k~:) t],

p(E~,)) ~p(E~)) (14)

For a spherical diffusor R(o()(t) = Rl~)(t) ~b~) (t) in (11) has to be replaced by ~bi()(t) = exp[-(6D~p~h)

+ k()t]

Z

= R(o~)(t) = "sph o~()(t)

= exp[-(6D~p~ + k~,) t]. Consequently,

C(,s(Aex)P2(O(A~S)),

(15)

s=a,b ....

where VAEa(¢'s)is the angle between absorption dipole moment to an excited state s and the emission dipole moment for species (. The parallel and perpendicular components of the fluorescence decay are obtained from the general expression (11) after substituting 0if = 0 ° (III) and 0if = 90 ° ( I L ) , i.e. N

lll(_l_)(t, Aex, Aem) =C(Aex,Aem) EO),,l~(Aex)K,(Aem)[exp(-k(t)-t-4(-2)~p(~()(t)]

.

(16)

When the fluorescence decay is collected at 0if = 54.7 ° (the so-called "magic" angle), the measured fluorescence decay corresponds to N lmag(t, Aex, '~em) -- C (,~ex, ,'~em) ~ w~:,E(,h.ex) (=1

K((~-em) exp ( - k ( t ) ,

(17)

and its time-evolution has a purely kinetic character. From the polarized components of the fluorescence decay the time-dependent excitation and/or emission multiwavelength emission anisotropy surface can be reconstructed, i.e. r(t, Aex, Aem) =0.4

}--~e ¢'O~,'~('~ex) K~'(aem) ~(~:) (t) EsC,k (.O(,£(~.ex) Kse(,,~em) e x p ( - k ( t )

From the relation - 0 . 2 < r0 < 0.4, where r0 = obtained: -:<2- Z Z q=0,1,2 s=a,b ....

c¢,s(&x) T(q¢'')<1-

,

r(t

-:<2-

(18)

= 0, aex, Aem), the following constraints can be easily

Z

C(,s(&x) P2(Oi~S))
(19)

s=a,b ....

for symmetric and spherical diffusors, respectively. 3. Global analysis of the fluorescence decay surface

The components of polarized fluorescence and the decay collected at the "magic" angle can be rewritten in the form

584

J.J. Fisz/Chemical Physics Letters 259 (1996) 579-587

~-~ [af,s(~ex, ,~em)exp(--//~'F,()+ 4(--2) ~-~af, s( Aex~em) T(q"s) e x p ( - t / T f , q)] ,

l[l(_k)(t,,~ex,Aem) = ~ (

s

q

(20) and

s where 1/~'F4, = k~,, 1/rf, q = 6 D ~ ) + q2(Dlf) - D ~ )) + k¢ and af, s(Aex, ,~em) = C(Aex, Aem) o.)f,s(Aex) Kg(Aem). Note that 1/rf, q can be replaced by l/yf, q = 1/q'F,( + 1/YR,f.q, where rR4',q are the correlation times for the rotational motion of the molecules and are expressed by the components of the diffusion tensor alone. In the analyses of the measured polarized and polarization-free fluorescence decay surfaces, the decays l(meas)(t ,"~ex, Aem), /(meas)(t, Aex, Aem) and l(meas)tt ~ "mag ~....~ ex,-,emJ are the subject of a simultaneous analysis "it ~ "' "Jperformed globally for different excitation and emission wavelengths. Because the fluorescence decay times ~'F,~ and the correlation times for the rotational motion rR,~,,q can correspond to different timescales, the same excitation and emission multiwavelength measurements should be repeated for different two or more time windows of the instrument. All these decay files should be analyzed globally with the same set of fitted parameters linked correspondingly. In the first step of the experimental data analysis, the following set of fitted parameters can be assumed: af, s(,~ex,,~em), 7"F,f, 7(,q (or 7"R,(,q) and T(q('s). The preliminary information about the system under study obtained from this algorithm can be useful in the analysis of the same files, in which the fitted parameters are: a¢,s(Aex, Aem), rF,¢ (or k(), DIe), D ~ ~, 0(A~'s~, 0(E¢~ and ~O(A ¢'s~--~O(E~ . Note that from the amplitudes a(,s(aex, Aem) the time-resolved excitation and emission multiwavelength fluorescence surfaces can be decomposed into the contributions from different species, i.e. by calculating the coefficients a(,s (Aex, Aem) C¢,s(Aex) =

~saf.s(aex, Aem )

_

¢Os(aex) }--~sW(.s(Aex) ,

a~,,~( aex, aem) w,f,s (Aex) K((Aem) df(Aex, Aem) = Y'~f,s ag:,s(hex, ~-em) = ~-~es ¢'°es(~'ex) K~e(~.ern)

(22)

(23)

In the global analysis of the fluorescence decays the components of the diffusion constants and the polar angles defining the directions of the absorption and emission dipole moments are linked for different excitation and emission wavelengths for each species. Along the emission wavelength axis all species having different directions of emission dipole moments will contribute to the polarization effect differently, even if their rotational dynamics differ slightly. Also the opposed situation can differentiate the species. The same concerns the excitation wavelength axis. In this case, however, if the species possess two or more absorption bands with a different direction of the transition dipole moments, a change of the excitation wavelength will change the polarization effects (a contribution of two or more polarizations in the absorption process) for a single species. This can be recognized from the approach discussed in this work. During the analysis of the data the constraints (19), implemented into the global analysis program, can considerably improve the reliability of the analysis procedure. The linking scheme for the parameters in the case of the two-species problem is shown in Scheme 2. It is assumed that the polarized fluorescence experiments were performed for N different excitation wavelengths and the polarized fluorescence was collected for M different emission wavelengths. In Scheme 2 the decays collected at the "magic" angle and the polarized decays are analyzed with the proper number of parameters (not all fitted parameters contribute to the isotropic decays; compare case (A) with (B) and (C) with ( D ) ) . The blocks indicated in Scheme 2 mean that for different excitation and/or emission wavelengths the fitted parameters are physically indistinguishable and they refer to the same logical numbers, correspondingly. These

J.J. Fisz/ Chemical Physics Letters 259 (1996) 579-587

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blocks represent the linking scheme for the adjustable parameters over all experimental files being the subject of a simultaneous numerical analysis. The blocks in Scheme 2 which are denoted by the same names of the fitted parameters also correspond to the same logical number of the fitted parameter. In a numerical program Scheme 2 is represented by a matrix with logical numbers identifying the independent fitted parameters, for each fluorescence decay detected. In Scheme 2 is shown an example of the linkage matrix for the fitted parameters in the case of two emission and three excitation wavelengths. It is assumed that for all six pairs of (Aex, Aem) both polarized fluorescence decays were collected (i.e. to each row in the linkage matrix there correspond two polarized fluorescence decays). The fitted parameters which are the same for all the decays are described by the same logical number, while to all those parameters which are excitation or emission wavelength-dependent, different logical numbers are assigned (e.g. for the amplitudes a~,s(Aex,/~em) different logical numbers are assigned as in the example shown in Scheme 2). 6ag(~-~,~-~m,t)

I.(~,ex,~,e~,,t) and K(X~.,K~,,t)

I~.,(~.e.. x,,~. t)

I,,(~e..~.~,t) and

I L(z...,~.~m,t)

Example ~

~

1

2 3 4

5

6

7

8

9

10

11 12 13 14

15

16

17

18

19

20

1

2 3 4

21

6

7

22

9

10

11 12 13 14

23

16

17

24

19

20

1

2 3 4

25

6

7

26

9

10

11 12 13 14

27

16

17

28

19

20

1

2 3 4

29

6

7

30

9

10

11 12 13 14

31

16

17

32

19

20

1

2 3 4

33

6

7

34

9

10

11 12 13 14

35

16

17

36

19

20

1

2 3 4

37

6

7

38

9

10

11 12 13 14

39

16

17

40

19

20

!

Scheme 2 The globally and locally adjustable parameters can be distinguished by the elements of another matrix of the same dimension as the linkage matrix, with the elements taking one of the following values: - 1, 0 and 1. If a given parameter is adjustable for a given fluorescence decay, the value 1 is assigned to this parameter. If this parameter is kept constant, the value - 1 is assigned. Value 0 corresponds to a situation when a given parameter is neglected in the calculations (e.g. the diffusion coefficients and polar angles describing the directions of absorption and emission dipole moments for isotropic fluorescence decays). The structure of the linkage matrix shown in Scheme 2 can be further extended by accounting for the decays collected at the same excitation and/or emission wavelengths but for different time windows of the instrument.

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J.J. Fisz/Chemical Physics Letters 259 (1996) 579-587

The advantage of using the components of the diffusion tensors as the fitted parameters, instead of the correlation times 7"R,j which are the fitted parameters in the method based on relations (1) or (4) and (5), can be displayed in the following example for a two-species case. In this case, in the matrix of logical numbers linking the fitted parameters and associating the fluorescence decay times with appropriate correlation times for rotational motion, a sequence of the logical numbers for the following sequence of the fitted parameters r)(1) ''-'-tr)(1) ,kl ,D~I2),D~2),k2] can be [ 1 , 2 , 3 , 4 , 5 , 6 ] (case A), which means that the system is composed of ~11 two species of different rotational mobility and two fluorescence lifetimes associated to both diffusion tensors. A sequence [ 1,2, 3, 1,2,4] (case B) means that both species are indistinguishable from the point of view of their rotational dynamics, but they are differentiated by the fluorescence lifetimes. These species can be distinguished by the polarizations of absorption and emission dipole moments. The sequence [ l, 2, 3, 4, 5, 3] (case C) assumes that both species have similar (indistinguishable) fluorescence decays but differ in the rotational dynamics. The same cases considered in terms of the parameterization in relations (1) or (4) and (5) would need the following sequence of the fitted parameters [7"R,I,7"R,2,7"R,3,TFA, 7"R,4,7"R,5,TR,6]. Note that this parameterization introduces two additional fitted parameters as compared to our case. The sequence of the values of Yi,j, associating rF, i with 7"a,j if Yi,j = 1 ( omitting the Yi,j = 0 values), and corresponding to our case A reads: [~/l,l, '~1,2, ~/13,')/2,4, ~/2,5, ~2,6] However, this sequence can also mean that although both relaxation times are distinguished as in our case, from the point of view of rotational dynamics one can conclude that there are six species each rotating with a single correlation time (spherical diffusors) or any other possibilities involving symmetric and spherical rotor models. Note that the pre-exponential coefficients bj are also independently fitted parameters and that in the case considered here they will be grouped into two sets each of three bj's. Therefore, they cannot help in a more precise identification of the correlation times rR,j. This problem is evidently clear if the absorption and emission dipole moments have similar directions in both species and when the absorption and emission bands differ slightly for both species. The same possibility can also be considered for the cases B and C. Both approaches become more similar to each other for spherical rotors. In our parameterization the sequence of logical numbers for two species case can be: (a) [ 1 , 1 , 2 , 3 , 3 , 4 ] (two spherical rotors with different fluorescence lifetimes), (b) [ 1,1,2, 1, 1,3] (two spherical rotors rotating with the same correlation times but of different fluorescence lifetimes) and (c) [ l, l, 2, 3, 3, 2] (two spherical rotors with different correlation times but having similar fluorescence lifetimes). In the parameterization used in relations ( l ) or (4) and (5) the matrix elements Yi,j will correspond to the sequence of the correlation and fluorescence decay times [7"R,1, TF.1,TR,2,9"F,2]. The sequences of the ")/i,j = 1 corresponding to cases (a), (b) and (c) are: (a) [Tin, Y2,2], (b) [Yl,l,')/2,1 ] and (c) [yl,l,yl,2]. However, in the case when the species can be excited to higher excited states and the corresponding absorption dipole moments have different orientations, our approach enables for the proper identification of such cases, even for spherical rotors. The case of two or more species exhibiting multi-exponential photophysics can also be considered within Scheme 2. The multi-exponential photophysics of a given fluorophore can be treated as a case of a sub-population of the fluorophores that may have indistinguishable (or slightly different) dynamic and spectroscopic properties. In such cases, an appropriately greater number of species in Scheme 2 has to be considered with the same components of the diffusion tensors (if they do not differ in the rotational dynamics) and the same orientations of the absorption end emission dipole moments (if they do not differ also from this point of view). Also the excitation- and emission-wavelength dependent coefficients may be the same for these species if they exhibit the same (or slightly different ) absorption and emission bands. By considering different linking schemes for the fitted parameters one can try to identify each exponential constituent decay in the multi-exponential photophysics by trying to distinguish them from the point of view of the species-associated dynamics, absorption and emission spectra as well as from the point of view of the species-associated directions of the absorption and emission transition dipole moments.

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The a l g o r i t h m discussed in this work will be the subject o f studies based on the analyses of the c o m p u t e r generated synthetic fluorescence decays. The results o f these studies will be described elsewhere.

Acknowledgement This work was supported by the C o m m i t t e e for Scientific Research ( K B N , P o l a n d ) within the Project 2.P303.089.04.

References [ 11 D.V. O'Connor and D. Phillips, Time-resolved single photon counting (Academic Press, New York, 1984). [21 R.B. Cundall and R.E. Dale, eds., Time-resolved fluorescence spectroscopy in biochemistry and biology (Plenum, New York, 1983). [31 A. Jablofiski, Bull. Acad. Pol. Sci., Math. Astron. Phys. 8 (1960) 259. [41 E. Small and I. Eisenberg, Biopolymers 16 (1977) 1907. [5] J.M. Beechem and L. Brand, Photochem. Photobiol. 44 (1986) 323. [6] J.R. Knutson, L. Davenport and L. Brand, Biochemistry 25 (1986) 1805. [7] L. Brand, J.R. Knutson, L. Davenport, J.M. Beechem, R.E. Dale, D.G. Walbridge and A.A. Kowalczyk, in: Spectroscopy and the dynamics of molecular biological systems (Academic Press, London, 1985). [8] J.-E. l.dSfroth, Eur. Biophys. J. 13 (1985) 45. [9] M. Crutzen, M. Ameloot, N. Boens, R.M. Negri and EC. De Schryver, J. Phys. Chem. 97 (1993) 8133. [ 101 D.M. Brink and G.R. Satchler, Angular momentum (Oxford Univ. Press, Oxford, 1968). [ 111 B.J. Bern and R. Pecora, Dynamic light scattering with application to chemistry, biology and physics (Wiley, New York, 1976).