J. Phys. Chem. 1995, 99, 2353-2357
2353
Relations among Steady State, Time Domain, and Frequency Domain Fluorescence Quenching Rates. 2 Andrzej Molski Institute of Theoretical Dynamics and Department of Chemistry, University of Califomia, Davis, Califomia 95616, and Adam Mickiewicz University, Department of Physical Chemistry, Grunwaldzka 6, 60-780 Poznali, Polanddf Received: August IO, 1994; In Final Form: November 30, 1994@
A nonequilibrium statistical thermodynamic approach (Molski, A.; Keizer, J. J. Phys. Chem. 1993, 97, 8707) developed to study relations among rate coefficients for steady state, time domain, and frequency domain fluorescence quenching is extended to the case where static quenching is present. An exact, model-independent expression is derived for the mean-field, frequency-dependent rate coefficient k"f in the linear harmonic regime. This expression is a rigorous formulation of the relationship whose approximate form was derived in our previous work. It is argued that the relations between various quenching rate coefficients are analogous with those valid in the absence of static quenching. Treatments of fluorescence quenching kinetics by Sung et al. and by Zhou and, Szabo are briefly commented upon.
I. Introduction In this article we extend our previous results' on the kinetics of fluorescence quenching to include the effect of static quenching?-4 Thus, in addition to the simplest kinetic scheme?-' K(t)
F-F*
tr 1 F*-F
F*
+ Q-F + Q
(1.4
we allow for the ground state, reversible complex formation rea~tionz-~
F
+ Q ==
kcf
kcd
FQ, K, = kcJkcd
(1.e)
In system 1, the symbols ,z and zf represent the nonradiative and radiative lifetimes of excited fluorophore molecules F*, k, is the bimolecular quenching rate constant, and the rate constants for complex formation and dissociation, kCf and ked, determine the complex formation equilibrium constant K,. As in ref 1, we assume diffusive motion of spherical solute molecules with the diffusion coefficients DF, DF*, and DQ. Elementary quenching events 1.d occur upon F*Q collisions at the reaction distance R = RF* iRQ. The quenching rate is determined by the intrinsic bimolecular rate constant k2. Typical experimental situations correspond to
where @F* and @ F are the concentrations of the excited and ground state fluorophore molecules, and @ Q is the quencher concentration. Accordingly, we assume that the macroscopic + Permanent and correspondence address. @Abstractpublished in Advance ACS Abstracts, February 1, 1995.
concentrations @F and @ Q are practically constant and confine ourselves to a study of pseudo-first-order quenching kinetics. When the distribution of relative F*Q distances is an equilibrium one, the quenching constant k, in 1.d is equal to k2. However, we are interested in nonequilibrium diffusional effects when the formal kinetic description in terms of a rate constant is no longer adequate. In such a situation it is useful to introduce an appropriately defined molecular rate coefficient k(t) (see next section). Recently, Szabo7 developed a formula relating the rate coefficient P s for steady state fluorescence quenching to the Laplace transform P ( z ) of the time-dependent rate coefficient ko(t) for quenching after a short (6) excitatio! pulse. Then, Zhou and Szabo8 introduced a relation between k"(z) and the meanfield rate coefficient kd describing frequency domain kinetics. In ref 1, Szabo's formula was derived using the methods of nonequilibrium statistical thermodynamics. However, the formula we obtained there for kd was slightly different than that given by Zhou and Szabo in ref 8. We note that our formula was only approximate and not an exact consequence of the statistical thermodynamic formalism. Very recently, Sung et al.9 showed how the results of refs 1 and 7 can be obtained using their Green function technique. Moreover, they produced an explicit theory for kd in the case of the SmoluchowskiCollins-Kimball model. In the present paper we extend our previous work in two directions. First, we derive an exact formula linking kd and @(z) in the linear harmonic regime. This formula follows exactly from ow formalism, and our previous result in ref 1 can be regarded as a useful approximation. We also discuss the relationships between the various approaches and show that our treatment of frequency domain fluorescence quenching is consistent with that by Sung et al? In fact their results for the Smoluchowski-Collins-Kimball reactivity model are demonstrated to be a special case of our formalism. Second, we allow for the possibility of static quenching. We argue that the formation of a ground state complex does not change the relations between various rate coefficients Ps,ko(t), and kd, so that the contribution to the positive deviation of the StemVolmer plot coming from static quenching is purely stoichiometric, i.e. not coupled to the nonequilibrium effects on Ps.
0022-365419512099-2353$09.00/0 0 1995 American Chemical Society
Molski
2354 J. Phys. Chem., Vol. 99, No. 8, 1995 11. Formal Kinetic Treatment and Its Extension The formal kinetics analysis of system 1 based on the rate equation for @F* is
where tois the lifetime of the fluorophore in the absence of the quencher. ti1 = t , '
+ t;'
where the rate V gives the number of quenching events per unit volume per unit time. The rate equation (3) becomes
The steady state molecular quenching coefficient is denoted kss, and now the Stem-Volmer equation (6) is 10
I = (1
(4)
The excitation rate K(t) is proportional to the macroscopic concentration, @F, of the ground state fluorophore so that
Note that eq 13 and 14 are formally exact, but we still need to evaluate k(t) and kss. For oscillatory input (7), one has' k(t) = kss
(5) where @(tj is the excitation rate in the absence of complex formation 1.e (i.e. when K, = 0 or @Q = 0). Thus, at steady state when K(tj = Kss = const, the ratio of the intensity in the absence ( l o ) and in the presence ( I ) of the quencher becomes
'OI = ( l + @QkqT,)(l+
+ @QkssTo)(l+ Kc@,)
Kc@Q>
(6)
where the second factor on the rhs modifies the well-known Stem-Volmer equation to accommodate reaction 1.e. This factor leads to a positive deviation of the Stem-Volmer plot, which becomes parabolic rather than linear. When the illumination is oscillatory,'0-'2
+
K(t) = F S AK exp(iwt)
(7)
so is the concentration @F*(tj,
+ Ak exp(iwt)
(15)
which reflects the fact that nonequilibrium spatial effects depend on the input rate K(t). Note, however, that the intrinsic bimolecular rate constant k2 does not oscillate. When working in the frequency domain, it is useful to introduce the meanfield, frequency-dependent rate coefficient kmf defined via the relation'
+ + ~ ~ k & ( w ) ) - '(16)
J ( w ) = AeF,/AK = (io
Introducing expression 15 into the rate equation (13) and linearizing, we get an alternate definition of the mean-field coefficient,
which is valid in the linear, harmonic regime. We note that, unlike in ref 1, here we do not approximate k" any further and use eq 17 as the working definition of kmf.
111. Rate Coefficients ko, kss, and kmf where o in the angular frequency. Using eq 3, we find
J ( w ) = AeF,/AK = (io4-
TO'
eQkq)-'
(9)
In frequency domain fluorometry, one measures the modulation M ( o ) and the phase angle @(oj, which are related to J ( o ) by
In the statistical thermodynamic approach, the description of the bimolecular quenching process 1.d is given in terms of the radial distribution function g.14-16 In a uniform system, the radial distribution function depends on the relative distance r = Iri = lr' - r"l. For quenching occurring at the distance R = RF* RQ, the relation
+
k(t) = kzg(RJ) and
where (zI, Re(z), and Im(z) are the absolute value and the real and imaginary parts of the complex number z, respectively. In the present paper we study frequency domain kinetics in the linear harmonic regime; that is, we assume that the quantities AK and heF*do not depend on time and that they are small so that we can linearize the corresponding evolution equations. Now we modify the formal kinetic description of system 1 to accommodate nonequilibrium spatial effects.l4-I6 To this end we replace the quenching constant kq in the above formulas with the molecular quenching rate coefficient k(tj defined through the rate V(t)of the elementary quenching events,
(18)
links the molecular rate coefficient k(t) and the radial distribution function g. In order to derive an evolution equation for the radial distribution function, we rewrite system 1 as 0
K(t)
F*
(19.a)
r iI
F* -0
F*
+ Q -40 + Q
(19.b)
(19.c)
where the zeros replace the species whose spatial correlations with F*'s can be ignored. This is made possible by the disparity in the concentration as indicated by the inequalities 2. Using the principles of the statistical thermodynamic approach,14-16 one can derive the following equation satisfied by the radial distribution function (for details see the Appendix of ref 1j:
Fluorescence Quenching Rates
J. Phys. Chem., Vol. 99, No. 8, 1995 2355
+
where the relative diffusion coefficient D = Dp DQ. The set of equations 5, 13, 18, and 20 determine the evolution of the system. We have cast this system in the form analogous to that found in the absence of static quenching' to show that complex formation 1.e has a quantitative rather than qualitative effect on nonequilibrium spatial effects reflected in the radial distribution function g . Thus, under standard experimental conditions as expressed in eq 2, one can expect that the relations between the rate coefficients do not depend on whether a ground state complex is formed. Below, we show this explicitly, modifying our approach as compared with ref 1. This enables us to derive an exact Laplace transform relationship for the frequency domain coefficient in the linear harmonic regime. First, however, we need to recall the definition of the time domain (6 pulsed) quenching coefficient k", which is a quantity of fundamental importance in our treatment. The rate coefficient k" is determined by the following evolution equation for g combined with relation 18: 0
!k = DV2go - ko(t) 6(r) at where the superscript 0 is used to indicate the initial condition go(r,O) = 1. The Laplace transform of this equation can be written as D V ~ ~ O (Z >zgo(z)
+ 1 - io(z) 6(r) = o
+ gss- io(zlss) 6(r) = 0 (23)
Thus, both go and go(ss) describe fluorescence decay when no new excited fluorophores are produced, K ( t ) = 0. The difference between the radial distribution functions go and go(ss) results from different initial spatial distributions of reactants, created either by a 6 pulse (go) or by a steady state illumination which is suddenly turned off (go(ss)). As we will shortly see, these two radial distribution functions, as well as the corresponding molecular rate coefficients k" and ~ " ( s s ) , are interrelated. A. Relation between kSSand ko(z). At a steady state we have from eq 20
DV2gss - z*gss
go(r,t) = Jdr' S ( r , t l J ) go(r',O), gO(r,t)ss)= l d r ' S(r,tlr') g"(r') (27) The function S(r,tlr') can be interpreted as a survival probability density. Using eq 25 and the Markov property of the transition function S
S ( r , t l J ) = Jdr" S(r,t - t"1r") S(J',t''Ir')
(28)
we find
which after differentiating with respect to time, Laplace transforming, and rearranging gives
and the corresponding relationship for the molecular rate coefficients
(22)
We find it useful to introduce a closely related quantity, gO(r,tlss),which solves eq 21 with the initial condition gO(r,O1ss) = gss corresponding to a steady state. A Laplace transform equation for gO(r,tlss)is
DV2&o(zlss)- zgo(zlss)
Volmer plot is due to (i) purely stoichiometriceffects of reaction 1.e and (ii) nonequilibrium diffusional effects expressed via kss. Using formula 25 we can link the radial distribution functions go and go(ss). First, we introduce the transition function S(r,tlr') related to go and go(ss) as
+ z* - kss 6(r) = 0, z* = ti*+ eQkSs (24)
kss - z*i0(z) i0(ZlSS) =
z - z*
(3 1)
To the best of the author's knowledge formula 30 linking evolution of the system for the two different initial conditions, go(0) = 1 and go(Olss) = gss, has not been presented in the literature before. B. Relation between kd(w) and &z). We begin by noting that the oscillatory illumination 7 causes oscillations not only of the concentration @F*(t)but also of the radial distribution function
g(r,t) = g"(r)
+ A g ( r ) exp(iwt)
(32)
which implies the following relation between Ak and A g :
A k = k,Ag(R)
(33)
It is useful to introduce the mean-field distribution function gmf, (34)
It follows from a comparison of eqs 22 and 24 that the ratio gss/z* is equal to ko(z*), so that which is related to the mean-field coefficient kmfas
gss(r)= z*jo(r,z*)
(25)
kmf = k2gd(R)
Specializing eq 25 to r = R and recalling eq 18, we find
kss = z*ko(z*)
(26)
Equation 26 was first derived by Szabo' and rederived in ref 1 for the case with no static quenching. However, the fact that eq 26 is also valid in the present case should not be taken to imply that the macroscopic characteristics of fluorescence quenching are not affected by reaction (1.e). To see this, we recall eq 14, which shows that the positive curvature of a Stem-
(35)
Now we substitute eqs 15 and 32 in eq 20 and linearize to obtain iwAg = D V 2 A g - z*Ag -
Combining eqs 24 and 36 according to definition 34, we get
Molski
2356 J. Phys. Chem., Vol. 99, No. 8, 1995 DV2gmf- ( i o
+ z*)gmf + (lw -k ti' + eQk&) QQ(k& - kSS)gSS - kd d(r) = 0 (37)
Now we multiply eq 22 by the factor iW
explicit formula for the ratio J ( w ) = A@F*/AKdetermining the modulation and the phase difference of a frequency domain e~periment:~ J-'(w) = iw
+ 'Ti'+ @Qk&(U)
+ ti' + QQkmf(w)= z* + iw ( 1 - 4 n : y y y
(43)
and eq 23 by the factor where
-QQ(kd(w) - kss) and sum them up. Comparing the result with eq 37, one finds
k m f ( o )= [(iw
+ ti' + e Q k d ( w ) ) i O ( z )-
M, =
k2
k2
k,
+ (1 + W)k,' M1= k2 + (1 + alR)kD' k D = 4 n R D (44)
@Q(k,(w) - kSS)i0(ZISS)l,=iw+~~-,_oykrr (38) and and after rearranging
k m f ( w )= [zi0(z)
= (z*/D)li2, a, = [(iw
+ QQ(kmf(w)- kss)(ko(z)io(Z/SS))lz=iw+z*, Z* = ti1
+
zio(z) - kss
, z = io i o - eQ(zio(z) - kss)
+ z*
(40)
The result of ref 1 can be recovered if the second term on the rhs of eq 39 is dropped. This can be a good approximation that becomes exact at low quencher concentrations (see the Comments section). C. Real Time Analysis. An analysis of picosecond fluorescence quenching experiments requires a solution of evolution equation 20, where full account is taken of the time dependence of the excitation K(t). This can be accomplished numerically.2 For practical applications, however, it is useful to deal with eq 20 transformed into an integral rather than differential form. Following Szabo,17one can use the change of variables cF*(r,t) = e ~ * ( g(r,t) t) to transform eq 20 into a linear equation which can be formally solved to give the following integral equation: k(t) gF*(t)= J f K(f - t)exp(-LL$r;'
+
@Qk('$))d'$)ko(Z) dz (4 which is coupled with the rate equation 13. Here, we do not pursue the problem of applications of eq 41, but only note that this equation is consistent with eqs 26 and 39. To see this, we substitute eqs 7, 8, and 15 in eq 41 and linearize to get eq 26 and the following relation: kss
+
Ak -eFs A@,*
ss
-AK L-exp(-t(iw
-Fs Ak
A@F*
A@F*
J-exp(-zz*)
ko(t)
-1
(45) (39)
Equation 39 is the main result of this section. It can be used in conjunctio? with eq 31 to calculate kmf, given the Laplace transform P(z) and the steady state rate coefficient kss. For instance, combining eqs 39 and 31 and rearranging, one gets kmf= kss + iw
+ Z * ) / D ] ~z*/ ~=, to + qQkSS
+ z*)) k o ( t )d t -
(' exp(iwr)) d t (42) io -
which is equivalent to eq 39. IV. Comments
The treatment presented in this paper compliments and extends those in refs 1 and 9. Assuming the SmoluchowskiCollins-Kimball reactivity model, Sung et al. obtained an
The Smoluchowski-C$lins-Kimball the Laplace transform ko(z):
model is determined by
We stress that the present theory is not equivalent to that of Sung et aL9 This is because our eq 21 ignores the excluded volume effects and, therefore, cannot produce the same result for ko. However, one can repeat the derivation of eq 40 for a more general reactivity model:
k(t) = S k , ( r ) g(rA d r
(47)
where the distance-dependent reactivity function k2(r) can be long-ranged. Using
k2(r)= k,d(r - R)/4nR2
(48)
one gets eq 18. Thus, eq 40 is valid fo: both contact and longranged reactivity models as defined by ko(z) and is, in this sense, model-independent. This fact enables one to make a consistent comparison between different theories using an explicit form of ko(z). It is not difficult to see that eq 46 in combination with eqs 26 and 40 leads to eq 43, so that one recovers the result of Sung et al. In ref 1 we derived the expression
kmf = zio(z) for z = iw
+ to' + eQkSs
(49)
which can be regarded as a simple approximation to the rigorous formula 40. How good this approximation is depends on the particular reactivity model. As shown in ref 9, the modulation M ( w ) and phase angle Q(u)predicted by the two expressions are quite close for the Smoluchowski-Collins-Kimball kinetics. This is not unexpected since the modulation and the phase angle are monotonic functions of frequency, and the two meanfield rate coefficients show the same low- and high-frequency asymptotics:
k&(O) = kss,kmf(-) = k,
(50)
This is also the reason why the formula suggested by Zhou and
Fluorescence Quenching Rates
J. Phys. Chem., Vol. 99, No. 8, 1995 2357
Szabo,8
k"f = zio(z)
for
z = iw
+ TO' + @Qkd
(51)
gives similar results. As noted by Szabo," when one makes the formal substitutions mf
~ ( t=) exp(iwt)Ko, &(t) = exp(iwr) &w>, e,*(@) = iw
+ TO' + @Qkmf,k ( f ) = k"f(w) (52)
in eq 41, then one recovers eq 51. However, this procedure is unphysical, since the illumination intensity should be represented as K = Ps-I- Ak exp(iwf), rather than as K = KOexp(iwf), the latter admitting negative values. Similarly, one should write @F*(f) = A&* A@F*exp(iwf) rather than @F*(f) = exp(iwt) @,"f(w). Interestingly, the two representations are operationally equivalent for linear theories, for instance in the standard approach to fluorescence quenching where
+
&(t) = JrK(f - Z) s(Z) dz, s(t)= eXp(-t/Zo -
@QLrk(Z) dZ) (53) Thus, negative intensities and concentrations can be tolerable in the linear case. However, for a nonlinear theory such as the one considered here, the formal representation 52 is not equivalent to the physical one in eqs 7 and 8, and, therefore, we find no rationale for using eq 52 in the present context. We conclude that the statistical mechanical origin of eq 51 needs to be established in a different way. In this paper we have used the methods of nonequilibrium statistical thermodynamics to study relations between various fluorescence rate coefficients. The present results suggest some problems to be addressed in future work. First, in the formulation presented above, the theory does not allow for the effect of interparticle forces. Using a generalized Smoluchowski approach,18 one can show that the relations 26 and 40 are also valid for the case of a nonvanishing potential.20 A natural question is whether the nonequilibrium statistical thermodynamic approach can be generalized to include the effects of excluded volume and of long distance interparticle forces. A second
problem is how the results in the frequency domain can be extended beyond the linear harmonic regime (e.g. to the case when [AKl Psand A@* = &), as well as to the case when the back reaction is present (e.g. monomer-excimer kinetics).lg A third problem is the relationship between formula 40 and that by Zhou and Szabo, 5 1, and the derivation of the latter. Work along these lines is in progress.
Acknowledgment. I wish to thank Professor Attila Szabo and Professor Sangyoub Lee for sharing their results prior to publication, and Professor Joel Keizer for his inspiring comments and critical reading of the manuscript. This work was supported by the Department of Chemistry and the Institute of Theoretical Dynamics of the University of Califomia at Davis. References and Notes (1) Molski, A.; Keizer, J. J. Phys. Chem. 1993, 97, 8707. (2) Sung, J.; Shin, K. J.; Lee, S. Chem. Phys. 1992, 167, 17; 1994, 179, 23. (3) Nemzek, T. L.; Ware, W. R. J. Chem. Phys. 1975, 62, 477. (4) Baird, J. K.; McCaskill, J. S.; March, N. H. J. Chem. Phys. 1981, 74, 6812; 1983, 78, 6598. (5) Rice, S. A. Diffusion-Limited Reactions In Comprehensive Chemical Kinetics; Bamford, c. H., Tipper, c. F. H., Compton, R. G., Eds.; Elsevier: Amsterdam, Vol. 25, 1985. (6) Keizer, I. Chem. Rev. 1987, 87, 167. (7) Szabo, A. J . Phys. Chem. 1989, 93, 6929. (8) Zhou, H. X.; Szabo, A. J. Chem. Phys. 1990, 92, 3874. (9) Sung, I.; Shin, K. J.; Lee, S. J . Chem. Phys. 1994, 101, 7241. (10) Lakowicz, J. R. Principles of Fluorescence Spectroscopy; Plenum: New York, 1983. (1 1) Lakowicz, J. R.; Johnson, M. L.; Gryczynski, I.; Joshi, N.; Laczko, G.J . Phys. Chem. 1987, 91, 3277. (12) Lakowicz, J. R.; Szmacinski, H.; Gryczynski, I.; Wiczk, W.; Johnson, M. L. J . Phys. Chem. 1990, 94, 8413. (13) Eads, D. D.; Dismer, B. G.; Fleming, G. R. J. Chem. Phys. 1990, 93, 1136. (14) Keizer, J. J. Phys. Chem. 1982, 86, 5052. (15) Keizer, J. Statistical Thermodynamics of Nonequilibrium Processes; Springer: New York, 1987. (16) Molski, A.; Keizer, J. J. Chem. Phys. 1991, 94, 574; 1992, 96, 1391. (17) Szabo, A. Private Communication. (18) Molski, A. Chem. Phys. 1994, 182, 203. (19) Naumann, W.; Molski, A. J. Chem. Phys. 1994,100, 1511; 1520. (20) Molski, A,; Naumann, W. In preparation. JP942125U
