8707
J. Phys. Chem. 1993,97, 8707-8712
ARTICLES Relations among Steady-State, Time Domain, and Frequency Domain Fluorescence Quenching Rates Andrzej Molski Adam Mickiewicz University, Department of Physical Chemistry, Grunwaldzka 6, 60- 780 Poznah. Poland
Joel Keizer' Institute of Theoretical Dynamics and Department of Chemistry, University of California, Davis, California 95616 Received: February 2, 1993; In Final Form: April 22, 1993
Methods of nonequilibrium statistical thermodynamics are used to study relations among rate expressions for steady-state, time domain, and frequency domain fluorescence quenching rates. Equations are derived that relate the Laplace transform, @ ( z ) ,of the quenching rate coefficient in the time domain, P(t),to the steadystate rateconstant, k", and to themean field rate coefficient in the frequency domain, km'(o), These relationships can be useful in calculating steady-state and frequency domain results when the Laplace transform, @(z), is given. The equation linking @ ( z ) with k" is a rigorous consequence of the statistical nonequilibrium thermodynamic theory and is equivalent to an equation derived by Szabo (J.Phys. Chem. 1989,93,6929). The equation relating @ ( z )to kmf(w) is obtained for the case of low illumination intensity and is different from that conjectured by Zhou and Szabo (J. Chem. Phys. 1990,92,3874). The relationships between the steady-state, time-dependent, and frequency-dependent quenching rates illustrate a more general principle: the molecular rate coefficient, k(t),of a diffusion-controlledbimolecular process is coupled to the rates of concurrent unimolecular processes, e.g., particle generation and decay.
I. Introduction and the Main Result The mechanism of fluorescence quenchingis often modeled by the following kinetic scheme:'-'
F
-
KO)
F*
fluorophore in the absence of the quencher
7&'
F*
+
= 7,;'
(3) The fluorescence intensity I is proportional to p p ( t ) . Thus, at steady state when K ( t ) = Kss = const, the ratio of the intensity in the absence (IO)and in the presence (I) of the quencher is given by 7;'
7;'
+
Io/Z = 1 pQkq70 (4) which is the well-known Stern-Volmer equation. When the illumination is oscillatory with the angular frequency w , i.e.
F
+
K ( t ) = K" AK exp(iwt) the concentration p p ( t ) is also oscillatory where T~~and rfstand for the nonradiativeand radiative lifetimes of excited fluorophore molecules F*,respectively, and kq is the bimolecular quenching rateconstant. Reactions la-ld represent, respectively, excitation of the fluorophore F with rate K ( t ) , nonradiative decay of the excited fluorophore F*, fluorescent decay of F*,and collisonal quenching by the quencher Q. In the present paper, we are concerned with relations between the rates of the bimolecular quenching reaction Id in steady-state, timeresolved, and frequency-resolved experiments. The simplest quantitative descriptionsof the scheme composed of reactions la-ld is given by phenomenological equations of formal chemical kinetics d
x ~ F * ( t )= -(TO'
+PQ~JPF*(~)+~
( t )
(2)
where p p is the concentration of the excited fluorophore F*, p~ is the quencher concentration, and TO is the lifetime of the
(5)
pF*(t) = + A+ exp(iwt) (6) Using eq 2, the factor J ( w ) = App/AK can be expressed as J(W)
= (iw
+
+ pQkq)-'
7;'
(7)
In frequencydomain flu0rometry,4*~.~ one measures the modulation M ( u ) and the phase angle @ ( w ) , which are related to j ( w ) by
and
1
I
Im J ( w ) (9) Re J ( w ) where 121, Re(z), and Im(z) are the absolute value and the real and imaginary parts of the complex number z, respectively. @(a) = arctan
-
0022-3654/93/2097-8707$04.00/0 0 1993 American Chemical Society
8708 The Journal of Physical Chemistry, Vol. 97, No. 34, 1993 In standard chemical kinetics, one usually assumes that the rate constant k, is a "true" constant, Le., kq is thought of as independent of time and concentration. However, standard chemical kinetics fails to provide an adequate description of scheme la-Id when the intrinsic quenchingprocess is sufficiently rapid. This is because the intrinsic bimolecular rate is modified by nonequilibrium effects caused by the slow diffusive motion of the reactants. One can, however, retain the form of eqs 4 and 10 by formally replacing k, with an effective, mean field rate coefficient k d and treating the resulting relation
+
J(w) = App/AK = ( i o 70' + pqKnf(o))-' (10) as the definition of k d ( ~ ) Defined .~ in this way, the rate coefficient P i s n o t a true rateconstant but is a complexquantity depending on concentration ( p ~ and ) angular frequency ( w ) . In the present paper, we apply the methods of statistical nonequilibrium thermodynamics to explore relations between three rate coefficients: P(t),ku, k d ( o ) . The molecular rate coefficient P(t) describes fluorescence quenching when the fluorophore is excited with an infinitely short, initial pulse of light. The molecular rate coefficient ku describes steady-state fluoroescence quenching. The mean field rate coefficient kd(w) is defined by eq 10. In the following, we derive a simple relationship between &"'(a) and the Laplace 'transform @(o) of the time domain rate coefficient P(t)
where the molecular, steady-state rate coefficient ku satisfies
P = (TO-'+ pQk")k'(~ = (7;'
+ pQk")k00(r,7i)-' + PQp5)
(25)
Specializing eq 25 to r = R and recalling eq 17, one verifies the truth of eq 12. B. Relation between P ( w ) and &T). We begin by noting that the oscillatory illumination (eq 5) causes oscillations not only of the concentration p p (eq 6) but also of the radial distribution function
and of the molecular rate coefficient k ( t ) = k,g(R,t) = k" + Ak exp(iot)
(27)
Note that eq 14 implies the following relation between Ak and Ag:
kd(w) = k"
+ (iw + 70' + p Q k " ) p Ak F*~
(31)
Because the concentration pF* is proportional to the illumination intensity Kss, this approximation is consistent with the low illumination intensity limit. It is convenient to introduce the function s"'defined by ef(w) =f
Ag
+ (iw + 7;' + p Q k " ) p F * ~
(32)
so that we have P f ( w ) = k,g""(w)
(33)
Now we substitute eqs 26 and 27 into eq 15 and linearize to obtain
+ pQk")Ag -
DV2Ag - (iw + 7;'
Combining eqs 18 and 34 according to the approximation in eq 3 1, we find the following equation involving pfand kmf:
+ 701+ pqKnf)(pf- 1) -
D V 2 f f- (io
f ? ' p ~ ( A k / h K ) ( f - 1) - k"6(r) = 0 (35) We should stress that the mean field rate coefficient k d ( w ) and the mean field distribution g"' are defined in a formal way (cf. eqs 10,3 1,32) and are complex numbers having no clear molecular interpretation. Their definitions have been chosen so that they may be used to determine the modulation M ( w ) and the phase shift @(a)through eqs 8 and 9. In the low illumination intensity limit, the third term in the right-hand side of eq 35 can be dropped. Comparing the result with eq 23, we find ff(w)
for
= z&'(z)
z = io + 7;'
+ pQk"
(36)
Writing eq 36 explicitly as Ak = k,Ag(R)
(28)
Although not indicated explicitly, the quantities Ak and Ag are complex functions of w determining the phase shift and the amplitude relation between the oscillatoryportion of the excitation (eq 5 ) and the oscillatory portions of the radial distribution function g and the molecular rate coefficient k ( t ) . Note that standard chemical kinetics presumes the rate constant k, to be independent of the excitation frequency. Becuase eqs 15 and 16 are nonlinear, it is important that k ( t ) and g(r,t) be expressed as in eqs 26 and 27. This allows us to obtain the correct linear response around the steady state, which permits us to compare with experiments carried out using similar modulation techniques.4~5Substituting eqs 26 and 27 into eq 16 and linearizing, we get
g"'(w) = (io
+
+ pQk")&'(iw + TO1 + PQP)(37)
and recalling eq 33, one recovers the relationship promised in eq 11. 111. Comparison with Previous Work
In order to carry out calculations of the modulation and phase angle with eq 11, it is necessary to calculate both the steady-state rate constant, k", and the Laplace transform, To do this, we need to solve eqs 17-19 and eqs 20-22, respectively. Both sets of equations have been solved in other contexts, the first via Fourier transform technique^^^^ and the second using integral equations.10 Using these methods, it is easy to show that the solution of eqs 17-19 for kss can be written
e.
k"5
=
4?rDRk2 4 r D R + exp(-tR)k,
Molski and Keizer
8710 The Journal of Physical Chemistry, Vol. 97, No. 34, 1993
where the correlation length is determined by
+
= (D/TC1 PQP)'''
(39)
Note that eq 38 can be solved iteratively since ( depends on h x c e p t in the limit that p q goes to zero. In general, the term p ~ k uis important only for concentrations above 10-3 M.*J2At low densities, eq 38 provides an explicit expression of k". Solving eqs 20-22 for the Laplace transform, ko, is somewhat more difficult.10 However, eq 21 is uncoupled from eq 22 and has thesame formas theequation for the time-resolveddistribution function for an irreversible bimolecular reaction,l0 for which we already have the solution. Taking advantage of this fact, we can write immediately P ( z )=
'k (40) z ( l + exp(-R(z/D)'/')k2/4?rDR)
As a check, note that the expressions in eqs 38-40 satisfy eq 12. The expressions in eqs 38 and 40 provide all that is needed to calculate k d ( o )in eq 11, which using eq 10 allows us to calculate the modulation, M(w), and phase angle, a(@), using eqs 8 and 9. We see that the nonequilibrium statistical thermodynamic theory leads not only to the relations of eqs 11 and 12 but also to be explicit expression (eq 40) for the Laplace transform ,@. It is important, however, to stress that the range of validity of relations 11 and 12 is not confined to this particular model of fluorescence quenching. Since the derivation of eqs 11 and 12 does not make use of the explicit form of the Laplace transform ko, these relations can be applied to different models of fluorescence quenching defined by various specific forms of ,@. For example, the statistical nonequilibrium thermodynamic model (eq 40) describes collisional quenching, but ignores the excluded volume effects. Such effects are taken into account in the Smoluchowski-Collins-Kimball model determined by
ko(z)= k,k,/(k,
+ zk,)
(41)
where
80.0
I
. I
c,
ert
I
3 60.0 -
BE
' a
40.0 -
I
0.0 0.0
5.0
10.0
47rDR( 1
+ (R(z/D)'/'))
(42)
Still moregeneral is the step-function nonradiative lifetime model of Zhou and Szabo' lifetime T when the fluorophore-quencher separation lies between the distance of closest approach R and some maximum quenching separation a. For this model
where u = 4?r(a3- R 3 ) / 3
(44)
CX cosh X - (e - X'aR/e) sinh X K ( z ) = 4.rrDa RX cosh X + e sinh X
(45)
with
x = ( ~ ' ( 1 + TZ)/DT)'/' e=a-R
(46) (47)
Equations 11 and 12 can be used, in conjunction with various forms of &z), to compare the theories of diffusion-influenced fluorescence quenching. We plan to present such an analysis in a separate publication. Here, we confine ourselves to some illustrative calculations with Smoluchowski-Collins-Kimball model as defined by eqs 41 and 42. This model describes diffusion-
20.0
dimensionless frequency Figure 1. Comparison of the modulation and phase angle calculated
using standard chemical kinetics (solid lines, k, = 4rRDkz/(4rDR + kz)),the present theory (dotted lines), and the theory of Zhou and Szabo (broken lines) for k2/4rRD = 10.0, R ~ / D T=o 0.1, and Q = 4rR3ppq/3 = 0.5. The dimensionlessfrequency Y = w r 0 / 2 r , where w is the angular velocity.
influenced collisonal quenching and seems to be the one that is most often used in the literature. In ref 7 , Zhou and Szabo propose the following equation for the mean field frequency domain rate coefficient:
kmf(0)= (io + 7;'
+ pQkmf(0)) x k0(iw
k,
15.0
+ TO' + p Q k m f ( 0 ) )
(48)
which should be compared with our eq 11. In both approaches, the modulation and phase angle can be written as functions of the dimensionless frequency v = 0~0/21r,where TOis the natural lifetime of the fluorophore and w is the angular frequency of the modulation. These functionsaredetermined by thedimensionless parameters k2/4.rrDR,R2/DT0, and 4 = 4.1rR3pq/3. Figure 1 shows the modulation, M(w), and phase angle, a(w), calculated using eq 11 and eq 48 for values corresponding to a diffusioncontrolled reaction at high quencher density, Le., k2/4.rrDR = 10, R2/DT0 = 0.1, and 4 = 4uR3pq/3 = 0.5. For the sake of comparison, Figure 1 also shows the modulation and phase angle calculated using standard chemical kinetics (cf. eq 7 with the quenching constant given by the formula k,, = 41rDRk2/(4*RD + kz)). While it is clear from the figure that both theories differ significantly from the result based on a constant value of k,, differences between Zhou and Szabo's expression (eq 48) and ours (eq 11) may be difficult to detect with the present experimental resolution. Recently, there has been much interest in interpreting transient, nonexponential effects in fluorescence quenching as measured by different techniques,e As an example, we take the quenching of indole by iodide in aqueous solutionsinvestigated in ref 4 using frequency domain fluorometry. The transient effects were interpreted in terms of diffusion-influencedcollisonal quenching. Figures 2 and 3 show the Stern-Volmer plots and the modulation and phase angle, respectively, calculated using the present theory and standard chemical kinetics (cf. eq 7). The following parameters, which are representativeof the aqueousiodide/indole
Relations among Rate Expressions
The Journal of Physical Chemistry, Vol. 97, No. 34, 1993 8711
20.0
of the intrinsic reaction rate k2 obtained from the Stern-Volmer plots and frequency domain data are sometimes different.' A comparison of the results in Figure 3 for the modulation and phase angle based on eq 11 with those based on eq 7 shows significant transient nonequilibrium effects. This conclusion is in general agreement with the experimental observations in ref
15.0
4.
IV. Concluding Remarks
-e 10.0 CI
5.0
0.0
volume fraction Figure2. Comparison of the Stern-Volmer plots calculatedusing standard chemicalkinetics (solid line, k, = 4rRDk2/(4rDR + k2))and the present theory (dotted line) for kz = 0.2 X 109 M-l s-l, TO = 4.5 ns, R = 7 A, and D = 1.8 X lV5cmz s-l. 100.0
v)
8 80.0
.R
c,
Q
I I
a
3E 60.0 &
%
In the present paper, we have employed statistical nonequilibrium thermodynamic theory of diffusion-influencedreactions to derive equations linking the molecular rate coefficients for steady-state, time domain, and frequency domain fluorescence quenching. Equation 12 relates the steady-state molecular rate coefficient k" to the Laplace transform, &z), of the time domain rate coefficient ko(t). This equation can be solved iteratively for k",once @(z) is known. The mean field rate coefficient W(u) pertinent to frequency domain fluorometry is expressed explicitly in terms of k" and &z) by eq 11. This relation is valid in the low-illumination limit. Since the derivation given in the previous section does not make use of the dimensionality of the system, formulas 11 and 12 are valid for one, two, and three dimensions. Thus, they can be applied equally well to systems confined to linear arrays, membranes, or solution. The work of Szabo and Zhou7.11 illustrates that Laplace transform relations can be useful in connecting different types of fluorometry. The present paper shows how such relations can be derived using the methods of nonequilibrium statistical thermodynamics. Since the Laplace transform @(z) is known analytically for a variety of models of fluorescence quenching," eqs 11 and 12 should be useful in steady-state and frequency domain fluorometry. Moreover, they can be used to check the consistencyof experimental data when steady-state, time domain, and frequency domain data are available for the same system. It is important to see the results of the present paper in a broader perspective. First, we note that, for the sake of concreteness, in the previous section we focused on the Smoluchowski-collins-Kimball-type model of fluorescence quenching. This model assumes reaction upon collisions at the distance of closest approach (cf. eqs 14, 15, 20). The relations 11 and 12 are valid, however, for a much broader class of reactivity models. To see this, one can consider the relation
k ( 0 = Jdr k2(r)&,t)
(49)
40.0
defining the molecular rate coefficient in terms of the intrinsic reactivity function kz(r). The derivations of the previous section remain valid if we replace eq 14 with its more general counterpart eq 49. One can recover the Smoluchowski-Collins-Kimball reactivity by taking
n
W E Q Q)
20.0
E
a 0.0 (
5.0
10.0
15.0
I
dimensionless frequency Figure 3. Comparison of the modulation and phase angle calculated using standard chemical kinetia (solid lines, k, =: 4r-k2/(4rDR + k2)) and the present theory (dotted lines) for kz = 9.2 X lo9 M-' s-l, TO = 4.5 ns, R = 7 A, D = 1.8 X lV5cmz s-l, and PQ = 0.5 M. system? were used: reaction radius R = 7 A, relative diffusion coefficient D = 1.8 X 10-5 cmz s-1, intrinsic reaction constant k2 = 9.2 X 109 M-1 s-1, intrinsic lifetime 70 = 4.5 ns, and I-quencher
concentration PQ = 0.5 M. An interesting feature in Figure 2 is the positive deviation of the Stern-Volmer plot (dotted line) from the standard kinetic prediction (solid line), which is similar to that found in earlier work.12 Although the Stern-Volmer plot looks linear, it is not. Therefore, it should not be interpreted in terms of the standard formula. This may explain why the values
where k2 is the intrinsic (equilibrium) reactivity. Second, eqs 11 and 12 can be considered as examples of the Laplace transform relations of chemical kinetics. Such relations have been known (but perhaps not widely appreciated) in chemical kinetics for a long time. For instance, in radiation chemistry one connects scavengerexperiments and timsdependent kinetics using a Laplace transform technique.13 In photochemistry, Najbar" recently proposed to use luminescence decay function measured at different quencher concentrations to obtain the Laplace transform of the time-dependent bimolecular rate constant. Finally, we should note that regardless of the method of excitation involved in eqs 15 and 16 only the basic elementary processes of the quenching reaction, diffusion, and intrinsic fluorescence are involved in the calculation of the response of the system. The basic transport coefficientsthat are involved in these are the diffusion constant (D), the lifetime in the absence of
8712 The Journal of Physical Chemistry, Vol. 97, No. 34,1993
where the fluctuations 6 p =~ p~ - ( p ~ and ) ~ P F *= PF. - (PF*) are defined as the deviations of the instaneous values from the average values. The terms fF* and f Q are random sources, that, at the hydrodynamic level of description, are purely random,Gaussian, white noise stochastic processes. Only two of the correlation functions of thej's are required to derive eq 13; they are9
quenching (7& and the intrinsic quenching rate constant (kz). Nonetheless,dependingonthe mode of excitation (steady, a pulse, or a small increment of frequency dependence to steady illumination) the measured rate coefficients are different. Thus in the steady-state Stern-Volmer-type experiment, where the theory has been successfully compared with experiment,12one gets the expression of the bimolecular rate constant, e, given in eq 38. In a pulsed experiment, one gets the complicated time dependence, kO(t), given in ref 10 (and recently verified experimentallyin ref 15). Finally, for small periodic perturbations around a steady state one obtains a mean field rate coefficient, kmf(w),that can be used to calculate the phase response and modulation responses (Figures 1 and 3). Nonetheless, as shown in eqs 11 and 12 these three rate coefficients are determined by a single quantity: the Laplace transform of kO(t).
Appendix In this Appendix, we briefly discuss how eq 15 can be derived within the framework of nonequilibrium statistical thermodynamics.8-10 Since the concentration of the excited fluorophore p ~ is*much smaller than that of the ground-state fluorophorep ~ we can consider the following simplified version of the kinetic scheme (la-ld):
-
F* + Q
F*
-+ 4
0 Q
(51c)
where we have replaced F's by the zeros to stress that we neglect the correlations of F's and F*'s and Q's. This is a formal trick simplifying the presentation. The results following from the scheme 51a-51c can be obtained from the "complete" scheme la-ld in the limit p p
