448 NUMERICAL COMPUTER METHODS [21]

[2 1] F l u o r e s c e n c e Q u e n c h i n g Studies : Ana lys i s o f N o n l i n e a r S t e r n - V o l m e r D a t a

By WILLIAM R. LAWS a n d PAUL BRIAN CONTINO

Introduction

Many parameters can be obtained by fluorescence spectroscopic meth- ods to provide insights into the environment, structure, and dynamics of a fluorescent probe that is either covalently bound or liganded to a biological molecule. One important, commonly used method is the addition of a quenching agent to reduce the fluorescence quantum yield of the probe. By comparing the quenching efficiency of different types of quenching agents under various conditions, the environment of the probe, and thus a specific region of the biomolecule, can be characterized in terms of neighboring ionic groups and solvent accessibility. Often, however, steady-state fluorescence quenching data do not follow the standard linear Stern-Volmer expression. 1 These deviations mean that other processes are occurring besides dynamic (collisional) quenching. If the nonlinear data are analyzed in a systematic way, it is possible that even more information may be obtained about the system.

Many situations can cause steady-state quenching data to be nonlinear when plotted by the standard Stern-Volmer relationship; these fluores- cence quenching mechanisms have been reviewed by Eftink and Ghiron. 2 In this chapter, we show how two of these situations, static quenching and multiple species, cause the nonlinear deviations. We then evaluate the ability of the commonly used Marquardt nonlinear least-squares algorithm 3 to recover known parameters from synthetic data. The results of this study point out the experimental and analysis criteria that must be met to obtain optimal information from fluorescence quenching studies. 4 In particular, we demonstrate the need to perform time-resolved fluorescence quenching studies. In this way, the dynamic quenching parameters are either con-

I O. Stern and M. Volmer, Phys. Z. 20, 183 (1919). 2 M. R. Eftink and C. A. Ghiron, Anal. Biochem. 114, 199 (1981). 3 p. R. Bevington, "Data Reduction and Error Analysis for the Physical Sciences." McGraw-

Hill, New York, 1969. 4 Inherent in this discussion is the assumption that all experimental corrections to the steady-

state intensities have been made and are not contributing to the nonlinear behavior of the data. This includes any dilution corrections as a result of the titration, as well as corrections for both primary and secondary inner filter effects due to the absorption of light at the excitation and emission wavelengths, respectively, by the quenching agent (see Ref. 8).

Copyright © 1992 by Academic Press, inc. METHODS IN ENZYMOLOGY, VOL. 210 All fights of reproduction in any form reserved.

[ 2 1 ] ANALYSIS OF NONLINEAR FLUORESCENCE QUENCHING 449

firmed or evaluated, the distinction between single and multiple species is verified, and the existence of static quenching, or other process, is established.

Theory

Single Species with Dynamic Quenching

The fluorescence intensity, F0, of a chromophore, A, in the absence of added quenching agents depends on the initial concentration of the excited state, [A~], and the rates of the depopulation processes. Assuming that a quencher, Q, does not absorb light or affect the extinction of A, then the fluorescence intensity, F, of A in the presence of added quenching agents also depends on [A~']. In this case, F will be less than F0 since the decay kinetics now include the collisional quenching process. The ratio of these two intensities is given in Eq. (1):

FolF = ([A~]rolr')l([A~]r/r') = rolz (I)

where r ' is the natural (in vacuo) lifetime of A*, z 0 is the lifetime in an interacting system without added quencher, and z is the lifetime at a particular [Q]. Both ~'0 and ~- are defined in terms of sums of rate constants: "r 0 ---- (knr + k f ) -1 and r = (knr + kf + kq[Q])-l, respectively, where kf is the rate of fluorescence in the absence of any interactions, knr is the sum for all the nonradiative rates in interactive systems lacking the quencher, and kq is the rate of collisions with the quencher that result in the radiation- less deactivation of A*. The intensity ratio can therefore be expressed [Eq. (2)] in terms of the rate constants:

"Fo/F = (kf + knr)-1/(kf + knr + kq[Q])-1 (2)

By rearranging and defining Ksv = kqTo = kq/(kf + n.r), Eq. (3) is obtained:

Fo/F = 1 + Ksv[ Q] (3)

which is the well-known, linear Stern-Volmer relationship I with Ksv the Stern-Volmer constant. An example of this dynamic quenching of a single species is shown by curve A in Fig. 1 where Ksv = 8 M -1.

Single Species with Dynamic and Static Quenching

Static quenching mechanisms are directly implicated when the quan- tum yield of the probe as measured by steady-state methods (number of photons emitted divided by the number of photons absorbed) is less than the quantum yield as measured by time-resolved methods (kinetics of the

450 NUMERICAL COMPUTER METHODS [21]

12

10

8

o 6

4

2

0 0.0

D /

/ / C

/ ..-'. B / " . - J

. ~ ' ~

I I I I I

0.1 0.2 0.3 0 .4 0.5 0 .6

[Quencher], M

FIG. 1. Curves representing quenching functions for a single chromophore with Ksv = 8 M -1. (A) Equation (3); (B) and (D) Eq. (12) with Ka values of I and 2 M -I, respectively; (C) Eq. (6) with V = 1 M-~.

decay of the excited state). A static quenching mechanism is a very rapid process that removes a fraction of the A* population before it can be dynamically quenched. Therefore, time-resolved fluorescence measure- ments are not dependent on [A~], and static quenching processes are only observed by steady-state intensity measurements. Consequently, the Fo/F ratio will (1) increase at a rate greater than that due to dynamic quenching alone and (2) have upward curvature since the static quenching mechanism is also a function of [Q]. The sphere of action and the ground- state complex static quenching models are discussed here.

Sphere of Action Static Quenching Model. Any quenching interaction requires that the chromophore and the quencher be within a certain dis- tance of one another. In solution, this critical distance defines an interac- tion sphere of volume V. On excitation of the chromophore, a quencher molecule may already be within this volume and thus be able to quench without the need for a diffusion-controlled collisional interaction. 2 The probability of the quencher being within this volume at the time of excita- tion depends on the volume and on the quencher concentration. Assuming that the quencher is randomly distributed in solution, the probability of static quenching is given by a Poisson distribution, e-V[Q]; the corrected fluorescence intensity ratio is expressed in Eq. (4):

Fo/F = ([A~']7.o/~")]/([A~'] e - V[Ql "/'/7'') (4)

On substitutions and rearrangements, Eq. (5) can be obtained:

(FolF) e-VtQ] = ,To/T (5)

Combination of this static quenching mechanism with the dynamic quench- ing described by Eq. (3) results in Eq. (6):

[21] ANALYSIS OF NONLINEAR FLUORESCENCE QUENCHING 451

Fo/F = (1 + Ksv[Q])e viol (6)

An example of the curvature resulting from a single chromophore undergoing both dynamic quenching and sphere of action static quenching [Eq. (6)] is shown by curve C in Fig. 1 where V = 1 M- 1. For comparison, it has been generated with the same Ksv value (8 M-1) used for the linear Stern-Volmer plot (curve A). Both Ksv and V must be evaluated when nonlinear steady-state fluorescence quenching data with upward curvature are analyzed by Eq. (6). The dynamic aspects of the decay of the excited state of the chromophore are still represented by Ksv. Thus, the value of K~v should be consistent with the known lifetime in the absence of quencher, ~'0, and a reasonable value for the bimolecular quenching con- stant, kq. The Vterm must be greater than zero since it represents a volume in which quenching occurs without the need for diffusion. 5 Furthermore, V must be within a range of values that denotes a radius that is not much larger than van der Waals radii; a range of 1-3 M-l yields acceptable radii (< 10 A) for diffusionless interactions. 2 It has been previously noted that V tends to be about 10% of Ksv for acrylamide quenching of indole com- pounds 2 and a fluorescent estrogen analog. 6 However, this is most likely just a function of the individual lifetimes and quenching rates for these two systems. In general, V is independent of the Stern-Volmer constant and should be within a range for an effective radius of interaction.

Ground-State Complex Static Quenching Model. A quencher can inter- act with the chromophore to form a ground-state complex, A • Q, as detailed in Eq. (7):

A + Q,~-A. Q (7)

This complex is able to reach the initial Franck-Condon state; however, because the quencher is part of the complex, quenching occurs "instanta- neously" and efficiently without the need for a diffusion-controlled inter- action.

In the absence of quencher, F 0 will be proportional to the entire chro- mophore concentration, [Atotal]. F, however, will be proportional to the free chromophore concentration, [Af], and will be a function of [Q]. The equilibrium expression for this reaction is given in Eq. (8), where Ka is the association constant:

K a = [n . Q]/[Af][Q] (8)

5 To account for units, V represents the volume per mole of the chromophore and the quencher at the critical interaction distance; division by Avogadro's number yields the volume for one chromophore and one quencher.

6 E. Casali, P. H. Petra, and J. B. A. Ross, Biochemistry 29, 9334 (1990).

452 NUMERICAL COMPUTER METHODS [21]

By conservation of mass, [Atotal] = [Af] + [A. Q], and, with rearrangement of Eq. (8), an expression for [Af] can be derived [Eq. (9)]:

[Af] = [AtotJ/(1 + Ka[Q]) (9)

Thus, the excited-state concentration of chromophore in the presence of quencher will be less than in its absence, and the ratio of fluorescence intensities becomes

Fo/F = ([A~,]~.0/~.,)/([A~,]( 1 + Ka[Q])-1~./,/.,) (10)

Equation (10) can be rearranged in a manner similar to the previous case:

(F0/F)(1 + Ka[Q]) -l = ~-0/~- (11)

Therefore, by combining this type of static quenching mechanism with the dynamic quenching described in Eq. (3), the expression in Eq. (12) is derived:

Fo/F = (1 + Ksv[Q])(1 + Ka[Q] ) (12)

Equation (12), which represents a single species undergoing both dynamic quenching and ground-state complex static quenching, is a quadratic ex- pression with upward curvature. As shown in Fig. 1, Eq. (12) (curve B) has a similar amount of curvature with a K~ of 1 M - 1 as that generated by Eq. (6) (curve C).

Both Ksv and Ka need to be determined in an analysis of nonlinear steady-state fluorescence quenching data by the ground-state complex static quenching model detailed in Eq. (12). The previous criteria still apply to the value for Ksv, which represents the dynamic quenching portion of the interaction. As shown by curves B and D in Fig. I, Ka does not have to be very large to induce significant curvature. The K a of 1 M-* in curve B and the Ka of 2 M - 1 in curve D are of a magnitude expected for very weak or nonspecific binding. This range of values for Ka is obviously too small to be easily measured by most experimental techniques and is in the range suggested for the affinity of acrylamide to proteins. 7 Note that the additional term in Eq. (12) is actually linear with respect to [Q] in contrast to the experimental term in Eq. (6); under the special circumstance of no dynamic quenching, this may allow the two static quenching mechanisms to be distinguished. 6

Multiple Species with Dynamic Quenching

Multiple chromophores, each with its own distinct K~v term, can also lead to nonlinear Fo/F plots. Multiple chromophores can arise in several ways even in a chemically pure system. If a protein has more than one

7 M. R. Eftink and C. A. Ghiron, Biochim. Biophys. Acta 916, 343 (1987).

[21] ANALYSIS OF NONLINEAR FLUORESCENCE QUENCHING 453

tryptophan residue, for example, each one could have a different Ksv term due to either a different kq, ~'0, or both. If a macromolecule has been labeled with an extrinsic probe, either covalently or by adsorption, there could be more than one site of interaction. Finally, there is the possibility that a single chromophore, either intrinsic or extrinsic, located at a single site might have several different environments due to different structural conformations in the ground state. For example, based on our previous tyrosine studies, s-a° the three rotamers for the tyrosine side chain about its C~-C t~ bond are a source of ground-state heterogeneity, Therefore, multiple species in the ground state lead to multiple excited states, and these unique excitedstate species could have different environments re- sulting in distinct quenching constants.

If each species can be spectrally isolated, then the normal Stern-Volmer expression, the sphere of action model, or the ground-state complex model [Eqs. (3), (6), or (12)] would be appropriate for an analysis depending on the linearity of the Fo/F versus [Q] plot. However, often the excitation and/or emission energies of each species overlap. The measured F 0 and F values, therefore, consist of weighted sums of the F 0 and F values for each species. This means that the fluorescence intensity ratio will be a function of E(1 + Ks~(i)[Q]), which is an extension of Eq. (3) where i denotes a specific quenching constant/species. Assuming that the individ- ual species do not interact with one another, the complete expression for the dynamic quenching of multiple species must weight each element of this summation by the fractional intensity, f(i), of each emitting species. These f(i) are a function of the concentration, extinction, quantum yield, and emission spectrum of each species; by definition, E f(i) = 1. The complete expression for multiple (n) species undergoing only dynamic quenching is given in Eq. (13):

[f'~=l f(i) ] -1 F°/F = {1 + Ksv(i)[Q]}J

(13)

Depending on the values of the K~v(i) andf(i) terms, the Fo/F expression may be linear or curved downward. 2 Generally, downward curvature will be observed since the initial slope of the intensity ratio is influenced by the largest Ksv. As [Q] increases, the effect of the smaller Ksv terms will begin to be seen and the "average" Ksv will be less, resulting in the

8 p. B. Contino and W. R. Laws, J. Fluoresc. 1, 5 (1991). 9 W. R. Laws, J. B. A. Ross, H. R. Wyssbrod, J. M. Beechem, L. Brand, and J. C.

Sutherland, Biochemistry 25, 599 (1986). l0 j. B. A. Ross, W. R. Laws, A. Buku, J. C. Sutherland, and H. R. Wyssbrod, Biochemistry

25, 607 (1986).

4 5 4 NUMERICAL COMPUTER METHODS [21]

LL

A

B

0 I I I I I

0 .0 0.1 0.2 0.3 0.4 0.5 0.6

[Quencher], M

FIG. 2. Effect of multiple species on the Fo/F ratio. In all cases, f ( l ) = f(2) = 0.5 and Ksv(2) = 8 M - i . (A) Equation (13) with K~v(l) = 5 M - i ; (B) Eq. (13) with K~v(l) = 1 M-I ; (C) Eq. (14) with Ksv(l ) = 1 M - l and V(I) = V(2) = 1 M -1.

downward curvature. This dependence of the degree of curvature on the extent of the differences in the Ksv(i) terms is shown in Fig. 2. For two similar Ksv values, the expression appears to be linear (curve A). The downward curvature in curve B is a result of the Ksv values being signifi- cantly different.

Multiple Species with Dynamic and Static Quenching

Each species may also undergo static quenching by either the sphere of action or the ground-state complex model.

Sphere of Action Static Quenching Model. If we include an e v[Q] term for each species, the development will be the same as for one chromo- phore. The resulting expression for the multiple species/sphere of action model is Eq. (14):

Fo/F = {1 + Ksv(i)[Q] } e v(i)[e]] (14)

The trend in any data following the quenching mechanism detailed by Eq. (14) will of course depend on the specific values off(i), Ksv(i), and V(i). It is important to note that the addition of static quenching can override any downward curvature due to different K~v values and linearize the data or even cause plots to have positive curvature. This is shown by curve C in Fig. 2. The same f(i) and K~v(i) values used in curve B were used in Eq. (14), but a V term of 1 M -j for each species is able to induce upward

[ 2 1 ] ANALYSIS OF NONLINEAR FLUORESCENCE QUENCHING 455

curvature. In fact, curve C is actually a complex function since the upward curvature occurs after a small amount of downward curvature.

Ground-State Complex Static Quenching Model. As shown in Eq. (15), nonlinear quenching data could be due to the multiple species/ground- state complex model:

[~=i f(i) ]-1 F°/F = {1 + K~v(i)[Q]}{1 + Ka(i)[Q]} (15)

This expression is also able to cause upward curvature even when the Ksv terms are different enough to have downward curvature.

Analysis

In this section, we show that unless steady-state quenching experi- ments are performed to sufficiently high quencher concentrations, an in- correct analysis can occur because data that appear linear are actually nonlinear. We also show that it is difficult to distinguish between the two static quenching models for both single species and multiple species systems. We further demonstrate that it is necessary to obtain the dynamic quenching parameters as well as the fractional intensities to enable an analysis of multiple species quenching data.

Concentration Range and Linearity versus Nonlinearity

A typical upper limit for quencher concentration used for many fluo- rescence quenching studies in the literature is approximately 0.2 M; this concentation range usually requires minimal experimental corrections, 4 and the signal-to-noise ratio does not become a problem for the quenched sample. Considering the nonlinear examples displayed in Figs. 1 and 2, most could be interpreted as linear over this limited concentration range, particularly if experimental data with the inherent noise are considered. To demonstrate this point, we synthesized four data sets with noise (see Appendix at the end of this chapter) consisting of 11 concentrations equally distributed between 0 and 0.2 M. These data sets were generated using the respective equations and parameters for the curves shown in Fig. 1. Each data set was then analyzed by linear least-squares regression assum- ing a fixed intercept of 1.0 since, by defintion, Fo/F = 1.0 at [Q] = 0. In each case, the correlation coefficients for the regressions were around 0.99, which could be interpreted as an acceptable fit to a straight line. This was true even for the situation represented by curve D in Fig. 1, which is clearly nonlinear by 0.2 M.

The apparent linearity of these expressions over this small concentra-

456 NUMERICAL COMPUTER METHODS [21]

0.4 A

0.0

¢n I

- 0 . 4 "10 ¢n 0.4 t r

° ; t ' ~t o o o

o

t C oo

o ; o

o.o f,,o - 0 . 4

0.0 0.1 0.2

B o

J 8 o o •

o ^ I I t~ o ~ ¢ . . . . |

~lt~--~--t----~-- 8 0 o°mo o

oo 8 o o o

o o o o = /

i O__oI.

0.0 0.1 0.2

[Quencher ] , M

FIG. 3. (A) through (D) represent the residuals (data - fit) for curves A through D of Fig. 1, respect ively, result ing from fitting data generated by that funct ion to a straight line over a [Q] range o f 0 to 0.2 M. The smooth line denotes the residuals for noiseless data. Open circles are the residuals for the entire da ta set. Solid circles are the averages of the residuals at each [Q].

tion range can also be demonstrated by the residuals of the synthetic data and the linear fit. For data without added noise (represented by curves B, C, and D in Fig. 1), the residuals calculated from a linear fit, as expected, exhibit systematic deviations as shown by the smooth lines in the corre- sponding panels of Fig. 3. For the residuals of the data with noise, these systematic trends are lost; this is true even for the averages of the residuals at each [Q]. It could be argued that there is a trend in the average residuals for plots B, C, and D (Fig. 3) when contrasting them against the resid- uals for noiseless data or by comparing them to the randomness of the re- siduals for the normal Stern-Volmer situation (panel A), but this is an artificial comparison resulting from synthetic data. Consequently, the non- linear examples shown in Fig. 1 cannot be distinguished from a straight line over this quencher concentration range; a similar situation exists for the examples shown in Fig. 2 which represent the quenching of multiple species.

If the quenching of a single species involves more than collisional interactions, a consequence of such a linear analysis over this small con- centration range is the incorrect estimation of Ksv. For the synthetic data representing curves B and C in Fig. 1, the Ksv obtained by fitting to a straight line was 10.5 M-1 instead of the value of 8 M-1 used to generate

[21] ANALYSIS OF NONLINEAR FLUORESCENCE QUENCHING 457

the data. An analysis of the data representing curve D in Fig. 1 gave a K~ value of 12.1 M-1. Because the characterization of the environment of a chromophore requires a comparison of the kq terms for different quenchers under different conditions, and because kq = K~v/rO, an incorrect K~v value could lead to an incorrect hypothesis concerning the nature of the environment.

Quenching studies carried out to higher quencher concentrations and with a greater data density may permit the linear/nonlinear behavior of the data to be resolved. These studies could require large corrections owing to dilution and the extinction coefficient of the quencher, but careful experimental procedures and evaluation of these corrections permit highly reproducible data to be obtained. To further test the linearity/nonlinearity of the data, or to examine the effectiveness of any quenching model to fit the data, residuals should always be examined to ensure against systematic behavior.

Single Species with Dynamic and Static Quenching

Sphere of Action Static Quenching Model. To test the ability of non- liner least-squares to recover Ksv and V in the sphere of action model, we synthesized data sets with noise (see Appendix) consisting of 17 quencher concentrations equally spaced over 0 to 0.5 M. These data sets were generated by Eq. (6) with V equal to 1.0 M -~ and Ksv equal to either 2, 4, 8, or 16 M-~. In all cases, both recovered parameters were consistently within 2 to 3% of the correct value, and the random distribution of the residuals for all points and for the averaged residuals indicated excellent fits. The same convergence point (same values for the parameters and the same minimum on the X 2 surface 3) was obtained for iterations starting with different guesses spanning physically relevant ranges for both parameters.

Ground-State Complex Static Quenching Model. Data sets with noise (see the sphere of action model above) were generated by Eq. (12) with Ka equal to 1 M -~ and Ksv equal to either 2, 4, 8, or 16 M -1 and then analyzed by the ground-state complex model using nonlinear least- squares. Although excellent fits were obtained for each data set, and the convergence point was not particularly sensitive to the initial guesses for the parameters, there were problems in recovering the parameters. These problems are a direct result of the symmetry of Eq. (12) since there is no mathematical difference between Ksv and Ka- Recovery of the parameters depended on the relative magnitudes of the parameters used to generate the data. If K~v and Ka were similar, as for the case of Ksv equal to 2 M-1 and Ka equal to 1 M - 1, then K~v equaled Ka. If Ksv and Ka were different, as for the case of K~v equal to 4 M-1 and Ka equal to 1 M -1, then the

4 5 8 N U M E R I C A L C O M P U T E R M E T H O D S [ 2 1 ]

correct values were recovered. However, the symmetry of Eq. (12) made the assignment of the iterated parameters to a specific variable (Ksv or Ka) ambiguous.

Cross Analyses. To test whether the data sets generated by either of the two static quenching models uniquely describe that particular model, we analyzed each data set by the other model. For data generated by either model, equivalent fits were achieved by the other model, and the values for the parameters were physically reasonable. Therefore, these two static quenching models applied to a single species cannot be resolved with this data density over this concentration range by these statistical criteria. If this approach is experimentally feasible, data collected to even higher quencher concentrations might be able to resolve the different curvatures of the two models.

Multiple Species with Dynamic Quenching

If multiple species are known to exist, or if the Fo/F data have down- ward curvature, then the steady-state fluorescence quenching data must be analyzed by Eq. (13) or some form of Eq. (13) that includes static quenching and/or other processes. Analysis by the multiple species model requires that K~v(i) be determined for n distinct species and f(i) be deter- mined for n - 1 species.ll Therefore, the minimum number of parameters that must be iterated for two species is three.

To evaluate the ability of nonlinear least squares to recover parameters from a multiple species quenching situation, we simulated data as above using the same parameters used to generate curve A in Fig. 2 [f(1) = f(2) = 0.5, K~v(1) = 5 M -l, and Ksv(2) = 8 M -1 in Eq. (13)]. Using various starting guesses, different convergence points were reached if the algorithm converged. These convergence points yielded equivalent X 2 minima and adequate residuals but gave different values for the parame- ters. At no time, however, were the original values for the parameters recovered; this was true even when the known values were given as the guesses. It is often difficult to obtain a unique solution in a multiple parameter fitting problem if two or more parameters are similar in magni- tude. To check whether the near equivalence in the two K~v values was a factor, we generated data using the parameters for curve B of Fig. 2 [f(1) = f(2) = 0.5, K~v(1) = 1 M-J, and Ksv(2) = 8 M-1]. This near-equivalence was not a factor; similar analysis problems were observed. As would be expected, data synthesized for the quenching of three species had similar

n - 1

u O n l y n - I f ( i ) t e r m s m u s t b e e v a l u a t e d s i n c e f ( n ) = 1 - ~ f(i). i ~ l

[ 2 1 ] ANALYSIS OF NONLINEAR FLUORESCENCE QUENCHING 459

analysis problems. These problems are symptomatic of trying to fit for too many cross-correlated parameters.

Multiple Species with Dynamic and Static Quenching

Each species could also be quenched by a static mechanism. For both the sphere of action and the ground-state complex models discussed here, 3n - 1 parameters must be iterated for n species. Fitting for too many fitting parameters was already shown to hinder the analysis of multiple species undergoing only dynamic quenching. Consequently, as expected, our attempts to analyze data generated by either Eqs. (14) and (15) were futile.

Independently Determined Parameters

If steady-state quenching data are not collected to high enough concen- trations of quencher, then a linear regression could adequately fit the data and an incorrect value would be obtained for the Stern-Volmer constant. If the trend in the Fo/F data for a single species is nonlinear, there is a problem deciding which static quenching process is causing the effect since both the sphere of action and the ground-state models can fit the data equivalently. If multiple species are involved, then it is difficult to find a unique solution since too many parameters have to be evaluated.

One possible solution to these problems is to obtain additional informa- tion about the system by independent means. With this information, the fitting algorithm may be aided by reducing the number of dependent param- eters and/or restricting the search to a specific set of conditions. This additional information can be easily determined by time-resolved fluores- cence intensity decay measurements which can provide Ksv for a single species system and, depending on the number of species, may be able to obtain Ksv(i) and f(i) for a multiple species system.

From Eq. (1), Fo/F = Zo/Z; therefore, ro/Z can be substituted into Eq. (3), and the slope of the lifetime ratio versus [Q] provides Ks~. This plot is linear since static quenching does not affect the lifetime of the excited state. This analysis requires that the fluorescence intensity decay of a single species obeys a single exponential decay law, and that only the lifetime is affected by the addition of quencher.

Additional information about a presumed single species system can be obtained from time-resolved fluorescence quenching studies. If the system consists of a single species with only dynamic quenching, then Ksv as determined by Fo/F must be the same as K~v determined by Zo/Z. However, if the K~v term obtained from steady-state quenching data is larger than the K~v term obtained from time-resolved data, then a static quenching

460 NUMERICAL COMPUTER METHODS [2 1]

mechanism is implicated even though the Fo/F data appear to be linear. Finally, if on the addition of quencher the intensity decay law is no longer a single exponential, then the presumed single species system is actually a multiple species system. This situation will occur when the ~'0 values for the multiple species are similar (unresolvable) and the Ksv(i) values are different.~2 As a result, the fluorescence lifetime of one species (assuming only two for this example) will be affected to a greater extent than the other, and the lifetimes become resolvable.

If a single species has been verified by time-resolved quenching studies but static quenching is occurring, the K~v determined from the "r0/r plot represents the dynamic aspect of the quenching interaction. This term can now be used as a constant in the fitting process. This approach was applied to the data generated by either Eq. (6) or Eq. (12) which were previously shown to be equally analyzed by both static quenching models. For both sets of data, inclusion of the known value for Ksv as a constant in the algorithm did not differentiate the static quenching mechanisms; both models still fit data generated by either model equally well.

We have shown above that the number of iterated parameters hinders the analysis of multiple species sysetms. To estimate how many Ksv(i) and f(i) values need to be independently known to allow the fitting algorithm to recover the remaining dependent parameters, we reduced the number of dependent parameters one at a time by making that variable a constant in the analysis of the above data sets. All possibilities of known and iterated parameters were considered; from these analyses, two basic conclusions emerged. If thef(i) values were provided, all but one of the Ksv(i) values also had to be known to recover the other K~v(i). If all the K~v(i) values were used as constants, then all but two of the f(i) were required: one iterated and the other a difference, t~ This was demonstrated for the multi- ple species model where Eq. (13) was used to generate data for both two and three species. If data simulating multiple species with either static quenching model were evaluated, all the Ks~(i) and f(i) terms had to be constants to recover the static quenching parameters.

Time-resolved fluorescence intensity decay studies can provide many of the parameters necessary for the analysis of quenching data from multi- ple species. If each species has a unique, resolvable lifetime and if the species do not interact, then the fractional intensities, f(i), for each species can be determined by obtaining the intensity decay parameters for the sample in the absence of quencher. From this experiment, the fractional intensities are calculated according to Eq. (16):

12 It is assumed that the quencher does not alter the basic environment of each species.

[(21] ANALYSIS OF NONLINEAR FLUORESCENCE QUENCHING 461

f( i) = a(i)%(i) a(i) %(i) (16)

where %(/) represents the unquenched lifetime of species i and a(i) is the preexponential amplitude term for each species when the intensity decay is analyzed by a sum of exponentials. ~3'~4

If the species have unique, resolvable lifetimes, then time-resolved quenching studies will provide the Ksv(/) terms. These values are obtained by extending Eq. (3) (for lifetime ratios) to multiple species as given in Eq. (17):

T0(0/r(/) = 1 + K~v(/)[Q] (17)

Time-resolved experiments can provide all the dependent parameters in Eq. (13); if the steady-state data deviate from a fit generated by Eq. (13) with these parameters, then static quenching is implicated. These same parameters may also be used in an analysis of multiple species with both dynamic and static quenching as represented by Eqs. (14) and (15). This approach has been used to explain the nonlinear Fo/F acrylamide quench- ing data for tyrosinamide. 8 By obtaining the Ksv(/) and f(/) values for the three tyrosine rotamers about the C~-C ~ bond, it was shown that each rotamer experienced both dynamic and static quenching by acrylamide. Unfortunately, differentiation between the two static quenching models could not be achieved. The same types of analysis problems which existed for a single species with static quenching were also found to exist for multiple species. We have verified this analysis behavior with synthetic data. Both the sphere of action and the ground-state complex models [Eqs. (14) and (15)] fit the data generated by either model equally well once the Ksv(i) and f( /) terms are made constants in the analysis.

Conclusions

We have shown that it is essential to have time-resolved fluorescence quenching parameters to analyze steady-state fluorescence quenching studies properly. In the case of a single chromophore in a single environ- ment, the time-resolved studies can verify that in fact there is a single species. Furthermore, if the Ksv values determined from both the time- resolved and the steady-state quenching studies do not agree, then other quenching processes such as static interactions have to be included in the overall mechanism. In the case of multiple species, the only way an

13 A. E. W. Knight and B. K. Selinger, Chem. Phys. Lett. 10, 43 (1971). 14 A. Grinvald and I. Z. Steinberg, Anal. Biochem. 59, 583 (1974).

462 NUMERICAL COMPUTER METHODS [21]

analysis of steady-state data can be attempted is to first determine the K,v(i) and thef(i) values from time-resolved experiments. Otherwise, there are too many iterated parameters, and a unique solution cannot be found.

Without time-resolved quenching experiments, incorrect conclusions about a system are easily made. For example, the environment of the steroid binding site for the serum sex steroid-binding protein was examined using just steady-state quenching of a fluorescent steroid analog, equilenin. A quenching mechanism that was consistent with the data was then used to develop a hypothesis about the hormone-protein interaction and the accessibility of the steroid binding site.15 Subsequent studies on this sys- tem, which included time-resolved quenching experiments, demonstrated that an entirely different quenching mechanism was operating. As a result, the steroid-protein binding interaction had to be reevaluated, and different conclusions were drawn regarding the nature of the steroid binding site. 6

The quenching parameters that are obtained by time-resolved studies can be included in the Fo/F expression either for a single species or for multiple species to detect the existence of additional quenching processes. We have shown, however, that if static quenching is involved it is difficult to determine the particular static quenching mechanism even under opti- mal experimental conditions. This was found for both single and multiple species systems. It should be possible to resolve the different curvatures induced by the sphere of action and ground-state complex static quenching models by going to quencher concentrations above 0.5 M. This will be experimentally feasible, however, only for chromophores that have excita- tion and emission bands well removed from the absorption bands of the quencher itself. This situation diminishes the magnitude of the inner filter corrections and leaves only the problems associated with measuring the weak fluorescence intensity of the sample at high quencher concentrations.

In this chapter, we have examined several quenching situations with synthetic data sets. Before drawing conclusions about the environment of a chromophore based on fluorescence quenching studies, similar simulation studies should always be performed to ensure (1) that the parameters can be recovered from steady-state quenching data and (2) that such conclusions are reasonable. If problems occur in the analysis of quenching data, such as multiple solutions, nonconvergence, or recovery of parame- ters with no physical relevance, it is possible that other nonlinear least- squares algorithms ~6 could be more sensitive or that a different formaliza- tion of the quenching expression other than the Fo/F ratio may be a

15 A. Orstan, M. F. Lulka, B. Eide, P. H. Petra, and J. B. A. Ross, Biochemistry 25, 2686 (1986).

16 M. L. Johnson and S. G. Frasier, this series, Vol. 117, p. 301.

[ 2 2 ] S I M U L T A N E O U S A N A L Y S I S O F M O D E L S A N D P A R A M E T E R S 463

better way to represent the data. But regardless of which chromophores, quenchers, conditions, algorithms, and formulations are employed, to examine a system and obtain the most information possible it is imperative that both steady-state and time-resolved fluorescence quenching studies be performed.

Appendix

To simulate experimental data for a single titration, Fo/F values were calculated for the stated number of concentrations evenly distributed over the given [Q] range. Each data point for [Q] > 0 then had Gaussian- distributed noise added with a standard deviation of 0.05, a representative error based on our experimental experience. Finally, this process was done a total of six times, representing six separate titrations, to give the entire data set used for the analysis of a particular quenching model. This number of data points and concentration range are based on our ability to recover the quenching parameters from nonlinear synthetic data without noise [Eq. (6)] for a single titration to within 10% of the known values. The overall data density provided by six titrations is our present standard experimental protocol for steady-state quenching studies.

Acknowledgments

This work has been supported by National Institutes of Health Grants DK-39548 and GM-39750. Our discussions with Drs. Carol A. Hasselbacher and J. B. Alexander Ross are greatly appreciated. We also wish to thank Evan Waxman for programming assistance and helpful discussions.

[22] S i m u l t a n e o u s Ana lys i s for T e s t i n g o f M o d e l s and

P a r a m e t e r E s t i m a t i o n

By DONALD F. SENEAR and DAVID WAYNE BOLEN

Introduction

Physical measurements on biological systems are designed to extract some thermodynamic quantity or molecular property which describes the molecule or system of interest. The property or quantity in question generally originates from some theoretical framework or model imagined to emulate the real system. The experimentalist is then faced with the problem of obtaining realistic estimates of the parameters of interest while,

Copyright © 1992 by Academic Press, Inc. METHODS IN ENZYMOLOGY, VOL. 210 All rights of reproduction in any form reserved.