Fluorescence Correlation Spectra. Conceptual Basis and Theory

BIOPOLY MERS VOL. 13, 1-27 (1974)

Fluorescence Correlation Spectroscopy. I. Conceptual Basis and Theory

ELLIOT L. ELSON, Department of Chemistry, Cornell University, Ithaca, New York 14850 and DOUGLAS MAGDE,* Department of

Applied Physics, Cornell University, Ithaca, New York 14850

synopsis We describe a method for determining chemical kinetic constants and diffusion coeffi-

cients by measuring the rates of decay of spontaneous concentration fluctuations. The equilibrium of the system is not disturbed during the measurement. We measure the number of molecules of a specified type in a defined open volume as a function of time and compute the time course of the deviations from the thermodynamic mean concentration. The method is based on the principle that the rates of decay of spontaneous microscopic fluctuations are determined by the same phenomenological rate coefficients as those of macroscopic departures from equilibrium which result from external perturbations. Hence, an analysis of fluctuations yields the same chemical rate constants and diffusion coefficients as are measured by conventional procedures. In practice the number of the specified molecules is measured by a property such as absorbance or fluorescence which is specific and sensitive to chemical change. The sample volume is defined by a light beam which traverses the cell. As the molecules appear in or disap pear from the light beam, either due to diffusion or chemical reaction, their concentration fluctuations give rise to corresponding fluctuations of the intensity of absorbed or emitted light. This paper presents the theory needed to derive chemical rate constants and diffusion coefficients from these fluctuations in light intensity. The theory is applied to three examples of general interest: pure diffusion in the absence of chemical reaction; the binding of a small rapidly diffusing ligand to a larger slowly diffusing macromolecule; and a unimolecular isomerization. The method should be especially useful in studying highly cooperative systems, relatively noncooperative systems with intermediate states closely spaced in free energy, small systems, and systems not readily subject to perturbations of state.

INTRODUCTION This paper describes a method for characterizing the kinetics of chemical

reactions and the diffusion coefficients of the reactants by measurements made a t thermodynamic equilibrium. The components of the system, even a t equilibrium, are continually fluctuating about their thermodynamic mean concentrations. The fluctuations are due to random variations in the numbers of molecules which are formed or eliminated by the chemical reactions and, if the observed region of the solution is open,

* Present address: Department of Chemistry, Washington State University, Pull- man, Washington 99163.

1

@ 1974 by John Wiley & Sons, Inc.

2 ELSON AND MAGDE

which enter or leave the region by diffusion. (We disregard conccntra- tion fluctuations resulting from fluctuations in temperature or volume. These will be of negligible magnitude for all situations considered.) Ob- servation of these spontaneous fluctuations can yield a complete kinetic description of the system equivalent to that obtained by conventional means. This is because the same phenomenological diffusion coefficients and chemical rate constants which describe the rates at which the system returns to equilibrium after a macroscopic perturbation of its state also describe the rates at which spontaneous fluctuations decay.' With conventional methods for determining kinetic coefficients, for example using mixing procedures or temperature jumps, one observes the rates at which a system returns to equilibrium after a macroscopic perturbation. With the method we describe, however, these same coefficients are determined by observing the decay of spontaneous fluctuations without disturbing the equilibrium of the system. In principle we simply measure the number of molecules of a specified type in a defined open volume of solution as a function of time. The average time behavior of spontaneous concentration fluctuations is then computed. We call this method concentration correlation spectroscopy, or CCS.

In this paper the general principles and theoretical basis of CCS are presented along with a detailed application of the theory to examples of general interest. A consideration of experimental methods and problems as well as a description of a particular experiment is presented in the accompanying paper.2

A sensitive and specific index of concentration such as optical absorption or fluorescence is used to measure the number of molecules of a specified type in a defined system volume. We devote this discussion to an experimental design in which this volume is open and is defined by a beam of light which traverses the solution. For example, if fluorescence is used (fluorescence correlation spectroscopy, or FCS), the emitted intensity is proportional to the number of fluorescing molecules in the illuminated region. As the number of these molecules in the light beam changes through diffusion and reaction, the fluctuations in number are represented as fluctuations in fluorescence intensity. These are recorded as fluctuations in the photocurrent of a photomultiplier exposed to the fluorescent radiation. An alternative arrangement would have a closed system volume uniformly illuminated. Then fluctuations of light intensity could arise only through the chemical reaction. Interpretation of experiments with this alternative design can be carried out by a straightforward modification of the theory presented in this paper.

CCS should prove valuable in the investigation of many problems of biophysical interest. For example, when a protein or nucleic acid molecule is transformed from an ordered to a disordered conformation or vice versa it appears to pass through many partially ordered intermediate states closely spaced in free energ~.~-6 Kinetic studies by CCS could provide important information on the mechanism of these transformations. If

The concept of the experiment is extremely simple.

FLUORESCENCE CORRELATION SPECTROSCOPY. I 3

each step of the process takes place more or less independently of the next (SO that the transformation is relatively noncooperative), the time course of spontaneous conformational fluctuations of partially disordered molecules should reflect transition rates among small numbers of adjacent intermediate states. This should be simpler to interpret than the response to a finite perturbation during which the molecules traverse a large range of intermediate states. CCS should also be useful in studies of the kinetics of highly cooperative aggregation phenomena. The rate of the aggregation may depend nonlinearly on the magnitude of a finite perturbation of state. Mechanistic interpretation of an experiment of this sort is difficult. Rieasuring the kinetics with no perturbation by observing spontaneous fluctuations, however, ensures linear behavior. Furthermore, the methods described here allow the observation of systems containing relatively few of the molecules under study. This is necessary for the measurement of spontaneous concentration fluctuations. However, this capability could also facilitate investigations of surface phenomena, for example, in mem- branes, or studies of the behavior of individual cells which have a small capacity for optically labeled molecules and may be difficult to subject to controlled perturbations.

Much theoretical' and some experimentals work has been devoted to the study of reaction kinetics in systems a t equilibrium by measurements of quasi-elastic light scattering. While this technique has been extremely successful in the determination of diffusion coefficients of macromolecules,g its application to the measurement of chemical reaction rates has proved difficult. This is because the refractive index changes which are the indicators of reaction progress are usually quite small. Differences in diffusion coeficients of reactants and products can also serve as reaction indicators.' Here too the experiment is difficult unless the changes in diffusion coefficients are relatively large as in cooperative aggregation equilibrium. lo

In contrast, a key feature of FCS is that it takes advantage of the large fluorescence changes which can result from chemical reactions and so increases the feasibility of observing the decay of concentration fluctuations via these reactions. Another advantage of fluorescence detection is its chemical specificity which will prove useful in interpreting complex kinetic measurements and in observing particular reactions in complex mixtures such as biological systems.

Studies of fluctuations in the number of particles in a defined open volume of solution have recently been carried out using quasi-elastic light scattering. *l.12 These fluctuations result from diffusion. Chemical reaction was not considered. Although the method of detection is different, these fluctuations are analogous to those seen by FCS.

Non-optical techniques may also be used to observe fluctuation relaxation. Feher and Weissman have proposed the use of conductivity measurements as indicators of chemical kinetics and have investigated the rate of the ionization reaction of beryllium sulfate by observing conductivity fluctuations. ~ 1 4

4 ELSON AND MAGDE

Our development of a theory for FCS begins with a phenomenological discussion of the variation over space and time of spontaneous concentration fluctuations in ideal diffusive, reactive systems. This general theory is next adapted to an analysis of the optical measurements to be obtained as results in a fluorescence experiment. The theory is then applied to two examples: pure diffusion in the absence of chemical reaction and the binding of a small rapidly diffusing ligand to a large, slowly diffusing macromolecule. (A simpler chemical example, a unimolecular isomerization, is discussed in Appendix 3.) Finally, the results and properties of the method are discussed. A preliminary account of this work has already appeared.15

CORRELATION OF CONCENTRATION FLUCTUATIONS

The extension in space and duration in time of spontaneous concentration fluctuations can bc characterized in terms of phenomenological rate coefficients and the equilibrium properties of the system. This is most readily done in terms of a concentration correlation function, +jz(r,r‘,T),

which compares the size of the fluctuation of component j at one position and time to the size of the fluctuation of component 1 a t another position and a t a time T later. Let the concentration (in number of molecules per unit of volume) of the j t h chemical component at position r and time t be C,(r,t) . The thermodynamic mean concentration of this component, c3 = (Cj(r, t)) , is independent of position and time. (( ) denotes ensemble average.) The fluctuation a t r and t is 6C3(r,t) = C,(r,t) - c,. Then the concentration correlation function is defined as

q+&,r’,T) = (6Cj(r,t)6Cc(r’,t + T ) ) ( 1 4

(The principle of microscopic reversibility requires that &&,r’,T) =

+jt(r,r’,-T) = &j(r,r’,lT\). For simplicity 1.1 is written as T in the following.) The system is supposed to be stationary, and so its average properties are independent of time. Hence, + depends on the interval T but not the absolute time t :

4dr,r’, 7 ) = (6Cj(f,0)8CE(r’, 7 ) ) (1b)

For intervals which are short compared to the characteristic times for diffusion and reaction +jc(r,r’,T) = +jl(r,r’,O), which is independent of rates of change and can be calculated from equilibrium properties alone. For intervals long compared to the characteristic times, fluctuations are uncorrelated: +jl(r,r‘,co) = 0. The rates a t which &z decays to zero with increasing T give a measure of how quickly diffusion and chemical reaction change the configuration of the system.

The interpretation of experimental measurements requires that correlation functions +jl(r,r’,T) be expressed in terms of the equilibrium properties and rate coefficients of the system. This in turn requires that 8Cz(r ,T) be expressed in those terms. The number of molecules of a specified type


in an open region can change by diffusion or by chemical reaction. For changes due entirely to diffusion, SC,(r,T) is determined by the diffusion equation :

where D, is the diffusion coefficient of component 1. to chemical reaction are described by

Changes due entirely

where m is the number of reactive components and the elements of the matrix Tzk are composcd of chemical rate constants and the equilibrium concentrations of the reactants. l6 Conventional linearized rate equations hold here because the deviation from equilibrium of the concentration of any of the components must be small. When both reaction and diffusion occur, these processes are coupled so that the rates of relaxation of concentration fluctuations are determined for ideal solutions by the system of equations :I

The spatial variation of SC, results from diffusion. Since the type of experiment discussed here uses an open sample volume, diffusion will always be a contributing process.

Eqs. (4) have been treated previously in discussions of the possible application of quasi-elastic light scattering to chemical kinetics.’ Our analysis differs from this earlier work both by being cast in terms of space-time concentration fluctuation correlation functions and by our having to take into account the size and shape of the sample volume.

Development of the correlation function, &,(r,r’,T), requires that SC,(r,T) be expressed as a solution of Eqs. (4). The boundary conditions for the equations are determined by the spatial constraints on diffusion. We suppose that the sample solution is contained in a cell with plane faces per- pendicular to the beam of incident light which excites the fluorescence. The z axis of coordinates is chosen to be along the axis of propagation and in the center of the incident beam of light. The dimensions of the sample cell in the x and y directions are much larger than the diameter of the beam. Hence diffusion may be supposed to range from -a to +a in any x,y plane in the cell, and the fluctuation amplitude may be supposed to vanish at those limits. The optical path length in the z direction is L, defined by the distance between the cell windows. Hence, there are impermeable walls normal to the z axis at z = 0 and z = L which cause the diffusive flux to vanish on these planes. Therefore, the boundary conditions for diffusion are

8C(r,T) = 0 a t z,y = fa (58)

6 ELSON AND MAGDE

d&C(r,r) a2

A conventional Fourier transformation is applied

~- - 0 a t z = 0,L

to GCl(r,t) along the (5b) it is appropriate x and y axes. However, because of the condition Eq.

to develop GC,(r,t) in terms of a Fourier cosine series along the z axis. transformation of GC,(r,t) takes the form

The

where F,[ and n = 0,1,2, . . . . . The inverse transformation is

8Cl(r,7) = F,-l[CZ(v,7) 1

] indicates a transformation to the v-variabIes: v = (vX,vy,n);

Transformation of Eq. (4) yields

where M,, = T,, - DZv26,, and v2 = vx2 + vvz + (nIr/L)2. Eqs. (8) constitute a system of linear first-order differential equations

with constant coefficients for the Fourier components of the concentratio4 fluctuations. The solution of this system may be obtained by standard methods17 and may be expressed as

s= 1 k = 1

where the X z ( s ) are the components of the right eigenvector corresponding to the eigenvalue, The (X-l)t(a) are the components of the inverse matrix of the eigenvectors. are the left eigenvectors of the matrix M.

C,(v,O). function +jI(r,r’,T) in terms of Cz(v,r ) using Eq. (7).

of the matrix M. It is easy to see that the

Further development of Eq. (9) requires specification of the initial values This may be accomplished by first expressing the correlation

+jt(r,r’,T) = @CA~,OF,~-~{ G ( v , ~ ) f )

Since ensemble averaging and Fourier transformation are linear operations, their order may be interchanged:

‘#Jjdr,r’,r) = FT#-’{ (~j (r ,O>~dv,T)) )


Substituting Eq. (9) into this expression,

4j,(r,r’,T) = F,,-I C x,(s) exp(A(s)T) C (x-~),(~) (K’j(r,O)G(v,O))

Using Eq. (6) and again interchanging ensemble averaging and Fourier transformation,

L k

= ~ ~ ~ - 1 C Xz(s) exp(x(s)T) C (x-l),(s)F,[~jlr(r,cr’,o) 1 1 i s k (10)

The correlation function 4ik(rlr”10) is obtained by taking account of the fact that the experiments are carried out with very dilute solutions which are, therefore, essentially ideal. Hence, the statistics of solute molecules are independent, and the fluctuations of concentrations of different species must be uncorrelated. Similarly the spatial correlation length of concentration fluctuations of a particular species must be very small. Therefore

4,(r,r”,O) = (K’j(r,O)sC,(r’’,O)) = Cj6jks(r - r”) (11)

(Poisson statistics determine that the mean-square fluctuation of C j in a unit of volume is cj; cf. Kirkwood and Goldberg18). Substitution of Eq. (11) into Eq. (6), execution of the indicated integrals, and substitution of the result into Eq. (10) yields

tPil(r,r’,T)

+ 2 5 [ CX2(s) exp(X(s)T)(X-l)j(s) cosE cosnd]} L (13) n = l L

This correlation function has the expected symmetry property :

4jz(r,r’,T) = 4ij(r,r’,T) (14)

This is demonstrated in Appendix 1. The correlation function may now be evaluated for specific diffusive and reactive systems for which the X(s) and However, as will be seen, it is simplest to sub- stitute Eq. (13) directly into the expression for the photocurrent correlation function developed below.

may be determined.

8 ELSON ANI) MAGDE

PHOTOCURRENT CORRELATION FUNCTION

Fluctuations in concentration arc seen cxpcrimentally as fluctuations in a photomultiplier current. The time course of thcse fluctuations is represented in a photocurrent autocorrclat.ion function. Either absorbance or fluorescence could be used to measure conccmtrations. Fluorcsccnce is emphasized here to accommodate the cxpctriment described in t.hc accompanying paper.2 'However, the modifications necessary for application to other means of detection are simple and obvious.

It is convenient to use a laser to provide thc incident exciting light in an FCS experiment. because of the possibility of obtaining high intensity which can be focuscd into a very narrow beam from this source. How- ever, it should be emphasizcd that., in contrast to quasi-elastic light- scattering experiments, the spatial coherencc of the laser light is not a factor in FCS experiments. Thc coherence time of the broad (:mission spectrum is sec uncouplcs the fluorescence from the oohcrent exciting light. Hence, thew arc no systematic interference ttffccts in the fluoroscent radiation, and so fluorescent intensities are simply added. Therefore, tho measured phot.omultiplier current, i ( t ) , resulting from the fluorescent radiation is proportional to the numbers of molecules of various kinds in the illuminated sample volume. If the opt.ica1 extinction coefficient and fluorescerit quantum yield of the j t h component, measured at the excitation arid emission wavelengths appropriat.e to the experiment are tj and Q j , respec- t.ively, then

sec arid a random fluorescent emission delay of

3 J

The factor 9 accounts for the quantum c.fficicnc*y arid gain of the photomultiplier as well as geometrical and optical filtering losses inherent in the experimental arrangement. I ( r ) is the intensity of thc incident laser light at position r. The low concentration of absorbing solutes ensures that the incident intensity is essentially constant along the z axis. In a transverse plane the 1asc.r beam is assumed to have a Gaussian intensity profile: Z(r) = 10 exp [ -2(z2 + y2)/zuZ] where l o is the maximum intensity at the center of the beam and w is the radius at which ] / I o = c>xp(-2). Since u: is small compared to the transverse dimensions of the cell, the integrations can be extended over the ranges --co 2 x 5 00 and -a 5 y 5 -co.

Concentration fluctuations camp fluctuations in the photocurrent:

ai(t) = z ( t ) - (2) = 9 c C , Q , J 1(r)~ , (r , t )d3r 3

Hence, the expwimental manifestation of the correlation of concentration fluctuations is the temporal autocorrelation of the photocurrent, G(7) :

m m

3 - 1 1 - 1 G(7) = (6i(0)6i(7)) = c c GJ7) (16a)


where we have defined the partial contribut.ion of +jl(r,r’,T) to G ( T ) as

G , ~ ( T ) = g2q&Ql s I(r)I(r’)+jl(r,r’,~)d3r d3r’ (16b)

Here we have omitted the shot noise and noise from other extraneous sources for simplicity. The symmetry condition, Eq. (14), reduces the number of distinct partial contributions to the photocurrent correlation function from m2 to ‘12m(m + 1). Hence, the number of terms in the summation in Eq. (16a) may be correspondingly reduced by combining the identical contributions of G , ~ ( T ) and G,(T) to give

m

A direct evaluation of G(0) provides a check on later calculations and Using Eq. (11) it is easily serves as a guide in designing experiments.

shown that

G(0) = g2L (“> c ( c ~ Q ? ) ~ ~ , m u 2 j

where P, the laser power, is

P = J:m dx J’-mm dyl(r) = 1,,aw2/2

Since the mean photocurrent is (i) = gPL C ejQiCj, the relative root mean

square fluctuation amplitude is j

Strongly absorbing and fluorescing compounds contribute dispropor- tionately to the total amplitude. In a system with one fluorescent component the relative signal is

Therefore the relative photocurrent fluctuation is inversely proportional to the square root of the number of fluorescent molecules in the sample volume, defined as *w2L. This fact, expected for a Poisson distribution, illustrates the requirement for small numbers of molecules in the sample volume.

When Eq. (13) is substituted in Eq. (16b) and the integrations over x and z’ executed, it is seen that only terms with n = 0 contribute to G ( T ) . After integration also over x, x‘, y, and y‘ one obtains:

10 ELSON AND MAGDE

where

Hj1 = g2Eje1&jQ~P2L€ji/rw2 (20b) The interpretation of the measured G ( r ) in terms of the elementary

processes of diffusion and chemical reaction is now resolved into, first, obtaining the eigenvalues and eigenvectors of the matrix M and, then, exe- cuting the integrals in Eq. (20a). The development of expressions for the X(s) and X(a) must be accomplished individually for the specific reaction mechanisms which determine the form of T. A simple example, diffusion of a single fluorescent solute which undergoes no chemical reactions, clarifies the calculation and also provides a result of practical utility.

PURE DIFFUSION

We consider an example in which there is a single nonreactive fluorescent Since concentration fluctuations of this solute

Application solute, say component 1. can relax only by diffusion, GC,(r,t) is determined by Eq. (2). of the Fourier transformation, Eq. (6), yields

Hence, C,(v,r) = C,(v,O) exp [ - Dlv2r]

In this example the fluctuations of component j are not coupled to the fluctuations of any other component. Then the photocurrent correlation function may be obtained from Eq. (20) by setting the components of X(s) and (X-l)(s) equal to unity and by omitting the summation over s:

Integration over the Fourier components yields the final result for a Gaussian beam profile :

G(7) = Hii[1 + ( ~ / r J l - ~ (21)

where r , = w2/4D2, a characteristic diffusion time for a Gaussian laser beam (sample volume) of radius w. This result is consistent with Eq. (19) for 7 = 0.

It may also be of interest to observe that when the integrations in Eq. (13) are carried out for this example, the correlation function for simple diffusion is found to be


Then Eq. (21) may be obtained by substitution of Eq. (22) into Eq. (16b). It is a noteworthy property of Eq. (21) that the time course of G ( T ) is

determined by the factor (1 + T / T J - ~ even though the concentration correlation function, Eq. (22), has the expected exponential time dependence. This property, which will also appear in more complex examples, results from the convolution of the exponential Fourier components of diffusion with the Gaussian laser beam intensity profile.

AN EXAMPLE OF COUPLED DIFFUSION AND CHEMICAL

MACROMOLECULE REACTION: BINDING OF A SMALL LIGAND TO A

The example to be discussed in detail is of general interest and applies to the experiment described in the accompanying paper.2 (A simpler example, the unimolecular isomerization, is presented in Appendix 3.) The reaction is assumed to be a simple biomolecular process with rate constants kf and kb:

kr

k b A + B - C (23)

The equilibrium constant is K = kf/kb = CC/CA.CB. The matrix M is

I - ( v ~ D A + k f c ~ ) - k f C A kb

M = - k C - ( ~ D B -k kfC.4) kb

[kf; kf C A -(V2Dc + kb) The ligand is very small compared to the macromolecule. calculation may be simplified by setting DA = DC = D. eigenvalues of M are

A‘O’ = - V D 2

Hence, the Then the three

2X‘” = - [ u ( D -I- DB) 4- R ] f [(.‘A + R)2 - 4v2AkrC~]*” (24)

where A = D - DB. the reaction (23):16 R = kf(CA + CB) + kb.

R is the conventional chemical relaxation rate for The right eigenvectors are

1 N ( - , XC-, = - [l,F(-), -1 ] (25c)

The components are listed in the order A, B, C. Also

12 ELSON AND MAGDE

and N(O), N ( + ) , and N ( - ) are normalization constants which ensure that X(S).(X-')(~) = 1. The left eigenvectors, with components listed in the same order, are

Because of the symmetry property, Eq. (14), there are a t most six independent correlation functions &&,r',T). The quantities required for the calculation of these functions or for the direct evaluation of their contributions to G(T), Eq. (20), are the

zjl(7) = [C 8 x~(*) (x-~)~(s ) exp(X ( s ) T ) L o (27)

When these quantities are written using Eqs. (24), (25), and (26), three relationships become immediately apparent :

ZBA = -ZBC (28%)

(28b)

(28c)

Zcc + Z C A = exp(X(O)T)

Z A A - K C B Z C ~ = (1 - KC^) exp(X(O)T)

Hence, the following three Z,,(T) are sufficient for the further development of G(T) :

(29a) ZBB(T) = [F(- ) - F(+)]-'[-F(+) exp(X(+)T) + F(-) exp(X(-)~)]

Z c c ( ~ ) = (1 + KCB)-1

ZBC(T) = [F-) - F(+)]-'[exp(X(+)~) - exp(X(-)~)] (29c)

It is apparent that Z,,(O) = tijl which is a consequence of the orthogonality and normalization of the eigenvectors. The contributions to G(T) from GBB(T), Gcc(T), and GBC(T) may be calculated explicitly using Eqs. (29). Then the contributions from GBA(T), GcA(T) , and GAA(T) may be computed from these results using Eqs. (28).

In order to evaluate Eq. (20) in closed form it is necessary to simplify the expressions for the Z,,(T) by introducing approximations. The approximations used here are valid for the experiment described subsequently.2 Since the macromolecule is much larger than the ligand, DB >> D. Experi- ments are carried out under conditions such that CB - CC << %A - K-I. Finally, the characteristic rate of the chemical relaxation is greater than


the rate of diffusion across the laser beam. we can use a -= v2A/R as an expansion parameter. kfC,/R.

Hence, DB/Rw’ << 1. Hence, Also we set /3 =

Then to first order in a:

A‘+’ = -?[DB + PA]

A(-) = -R - v2[D - P A ]

ZBB

ZCC

(1 - p)[l + 2pa] exp(k(+)T) + p[1 + 2(p - l ) a ] exp(X(-).r)

K ~ B exp(X(O’7) + p[1 + 2(p - l ) a ] exp(X(+)7)

+ (1 - @(l + 2/34 exp(X(-)T)

ZBC % p[1 - (1 - 2p)a][exp(X(+)7) - exp(X(-).r)l

from Eq. (20): The contributions of the various +,i(r,r’,7) to G(T) may be evaluated

GjZ(7) = H j l w’ 4u dv,dv, exp[ - (vz2 + vv2) :] [Zj1(7) ],+a (30)

14 ELSON AND MAGDE

From Eq. (22) we have also

GBA = - (HBA/HBC)GBC (334

(33b)

(33~)

Substitution of the expressions for G,,(T) into Eq. (16c) then completes the evaluation of G(7) in a form which is suitable for the interpretation of experiments.

GCA = HCA(1 + 7 / 7 A ) - l - (HCA/HCC)GCC

GAA = H A A ( ~ - K C B ) ( ~ + T / T A ) - ~ + (KCBHAA/HCC)GCC

RESULTS AND DISCUSSION

We have described a method for determining diffusion coefficients and chemical rate constants with an experiment which is very simple in concept-a measurement of the variation with time of the number of molecules of specified solutes in a defined open volume. The system remains at equilibrium, undisturbed during the measurement. Since the molecules can enter and leave the volume only by diffusion and chemical reaction, the rate coefficients for these processes determine the rate of evolution of spontaneous concentration fluctuations. However, the interpretation of the experimental observations in terms of the elementary processes, the individual chemical reactions and the diffusion of particular solutes, may require an involved train of calculation. As in any study of complex chemical kinetics it is first necessary to postulate a model reaction scheme from which the matrix T, Eq. (3), can be derived. Comparison of the experimental results with the behavior predicted from the model verifies the suffciency of the latter, but not its necessity. The task of interpreting coupled reaction schemes of this kind is a familiar one to chemical kineti- cists. l 6 However, the experiment proposed here involves not only the coupling together of individual chemical reactions but also the coupling of these reactions to diffusion. As Eq. (8) shows, these elementary processes are coupled together in the matrix M. Hence, the theory yields an analysis of the correlation functions in terms of “normal modes” of flue- tuation with characteristic rates, and amplitudes X l ( s ) ( X - l ) j(’)Cjj.

These in turn are functions of the diffusion coefficients and chemical rate constants which describe the elementary processes. Ultimately the dependence of the correlation functions on the spatial distribution of solutes (and therefore on w) results from diffusion while dependence on concentrations results from chemical reactions. However, because of the coupling of diffusion and reaction a particular X(S) depends in general both on w and concentrations. (An exception is the mode, Eq. (32b), which for this particular kinetic model reflects only one process, the diffusion of macromolecule and complex.) Hence, an interpretation of experimental results requires a study of the dependence of the various normal modes on w and concentration.

FLUORESCENCE CORRELATION SPECTROSCOPY. I 1.5

Because of the symmetry condition, Eq. (14), an open system yields at most l12m(m + 1) independent concentration correlation functions. Of these m are the autocorrelation functions 4jj(r,r’,T). These functions have initial amplitude $&,r’,O) = C$(r - r’) in an ideal system. The cross correlations 4j2( j + 1) are different from the autocorrelation functions in an important respect. In a system in which diffusion but not chemical reaction takes place the cross correlation functions vanish at all 7. In a reactive system the cross-correlation functions have zero amplitude initially [&&,r’,O) = 01; however, they develop finite amplitudes a t later times. This is because a spontaneous fluctuation in the concentration of any one reactant will drive reactions connecting it to another reactant thereby causing a correlation in their concentration changes. Of course, all correlation functions vanish as 7 + 05.

The number of independent concentration correlation functions may be reduced from I12m(m + 1) by relationships which are specific to particular reaction mechanisms. This is illustrated for the bimolecular combination reaction, Eq. (23), by Eqs. (28). These three equations reduce the number of independent concentration correlation functions for this three component system from six to three. Eqs. (28) are consequences of the assumption that DA = DC and of the stoichiometry of the reaction.

The effects of reaction and diffusion in this particular coupled system may be seen in greater detail in the approximate expressions, Eqs. (31) and (32). Comparison of Eqs. (32) and (21) show that the relaxation of fluctuations by diffusion is described by the same function of time as in a system with no reaction. The normal mode corresponding to Eq. (32b), represents simple diffusion of the macromolecule and complex, with characteristic time T A . The other two normal modes depend on the characteristic times d+) and d-) . These more complex characteristic times depend on the chemical equilibrium. For example, T ( + ) = T B ( ~ + C C / ~ B ) is determined by the rate of diffusion of the small ligand as it is slowed by interaction with the larger macromolecule. The term A(- ) , Eq. (32d), is dominated by the exponential time dependence of chemical relaxation. The characteristic time for chemical relaxation, R-I, is the same as that seen in conventional perturbation experiments of bimolecular combination reactions.16 The fact that simple diffusion times depend on beam radius but not on reactant concentrations, that R depends on concentrations but not on w2, and that the composite times T( * ) depend on both w 2 and concentrations prove to be useful in disentangling the various contributions to an experimentally observed fluctuation correlation spectrum.

The contributions to G(T) of the autocorrelation GCC(T) and the cross- correlation function, GBC(T) are shown in Figures 1 and 2, respectively, together with the individual modes A ( + ) and A ( - ) for specific conditions which are given in the figure captions. These parameters are compatible with the approximations used to derive Eqs. (31) and (32): The autocorrelation function Gcc(r) shows a rapid component of relatively large amplitude which is due to the chemical relaxation. The A ( + ) and A(-)

16 ELSON AND MAGDE

time (seconds)

Fig. 1. Partial correlation function GCC(T) and components A(+) and A(-) . In this example, with relatively low ZA, the observable correlation function GCC(T) is dominated by the chemical kinetic effects in A(-) . (The component A(O) is smaller than on this scale over the entire time range.) The solid curves were computed by numerical integration of Eq. (30) using the exact amplitude functions of Eq. (29b). The points are calculated from the approximate expressions Eqs. (31) and (32). The approximation is fairly accurate in this example for which @ = 0.2 and D B / R w ~ = 0.048. The conditions used in the computation are K = 5 X lo6 M-1, k b = 100 sec-1, DA = 1.5 x 10-7 cm*/sec, DB = 10 X DA, CA =

5 X

The influence of diffusion is smaller.

M, CB = 2 X 10-9 M, and w = 5 x cm.

contributions to G B C ( 7 ) are equal and of opposite sign at r = 0 so that the cross-correlation function has zero amplitude initially. The rapid increase in G B C is due to the chemical relaxation in A(- ) . The slower decay is determined by the diffusion time r(+).

In principIe, fluctuation relaxation experiments provide an experimental decomposition of the amplitude functions which are observed in conventional perturbation relaxation experiments. This decomposition results

5 time (seconds)

Fig. 2. Partial correlation GBC and components A(+) and A(-) . This figure shows the rapid increase in GBC due to chemical relaxation ( A ( - ) ) and the subsequent slow, diffusive decay (A(+)) . Methods of computation, symbols, and conditions are as in Figure 1.


from the condition expressed in Eq. (11): the average initial time correlations of unlike species in an ideal system at equilibrium vanish. Con- sequently, Gj2(.) can be computed using a single component of each eigenvector (X-l)(s) as is expressed in Eq. (20). In contrast, a perturbation provides a correlated displacement of the equilibrium concentrations of all components. Hence, the deviation of the concentration of component j from its equilibrium value as a function of time after the perturbation is given as

SC,(.) = C Yj(s) exp(A(s'7) C (Y-')2("~C,(~) where 6C,(O) is the initial deviation of component I and Y(s) and (Y-1)(8)

are the eigenvectors of an appropriate matrix, e.g., T in a typical chemical relaxation experiment. Hence, all components of the (Y-l)(s) are required to interpret the simplest datum of a perturbation experiment. Of course, practical application of this fact requires experimental observation of individual cross-correlation functions Gig(.) which may prove difficult to accomplish.

The calculation of the spectrum of fluctuation relaxations in open systems has the additional difficulty, not encountered in perturbation experiments in closed systems, of evaluating the integral over Fourier components in Eq. (20). This difficulty is avoided in measurements in closed systems with uniform illuminating beam profile. Photocurrent fluctuations are then due only to chemical reaction; an integration over spatial Fourier components is not required. An experimental arrangement of this type avoids the potential complications of interpretation presented by the coupling of reaction and diffusion and so should be useful in studies of chemical kinetics by FCS.

The photocurrent correlation function observed for simple diffusion in an ideal solution using an incident beam with Gaussian intensity profile is given in Eq. (21). The time 7, = W2/4D1 is the characteristic time necessary for diffusion of component I the root mean square distance w/d2.lg The time dependence, (1 + T / T ~ ) - ~ , is characteristic of the Gaussian beam profile. A different beam profile will in general yield a different time dependence. For example, the uniform circular beam with I(x,y) = lo (a constant) for 0 5 x 2 + y2 5 w2 and I = 0 elsewhere yields the following photocurrent correlation function for simple diffusion of component 1:

S 1

G ( T ) = HH b exp(-2b) [10(2b) + 12(2b) J

} (34) (- 1)'(2 + 2k)!(k + 1) !(b"')

[(2 + k)!12(k!)2 + 2 C k = O

In this equation I 0 (2b) and I2 (2b) are modified Bessel functions of zero and second order, and b = T J T . This result is derived in Appendix 2. Substitution of Eq. (11) into Eq. (16b) with I(x,y) given as the uniform

18 ELSON AND MAGDE

circular profile shows that the root mean square fluctuation amplitude for T = 0 is the same as for the Gaussian profile. A comparison of the time course of the photocurrent correlation functions for the Gaussian and the uniform beams is shown in Figure 3. It is gratifying to observe the close similarity in the time course for diffusion for the two beam profiles. This suggests that the results obtained by fluorescence correlation spectroscopy do not strongly depend on the exact shape of the beam.

A distinctive feature of our present technique is the use of fluorescence or absorbance as the measure of concentration. The feasibility of detect- ing relaxation of fluctuations via the chemical reaction is enhanced by the fact that the reaction can cause large changes in these properties. One might ask, however, whether chemical relaxation can be observed even when the reaction has no effect on the observed optical properties. Sup- pose, for instance, that the combination of the 1,igand and the macromolecule does not change the extinction coefficient or fluorescence quantum yield of the former. Then at a wavelength a t which the macromolecule does not absorb EA = 0 and &A = 0 while EB and E C = E and QB = QC 5 Q,

r/rD Fig. 3. Comparison of photocurrent correlation function for free diffusion for Gaussian

The solid curve was calculated from Eq. (21) for The circles were calculated from Eq. (34) for uniform circular

and uniform circular beam profiles. the Gaussian profile. profile.


It is readily shown that the contribution to G(T) corresponding to T ( - )

vanishes to first order in the expansion parameter a. Hence, to this order of approximation relaxation of fluctuations via the chemical reaction cannot be detected without a chemical effect on the optical properties of the chromophore. Nevertheless it is not true in general that observation of relaxation via the reaction necessarily requires a chemical effect on the optical properties. This is demonstrated explicitly for the unimolecular isomerization in Appendix 3. (However, detection of the chemical relaxation may be impractically difficult without a large chemical effect on optical properties.) A potentially important advantage in the use of absorbance or fluorescence as fluctuation indicators is the chemical specificity of these properties. This feature may be useful both in resolving partial contributions to the photocurrent correlation function in well-defined systems and in observing fluctuations in the concentration of particular species in complex mixtures.

Finally, we should make some comment on factors neglected in this discussion. We have not accounted for effects of thermodynamic non- ideality or for the contributions of rotational diffusion to the observed correlation functions. Since the experiments are designed to be carried out at very low concentration, the former should be small. Correlation of rotational fluctuations should be observable in principle with the addition of a polarization analyzer in the fluorescence detection optics and the use of polarized exciting light. However, rotational relaxation times are expected to be much shorter than the characteristic times for translational diffusion and reaction in the proposed experiments. We have also neglected the effects of extraneous noise such as shot noise and competing processes such as photolysis of the observed chromophores. These sub- jects will be considered in detail in the accompanying article.2

SUMMARY

1. Measurements of the temporal correlations of spontaneous concentration fluctuations can be used to determine chemical rate constants and diffusion coefficients in reactive systems without disturbing their equilibrium. We present a theory for the interpretation of experiments in which fluctuations in the concentration of specified reactants are observed by fluorescence.

The contributions of diffusion to the temporal photocurrent autocorrelation function, G(T), are factors of the form (1 + T/TD)- ' for an incident laser beam of Gaussian intensity profile. The characteristic diffusion time, TD, depends on the radius of the laser beam. There is a different functional dependence on T for other beam shapes. Nevertheless, the overall behavior does not seem to depend strongly on the details of the beam shape (Fig. 3).

Since there is no spatial dependence of the chemical kinetic equations, the factors in G(T) which represent chemical kinetics retain their

2.

3.

20 ELSON AND MAGDE

usual exponential character. The Characteristic times for chemical reactions are independent of beam size and shape.

The coupling of diffusion and chemical reaction requires that the time course of the fluctuations be analyzed in terms of normal modes of relaxation each of which can contain contributions from many different elementary processes. Hence, there are characteristic times for particular normal modes which refer simultancously to diffusion and reaction. For example, in the discussion of the bimolecular combination reaction we showed that the A ( + ) mode represents the diffusion of the small ligand slowed by interaction with thc slowly diffusing macromolecule.

The relative magnitude of the fluctuation signal varies inversely with the number of observed molecules in the sample volume. Hence, low concentrations, small sample volumes, and high sensitivity of detection are required.

The phenomenological rate coefficients determined by observation of spontaneous concentration fluctuations in systems in equilibrium are precisely those sought in conventional perturbation experiments. Whereas the interpretation of perturbation kinetic studies of cooperative nonlinear systems may be difficult, the interpretation of fluctuation experiments on these systems is simplified by the necessary linearity of the equations describing the rate of fluctuation decay.

A large change in the fluorescence as a result of the chemical reaction increases the possibility of observing the decay of concentration fluctuations via the reaction and therefore of determining the chemical rate constants.

4.

5.

6.

7.

APPENDIX 1

Symmetry of Concentration Fluctuation Correlation Functions

We demonstrate here the validity of the symmetry condition, Eq. (14). Symmetrization of the matrix M, Eq. (8), is the first step. Wei and Praterz0 have

shown that the matrix of kinetic constants of a complex monomolecular system can be symmetrized with the similarity transformation: Here S' is a symmetric matrix, K is the original matrix of kinetic rate constants, analogous to T [Eq. (3)] but for a system of fir$-order reactions, and A is the diagonal matrix Ail = d&. (Then ( A - l ) j l = (l /dcj)8j~.) The result is a consequence of the principle of detailed balance whereby K& = Kj&. The same transformation symmetrizes the matrix T, which refers to a system of reactions of general order in a region near equilibrium in which the reaction rates depend linearly on the deviations of the reactant concentrations from their equilibrium values. This is true because detailed balance requires that Tijej = Tjici. NOW M differs from T only in the diagonal terms. Hence, it must be true that S = A+MA is a symmetric matrix.

The correlation function +&,r',T) may be expressed in terms of the eigenvectors of S. These eigenvectors, Z(k) , are defined by

S' = A-lKA.

Hence, the argument of Wei and Prater applies also to T.


and similarly

Hence, = + ~ j as was to be shown.

APPENDIX 2

Photocurrent Correlation Function for Pure Diffusion with a Uniform, Circular Incident Beam

We derive here an expression for G(7) for simple diffusion of component 1 with an incident uniform, circular intensity profile given by

Z(r) = lo(a constant) for 0 5 x2 + y2 5 w2

= 0 elsewhere

Eq. (13) may be used to express the concentration correlation function for simple diffusion as

exp[ -tf)2 7 1 cos 7 cos "-") L (A2.1)

When this expression is substituted into Eq. (16b) and the integrations over z and z' carried out, we obtain

where &(v,,u,) = J:- d~ J''mm dy ~ ( w ) exp[i(v,z + w ) ~ .

Now let x = p cos 8; y = psin e X' = p'cose'; y' = p'sine'

U, = u cos+; U, = usin +

22 ELSON AND MAGDE

Then

It is readily shown21 that

&(v=,vu) = &(a) = 10% P d d o ( u p ) 1 In this and subsequent equations J,(z) is the Bessel function of the first kind of orderp. The integration over p may be carried out to givezz

&(u) = (27rw/u)Jl(wu)ro

Similarly, &(- v z , v ~ ) = &(u).

function is Therefore, after carrying out the integration over +, the photocurrent correlation

Now we use the recurrence relation23

Hence

= (') 2017 exp[-w2/2Dlr]

Here I&) and I&) are the modified Bessel Functions of zero and second order. We now use the following relationshipz5 to transform the product Jo(wa)J2(wo):

Hence the remaining integral in Eq. (A2.4) is transformed to a series in terms of standard Gaussian integrals. Thus,

~2 (-l)k(2 + 2k)!(k + 1)!(~' /4)~ exp(-D17u2)Jo(wu)J,(wu)u du = -

8 k = O [(2 + 2k)!]2(k!)2((Dr~)k+2

(A2.6)

NOW we let 7 j = w2/4Dj. Finally, substituting Eqs. (A2.5) and (A2.6) into Eq. (A2.4) we obtain the desired result Eq. (34).


APPENDIX 3

Unimolecular Isomerization We consider here a reaction of the type

kf A-B

kb (A3.1)

with rate constants kf and h and equilibrium constant K = kf/kb = Z B / ~ A . This reaction is interesting because of its simplicity and the possibility of its appearance in actual experimental systems. In addition to presenting the photocurrent correlation function for coupled diffusion and chemical reaction for this example, we shall also consider whether or not we can measure chemical kinetics by FCS even if the chemical reaction has no effect on the optical properties of the reactants.

The matrix, M, Eq. (8), is

The eigenvalues are given by

A(') = -'/2[v2(D~ + &) + R ] f 1/2[~4Aa + 2v2A(kf - kb) + R2I1/' (A3.2)

where A = DA - DB and R = kf + kb, the conventional chemical relaxation rate for this reaction.16 The right eigenvectors with components in the order A, B are

X(+) = ( l / N ( + ) ) [ l , F ( + ) ] (A3.3a)

X(-) = (I/N(-))[l,F(-)] (A3.3b)

where

Dav2 + kr + A(*) Fc') = kb

and N ( + ) and N ( - ) are normalization constants. ponents listed in the same order are

The left eigenvectors, with com-

Then the Zji(~) , Eq. (27), are given as

(A3.4a)

(A3.4b)

(A3.5a)

- exp(x( +b) exp(X(-b) ZBA(T) = F(-) - F ( + , + F ( - , - F(+,

The symmetry condition ~ B Z B A = ~ A Z A B leads to

$"(-IF(+) = - c B / c A = - K

This relationship may also be verified by direct evaluation.

(A3.5~)

(A3.6)

24 ELSON AND MAGDE

At this point it is instructive to consider the special case in which DA = DB 3 D. Then it is readily seen, by setting A = 0 in the equations above, that

A(+) = - V D 2

A(-) = - ( $ D + R )

F ( + ) = K

F ( - ) = -1

and

exp(- v ~ D T ) 1 + K

[ I - exp(-R~)] ZBA(T) =

The partial contributions to the photocurrent correlation functions may be evaluated exact.ly from Eq. (30).

(A3.7a)

(A3.7b)

(A3.7~)

where T D = w2/4D. We note that G ( T ) = GAA(T) + GBB(T) -+- ~ G B A ( T ) and therefore, G(0) = HAA + HBB as was to be expected from Eq. (17).

Further development of the general case, in which DA # DB, requires simplification of Eqs. (A3.5) to allow the integration over Fourier components indicated in Eq. (30). As before we suppose that the characteristic rate of chemical relaxation is greater than all rates of diffusion across the laser beam. Hence, IA/Rw21 << 1, and so we again use the expansion parameter a = u2A/R. Then,

A(+) = - ,,2D( +)

A(-) = -((v*D(-) + R )

F ( + ) = K(a + 1)

F ( - ) = (a - 1)

(A3.8a)


X (,[I + eK] eXp[-u2D(+)~] + [ 1 - - 12i:] exp[ - Ri- - 9D(-)i-]

(A3.8b)

(A3.8~)

After execution of the integrations over 9 in Eq. (30) we obtain

G j i ( ~ ) = HjZ[Ajt'+)(T) + AjZ(-)(T)] (A3.9a)

VBA'-' = - VBA'+' =

and

As in the previous example there are composite characteristic diffusion times which depend on the chemical equilibrium and also the characteristic diffusion time T A which depends on the relative rates of diffusion of A and B. The rate of chemical relaxation is reflected only in the A(-) mode.

We now consider whether or not it is possible to observe chemical relaxation even when the reaction has no effect on the optical properties of the reactants. Then = CB = e and QA = QB = Q. If DA = DB, the photocurrent correlation function may be expressed exactly using Eqs. (A3.7) and (20b) as

Since A and B are distinguishable neither by their diffusion coefficients nor by their optical properties this is the result to be expected from Eq. (21). The chemical relaxation vanishes exactly for this example, This result is analogous to that obtained by Bloomfield and Benbasat' for the study of the unimolecular isomerization by quasi- elastic light scattering when the refractive index increments of A and B are the same.

26 ELSON AND MAGDE

When DA # DB the time course of chemical relaxation may still be observable in principle even if A and B are optically indistinguishable. This will be so whenever, for some finite value of T ,

G(-)(T) = dv& exp[ - V2w2/4]z(-)(T) # 0 S Here Z( - ) (T ) = ~ A ~ A A ( - ) ( T ) + ~B"ZBB(-) (T) + ~ZBA(-)(T)]

and using the exact expressions (A3.5),

(A3.10)

with Eq. (A3.6) we obtain

Evidently, E ( - ) ( T ) > 0 except when 7 -c m or when F(-) = -1. The condition that F(-) = -1 prevails when DA = DB so that G(-)(T) = 0 as was shown before. Excepting this special case, however, G(-)(T) # 0 for finite 7. Hence, chemical relaxation should be observable in principle even when the optical properties of A and B are identical (as long as DA # DB). However, G(-)(T) may be so small that its observation is difficult in practice. This is illustrated in particular by the approximate solution discussed above. Substitution of the appropriate terms from Eq. (A3.9) into Eq. (A3.10) shows that Z( - ) (T ) vanishes to first order in O( for all T.

Finally, it should be pointed out that the equations developed here are, with appropriate redefinition of certain quantities, also sufficient to describe the dimerization reaction:

kr 2A Az

k b (A3.11)

Bloomfield and Benbasat' discuss the relationship between the results for this reaction and that of Eq. (A3.1).

The authors are very grateful to Dr. W. W. Webb for his active participation in the work described here. His advice, encouragement, and support were of great help throughout the course of this project. The authors are also grateful to Dr. M. Nelkin for illuminating discussions and to Drs. W. W. Webb, C . M. Elson, and R. L. Baldwin for criticism of the manuscript. This work was supported by grants from the National Science Foundation (to W. W. Webb) and from the National Institutes of Health (to E. L. Elson, GM #16927). E. L . Elson holds a Career Development Award from the National Institutes of Health.

References 1. S. R. deGroot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland

2. D. Magde, E. L. Elson, and W. W. Webb, Biopolymers, 13,29 (1974). 3. R. B. Inman and M. Schnos, J . Mol. Biol., 49,93 (1970).

Publishing Co., Amsterdam (1962).


4. D. M. Crothers, Bwpolymers, 6 , 1391 (1968). 5. T. Y. Tsong, R. L. Baldwin, P. McPhie, and E. L. Elson, J . Mol. Bwl. 63, 453

6. E. L. Elson, Biopolymers, 11,1499 (1972). 7. See V. A. Bloomfield and J. A. Benbasat, Macromol., 4,609 (1971), and references

cited, particularly the work of B. J. Berne, H. L. Frisch, L. Blum, Z. W. Salzburg, and co-workers.

8. Y. Yeh and R. N. Keeler, J . Chem. Phys., 51, 1120 (1969); Y. Yeh, J . Chem. Phys., 52,6218 (1970); Y. Yeh and R. N. Keeler, Quart. Rev. Biophys., 2,315 (1964).

9. H. Z. Cummins and H. L. Swinney, Progr. Opt., 8,133 (1970); G. B. Benedek, in Polarisation, Matiire et Rayonnement, Livre de Jubild l'honneur du Professor A . Kastler, edited by the French Physical Society (Presses Universitaire de France, Paris, 1969), p. 49.

(1972).

10. V. A. Bloomfield and J. A. Benbasat, Biophys. Soc. J . Abstract, 13,315a (1973). 11. D. W. Schaefer and B. J. Berne, Phys. Rev. Lett., 28,475 (1972). 12. D. W. Schaefer and P. N. Pusey, Phys. Rev. Lett., 29, 843 (1972). 13. G. Feher, Biophys. SOC. J . Abstract, 10, 118 (1970). 14. G. Feher, M. Weissman, Proc. Natl. Acad. Sci. ( U S . ) , 70,870 (1973). 15. D. Magde, E. Elson, and W. W. Webb, Phys. Rev. Lett., 29,705 (1972). 16. M. Eigen and L. C. de Maeyer, Techniques of Organic Chemistry, S . L. Freiss,

E. S. Lewis, and A. Weissberger, Eds., Vol. 8, Part 11, Interscience Publishers, New York.

Matrices and Their Application, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1966).

17. N. R. Amundson, Mathematical Methods in Chemical Engineering:

18. J. G. Kirkwood and R. J. Goldberg, J . Chem. Phys., 18,54 (1950). 19. S. Chandrasekhar, Rev. Mod. Phys., 15, 1 (1943). 20. J. Wei and C. I). Prater, Advances in Catalysis, 13, 203 (1962). 21. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge

(1927), p. 395. 22. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the

Special Functions of Mathematical Physics, Springer-Verlag, Berlin, Heidelberg, New York (1966), p. 86.

23. Ibid., p. 67. 24. Ibid., p. 93. 25. Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, Eds.,

Dover, New York (1965), p. 360.

Received June 26, 1973 Revised September 10, 1973

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Fluorescence Correlation Spectra. Conceptual Basis and Theory