Fluorescence correlation spectroscopy. II. An experimental realization

VOL. 13, 29-61 (1974)

BIOPOLY MERS

Fluorescence Correlation Spectroscopy. 11. An Experimental Realization DOUGLAS A/IAGDE,* Departments of Chemistry and Applied Physics, ELLIOT L. ELSON, Department of Chemistry, and WATT W. WEBB, Department of Applied Physics, Cornell University, Ithaca, New York 14860

synopsis This paper describes the first experimental application of fluorescence correlation spectroscopy, a new method for determining chemical kinetic constants and diffusion coefficients. These quantities are measured by observing the time behavior of the tiny concentration fluctuations which occur spontaneously in the reaction system even when it is in equilibrium. The equilibrium of the system is not disturbed during the experiment. The diffusion coefficients and chemical rate constants which determine the average time behavior of these spontaneous fluctuations are the same as those sought by more conventional methods including temperature-jump or other perturbation tecliniques. The experiment consists essentially in measuring the variation with time of the number of molecules of specified reactants in a defined open volume of solution. The concentration of a reactant is measured by its fluorescence; the sample volume is defined by a focused laser beam which excites the fluorescence. The fluorescent emission fluctuates in proportion with the changes in the number of fluorescent molecules as they diffuse into and out of the sample volume and as they are created or eliminated by the chemical reactions. The number of these reactant molecules must be small to permit detection of the concentration fluctuations. Hence the sample volume is small (10-8 rnl) and the concentration of the solutes is low (-10-9 M ) . We have applied this technique to the study of two prototype systems: the simple example of pure diffusion of a single fluorescent species, rhodamine 6G, and the more interesting but more challenging example of the reaction of macromolecular DNA with the drug ethidium bromide t o form a fluorescent complex. The increase of the fluorescence of the ethidium bromide upon formation of the complex permits the observation of the decay of concentration fluctuations via the chemical reaction and consequently the determination of chemical rate constants.

INTRODUCTION This paper describes experimental considerations of fluorescence correlation spectroscopy (FCS) and reports its first experimental application. FCS is a new method for measuring the kinetic properties of chemical reaction systems in thermodynamic equilibrium by observing the rate of decay of spontaneous concentration fluctuations. We have used i t t o study the combination of macromolecular DNA with the smaller molecule ethidium bromide (EtBr) t o form a fluorescent complex and have determined

* Present address: Department of Chemistry, Washington State University, Pullman, Washington 99163. 29

@ 1974 by John Wiley & Sons, Inc.

30

MAGDE, ELSON, AND WEBB

the chemical rate constants for the reaction as well as the diffusion coefficient of the EtBr. We have also measured the diffusion coefficient of another fluorescent compound, rhodamine 6G (R6G). This example of simple diffusion, because of its simplicity, provides a clear illustration of the underlying principles and mode of operation of the method. The results obtained in these two prototype experiments demonstrate the feasibility of the new technique and provide practical experience to aid in the design of future applications. A brief summary has been reported earlier.' The conceptual basis and theory of FCS has been presented in the preceeding paper.2 Underlying the FCS approach is the principle that the rates of relaxation of a system t o equilibrium after a small macroscopic perturbation of its state and the time correlation of spontaneous fluctuations of the system when undisturbed a t equilibrium are described by the same phenomenological rate coefficient^.^ I n an open volume of solution the concentration of a reactant fluctuates about its thermodynamic mean value as a result of random variations both in the number of molecules formed or eliminated by the chemical reaction and in the numbers which enter or leave the region by diffusion. Hence, the conventional phenomenological rate coefficients for diffusion and reaction may be determined from observations of the rates of decay of spontaneous concentration fluctuations without disturbing the equilibrium of the reaction system. This contrasts with conventional methods for measuring these coefficients (e.g., temperature-jump) in which the rate of return of the system to equilibrium is observed after a macroscopic perturbation of state. The primary experimental task in an FCS experiment is t o measure the number of molecules of a specified reactant in a defined open volume of the solution. The average time behavior of deviations from the thermodynamic mean number is computed from this stochastic time record. I n the experiments described in this paper the concentration of a reactant is determined from its fluorescence. The sample volume is defined by an incident beam of light. The measured fluorescent intensity fluctuates in proportion to the number of fluorescent molecules in the illuminated volume. The fluorescence is detected with a photomultiplier and converted t o a fluctuating current, i(t). If the mean photocurrent, which is independent of time, is (i) (where ( ) indicates ensemble average), the photocurrent fluctuation a t time t is 6i(t) = i(t) - (i). The time behavior is then computed in the form of a photocurrent autocorrelation function, G(T). Thip function represents the time average of the product of a photocurrent fluctuation a t some reference time, 6i(t), with the photocurrent fluctuation at an interval of time T later: G(T) = lim T+m

f SOT

-

6i(t).6i(t

+ ~)dt

(1)

It is evident that as long as the processes giving rise t o the fluctuations 6i(t) are purely random, the correlation between 6i(t) and 6i(t T) must

+

FLUORESCENCE CORRELATION SPECTROSCOPY. I1

31

vanish as T becomes infinite. Therefore G ( T ) decays to zero with increasing 7 . The rate of decay of G ( T ) is governed by the rate of decay of fluctuations in the concentration of the fluorescent solute, which, in turn, can be described in terms of chemical rate constants and diffusion coefficients. The theory required for interpreting G ( T ) in terms of these phenomonological coefficientsis presented in the accompanying paper.2 In order to determine chemical rate constants by FCS it is necessary under most circumstances that the chemical reaction directly affect the optical property used to measure concentrations. Otherwise the experimental data contain information about diffusion coefficients and chemical equilibrium constants, but chemical kinetic effects become imperceptible.2 The experiment becomes easier the larger the effect of the reaction on the optical property. It is possible in principle to deduce chemical kinetics even when there is no chemical effect on optical properties as long as reaction progress is reflected in changes in the diffusion coefficients of the reactant^.^^^ Nevertheless, w-e expect that it will be difficult in practice to extract chemical kinetic constants under these conditions. Several optical properties including absorbance, optical rotation, and Raman scattering, as well as fluorescence, are highly specific indicators of the concentration of particular chemical species. Of these fluorescence is the most sensitive allowing measurement of very small numbers of molecules, fewer than lo-’’ moles in our experiments. Since observation of small numbers of molecules is necessary for detection of spontaneous fluctuations, fluorescence is most suitable for these first experiments. The choice of a specific and sensitive indicator of reaction progress is one feature which distinguishes FCS from quasi-elastic light scattering, another method with potential for studying reaction kinetics in systems at equilibrium. Light-scattering has been successfully employed in measuring diffusion constants of macromolecules5and considerable t h e o r e t i ~ a l ~ + and ~ some experimental workghas been devoted to extending the method to the measurement of chemical kinetic parameters. A major difficulty has been that, unlike fluorescence, the refractive index increments, which are the measure of reactant concentrations in a light-scattering experiment, are quite insensitive to chemical reaction. All molecules of similar size scatter light to more or less the same extent so long as the frequency of the incident light is not near a resonant absorption. Therefore, the important advantage in measuring concentration with properties like fluorescence and absorbance is the chemical specificity which results from their spectral properties. This characteristic is likely to be useful in interpreting complex reaction mechanisms and in studying particular reactions in complex mixtures of substances such as might occur in biological systems. Other methods for studying chemical kinetics in reaction systems at equilibrium are available. Magnetic resonance has served for a number of years to investigate reaction kinetics over a wide range in rates.1° Reaction rates are usually measured by broadening of a resonance line shape. It is interesting to note that the FCS measurements reported here

32

MAGDE, ELSON, AND WEBB

correspond to a broadening of 1-100 Hz on a line of intrinsic width 1013Hz whereas in a typical nmr experiment the resolution is about 1 Hz in 108 Ha. Recently a technique has been introduced for observing the conductivity of an electrolyte at equilibrium and interpreting the random current fluctuations in terms of the fluctuations in the concentrations of ions." This method should be useful for determining the rate constants for reactions which change the number of charged particles in solution. The fmdamental consideration in the design of an FCS experiment is the detection of the very small spontaneous concentration fluctuations. The number of molecules of species j in the sample volume is governed by a Poisson distribution. Therefore, the relative root mean square fluctuation (which is proportional to the observed fluctuation signal) is [((6 N,) 2> I l/'/(NJ> = (Nj>where 6Nj = N j - ( N j ) . Since the relative fluctuation signal increases with decreasing N , it is important to minimize N j . A lower limit, however, is set by the requirement that the fluorescence of the reactants be at least comparable to any residual blank fluorescence in the solution or optical system. Usable values of N , decrease (and the fluctuation signal increases) as the absorbance coefficient and fluorescence quantum yield of the j t h species increase. Furthermore, the experimental estimate of the relative fluctuation amplitude may be improved by increasing the length of the time record processed. The improvement, however, varies only as the square root of the duration of the experiment. Another critical concern is that enough fluorescent photons must be detected per unit of time to define the rate of fluctuation relaxation. It. is desirable to have many photons per relaxation time. The optimal photon counting rate is discussed below. Aside from an optimal choice of wavelength for emission and absorption the fluorescent intensity may be increased by increasing the incident intensity of exciting light. The excitation intensity, however, is limited by the rate of photochemical destruction of the chromophores. Indeed photochemical interference is at present the most severe practical limitation of FCS. The choice of the EtBr-DNA system for our first experiments was motivated both by the intrinsic interest of the system and by its suitability as a test case for the FCS approach. Ethidium bromide is a trypanocidal drug12 and inhibitor of nucleic acid synthesisl3 which appears to bind to DNA in at least two different ways: (1) by binding to the outside of the double helix and (2) by intercalation between the stacked base pairs of the double helix. The latter mode of binding has made EtBr useful as a probe of the supercoiling of closed "circular" DNA molecules.14 Under the conditions of our experiments, high salt concentration and low EtBr concentration, the binding is predominately by intercalation115although one can imagine that the exterior site may be occupied as a transitory state between the free state and full intercalation. We have analyzed our results using the simplest possible model for the binding mechanism: kr

A+B$C kb

(2)

FLUORESCENCE CORRELATION SPECTROSCOPY. I1

33

where A, B, and C represent DNA binding site, free EtBr, and bound complex, respectively. Since our current experiments have relatively low resolution, this analysis serves well most requirements of the investigation. The simple mechanism is, however, almost certainly incomplete. We will point out below that certain apparent anomalies in our results might be traced to complexities not accounted for at the level of approximation of Eq. (2). Conventional measurements16also reveal puzzling complexities not yet understood in detail. The property of the EtBr-DNA reaction which suggests its suitability for study by FCS is the twentyfold increase in the fluorescence of the drug when it binds to DNA.15 It is also helpful that the absolute values of the absorbance and fluorescent quantum efficiency are reasonably large in the complexed state and that the fluorescent emission can be excited at a readily available laser wavelength and detected with a high-efficiency photocathode. Finally, it is important that the kinetic parameters occur in a suitable time range. In the remaining three sections of this paper, we will, first, discuss in detail the somewhat stringent experimental conditions required to carry out an FCS measurement; second, present the experimental data for our two prototype examples of FCS and interpret these data in terms of the theory developed in the companion article;2 and finally, conclude with our analysis of the potential significance of FCS and particular suggestions for further development and application of the method.

EXPERIMENTAL PROCEDURES The basic ingredients of the experimental apparatus are illustrated schematically in Figures 1 and 2. A laser beam to provide the exciting radiation was filtered, dressed by a spatial filter, and focused to a very small diameter at a thin optical cell holding the solution to be studied. The laser intensity transmitted through the cell was monitored by a photodiode. The illuminated sample volume was defined by the laser beam and the space between the entrance and exit windows of the cell. A light-tight enclosure with a parabolic reflector surrounded the cell to exclude extraneous light and establish a uniform constant temperature. A large window passed light from the enclosure to filters that stopped the exciting laser radiation and passed the fluorescence radiation to a photomultiplier. The total photomultiplier current providing a linear measure of the light passing the filter was amplified and corrected for laser power variations in a differential amplifier, electronically filtered to remove the time-independent and highfrequency parts and supplied to a special purpose hardwired computer that carried out the analysis of the temporal correlations of the current fluctuations. The following sections describe crucial design features and experimental problems. First the optical system, then the electronics are described in detail. Next the all-important error analysis is presented, first the essential statistical analysis and then the discussion of possible systematic errors.

MAGDE, ELSON, AND WEBB

34

Finally, the materials and methods of the particular chemical system studied are given.

Optical A schematic diagram of the optical apparatus is shown in Figure 1. This apparatus is essentially a filter fluorimeter designed t o observe a welldefined, reproducible, precisely characterized, and very small sample volume, to maximize the collection and detection of fluorescent photons, to minimize blank fluorescence, and t o reduce as far as possible all sources of “extraneous” fluctuations of the fluorescence intensity which could mask the very small fluctuations expected from the thermodynamic concentration fluctuations . The excitation light source was the 514.5-nm line of a Coherent Radiation Model 52 G Argon ion laser, the intensity of which was stabilized by a feedback circuit monitoring a small fraction of the light output. When the laser was operating a t the output intensity which gave optimum stability, well below maximum output, the total rms variation of the light output was measured to be less than 0.1%over the bandwidth of interest (0.1-1000 Hz). Most variations occurred at discrete frequencies of 20 and 180 Hz and could have been distinguished and removed in the final data analysis. It was considerably more efficient, however, t o discriminate against most of this modulation by performing a differential measurement as described later. The laser output was attenuated to about 1 mW by colored glass filters which also eliminated the spontaneous red fluorescence of the argon plasma which otherwise could make its way t o the photomultiplier to be confused with the weak signal expected from the red-orange EtBr fluorescence. The beam next passed through a spatial filter and beam expander which guaranteed a TEMW Gaussian beam.” The intensity profile of this beam is y2)>/W2]where I(r) is the intensity a t described by I(r) = loexp [ -2(x2 position r ; lois the maximum intensity at the beam center which coincides with the z axis of coordinates; and W is the radius at which I = loexp( -2). The depth of focus was much greater than the thickness of the oell so that along the z direction the intenisty was constant to within 5% as long as the solution was optically thin, i.e., had negligible absorption. By thnslating a

+

c

LASER

Fig. 1. Schematicof the optical apparatus for an FCS experiment. The symbols are: F, colored filter; SF, spatial filter; L, lens; MON, laser intensity monitor; PM, photomultiplier. The sample cell and the parabolic fluorescence-collecting mirror are also shown.

FLUORESCENCE CORRELATION SPECTROSCOPY. I1

3-5

pinhole across the beam and measuring the transmitted intensity we verified that the expanded beam had a Gaussian radial intensity distribution with W = 0.26 cm. Since the effect of a lens is t o produce an intensity distribution at the focal plane which is the two-dimensional x,y Fourier transform of the distribution at the lens, this careful definition of the input guaranteed that the intensity distribution at the focus in the fluorescence cell was Gaussian. The 9-cm focal length fused silica aspheric condensing lens produced an expected focal spot size of w = 5.7 pm. A dielectric coated mirror, not shown in Figure 1, between the lens and the cell reflected the beam t o a vertical direction and into a chamber machined in a solid copper cylinder about 5 cm in diameter and 12 cm long. The copper provided thermal inertia about the sample t o inhibit rapid temperature variations. Active thermostatic control t o better than 1 mC was also incorporated into the block but has not been used in the experiments reported here. The sample cells with inside dimensions 20 mm X 10 mm X 0.15 or 0.025 mm were constructed of Suprasil fused silica by Hellma Cells and by Precision Cells. Capillary tubing on both ends of the cells permitted filling and cleaning as well as mounting in the copper block. The cell was positioned horizontally along the axis of the copper cylinder at the focus of the lens with the beam traversing the narrow dimension in a vertical direction in order t o eliminate any possibility of convection currents. Calculation predicts that convection would be impossible in such small cells in any position. An aluminized parabolic reflector was positioned so that the illuminated portion of the cell occupied its focal center and beamed most fluorescent as well as scattered light horizontally onto the 2.54 cm 5-20 photocathode of a selected EM1 9556 A photomultiplier. Entrance and exit apertures for the laser beam were bored in the reflector and in the copper cylinder. After traversing the cylinder, the laser beam was deflected t o the horizontal and, after attenuation, onto an RCA 1P39 vacuum photodiode, which served as the laser intensity monitor for the differential measurement. The fluorescent light was co'lected with better than 50% efficiency. Since the entire sample cell was in view of the photomultiplier and since light collection was so efficient, considerable care must be taken to minimize blank fluorescence. Here we use the term very broadly t o include all undesired light reaching the photomultiplier. Certainly the cell itself must not fluoresce. Glass fluoresces quite strongly. Most grades of quartz fluoresce significantly even with excitation in the green. Only Suprasil fused silica (among the grades of quar z t r ed) exhibited sufficiently weak fluorescence to be useful in the present experiment. The surfaces of the cell windows must be scrupulously clean. Be3 des fluorescence excited directly by the laser beam, one must minimize scattered light a t the exciting wavelength. Since EtBr and R6G fluoresce a t substantially longer wavelengths than the green exciting light, it was quite easy t o block the exciting light from the photocathode with sharp-cut colored filters. Unfortunately, almost anything used as a fi ter will show some weak fluorescence in the orange. Even with very little scattering of green light, we found i t neces-

MAGDE, ELSON, AND WEBB

36

sary t o use as the first filter element a saturated solution, 3 mm thick, of K2Cr207contained in a cell with, again, Suprasil windows. After this initial attenuation, a Corning 3-67 filter blocked a small residual amount of green light without emitting too much fluorescence. Once all these precautions were observed, the largest source of blank fluorescence proved t o be Raman scattering in the water. Despite the short path length, the Raman scattering was comparable in intensity t o the fluorescence of several thousand highly fluorescent molecules and thereby set a lower limit on the concentrations which could be used. The total blank emission finally achieved was the equivalent of about lo4molecules of R6G, lo5EtBr complex, or lo6 free EtBr. The most important characteristic in the choice of the photocathode is photoemissive quantum efficiency. (The dark current is swamped by blank fluorescence.) The 5-20 exhibits quantum efficiency averaging about 8% in the red-orange. The cumulative losses including geometrical collection efficiency, reflection loss in the prarabolic mirror, losses in the optical filters, and finally at the photocathode were such that about 2.5y0 of fluorescent photons were detected as photoelectrons. The major loss by far is in the photocathode. The FCS experiment is not particularly sensitive t o mechanical vibrations, and so routine vibration isolation was adequate. All components except the photomultiplier were mounted on an 8-in. steel channel which rested on a simple composition table top supported on small inner tubes connected to large drums serving as air reservoirs. The photomultiplier was mounted on an independent steel channel on the table beside the other optical components. A potentially more troublesome source of “noise” was instability in the direction of propagation of the laser beam. Some of this seemed t o be inherent in the laser; a larger component was traced t o convective air currents causing refractive index changes in the light path. The latter could be eliminated by simply enclosing all optical beams within small tubes. The effect of beam wander from any cause is greatly reduced by mounting the fluorimeter optics as close as practical t o the laser.

Electronic A block diagram of the apparatus appears in Figure 2. The laser light impinging on the monitor photodiode was attenuated t o limit the photocurrent drawn t o about 1 PA. The fluorescent radiation falling on the

MON

LPF2

COR2

Fig. 2. Block diagram of the electronic apparatus for an FCS experiment. The symbols are: PM, detector photomultiplier; MON, laser intensity monitor; DA differential amplifier; HPF, high-pass filter; LPF, low-pass filter; COR, correlator; XY, data plotter.

FLUORESCENCE CORRELATION SPECTROSCOPY. I1

37

photomultiplier cathode produced a cathode photocurrent which was amplified a controllable amount in the dynode chain t o yield a mean anode current equal t o that of the monitor. The two currents were converted t o voltages with 105 ohm load resistors. The voltages were then compared in a PAR 113differential amplifier which responds by amplifying the difference between the two input signals. When small enough, the difference is a n excellent approximation t o a normalized ratio. The amplifier was set typically a t a voltage gain of 100. It was also used to perform some signal conditioning; by operating as an a.c.-coupled amplifier with a selected passband of 0.03-1000 Hz i t discriminated against shot noise and very slow drifts of the signal away from the monitor. The output of the amplifier was delivered in parallel t o two signal correlators (SAICOR Model 42) adjusted to sample at time increments differing by a factor of ten. Each correlator was preceded by a section of a Kron-Hite 3322 filter operating in the low-pass R C or transient-free mode. For most of the measurements reported below, the time increments were 2 msec and 20 msec and the filter cutoffs 2000 and 200 Hz, respectively. (There was some redundancy in the filtering, but the different filter shapes of the PAR and the Kron-Hite devices made the arrangement useful.) Each correlator simulates a machine which, in effect, carries out analogto-ditigal conversion on the input signal a t 256 levels of resolution a t each sample time increment. This datum is stored in temporary memory t o be multiplied by itself and 99 subsequent samples measured and stored in the same way. This is done with full efficiency: sample 1 is multiplied by samples 1though 100, sample 2 by 2 through 101, and so on. The products are then summed in storage locations corresponding t o the lag times for zero through 99 increments. The “simulation” introduced by our estimate, a loss of efficiency of a factor of about four compared with an ideal correlation. The method of accomplishing the simulation is treated as proprietary by the manufacturer, but appears t o be similar t o previously published principles, l8 and t o a very recently proposed device.19 Some integrations were continued for periods as long as 24 hr. An excellent test of all the electronics and part of the collection optics including especially the gain stability of the photomultiplier and preamplifier was provided by positioning a n ordinary flashlight bulb a t the focus of the parabolic light collector in place of the fluorescent solution and driving i t with a stable current with a very weak superimposed modulation. I n this way we ascertained that fluctuations a t least as small as 100 parts per million (ppm) of the d.c. intensity could easily be measured. By substituting a strictly constant and very small current, provided by a battery, we could systematically check the response of the apparatus in the shot noise limit, that is under the condition that the light was so weak that the quantum nature of photons was detected as random fluctuations in a n otherwise “constant” light flux. I n this case, there should be a contribution t o the photocurrent correlation function G(T)only in the r = 0 channel. The magnitude of this shot noise is discussed below. Since no photon

38

MAGDE, ELSON, AND WEBB

is correlated with any other, the value of G(l) to G(99) should be constant and equal to zero within statistical uncertainty. What was actually observed in G(7) for short lag times was the response of the low pass filters, which imposed a correlation on the current. The results of a particular test with a mean photocurrent designed to duplicate that observed in the actual chemical reaction experiment are shown in Figure 3. The normalized outputs of both correlators are presented. All electronic and optical “adjustments” are identical to those used in the actual measurements of the chemical relaxations. The values for G(0)/(i(t))2, which have registered offscale, were 1.4 X for Figure 3a and 1.7 X for Figure 3b. The number of samples per lag time was 30 X lo6 in (a) and 2.9 X lo6 in (b). The important conclusion is that even for integration times as long as these, about 16 hr, there were no artifacts imposed on the correlation function, except for a small d.c. offset. This offset was reproducible and primarily due to the correlators, as evidenced in Figure 3, by the fact that it was different for the two machines. It can besubtracted from the observed correlation functions and this has been done for the experimental data reported later in this paper, but the correction was often negligible and never amounted to more than 10% of the signal contribution at T = 0. This offset is not really a defect in the machines, which were being operated here for integration times much longer than their designers intended.

0.5

I.o 1.5 T (seconds)

Fig. 3. The correlation function obtained by the two correlators for a purely random white noise source under conditions similar to those of Figure 7. The 7 = 0 datum is off-scale. The observed lack of correlation proves that the detection optics and the electronics are free from spurious correlations which could distort the experimental measurements.

FLUORESCENCE CORRELATION SPECTROSCOPY. I1

39

Precision of Measurements Since the fluctuation signal which we measured was always small, it is important t o consider carefully the types of experimental uncertainties which could obscure this signal. These uncertainties arose from two kinds of effects: first, uncertainties both in the measurement of the light intensity and in the characteristic fluctuation behavior and, second, irreducible systemat.ic uncertainties in the experiment (which appeared here, as in most kinetic experiments, as a “baseline” uncertainty).

Random Errors The experimental determination of G(r) defined by Ey. (l), can be imagined to occur in two stages: (1) the measurement of the time correlation of individual fluorescence fluctuations and (2) the averaging of these individual fluctuations to yield G(r). Random errors can occur at both stages. This separation cannot be maintained rigorously for the data processing used in these experiments, but it does provide qualitative insight into the sources of random uncertainties and the means for reducing them. Individual fluorescence measurements are subject to a statistical uncertainty due to the random emission of photoelectrons from the photocathode. We term this “shot noise.” The relative error due to the shot noise may be reduced by increasing the number of photons detected either by increasing the intensity of fluorescent emission or by increasing the time of observation. I n principle, therefore, the behavior of a single fluctuation could be obtained to any desired degree of precision by increasing the excitation intensity and consequently the fluorescent intensity. Nevertheless, a precise value of G ( r ) cannot be constructed from this single precise observation since the concentration fluctuations are themselves random processes. Many concentration fluctuations must be observed to define their average properties precisely. Therefore, although the shot noise may be reduced to negligible proportions by increasing the excitation intensity, this will eliminate only one source of random uncertainty in G ( 7 ) . Furthermore, an increase in excitation intensity causes problems which will be discussed below in the section on systematic noise. A practical goal in the experimental design would be to have the uncertainties from shot noise and from the stochastic character of the fluctuations approximately equal. Quantitative calculation of the uncertainty expected in an experimental measurement of G(r) requires a detailed specification of both the form of G ( r ) and of the experimental signal detection and processing. With this information one can calculate the signal-to-noise ratio, which we define as

S / N = lim G(r)/[Var Gm(r)1”’ r-4

(3)

Here Var x indicates the variance of x. Gm(r) is the experimentally measured value of G(r). Although in this paper and in Ref. 2 we have been concerned only with the r-dependent part of the correlation function, it is important to realize that there is also a much larger (by lo4to lo6)constant

MAGDE, ELSON, AND WEBB

40

portion ((i)2).The experimental apparatus electronically filters out this d.c. current; however, the uncertainty in the current cannot be filtered out and this uncertainty has a determining effect on S I N . KoppelzOhas analyzed the problem of evaluating Eq. (3), and his results can be applied to our case with high accuracy. His analysis is developed for pulse correlation methods but can be carried over to our continuous current correlation procedure. The relevant assumptions in his evaluation of S I N are: 1. The correlation function is calculated by detecting and processing in a digital format single photons so that Gm(T)

=

+

l M Gm(lCA7) = - C & L ( ~ A T > ~ ( ~ AT) AT

M

j-1

where AT) = n ( j A ~ ) A, n ( j A r ) is the number of photons detected in thelth interval AT, A is the estimated mean value of n and M is the total of intervals observed. 2 . M A T is much larger than any other times of interest. 3. The T-dependent part of the correlation function exhibits a single exponential decay with characteristic time r -1 and amplitude p(@. Thenz0

The parameters (n}and p appear only in combination. The product is simply the number of photons detected per molecule per time increment AT. (This corresponds to the important parameter of quasi-elastic light-scattering experiments, the number of photons detected per coherence volume per time increment. In FCS, there is no intermolecular coherence in the emitted intensity because of the long and random delay between the absorption of an excitation photon and the emission of a fluorescent photon.) The product FAT is the ratio of the sample time increment t o the characteristic time of the correlation function. Usually, to measure the shape of G ( T ) , rA7 = 0.1. Two limiting cases of Eq. (4) are apparent: When the detection rate of photons is sufficiently high (n)p >" 1, then the ratio SIN becomes independent of light flux

M r A T is simply the ratio of the length of observation time to the characteristic time of the fluctuations. Hence, the S I N ratio improves as the square root of the number of fluctuations observed. In the other limit (n)p