principles of fluorescence correlation spectroscopy - … · · 2014-11-19principles of...

______________________________________________________________________________________ Principles of Fluorescence Correlation Spectroscopy

Principles of Fluorescence Correlation Spectroscopy Lisa J. Carlson

Department of Chemistry, University of Rochester, Rochester, NY 14627

I. INTRODUCTION

All processes in nature tend to progress toward a state of equilibrium, and the temporal

quantification of microscopic nonequilibrium processes that play a role in disrupting this pattern can be

used to predict the macroscopic behavior of a system as it approaches equilibrium.1 In order to achieve this

goal, fluorescence correlation spectroscopy (FCS) and autocorrelation statistical analysis are used to study

molecular diffusion and the kinetics of chemical equilibria by monitoring concentration fluctuations in a

small ensemble of molecules. The historical development of this analytical technique, its principles, and its

employment as a means of determining the volume and mass of an unknown protein and DNA sample are

presented.

Fluorescence correlation spectroscopy is used to study both the Brownian diffusion and the

equilibrium kinetics of a chemical system.2 As molecules diffuse through

a small, open detection volume, the number of particles in this space

varies around its equilibrium value (Figure 1). Two major results can be

extracted by tracking the emission from these particles: the temporal

autocorrelation defines the time scale of diffusion and its variance

provides the average number of fluorescent molecules in the detection

volume.2 Hence, when changes in the fluorescence are measured over

time, these data provide information that can be used to determine

diffusion coefficients, rate constants, and sample concentrations; aggregation and dynamics linked to

rotation and translation are also of interest and can be studied in this manner.3

Historically, fluorescence correlation spectroscopy is the mathematical descendant of quasi-elastic

light scattering (QELS) spectroscopy.2 While both FCS and QELS use a small sample volume to

noninvasively probe concentration fluctuations, it is the enhanced sensitivity of fluorescence to

conformational, environmental, and chemical changes in a system that allows FCS to be more useful in

these scenarios than tracking scattered light. The roots of fluorescence correlation spectroscopy trace back

to 1903, when Smoluchowski first outlined the relationship between Brownian movement and

autocorrelation, which is used in the statistical analysis of FCS data.4 However, it wasn’t until 1972 that

Magde, Elson, and Webb applied these ideas to the design of FCS spectrometers at Cornell University in

order to study the kinetics of the reversible binding between ethidium bromide (a fluorescent nucleic acid

synthesis inhibitor) and DNA.4 The notion of incorporating a confocal microscope into a typical FCS setup

was introduced by Koppel et al. in 1976;2 since that time, advances in detection, identification, and

characterization of single molecules in dilute solution have catalyzed a Renaissance in fluorescence

correlation studies.1

Figure 1: The confocal Gaussian detection volume.1

December 2005 Carlson - 2


II. THEORY

The primary observable in FCS is fluorescence, and

changes in fluorescence intensity reflect the concentration

fluctuations of a molecular system (Figure 2a).1 In FCS, the

autocorrelation of this fluorescence variation is used to evaluate

the temporal progression of a system around its equilibrium

state. The autocorrelation is the cross-correlation of a signal

with itself and is obtained by comparing a measured value at a

time t with that at a later time (delayed by τ), as shown in

Figure 2b. In this sense, one would expect two signals taken at

nearly the same time to have a high correlation value and those

taken farther apart to result in a lower correlation value (Figure

2c). The amplitude of the autocorrelation function is influenced

by the number of molecules in the detection volume. The

relative effect of one particular molecule on the total measured

fluorescence decreases as the number of molecules increases,

and the normalized amplitude of the autocorrelation function

declines accordingly.2 It is for this reason that extremely dilute

concentrations are used for FCS studies, such that

approximately five molecules are desired in the detection

volume at one time.

The normalized autocorrelation function can be

expressed as the product of the fluorescence fluctuation at a

given time t and a later time t + τ, normalized by the square of

the average fluorescence:1,2

(1) G(τ) = <δF(t)δF(t+τ)>/<F(t)>2,

where δF(t) is the difference between the fluorescence intensity at time t and its average value. More

specifically, if chemical kinetics are neglected and only one species is being detected, then the

autocorrelation function for the confocal observation volume that takes the shape of a prolate ellipsoid is

given by1,2

(2) GD(τ) = [N(1+τ/τD)(1+τ/(ω2τD))1/2]-1.

Here, N is the average number of fluorescent molecules, τD is the typical time that a molecule spends in the

observation volume, and ω is the axial to lateral ratio of this space. Finally, this idea can be extended to

account for more than one diffusing species in the observation volume when chemical kinetics are ignored

by taking a weighted linear combination of the correlations for each non-interacting species:2

(3) GD(τ) = ! != =

m

i

m

i iiiFGF

1 1

22 )/()(" .

Figure 2: (a) A typical fluorescence signal measured in time. (b) A binned portion of the fluorescence in (a) indicating the average fluorescence. (c) The autocorrelation of fluorescence, which characterizes fluctuations caused by diffusion.2



The autocorrelation function reveals the time required for a molecule to diffuse through an open

observation volume. Diffusion is the macroscopic result of random thermal motions that occur on a

molecular level, and the collisions between solute particles and solvent molecules experiencing thermal

movement are responsible for this phenomenon.5 Fick’s first and second laws of diffusion describe the rate

of diffusion of a solute across an area in three and one dimensions, respectively. However, the work of

Brown, Guoy, Einstein, and Smoluchowski is responsible for the development of the random walk

equation, which uses a diffusion coefficient, D, to relate the average time required for a molecule to travel

an average distance, <x>, in time t.5,6

(4) <x>2 = 2Dt

Once the diffusion coefficient is known, the Einstein relation bridges this value and the drag coefficient,

γ:5,6

(5) D = kBT/γ.

For a sphere with radius r, the drag coefficient is calculated using the Stokes equation5,6

(6) r = γ/6πη,

where η is the viscosity of the solvent. In this manner, the radius, mass, and volume of a diffusing particle

may be determined.

III. EXPERIMENT

These ideas were implemented in a fluorescence correlation spectroscopy experiment designed to

target the volume and mass of two samples: an unknown protein antibody and DNA. A block diagram of

the Nikon inverted confocal microscope that was used as the foundation

for this experimental setup is depicted in Figure 3. An Nd:YAG laser

emitting at λ = 532.0 nm was passed through a telescope in order to

overfill the back aperture of an NA 1.4 100x oil immersion objective (η

= 1.515). The fluorescence emitted by the excited sample was collected

using the same objective and was passed through a notch filter and

pinhole before being detected by an avalanche photodiode.

A preliminary estimate of the concentration required to detect

up to ten single molecules, as well as the probability of detecting one

such particle in a 2.31 x 10-20 L detection volume is displayed in Tables

1 and 2.2,7 The concentration of an 81 nM stock solution of unknown

protein was adjusted to 0.0081 nM by a two step dilution with a buffer

composed of 50 nM K3PO4, 0.1% OG buffer, and 0.2% bovine serum

albumin in water (prepared June 1, 2005). One hundred microliters of this sample was injected into the

sample holder and the focus of the laser (P = 230 µW) was raised into the liquid. Data were collected for

15 seconds at a 10 µs time resolution and were binned every 2 milliseconds (Table 4). Three initial trials

were carried out for the protein, corresponding to the following temperatures (listed in sequential order):

Figure 3: Experimental setup.2



-2.4°C, 46.1°C, and 26.2°C. The temperature was adjusted in this order because light was not directed to

the detector during the first trial for the intermediate temperature.

The unknown 0.215 nM DNA sample was prepared from a 2.15 µM stock solution and an aqueous

tris chloride/EDTA buffer by following the two-step dilution process used for the unknown protein. A

series of six trials were carried out to study the effect of increasing temperature (-3.9°C, 10.0°C, 20.4°C,

29.7°C, 40.6°C, and 59.2°C) on the diffusion rate and the average number of molecules in the observation

volume. The laser power was 200 µW and the data were collected by using 25 second collection times at

10 µs time resolution and then binning data every 200 milliseconds (Table 4). A similar procedure was

followed to collect data for the blank tris chloride/EDTA buffer at -3.9°C.

The construction of autocorrelation curves for these trials was completed in MATLAB and the

curves were fitted with equation (2) in Origin.

IV. ANALYSIS

The mass and volume of the unknown protein and DNA were calculated using information

available from a plot of the autocorrelation function versus time for each species. Plots were constructed

for the background (T = -3.9; Plot 1), the protein at three temperatures (T = -2.4°C, 26.2°C, 44.8°C; Plots

2-4), and DNA at six temperatures (T = -3.9°C, 10.0°C, 20.4°C, 29.7°C, 40.6°C, 59.2°C; Plots 5-10). The

MATLAB software package was used to calculate the autocorrelation of the temporal data collected during

the experiment; the graphing software package Origin was then used to fit the data to the normalized

autocorrelation function, equation (2). During the fitting procedure, the value of ω was held constant (ω =

3.54 for all trials; Table 1), while τD and N were allowed to vary. An initial approximation of the average

number of fluorescent molecules, N, in the observation volume was obtained by taking the reciprocal of the

limit of G(τ) as τ approached zero:

GD(0) = [N(1+(0)/τD)(1+(0)/(ω2τD))1/2]-1 = N-1.

Typically, one hundred Levenberg-Marquardt iterations were carried out for both N and τD until their

values remained unchanged between iterations. The average diffusion time was then obtained from the

fitted value of τD, equal to the time at the inflection point of the fitted curve.

Temporal emission data for the non-fluorescent DNA buffer was collected as a point of

comparison between samples with and without

labeled molecules (Plot 1). As expected, the

autocorrelation function has no amplitude for this

trial. Oppositely, the autocorrelation function

provides a good fit for the protein and DNA data, and

illustrates the behavior of these molecules around

their equilibrium states (Plots 2-10). Since the

amplitude of the autocorrelation function is directly

linked to the number of fluorescent molecules within

the detection volume, it is noted that the amplitude of 1E-3 0.01 0.1

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Plot 1: TrisCl/EDTA Buffer (T = -3.9 C)

O r i g i n D e m o O r i g i n D e m o O r i g i n D e m o









the autocorrelation function for the DNA samples is quite low, especially for trials DNA 2 and 6, whose

average number of fluorescent molecules is approximately 988 and 281, respectively. Lower

concentrations of DNA and protein would be used if the experiment were to be repeated.

The diffusion times obtained through this fitting procedure are summarized in Tables 5 and 6. By

implementing equations (4), (5), and (6), the average diffusion time of 457 microseconds yielded the

average radius and volume of the unknown protein as 0.371 pm and 0.297 pm3, respectively. The mass of

the protein was determined to be about 2.472 x 10-10 kD; this molecule was an antibody whose mass is

known to be near 150 kD. As the temperature increased, the diffusion time for one molecule decreased,

which agrees with the idea that molecules move faster when more thermal energy is available. The average

number of fluorescent molecules in the detection volume was 20, which is slightly high for single molecule

detection. One consideration in this determination is that the possibility of protein denaturation was

introduced for the Protein 2 and 3 trials, since some metastable proteins denature in this temperature

regime. The temperature was first ramped to 44.8°C before being reduced to 26.2°C for the Protein 2 trial,

and so some uncertainty was introduced due to this nonlinear temperature variation.

The calculated radius of the DNA sample was 0.578 pm (V = 1.790 pm3) and its mass was 1.488 x

10-9 kD. The average diffusion time was found to be 703 microseconds and the mean number of

fluorescent molecules in the detection volume was 77. The magnitude of N is cause for some concern

because as N increases, the amplitude of the autocorrelation curve decreases and makes accurate curve

fitting more difficult. In contrast to the protein experiment, the diffusion time of the DNA and temperature

were directly related here. The trials DNA 2 and DNA 6 were excluded from the radius and mass

determinations because the amplitude of the autocorrelation function was such that a signal could not be

reliably distinguished.

Because the results of this study are not realistic for the unknown protein or DNA, it is worthwhile

to account for these deviations from expected results. Sample preparation, data collection, and theoretical

modeling are three areas that could be adapted to improve the validity of these results. The samples were

typically too concentrated for single molecule detection since the average number of molecules in the

observation volume was 20 for the protein and 77 for the DNA. If these values were limited to less than

ten molecules, then both the amplitude of the autocorrelation function and the signal to noise ratio would be

much higher. Also related to sample preparation, the buffer used in the protein trials included OG, which is

a detergent. The viscosity of the unknown solutions was assumed to be equal to that of water (Table 1).

However, detergents typically decrease the viscosity of a solution and enable faster diffusion of a molecule

throughout that solvent. This trend was displayed for both solutions, and viscosity may have played a role

in both the protein and DNA trials.

The experimental setup and data collection techniques could be adjusted to improve the quality of

the data. Glass cover slips exhibit autofluorescence, and while this constant background does not strongly

affect the autocorrelation fits, it does lead to a decreased signal to noise ratio. As a substitute, quartz cover

slips could be used to eliminate this additional background noise. The setup also incorporated an Nd:YAG

laser operating at a power near 200 µW (Table 4). It is possible that the dyes used to label the protein and



DNA experienced photobleaching at this excitation intensity. In order to confirm or dismiss this

supposition, the experiment could be carried out as a function of varying laser intensity in order to

determine the rate at which the dyes bleach. In addition, an examination of the time resolution used in data

collection indicates that although we could achieve only ten-microsecond time resolution, the calculated

diffusion times of 457 µs (protein) and 703 µs (DNA) were well above the threshold limits for this value.

As such, improving the time resolution would not largely influence the calculated diffusion times.

Oppositely, if the experiment were to be repeated, the number of data points collected during a time scan

would be increased in order to improve the SNR.

Finally, the fitting of autocorrelation curves may have been responsible for some error in the

results of this study. Inaccuracy related to estimating the number of fluorescent molecules in the detection

volume may have contributed to some variation in the determination of N and τD from the autocorrelation

curve fits. Modifying the sample preparation and data collection as described above would increase the

signal to noise ratio, making curve fitting more accurate and reliable.

V. CONCLUSIONS

Fluorescence correlation spectroscopy is an optical technique that makes use of diffusion and

kinetics to aid in the understanding of equilibrium processes. As an example of this idea, FCS was used to

determine the mass and radius of an unknown protein (m = 2.472 x 10-10 kD; r = 0.371 pm) and DNA (m =

1.488 x 10-9 kD; r = 0.578 pm). The approximate mass of the protein is known to be near 150 kD, and so

the calculated results for the unknown protein and DNA are not valid. Nevertheless, meaningful

information can still be extracted from this study by accounting for improvements that could be made in the

sample preparation, experimental setup, data collection, and curve fitting. As advances in experimental

techniques are made, it is likely that fluorescence correlation spectroscopy coupled with autocorrelation

analysis will provide a meaningful approximation of the mass and size of these diffusing particles.



Appendix

Table 1: Constants and Associated Symbols Quantity Symbol Value Units Laser Wavelength λ 532.00 nm Magnification M 100 x Numerical Aperture NA 1.4 - Refractive Index n 1.515 - Beam Waist Δx 231.724 nm Depth of Focus Δz 822.429 nm Confocal Detection Volume Vdet 2.31227E-20 L Axial:Lateral Ratio ω 3.549 - Avogadro's Number NA 6.022E+23 molecules/mole Number of Fluorescent Molecules N - - Number of Molecules of Interest x 1 - Poisson Probability of Detection Px - - Typical Protein Density ρ 1,380 kg/m3 Viscosity of Water η 1.000E-03 Pa*s Boltzmann Constant kB 1.381E-23 J/K Temperature T 298 K Radius of a Globular Protein r - nm Drag Coefficient γ N*s/m Diffusion Coefficient D - m2/s Drift Time t1D - s Root Mean Square Velocity vrms - m/s Sample Calculations Δx = ((0.6098)(λ))/(NA) = ((0.6098)(532.00 nm))/(1.4) = 231.724 nm Δz = ((2)(n)(λ))/(NA)2

= ((2)(1.515)(532.00 nm))/(1.4)2 = 822.429 nm Vdet = ((4/3)(π)(Δx/2)2(Δz/2))(1 m/1 x 109 nm)3

= ((4/3)(π)(23,172.4 nm/2)2(8,224,285.714 nm/2))(1 m/1 x 109 nm)3(1 L/1 m3) = 2.3123 x 10–20 L ω = Δz/Δx = 822.429 nm/231.724 nm = 3.549



Table 2: Protein concentration and probability of detection as a function of N N x [Protein] (M) [Protein] (nM) Probability 1 1 7.1816E-13 0.00072 0.3679 2 1 1.4363E-12 0.00144 0.2707 3 1 2.1545E-12 0.00215 0.1494 4 1 2.8726E-12 0.00287 0.0733 5 1 3.5908E-12 0.00359 0.0337 6 1 4.3090E-12 0.00431 0.0149 7 1 5.0271E-12 0.00503 0.0064 8 1 5.7453E-12 0.00575 0.0027 9 1 6.4634E-12 0.00646 0.0011 10 1 7.1816E-12 0.00718 0.0005 Sample Calculations N = 1 [Protein] (M) = N/(NA)(Vdet)) = (1 molecule)/((6.022 x 1023 molecules/mole)( 2.3122716 x 10–12 L)) = 7.1816 x 10-13 M [Protein] (nM) = ([Protein] (M))(1 x 109 nM/1 M)) = (7.1816 x 10-13 M)( 1 x 109 nM/1 M)) = 0.0007181 nM Px = ((Nx)(e-N))/(x!) = ((11)(e-1))/(1!) = 0.36788 Parallel calculations were carried out for N = 2-10.



Table 3: Drift time and vrms as a function of protein size

Protein Mass (kD) r (nm) γ (N*s/m) D (m2/s) tD (s) vrms (m/s) a 10 1.4216 2.6796E-11 1.5358E-10 1.7477 27.2738 b 30 2.0502 3.8646E-11 1.0649E-10 2.5207 15.7465 c 50 2.4308 4.5820E-11 8.9816E-11 2.9886 12.1972 d 70 2.7193 5.1258E-11 8.0287E-11 3.3433 10.3085 e 100 3.0626 5.7730E-11 7.1287E-11 3.7654 8.6247 f 130 3.3426 6.3006E-11 6.5318E-11 4.1095 7.5644 g 150 3.5059 6.6084E-11 6.2275E-11 4.3103 7.0421

Sample Calculations m = 10 kD r = ((m/NA)/((4/3)(π)(ρ)))1/3(1 x 109 nm/1 m) = (((10 kg/mol)/( 6.022 x 1023 molecules/mole))/((4/3)(π)(1.380 x 103 kg/m3))1/3(1 x 109 nm/1 m) = 1.4216 nm γ = 6πηr = (6)(π)(1 x 10-3 Pa*s)(1.4216 nm)((1 kg/ms2)/1 Pa)(1 N/(1 kgm/s2))(1 m/1 x 109 nm) = 2.6797 x 10-11 N*s/m D = kBT/γ =((1.381 x 10-23 J/K)((1 kgm2/s2)/1 J)(298 K))/((2.6797 x 10-11 N*s/m)((1 kgm/s2)/1 N)) = 1.5358 x 10-10 m2/s tD = (Δx)2/(2D) = ((23,172.4 nm)(1 m/1 x 109 nm))2/((2)( 1.5358 x 10-10 m2/s)) = 1.7481 s vrms = ((3kBT)/(m/NA))1/2

= (((3)(1.381 x 10-23 J/K)((1 kgm2/s2)/1 J)(298 K))/(10 kg/mol/(6.022 x 1023 molecules/mol)))1/2 = 27.2670 m/s Parallel calculations were carried out for all other masses of protein.



Table 4: Data Collection Parameters Sample Concentration

(nM) Temperature (°C)

Power (µW)

Collection Time (s)

Time Resolution (µs)

Binning (ms)

Protein 1 0.0081 -2.4 230 15 10 2 Protein 2 0.0081 46.1 230 15 10 2 Protein 3 0.0081 26.2 230 15 10 2 Tris/Cl EDTA Buffer - -3.9 200 20 10 200 DNA 1 0.215 -3.9 200 25 10 200 DNA 2 0.215 10.0 200 25 10 200 DNA 3 0.215 20.4 200 25 10 200 DNA 4 0.215 29.7 200 25 10 200 DNA 5 0.215 40.6 200 25 10 200 DNA 6 0.215 59.2 200 25 10 200



Table 5: Summary of Protein Results Protein 1 2 3 Average T (°C) -2.4 26.2 44.8 - T (K) 270.75 299.35 317.95 - N 19.22952 22.92411 17.32037 20 tD (s) 3.200E-04 6.700E-04 3.800E-04 4.567E-04 tD (ms) 0.320 0.670 0.380 0.457 D (m2/s) 8.390E-07 4.007E-07 7.065E-07 6.487E-07 γ (Ns/m) 4.457E-15 1.032E-14 6.215E-15 6.996E-15 r (m) 2.364E-13 5.473E-13 3.297E-13 3.711E-13 r (pm) 0.236 0.547 0.330 0.371 Mass (kD) 4.601E-11 5.707E-10 1.248E-10 2.472E-10 V (pm3) 0.055 0.687 0.150 0.297

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Data: Protein2Model: ACF Chi 2/DoF = 0.00001R^2 = 0.48076 N 22.92411 ±0tD 0.00067 ±0.00001w 3.54 ±0

Plot 3: Protein 2 in Buffer (T = 26.2 C)

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Data: Protein1Model: ACF Chi 2/DoF = 0.00001R^2 = 0.39431 N 19.22952 ±0tD 0.00032 ±6.2742E-6w 3.54 ±0

Plot 2: Protein 1 in Buffer (T = -2.4 C)

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Data: Protein3Model: ACF Chi 2/DoF = 0.00001R^2 = 0.48272 N 17.32037 ±0.18018tD 0.00038 ±0w 3.54 ±0

Plot 4: Protein 3 in Buffer (T = 46.1 C)

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Table 6: Summary of DNA Results

DNA 1 2 3 4 5 6 Average Average (Excluding 2 & 6)

T (°C) -3.9 10 20.4 29.7 40.6 59.2 - - T (K) 269.25 283.15 293.55 302.85 313.75 332.35 - - N 66.51482 987.89306 104.68294 69.14913 67.27686 281.21719 263 77 tD (s) 3.200E-04 7.500E-04 3.000E-04 1.300E-03 8.900E-04 6.000E-04 6.933E-04 7.025E-04 tD (ms) 0.320 0.750 0.300 1.300 0.890 0.600 0.693 0.703 D (m2/s) 8.390E-07 3.580E-07 8.949E-07 2.065E-07 3.017E-07 4.475E-07 5.079E-07 5.605E-07 γ (Ns/m) 4.432E-15 1.092E-14 4.530E-15 2.025E-14 1.436E-14 1.026E-14 1.079E-14 1.089E-14 r (m) 2.351E-13 5.795E-13 2.403E-13 1.074E-12 7.620E-13 5.442E-13 5.726E-13 5.779E-13 r (pm) 0.235 0.580 0.240 1.074 0.762 0.544 0.573 0.578 Mass (kD) 4.524E-11 6.775E-10 4.831E-11 4.317E-09 1.540E-09 5.609E-10 1.198E-09 1.488E-09 V (pm3) 0.05444 0.8152 0.05814 5.194 1.853 0.6749 1.442 1.790

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Data: DNA1Model: ACF Chi 2/DoF = 9.5869E-6R^2 = 0.04934 N 66.51482 ±2.90794tD 0.00027 ±0w 3.54 ±0

Plot 5: DNA 1 in Buffer (T = -3.9 C)

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Data: DNA2Model: ACF Chi 2/DoF = 9.5736E-6R^2 = 0.00065 N 987.89306 ±382.00404tD 0.00075 ±0w 3.54 ±0

Plot 6: DNA 2 in Buffer (T = 10.0 C)

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Data: DNA3Model: ACF Chi 2/DoF = 9.8119E-6R^2 = 0.01307 N 104.68294 ±0tD 0.0003 ±0.00003w 3.54 ±0


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Data:DNA4Model: ACF Chi 2/DoF = 0.00001R^2 = 0.16859 N 69.14913 ±1.4859tD 0.0013 ±0w 3.54 ±0


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Data: DNA5Model: ACF Chi 2/DoF = 0.00001R^2 = 0.06584 N 67.27688 ±0tD 0.00089 ±0.00004w 3.54 ±0


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Data: DNA_DNA6Model: ACF Chi 2/DoF = 9.4131E-6R^2 = 0.00615 N 281.21719 ±34.34875tD 0.0006 ±0w 3.54 ±0


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Acknowledgements The author is indebted to John Lesione for his assistance in planning, collecting, and analyzing data for this

project. She also thanks Katie Leach for helpful revisions and discussions related to this work.

References (1) Maiti, S.; Haupts, U.; Webb, W. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, 11753-11757. (2) Hess, S. T.; Hunag, S.; Heikal, A. A.; Webb, W. W. Biochemistry 2002, 41, 697-705. (3) Aragón, S. R.; Pecora, F. J. Chem. Phys. 1976, 64, 1791-1803. (4) Microscopy from Carl Zeiss. www.zeiss.com/C1256D18002CC306/0/C86B08C84296799DC1256

D5900335186/$file/40-535_e.pdf (accessed November 2005).

(5) Laidler, K. J.; Meiser, J. H.; Sanctuary, B. C. Physical Chemistry; Fourth ed.; Houghton Mifflin Company: Boston, MA, 2003.

(6) Howard, J. Mechanics of Motor Proteins and the Cytoskeleton; Sinauer Associates, Inc.:

Sunderland, MA, 2001. (7) Földes-Papp, Z.; Demel, U.; Tilz, G. P. Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 11509-11514.

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