Abstract: Zero-Mode Waveguides were first introduced for FluorescenceCorrelation Spectroscopy at micromolar dye concentrations. We showthat combining zero-mode waveguides with fluorescence correlation spec-troscopy in a continuous flow mixer avoids the compression of the FCS sig-nal due to fluid transport at channel velocities up to ∼ 17 mm/s. We derive
an analytic scaling relationship∣∣∣
δkONkON
∣∣∣ =
∣∣∣
δkOFFkOFF
∣∣∣ ∼ kON+kOFF
kON
0.1DBDF−DB
e√SNR
converting this flow velocity insensitivity to improved kinetic rate certaintyin time-resolved mixing experiments. Thus zero-mode waveguides makeFCS suitable for direct kinetics measurements in rapid continuous flow.
OCIS codes: (170.2520) Medical optics and biotechnology: Fluorescence microscopy,(220.4241) Optical design and fabrication: Nanostructure fabrication, (230.7370) Optical de-vices: Waveguides, (300.2530) Spectroscopy: Fluorescence, laser-induced
References and links1. J. B. Knight, A. Vishwanath, J. P. Brody, and R. H. Austin, “Hydrodynamic Focusing on a Silicon Chip: Mixing
Nanoliters in Microseconds,” Phys. Rev. Lett. 80, 3863–3866 (1998).2. E. A. Lipman, B. Schuler, O. Bakajin, and W. A. Eaton, “Single-molecule measurement of protein folding kinet-
ics,” Science 301, 1233–1235 (2003).3. D. E. Hertzog, B. Ivorra, B. Mohammadi, O. Bakajin, and J. G. Santiago, “Optimization of a microfluidic mixer
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microsecond kinetic studies of protein folding,” Anal. Chem. 76, 7169–7178 (2004).5. D. Magde, E. Elson, and W. Webb, “Thermodynamic fluctuations in a reacting system–measurement by fluores-
cence correlation spectroscopy,” Phys. Rev. Lett. 29, 705–708 (1972).6. E. L. Elson and D. Magde, “Fluorescence correlation spectroscopy. I. Conceptual basis and theory,” Biopolymers
13, 1–27 (1974).7. D. Magde, E. L. Elson, and W. W. Webb, “Fluorescence correlation spectroscopy. II. An experimental realiza-
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by Single Molecule Fluorescence Correlation Spectroscopy,” Anal. Chem. 72, 3260–3265 (2000).9. K. K. Kuricheti, V. Buschmann, and K. D. Weston, “Application of fluorescence correlation spectroscopy for
velocity imaging in microfluidic devices,” Appl. Spectrosc. 58, 1180–1186 (2004).10. M. Levene, J. Korlach, S. Turner, M. Foquet, H. Craighead, and W. Webb, “Zero-Mode Waveguides for Single-
Molecule Analysis at High Concentrations,” Science 299, 682–686 (2003).
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11. M. Leutenegger, M. Gosch, A. Perentes, P. Hoffmann, O. J. Martin, and T. Lasser, “Confining the samplingvolume for Fluorescence Correlation Spectroscopy using a sub-wavelength sized aperture,” Opt. Express 14,956–969 (2006).
12. J. Wenger, D. Gerard, P.-F. Lenne, H. Rigneault, J. Dintinger, T. W. Ebbesen, A. Boned, F. Conchonaud, andD. Marguet, “Dual-color fluorescence cross-correlation spectroscopy in a single nanoaperture: towards rapidmulticomponent screening at high concentrations,” Opt. Express 14, 12,206–12,216 (2006).
13. S. Maiti, U. Haupts, and W. W. Webb, “Fluorescence correlation spectroscopy: Diagnostics for sparse molecules,”Proceedings of the National Academy of Sciences 94, 11,753–11,757 (1997).
14. J. Widengren and U. Mets, Single Molecule Detection in Solution, chap. 3, pp. 69–120 (Wiley-VCH, 2002).15. J. Enderlein and C. Zander, Single Molecule Detection in Solution, chap. 2, pp. 21–67 (Wiley-VCH, 2002).16. P. G. Gloersen, “Ion-beam etching,” J. Vac. Sci. Technol. 12, 28–35 (1975).17. R. Lee, “Microfabrication by ion-beam etching,” J. Vac. Sci. Technol. 16, 164–170 (1979).18. K. Samiee, M. Foquet, L. Guo, E. Cox, and H. Craighead, “λ -Repressor Oligomerization Kinetics at High Con-
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1. Introduction
Our paper illustrates how the use of a floor of zero-mode wavegtuides (ZMW) sustains sen-sitivity to diffusion measurements for Fluorescence Correlation Spectroscopy (FCS) in highvelocity flow channels, as occur in Continuous Flow Microfluidic Mixer (CFMM) designs. Thebasic idea is very simple: the floor of a CFMM is carpeted with an array of ZMWs which samplethe local concentration of molecules at a particular region of the flow pattern but are shieldedfrom the advection of the flow by the walls of the ZMW. Single molecules within a ZMW havea characteristic residence time given by their diffusion coefficient and the effective volume ofthe ZMW. Although above the entry of the ZMW the fluid is advecting, within the ZMW thereis no advection and hence we expect the mean residence times in the ZMW waveguide, andhence the determination of the diffusion coefficient of the molecule, to be independent of thespeed of the external flow.
This result has important consequences. CFMM designs allow studies of biological reactionand mixing kinetics with low reagent consumption and microsecond time resolution [1, 2, 3].The flow velocity profile assigns reaction times to different distances from inlets. Hydrody-namic focusing achieves submicrosecond time resolution and mixing times less than 10 μs,enabling protein folding kinetic measurements [4]. Magde et al. developed FCS [5, 6, 7] forstudying chemical kinetics by measuring the average duration, or correlation time, of fluo-rescence intensity bursts for a single chromophore as it passed through a small sample vol-ume. The time scales obtained correspond to molecular processes including diffusion, rotation,and quenching and macroscopic processes such as advective flow inside microfluidic channels[8, 9]. However, the sensitivity of FCS to the advection time in high velocity streams meansthat it cannot be used to obtain diffusion coefficients in a CFMM device if the mean diffusionaltime of the chromophore out of the sample volume is greater than the time to advect the chro-mophore out of the sample volume. Improving diffusion constant sensitivity of FCS at highvelocity would allow FCS to characterize the time evolution of species populations during achemical reaction in a CFMM device.
Diffraction-limited FCS collects intensity fluctuations from the passage of single chro-mophores through fL (10−12 L) volumes, requiring nanomolar concentrations for single mole-cule correlation statistics. Zero-mode waveguides reduce observation volumes to aL (10 −15 L)for studies at micromolar concentrations by illuminating sub-wavelength apertures (SWAs)–openings in metal films roughly 100 nanometers thick [10, 11, 12] with diameters less than thewavelength of incident light. Workers refer to a subset of SWAs by the name ZMW. A ZMWwould be too narrow to propagate the incident wavelength if the metal film were extended to
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10078#94441 - $15.00 USD Received 31 Mar 2008; revised 14 May 2008; accepted 19 Jun 2008; published 23 Jun 2008
infinite thickness. Levene et al. studied interactions between fluorescent nucleotides and poly-merases immobilized in ZMWs to observe incorporation events and photobleaching [10], butthe possibility of using ZMWs with FCS for reaction studies in continuous flow has not beendiscussed.
This paper compares FCS sensitivity for diffraction-limited and ZMW methods in rapid flow.In section 2 we present standard diffraction-limited FCS, our data collection techniques, signalfunctions calculated, and noise functions measured. Section 3 repeats the discussion in section2 for ZMWs. We use the signal and noise functions from sections 2 and 3 to calculate SNRsfor determination of diffusion coefficients as a function of flow speed in section 4. Finallysections 4 and 5.1 show that, in flow channels, ZMWs improve SNR for distinguishing diffusioncoefficients, thus ZMWs improve uncertainty in measuring kinetic rate coefficients.
2. Signal and noise in diffraction-limited FCS
2.1. Sample construction for fluorescence collection
Introductory reviews of the FCS literature covering its inception in the 1970s and modern appli-cations can be found in [13, 14]. Figure 1 illustrates a general arrangement using a diffraction-limited observation profile. The high numerical aperture objective focuses an excitation beam,coded with a dashed line, to a waist of characteristic radius ωxy. The detection profile has acharacteristic depth ωz because a confocal pinhole precedes the detector to reject out-of-focusemission light, coded with an unbroken line. A digital autocorrelator analyzes the detector pho-tocurrent I(t) for correlations
G(τ) = 1+< δ I(t)δ I(t + τ) >
< I >2 (1)
in time. The correlation function G(τ) exceeds unity for finite time delays τ because fluores-cence persists while a single chromophore diffuses into the observation volume.
The confocal observation profile S resembles roughly a Gaussian ellipsoid
S(r) = S0 exp
(
−2x2 + y2
ω2xy
−2z2
ω2z
)
(2)
where the characteristic lengths ω are chosen to be e−2 radii [14, p. 76]. If the fluid has a localvelocity v in the xy-plane which advects the molecule the photocurrent correlation functionis compressed by the advection of the molecule. The photocurrent correlation function for asingle species with diffusion coefficient D and average speed v is described by a normalizedcorrelation function [15]
g(τ) =
(
1+4Dτω2
xy
)−1(
1+4Dτω2
z
)−1/2
exp
⎛
⎝−(
vτωxy
)2(
1+4Dτω2
xy
)−1⎞
⎠ (3)
where the full correlation function is G(τ) = 1 + g(τ)/N. The time-average number of mole-cules N in the observation volume
Vol =
(∫S(r)dr3
)2
∫
(S(r))2 dr3(4)
here, π3/2ω2xyωz, appears in a denominator below g revealing an underlying Poisson process. As
we will show in the Data Collection section, the influence of the vτ term can be quite dramatic.
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10079#94441 - $15.00 USD Received 31 Mar 2008; revised 14 May 2008; accepted 19 Jun 2008; published 23 Jun 2008
CCD
EL
LP
60×
DCRe
BS
DCRx
PH
BP
Sample
v
ωxy
ωz
(a) (b)
Fig. 1. Diffraction-limited FCS collects fluorescent emission from molecules within a con-focal detection profile. The duration of the fluorescence signal from a single molecule islimited by the characteristic time to diffuse or flow advectively out of the observation pro-file. (a) Optical path in a custom instrument. The polychroic beam splitter DCRe, 500-nmlong pass filter LP, and band-pass filter BP excluded light at the excitation wavelength488 nm from the detection path. The second dichroic mirror DCRx gives the option to de-tect either red or green fluorescence. The pinhole PH is a fiber end. A beam splitter BS andcamera CCD assist sample alignment using the 60× objective.
2.2. Data collection
We performed observations directing 270 μW of 488 nm excitation from an Ar-Kr laser (Spec-tra Physics, Mountain View, CA) into a microscope of custom design. The line illustration inFig. 1 identifies the primary features of our instrument. A high numerical aperture (Nikon 60×)water-immersion objective focused the beam onto our samples. An 8 μm fiber (Corning, Corn-ing, NY) implented confocal rejection. We monitored intensity statistics at a GaAsP photoncounting head (Hamamatsu, Bridgewater, NJ) which fed a real-time USB interface autocorre-lator (Correlator.com, Bridgewater, NJ).
The stage held sample flow channels. Fused silica coverslips adhered to microscope slideswith melted Parafilm stencils produced cavities 0.5 cm wide by 120 μm thick by a few centime-ters in length. A pair of sandblasted apertures, along with punctured poly(dimethylsiloxane)blocks provided inlet and outlet ports for connection to a microsyringe pump. We diluted 44-nm diameter microsphere stock (G40 468 nm/508 nm ex./em., Duke, Fremont, CA) 7600× in18 MΩ water and sonicated hours preceding measurement to break up aggregates that formduring storage. We refurbished silica chips and glass flow mounts for reuse with a solvent rinseand low-power oxygen plasma.
We focused the observation profile 10 μm below the glass interface and varied the localvelocity by setting the pump at rates from 0 mL/hr to 30 mL/hr. Each correlation function took30 s to measure, and we fit each function using Eq. 3. Figure 2 presents the averages of 5normalized measured functions and the averages of their corresponding fits. The model usede-squared radii ωxy = 0.35 μm and ωz = 2.4 μm, which correspond to an observation volumeof 1.7 fL. The fitted population N ∼ 1.1 corresponds to a concentration of 1.1 nM.
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10080#94441 - $15.00 USD Received 31 Mar 2008; revised 14 May 2008; accepted 19 Jun 2008; published 23 Jun 2008
10-4
10-3
10-2
10-1
1
1.2
1.4
1.6
1.8
2
2.2
τ (s)
G
0 mm/s
5.5 mm/s
11 mm/s
17 mm/s
0 mm/s
5.5 mm/s
11 mm/s
17 mm/s
Fig. 2. Normalized autocorrelation curves acquired from diffraction-limited FCS with localfluid flow. Data are identified through markers, and calculations are distinguished by linewidth (thick or thin) and line color (black or gray). The legend tabulates velocities at thecenter of the channel.
2.3. Signal and noise
To quantify FCS’s ability to distinguish diffusive species, we subtract the correlation functionsexpected for two species whose diffusion coefficients differ, for example by 10-percent. Wedefine a signal ΔG as
ΔG(τ) = GD(τ)−G1.1D(τ) (5)
the difference between correlation curves at a reference diffusion coefficient D and at 1.1D.Panel (a) of Fig. 3 plots this signal function for the microspheres studied, for all the fluidvelocities explored in Fig. 2, according to the Gaussian ellipsoid model.
One striking feature of the signal is that it moves toward shorter time delays τ with in-creasing velocity. To see this analytically, we calculate the zero of the signal function. For ourmicroscope, ωz is significantly longer than ωxy, so our system resembles a 2-d system such thatωz → ∞. The signal
ΔG ∼ ∂G∂D
ΔD =4gτω2
xy
(
1+4Dτω2
xy
)−2[
1+4Dτω2
xy−(
vτωxy
)2]
0.1D (6)
has an analytic zero-crossing at
Dτx
ω2xy
=2+√
4+Peg2
Pe2g
(7)
time τx. The FCS observation radius ωxy provides a characteristic length scale for defining thePeclet number Peg = ωxyv/D.
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10081#94441 - $15.00 USD Received 31 Mar 2008; revised 14 May 2008; accepted 19 Jun 2008; published 23 Jun 2008
Panel (b) of Fig. 3 presents the standard error of the normalized correlation functions aver-aged in Fig. 2 as an estimate of uncertainty in our correlation curves. Since the signal movestoward short delays as indicated in Eq. 7 and in Fig. 2, high velocities collapse the signal func-tion ΔG(τ) in vertical amplitude and time scale τx, burying signal under noise.
10-4
10-3
10-2
10-1
0
10
20
x 10-3
τ (s)
D
L ΔG
10-4
10-3
10-2
10-1
10-3
10-2
10-1
τ (s)
D
L δG
0 mm/s
5.5 mm/s
11 mm/s
17 mm/s
0 mm/s
5.5 mm/s
11 mm/s
17 mm/s
(a)
(b)
Fig. 3. Diffraction-limited FCS’s ability to distinguish diffusion constants becomes buriedin noise at high fluid velocity. (a) The signal function ΔG is the difference between thecorrelation function at diffusion constant D = 5.1× 10−12 m2/s and 1.1D for varied ve-locities. The signals at 5.5 mm/s, 11 mm/s, and 17 mm/s are multiplied by a factor of 10for clarity. (b) Uncertainties in the correlation data from Fig. 2.
3. Signal and noise in zero-mode waveguides
3.1. Nanofabrication and sample assembly for reduced fluorescence collection volume
Clearly, at high flow speeds v FCS cannot measure diffusion coefficients accurately. Since thereis no flow inside a ZMW, we next tested the ability of the ZMW to distinguish species with dif-ferent diffusion coefficients in flows. Standard electron beam lithography and argon ion etchingtechniques produced the ZMWs in Fig. 4 [16, 17]. After electron beam evaporation deposited160 nm-thick films of gold on fused silica chips, we applied poly(methyl-methacrylate) resist toprepare an aperture array with 2 μm pitch. Apertures opened to ∼ 200 nm radii at the sample-gold interface, with the ∼ 25 nm radii at the silica-gold interface providing sub-wavelengthscale.
These silica-gold chips served as the coverglasses for microslide modules as discussedin section 2.2. We again adjusted pump rate between 0 mL/hr and 30 mL/hr to adjust thechannel-center fluid velocity. A 10× dilution of stock spheres corresponded to a concentra-tion ∼ 0.80 μM.
3.2. Data collection
Approximately 300 μW of laser power was fed into our microscope for correlation functionmeasurements each lasting 25 s. In analogy to section 2.2, we fit each correlation function, thenplotted the averages of normalized data and fits in Fig. 5.
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10082#94441 - $15.00 USD Received 31 Mar 2008; revised 14 May 2008; accepted 19 Jun 2008; published 23 Jun 2008
CCD
EL
LP
60×
DCRe
BS
DCRx
PH
BP
Sample
v
(a) (b)
R
r
Gold
Fused silica
Water
(c) (d)
h
Fig. 4. (a) and (b) A zero-mode waveguide method confines intensity fluctuations measure-ment to diffusers proximate to the metal substrate. (c) and (d) Repeated scanning elec-tron micrographs including those shown indicate typical scales h = 163.3 ± 8.8 nm,R = 209.3 ± 4.4 nm, and r = 26.1 ± 1.3 nm.
There are difficulties in deriving fundamentally the observation profile in ZMWs [10].Samiee and Levene enumerated assumptions for a simplified model [18]. In Samiee and Lev-ene’s work a narrow cylindrical ZMW rendered the excitation profile highly radially uni-form. Assuming that the detection profile would also be radially uniform gave effective one-dimensional diffusion in the axial direction. A simple exponential S(z) = exp(−z/z 0) axialobservation profile was chosen. We have not seen an exponential observation profile measuredindependently of the correlation function nor fundamentally derived, and literature calculationsof ZMW excitation profiles always demonstrate finite axial intensity variation [10]. Thus, wetake Samiee and Levene’s model as an empirical fitting function despite the complicated obser-vation profile and diffusion that our concave ZMWs might present [19].
The fitting function
gZMW (τ) ∝π4
(
2
√
Tπ
+(1−2T)exp(T )erfc(√
T )
)
− R(1+R2)2
√π
2erf(R
√T )
R√
T(8)
with GZMW = 1 + AgZMW has normalization chosen such that gZMW (0+) = 1. The definitionsT = Dτ/z2
0 and R = z0/h reinstate dimensions with h identifying the cavity depth. The ampli-tude A becomes the reciprocal of the population N in the observation profile that Eq. 4 definesonly in the absence of autofluorescence. Accounting for significant detection of backgroundlight reflected off the gold surface requires a correction
N =1A
(IGOLD+SPHERES − IGOLD
IGOLD+SPHERES
)2
(9)
factor. The photocurrent must be measured while illuminating gold and microspheresIGOLD+SPHERES, and while the beam is displaced, exciting only the gold surface IGOLD.
We fit Eq. 8 with parameters h = 163.3 nm and z0 = 46.4 nm obtaining D and A. Approximate
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10083#94441 - $15.00 USD Received 31 Mar 2008; revised 14 May 2008; accepted 19 Jun 2008; published 23 Jun 2008
the scanning electron micrographs in Fig. 4 with parabolic radial functions
r(z) ≈ R− (R− r)(
h− zh
)2
(10)
where the aperture has radius r at the silica-gold interface and radius R at the solution entrance.Assuming a radially uniform axially decaying observation profile throughout the ZMW, Eq. 4yields an observation volume of 6.5 aL. This volume is probably overestimated since the exci-tation profile decays in the radial direction for narrow apertures in optically thin metallic films[19]. Our measured correlation functions correspond to an observed population of N ∼ 1.1.Eq. 9 thus gave a lower bound concentration of � 0.28 μM consistent with the 0.80 μM con-centration of the solution prepared.
10-5
10-4
10-3
10-2
1
1.2
1.4
1.6
1.8
2
2.2
τ (s)
G
0 mm/s
5.5 mm/s
11 mm/s
17 mm/s
0 mm/s
5.5 mm/s
11 mm/s
17 mm/s
0 10 20
4.6
4.8
5
5.2
5.4
5.6
5.8
VC
(mm/s)
D (μ
m2 /s
)
(a) (b)
Fig. 5. ZMW correlation curves. (a) The averages of seven normalized correlation dataseries and the averages of their fits. (b) The fitted diffusion constants at channel-center ve-locities of 0 mm/s, 5.5 mm/s, 11 mm/s, and 17 mm/s. Markers indicate data. Lines indicatecalculations.
3.3. Signal and noise
The right panel of Fig. 5 shows the average fitted diffusion coefficient D from each flow rate. Afit of the constant function D(vc) = D gives D = 5.1 × 10−12 m2/s. The reduced chi-squaredvalue χ2
ν = 0.59 shows that the data are consistent with the claim that the ZMW correlationfunctions are independent of channel-center velocity v c, so we calculated a single differencesignal ΔG defined in Eq. 5 using Eq. 8. To estimate the uncertainty in correlation functions inanalogy to section 2.3, Fig. 6 plots the standard error of the normalized correlation curves aver-aged to produce Fig. 5. Because the ZMW difference signal ΔG does not collapse in amplitudeor characteristic delay time τ , signal remains above noise at high channel fluid velocity.
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10084#94441 - $15.00 USD Received 31 Mar 2008; revised 14 May 2008; accepted 19 Jun 2008; published 23 Jun 2008
10-5
10-4
10-3
10-2
0.005
0.01
0.015
0.02
τ (s)
ZM
W ΔG
10-5
10-4
10-3
10-2
10-2
10-1
τ (s)
Z
MW
δG
0 mm/s
5.5 mm/s
11 mm/s
17 mm/s
0 mm/s
5.5 mm/s
11 mm/s
17 mm/s
(a)
(b)
Fig. 6. ZMW FCS’s ability to distinguish diffusion constants persists at high channel-centervelocity. (a) The signal function ΔG is the difference between the correlation function atdiffusion constant D = 5.1 × 10−12 m2/s and 1.1D. (b) Uncertainties in the correlationdata from Fig. 5.
4. Results
We define a signal-to-noise ratio for the overall signal function ΔG
SNR = ∑τ
(ΔG(τ)δG(τ)
)2
(11)
as a standard sum of squared ratios. The summation runs over discrete data points rather than thecontinuous variable τ . Applying Eq. 11 to calculated signals and measured noise in Fig. 3 andFig. 6 gives the sensitivity plots in Fig. 7. The SNR can be interpreted in terms of a minimumdiscernible diffusion coefficient difference. Two correlation curves are barely resolved when theSNR equals unity. Under a linear regime, for example as in Eq. 6, the barely resolved diffusioncoefficient difference is proportional to
ΔDCRIT ICAL ∼ 0.1D√SNR
(12)
the reciprocal of the square-root of the SNR.The SNR in the diffraction-limited setup degrades by 47 dB by the time the channel center ve-
locity has increased to 17 mm/s. Already 34 dB of degradation occur by 5.5 mm/s. Diffraction-limited FCS becomes insensitive to diffusion coefficient when advective flow moves fluorescentprobes out of observation faster than they can diffuse out of the observation profile. In microflu-idics velocity profiling, the insensitivity of diffraction-limited FCS to the diffusion coefficientis cited as a convenience for fitting Eq. 3 [9]. In contrast, the SNR from our ZMWs remainsconstant over the same range of channel-center velocities.
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10085#94441 - $15.00 USD Received 31 Mar 2008; revised 14 May 2008; accepted 19 Jun 2008; published 23 Jun 2008
0 2.5 5 7.5 10 12.5 15 17.5 20-30
-20
-10
0
10
20
30
VC
(mm/s)
SN
R (d
B)
ZMW
DIFFR. LIM.
Fig. 7. Applying Eq. 11 to signal and uncertainty functions in Fig. 3 and Fig. 6 showsthat diffraction-limited FCS loses sensitivity as channel-center fluid velocity increases. Incontrast, ZMW FCS retains sensitivity at high values of channel-center fluid velocity.
5. Discussion
5.1. Reaction rate uncertainty
Figure 8 illustrates a possible technique for measuring kinetics for a ligand-substrate reactionusing a continuous flow mixer. For reversible binding with substrates available in excess, thebound fraction of fluorescent probe pB
dpB
dt= kON pF − kOFF pB (13)
increases with rate coefficient kON and decreases with rate coefficient kOFF . The unbound pop-ulation of probes is denoted pF . The equilibrium population fraction of bound probes
pB(∞) =kON
kON + kOFF(14)
and the difference between equilibrium and instantaneous population fractions
ΔpB(t) = pB(∞)− pB(t) (15)
express the integral of equation 13
ΔpB(t)pB(∞)
= exp [−(kON + kOFF)t] (16)
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10086#94441 - $15.00 USD Received 31 Mar 2008; revised 14 May 2008; accepted 19 Jun 2008; published 23 Jun 2008
in compact form. In Eq. 16 we represented the simple experiment in which all labels are free atthe inlets (t = 0). The population fraction measured at various local reaction times throughoutthe channel reveals the sum of reaction rates kTOT = kON + kOFF . Assuming that the equilib-rium population fraction pB(∞) is otherwise measured to arbitrary precision using steady-statemethods, kTOT determines the binding
kON = pB(∞)kTOT (17)
and unbindingkOFF = [1− pB(∞)]kTOT (18)
rates. Fractional uncertainties in t and ΔpB both contribute to the fractional uncertainty in kTOT .Diffusion parallel to flow in a plug profile broadens a mixture localized at the inlets into
a Gaussian distribution of position width√
2Dt during the time t it requires to reach an ob-servation point. At high velocity, the point at t typically samples molecules from distributionscentered as far as
δx = vδt =√
2Dt (19)
ahead or behind, so the average time 〈t〉 associated with a fluorescence collection spreads inproportion to
δ〈t〉〈t〉 ∝
√2Dtvt
=
√
2Pe‖
(20)
where we define the “parallel” Peclet number
Pe‖ =(vt)v
D(21)
identifying the distance from the inlets as a characteristic length. A high-velocity experimentexplores the regime of high Peclet numbers that suppresses diffusive broadening parallel toflow. A plug flow system operating at channel velocities in the mm/s range would achieve ahigh Peclet number of 200 or δ 〈t〉/〈t〉 = 1/10 after 5 μs.
Diffusion perpendicular to flow in a non-plug profile distributes the times of flight of particlestraveling from inlets to a specific point in a mixing channel. One particle can spend moretime than another near the high velocity streams toward the center of the channel. Narrowingthe channel increases the number of times that each particle diffuses through high and lowvelocity streams, reducing the deviation of the time of flight of a given particle from the velocityaveraged over the channel cross section. Thus Taylor dispersion can be reduced without a plugflow profile.
With negligible time-of-flight uncertainty, the fractional uncertainties
∣∣∣∣
δkON
kON
∣∣∣∣=∣∣∣∣
δkOFF
kOFF
∣∣∣∣=[
ΔpB
pB(∞)ln
(pB(∞)ΔpB
)]−1 δ pB
pB(∞)(22)
in the reaction rates become linearly dependent on the measured uncertainty δ p B of the boundfraction of probe. The fractional error
∣∣δkON/OFF/kON/OFF
∣∣BEST
= eδ pB/pB(∞) measured at
tBEST =1
kON + kOFF(23)
in other words, when ΔpB/pB(∞) = 1/e, is the minimum uncertainty afforded by a measure-ment at a single reaction time t.
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10087#94441 - $15.00 USD Received 31 Mar 2008; revised 14 May 2008; accepted 19 Jun 2008; published 23 Jun 2008
+
Δt
Fig. 8. Substrates and fluorescent labels injected at opposite inlets of a T-mixer form boundcomplexes at the outlet. The present example is transcription factor binding. A fluorescentlylabeled transcription factor serves as the ligand, and DNA containing the binding site takesthe role of the substrate. Flow velocity maps a spatial interval to a reaction time interval Δt.
The autocorrelation function for two non-interacting fluorescent species labeled free (F) andbound (B) is a normalized “sum over variances”
G(τ) = 1+NF Q2
FgF(τ)+NBQ2BgB(τ)
(NF QF +NBQB)2 (24)
where each species of average observed population NF or NB has a fluorescence yield QF orQB, and the normalized correlation functions gF and gB are calculated using the diffusion coeffi-cients DF and DB along with the parameters of the observation profile. Eqs. 3 and 8 provide twoexamples of g. The autocorrelation curve determines the bound fraction p B. If the fluorescenceyields for free and bound labels are equal, the correlation curve
G(τ) = 1+1N
[gF(τ)− pB(gF(τ)−gB(τ))] (25)
simplifies, where N is now the sum of the bound and unbound populations. Rearranging Eq. 25
pB ≈ f (τ) =gF(τ)−N [G(τ)−1]
DB−DF0.1DB
ΔG(26)
gives a constant “function,” where ΔG refers to the signal in Eq. 5. Correlation data g F , G, andfitted population N, along with a calculated signal ΔG and known diffusion coefficients, yieldan estimate of pB at each correlation delay τ . The experimental estimate of pB
pB,EST =∑τ f (τ)/δ f (τ)2
∑τ 1/δ f (τ)2 (27)
averages over the pB estimates at each τ , weighted according to their uncertainties, δ f (τ).Estimating the uncertainty in the numerator of f (τ), δ [gF(τ)−N(G(τ)−1)], with the uncer-tainties δG in Figs. 3 and 6 gives
δ pB,EST ∼ 0.1DB
DF −DB
1√SNR
(28)
an uncertainty for the unbound fraction, thus a fractional uncertainty∣∣∣∣
δkON
kON
∣∣∣∣BEST
=∣∣∣∣
δkOFF
kOFF
∣∣∣∣BEST
=kON + kOFF
kON
0.1DB
DF −DB
e√SNR
(29)
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10088#94441 - $15.00 USD Received 31 Mar 2008; revised 14 May 2008; accepted 19 Jun 2008; published 23 Jun 2008
in the reaction rates kON and kOFF , which can be written in terms of the minimum resolvablediffusion coefficient difference ΔDCRITICAL
∣∣∣∣
δkON
kON
∣∣∣∣BEST
=∣∣∣∣
δkOFF
kOFF
∣∣∣∣BEST
=kON + kOFF
kONe
ΔDCRITICAL
DF −DB(30)
using Eq. 12.Eq. 29 shows that the rate uncertainties depend on the ability to distinguish initial and final
reaction population fractions. The rate uncertainty increases when the correlation curve signalbecomes insignificant compared to noise, as occurs with diffraction-limited FCS. In contrast,ZMW maintains SNR and rate certainty. Eq. 29 also shows that fast unbinding k OFF degradesthe uncertainty in measured rate coefficients. When kOFF is large compared to kON , the initialpopulation fraction pB(0) = 0 approximates the equilibrium value pB(∞) ∼ 0, so the time-evolution of the correlation curve vanishes. Similarly, rate uncertainties increase as the diffusioncoefficients of the bound and unbound state coincide.
5.2. Physiological samples
Eq. 23 shows that in ZMWs and diffraction-limited systems alike, it is necessary to build fastmixers to observe the most informative stages of biological reactions. Eq. 23 shows that in-creases in either the binding or unbinding rate coefficients require decreases in mixing time. Ifthe two coefficients correspond to realistic timescales of a few microseconds, for example, themixing time should be reduced to a couple microseconds. Even if one of the rate coefficients issignificantly slower, the exponential decay in Eq. 16 continues to occur over microseconds. Thisis roughly the fastest reaction time scale practically accessible because modern CFMM requirea few microseconds to achieve thorough mixture. We hope that future improvements in CFMMdesign will provide access to binding and unbinding time scales faster than microseconds.
Biological molecules often have diffusion coefficients an order-of-magnitude larger thanthose of the microspheres we used. The diffraction-limited correlation curve should qual-itatively become determined by diffusion when diffusion times become shorter than ad-vection times. For our experiment, however, even biologically typical diffusion coefficients∼ 10−10m2/s would remain in a significantly advective regime for flow velocities above∼ 10 mm/s. The generic shape of the signal would remain unchanged, while the amplitudewould increase 100 fold, lifting SNR by 20 dB. Increasing the diffusion coefficient by 100-foldwould also decrease the SNR by about 10 dB for low- or zero-velocity flow since the FCS noiseincreases with decreasing correlation time τ .
For the ZMWs, the signal would translate horizontally to times 100-fold shorter otherwise re-taining functional form. Estimating from measured noise, this could increase the relevant noiseby about 10-fold, decreasing the SNR by 10 dB. The ZMW SNR would still be a horizontalline. In the advection-dominated regime, diffraction-limited SNR would still be a decreasingfunction of velocity, though the SNR advantage of ZMW at vC ∼ 17 mm/s might be 17 dBinstead of 47 dB. Reduced flow speed would help diffraction-limited FCS distinguish speciesof different diffusion coefficients but also make fast reactions difficult to observe.
5.3. Protected ZMW observation profile
A diffraction-limited beam focused at the solution interface and TIR FCS can take advantageof stick boundary conditions to achieve a degree of protection from channel flow. ZMWs offertwo levels of additional protection. ZMWs are protected from flow because they are etchedbeneath the surface. Additionally, observation volumes in ZMWs are smaller than volumesin diffraction-limited arrangements. Volumes in our studies were ∼ aL and ∼ fL in ZMWs
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10089#94441 - $15.00 USD Received 31 Mar 2008; revised 14 May 2008; accepted 19 Jun 2008; published 23 Jun 2008
and diffraction-limited FCS, respectively. Fluorescent molecules diffuse more rapidly throughsmaller volumes, reducing response of ZMW correlation curves to local velocity.
6. Summary
The ability to measure accurately diffusion coefficients using FCS within a flow mixer opens upthe measurement of reagent populations and thus reaction rates in out-of-equilibrium reactions.We have shown that by using ZMWs in the floor of a diffusional mixer, advection effects inFCS can be canceled out, and out-of-equilibrium reactions can be analyzed. We have shownthat a repertoire of reactions accessible to ZMWs is not accessible to diffraction-limited FCS.While the investment of time and materials in nanofabrication of ZMWs can be substantial, thebenefts are also substantial.
Acknowledgments
This work was supported in part by the Nanobiotechnology Center (NBTC), an STC Programof the National Science Foundation under Agreement No. ECS-9876771. Part of this workwas performed at the Cornell NanoScale Facility, a member of the National NanotechnologyInfrastructure Network, which is supported by the National Science Foundation (Grant ECS 03-35765). This work was supported in part by the Department of Defense through the NationalDefense Science and Engineering Graduate Fellowship.
(C) 2008 OSA 7 July 2008 / Vol. 16, No. 14 / OPTICS EXPRESS 10090#94441 - $15.00 USD Received 31 Mar 2008; revised 14 May 2008; accepted 19 Jun 2008; published 23 Jun 2008
