## Abstract

Zero-Mode Waveguides were first introduced for Fluorescence Correlation Spectroscopy at micromolar dye concentrations. We show that combining zero-mode waveguides with fluorescence correlation spectroscopy in a continuous flow mixer avoids the compression of the FCS signal due to fluid transport at channel velocities up to ~17 mm/s. We derive an analytic scaling relationship $\mid \frac{\delta {k}_{\mathrm{ON}}}{{k}_{\mathrm{ON}}}\mid =\mid \frac{\delta {k}_{\mathrm{OFF}}}{{k}_{\mathrm{OFF}}}\mid ~\frac{{k}_{\mathrm{ON}}+{k}_{\mathrm{OFF}}}{{k}_{\mathrm{ON}}}\frac{0.1{D}_{B}}{{D}_{F}-{D}_{B}}\frac{e}{\sqrt{\mathrm{SNR}}}$ converting this flow velocity insensitivity to improved kinetic rate certainty in time-resolved mixing experiments. Thus zero-mode waveguides make FCS suitable for direct kinetics measurements in rapid continuous flow.

©2008 Optical Society of America

## 1. Introduction

Our paper illustrates how the use of a floor of zero-mode wavegtuides (ZMW) sustains sensitivity to diffusion measurements for Fluorescence Correlation Spectroscopy (FCS) in high velocity flow channels, as occur in Continuous Flow Microfluidic Mixer (CFMM) designs. The basic idea is very simple: the floor of a CFMM is carpeted with an array of ZMWs which sample the local concentration of molecules at a particular region of the flow pattern but are shielded from the advection of the flow by the walls of the ZMW. Single molecules within a ZMW have a characteristic residence time given by their diffusion coefficient and the effective volume of the ZMW. Although above the entry of the ZMW the fluid is advecting, within the ZMW there is no advection and hence we expect the mean residence times in the ZMW waveguide, and hence the determination of the diffusion coefficient of the molecule, to be independent of the speed of the external flow.

This result has important consequences. CFMM designs allow studies of biological reaction and 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. Hydrodynamic focusing achieves submicrosecond time resolution and mixing times less than 10 *µ*s, enabling protein folding kinetic measurements [4]. Magde *et al.* developed FCS [5, 6, 7] for studying chemical kinetics by measuring the average duration, or correlation time, of fluorescence intensity bursts for a single chromophore as it passed through a small sample volume. 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 means that it cannot be used to obtain diffusion coefficients in a CFMM device if the mean diffusional time of the chromophore out of the sample volume is greater than the time to advect the chromophore out of the sample volume. Improving diffusion constant sensitivity of FCS at high velocity would allow FCS to characterize the time evolution of species populations during a chemical reaction in a CFMM device.

Diffraction-limited FCS collects intensity fluctuations from the passage of single chromophores through fL (10^{-12} L) volumes, requiring nanomolar concentrations for single molecule 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 the wavelength of incident light. Workers refer to a subset of SWAs by the name ZMW. A ZMW would be too narrow to propagate the incident wavelength if the metal film were extended to infinite thickness. Levene *et al.* studied interactions between fluorescent nucleotides and polymerases immobilized in ZMWs to observe incorporation events and photobleaching [10], but the possibility of using ZMWs with FCS for reaction studies in continuous flow has not been discussed.

This paper compares FCS sensitivity for diffraction-limited andZMW methods in rapid flow. In section 2 we present standard diffraction-limited FCS, our data collection techniques, signal functions calculated, and noise functions measured. Section 3 repeats the discussion in section 2 for ZMWs. We use the signal and noise functions from sections 2 and 3 to calculate SNRs for determination of diffusion coefficients as a function of flow speed in section 4. Finally sections 4 and 5.1 show that, in flow channels, ZMWs improve SNR for distinguishing diffusion coefficients, 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 applications 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 *ω _{xy}*. The detection profile has a characteristic depth

*ω*because a confocal pinhole precedes the detector to reject out-of-focus emission light, coded with an unbroken line. A digital autocorrelator analyzes the detector photocurrent

_{z}*I*(

*t*) for correlations

in time. The correlation function *G*(*τ*) exceeds unity for finite time delays *τ* because fluorescence persists while a single chromophore diffuses into the observation volume.

The confocal observation profile *S* resembles roughly a Gaussian ellipsoid

where the characteristic lengths *ω* are chosen to be *e*
^{-2} radii [14, p. 76]. If the fluid has a local velocity *v* in the *xy*-plane which advects the molecule the photocurrent correlation function is compressed by the advection of the molecule. The photocurrent correlation function for a single species with diffusion coefficient *D* and average speed v is described by a normalized correlation function [15]

where the full correlation function is *G*(*τ*)=1+*g*(*τ*)/*N*. The time-average number of molecules *N* in the observation volume

here, *π*
^{3/2}
*ω*
^{2}
_{xy}*ω _{z}*, 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.

#### 2.2. Data collection

We performed observations directing 270 *µ*W of 488 nm excitation from an Ar-Kr laser (Spectra Physics, Mountain View, CA) into a microscope of custom design. The line illustration in Fig. 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 *µ*m fiber (Corning, Corning, NY) implented confocal rejection. We monitored intensity statistics at a GaAsP photon counting head (Hamamatsu, Bridgewater, NJ) which fed a real-time USB interface autocorrelator (Correlator.com, Bridgewater, NJ).

The stage held sample flow channels. Fused silica coverslips adhered to microscope slides with melted Parafilm stencils produced cavities 0.5 cm wide by 120 *µ*m thick by a few centimeters 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× in 18 MΩ water and sonicated hours preceding measurement to break up aggregates that form during storage. We refurbished silica chips and glass flow mounts for reuse with a solvent rinse and low-power oxygen plasma.

We focused the observation profile 10 *µ*m below the glass interface and varied the local velocity by setting the pump at rates from 0 mL/hr to 30 mL/hr. Each correlation function took 30 s to measure, and we fit each function using Eq. 3. Figure 2 presents the averages of 5 normalized measured functions and the averages of their corresponding fits. The model used *e*-squared radii *ω _{xy}*=0.35

*µ*m and

*ω*=2.4

_{z}*µ*m, which correspond to an observation volume of 1.7 fL. The fitted population

*N*~1.1 corresponds to a concentration of 1.1 nM.

#### 2.3. Signal and noise

To quantify FCS’s ability to distinguish diffusive species, we subtract the correlation functions expected for two species whose diffusion coefficients differ, for example by 10-percent. We define a signal Δ*G* as

the difference between correlation curves at a reference diffusion coefficient *D* and at 1.1*D*. Panel (a) of Fig. 3 plots this signal function for the microspheres studied, for all the fluid velocities 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 increasing velocity. To see this analytically, we calculate the zero of the signal function. For our microscope, *ω _{z}* is significantly longer than

*ω*, so our system resembles a 2-d system such that

_{xy}*ω*→∞. The signal

_{z}has an analytic zero-crossing at

time *τ _{x}*. The FCS observation radius

*ω*provides a characteristic length scale for defining the Péclet number Pe

_{xy}_{g}=

*ω*/

_{xy}v*D*.

Panel (b) of Fig. 3 presents the standard error of the normalized correlation functions averaged in Fig. 2 as an estimate of uncertainty in our correlation curves. Since the signal moves toward short delays as indicated in Eq. 7 and in Fig. 2, high velocities collapse the signal function Δ*G*(*τ*) in vertical amplitude and time scale *τ _{x}*, burying signal under noise.

## 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 there is no flow inside a ZMW, we next tested the ability of the ZMW to distinguish species with different diffusion coefficients in flows. Standard electron beam lithography and argon ion etching techniques produced the ZMWs in Fig. 4 [16, 17]. After electron beam evaporation deposited 160 nm-thick films of gold on fused silica chips, we applied poly(methyl-methacrylate) resist to prepare an aperture array with 2 *µ*m pitch. Apertures opened to ~200 nm radii at the sample-gold interface, with the ~25 nm radii at the silica-gold interface providing sub-wavelength scale.

These silica-gold chips served as the coverglasses for microslide modules as discussed in section 2.2. We again adjusted pump rate between 0 mL/hr and 30 mL/hr to adjust the channel-center fluid velocity. A 10× dilution of stock spheres corresponded to a concentration ~0.80 *µ*M.

#### 3.2. Data collection

Approximately 300 *µ*W of laser power was fed into our microscope for correlation function measurements each lasting 25 s. In analogy to section 2.2, we fit each correlation function, then plotted the averages of normalized data and fits in Fig. 5.

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 Levene’s work a narrow cylindrical ZMW rendered the excitation profile highly radially uniform. 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}) axial observation profile was chosen. We have not seen an exponential observation profile measured independently of the correlation function nor fundamentally derived, and literature calculations of ZMW excitation profiles always demonstrate finite axial intensity variation [10]. Thus, we take Samiee and Levene’s model as an empirical fitting function despite the complicated observation profile and diffusion that our concave ZMWs might present [19].

The fitting function

with *G _{ZMW}*=1+A

*has normalization chosen such that*

_{gZMW}*g*(0

_{ZMW}^{+})=1. The definitions

*T*=

*Dτ*/

*z*

^{2}

_{0}and

*R*=

*z*

_{0}/

*h*reinstate dimensions with

*h*identifying the cavity depth. The amplitude

*A*becomes the reciprocal of the population

*N*in the observation profile that Eq. 4 defines only in the absence of autofluorescence. Accounting for significant detection of background light reflected off the gold surface requires a correction

factor. The photocurrent must be measured while illuminating gold and microspheres *I*
_{GOLD+SPHERES}, and while the beam is displaced, exciting only the gold surface *I _{GOLD}*.

We fit Eq. 8 with parameters *h*=163.3 nmand *z*
_{0}=46.4 nm obtaining *D* and *A*. Approximate the scanning electron micrographs in Fig. 4 with parabolic radial functions

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. 4 yields an observation volume of 6.5 aL. This volume is probably overestimated since the excitation 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 *µ*M consistent with the 0.80 *µ*M concentration of the solution prepared.

#### 3.3. Signal and noise

The right panel of Fig. 5 shows the average fitted diffusion coefficient *D* from each flow rate. A fit of the constant function *D*(*v _{c}*)=

*D*gives

*D*=5.1×10

^{-12}m

^{2}/s. The reduced chi-squared value χ

^{2}

_{v}=0.59 shows that the data are consistent with the claim that the ZMW correlation functions are independent of channel-center velocity

*v*, so we calculated a single difference signal Δ

_{c}*G*defined in Eq. 5 using Eq. 8. To estimate the uncertainty in correlation functions in analogy to section 2.3, Fig. 6 plots the standard error of the normalized correlation curves averaged to produce Fig. 5. Because the ZMW difference signal Δ

*G*does not collapse in amplitude or characteristic delay time

*τ*, signal remains above noise at high channel fluid velocity.

## 4. Results

We define a signal-to-noise ratio for the overall signal function Δ*G*

as a standard sum of squared ratios. The summation runs over discrete data points rather than the continuous variable *τ*. Applying Eq. 11 to calculated signals and measured noise in Fig. 3 and Fig. 6 gives the sensitivity plots in Fig. 7. The SNR can be interpreted in terms of a minimum discernible diffusion coefficient difference. Two correlation curves are barely resolved when the SNR equals unity. Under a linear regime, for example as in Eq. 6, the barely resolved diffusion coefficient difference is proportional to

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 velocity 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 flowmoves fluorescent probes out of observation faster than they can diffuse out of the observation profile. In microfluidics velocity profiling, the insensitivity of diffraction-limited FCS to the diffusion coefficient is cited as a convenience for fitting Eq. 3 [9]. In contrast, the SNR from our ZMWs remains constant over the same range of channel-center velocities.

## 5. Discussion

#### 5.1. Reaction rate uncertainty

Figure 8 illustrates a possible technique for measuring kinetics for a ligand-substrate reaction using a continuous flow mixer. For reversible binding with substrates available in excess, the bound fraction of fluorescent probe *p _{B}*

increases with rate coefficient *k _{ON}* and decreases with rate coefficient

*k*. The unbound population of probes is denoted

_{OFF}*p*. The equilibrium population fraction of bound probes

_{F}and the difference between equilibrium and instantaneous population fractions

express the integral of equation 13

in compact form. In Eq. 16 we represented the simple experiment in which all labels are free at the inlets (*t*=0). The population fraction measured at various local reaction times throughout the channel reveals the sum of reaction rates *k _{TOT}*=

*k*+

_{ON}*k*. Assuming that the equilibrium population fraction

_{OFF}*p*(∞) is otherwise measured to arbitrary precision using steady-state methods,

_{B}*k*determines the binding

_{TOT}and unbinding

rates. Fractional uncertainties in *t* and Δ*p _{B}* both contribute to the fractional uncertainty in

*k*.

_{TOT}Diffusion parallel to flow in a plug profile broadens a mixture localized at the inlets into a Gaussian distribution of position width
$\sqrt{2Dt}$
during the time *t* it requires to reach an observation point. At high velocity, the point at *t* typically samples molecules from distributions centered as far as

ahead or behind, so the average time 〈*t*〉 associated with a fluorescence collection spreads in proportion to

where we define the “parallel” Péclet number

identifying the distance from the inlets as a characteristic length. A high-velocity experiment explores the regime of high Péclet numbers that suppresses diffusive broadening parallel to flow. A plug flow system operating at channel velocities in the mm/s range would achieve a high Péclet number of 200 or *δ*
_{〈t〉}/〈*t*〉=1/10 after 5 *µ*s.

Diffusion perpendicular to flow in a non-plug profile distributes the times of flight of particles traveling from inlets to a specific point in a mixing channel. One particle can spend more time than another near the high velocity streams toward the center of the channel. Narrowing the channel increases the number of times that each particle diffuses through high and low velocity streams, reducing the deviation of the time of flight of a given particle from the velocity averaged over the channel cross section. Thus Taylor dispersion can be reduced without a plug flow profile.

With negligible time-of-flight uncertainty, the fractional uncertainties

in the reaction rates become linearly dependent on the measured uncertainty *δ _{pB}* of the bound fraction of probe. The fractional error |

*δk*

_{ON/OFF}/

*k*

_{ON/OFF}|

_{BEST}=

*eδp*/

_{B}*p*(∞) measured at

_{B}in other words, when Δ*p _{B}*/

*p*(∞)=1/

_{B}*e*, is the minimum uncertainty afforded by a measurement at a single reaction time

*t*.

The autocorrelation function for two non-interacting fluorescent species labeled free (*F*) and bound (*B*) is a normalized “sum over variances”

where each species of average observed population *N _{F}* or

*N*has a fluorescence yield

_{B}*Q*or

_{F}*Q*, and the normalized correlation functions

_{B}*g*and

_{F}*g*are calculated using the diffusion coefficients

_{B}*D*and

_{F}*D*along with the parameters of the observation profile. Eqs. 3 and 8 provide two examples of

_{B}*g*. The autocorrelation curve determines the bound fraction

*p*. If the fluorescence yields for free and bound labels are equal, the correlation curve

_{B}simplifies, where *N* is now the sum of the bound and unbound populations. Rearranging Eq. 25

gives a constant “function,” where Δ*G* refers to the signal in Eq. 5. Correlation data *g _{F}*,

*G*, and fitted population

*N*, along with a calculated signal Δ

*G*and known diffusion coefficients, yield an estimate of

*p*at each correlation delay

_{B}*τ*. The experimental estimate of

*p*

_{B}averages over the *p _{B}* estimates at each

*τ*, weighted according to their uncertainties,

*δf*(

*τ*). Estimating the uncertainty in the numerator of

*f*(

*τ*),

*δ*[

*g*(

_{F}*τ*)-

*N*(

*G*(

*τ*)-1)], with the uncertainties

*δG*in Figs. 3 and 6 gives

an uncertainty for the unbound fraction, thus a fractional uncertainty

in the reaction rates *k _{ON}* and

*k*, which can be written in terms of the minimum resolvable diffusion coefficient difference Δ

_{OFF}*D*

_{CRITICAL}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 signal becomes 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}* degrades the uncertainty in measured rate coefficients. When

*k*is large compared to

_{OFF}*k*, the initial population fraction

_{ON}*p*(0)=0 approximates the equilibrium value

_{B}*p*(∞) ~0, so the timeevolution of the correlation curve vanishes. Similarly, rate uncertainties increase as the diffusion coefficients of the bound and unbound state coincide.

_{B}#### 5.2. Physiological samples

Eq. 23 shows that in ZMWs and diffraction-limited systems alike, it is necessary to build fast mixers to observe the most informative stages of biological reactions. Eq. 23 shows that increases in either the binding or unbinding rate coefficients require decreases in mixing time. If the two coefficients correspond to realistic timescales of a few microseconds, for example, the mixing time should be reduced to a couple microseconds. Even if one of the rate coefficients is significantly slower, the exponential decay in Eq. 16 continues to occur over microseconds. This is roughly the fastest reaction time scale practically accessible because modern CFMM require a few microseconds to achieve thorough mixture. We hope that future improvements in CFMM design will provide access to binding and unbinding time scales faster than microseconds.

Biological molecules often have diffusion coefficients an order-of-magnitude larger than those of the microspheres we used. The diffraction-limited correlation curve should qualitatively become determined by diffusion when diffusion times become shorter than advection times. For our experiment, however, even biologically typical diffusion coefficients ~10^{-10}m^{2}/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 amplitude would increase 100 fold, lifting SNR by 20 dB. Increasing the diffusion coefficient by 100-fold would also decrease the SNR by about 10 dB for low- or zero-velocity flow since the FCS noise increases with decreasing correlation time *τ*.

For the ZMWs, the signal would translate horizontally to times 100-fold shorter otherwise retaining functional form. Estimating from measured noise, this could increase the relevant noise by about 10-fold, decreasing the SNR by 10 dB. The ZMW SNR would still be a horizontal line. In the advection-dominated regime, diffraction-limited SNR would still be a decreasing function of velocity, though the SNR advantage of ZMW at *v _{C}*~17 mm/s might be 17 dB instead of 47 dB. Reduced flow speed would help diffraction-limited FCS distinguish species of 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 advantage of stick boundary conditions to achieve a degree of protection from channel flow. ZMWs offer two levels of additional protection. ZMWs are protected from flow because they are etched beneath the surface. Additionally, observation volumes in ZMWs are smaller than volumes in diffraction-limited arrangements. Volumes in our studies were ~ aL and ~ fL in ZMWs and diffraction-limited FCS, respectively. Fluorescent molecules diffuse more rapidly through smaller 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 up the 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 in FCS can be canceled out, and out-of-equilibrium reactions can be analyzed. We have shown that 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, the benefts are also substantial.

## Acknowledgments

This work was supported in part by the Nanobiotechnology Center (NBTC), an STC Program of the National Science Foundation under Agreement No. ECS-9876771. Part of this work was performed at the Cornell NanoScale Facility, a member of the National Nanotechnology Infrastructure 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 National Defense Science and Engineering Graduate Fellowship.

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