Measurement of the Surface Concentration for Bioassay Kinetics in

sivity at the surface 3-4 orders of magnitude greater than molecular diffusion alone, even close to the point of injection. We directly measure the ef...
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Anal. Chem. 2005, 77, 833-839

Measurement of the Surface Concentration for Bioassay Kinetics in Microchannels Aure´lien Bancaud, Gaudeline Wagner, Kevin D. Dorfman, and Jean-Louis Viovy*

Laboratoire Physico-Chimie Curie, CNRS/UMR 168, Institut Curie, 26 rue d’Ulm, F-75248, Paris Cedex, France

We present a simple and versatile method, based on fluorescence microscopy, to reliably measure the concentration of advected molecules in the vicinity of surfaces in microchannels. This tool is relevant to many microfluidic applications such as immunoassays and singlemolecule experiments, where one probes the kinetics of a reaction between an immobilized target and a reactant carried by the bulk flow. The characterization of the surface concentration highlights the dominant role of transverse diffusion, which generates an apparent diffusivity at the surface 3-4 orders of magnitude greater than molecular diffusion alone, even close to the point of injection. We directly measure the effects of the longitudinal position along the channel and of the flow rate on the concentration front and develop a simple analytical model that compares well with the data. Finally, we propose a method to properly account for concentration fronts in single-molecule measurements and use it to directly access the kinetics parameters of protamineinduced condensation of DNA. Interest in microfluidic devices and components has been stimulated recently by their potential application in analytical and bioanalytical chemistry. Standard microfluidic bioassays use an immobilized biospecific layer and take advantage of the inherent high surface-to-volume aspect ratio to directly observe immobilization of immunoreagents.1 The flow of solutions is mainly controlled by use of electroosmosis or pressurized flow.2 In principle, enzyme activity can be directly measured if, for instance, a nondetectable species is converted to a detectable one.3-4 Many single-molecule experiments also consist of attaching a DNA molecule to a coated surface in a microchannel. Specific enzymes (or one of their cofactors) are advected to the molecule by a controlled flow and, from real-time visualization, their activity and kinetic parameters are measured quantitatively.5-7 Neverthe* Corresponding author. Telephone: (33)1 42 34 67 52. Fax: (33)1 40 51 06 36. E-mail: [email protected]. (1) Dodge, A.; Flury, K.; Verpoorte, E.; de Rooij, N. F. Anal. Chem. 2001, 73, 3400-3409. (2) Bilitewski, U.; Genrich, M.; Kadow, S.; Mersal, G. Anal. Bioanal. Chem. 2003, 377, 556-569. (3) Yakovleva, J.; Davidsson, R.; Bengtsson, M.; Laurell, T.; Emneus, J. Biosens. Bioelectron. 2003, 19, 21-34. (4) Yakovleva, J.; Davidsson, R.; Lobanova, A.; Bengtsson, M.; Eremin, S.; Laurell, T.; Emneus, J. Anal. Chem. 2002, 74, 2994-3004. (5) Ladoux, B.; Quivy, J. P.; Doyle, P.; du Roure, O.; Almouzni, G.; Viovy, J. L. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 14251-14256. (6) Taekjip H.; Rasnik, I.; Babcock, H. P.; Gauss, G. H.; Lohman, T. M.; Chu, S. Nature 2002, 419, 638-641. 10.1021/ac048996+ CCC: $30.25 Published on Web 12/29/2004

© 2005 American Chemical Society

less, for bioassays or single-molecule experiments, the surface concentration of the reactant that is injected in the channel has to be well characterized in order to reliably access the kinetics. Indeed, the spreading of the reactant can be large, as Taylor diffusion greatly enhances the apparent diffusivity. Shear dispersion, originally elucidated by Taylor,8 arises when a diffusing substance is advected in a nonuniform velocity flow. It naturally occurs in microchannels, as the flow profile is parabolic. Taylor showed that, once the tracer molecule has had time to diffuse over the whole cross section, the transverse concentration is homogeneous and the front spreads with an apparent longitudinal Gaussian diffusivity. In the Taylor dispersion regime, a simple bulk concentration measurement gives access to the surface concentration. However, in many practical applications, the time to reach the Taylor regime is very long and the Taylor’s Gaussian approximation is not accurate. The transient concentration is strongly inhomogeneous, especially close to the surface, where the flow velocity decreases to zero. Thus, we propose a simple and versatile technique, well-suited for a whole class of analytes and microchannel geometries, to effectively characterize concentration fronts at the walls using fluorescence microscopy. Our strategy uses a fluorescently labeled tracer. To probe specifically the vicinity of the surface, we add a nonfluorescent dye that is strongly absorbing the excitation wavelength and weakly absorbing the emission one. The fluorescent intensity is collected by a video camera, permitting a direct measurement of the surface concentration. In what follows, we measure the behavior of the surface concentration front for different flow rates and longitudinal positions along the channel. Although the velocity near the surface is close to zero, transverse molecular diffusion brings analytes from higher velocity streamlines, creating an apparent velocity, together with a large broadening of the front. To interpret our results at the scaling level and understand the underlying physics, we propose an analytical model for which the surface concentration is the result of Taylor diffusion in an effective layer whose depth depends on the flow rate and the longitudinal position. We conclude by demonstrating that accurate kinetics measurements at the single-molecule level require a proper accounting of surface concentration. We also propose a technique to incorporate concentration fronts directly into the data analysis and to improve strongly the accuracy of kinetic parameters determination. We (7) Van Oijen, A. M.; Blainey, P. C.; Crampton, D. J.; Richardson, C. C.; Ellenberger, T.; Xie, S. Science 2003, 301, 1235-1238. (8) Taylor, G. I. Proc. R. Soc., London A 1953, 219, 186-203.

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Figure 1. Orange G absorbance at 450, 490, 520, and 550 nm.

use the example of protamine-induced condensation of DNA.9-10 This protein maintains the genomic DNA in mammalian sperm in a highly compact and transcriptionally inactive state.11-12 EXPERIMENTAL SECTION Experimental Strategy. Our fluorescent probe is casein (a milk protein of molecular weight roughly 25 000), which is coupled to fluorescein (Sigma) to make it observable. Its excitation wavelength is 490 nm and emission is at 540 nm. To probe the local concentration close to the surface, we add a nonfluorescent dye, Orange G (Sigma), absorbing strongly at the excitation wavelength (490 nm) and weakly at the emission one (540 nm) (Figure 1). At 490 nm, we measure an extinction coefficient of β(490) ) 17 620 L mol-1 cm-1. The absorbance at 550 nm is negligible in comparison, β (550) ) 204 L mol-1 cm-1. The Beer-Lambert law predicts an exponential decay of the excitation intensity (E) proportional to the dye molar concentration (cdye) multiplied by the extinction coefficient, which leads to the following expression:

E(z) ) E0 exp(- z/ζ), ζ-1 ) ln (10)f (NA)βcdye

(1)

where z is the beam direction (Figure 2C) and f (NA) accounts for the numerical aperture of the objective. With simple geometric arguments, one obtains f (NA) ) -ln(cos θ0)/(1 - cos θ0) with NA ) n sin θ0, which gives 1.61 for an oil immersion 1.4 NA objective. From eq 1, the penetration length ζ can be estimated to 3.4 µm at 0.045 M Orange G. When a solution of fluorescent casein is in the field of the objective, the fluorescent intensity I(t) measured by the detector is simply defined by the convolution of the intensity and the exponential decay due to the absorbing dye,

I(t) ) R

∫

∞

0

E0 exp(- z/ζ)c(z,t) dz

(2)

with R a numerical factor accounting for the quantum yield of the fluorophore and c(z,t) the local concentration of fluorescent casein. (9) Brewer, L.; Corzett, M.; Balhorn, R. Science 1999, 286, 120-123. (10) Brewer, L.; Corzett, M.; Lau, E. Y.; Balhorn, R. J. Biol. Chem. 2003, 43, 42403-42408. (11) Sassone-Corsi, P. Science 2002, 296, 2176-2178. (12) Meistrich, M. L. In Histones and Other Basic Nuclear Proteins; Hilnica L. S., Stein, G. S., Stein, J. L., Eds.; CRC Press: Boca Raton, FL, 1989; pp 165182.

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Figure 2. Schematic of the experimental apparatus: (1) syringe pump, (2) electrovalve, (I) inlet channel, (O1) outlet channel 1, (O2) outlet channel 2, (OP) objective position. (A) Sample is aspirated in microchannel O1 until the concentration is homogeneous at the T-junction. (B) The electrovalve switches the aspiration to channel O2, where the objective is positioned. (C) Side view of the PDMS chip with a representation of the region of collected intensity (E(z)).

Microfluidic Chip. Chips were prepared by rapid prototyping and PDMS technology.13 Starting with a master composed of a positive relief of SU-8 resin on a glass wafer made by photolithography, a PDMS replica of the master was formed. Once cured, PDMS was peeled away, punched with holes, and sealed on a coverslip. The channels were 500 µm wide, 165 µm high and constructed in the shape of a T with one inlet (I) and two outlets (O1, O2) as depicted in Figure 2. The two outlets were connected by silicone tubing to an aspirating syringe pump (Kd Scientific). To switch the flow between outlets, a pinch electrovalve (NResearch) was used. The PDMS chip was mounted on an inverted microscope (Zeiss) with laser excitation (488 nm) (Coherent) equipped with an oil immersion objective (100×, 1.4 NA, Olympus) (Figure 2C). The area probed by the objective was a circle of diameter roughly 50 µm precisely centered in the channel. The depth of the collected intensity region was adjusted with the protocol described in the previous paragraph. Images were then grabbed using an intensified CCD camera (LHESA, Les Ulis, France), digitized, and analyzed in real time. Surface Concentration Measurement. First the microchannels were filled with 1× TBE buffer (Tris base 89 mM, boric acid 89 mM, and EDTA 2 mM (pH 8.3)) containing the desired amount of Orange G and 1 mg/mL casein to coat surfaces and make them hydrophilic. We then aspirated into channel O1 the same solution, complemented with 0.05 mg/mL fluorescent casein, until a homogeneous concentration was reached at the T-junction (Figure 2A). In the meantime, the objective was accurately centered transversally in channel O2 and positioned between 2 and 4.5 mm (13) Whitesides, G. M.; Ostuni, E.; Takayama, S.; Ingber, D. E. Annu. Rev. Biomed. Eng. 2001, 3, 335-373.

away from the T-junction by an electric translational stage (Newport). In the following, we neglect the influence of the width of the channel, defining the x-axis as the longitudinal position and the z-axis as the direction transverse to the flow, collinear to the excitation beam. The electrovalve was then switched to channel O2, and acquisition was started simultaneously (Figure 2B). Initially, the concentration profile was a step at the T-junction and we measured I(t) in real time. Each experimental curve is the average of at least two measurements. Protamine Condensation Assay. We propose to use the setup described above to obtain a reliable assay for kinetics of the compaction of DNA by protamine from salmon (MW 5100, Sigma). This protein contains a large number of basic amino acids and strongly interacts with DNA. Its binding results in condensation. This process has already been observed in real time on a single DNA molecule with an optical tweezers setup.9 Our strategy simply consists of attaching a streptavidin end modified λ-phage DNA (48.5 kbp) to a biotin-coated glass coverslip, in the PDMS microchannel (O2). The flow stretches the molecule.14 Using YOYO-1 (Molecular Probes), DNA is fluorescently labeled and its contour length is measured in real time with simple video microscopy. We then introduce protamine at a concentration of 50 nM, solubilized in a buffer containing 50 mM NaCl, 10 mM Tris-HCl (pH 7.5), and 2% β-mercaptoethanol, into the flow channel (O1). The electrovalve switches to channel O2, and the kinetics of condensation are directly measured in real time on a single molecule from the decrease of its length.5,9 THEORETICAL BACKGROUND Taylor-Aris dispersion8,15 arises from the combination of longitudinal advection and transverse diffusion in the channel. Once the time scale of flow advection L/uj (with L the position of the detector along the channel and uj the average speed of the flow) is much greater than the transverse diffusion time H2/D (where H is the channel height and D the molecular diffusivity of the analyte), the concentration profile is transversally homogeneous and obeys Fick’s law with a Taylor dispersion coefficient:16

DTaylor ) D(1 + (1/210)Pe2)

(3)

where Pe ) uj H/D is the Peclet number. To put in some numbers, we consider the case of bovine serum albumin (BSA; MW 67 000), whose molecular diffusion coefficient is 6 × 10-11 m2/s.17 For a channel of 165 µm high and a typical flow velocity of 0.4 mm/s, we find that the Peclet number is roughly 1000 and the Taylor dispersion coefficient is 3 × 10-7 m2/s without taking into account the channel’s width. This dimension further increases the dispersion.18 Importantly, once this regime is reached, the surface concentration can be directly measured from the bulk value, as the transverse distribution is homogeneous. However, we estimate L ) 8HPe,19 the length at which the Taylor regime is attained, to be 1 m in our case. This is very long (14) Doyle, P.; Ladoux, B.; Viovy, J. L. Phys. Rev. Lett. 2000, 84, 4769-4772. (15) Aris, R. Proc. R. Soc., London A 1956, 235, 67-77. (16) Wooding, R. A. J. Fluid Mech. 1960, 7, 501. (17) Johnson, E. M.; Berk, D. A.; Jain, R. K.; Deen, W. M. Biophys. J. 1995, 68, 1561-1568. (18) Guell, D. C.; Cox, R. G.; Brenner, H. Chem. Eng. Commun. 1987, 58, 231. (19) Taylor, G. I. Proc. R. Soc., London A 1954, 225, 473-477.

compared to the channel size and definitely not in the range of classical microfabrication. For smaller distances, the transverse concentration is strongly inhomogeneous. Close to the walls, the flow vanishes and the arrival of analytes is dominated by transverse diffusion that brings material from higher in the channel. The exact analytical solution of the differential equations describing the concentration for L < 8HPe is a complex mathematical problem. Asymptotic solutions at the walls have been derived only for very short times.20 The governing equations may also be solved numerically, but we are more interested in a simple and analytical model that predicts the scaling behavior of the surface concentration. Since we are only looking at the region close to the surface, we propose that the concentration front at the surface arises primarily from an effective layer of thickness H whose solute molecules can reach the surface by transverse diffusion before the arrival of the front. We neglect the contribution from molecules initially at z > H. To simplify the problem, we assume that there is no flux at z ) 0 and z ) H for the solute particles of interest. As the analytes have time to diffuse across the considered fluid layer, the situation is equivalent to Taylor dispersion between parallel plates with a truncated parabolic flow, whose details are considered in the Appendix. The Taylor condition gives a crude evaluation of the effective thickness:

 ≈ (H/L)-1/3Pe-1/3

(4)

The concentration front arising from this problem is an error function with an effective velocity uj eff and diffusivity Deff. We calculate them in the Appendix, finding to the leading order term in 

u j eff ≈ u j ≈ u j (H/L)-1/3Pe-1/3

(5)

Deff ≈ D4Pe2 ≈ D(H/L)-4/3Pe2/3

(6)

We note that the diffusive layer responsible for the surface concentration  increases with the longitudinal position of the detector L, as do the effective diffusivity and effective velocity. Importantly, uj eff and Deff depend weakly on the protein molecular weight (an exponent of -1/9 when combining Einstein’s law with eqs 5 and 6). In a typical experiment, we have H/L ≈ 0.05 and Pe ≈ 1000, so that  ≈ 0.25 (eq 4). Clearly, the form of the concentration front near the surface is strongly affected by the bulk flow. Moreover, since  is not small compared to 1, we would expect some error in our leading order solutions eqs 5 and 6, but we still anticipate that the scaling will be a reasonable approximation of that observed in the experiments. RESULTS Are We Measuring a Surface Concentration? We probed the intensity profile as a function of the concentration of Orange G. The flow rate was set at 30.6 nL/s, and the objective position (20) Phillips, C. G.; Kay, S. R. Proc. R. Soc., London A 1997, 453, 2669-2688. (21) Brenner, H.; Edwards, D. A. Macrotransport Processes; Butterworth-Heinemann: Boston, 1994.

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Figure 3. Normalized intensity of fluorescent casein at 2.5 mm away from the T-junction, with a flow rate of 30.6 nL/s, as a function of Orange G concentration (0, 0.0045, 0.0225, 0.045, and 0.09 M).

was 2.5 mm downstream from the T-junction. We used Orange G concentrations of 0, 0.0045, 0.0225, 0.045, and 0.09 M, corresponding to estimated penetration lengths of ∞, 35, 6.8, 3.4, and 1.7 µm (Figure 3). From the curves in Figure 3, we see that the intensity profile changes strongly when the Orange G concentration increases from 0 to 0.0225 M. This behavior arises from transverse heterogeneity in casein concentration created by the Poiseuille flow. With a large penetration length, our detector integrates over streamlines where analytes are quickly advected. Therefore, the lag time before the proteins arrive appears to increase with the Orange G concentration. The concentration profile appears to reach a limit when the penetration length is less than 5 µm (concentrations of 0.045 and 0.09 M in Figure 3). The steady-state concentration is attained within roughly 20 s whereas the time scale for proteins such as BSA to diffuse across a 3-µm layer is ∼0.1 s (For BSA, the time to diffuse across a 3-µm layer is roughly 0.15 s.). Moreover, the diffusion is fast relative to the acquisition time (6 Hz), so we can assume that concentration is roughly homogeneous across the penetration length at each data point. Thus, eq 2 can be linearized

I(x,t) )

∫

∞

0

( ζz) dz

c(x,z,t) exp -

c(x,0,t)

∫

∞

0

Figure 5. Time versus normalized intensity. Crosses represent the normalized intensity at 2.5 mm from the T-junction with a flow rate of 30.6 nL/s and 0.045 M Orange G. The line represents an error function fit from which ueff and Deff are deduced. Table 1. Fitted Values of u j eff and Deff for Casein Concentration Fronts as a Function of Flow Ratea flow rate (nL/s) u j eff (mm/s) ( 0.005 Deff (10-8 m2/s) ( 0.3

30.6 0.230 5.3

61.2 0.350 8.7

91.8 0.476 16

a The Orange G concentration is 0.045 M, and the objective position is set 2.5 mm away from the T-junction.

ζ0.99, data not shown). Deff and uj eff (which depend on the flow rate) are the two fitting parameters, whose values are summarized in Table 1. The effective velocity remains high even close to the surface (0.23 mm/s), ∼60% of the average velocity of the flow (0.37 mm/ s). This result highlights the dominant role of transverse diffusion, which brings analytes from high-velocity streamlines. The effective velocity increases nonlinearly with the flow rate, with an exponent of 0.66. This attenuated effect is well predicted by our model; the flow rate decreases with the effective layer thickness H with an

underlines that the surface concentration is dominated by the mixing in an effective fluid layer with a thickness dependent on the flow rate and on the distance from the T-junction.

Figure 6. Casein concentration front for three different longitudinal positions of the objective downstream in channel O2. Orange G concentration is 0.045 M (3.5 µm), and flow rate 30 nL/s. Table 2. Fitted Values of u j eff and Deff for Casein Concentration Fronts as a Function of Longitudinal Position from the T-Junctiona position in mm u j eff (mm/s) ( 0.003 Deff (10-8 m2/s) ( 0.3

2 0.215 3.8

2.5 0.230 5.3

3 0.235 6.3

3.5 0.244 7.0

4 0.250 8.4

4.5 0.253 8.7

a The flow rate is set at 30.6 nL/s and the Orange G concentration at 0.045 M.

Pe1/3

exponent of (eq 3). The effective diffusion increases roughly linearly with the flow rate with an exponent of 0.97. This increase is also consistent with the theory. For a flow rate of 91.8 nL/s, Deff reaches a value as high as 4 orders of magnitude greater than molecular diffusion (1.6 × 10-7 m2/s). We next moved the microscope objective along the channel (between 2 and 4.5 mm) with a constant flow rate (30.6 nL/s, Pe ) 1000) and an Orange G concentration of 0.045 M. Figure 6 presents the surface concentration profiles for three different positions. Both the effective velocity and diffusion rise continuously in the considered interval. Even close to the T-junction (2 mm), the effective diffusivity is 3 orders of magnitude higher than molecular diffusivity (3.8 × 10-8 m2/s, Table 2). Moreover, its value more than doubles from 3.8 × 10-8 m2/s at 2 mm to 8.7 × 10-8 m2/s at 4.5 mm, with a roughly linear increase (scaling of 1.02). The apparent surface velocity also rises, as mixing with streams of stronger flow rates becomes more efficient, but the effect is smaller in amplitude (from 0.216 mm/s at 2 mm to 0.253 mm/s at 4.5 mm) and has a scaling of 0.2. Our model captures this behavior. As the detector moves downstream (L increases), the time to reach it increases and transverse diffusion enlarges the thickness of the effective layer , as well as the apparent velocity and diffusivity. To summarize, the scaling from fitting our measurements with an error function is

u j eff ≈ u j (H/L)-0.2Pe-0.33

(8)

Deff ≈ D(H/L)-1.02Pe0.97

(9)

The exponents compare well with the predictions of the model (eqs 5 and 6). Although the assumptions are crude, it nonetheless

APPLICATION We now characterize the kinetics of protamine condensation in situ. To address this issue, we have already developed an original instrumentation based on microfluidic techniques to stretch a DNA molecule tethered on a surface in a laminar flow,5 as described in the Experimental Section. From one experiment to another, the molecules are located at different positions downstream from the T-junction. The apparent kinetics are strongly biased by the position along the channel, as we show for two molecules, respectively, at 2 or 4 mm (Figure 7A). We expect the true biological kinetics to be independent of the longitudinal position, and we will now demonstrate that the surface concentration is responsible for this experimental artifact. To account for the front in our kinetics measurements, let us first consider the simple problem of a first-order kinetics where A and B react to produce C. k1

A + B 98 C

(10)

In our experiment, A is the fraction of naked DNA, immobilized on the surface at the longitudinal position x, and B are the flowing proteins, whose concentration is time dependent. We define very generally f (x,t) as the normalized concentration of B, i.e., B ) B0f (x,t). C(x,t) is the fraction of the molecule condensed. Initially, the DNA molecule is uncompacted so A(x,0) ) 1 and C(x,0) ) 0. The evolution of the concentration of A is given by the kinetic equation

∂A/∂t ) -k1AB ) -k1B0f (x,t)A

(11)

One deduces from this differential equation that A(x,t) ) exp(k1′τ(x,t)) with τ(x,t) ) ∫0t f (x,u) du and k1′ ) k1B0. It is straightforward to show that the fraction of the molecule compacted at position x and time t is given by

C(x,t) ) 1 - exp[-k1′τ(x,t)]

(12)

As protamine is responsible for the compaction, the observed length of the molecule is related to C(x,t). Therefore, we should be able to fit our length versus time data with an exponential whose pertinent time scale, τ(x,t), accounts for the time dependence of the concentration of B. (Even if the reaction is a succession of first-order events, it is straightforward to demonstrate that the solution remains a sum of exponentials with the same pertinent time scale.) To do this, we examine whether the parameter τ(x,t), computed from the casein concentration front data (Figure 6), can collapse the different apparent kinetic data measured from DNA at different points along the channel. We define a new function g: l(t) ) l(τ-1τ(x,t)) ) lτ-1(τ(x,t)) ) g(τ(x,t)), which is plotted as the molecule length as a function of τ(x,t) (Figure 7B). The fit of the function g gives direct access to the real kinetic parameters. Analytical Chemistry, Vol. 77, No. 3, February 1, 2005

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Figure 7. (A) Protamine-induced condensation at a concentration of 50 ng/µL for two different DNA molecules measured by their decrease in length (rounds 2 mm, crosses 4 mm from the T-junction). The dotted lines represent the casein concentration profile at 2 and 4 mm (from Figure 6). (B) Same kinetics represented as a function of their pertinent time scale (eq 12).

Finally, we reproduce identical experiments for a set of protamine concentration and plot them as in Figure 7B. We deduce the accurate kinetic parameters of the reaction. We can conclude that this process behaves as first-order kinetics, with a rate constant of 5 × 106 L mol-1 s-1. We also measure an initial linear slope of 38 ( 2 µm s-1 µM-1. It compares well with the value 9.2 ( 1.8 µm s-1 µM-1, for bull protamine-1, and 2.6 ( 1.8 µm s-1 µM-1, for salmon protamine, reported by Brewer et al.9,10 Our result is slightly higher because the tension induced by the flow is not limiting the rate of the process. CONCLUSIONS We have presented a tool to probe surface concentration in microchannels. Our technique uses simple fluorescence microscopy and is highly adaptable to a whole class of analytes. It can also be performed in any microfluidic geometry. It is especially well suited to directly measure the surface concentration when the Taylor dispersion regime is not reached, which is relevant to many microfluidic applications. We characterized quantitatively the evolution of the surface concentration as a function of flow rate and longitudinal position. The concentration at the surface has a few striking characteristics. Although the Poiseuille velocity profile decreases to zero at the surface, the effective velocity of proteins is ∼50-60% of the average velocity of the flow. The apparent diffusivity is 3 orders of magnitude higher than the molecular diffusivity, even close to the injection. A simple analytical model was developed and captures well the characteristics of the surface concentration. To prove the utility of our technique, we characterized the assembly reaction of DNA condensation by protamine in a microchannel. We showed that the concentration front has to be taken into account to reliably access the kinetics. Although, we considered a particular example, our method is a framework for a whole class of microfluidic applications and DNA/protein interaction with single-molecule experiments. Indeed, both the kinetics scheme and the technique to measure surface concentration are very general. APPENDIX The velocity profile between two parallel plates is

u(z) ) 6(u/H2)z(H - z)

(13)

where z ) 0 is the location of the bottom plate. We want to 838

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compute the dispersion caused by an effective layer in which the molecule can reach the surface by transverse diffusion before the arrival of the front. In other words, the transverse diffusion time in this effective layer is of the same order as the effective advection time:

2(H2/D) ≈ L/ueff()

(14)

with ueff the effective velocity in this layer. We can evaluate this quantity from the generic scheme presented by Brenner and Edward.21 Since there is no bias in the z direction, the effective velocity is just

ueff ) (1/H)

∫

H

0

u(z) dz ) u j (3 - 22)

(15)

Using only the leading order terms in  in eqs 14 and 15, we can readily produce eqs 4 and 5. The effective diffusivity is given by the integral

Deff )

D H

∫

H

0

dz (dB dz ) 2

(16)

where the so-called B-field is the solution of

d2B/dz2 ) ueff - u(z)

(17)

subject to the boundary conditions

dB/dz ) 0

at

z ) (0,H)

(18)

eq 17 can be integrated to yield

[

(

)]

dB z2 z  2 z3 + ) 6Pe 3 2 dz H 2 3 3H 2H

(19)

where the constant of integration is zero to satisfy the upper and lower boundary conditions. Then we compute the effective diffusivity from eq 16:

Deff/D ) (Pe2/210)(646 - 1265 + 634)

(20)

We notice that for  ) 1, we get back the expected full-channel dispersion result (3), and to leading order in  we reproduce eq 6. ACKNOWLEDGMENT The authors thank G. Almouzni and J. P. Quivy for many helpful discussions. A.B. and G.W. thank the French Ministry of Research and Technology for the predoctoral fellowship “Alloca-

tion Couple´e”. K.D.D. acknowledges a postdoctoral fellowship from the Human Frontier Science Program. This work was partly supported by CNRS-MENRT ACI “DRAB”. Received for review July 9, 2004. Accepted November 2, 2004. AC048996+

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