Localized Single Molecule Isotherms of DNA Molecules at Confined

Feb 17, 2009 - The local linear velocity of hydrodynamic flow was calculated by the Hagen−Poiseuille equation in different microregions with a local...
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Anal. Chem. 2009, 81, 2059–2066

Localized Single Molecule Isotherms of DNA Molecules at Confined Liquid-Solid Interfaces Heng Liang,†,‡ Xiaoliang Cheng,† and Yinfa Ma*,† Department of Chemistry and Environmental Research Center, Missouri University of Science and Technology, Rolla, Missouri 65409, and Separation Science Institute, The Key Laboratory of Biomedical Information Engineering of Education Ministry, Xi’an Jiaotong University, Xi’an 710049, China The study of dynamics and thermodynamics of single biological molecules at confined liquid-solid interfaces is crucially important, especially in the case of low-copy number molecules in a single cell. Using a high-throughput single molecule imaging system and Lagrangian coordinates of single molecule images, we discovered that the local equilibrium isotherms of single λDNA molecules at a confined liquid-solid interface varied from a stair type for the regions of single or double molecular DNA to a mild “S” type for the regions of triple molecular DNA spots, which does not agree with the conventional equilibrium isotherms in the literature. Single molecule images in time sequence for different λDNA concentrations were statistically analyzed by measuring preferential partitioning from shearing effects, which were used to measure the local velocity of DNA molecules by directly observing the migration of DNA fluorescence spots for the 12 continuous images. The local linear velocity of hydrodynamic flow was calculated by the Hagen-Poiseuille equation in different microregions with a local Lagrangian approach. The local single molecule isotherms for the tracked molecules in the regions of single, double, or triple molecular DNA layers within the laminar flows were obtained according to the average local velocities of both the stochastic molecule events and the corresponding local Poiseuille flows. A millisecond and microvolume approach to directly determine local single molecule isotherms at confined liquid-solid interfaces was established, and the microspace scale effects on the types of isotherms were discovered. This study may have significant impact on preparations of low-copy number proteins in a single cell, membrane separations, and other bioseparation studies. Low copy number of molecules in a cell, such as proteins, presents at fewer than 1000 molecules per cell. It plays an important role in cell functioning, including signaling and the regulation of gene expression. Obtaining detailed dynamic and thermodynamic data of low-copy number molecules at confined liquid-solid interfaces is of crucial importance in biology, medi-

cine, material science, mechanics, and engineering.1 Studies of biomolecular adsorptions/desorption probabilities at confined liquid-solid interfaces are directly related to the behaviors of biomolecular or pharmaceutical molecules at cell surfaces and the applicability of biocompatible materials. This will allow one to obtain the basic parameters of chromatography and electrophoresis, understand separation mechanisms in microscales or preparation-scales, and manipulate particles or large biomolecules to facilitate separation and identification. In this study, we used a high-throughput single molecule imaging system (SMIS) and a local Lagrangian approach2-4 based on a particle-in-cell method5 to directly obtain local equilibrium isotherms of low-copy number molecules6-8 at confined liquid-solid surfaces with the statistics of real-time, local velocities, and local positions of single molecules, using λDNA (48 502bp) as model molecules. The statistical results of individual molecule with a SMIS in our experiments demonstrated that the local equilibrium isotherms varied from the stair type to the “S” type, depending on the defined microradial regions at the near-wall interface for the same labeled λDNA molecules at a given liquid-solid interface. The local isotherms for tracked molecules in the regions of the single, double, or triple molecular DNA layers within the laminar flows were obtained according to the rate of local velocities of stochastic molecule events and corresponding local Poiseuille flows. A millisecond and microvolume appoach to directly determine local single molecule isotherms at the liquid-solid interface was established, and microspace scale effects on single molecule isotherms were discovered through a novel combination of experimental and statistical approach of single molecular detection (SMD). Exciting SMD studies9-12 have propelled researchers into new areas, including biological, chemical, physical, and material sciences. One of the important applications of SMD is to serve as detection devices to count molecules, such as real-time observation (1) (2) (3) (4) (5)

(6) * Corresponding author. Yinfa Ma, Department of Chemistry and Environmental Research Center, Missouri University of Science and Technology, Rolla, MO 65409. Phone: 573-341-6220. Fax: 573-341-6033. E-mail: [email protected]. † Missouri University of Science and Technology. ‡ Xi’an Jiaotong University. 10.1021/ac801800u CCC: $40.75  2009 American Chemical Society Published on Web 02/17/2009

(7) (8) (9) (10)

LeDuc, P.; Haber, C.; Bao, G.; Wirtz, D. Nature 1999, 399, 564–566. Liang, H.; Lin, B. J. Chromatogr., A 1998, 828, 3–17. Liang, H. Chin. J. Chromatogr. 2007, 25, 664–680. Liang, H.; Liu, X. Sci. China, Ser. B: Chem. 2004, 47, 443–452. Grigoryev, Y. N.; Vshivkov, V. A.; Fedoruk, M. P. Numerical “Particle-inCell” Methods: Theory and Applications; VSP BV: Utrecht, The Netherlands, 2000; pp 1-207. Huang, B.; Wu, H.-K.; Bhaya, D.; Grossman, A.; Granier, S.; Kobilka, B. K.; Zare, R. N. Science 2007, 315, 81–84. Navratil, M.; Whiting, C. E.; Arriaga, E. A. Sci. STKE 2007, 388, 1–2. Long, C.; Nir, F.; Sunney, X. X. Nature 2006, 440, 358–362. Betzig, E.; Chichester, R. J. Science 1993, 262, 1422–1425. Xie, X. S.; Dunn, R. C. Science 1994, 265, 361–364.

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of low copy number proteins in a single cell.6-8 The vision fields and high-throughput characteristics of SMD bring a profound influence for exploiting the molecular events in the regions near confined liquid-solid interfaces. A variety of SMD approaches13-21 were developed to explore detailed adsorption behavior at confined liquid-solid interfaces in the past decade. Ultimately, SMD can be used in trace chemical analysis and for discovering the details of molecular environments that are indiscernible in bulk measurements. Current applications of SMD focus mainly on detection and counting of molecules.6-8 There remains a tremendous challenge in directly acquiring equilibrium isotherms of low copy number molecules at confined liquid-solid interfaces. The local characteristics of adsorption behaviors of low-copy number molecules at confined liquid-solid interfaces at different microspatial scales cannot be directly determined by ensembleaveraged measurements. Even with single molecule images, there still exists key conceptual obstructions in obtaining local single molecule isotherms because of the following two main reasons. First, single molecule images (the spatial distribution of the solute molecules at a designated time) cannot be easily dealt with using the current chromatography theoretical framework,22-25 which is based on convection-diffusion partial differential equations (PDEs) with Eulerian description in fluid mechanics, since the theoretical framework is used to acquire the temporal distribution of the solute molecules at a specific position. Second, the numbers of solute molecules are not constant in designated space-time grids since the convective flux crosses the space-grid interface. The result is that all physical parameters (e.g., local velocity, local concentration, local equilibrium isotherms) are not completely tenable in a complete space-time grid.22-25 The Lagrangian description, however, considers the positions and other properties of fluid particles identified as functions of time, such as their initial positions and material functions of fluid particles.26 In the Lagrangian coordinates, each fluorescent molecular spot can be tracked to obtain the accurate positions and local velocities of individual DNA molecules, and all physical parameters and relationships among them are completely tenable in each complete space-time grid on a continuous time sequence. It is normally considered that one type of equilibrium isotherm reflects one kind of adsorption property of biomolecules at the (11) Gai, H.; Wang, Q.; Ma, Y.; Lin, B. Angew. Chem., Int. Ed. 2005, 44, 5107– 5110. (12) Armani, A. M.; Kulkarni, R. P.; Fraser, S. E.; Flagan, R. C.; Vahala, K. J. Science 2007, 317, 783–786. (13) Xu, X. H.; Yeung, E. S. Science 1998, 281, 1650–1653. (14) Yeung, E. S. Annu. Rev. Phys. Chem. 2004, 55, 97–126. (15) Kang, S. H.; Yeung, E. S. Anal. Chem. 2002, 74, 6334–6339. (16) Xu, X. H.; Yeung, E. S. Science 1997, 275, 1106–1109. (17) He, Y.; Li, H.-W.; Yeung, E. S. J. Phys. Chem. B 2005, 109, 8820–8832. (18) Zheng, J.; Yeung, E. S. Anal. Chem. 2003, 75, 3675–3680. (19) Shortreed, M. R.; Li, H.; Huang, W.-H.; Yeung, E. S. Anal. Chem. 2000, 72, 2879–2885. (20) Li, J.; Lee, J.-y.; Yeung, E. S. Anal. Chem. 2006, 78, 6490–6496. (21) Fang, N.; Zhang, H.; Li, J.; Li, H.-W.; Yeung, E. S. Anal. Chem. 2007, 79, 6047–6054. (22) Wilson, J. N. J. Am. Chem. Soc. 1940, 62, 1583. (23) Giddings, J. C., Unified Separation Science; Wiley-Interscience: New York, 1991; pp 1-290. (24) Guiochon, G.; Shirazi, S. G.; Katti, A. M. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: Boston, MA, 1994; pp 1-667. (25) Cazes, J.; Scott, P. P. W. Chromatography Theory; Marcel Dekker, Inc.: New York, 2002; pp 3-465. (26) Schamel, H. Phys. Rep. 2004, 392, 279–319.

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liquid-solid interface under given conditions, such as the same biomolecules, same confined liquid-solid interface, same laminar flow, same buffer components, and same temperature, etc. Can we directly obtain equilibrium isotherms from single molecule images at microregions at confined liquid-solid interfaces? Do the types of local equilibrium isotherms depend on the microradial regions at the near-wall interface under given conditions? Answers to these questions are vital in studying the molecular behaviors at cell surfaces, membrane separation, and other bioseparation studies. Single biomolecular adsorption behaviors at the liquid-solid interface have been studied by using a single molecule imaging system,18,19 and the mobility-based wall adsorption isotherms were reported to compare capillary electrophoresis with single-molecule observations.21 However, local equilibrium isotherms and the effect of microradial regions at the near-wall interface with Poiseuille flows on the local equilibrium isotherms are still not well understood. In this work, a novel appoach was established to directly determine single molecule isotherms at the confined liquid-solid interface. EXPERIMENTAL SECTION Buffer Solutions. An aqueous solution of 50 mM 8.2 pH Gly-Gly buffer (Sigma Chemical Co., St. Louis, MO) was used to prepare all samples and solutions. To aid the background reduction, all the buffers and solvents were filtered through a 0.2 µm filter and photobleached overnight under a UV lamp before use. Preparation of Samples. λDNA (48502bp) was obtained from Life Technologies (Grand Island, NY). All DNA sample were prepared in the photobleached Gly-Gly buffer described above. DNA samples were labeled with YOYO-I intercalating dye (Molecular Probes, Eugene, OR) at a ratio of one dye per five base pairs. Briefly, DNA stock solutions were prepared in the concentration range of 40∼200 pM that were further diluted prior to the start of the experiment. The concentrations of DNA sample used in the single molecule detection experiment were 0.1, 0.5, 1, 2, 3 pmol L-1. Dye-DNA samples were allowed to incubate for 2 h before further dilution and use. Capillary Column Pretreatment. A 20 cm long fused-silica square capillary (50 µm × 50 µm) (Polymicro, Phoenix, AZ) was used for all experiments with a 1 cm window cleared in the middle. TFE tubing was connected to each end of the capillary, and one end of the tubing was connected to a syringe needle to facilitate linking to the buffer and sample reservoir, which was held to a microstage with a constant height to apply a desired hydrodynamic flow, and the other end was inserted into a plastic vial as buffer reservoirs. Since the purpose of this work is to observe the single molecule migration, the capillary column was pretreated for 20 min with 0.3% (w/v) poly(vinylpyrrolidone) (PVP) 1 000 000 Mr in the Gly-Gly buffer prior to the start of the experiment, the capillary was filled with DNA samples instead of injecting a short injected plug. Single Molecule Detection System. The experimental setup for this study was shown in Figure 1. A 488 nm Ar+ laser (40 × NA, 0.75 objective) was used as the excitation beam. Unless specified, the laser power was set at 5 mW before reaching the capillary. The ICCD camera was operated at -20 °C; the gain and exposure time were set at 80 and 100 ms, respectively.

Figure 1. Schematic illustration of the experimental setup and optical arrangement.

The delay of the shutter driver was set to 10 ms. Data acquisition was through the WinView software provided by Princeton Instrument. RESULTS AND DISCUSSION Single Molecule Imaging and Lagrangian Coordinates. The principle of experimental design for single molecule imaging and its Lagrangian coordinates are shown in Figure 2. Figure 2A shows the square capillary and ICCD camera arrangement for obtaining a series of single molecule images in time sequence. The focusing plane with a certain depth in the vertical coordinates (Z) was sketched as a thin cuboid with green lines in the square capillary. The volume (V ) 2.607 × 104 µm3) of the focusing plane in eq 1 in the next section was obtained by experimental single molecule counting from the linear slope of the observed labeled DNA molecules vs sample concentrations (referring to Figure 3). Its inboard and outboard sides along the radial direction (r) were the confined surface of the PVP coated square fused-silica capillary, and the upper and lower sides were the free buffer solution. Thus, the labeled DNA molecules in this thin cuboid (the focusing plane) were only observed in mobile (liquid) phases, and the λDNA molecules at the above or below boundaries of the liquid-internal wall of the capillary in the Z direction were not observed, since they were out of the focusing plane. The effective focal region in the Z direction was estimated between Z∼Z + 2 µm. Figure 2B represents the focusing plane in the x and r coordinates and divided regions at the near-wall interface depending on the approximate diameter of single DNA fluorescent spots. The width ranges of the studied regions (parts B and C of Figure 2) were 40.5∼47.5 pixels (W1), 33.5∼47.5 pixels (W2), 27.5∼47.5 pixels (W3), and -27.5∼27.5 pixels (W4), respectively. W1, W2, and W3 just indicate the regions of fluorescent spots of single, double, or triple molecular DNA in the radial direction (r), respectively. W4 indicates the middle region of the focusing plane in the radial direction (r). Figure 2C shows Lagrangian coordinates within a real single molecule image, it also shows the local regions of one frame from the ICCD image of fluorescent DNA molecules with the concentration of 0.5 pmol L-1 DNA sample. The each image has the visible region of 95 × 500 pixels, and each pixel represents 0.53 µm × 0.53 µm of real space. The distance from the center to the wall of the capillary is 47.5 pixels. The origin of x coordinates in Lagrangian coordinates is at the axial midline of the square capillary. The coordinate is always unchangeable when the DNA imaging spots move along in consecutive images.

Figure 2. Localized probing region of the single molecule imaging system (SMIS) and its Lagrangian coordinates. (A) The focusing plane (a thin cuboid). Its volume (2.607 × 104 µm3) in the vertical coordinates (Z) was obtained from the linear slope of the observed labeled DNA molecule vs sample concentrations (referring to Figure 3B). Its inboard and outboard sides along the radial direction (r) were the confined surface of the PVP coated square fused-silica capillary (50 µm × 50 µm), and the upper and lower sides were the free solution. (B) The enlarged focusing cuboid as the ICCD visual field, showing the four regions of molecular layers: single molecule layer (W1), double molecule layer (W2), triple molecule layer (W3), and middle molecule layer (W4) in the radial direction (r). (C) Lagrangian coordinates of single molecule images, showing the local regions of one frame from the CCD image of fluorescent DNA molecules (0.5 pmol/L). The Lagrangian coordinates and the local regions were fixed even if each identified individual molecule moves along with the hydrodynamic flow (x coordinates) for a countable time-set (K) of the continuous images.

In the local Lagrangian approach (LLA),2-4 i ) 1, 2, . . . , I is assigned to identify individual molecules in the kth single molecule image. The time that the molecules evolve at confined liquid-solid surfaces is divided into a series of time intervals, ∆t or ∆tk ) tk+1 - tk, where the subscript k is the serial number of the time grids or images. ∆t is 110 ms in our experiments. Thus a given sequence of images is marked by K ) { 1, . . . , k, . . . , K }, k ∈K, where K is a countable time-set. Through the continuous observation of single-molecule trajectories (fluorescence spots) in the studied regions, each single molecule is identified and tracked. The front spot in these images is not permitted to overstep the visible scale of the last (Kth) image (Figure 2C). Thus, all studied molecules are ensured to satisfy the Lagrangian coordinates in LLA. In our studies, the time-set K equals 13 at current technological conditions. The local position, (xik, rik), of the individual molecule i is accurately obtained from the kth Analytical Chemistry, Vol. 81, No. 6, March 15, 2009

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Figure 3. Collection of fluorescent images of YOYO-I-labeled λDNA and statistical results. (A) In each case, the concentration of λDNA from leftmost to rightmost equaled 0.1, 0.5, 1, 2, 3 pmol/L, respectively. (B) Linear relationship between the total number (N) of observed labeled DNA molecules in a whole frame and DNA sample concentrations (C).

frame (image), where xik and rik are the local axial and radial positions for the ith DNA molecule in the kth single molecule image as shown in Figure 2C. All observed individual molecules in the studied regions were employed as our statistical objects. Since none of coordinates of DNA molecules in the studied regions overlap the boundary of two regions, the molecule i with its position, (xki , rki ), was assigned clearly to a special region of the kth image. In our single molecule images, the diameter of the fluorescence spot of each labeled λDNA molecule was approximately 7.0 pixels. It is clear that W1, W2, and W3 in the radial direction correspond to 1, 2, and 3 multiple diameters of a DNA fluorescent spot. It must be emphasized that W3 includes W1 and W2 from the near-wall interface. W2 includes W1, and W1 is at the near-wall interface. W4 indicates the middle region of the focusing plane of square capillary, which is far from the near-wall interface. The regions, W1, W2, and W3, are considered as the divided local regions for counting the local molecule numbers, their local velocities of single DNA molecules, and the local velocities of Poiseuille flow at special radial positions. The region (W4) were employed ¯ cav) of synchronously measuring the average linear velocity (U the hydrodynamic flow. The Lagrangian coordinates and the local regions were fixed even if each identified individual molecule moves along with the hydrodynamic flow (x coordinates) for a countable time-set (K) of the continuous images. Correlations between Observed Molecules and Sample Concentrations. A set of single molecule images from our measurements were depicted in Figure 3A. In our study and other reports,11 a linear relationship existed (as shown in Figure 3B) between the total number (N) of observed labeled λDNA molecules in a whole frame and λDNA sample concentrations (C, molecular number µm-3) in a certain concentration range,

the linear equation, eq 1. It equals 2.607 × 104 µm3 in our study, which is the volume of the focusing plane with this excitation mode, only inside of which the labeled λDNA molecules can be excited and detected. Motion of DNA Molecules in the Capillary under Hydrodynamic Flow. The observed trajectories of all observed λDNA molecules were parallel to the x coordinates (Figure 4A), indicating that the hydrodynamic flow inside the square capillary channel was a laminar flow. The laminar motion of λDNA molecules based on the observed fact (Figure 4A) is the physical basis that we measure the preferential partitioning from shearing effects associated with the change in flow velocity from 0 at the wall to the average value in the center of the capillary with the local Lagrangian approach.2-4 The maximal fluctuation of λDNA molecules in the radial direction for the 13 continuous frame images was also determined, which was less than 1/3 of the diameter of the λDNA molecule spot self in the laminar flow (Figure 4B). The laminar movements of observed λDNA molecules were valuable to study the local statistical behaviors concerning the local migrations of individual molecules within the different regions, W1, W2, and W3, at the near-wall interface. Because of the laminar movements of observed λDNA molecules27,28 and the Lagrangian characteristics,5,26 the radial positions (rik) and the local number of tracked DNA molecules were kept constant for a countable time-set (K) of the continuous molecule images. Local Distributions of Molecular Numbers and VelocityBased Probabilities on Confined Liquid-Solid Interfaces. Within the K single molecule images, the average probabilities m s in the mobile phases (Pi,r ) and stationary phases (Pi,r ) for the i i k k tracked DNA molecule i at the position (xi , ri ) can be obtained by m ) Pi,r i

N ) VC + b

(1)

where N ) [1/(K N was the observed total molecule number in the whole kth image for a sample concentration. Thus N is the average number of observed labeled λDNA molecules for (K - 1) images. V is the slope of K-1 k 1)]∑k)1 N,

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k

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1 K-1

∑

K-1 k)1

s m uik ⁄ Urk,m and Pi,r ) 1 - Pi,r i i i

(2)

where uik ) (xik+1 - xik)/∆tk, uik is an average of measured axial velocity of the identified molecule i during ∆tk with Lagrangian (27) Taylor, J. A.; Yeung, E. S. Anal. Chem. 1993, 65, 2928–2932. (28) Tallarek, U.; Rapp, E.; Scheenen, T.; Bayer, E.; As, H. V. Anal. Chem. 2000, 72, 2292–2301.

Figure 4. The laminar flow behaviors of identified single DNA molecules in Lagrangian coordinates. (A) Radial fluctuations of DNA fluorescence spots (i ) 1-9) (0.5 pmol/L) in the studied region (W3) for 12 continuous images (k ) 1-12). (B) The maximal radial fluctuation of DNA spots and corresponding diameters of DNA spots at the radial positions (ri) for the 12 continuous images.

description based on particle approach.5,26 The Uk,m ri is the local linear velocity of the mobile phases at the radial position, ri, during the time interval, ∆tk, in the laminar flows. Equation 2 s implies that Pi,mri ) 1 if uik ) Urk,i m; Pi,r ) 1 if uik ) 0; and 0 < i m s k Pi, ri < 1 and 0 < Pi,ri < 1 if 0 < ui < Urk,i m. The Urk,i m is a function of radial positions in the square capillary with the Hagen-Poiseuille equation,27,28 c ¯ av Urk,m ) 2U [1 - (rik ⁄ rc)2] i

(3)

where rc is the half-length of the inner side of the square capillary ¯ cav the average linear velocity of the hydrody(23.75 µm), and U namic flow for a certain λDNA concentration. In this study, k c K-1 k nj)4 k k k Uav ) [1⁄(K - 1)]Σk)1 [(1⁄nj)4 )Σi)1 ui,j)4 ], where ui,j)4 and nj)4 are the measured local axial velocity of the molecule i and the measured local molecule number in the region (Wj)4 or W4) and during ∆tk, respectively. Equation 3, as one type of Hagen-Poiseuille equations,27,28 can be employed in the square capillary, since the ¯ cav (in eq 3), which can be experimentally maximum flow velocity, U measured, has included the contributions of the geometrical factor. An important assumption on the flow conditions of the HagenPoiseuille equations is no fluid-wall interaction (except purely viscous). It permits us to employ eq 3 as a standard asymptotic parabolic velocity profile to scale the effect of the wall adsorption

Figure 5. The local average velocities of DNA fluorescence spots for quantifying local bulk flow rates and the local velocity gradients adjacent to the confined surface in Lagrangian coordinates. (A) The local average velocities of fluorescent DNA spots observed by SMIS in the studied regions (Wj ) 1, Wj ) 2, and Wj ) 3) for the 12 continuous images (0.5 pmol/L), and the local average linear velocities of hydrodynamic flow calculated by eq 3 in the same regions. (B) The local average velocities of fluorescent DNA spots observed by the SMIS at the radial positions (ri) for the 12 continuous images (0.5 pmol/L), and the local linear velocity of hydrodynamic flow calculated by eq 3 at the same ri.

interaction. Although the abstract concept of the velocity-based probabilities in mobile and stationary phases has been accepted, it is still difficult to understand eq 2 in the case of single molecule images. The quantification of the bulk flow rates and the local velocity gradients adjacent to the surface from the single molecular images of the 0.5 pmol L-1 DNA sample were demonstrated in Figure 5. Figure 5A shows the local average velocities of fluorescent DNA spots observed in the studied regions (Wj ) 1, Wj ) 2, and Wj ) 3) and the local average linear velocities of hydrodynamic flow calculated by eq 3 in the same regions. Figure 5B shows the local average velocities, uik, of fluorescent DNA spots observed at the radial position (ri) and the local linear velocity, Urk,i m, of hydrodynamic flow calculated by eq 3 at the same ri. It is obvious that Pi,mri with eq 2 can be obtained with the data of uik and Urk,i m in Figure 5B. In each studied region, with Wj, j ) 1, 2, 3 (referring to Figure 2B,C), and the K single molecule images, the local average probabilities of λDNA molecules in the mobile phases (Pjm) and stationary phases (Pjs) can be obtained from Analytical Chemistry, Vol. 81, No. 6, March 15, 2009

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Pm j ) (1 ⁄ nj)

∑

nj i)1

m P i,r and Psj ) 1 - Pm j i

(4)

where nj is the observed molecule number in the studied region Wj. nj is constant in the K single molecule images due to both the laminar flow27,28 and Lagrangian characteristics.5,26 Pjm indicates the integral status showing how the confined liquid-solid interface affects the tracked DNA molecules in the different microregions. Therefore, with a series of SMD images we can dirctly determine Pjm, the average probability of the microregion Wj for the statistical time, (K - 1)∆tk. Therefore, in the each studied region Wj, j ) 1, 2, 3, and at the given confined interface, the number of λDNA molecules in the mobile phases (Njm) and in the stationary phases (Njs) can be obtained by m Nm and Nsj ) Nj(1 - Pm j ) NjPj j )

(5)

K where Nj ) N0Σk)1 (nj/Nk), Nj is the original molecule number of λDNA molecules in the observed confined region Wj. The original molecule number (N0), N0 ) V · C, is for the λDNA K molecules in the volume, V, of the focusing plane. Σk)1 (nj/Nk) is the proportion of observed molecule number in the studied region, Wj, to the total number of molecules in the whole frame for the image sequence. For the given biomolecules at a specific confined liquid-solid interface, the local singlemolecule equilibrium isotherms can be obtained by using single molecule images and the local Lagrangian approach2-4 with eq 5. According to a series of continuous single molecule images, we can obtain local distributions of molecular numbers and velocity-based probabilities by the statistical method with the eqs 2-5. The data in Figure 6 were obtained from three independent experiments, which mean that the error bars are generated from triplacated data points, and they represent 95% confidence intervals. As shown in Figure 6A, a linear relationship was observed between the local molecule number (Nj) and sample concentrations (C) in the region W1, and two parabolic curves in the region W2 and the region W3. The contributions of local λDNA molecules to the local equilibrium isotherms are different between the region W1 and the region W2 or the region W3. Thus, Figure 6A shows clearly that the contributions of λDNA sample concentrations to the local molecule number (Nj) are different within the different regions (e.g., W1, W2, and W3). This phenomenon was not observed or reported previously. Further, the correlations between the local probabilities (Pjm) of λDNA molecules in the mobile phases in different regions (Wj) and λDNA sample concentrations are shown in Figure 6B. The correlation is highly random. The randomness between the local probability (Pm i,ri) of the λDNA molecule i in the mobile phases and its radial position (ri) are also observed in Figure 6C. High randomness of these correlations originates from the highly random local motion of the single-molecule observation. m The high randomness of local probabilities, Pi,r or Pjm, is i contributed to by many factors, such as local regions at the near-wall interface, conformational fluctuations, viscosity changes, and molecule heat motions. However, the high randomness does not mean that their statistical results fluctuate with poor reproduction. A good example will be in temperature measure-

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Figure 6. Local distributions of molecular numbers and velocitybased probabilities in the mobile phases for tracked DNA molecules in Lagrangian coordinates. (A) Correlations between the original molecule number of λDNA molecules in the observed confined regions (Wj, j ) 1, 2, 3) and DNA sample concentration (molecular number µm-3). (B) Correlations between the local probabilities of DNA molecules (Pjm) in the mobile phases within three confined regions (Wj, j ) 1, 2, 3) and DNA sample concentrations. (C) Correlations m between the local probabilities (Pi,r ) of the DNA molecule i in the i mobile phases and their radial positions (ri).

ment. In a temperature measurement, the result still possesses good reproduction though the local velocities of singlemolecules, if we can measure them, have significant fluctuations. In addition, the flip movement of the λDNA molecules in the laminar flow was distinctly observed,29,30 which can also contribute to the randomness of local molecular velocity. The

Figure 7. Local single molecular isotherms at confined liquid-solid interfaces: (A) Correlations between the picomole number of λDNA molecules per µm2 area of the confined inner surface of the square fused-silica capillary, qj (pmol µm-2), and the picomole number of λDNA molecules per µm3 volume of the confined mobile phases, cj (pmol µm-3), at the region of the single molecular DNA layer (Wj ) 1). (B) Correlations between qj and cj in the region of the double molecular DNA layer (Wj ) 2). (C) Correlations between qj and cj in the region of the triple molecular DNA layer (Wj ) 3).

randomness in Figure 6C indicates that the local axial velocity of the λDNA molecules, relevant to the local laminar flow rate, does m not change significantly. Actually, the local probability (Pi,r ) i increases slightly as radial position (ri) increases, which means that the adsorption of DNA molecules onto the capillary wall is not serious. Even though the local velocity rate (Pjm) with ri and λDNA molecules i (Figure 6B,C) are highly random, the changes between Nj and C (as shown in Figure 6A) are very systematic. At the given confined liquid-solid surface in a hydrodynamic laminar flow, the local equilibrium isotherms between Njm and Njs (with a given phase ratio) depend on the contributions of both Nj and Pjm in eq 5 at different λDNA concentrations. Localized Single Molecule Isotherms at Confined LiquidSolid Interfaces. On the basis of eqs 2-5 and the statistical data (as shown in Figure 6) from a series of continuous single molecule images, the local single-molecule equilibrium isotherms of λDNA molecules at the confined liquid-solid interface are obtained in Figure 7. The vertical coordinates were defined as qj (pmol µm-2), which represents the picomole number of λDNA molecules per unit area (µm2) of the confined inner surface of the square fused-silica capillary. This confined inner surface serves as the confined stationary phase. The horizontal coordinates, cj (pmol µm-3), represents the picomole number of λDNA molecules per unit volume (µm3) in the three different regions, Wj, j ) 1, 2, 3. These three regions serve as the confined mobile phases. With the use of the SMIS (as shown as Figures 1 and 2), qj and cj were obtained by qj ) Nsj ⁄ (NASc) ) Nsj d ⁄ (NAV)

(6a)

cj ) Nm j ⁄ (NAV)

(6b)

where Sc and d are the area of the confined inner surface and the diameter observed of the thin cuboid (the focusing plane) (29) Lee, J. S.; Shaqfeh, E. S. G.; Muller, S. J. Phys. Rev. 2007, 75, 040801– 040804. (30) Wang, G. M.; Sandberg, W. C. Nanotechnology 2007, 18, 135701–135709.

in the square fused-silica capillary, respectively. NA is the Avogadro constant. In our case, V ) 2.607 × 104 µm3 through fitting the data of counting local molecule numbers with eq 1. Also d ) 47.5 µm from the SMD measurements. It is worth mentioning that the number of molecules shown on the isotherm curves are not the total number of DNA molecules actually observed from single molecule imaging. Instead, they are obtained from the probability of DNA molecules in the mobile phase or in the stationary phase, which depends on both local molecular number and its local velocity ratio. To compare the localized single molecule isotherms with the bulk measurement-equilibrium isotherms, experimental points from SMD measurements, (qj, cj), were fitted with the equations of sigmoidal dose-response (variable slope) with GraphPad Prism 4 software for three different microregions (Wj, j ) 1, 2, 3). The localized single molecule isotherms fitted with the 95% confidence (n ) 5) were obtained for Wj ) 1, Wj ) 2, and Wj ) 3 by qj)1 ) 0.6547 + {3.405[1 + 10778.9×[(log 1.196)-cj)1]]-1}; r ) 0.9731 (7a) qj)2 ) 0.6567 + {7.397[1 + 10223.9×[(log 1.293)-cj)2]]-1}; r ) 0.9792 (7b) qj)3 ) 0.6202 + {13.14[1 + 1032.42×[(log 1.479)-cj)3]]-1}; r ) 0.9394 (7c) With the comparison of the sigmoidal eqs 7a, 7b, and 7c, the lumbar highs of the sigmoidal curves increase from 3.405 × 10-15, 7.397 × 10-15, to 13.14 × 10-15 pmol µm-2, the slopes decrease from 778.9, 223.9, to 32.42 µm-1 for Wj ) 1, Wj ) 2, and Wj ) 3. This trend of change of lumbar highs and slopes were easily discovered by comparing the b1, b2, and b3 in sigmoidal curves in Figure 7. It shows that the local equilibrium isotherms of single λDNA molecules at the confined liquid-solid interface varied from the stair types for the regions of single or double molecular DNA to the mild “S” type for the region of triple molecular DNA. The microspace scale effects on the isotherm types were not reported Analytical Chemistry, Vol. 81, No. 6, March 15, 2009

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in the conventional description of equilibrium isotherms in the literature. In addition, the data to obtain thermodynamic information from the different microregions close to the wall could not be fitted by Langmuir or Freundlich isotherms, which may be due to the microspace scale effects or the low concentrations of DNA sample (0.1∼3 pmol) or other unknown reasons. We can find from Figure 7 that the isotherm shapes change as the increase of microregions from Wj ) 1, Wj ) 2, to Wj ) 3, although the confined liquid-solid interfaces and other conditions are fixed. It needs to kept in mind that the intraparticle pores of the chromatography packing materials and the dimensions of microchannels will have great effects on the equilibrium or nonequilibrium adsorptions on the confined liquid-solid interface. The hydrodynamic flow velocity was quite slow in this SMD experiment (as shown in Figure 5), since faster flows would likely produce streaks instead of dots in the images. However, as long as the laminar flow with relatively low Reynolds number is ensured, the Hagen-Poiseuille equation will be tenable; the approach of the single molecular isotherm should also be tenable, although some typical chromatography, electrophoresis separations, or other bioseparations are carried out with faster flows. This millisecond and microvolume approach will have significant impact on preparations of low-copy number proteins in the single cell, membrane separations, and other bioseparation studies. Of course, we will further improve the approach, such as decreasing imaging time and time interval and observing streaks instead of dots in the images, etc.

of single biological molecules at confined liquid-solid interfaces. With the use of λDNA (48 502bp) as model molecules, the local equilibrium isotherms at confined liquid-solid surfaces have been obtained from the statistics of real-time, local velocities, and positions of single molecules by measuring preferential partitioning from shearing effects associated with the change in flow velocity from 0 at the wall to the average value in the center of the capillary. The local equilibrium isotherms, which were fitted based on the statistical results of the confined surface molecule densities and the local concentrations in the confined volume, demonstrated that they varied from the stair type to the mild “S” type, depending on the defined microradial regions at the nearwall interface for the same labeled λDNA molecules at a given liquid-solid interface. These single molecule isotherms will have significant impact in many areas, such as quantitative determination of the adsorption of biological or pharmaceutical molecules at confined liquid-solid interfaces and micropreparative chromatography or electrophoresis of low-copy number of protein molecules in single cells.

CONCLUSIONS We have used the single molecule imaging system and a local Lagrangian approach to study the dynamics and thermodynamics

Received for review August 27, 2008. Accepted January 23, 2009.

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ACKNOWLEDGMENT The authors are thankful for financial support from the Department of Chemistry and Environmental Research Center at Missouri University of Science and Technology and the National Natural Science Foundation of China (Grants 20299030, 20175015, and 29775017).

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