Abstract

We have been investigating a microfluidics platform for high-speed, low-cost sequencing of single DNA molecules using novel “charge-switch” nucleotides. A significant challenge is the design of a flowcell suitable for manipulating bead-DNA complexes and sorting labeled polyphosphate molecules by charge. The flowcell is part of a single-molecule detection instrument, creating fluorescence images from labeled polyphosphates. These images would ultimately be analyzed by signal processing algorithms to identify specific nucleotides in a DNA sequence.

Here we describe requirements of the fluidics system for loading, identifying, tracking, and positioning beads. By dynamically modulating pressure gradients in the plenum chambers of a multi-channel network, we could guide individual beads with high precision to any desired coordinate and reversibly trap them in stepped channels. We show that DNA immobilized on pressure-trapped beads can be physically extended into a downstream channel under electric force for analysis. Custom dynamic algorithms for automated bead control are described.

© 2008 Optical Society of America

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  1. L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).
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  4. J. Stephan, K. Dorre, S. Brakmann, T. Winkler, T. Wetzel, M. Lapczyna, M. Stuke, B. Angerer, W. Ankenbauer, Z. Foldes-Papp, R. Rigler, and M. Eigen, "Towards a general procedure for sequencing single DNA molecules," J. Biotechnol. 86, 255-267 (2001).
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2002

L. T. Franca, E. Carrilho, and T. B. Kist, "A review of DNA sequencing techniques," Q. Rev. Biophys. 35, 169-200 (2002).
[CrossRef] [PubMed]

C. Gosse and V. Croquette, "Magnetic tweezers: micromanipulation and force measurement at the molecular level," Biophys. J. 82, 3314-3329 (2002).
[CrossRef] [PubMed]

A. C. Kak, M. Slaney, and G. Wang, "Principles of computerized tomographic imaging," Med. Phys. 29, 107 (2002).
[CrossRef]

2001

J. Stephan, K. Dorre, S. Brakmann, T. Winkler, T. Wetzel, M. Lapczyna, M. Stuke, B. Angerer, W. Ankenbauer, Z. Foldes-Papp, R. Rigler, and M. Eigen, "Towards a general procedure for sequencing single DNA molecules," J. Biotechnol. 86, 255-267 (2001).
[CrossRef] [PubMed]

2000

J. C. McDonald, D. C. Duffy, J. R. Anderson, D. T. Chiu, H. Wu, O. J. Schueller, and G. M. Whitesides, "Fabrication of microfluidic systems in poly(dimethylsiloxane," Electrophoresis 21, 27-40 (2000).
[CrossRef] [PubMed]

1999

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, "Single-molecule biomechanics with optical methods," Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

1997

H. Kojima, E. Muto, H. Higuchi, and T. Yanagida, "Mechanics of single kinesin molecules measured by optical trapping nanometry," Biophys. J. 73, 2012-2022 (1997).
[CrossRef] [PubMed]

1993

S. C. Kuo and M. P. Sheetz, "Force of single kinesin molecules measured with optical tweezers," Science 260, 232-234 (1993).
[CrossRef] [PubMed]

1972

C. M. Frey, J. L. Banyasz, and J. E. Stuehr, "Interactions of divalent metal ions with inorganic and nucleoside phosphates. II. Kinetics of magnesium(II) with HP 3 O 10 4-, ATP, CTP, HP 2 O 7 3-, ADP, and CDP," J. Am. Chem. Soc. 94, 9198-9204 (1972).
[CrossRef] [PubMed]

Anderson, J. R.

J. C. McDonald, D. C. Duffy, J. R. Anderson, D. T. Chiu, H. Wu, O. J. Schueller, and G. M. Whitesides, "Fabrication of microfluidic systems in poly(dimethylsiloxane," Electrophoresis 21, 27-40 (2000).
[CrossRef] [PubMed]

Angerer, B.

J. Stephan, K. Dorre, S. Brakmann, T. Winkler, T. Wetzel, M. Lapczyna, M. Stuke, B. Angerer, W. Ankenbauer, Z. Foldes-Papp, R. Rigler, and M. Eigen, "Towards a general procedure for sequencing single DNA molecules," J. Biotechnol. 86, 255-267 (2001).
[CrossRef] [PubMed]

Ankenbauer, W.

J. Stephan, K. Dorre, S. Brakmann, T. Winkler, T. Wetzel, M. Lapczyna, M. Stuke, B. Angerer, W. Ankenbauer, Z. Foldes-Papp, R. Rigler, and M. Eigen, "Towards a general procedure for sequencing single DNA molecules," J. Biotechnol. 86, 255-267 (2001).
[CrossRef] [PubMed]

Banyasz, J. L.

C. M. Frey, J. L. Banyasz, and J. E. Stuehr, "Interactions of divalent metal ions with inorganic and nucleoside phosphates. II. Kinetics of magnesium(II) with HP 3 O 10 4-, ATP, CTP, HP 2 O 7 3-, ADP, and CDP," J. Am. Chem. Soc. 94, 9198-9204 (1972).
[CrossRef] [PubMed]

Brakmann, S.

J. Stephan, K. Dorre, S. Brakmann, T. Winkler, T. Wetzel, M. Lapczyna, M. Stuke, B. Angerer, W. Ankenbauer, Z. Foldes-Papp, R. Rigler, and M. Eigen, "Towards a general procedure for sequencing single DNA molecules," J. Biotechnol. 86, 255-267 (2001).
[CrossRef] [PubMed]

Carrilho, E.

L. T. Franca, E. Carrilho, and T. B. Kist, "A review of DNA sequencing techniques," Q. Rev. Biophys. 35, 169-200 (2002).
[CrossRef] [PubMed]

Chiu, D. T.

J. C. McDonald, D. C. Duffy, J. R. Anderson, D. T. Chiu, H. Wu, O. J. Schueller, and G. M. Whitesides, "Fabrication of microfluidic systems in poly(dimethylsiloxane," Electrophoresis 21, 27-40 (2000).
[CrossRef] [PubMed]

Croquette, V.

C. Gosse and V. Croquette, "Magnetic tweezers: micromanipulation and force measurement at the molecular level," Biophys. J. 82, 3314-3329 (2002).
[CrossRef] [PubMed]

Davis, L. M.

L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).

Dorre, K.

J. Stephan, K. Dorre, S. Brakmann, T. Winkler, T. Wetzel, M. Lapczyna, M. Stuke, B. Angerer, W. Ankenbauer, Z. Foldes-Papp, R. Rigler, and M. Eigen, "Towards a general procedure for sequencing single DNA molecules," J. Biotechnol. 86, 255-267 (2001).
[CrossRef] [PubMed]

Duffy, D. C.

J. C. McDonald, D. C. Duffy, J. R. Anderson, D. T. Chiu, H. Wu, O. J. Schueller, and G. M. Whitesides, "Fabrication of microfluidic systems in poly(dimethylsiloxane," Electrophoresis 21, 27-40 (2000).
[CrossRef] [PubMed]

Eigen, M.

J. Stephan, K. Dorre, S. Brakmann, T. Winkler, T. Wetzel, M. Lapczyna, M. Stuke, B. Angerer, W. Ankenbauer, Z. Foldes-Papp, R. Rigler, and M. Eigen, "Towards a general procedure for sequencing single DNA molecules," J. Biotechnol. 86, 255-267 (2001).
[CrossRef] [PubMed]

Fairfield, E. R.

L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).

Foldes-Papp, Z.

J. Stephan, K. Dorre, S. Brakmann, T. Winkler, T. Wetzel, M. Lapczyna, M. Stuke, B. Angerer, W. Ankenbauer, Z. Foldes-Papp, R. Rigler, and M. Eigen, "Towards a general procedure for sequencing single DNA molecules," J. Biotechnol. 86, 255-267 (2001).
[CrossRef] [PubMed]

Franca, L. T.

L. T. Franca, E. Carrilho, and T. B. Kist, "A review of DNA sequencing techniques," Q. Rev. Biophys. 35, 169-200 (2002).
[CrossRef] [PubMed]

Frey, C. M.

C. M. Frey, J. L. Banyasz, and J. E. Stuehr, "Interactions of divalent metal ions with inorganic and nucleoside phosphates. II. Kinetics of magnesium(II) with HP 3 O 10 4-, ATP, CTP, HP 2 O 7 3-, ADP, and CDP," J. Am. Chem. Soc. 94, 9198-9204 (1972).
[CrossRef] [PubMed]

Gosse, C.

C. Gosse and V. Croquette, "Magnetic tweezers: micromanipulation and force measurement at the molecular level," Biophys. J. 82, 3314-3329 (2002).
[CrossRef] [PubMed]

Hahn, J. H.

L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).

Harger, C. A.

L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).

Higuchi, H.

H. Kojima, E. Muto, H. Higuchi, and T. Yanagida, "Mechanics of single kinesin molecules measured by optical trapping nanometry," Biophys. J. 73, 2012-2022 (1997).
[CrossRef] [PubMed]

Jett, J. H.

L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).

J. H. Jett, R. A. Keller, J. C. Martin, N. K. Seitzinger, and E. B. Shera, "Single molecule detection in flowing sample streams as an approach to DNA sequencing," abstracts of papers of the American Chemical Society 198, 41-ANYL (1989).

Kak, A. C.

A. C. Kak, M. Slaney, and G. Wang, "Principles of computerized tomographic imaging," Med. Phys. 29, 107 (2002).
[CrossRef]

Keller, R. A.

J. H. Jett, R. A. Keller, J. C. Martin, N. K. Seitzinger, and E. B. Shera, "Single molecule detection in flowing sample streams as an approach to DNA sequencing," abstracts of papers of the American Chemical Society 198, 41-ANYL (1989).

L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).

Kist, T. B.

L. T. Franca, E. Carrilho, and T. B. Kist, "A review of DNA sequencing techniques," Q. Rev. Biophys. 35, 169-200 (2002).
[CrossRef] [PubMed]

Kojima, H.

H. Kojima, E. Muto, H. Higuchi, and T. Yanagida, "Mechanics of single kinesin molecules measured by optical trapping nanometry," Biophys. J. 73, 2012-2022 (1997).
[CrossRef] [PubMed]

Krakowski, L. A.

L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).

Kuo, S. C.

S. C. Kuo and M. P. Sheetz, "Force of single kinesin molecules measured with optical tweezers," Science 260, 232-234 (1993).
[CrossRef] [PubMed]

Lapczyna, M.

J. Stephan, K. Dorre, S. Brakmann, T. Winkler, T. Wetzel, M. Lapczyna, M. Stuke, B. Angerer, W. Ankenbauer, Z. Foldes-Papp, R. Rigler, and M. Eigen, "Towards a general procedure for sequencing single DNA molecules," J. Biotechnol. 86, 255-267 (2001).
[CrossRef] [PubMed]

Marrone, B. L.

L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).

Martin, J. C.

J. H. Jett, R. A. Keller, J. C. Martin, N. K. Seitzinger, and E. B. Shera, "Single molecule detection in flowing sample streams as an approach to DNA sequencing," abstracts of papers of the American Chemical Society 198, 41-ANYL (1989).

L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).

McDonald, J. C.

J. C. McDonald, D. C. Duffy, J. R. Anderson, D. T. Chiu, H. Wu, O. J. Schueller, and G. M. Whitesides, "Fabrication of microfluidic systems in poly(dimethylsiloxane," Electrophoresis 21, 27-40 (2000).
[CrossRef] [PubMed]

Mehta, A. D.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, "Single-molecule biomechanics with optical methods," Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Muto, E.

H. Kojima, E. Muto, H. Higuchi, and T. Yanagida, "Mechanics of single kinesin molecules measured by optical trapping nanometry," Biophys. J. 73, 2012-2022 (1997).
[CrossRef] [PubMed]

Ratliff, R. L.

L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).

Rief, M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, "Single-molecule biomechanics with optical methods," Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Rigler, R.

J. Stephan, K. Dorre, S. Brakmann, T. Winkler, T. Wetzel, M. Lapczyna, M. Stuke, B. Angerer, W. Ankenbauer, Z. Foldes-Papp, R. Rigler, and M. Eigen, "Towards a general procedure for sequencing single DNA molecules," J. Biotechnol. 86, 255-267 (2001).
[CrossRef] [PubMed]

Schueller, O. J.

J. C. McDonald, D. C. Duffy, J. R. Anderson, D. T. Chiu, H. Wu, O. J. Schueller, and G. M. Whitesides, "Fabrication of microfluidic systems in poly(dimethylsiloxane," Electrophoresis 21, 27-40 (2000).
[CrossRef] [PubMed]

Seitzinger, N. K.

L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).

J. H. Jett, R. A. Keller, J. C. Martin, N. K. Seitzinger, and E. B. Shera, "Single molecule detection in flowing sample streams as an approach to DNA sequencing," abstracts of papers of the American Chemical Society 198, 41-ANYL (1989).

Sheetz, M. P.

S. C. Kuo and M. P. Sheetz, "Force of single kinesin molecules measured with optical tweezers," Science 260, 232-234 (1993).
[CrossRef] [PubMed]

Shera, E. B.

J. H. Jett, R. A. Keller, J. C. Martin, N. K. Seitzinger, and E. B. Shera, "Single molecule detection in flowing sample streams as an approach to DNA sequencing," abstracts of papers of the American Chemical Society 198, 41-ANYL (1989).

L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).

Simmons, R. M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, "Single-molecule biomechanics with optical methods," Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Slaney, M.

A. C. Kak, M. Slaney, and G. Wang, "Principles of computerized tomographic imaging," Med. Phys. 29, 107 (2002).
[CrossRef]

Smith, D. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, "Single-molecule biomechanics with optical methods," Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Soper, S. A.

L. M. Davis, E. R. Fairfield, C. A. Harger, J. H. Jett, J. H. Hahn, R. A. Keller, L. A. Krakowski, J. C. Martin, B. L. Marrone, R. L. Ratliff, N. K. Seitzinger, E. B. Shera, and S. A. Soper, "High-speed DNA sequencing - an approach based upon fluorescence detection of single molecules," abstracts of papers of the American Chemical Society 200, 75-ANYL (1990).

Spudich, J. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, "Single-molecule biomechanics with optical methods," Science 283, 1689-1695 (1999).
[CrossRef] [PubMed]

Stephan, J.

J. Stephan, K. Dorre, S. Brakmann, T. Winkler, T. Wetzel, M. Lapczyna, M. Stuke, B. Angerer, W. Ankenbauer, Z. Foldes-Papp, R. Rigler, and M. Eigen, "Towards a general procedure for sequencing single DNA molecules," J. Biotechnol. 86, 255-267 (2001).
[CrossRef] [PubMed]

Stuehr, J. E.

C. M. Frey, J. L. Banyasz, and J. E. Stuehr, "Interactions of divalent metal ions with inorganic and nucleoside phosphates. II. Kinetics of magnesium(II) with HP 3 O 10 4-, ATP, CTP, HP 2 O 7 3-, ADP, and CDP," J. Am. Chem. Soc. 94, 9198-9204 (1972).
[CrossRef] [PubMed]

Stuke, M.

J. Stephan, K. Dorre, S. Brakmann, T. Winkler, T. Wetzel, M. Lapczyna, M. Stuke, B. Angerer, W. Ankenbauer, Z. Foldes-Papp, R. Rigler, and M. Eigen, "Towards a general procedure for sequencing single DNA molecules," J. Biotechnol. 86, 255-267 (2001).
[CrossRef] [PubMed]

Wang, G.

A. C. Kak, M. Slaney, and G. Wang, "Principles of computerized tomographic imaging," Med. Phys. 29, 107 (2002).
[CrossRef]

Wetzel, T.

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Figures (9)

Fig. 1.
Fig. 1.

Charge-switch nucleotide. (A) dGTP is phosphate-labeled with a linker and a TAMRA dye. The linker has two quaternary amino groups (++) which confer an elementary charge of +2. The label is TAMRA (Invitrogen), a zwitterionic fluorophore contributing zero net charge. (B) Labeled pyrophosphate PPi-F is cleaved from the nucleotide by polymerase. This creates a“terminal phosphate” with an ionizable oxygen giving the PPi-F a stronger interaction with Mg++ as compared to the intact γ-dNTP.

Fig. 2.
Fig. 2.

Electrosorting on the basis of differential affinity for Mg++. Lane A=TAMRA amine (charge +1), lane B=TAMRA carboxylate (charge -1), lane C=g-dTTP, lane D=PPi-F. The structures of g-dTTP and PPi-F are similar to the dGTP analogs shown in Fig. 1. Samples were resolved by electrophoresis in a 5% agarose gel containing the indicated concentrations of Mg++ (0, 2 and 5 mM). As implied by Fig 1, the net charges of the g-dTTP and the PPi-F are differentially sensitive to Mg++ concentration.

Fig. 3.
Fig. 3.

Scheme for single-molecule sequencing by electrosorting. The target DNA strand is immobilized on a bead trapped in a microchannel (trap design not shown). Pressure-driven flow moves polymerase and all 4 γ-dNTPs past the DNA as indicated (vertical arrow). Nucleotide incorporation generates labeled pyrophosphate PPi-F. An electric field in the horizontal channel drives intact γ-dNTPs to the left and PPi-F to the right where it is detected by fluorescence. A shallow channel depth (0.5um) and total internal reflectance optics provide efficient detection of single molecules.

Fig. 4.
Fig. 4.

SEM picture of flowcell. Wheel flowcell network design cast from a silicon master.

Fig. 5.
Fig. 5.

Custom voltage/pressure controller. Left, front panel showing six channels. Right, PDMS-molded channel terminus showing platinum electrode.

Fig. 6.
Fig. 6.

Image processing. Flowcell image after edge detection (a), Cross-correlation result of binary-thresholded images (b), Result of Radon transform on binary-thresholded flowcell image (c).

Fig. 7.
Fig. 7.

Typical bead loading pressure settings. Numbers marked in Pa are pressures, and unlabeled numbers with accompanying arrows are volume flow rates in 10-15 m3/s. Unlabeled numbers over wheel spokes show spoke numberings. See Materials and Methods for description of flowcell design.

Fig. 8.
Fig. 8.

Bead docking. CCD fluorescence image of 10 beads docked within a wheel flowcell.

Fig. 9.
Fig. 9.

Images from bead-trapping experiment. Top left, fluorescence imaging of several beads, each with many DNA strands attached. Top right, brightfield image of bead approaching bead trap. Bottom left, brightfield image of bead trapped at intersection. Bottom right, fluorescence image of bead trapped at intersection with electric field extending DNA into the channel.

Equations (3)

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f ( x , y ) g ( x , y ) F ( u , ν ) { g ( x , y ) } ,
d dt p ( t ) = K ( ν a ( t ) ν t ( t ) )
ν ( t ) 1 e α θ 1 + e α θ

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