Physics of biomolecules and cells

Lecture 1b: Dielectrophoresis and Microfabrication

351

case for long polymers, we can use equation (5.6) to estimate the relaxation time of these polymers. Figure 11 shows the satisfactory agreement between the observed relaxation times and the ones predicted for a long polymer, considering the simplicity of the model used. Short polymers can be expected to have faster relaxation times. In the case of our 368 bp fragment, equation (5.7) predicts a relaxation time in water of about 10−5 s, substantially beyond the present 1 kHz bandwidth of our high voltage power supply.

Fig. 11. The measured relaxation time of 39.9 kB (T7 phage) DNA vs. viscosity (solid line) vs. the predicted relaxation time (dot-dash line).

Rough calculation of the force F felt by the polymer is more di cult as we have mentioned. From equation (3.2) we know that the dielectric force is proportional to the product of the polarizability of the molecule α times |E|dE/dz. The polarizability α is equal to Cx2, where C is the e ective capacitance of the molecule and x2 is the mean squared separation of the two charged ends of the molecule. In the rough approximation that the capacitance C is equal to oA/x, where A is the area of the charged ends of the molecule, we once again find that the force also depends on the statistical

mechanics of the polymer. If (1) L >> γ, we get that αlong = o[(2Lγ)]3/2, while if (2) L << γ we find that αshort = oAoL, where Ao is an area which characterizes the end area of rigid length of the molecule. These numbers are

rather poorly defined. The backwards way to do this is simply to calculate from the measured force at a given E and dE/dz the polarizability α. As we have shown, α is a strong function of length and conformation of the molecule so there is no single intensive parameter that characterizes DNA.

352	Physics of Bio-Molecules and Cells

There is still a problem with this analysis. Equations (3.2) and (5.1) together imply that the EDEP force is e ectively zero at high frequencies (which is not true because of other processes that come into play [18, 26]), rises at a frequency given by 1/τ and then remains constant down to DC. In fact, all our data show the apparent force falling to zero at DC frequencies. The problem is that we have ignored the electrophoretic force. The total force acting on a polyelectrolyte in an external electric field is the sum of the electrophoretic force Fe due to the net e ective linear charge density β of the polymer, and the dielectrophoretic force Fd due to the induced dipole moment p discussed above. The electrophoretic force Fe on a polyelectrolyte in the presence of an electric field is proportional to the local applied electrical field E and gives rise to a constant velocity ve:

ve = µeE; Fe = ζve = ζµeE

(5.8)

where µe is the electrophoretic mobility of the polymer, ve is the electrophoretic velocity and ζ is drag coe cient between the electrophoretic velocity and the force. The origin of the electrophoretic force Fe in polyelectrolytes has been intensively studied [27] and is characterized by the surprising fact that the electrophoretic mobility of a polyelectrolyte is basically independent of the length of the polymer in free solution, hence we can treat µe as a constant independent of length. We then have a final expression for the total force acting on a charged, polarizable polyelectrolyte:

dE

Ftot = ζµeE + α|E| d · (5.9) z

An interesting aspect of the dielectric force is that it is a nonlinear force as a function of E, and hence at su ciently high field strengths and su - ciently low ratios of µe/α a gradient can trap a molecule even in a static DC field, since the dielectrophoretic force will ultimately be greater than the linear electrophoretic force. By combining the electrophoretic and the dielectrophoretic response, we show in Figure 12 the forces and potential surfaces that charged, polarizable objects experience going through a gap similar to one of our devices. The parameters for the polarizability α and the electrophoretic mobility µe were chosen here to roughly correspond to our longest molecules studied, the 40 kB dsDNA. Note that the nonlinear dielectrophoretic component of the trapping force gives rise to a short-range trapping potential. If the field direction is switched, the electrophoretic potential surface will slope in the opposite way while the dielectrophoretic potential is invariant, so that only the dielectrophoretic component of the force serves as a trap.

Since the free flow electrophoretic mobility µe is basically independent of length of the DNA molecule, the e ect of electrophoresis of the molecule

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353

Fig. 12. The total force, electrophoretic and dielectrophoretic, experienced by a particle passing through the gradient trap shown in Figure 1 is presented as the dashed line in the figure. The potential U (z) surface that the particle moves along is shown by solid line.

is an apparent decrease in the force at low frequencies if the electrophoretic force is greater than the dielectrophoretic force, which seems to be the case for DNA. In fact, our entire model which we used to analyze the dielectrophoretic force acting on the molecules basically breaks down at low frequencies, since we do not have an equilibrium condition. At present, we have no way of disentangling the true dielectrophoretic force at low frequencies from the electrophoretic force.

6 Conclusion

We have used electrodeless EDEP to trap and concentrate single and double stranded DNA. The analytical simplicity of the field pattern in a electrodeless trap has allowed us to characterize the length and frequency dependence of the EDEP force. We showed the strong dielectrophoretic response of the DNA in the audio frequency range. We also demonstrated that for the given trapping voltage applied, the dielectrophoretic force dramatically increases with the increase of the length of the DNA molecule. There is actually a great dispersion in the force with length, hence by appropriate choice of parameters one can envision selectively trapping one range of DNA molecules while removing others. By measuring dielectrophoretic force under di erent solvent viscosity conditions, we were able to determine that movements

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of counterions in the Debye layer are responsible for the dielectrophoretic response of the DNA for 2 reasons: (1) a strong dependence of relaxation times on solvent viscosity indicates that the charge redistribution occurs via movement through the solvent; (2) the expected relaxation times due to di usion of ions across the radius of gyration of the polymer are in rough agreement with the observed relaxation times.

The dielectrophoretic trapping of the DNA in electrodeless traps has a great potential for use in biotechnology. The EDEP force may be adjusted accordingly by varying the shape and cross section of the constriction. Position of the constriction also can be controlled at will. Since EDEP trapping occurs in high field gradient regions, EDEP allows easy patterning of DNA by appropriate geometrical obstacle design. Other potential applications of EDEP method are selective trapping of specific ranges of DNA, concentration of DNA molecules to very tight bands before launch into a fractionating media, PCR cleanup, concentration of DNA in gene array chips to enhance sensitivity of the detection limit by increasing local S/N , or acceleration of gene hybridization rates by concentration of single stranded DNA, and in general for any reaction for which the rate scales with concentration or any power of the concentration greater than 1.

References

[1]H.A. Pohl, Dielectrophoresis: The Behavior of Neutral Matter in Nonuniform Electric Fields (Cambridge University Press, Cambridge, UK, 1978).

[2]M. Washizu and O. Kurosawa, IEEE Trans. Ind. Appl. 26 (1990) 1165-1172.

[3]T. Muller, G. Gradl, S. Howitz, S. Shirley, T. Schnelle and G. Fuhr, Biosens. Bioelectron. 14 (1999) 247-256.

[4]M. Washizu, S. Suzuki, O. Kurosawa, T. Nishizaka and T. Shinohara, IEEE Trans. Ind. Appl. 30 (1994) 835-843.

[5]C.L. Asbury and G. van den Engh, Biophys. J. 74 (1998) 1024-1030.

[6]N.G. Green and H. Morgan, J. Phys. Chem. B 103 (1999) 41-50.

[7]J. Rousselet, L. Salome, L. Adjari and J. Prost, Nature 370 (1994) 446-448.

[8]L. Gorre-Talini, J.P. Spatz and P. Silberzan, Chaos 8 (1998) 650-656.

[9]R. Pethig, Crit. Rev. Biotechnol. 16 (1996) 331-348.

[10]H. Morgan, M.P. Hughes and N.G. Green, Biophys. J. 77 (1999) 516-525.

[11]F.F. Becker, X.B. Wang, Y. Huang, R. Pethig, J. Vykoukal and P. Gascoyne, Proc. Nat. Acad. Sci. 92 (1995) 860-864.

[12]J. Yang, Y. Huang, X.B. Wang, F.F. Becker and P. Gascoyne, Anal. Chem. 71 (1999) 911-918.

[13]P. Chinachoti, M.P. Steinberg and D.A. Payne, J. Food Sci. 53 (1988) 580-583.

[14]K.R. Foster, F.A. Sauerm and H. Schwan, Biophys. J. 63 (1992) 180-190.

[15]J.D. Jackson, Classical Electrodynamics, 2nd Ed. (John Wiley & Sons, New York, 1975).

[16]K.F. Ren, G. Grehan and G. Gouesbet, Appl. Opt. 35 (1996) 2702-2710.

[17]S. Takashima, J. Mol. Biol. 7 (1963) 455-467.

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[18]S. Takashima, Electrical Properties of Biopolymers and Membranes (IOP, Philadelphia, PA, 1989).

[19]S.B. Smith, L. Finzi and C. Bustamante, Science 258 (1992), 1122-6.

[20]S.B. Smith, C. Yujia and C. Bustamante, Science 271 (1996) 795-799.

[21]B. Tinland, A. Pluen, J. Sturm and G. Weill, Macromolecules 30 (1997) 5763-5765.

[22]D. Porschke, Biophys. Chem. 22 (1985) 237-247.

[23]American Institute of Physics Handbook, Third Edition (American Institute of Physics, College Park, MD, 1972).

[24]W.D. Volkmuth, T. Duke, M.C. Wu, R.H. Austin and Attila Szabo, Phys. Rev. Lett. 72 (1994) 2117-2120.

[25]D.A. Hoagland, E. Arvanitidou and C. Welch, Macromolecules 32 (1999) 6180-6190.

[26]F. Oosawa, Biopolymers 9 (1970) 677-688.

[27]Capillary Electrophoresis: Theory and Practice, Paul D. Grossman, Joel C. Colburn (ed.) (Academic Press, 1997).

[28]C. Desruisseaux, D. Long, G. Drouin and G.W. Slater, Macromolecules 34 (2001) 44-52.

356	Physics of Bio-Molecules and Cells

Abstract

Now, I want to try and apply some of the principles I have outlined in the design of a real working device. Long DNA molecules in agarose gels and other polymer matrices, get hooked on many gel fibers simultaneously, exhibiting complex motion, and confounding theory and experiment alike. I’ll describe how a hex array allows better prediction of pulsed field parameters for a given range of molecular sizes, and a relatively simple theory describes the motion. The analytical nature of the motion is a real advantage of the technique since it may allow us to separate unlabeled molecules.

1 Introduction

This story is about 8 years long, and it is so long purely because of the stupidity of the author. Had I been a bit more subtle in my understanding of Nature the story would be shorter and more interesting I suspect.

As I originally mentioned, the shearing boundary at the zeta potential means that DNA molecules undergoing electrophoresis are free-draining, that is, the solvent e ectively passes through the random coil of the polymer. In “normal” hydrodynamics, if you move a polymer there is a velocity vector flow pattern V (r) radiating out from a point on the polymer that couples di erent parts of the object together, modifying the drag coe cient of the polymer from a strictly linear sum of terms (typically a 1/ln(L) correction), L = length of the object.

For a free-draining polymer, that is, a polymer undergoing electrophoresis, the linear sum works, and the drag is proportional to simply L and not L/ ln(L). Since the drag and the force both scale as L, the electrophoretic mobility is independent of the length and you cannot fractionate DNA molecules in bulk solution using electrophoresis. Thus, the ubiquitous presence of some sort of retarding medium (a gel) in most molecular biology labs. The retarding medium adds a length-dependent additional force as the random coil try to squeeze through the medium.

Thus, the basic idea of the microfabricated arrays was simply to simulate a gel by microfabricating obstacles which add another length dependent term to the drag on the polymer. However, there are two things at work here that kill the technology:

(1)The pore sizes (a = 1 micron) are much bigger than typical gels, where the pore sizes range from 10 nm to 500 nm or so.

(2)the self hydrodynamic forces acting on a random polymer in a “thin” slit of thickness d are MUCH greater than the self-forces felt by a polymer in bulk solution.

Lecture 2a: Hex Arrays

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Let’s now consider each of these points in turn. However, before we can launch into this subject we need to carefuly at how polymers move through structures.

The e ect of large pore sizes is to increase the e ective force acting at the contact points between polymer parts and the posts, these larger forces serve to enhance stretching of the polymer. That is, consider if you will that there is an electric field E in the solution. The applied force F on a stretched fragment spanning the pore of size a is aρE, ρ = linear charge density of the polymer. If the polymer is randomly coiled, the e ective amount of polymer in the pore is increased leading to an even greater force acting on the polymer. There is of course an entropic “spring” constant

The end-end distance Rz of a “hung” polymer in an electric field E is roughly:

Rz = κ ln[sin h(L/κ)/(L/κ)]

(1.1)

where L = total length of the polymer, made of N pieces of length 2p (p = persistence length of the polymer) and:

κ = 2ρpE/kbT

(1.2)

where ρ = charge/length of the polyelectrolyte.

If you play with this equation you learn a Big Lesson: in a “large” pore environment, DNA is highly aligned at low fields! A further fact, which can be ascertained from my earlier notes on hydrodynamics in 2.5 dimensions, is that in my thin “slit” of thickness h there is a hydrodynamic coupling to the surface via stick boundary conditions. The stick boundary condition slows down the entropic relaxation time of the polymer. This can be a big e ect, and the slow relaxation time enhances elongation. The bottom line is that elongation increases with decreasing etch depth.

Once the polymer is elongated, the mobility becomes length independent because the drag acting on the polymer is now proportional to the length. This is a disaster!

This is why you can’t run a gel at high voltages to speed up fractionation times, and why I didn’t believe the Human Genome Initiative could succeed as originally planned. Tom Duke in collaboration with our group has suggested two ways to get around this problem. I will first talk about the pulsed field/hex array idea. The idea here is if you can’t beat the physics, use the physics: that is, if the polymer physics wants to elongate the polymer, figure out a way to use elongated molecules. Tom’s idea was a take on Ed Southern’s idea for using PULSED TRANSVERSE fields to fractionate elongated polymers.

Transverse pulsed field electrophoresis in hexagonal arrays uses an array of 2 µm pillars with 2 µm spacings arranged in a hexagonal lattice and takes

358	Physics of Bio-Molecules and Cells

advantage of the DNA elongation that occurs in microfabricated arrays [5]. Application of a pulsed field along alternating axes of the array, separated by 120o, causes net motion of the DNA molecules along the bisector of the axes, with average migration speeds that depend on their length (Fig. 1).

Fig. 1. 2 µm wide pillars were used in these experiments. Cartoon DNA molecules are drawn to illustrate the motion of individual molecules of di erent lengths. Each period T consists of two pulses aligned along the channels created by the posts in the array, giving a net angle between the two field directions of 120o. Shorter molecules move farther in the array because once they have reoriented along the axis of the field they move in an unhindered straight line for the duration of the pulse. Longer molecules, on the other hand, spend most of the pulse period retracing their paths. In the example shown here, the longer of the two will never advance.

A useful separation device, in addition to using an e ective separation mechanism, must also collect and launch molecules in a narrow zone, since initial zone broadening destroys resolving power. In our device the DNA was entropically trapped and released as a band using the principle described by Han et al. [6]. We used an entropic barrier placed in the path of the DNA near the entrance to the array (Fig. 2). There is a small gap between the barrier top and the cover slip that seals the array. The gap between the top of the barrier and the cover slip was smaller than the radius of gyration of the DNA molecules to be fractionated (150 nm in these experiments). When very low DC fields are applied to move DNA molecules into the array, the molecules do not have enough applied force applied to them to squeeze through the gap, and hence they get trapped (Fig. 2A). Under higher fields they get stretched and move through the gap with mobilities that are independent of molecular length.

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Fig. 2. A) Sketch of the device. The diameter of a typical device was 1 cm to 3 cm. The pulsed field was applied through two pairs of external electrodes (C-D and E-F). The remaining electrode pair (A-B) was used for entropic trapping. The electrodes were insulated from each other by six silicone structures (lozenge shaped in panel A)). B) Cross section of the device showing entropic trapping. The arrows point in the direction of DNA motion, while their lengths correspond to the strength of the applied field. B-I): the beginning of DNA transport across the barrier using a high electric field. B-II): the concentration and cleanup step where the molecules are forced back against the barrier at low field before they are launched into the array.

In our device molecules were first transported into the array through the gap using a high electric field. They were then concentrated against the barrier by applying a low electric field oriented in the opposite direction. Concentrated molecules were then launched into the array by reversing the field direction. The alternating fields were not applied directly at the entropic barrier but rather were applied only after the DNA band had been moved several millimeters into the array. This was done to avoid the field curvature seen at the corners of the array and ensure that the fields were uniform in the directions necessary for predictable fractionation.

360	Physics of Bio-Molecules and Cells

2 Experimental approach

The devices were made of quartz using standard photolithography and reactive ion etching techniques. They contained an array of pillars oriented in a hexagonal lattice whose height was 2 µm. The arrays were sealed using glass coverslips with a spun on thin layer silicone elastomer [7] (RTV615A and RTV615B, GE Silicones, Waterford, NY). The silicone elastomer surface was treated for one minute in an oxygen plasma to make the silicone hydrophobic, necessary for wetting of the sealed device. The devices were shaped as hexagons 3 cm in diameter (Fig. 2B) to allow easy application of electric fields oriented at 120o [8]. They were mounted on a plastic holder that contained the outside electrodes. Two pairs of outside electrodes were used to apply pulsed fields (C-D, E-F Fig. 2) while the remaining pair was used for entropic trapping (A-B).

Fig. 3. Video clips of λ and T4 DNA pulsed at 244 V and with period T = 1 s after release from the entropic trap.

The DNA fluoresence stain TOTO-1 (Molecular Probes) was used at 1 µg/ml concentrations to stain T4 and λ DNA molecules which were loaded into the arrays in concentrations of 15 µg/ml and 5 µg/ml, respectively, and observed by epifluorescence using the 488 nm line of an Ar/Kr laser. The 0.5 × TBE electrophoresis bu er (45 mM Tris/borate, 1mM EDTA, pH 8.0) contained 0.1% POP-6 (Perkin Elmer Biosystems) to reduce electroendosmosis and 0.1 M DTT to reduce bleaching.

T4 (168.9 kbp) and λ (48.5 kbp) DNA were separated in a very short time with high resolution. The mixture (see Experimental Protocol) was resolved into two bands in 10 s (Figs. 3 and 4). The position of the peaks is plotted vs. time in Figure 4. From t1 = −10 s to t2 = 0 s, the applied voltage was 100 V and the pulse period T was 1 s. Under these conditions the bands separated by less than the bandwidth. At time t2 the voltage was increased to 244 V, keeping the period the same. The two bands were then