Physics of biomolecules and cells

Lecture 2b: The DNA Prism

371

data of Figures 4B and 5B. The bands remain discrete and well separated, with some di usional broadening, and the optical density trace in Figure 4C shows that the peaks remain symmetrical with time, and the areas under the peaks remain constant. Because there is unavoidable shearing during the preparation of BAC and PAC inserts from bacterial cultures, the baseline in Figure 4C is not zero, as one might expect; the baseline in the conventional separation shown for comparison in Figure 4A is also not zero. In fact an optical density scan of Figure 4A shows that the base-line is higher than Figure 4C. Thus the lack of shearing forces at low Reynolds numbers is an important general advantage.

Although band broadening (dispersion) in the prism mode is not yet fully understood, the various molecular sizes in Figure 5 are broader than can be accounted for by di usion alone. It seems likely that the angle of sample entry and, more importantly, the degree to which all molecules are fully stretched and therefore back track as required for high resolution (Fig. 2), currently limits resolution. Since an important strength of the microfabrication approach used here is the ability to define and modify parameters such as these, it is likely that resolution in the prism mode can be greatly improved with further work. At the same time, we note that the resolution required for many BAC and PAC sizing experiments, where the size of the insert is the criterion for further use, can now be easily carried out in seconds in the non prism mode (Fig. 4). This represents not only a vast saving in reagents and time sensitive costs, but in space as well.

These results may be compared to other reports in the literature, also designed to separate or analyze high molecule weight DNA by unconventional methods. In previous work from this laboratory, we showed in principle that pulsed field separations in a microfabricated array of posts was possible [3,4] and that di usion arrays could be used to separate DNA molecules in the many kb range [11]. Evidence for separation in both these cases was gathered by tracking individual molecules at high magnification. These devices are not yet practical tools, however, chiefly because the electric fields were shaped by a few discrete platinum electrodes which resulted in a highly spatially non-uniform electric field distribution. Further, the di usion arrays are very slow. Other valuable recent reports either operate in a much lower molecular weight regime and on a time scale of many minutes or hours using entropic trapping [5], or use a variant of FACS sorting, where size is judged by fluorescence [6]. These latter approaches are essentially single molecule devices, and although FACS sorting is fast, the total number of molecules that can be harvested per unit time is low, and the macroscopic versions su er from very high shear forces.

Finally, the success of these experiments has hinged largely on our discovery that microfluidic channels can be used to shape electric fields. In

372	Physics of Bio-Molecules and Cells

pulsed field gel electrophoresis the electric field must not only be switched on a programmable basis, but the array boundary conditions must be set so that the field lines are straight. This latter principle, first established by Davis and colleagues [1], led directly to the clamped homogeneous electric field device now widely used for the sizing and separation of DNA molecules greater than 25 kb. The electric field is set by clamping the electric potential at the gel boundary with a series of electrodes surrounding the gel. This principle cannot be carried over to micro and nanofabricated devices because these devices are sealed with a cover slip or, as reported here, a silicone coated cover slip, and even at low field strengths the electrodes evolve hydrogen and oxygen, which very quickly obscures the observational field. This problem is solved here, where microfluidic channels serve to shape the field by acting as large electrical resistors, and thus also act as current sources to inject current uniformly across the array boundaries, rather than set the potential at the array boundaries. There is no gas evolution in the viewing area, and this design principle can now be used wherever current must be carried by bu er in a microfabricated device, for example, if molecules are to be trapped by dieletrophoresis [7].

4 Conclusions

In summary, a new method for the continuous sorting of DNA in a microfabricated device has been realized using the principles outlined in this lecture series. Its distinctive features are its tiny size and consequent fluid volume, control of field shape by a new microfluidics principle, high speed, the complete replacement of standard sieving matrices and electronic control of field shape with structures fabricated on a wafer, and the ability to operate the device at high field strengths without the need for external cooling. We expect that the separation range can be extended both to much larger molecules using the current version, and to much smaller molecules, perhaps as small as 1 kb, in future versions.

The lower bound is set by the sieving power of the post array, where the clear channel has to be smaller than the DNA in its randomly coiled state. Although the current clear channel width is 1.5 micron, comparable to a 100 kbp randomly coiled molecule, modern fabrication facilities routinely mass-produce feature sizes smaller than 200 nm by optical lithography. When a DNA molecule of contour length L is randomly coiled, its radius of gyration RG is RG2 = pL/3, where p is the persistence length [8,9], p 50 nm for double stranded DNA. The width of a 1.5 micron channel is comparable to the size of a 100 kbp DNA molecule (2RG 1.5 micron). Thus in principle, a 200 nm post array could e ectively separate 1 kbp DNA molecules. Longer arrays allowing longer separation distances should

Lecture 2b: The DNA Prism

373

improve resolution for all molecular weights, since band broadening is a random process. This is clear from Figures 4 and 5. Using the same pulsing conditions used in Figure 5, resolution comparable to conventional pulsed field gel electrophoresis ( 10 kbp) should be achieved with a 2 cm-long array.

Theoretical analysis suggests that the resolution can be further improved with higher field strengths and shorter pulse times, currently limited by the power amplifier in our laboratory. Importantly, the methods used here to realize asymmetric pulsed-field fractionation, such as the generation of tunable uniform electric fields over larger arrays by current injection, will be practical tools for the realization of many lab-on-a-chip systems [10].

References

[1]G. Chu, D. Vollrath and R.W. Davis, Science 234 (1986) 1582-1585.

[2]J.P. Brody, P. Yager, R.E. Goldstein and R.H. Austin, Biophys. J. 71 (1996) 34303441.

[3]T.A. Duke, R.H. Austin, E.C. Cox and S.S. Chan, Electrophoresis 17 (1996) 10751079.

[4]O. Bakajin et al., Anal. Chem. 73 (2001) 6053-6056.

[5]J. Han and H.G. Craighead, Science 288 (2000) 1026-1029.

[6]H.P. Chou, C. Spence, A. Scherer and S. Quake, Proc. Natl. Acad. Sci. USA 96 (1999) 11-13.

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

[8]M. Doi and S.F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, Oxford, 1989).

[9]S.B. Smith, L. Finzi and C. Bustamante, Science 258 (1992) 1122-1126.

[10]M.A. Burns, B.N. Johnson and S.N. Brahamsandra, Science 282 (1998) 484-487.

374	Physics of Bio-Molecules and Cells

Abstract

We close this series of lectures on fractionation with this lecture on how to do brownian ratchets RIGHT, and it involves very careful consideration to the boundary value issues I talked about in Lecture 1.

1 The problems with insulators in rachets

It has been proposed that an array of spatially asymmetric obstacles could operate as a continuous-flow Brownian ratchet [1–3]. As particles drift down through the obstacle array, they are selectively displaced perpendicular to their drift direction based on di usion in the asymmetric structure. Experiments [4, 5] showed that such asymmetric obstacle arrays can fractionate DNA molecules of high molecular weight ( 105 base-pair). However, because there are discrepancies between the theory [1, 2] and the experimental results [4, 5], it is not clear how the fractionation e ciency scales with molecular weight. While early theories proposed that smaller molecules should fractionate faster [1, 2], a recent analysis [6] suggests that arrays of ion-impermeable obstacles could not serve as continuous-flow Brownian ratchets for point-like particles. Thus it remained questionable whether an ion-impervious obstacle array could in principle fractionate small macromolecules (oligonucleotides, proteins, etc.), which are of great biological importance.

The conventional understanding of continuous-flow sorting by Brownian motion using asymmetric obstacle arrays [1, 2] is depicted in Figure 1a. Consider particles emerging from gap A driven towards gap B by electric fields. Executing biased random walks towards gap B, the particles spread out over the parabolic shaded region. While particles taking path 1 are blocked and deflected back to gap B, particles taking path 2 are deflected to gap B+. The probability of being deflected depends on how likely a particle di uses past the corner of an obsticle (point C in Fig. 1a), and thus is a function of the ratio between the width of the parabolic shaded region and the characteristic obstacle dimension [1]. This ratio can be written mathematically as a dimensionless parameter D/va, where D is the particles di usion coe cient, v is its drift velocity, and a is the gap width between the obstacles. Since the deflection probability depends on the diffusion, particles of di erent di usion coe cients should migrate at di erent directions. Based on theoretical calculation, Duke et al. suggested that for the particular array geometry (Fig. 1a), the D/va of the molecules being separated should be between 0.02 and 0.3 for the best resolution, and the largest deflection probability should occur at D/va 0.7 [1].

Two major assumptions are made in the above model [1, 2]: (i) There is no deflection of the electric field lines by the obstacles. (ii) The particles are assumed to have no physical size (point-like particles).

Lecture 2c: Bigger is Better in Rachets

375

Fig. 1. a) Basic principle of continuous sorting in an asymmetric obstacle array. b) Electric field lines (and streamlines) in an ion-impermeable array. The boundary conditions used in this simulation is that all field lines flowing into gap A continue through gap B. c) Equipotential lines corresponding to the electric field under same condition.

In the actual implementation of these arrays by microfabrication techniques [4,5], however, the obstacles consist of fused silica or other materials impervious to the ions in the fluid. Because the ions flow around the obstacle and the electric field E is related to the ion flow J by Ohms law, J = σE, where σ is the conductivity of the electrolytic fluid, the electric field lines go around the obstacles (Fig. 1b), violating the first of the above assumptions.

To isolate deflection due to di usion, it is required that all field lines through an upper gap (A in Fig. 1b) map through a lower gap (B), which is aligned to the upper gap. If the field lines are misaligned so that some field lines through gap A leak to gap C or D, we will not be able to distinguish whether a particle migrating from gap A to C is by di usion or by following the field. This requirement has to hold over the entire array area. This occurs only for a single choice of the angle of the equipotential lines. The proper equipotential contours (Fig. 1c) in our array were determined by numerically solving Poissons equation using the said field requirement as boundary conditions. Note that although the average current flow is in the vertical direction (from A to B as shown in Fig. 1b), and the equipotential lines are always perpendicular to the local electric field by definition, they are not perpendicular to the average current direction and are not horizontal.

2 An experimental test

Assuming the upper and lower array edges are held at equipotentials, they should be along such an equipotential direction to generate the desired aligned current distribution in the array (in direction of gap A to gap B in Fig. 1b). Therefore we designed our structure with properly slanted top and bottom edges (Fig. 2). The array was etched in fused silica by

376	Physics of Bio-Molecules and Cells

Fig. 2. Schematic diagram of the device. The obstacles are 1.4 microns wide, 5.6 microns long, and 5 microns tall. The etched fused silica substrate was capped with a glass cover slip to form enclosed microfluidic channels. The array is 12 mm high and 6 mm wide.

standard microfabrication methods and sealed with a glass cover slip to form the channels. For fabrication convenience, the structure was designed so that the direction of the calculated equipotential lines was parallel to the rows of obstacles, which we found by modeling not to be the case in general. Further, in practice, the boundary condition is not implemented as a voltage source along the boundary, but rather as an array of current sources which inject (and extract) current along the top (and bottom) boundary. The current sources were implemented as an array of microfluidic channels with a high electrical resistance (compared to that of the array) connected to common fluid reservoirs (Fig. 2) [7]. Holes through the substrate allowed access to the reservoirs. The similar high voltage drop across all channels (compared to the small voltage drop in the array) leads to the same current flowing in each channel at the boundary. The microfluidic channel arrays ensure that in case of imperfect dimension control during microfabrication, the current pattern will still be highly aligned to the obstacles. A single extra channel connected to a special reservoir (with voltage calculated to

Lecture 2c: Bigger is Better in Rachets

377

give the same current as the other channels) was used to inject a 90-micron wide band of the molecules to be separated.

A mixture of Coliphage λ DNA (48.5 kbp, 5 µg/ml) and Coliphage T2 DNA (167 kbp, 2 µg/ml) in Tris-borate-EDTA bu er was injected into the array at various speeds using electric fields. At high fields (>5 V/cm), di usion was negligible (D/va < 0.05). In calculation of the di usion constants, we assume that Coliphage λ DNA and T2 DNA molecules adopt random coil conformations. According to the approach used in Reference 1 for calculating the di usion coe cients, Dλ = 0.64 micron2/s, and DT2 = 0.35 micron2/s. The D/va values of experimental data are calculated using the measured velocities and a = 1.4 microns. As expected no lateral separation occurred (Fig. 3a) for these di usion constants. The fact that the band did not curve even at the boundary of the array shows that the equipotential boundary conditions were properly imposed, and the current direction was well-aligned to the obstacle array. Lateral separation of the two species was observed at a field strength lower than 2 V/cm (D/va > 0.13 for Coliphage λ DNA), with λ molecules being deflected from vertical more than those of T2. The separation became larger ( 1.3o T) as the electric field was lowered to 0.8 V/cm (Fig. 3b, D/va 0.32). The drift velocity at fields less than 0.8 V/cm was so low (<1 micron/s) that the stability of the separated bands became hard to maintain. The two species could be separated into two cleanly resolved bands 11 mm from the injection point, and the density profile of these bands was well fitted by two Gaussian peaks (Fig. 3c). The resolution in the range of 50 kbp is30 kbp, or 60% [8]. Although we did observe separation of molecules in the D/va range proposed by Duke et al. [1], the measured separation was much smaller than the theoretical predictions (Fig. 3d).

To examine the scaling of the deflection to very small molecular sizes, a mixture of 411 bp (PCR product, 1 µg/ml) and λ DNA ( 20 ng/ml) was injected into the array, using electric fields ranging from 6 V/cm to 120 V/cm. At these field conditions, λ DNA molecules do not deviate from the field direction (Figs. 3a and 3d), and thus are used to label the field direction. The measured D/va for 411 bp molecules using 6 V/cm is 2.7. Because 411 bp DNA molecules are not random coils, the parameter D/va was obtained by experimentally measuring the band broadening instead of using the method in [1]. Assuming that band broadening is not influenced by the obstacles, we used the following equations: t = y/v and ∆x2 = 2Dt + ∆x2o, where y is the length of the band, ∆x2o is the initial width of the band, and ∆x is the width of the band after time t. Therefore, D/va = (∆x2 −∆x2o)/2ya. The measured D/va value at low field conditions (6 V/cm) is 2.7, which implies that the mean di usion distance during the time when a molecule moves from one row to the next is larger than

378	Physics of Bio-Molecules and Cells

Fig. 3. a) Fluorescence micrograph of Coliphage λ and T2 DNA stained with fluorescent dye TOTO-1 forming a band of 90 microns wide and 12 mm long at 12 V/cm. The slanted lines on the top and bottom mark the boundary of the obstacle array. Scale bar = 300 microns. b) Fluorescence micrograph of the two species separated into two band at 11 mm from injection using 0.8 V/cm. c) Fluorescence profile of b). Experiment data (thick black line) fitted with two Gaussian peaks. d) Separation angle between λ and T2 bands as a function of the dimensionless parameter D/va of λ molecules. The solid curve is the theoretical prediction from reference 1. Circles mark experiment data using electric fields of 12 V/cm, 1.8 V/cm, 1.2 V/cm, and 0.8 V/cm. The dash line is the theoretical curve calculated according to reference 1 using an e ective gap size 4.1 times larger than the physical gap size (5.7 microns instead of 1.4 microns). Note the log scale of the horizontal axis.

Lecture 2c: Bigger is Better in Rachets

379

the size of the obstacles. Therefore, the assumption holds. The D/va values for high field conditions were derived assuming constant mobility. so the corresponding D/va range tested is from 0.14 to 2.7. This covers the entire range that the theory suggests the maximum deflection (D/va 0.7) [1]. Therefore deflection should be observed for these smaller molecules. However, contrary to all expectations based on the theory [1, 2], absolutely no lateral deflection was observed.

We believe the reason the array failed to deflect small molecules (411 bp DNA) lies in the fact that small particles can precisely follow the electric field lines as they flow through the obstacle geometries (Fig. 4a). Contrary to the basic principles of di usion array [1, 2], where particles could widen out over the parabolic shaded region in Figure 1a only via di usion, small molecules will be spread out by the electric field. Particles are now drifted towards the vicinity of boundary L (via field line a in Fig. 4a) as well as to boundary R (via field line c). Thus small molecules injected from a gap will have a much higher chance to di use to the left than what the old theory suggested. In fact, a recent analysis showed that point-like particles are equally likely to di use in both directions [6]. We summarize the argument as follows. For small particles that precisely follow electric field lines, their

flux density Jparticle can be written as Jparticle = ρµE − D ρ, where ρ is the particle density, µ is the mobility, and D is the di usion coe cient. The

first term of the flux density is due to the electric field, whereas the second term is from di usion. According to the continuity equation, we have:

∂ρ	= • Jparticle = µE • ρ − D 2ρ.	(2.1)
∂t

Note we have used • E = 0 because the electrolytic solution is neutral. If there is a high field so that the second term in equation (1) becomes relatively small, we find:

E • ρ = 0

(2.2)

at steady-state. This says the particle density is approximately constant along any field line. Thus if one has a uniform concentration of particles arriving across all field lines entering a given gap (originating from a reservoir of uniform concentration), as the field lines (particle streamlines) widen out after the gap, the particle density will remain unchanged. This is illustrated in the fluorescence image of 411 bp DNA molecules in the array (Fig. 4a), which shows that DNA under high fields uniformly fills the entire space between rows of obstacles. Now, consider uniform injection of particles into all gaps at the top of the array using high fields, leading to uniform particle distribution in the array, and then we lower the field strength so that di usion becomes important. Since all spatial derivatives of ρ in

380	Physics of Bio-Molecules and Cells

Fig. 4. Schematic flow diagram and fluorescence image of particles in center of band for particles of size a) 411 bp at average field of 120 V/cm and b) 48.5 kbp (1.2 V/cm). The exposure times were 2 s so that the image brightness shows the particle density.

equation (1) are zero in our case of uniform density distribution, the particle density stays uniform according to equation (1), and thus the di usion flux of particles across any field line must be equal to the inverse flux. Combined with translational symmetry, this implies that the probability of a particle di using across boundary L in Figure 4a equals that across boundary R, a result which must hold for any distribution, not just for the assumed uniform distribution of particles. Given that there is no preferred direction of di usion, there is no physical basis for ratcheting.

When a much larger λ DNA molecule approaches a gap, it is physically deflected by the obstacle and centered to the gap, because of its finite size (a random coil of 1 micron) compared to the gap width ( 1.4 microns). Thus molecules initially following field lines a, b, and c in Figure 4b will all tend to leave the gap region on line b. The fluorescence image in Figure 4b clearly shows this shadowing in contrast to Figure 4a for the case of small molecules. Unlike 411 bp molecules, which are spread out in the space