Physics of biomolecules and cells

Fig. 4. The position and speeds of the band peaks of λ (squares), T4 (circles) and the mixed band (diamonds) vs. time. From t1 until t2, the pulsing parameters were 100 V and T = 1 s. The peak of the mixed band moved at 2.3 µm/s. At time t2 the voltage was increased to 244 V. The two bands were cleanly resolved in approximately 10 s. T4 DNA moved at 5.6 µm/s, while λ DNA moved at 14.9 µm/s. Speeds were obtained by fitting the straight lines shown through the band centroid positions.

resolved in approximately 10 s. Figure 5 confirms that under our pulsed field conditions the shorter molecules move faster than the longer molecules.

The widths of the bands after 11 min were approximately the same for both DNAs (FWHM was 200 µm). For a di usion constant D 1 µm2/s the expected di usional broadening ( [2Dt]1/2 due to self di usion in t = 11 min is 30 µm, while the observed broadening is 100 µm. This dispersive broadening possibly occurs because individual molecules of equal lengths get stretched by di erent amounts, or some times not at all, when encountering posts. The location of the bands after 11 min was consistent with the band separation derived from the microscopic migration velocities, ∆x = (Vλ − VT4)t = 6100 µm. The band capacity [9] nC in this experiment is estimated to be ∆x/[1.5[σ(λ) + σ(T 4)] = 20, i.e. 20 bands could be resolved in the 50 to 170 kbp range under the conditions used here. Since the separation is approximately linear in molecular weight (see below), this means that molecules which di er by 6 kbp can in principle be distinguished.

Observation of the microscopic dynamics confirms that the separation is a consequence of “switchback” motion of the DNA molecules for opposing field directions greater than 90o, as reported in preliminary investigations [10, 11] and illustrated in Figure 1. When the field direction is

Fig. 5. Images of single molecules taken at 11 minutes after the start of pulsing. These images were gathered 10 mm (left) and 4 mm (right) from the entropic barrier. In the images at the top, the field was turned o and the molecules were at thermal equilibrium, while the bottom panels show the separated molecules elongated under pulsed field conditions. Note that the radius of gyration of the relaxed molecules in the top panels clearly establishes that the faster moving species is λ DNA.

switched, each molecule moves o in the new field direction, led by the end which was previously trailing. As a result, the molecules retrace part of the path they have traveled. Since longer molecules backtrack further than shorter molecules, the rate of advance along the bisector of the field is slower.

The overall speed of migration in a pulsed hexagonal array is lengthdependent. Under the simplifying assumption that the molecules remain uniformly stretched during this motion, the net velocity VL of molecules of length L in a pulsed field array can be described by the simple equation [10]:

where the angle θ between transverse fields in our case is 120o and µo is the

continuous-field mobility, which is independent of length and L is a critical cut-o parameter. The critical cut-o length L arises from that fact that molecules which do not have time to reorient completely during a single pulse repeatedly retrace the same path and, since they make no progress, their velocity is e ectively zero. The reorientation time tor is to first order simply L/µ0E, or more empirically given by:

where c1 is a parameter which takes into account the fact that the DNA molecules are not fully aligned. Roughly, in order to fractionate molecules of length L the pulse period T should be set to tor. Thus, the upper limit of separation, L , is proportional to the pulse time T and to the speed µoE of a molecule along a free channel:

It is a strength of the analytical nature of the dynamics of DNA molecules in synthetic arrays that L and T can be predicted. Note that we can obtain expressions for µoE and L from our experimental data at 244 V with a pulse period T of 1 s for T4 and λ DNA. Some algebra gives:

These expressions can then be used to check the actual parameters used in the experiment. We can assume (somewhat incorrectly) that the lengths of the DNA in the above expression are the fully stretched lengths taking into account that one Kuhn length b contains 300 basepairs and is 130 nm long for TOTO-1 stained DNA [12], giving LT4 = 168.9 kbp = 73 µm and Lλ = 48.5 kbp = 21 µm. The measured values of the retarded velocities are VT4 = 5.6 µm/s and Vλ = 14.9 µm/s; from these parameters we get that µoE = 37 µm/s and L = 104 µm. The pulse periods T that should be used at L are then roughly tor = 3 s. Thus, pulse periods on the order of 1 s in this particular protocol are appropriate.

We can check the consistency of the experimental data at the two field strengths. Since the migration velocity in a continuous field is proportional to the applied voltage, we expect that at 100 V, µ0 E = 15 µm/s. Equation (3) then predicts that L = 43 µm. Equation (1) predicts that lambda DNA should move at velocity Vλ = 3.9 µm/s, which is what we

observe. Since the T4 DNA is longer than L , the theory predicts that it will not move, but in practice it advances at the low speed of 2 µm/s. This migration occurs because the backtracking motion is not ideal. We observe that the molecules do not remain uniformly extended, but usually get stretched by the field and subsequently relax during each pulse. Because of this inherent stochasticity, even the longest molecules do not retrace the same path indefinitely. Our measurements yields c1 = 0.18. Duke et al. [10] calculated the value c1 = 1.39, assuming that molecules are extended to their full contour length at all times. If the molecules are stretched to only a fraction of their length, we would expect the value of c1 to be proportionately reduced, because the reorientation time would be faster. The lower value of c1 that we measure is therefore consistent with our observations that, at the field strengths used, the molecules are rarely extended to more than 30–40% of their full contour length.

3 Conclusions

Separation in the hexagonal arrays will be even faster when higher electric fields with shorter periods are applied. In our experiments we were limited to relatively low electric fields and long periods because we wished to record single molecule images. Higher fields would cause greater molecular extension, which would enhance the regularity of the “switchback” motion and improve the discrimination between molecules of di erent size. Reduction of the depth of the device is also expected to increase the extension of the molecules [12].

We can compare these results with those of others. Pulsed field capillary gel electrophoresis [13, 14] achieves fast separation, but this method is severely limited by the tendency of high molecular weight DNA to form supramolecular complexes that interfere with separation. Chou et al. [15] have proposed a single molecule sizing device in which molecules in the 2 to 200 kbp range are sized one at a time. This method, however, cannot be used to separate many thousands of molecules simultaneously. Other methods, such as recently developed arrays that separate molecules based on their di usion coe cients [16], su er from rapidly deteriorating resolution as the molecules get bigger. Han et al. [6] have shown that entropic trapping can be used to separate molecules in reverse order, with the largest molecules moving the fastest. This method, although faster than conventional methods, is still at least an order of magnitude slower than separation in hexagonal arrays using quantitatively understood pulsed field parameters that we report here.

The separation principle that we demonstrate here is not limited to DNA molecules of a particular size. By appropriately adjusting the applied

electric field, the pulse time and array parameters such as the pillar size or spacing, the technique could be extended to the separation of polymers of all lengths. Nor is it limited to the separation of fluorescently-stained molecules. Because the motion of the molecules is predictable, it should be possible to operate the device “blind”, without the addition of dye stains which can contaminate subsequent processes such as polymerase chain reactions. Alternatively, the dye can be removed by standard techniques using ion exchange resins. In the future, it should be feasible to dispense with fluorescence-based detection methods and detect the DNA molecules electronically using nanosensors constructed on the chip [17].

References

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Abstract

The previous lecture discussed how using crossed fields greater than 90o can be used to quickly fractionate DNA. However there are problems with the technique we described in the past lecture: it requires running DNA as a single run of prepurified DNA, so that if one is lysing and extracting genomic DNA as part of a continuous process it is very di cult with the technology described. Further, we have not really exploited the ideas of 2.5 D hydrodynamics that we described earlier. Now it is time to correct that problem with what we call a DNA prism, which uses the idea of transverse pulsed field fractionation but with a twist: asymmetric pulsed fields. We now tie things together.

1 Introduction

The previous lecture discussed how using crossed fields greater than 90o can be used to quickly fractionate DNA. However there are problems with the technique we described inn the past lecture: it requires running DNA as a single run of prepurified DNA, so that if one is lysing and extracting genomic DNA as part of a continuous process it is very di cult with the technology described. Further, we have not really exploited the ideas of 2.5 D hydrodynamics that we described earlier. Now it is time to correct that problem with what we call a DNA prism, which uses the idea of transverse pulsed field fractionation but with a twist: asymmetric pulsed fields. We now tie things together.

The basic principle of prism separation is shown in Figure 1. As we have discussed, when an electric field is applied in one direction, molecules of all sizes migrate between the SiO2 posts with similar mobility (Fig. 1A). When the field is switched 120o, all molecules must backtrack through channels formed by the post array (Fig. 1B). The longer molecules take longer to reorient, and thus at each change in field direction, the separation between small and large molecules increases (Fig. 1C). This is the physical basis for pulsed field gel electrophoresis in microfabricated devices, and probably also for conventional gels. The prism device incorporates these principles and in addition allows continuous fractionation by biasing either the strength of the field from pulse to pulse, or the duration of the pulse at constant field strength (Fig. 1D). Because of the microfluidic design the field should be constant across the device.

2 Design

In our design the microfluidic channels of Figure 2 act as resistors and serve to shape a uniform electric field. Whether or not the device produces

Fig. 1. Schematic diagrams showing the di erent behavior of small and large DNA molecules in microfabricated arrays with asymmetric alternating-angle electric fields; A–C) illustrate the sequential motion of a long and a short molecule through a full cycle of alternating field. A) The high field moves both small and large molecules in a channel (arrow shows direction of motion). B) A low field rotated 120o causes reversal of the leading and trailing ends, and the low field (or short time) prevents the long molecule from sliding o the posts and reversing direction. C) The original field reapplied. The ends again reverse, the large molecule resumes its original track while the small molecule is now in a new track. D) Net motion after multiple cycles of a mixture of large and small molecules injected into the array at the same point. The small molecules follow the average field while large molecules follow the stronger field. Here and elsewhere the vectors point in the direction of DNA migration, rather than the direction of the electric field as traditionally defined.

uniform electric fields at the desired angles was tested by tracing the trajectory of fluorescently stained DNA. Predetermined voltages were applied to the reservoirs to create a DC field at −30o with respect to the vertical axis (Fig. 3 inset). A mixture of bacteriophage λ and T2 DNA was loaded into the DNA reservoir and then injected electrophoretically. The injected band should flow along the electric field line. A straight and narrow band50 microns wide was formed. If the electric field had not been uniform in direction, the band would have curved. If the electric field had not been uniform in strength, the band would have been tapered or dispersed. We therefore conclude that the electric field is uniform. The voltages were then switched to create a DC field 120o from the previous one, and the band moved at constant speed (Fig. 3).

3 Results

By tracking individual molecules in the bands, we found that the electric field was indeed horizontal. The field is uniform in strength as well, since if it were not, the band would curve or smile as it moves. The injected band is slightly S shaped, but this does not interfere with separation. The slight S-shape near the array boundary results because the non-infinite resistance

Fig. 2. Structure of the device illustrating the microfabricated sieving matrix integrated with the microfluidic channels. Insets are micrographs of 2 micron microposts with 2 micron spacing, microfluidic resistive channels connecting the post array to electrolyte bu ers, and a single special channel connecting the post array to the DNA reservoir.

of the channels makes them imperfect current sources, a well-understood design compromise. In a second experiment designed to test the separation power of the array, restriction enzyme digests of BAC and PAC preparations were injected and electrophoresed with constant pulse times, in a manner similar to conventional pulsed field electrophoresis [1]. A mixture of 61 kbp and 158 kbp BAC inserts was cleanly resolved in 7 s (Figs. 4B,C). The resolution in this experiment, defined as the full width at the half maximum of a band 7, was 77 kbp at 7 s, and 36 kbp at 14 s (Fig. 4). Although the resolution is not yet as sharp as can be achieved with standard methods (Fig. 4A), it is clear that the resolution increases with separation distance. Moreover, the current separation time was more than 4 orders of magnitude faster (7 s vs. 16 hr). We then continuously loaded and sorted a mixture of four BAC inserts using asymmetric pulsing (the prism mode) (Fig. 5A). The pulsing conditions empirically determine the angles and the widths of the bands. Although the resolution of the four species was achieved in a 1 mm long matrix in 4 s under certain pulsing conditions (Fig. 5A), the two smallest species (61 kbp and 114 kbp) were not resolved under other pulsing conditions tested (Fig. 5B). For molecules larger than 100 kbp, however, continuous separation does occur with high resolution (Figs. 5B,C).

The resolution achieved in this weight range is about 10 to 15 kbp, comparable to conventional methods. Note, however, that the separation

Fig. 3. Overlay of sequential fluorescent images of bacteriophage T2 DNA (164 kbp) mixed with l DNA (48.5 kbp) to show the spatial uniformity of the electric field. The band on the left (0 s) was imaged just after DNA injection using a 52 V/cm electric field −30o with respect to the vertical axis. The other four bands are from images taken at one second intervals later at a field strength of 38 V/cm in the horizontal (+90o) direction. The inset is a low power schematic of the device to orient the reader. The area of the field in Figure 3 is boxed with a dotted line.

is achieved in 10 s, using a 2.5 mm long array. These results can be understood with reference to Figure 2. The separation angle is simply the angular di erence between the band formed by very small molecules, which follow the average field direction, and the band followed by larger molecules, which tend more toward the direction of the stronger electric field pulse. As random coils, molecules smaller than the spacing between posts follow the average field direction, because they are too small to interact with the posts. For the device used here, with 2 micron post spacing, we observed a cut-o100 kb (Fig. 5C). The 61 kb insert in Figure 5C is below this cuto , while the inserts above the cuto ( 100 kb) interact with the posts and move in a manner depicted by Figure 2. Of course, a major reason for designing microfabricated devices of the kind discussed here is that we can control both the size of the posts in the array and their spacing, tuning both to suit the size range of the DNA molecules.

We note a second valuable feature of this device, its micron scale features. Apart from the advantages mentioned above small volumes, speed, a very shallow chamber depth and hence no heating at high field strengths, and

Fig. 4. Pulsed-field electrophoresis of BAC and PAC inserts in a microfabricated array. A) Characterization of the BAC and PAC inserts by PFGE. The first lane is a l-multimer marker, and the next lanes are NotI-digested artificial chromosomes of insert sizes 61, 114, 158, and 209 kbp, respectively. Running time, 16 hr at 6 V/cm. B) A time series showing the separation of a mixture of the 61 and 158 kb inserts. The initial DNA concentration was 10 micrograms/ml. The sample was injected at 50 V/cm along the vertical axis (0 s). The field was then pulsed symmetrically (60o with respect to the horizontal axis, 167 ms pulse duration, 50 V/cm field strength). C) Fluorescence profiles of bands at 0, 7, and 14 s.

Fig. 5. Prism separation. A) Four bands (61, 114, 158, 209 kbp, 10 micrograms/ml total) were resolved using 41 ms pulse times and asymmetric voltages of 231 V/cm and 137 V/cm. B) Profile of bands resolved using 100 ms pulse times and asymmetric voltages of 196 V/cm and 108 V/cm, observed 2.5 mm below the injection point fitted with Gaussian distributions. The horizontal axis is in units of degrees with respect to the vertical. C) Experimentally observed dependence of the peak deviation angle on molecular mass.

the ability to tune post size and placement there are very low shear forces in micron scale devices. Consequently, very large DNA molecules do not break as they move through them. This is because DNA shearing is a function of turbulence [8] and there is no turbulence at the low Reynolds numbers at which these devices operate [2]. The lack of shearing is evident from the