Entropic Trapping

Entropy can be used in other ways to effect a molecular separation based upon length. Extensive theoretical work was done in the 80s and 90s that described the effects of confining environments on polymer diffusion, entanglement, and mobility [48-52]. Experiments in gels confirmed that when polyelectrolytes such as DNA are forced to move through entropically restrictive environments, the polymer mobility is not accurately described by either the Ogston sieving model or the reptation model [53 -56]. Instead, a regime of entropic trapping exists where molecules are effectively trapped in pores larger than their relaxed blob size and only periodically hop through restrictive areas into new pores.

Well-characterized, "ideal" periodic structures were used in modeling the motion of polymers through restrictive environments, yet the experimental work involved random gel-matrices similar to those conventionally used to separate DNA molecules by length. It was not until the work of Han et al. that a lithographically defined array of entropic traps was used to show that DNA molecules are trapped and can be separated by length in a real "ideal" structure [25]. In their first work, Han et al. showed that 40 kb DNA molecules and 160 kb DNA molecules exhibit significantly different trapping times when forced to hop across barriers and into large "pores". Subsequent research showed that the mobility difference caused by the difference in delay times at each trap was sufficiently large to separate these molecules by length in a device 1.5 cm long containing thousands of traps. In a variety of experiments with so-called entropic trap arrays, Han et al. were able to separate a wide range of DNA lengths [57, 58]. T2 and T7 phage

DNA (160 kb and 40 kb respectively) were separated in 15 min in an entropie trap array. A comparable experiment using pulsed field gel electrophoresis would have required between 12 and 24 hours. Additional experiments showed that DNA from 5 kb to 40 kb could be separated in about 30 min with resolution comparable to that obtained using gel electrophoresis.

As originally described, entropie trapping relies upon an energy barrier and interfacial contact between a molecule and that energy barrier. DNA is a flexible polymer that has a characteristic radius of gyration set by the total number of bases in the polymer chain. When a relaxed blob of DNA is forced against a constriction in a microfluidic channel by an electric field, the blob makes interfacial contact with the restriction. This contact area is proportional to the radius of the blob: larger molecules "contact" the gap more than do small molecules. The molecule is subjected to two forces: an electric driving force and an entropic force discouraging it from entering the restriction. This free energy landscape is depicted in Fig. 3.13. The en-tropic barrier energy is sharp and significant. However, once a portion of the molecule overcomes the barrier, the large electric field in the restricted region pulls the entire molecule into the restriction and toward the next open region. Because the DNA molecule is "free draining" to both the electric field and whatever fluid flow is present, the process of entering the restriction is local in nature. Portions of the molecule randomly diffuse near the entrance to the restriction and when one happens to find itself on the downhill side of the energy landscape, it is pulled in and the rest of the molecule follows. To summarize: larger molecules have more interfacial area with the restriction and make more attempts to enter the high field shallow region. Consequently, larger molecules move between restrictions more readily than do smaller molecules.

Quantitatively, the following model was proposed to describe the entropic trapping process [59]. The energy landscape is composed of an electric and entropic component

where x is the distance the molecule has penetrated the shallow region, T is the temperature, and E is the electric field. Equation (3.4) represents the energy barrier that must be overcome to pass through the restriction. The probability of escape is then given by

where kB is the Boltzmann constant, and the prefactor a depends upon the number of escapes attempted (that is, the interfacial area). Equation (3.5) is the crux of the size-dependence in entropic trapping: the prefactor contains all of the size-dependence while the exponential term represents the thermodynamics of the energy barrier and does not depend on any aspect of the molecule. The time molecules remained trapped is given by t = rQ exp(-AF/kBT) = r0 exp(b/EkBT) , (3.6)

where t0 is the size-dependent prefactor, b is a geometric constant, and Es is the electric field at the edge of the shallow region. From the trapping time, the mobility can be written

where t is the time required for a molecule to move through a distance equal to the length of the restriction if the restriction were not there. The length

DNA motion

silicon oxide

Si substrate

thick region thin region f AE : entropic free energy --- difference thick region thin region f AE : entropic free energy --- difference

entropic traps

Figure 3.13 (a) A cross-sectional schematic of an entropic trap device, reprinted with permission from [59], Copyright 1999 American Physical Society. DNA is "trapped" in deep regions of high entropy and forced against shallow regions. The shallow regions are smaller than the radius of gyration of the molecule, and entry into those regions would require sacrificing configurational entropy. (b) The energy landscape of consecutive entropic traps. Molecules require considerable energy to overcome the barriers. The energy is attained when random, "beachhead" events extend a portion of the molecule a critical distance into the shallow region. Once a critical insertion length is obtained, the electrical force is sufficient to overcome the entropic force and the molecule jumps the barrier.

entropic traps

Figure 3.13 (a) A cross-sectional schematic of an entropic trap device, reprinted with permission from [59], Copyright 1999 American Physical Society. DNA is "trapped" in deep regions of high entropy and forced against shallow regions. The shallow regions are smaller than the radius of gyration of the molecule, and entry into those regions would require sacrificing configurational entropy. (b) The energy landscape of consecutive entropic traps. Molecules require considerable energy to overcome the barriers. The energy is attained when random, "beachhead" events extend a portion of the molecule a critical distance into the shallow region. Once a critical insertion length is obtained, the electrical force is sufficient to overcome the entropic force and the molecule jumps the barrier.

dependence in the mobility comes from t and the length dependence is such that longer molecules migrate faster than shorter ones. Later work by Han et al. confirmed the applicability of the model over a wide range of DNA lengths and discussed optimizing the device for maximizing resolution and separating chromosomal-length DNA [57, 58].

After the original entropic trap array separation experiments were published, a new wave of theoretical and experimental papers surfaced [60-62]. In the first of such theoretical comparisons, Tessier et al. confirmed that the model proposed by Han was essentially correct in that Monte Carlo simulations based upon Han's model produced results qualitatively similar to the experimental results [60]. The shapes of the experimental and theoretical mobility versus electric field graphs, for instance, are identical. Furthermore, the essential physical mechanism of escape - herniation of a portion of the molecule into the restrictive region leading to the entire molecule moving through the gap - is confirmed by the theoretical results. Chen and Escobedo examined wider ranges of molecule sizes and electric field strengths than were examined experimentally [61]. Even though their results indicate that the physical model proposed by Han et al. might not fully describe the system at low electric fields or for very short DNA molecules, their simulations match Han's results for the conditions used in experiments. Additional simulations by Streek et al. showed that there might be a second, independent physical mechanism contributing to the time molecules spend migrating between restrictions [62]. In their simulations, they show that while in transit across the deep regions, molecules can diffuse out of the high field region of the channel. That is, molecules can diffuse into the corners of the entropic traps where the electric field is weak. In the corners, molecules are effectively left to diffuse back into a region of higher electric field. This phenomenon operates on a much slower time scale than the mechanism proposed by Han et al. and was probably not observed because of the sizes of molecules and electric field strengths used in their experiments.

Entropic traps separate DNA molecules by length and have been used to separate quite long DNA molecules. This artificial sieving matrix is straightforward to fabricate and has great potential as a separation matrix within a ^TAS environment. The entropic trap array could potentially be used to handle the large DNA molecules that would be released from a cell lysed within a chip. Judicious use of entropic trap-like restrictions could also be used to separate or filter cellular debris and proteins from the DNA. Recent work has shown that by using very thin shallow regions in entropic trap array-like devices, one can separate very short DNA molecules and proteins [63, 64]. Thus, the general model of patterning very narrow restrictions followed by relatively large gaps can be used for biological applications across a wide range of biomolecular sizes. This broad-utility is exactly what is needed for ^TAS systems that start with whole cells and end with useful analytical results.

Was this article helpful?

0 0

Post a comment