14.2 DNA transport [DNA top]

DNA transport properties in bulk, in gels, and in nanostructured channels are required to predict the performance of fluid mechanical devices for DNA analysis. In this section, we summarize the existing experimental measurements of these properties.

14.2.1 DNA transport in bulk aqueous solution

We first consider DNA transport (including diffusion and electromigration) in bulk aqueous domains far from walls. DNA molecules can be large, so DNA’s diffusivity is relatively low as compared to smaller molecules, and the diffusivity is dependent on polymer length, as described below. DNA’s electrophoretic mobility is quite high compared to most macromolecules, owing to its highly charged sugar backbone, and DNA electrophoretic mobility is largely independent of polymer length.

DNA diffusivity in bulk Diffusionis the macroscopic description of Brownian motion caused by thermal energy; diffusivity was discussed in Chapter 4 for small molecules. For a macromolecule, we can describe several types of diffusion, including both translational diffusion and rotational diffusion. We focus in this chapter strictly on translational diffusion, i.e., diffusion of the center of mass of the DNA molecule. The translational diffusion is proportional to the thermal energy and thus proportional to k_BT , as well as the effective viscous mobility μ. Hydrodynamic diffusion of DNA in bulk aqueous solutions is reasonably well-characterized by treating DNA as anon-draining polymer that obeysZimm dynamics. The non-draining polymer assumption describes the motion of water near the polymer—the non-draining assumption entails assuming that the motion of water molecules in the region of the polymer is largely suppressed by the presence of the DNA molecule.

The Zimm dynamics approximation is closely related, but relates to the motion of various parts of the polymer—Zimm dynamics assumes that the motion of various parts of the polymer are tightly coupled to each other because of the viscous coupling. In the Zimm dynamics approximation (or in the non-draining approximation), the viscous coupling makes the DNA and surrounding water diffuse hydrodynamically as if it were a solid object. Experimentally, we observe that the Zimm model is accurate enough that diffusion of DNA is well approximated by modeling it as a rigid sphere with a radius equal to about ∕3.

Recall from Chapter 8, for comparison, that the mobility μ for a macroscopic particle of radius a in a liquid of viscosity η is given by the Stokes flow relation:

(14.6)

leading to the Stokes-Einstein relation for particle diffusivity:

(14.7)

and the viscous mobility for an ion with hydrated radius a is approximately given by

(14.8)

leading to an approximate relation between ion hydrated radius and diffusivity:

(14.9)

Using a Zimm dynamics model and assuming (as inferred from measurements) that DNA’s effective solid-particle radius can be approximated by ∕3, DNA diffusivity can be approximated as

The relation a ≃∕3, is an approximate value that corrects roughly for the assumptions embedded in the application of a macroscopic relation for flow over a rigid sphere to the transport of a macromolecule. The most important result of the analogy between DNA diffusion and solid particle diffusion is the conclusion that the diffusivity of DNA scales with  in the same manner that the diffusion of solid spherical particles scales with radius a. Zimm dynamics accounts for the fact that the parts of the DNA molecule are all hydrodynamically linked by the water, making the molecule diffuse as one cohesive body. Thus DNA in bulk solution acts diffusively (approximately, at least) as if it were a solid particle with a radius approximately equal to ∕3. The diffusivity D in free solution is independent of both the DNA sequence and the presence or absence of an applied electric field. Solution conditions need be considered only to the (minor) extent that solution conditions affect the radius of gyration.

Various models (see later sections in this chapter) predict  as a function of ℓ_c and other properties. Depending on the complexity of the model and the role of the solvent, these models typically predict that  is proportional to ℓ_c^(1∕2) or ℓ_c ^{3 5} , implying that the bulk diffusivity should be proportional to ℓ_c^-(1∕2) or ℓ_c^{- 3 5} and largely independent of other properties. For DNA in water, theoretical treatments lead to the conclusion that  should be proportional to ℓ_c^{- 3 5} . Experiments usually observe an exponent approximately equal to -0.57 (e.g., Figure 14.3), leading to approximate relations such as

which describes the data in [145] for DNA in free solution in a 1X TAPS buffer. Results for other buffers range from D ≃2–5 m²∕s^-0.57.

DNA electrophoretic mobility in bulk Electrophoresis, asdiscussed in Chapters 11 and 13 is the net electromigration of a molecule induced by Coulomb forces on a charged molecule or particle and, if present, its electrical double layer. Unlike hydrodynamic diffusion of DNA, electrophoresis of long DNA molecules in bulk aqueous solutions of at least modest electrolyte concentration is reasonably well-characterized by treating DNA as a free-draining polymer that obeys Rouse dynamics. Thefree-draining polymerassumption means that we assume that the motion of water molecules in the region of the polymer is unaffected by the presence of the DNA molecule. This implies that the distance over which the fluid velocity gradients decay (λ_D) is small as compared to the spacing of different components of the polymer, leading to free motion of the water with respect to the polymer. Rouse dynamics assumes that the coupling between polymer elements is minor. This phenomenon makes the DNA electrophorese hydrodynamically as if all polymer components were electrophoresing independently.

Despite the complexity of physics that governs DNA electrophoresis, we observe a relatively simple dependence experimentally. The electrophoretic mobility of long DNA in bulk electrolyte solutions is typically in the range μ_EP ≃ 2-5×10^-8 m²∕Vs, where the magnitude is a function of the electrolyte concentration and valence but, for N_bp > 400, is independent of on contour length (Figure 14.4). This simple relation breaks down for (a) low salt concentrations, in which λ_D becomes large, (b) confined geometries, in which spacing between polymer regions can be reduced by the geometries enforced by confinement, and (c) DNA oligomers, for which many of the geometric approximations of these models are invalid.

Increasing electrolyte concentration reduces the DNA electrophoretic mobility (Figure 14.5), as does the presence of multivalent cations, akin to that observed for the electrophoretic mobility of particles or the electroosmotic mobility of microdevice walls (Chapter 10).

Failure of Nernst-Einstein relation for polyelectrolytes The distinction between the size dependence of the electrophoretic mobility and the size dependence of the diffusivity is important. This contradicts the Nernst-Einstein relation(Equation 11.28), which links the diffusivity of a point charge to its electrophoretic mobility.

For an ion modeled as a point charge, the force that moves the ion (either a Coulomb force or random fluctuations caused by the thermal motion of the solvent) is balanced by the “drag” that the ion feels when it moves through the solvent. This “drag” is the same regardless of the source of the motion. Thus, the Nernst-Einstein relation illustrates that the electrophoretic mobility (when normalized by zF , a measure of the charge of a mole of ions) is equal to the diffusivity (when normalized by RT , related to the thermal energy of a mole of ions). The mean-field, point charge assumption means that each ion behaves independently from all other ions.

For DNA, we can think of the parts of the DNA molecule as a bunch of particles or rods. The hydrodynamic motion of all of the parts of a DNA molecule is akin to the motion of a collection of Stokes particles in close proximity—the surrounding water to a large extent moves along with the particles. The electrophoretic motion of all of the parts of a DNA molecule is akin to the electrophoresis of a collection of particles in close proximity—the surrounding water does not move along with the particles, because the Coulomb forces on the ions in the electrical double layer cause the hydrodynamic perturbations to decay with a characteristic length λ_D. So the different charge components of the backbone are to a great extent unaffected by each other. All feel the same electrophoretic force in the same direction, with minimal interference. These differences make DNA’s behavior in diffusion much different than electrophoresis. These differences also create many possibilities for novel DNA separation techniques, since Brownian ratchets and nanofilters (see Chapter 15) take advantage of the contour length dependence of diffusion and the contour length independence of electrophoresis.

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