INTRODUCTION

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The principal topic of the present work concerns the length dependence of the free solution electrophoretic mobilities of highly charged rigid rod models. The translational diffusion constants of the same models shall also be considered. Such structures are particularly relevant in biophysics because rods serve as realistic models for duplex DNA fragments. For example, the translational and rotational diffusion constants of short duplex DNA fragments consisting of n base pairs (bp) correspond to those of uncharged right circular cylinders of radius 1.0 nm and length L = 0.34n (in nm) (Eimer et al., 1990; Nuutero et al., 1994). Theory (Tirado and Garcia de la Torre, 1979; Garcia de la Torre and Bloomfield, 1981; Tirado et al., 1984) predicts a length dependence of the translational diffusion constant, D, that varies roughly as ln(n)/n. In the present study, we reconsider the translational friction/diffusion constants of rodlike models, but account for the effect of polyion and ion atmosphere charge. Physically, transport of a charged particle through a viscous medium causes the ion atmosphere around the particle to distort, and this, in turn, increases the frictional drag on the particle. This "electrolyte friction" effect has been examined theoretically (Booth, 1954; Schurr, 1980; Stigter, 1982a; Geigenmuller, 1984; Schurr, 1984; and Vizcarra-Rendon et al., 1990) and experimentally (Gorti et al., 1984) to a very limited extent.

Free solution electrophoretic mobilities of long duplex DNAs are available and the recent study by Hoagland and coworkers includes a summary of past work (Hoagland et al., 1999). Long duplex DNA mobilities are more difficult to interpret than those of short duplex DNAs due to their more complex global conformation. For DNAs substantially less than a persistence length, P, (which equals ~50 nm or ~150 bp under most salt conditions (Rizzo and Schellman, 1981)), the overall conformation can be modeled as a rigid rod. However, for fragments substantially longer than several hundred basepairs, the overall conformation is that of a random coil polymer although locally the DNA remains rodlike. Because long DNAs are locally rodlike, mobilities have been interpreted in terms of infinite rod models (Schellman and Stigter, 1977; Stigter, 1978a,b). Although this may be a good approximation for a polymer as "stiff" as duplex DNA, the influence of long-range conformation on electrophoretic mobilities is a point worth addressing, and this is done in the present work. With the development of boundary element methods that solve the coupled field equations (Allison, 1996), it is now possible to carry out electrophoretic mobility calculations on detailed model polyions such as short DNA fragment up to ~100 bp in length. Experimentally, less work has been done on short DNAs, but enough has been reported to demonstrate a rather weak dependence of mobility on length. In Tris buffers (Stellwagen et al., 1997) the free solution electrophoretic mobility displays a modest increase with length, but reaches a constant plateau value beyond ~400 base pairs.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS AVERAGE SURFACE POTENTIAL AND... REDUCED MOBILITIES COMPARISON WITH EXPERIMENT SUMMARY REFERENCES

The transport properties of rigid rod structures such as those shown in Fig. 1 are determined within the framework of the continuum hydrodynamic-electrodynamic model (Overbeek, 1943; Hunter, 1981; Allison and Stigter, 2000). The solvent is represented as an incompressible Newtonian fluid with viscosity  surrounding a rigid model polyion enclosed by hydrodynamic shear surface S_p. The success of continuum models in accurately predicting sedimentation and diffusion constants of detailed model structures has been recognized for over twenty years (Garcia de la Torre and Bloomfield, 1981). In addition, continuum models have been very successful in treating the equilibrium electrostatics of ionic solutions. This is true for a wide range of applications from the activity of dilute solutions of simple salts (Debye-Huckel theory) to such complex problems as the ionization states of detailed model proteins (Antosiewicz et al., 1996). Under certain conditions at least, it has also been established that detailed continuum models can successfully be applied to electrokinetic transport of DNA (Allison and Mazur, 1998; Mazur et al., 2001). Recent modeling of short 18- and 20-bp DNA fragments using detailed "atomic resolution" surface models (Mazur et al., 2001) has shown that it is not necessary to include a Stern layer of solvent that moves with the polyion as a rigid body to obtain translational diffusion constants that match experiment. Experiments on other systems have shown that the solvent-accessible surface and S_p coincide to within a resolution of 1 Å (Stigter, 1982b). For the DNA systems studied in the present work, solvent-accessible surface and S_p are used interchangeably. It should be emphasized that this might not be appropriate for a deeply grooved or holed surface. However, limited studies on short DNAs (Mazur et al., 2001) suggest that the grooves have no significant effect on transport, and such effects are ignored in the present study.

View larger version (32K): [in this window] [in a new window]   FIGURE 1   Representative straight-capped-cylinder models for 20-, 40-, and 80-bp duplex DNA. The number of plates making up these particular model structures are 112, 176, and 304, respectively.

The motion of the polyion is assumed small enough that the local fluid velocity at point y, v(y), and local pressure, p(y), are described by the linearized Navier-Stokes and solvent incompressibility equations,

(1)

(2)

where s(y) is the external force per unit volume acting on the fluid at y. For a polyion, the local charge density due to co- and counterion atmospheres, (y), interacts with the local electric field, (y) (where (y) is the local electrostatic potential) yielding s(y) = (y)(y). For the transport of model structures, translational or rotational friction coefficients are determined by translating or rotating the structures with uniform linear or angular velocities through a fluid otherwise at rest. Also, "stick" boundary conditions are normally assumed in which the particle and fluid velocities match on S_p. In the present work, we are interested in the translational motion of simple model structures such as those shown in Fig. 1 where translation/rotation coupling (Happel and Brenner, 1983) can be ignored. The more general transport problem is described in detail elsewhere (Garcia Bernal and Garcia de la Torre, 1980; Garcia de la Torre et al., 1982; Mazur et al., 2001). From the solution of Eqs. 1-2 (which, in the present work, is solved numerically) it is possible to compute the force, z⁽¹⁾, exerted on the fluid by the particle,

(3)

where u is the particle velocity and _t is the translational friction or resistance tensor. The (1) superscript on the force in Eq. 3 refers to case 1 transport, which means that the particle is translated through an otherwise stationary fluid with uniform velocity u, or equivalently, the particle is held stationary and placed in a flow field where the fluid velocity far from the particle is u (O'Brien and White, 1978; Allison and Tran, 1995). In case 2 transport, the particle is held stationary, but subjected to a constant external electric field, e, and the fluid is also stationary far from the particle. The force exerted on the fluid by the particle is

(4)

where q is the protonic charge (4.803 × 10¹⁰ esu) and Z is called the "tether force", or simply Z-tensor. In special circumstances, it is possible to fix, or tether, a large polyion in space and monitor its behavior in an external electric field (Smith and Bendich, 1990). For this reason, Z may be referred to as the tether force tensor. This point is discussed in more detail later. For a rodlike structure (Fig. 1) or a similar structure that is smoothly bent, it is straightforward to identify three orthogonal directions (i = 1 to 3) along which z⁽¹⁾ is parallel to u. In a reference frame whose axes coincide with these three orthogonal directions, _t is diagonal. The corresponding translational diffusion constant along direction i is simply

(5)

where k_B is Boltzmann's constant, T is absolute temperature, and the 1 superscript denotes inverse. The orientationally averaged translational diffusion constant, D, is simply the average of the three D_i (or (D + 2D)/3 for a straight rod diffusing in three dimensions where D and D are the diffusion constants parallel and perpendicular to the rod axis, respectively).

In general, nonequilibrium electric or flow fields distort a macromolecule's ion atmosphere from its equilibrium value (Booth, 1954; Stigter, 1982a). For a sedimenting particle (case 1 transport), this distortion or "ion relaxation" increases the frictional drag, or equivalently, reduces D_i. This electrolyte friction effect is expected to be small unless the charge on the polyion is large (Booth, 1954; Stigter, 1982a). For this reason, it is usually ignored, which is equivalent to neglecting s(y) in Eq. 1. Diffusion constants for right circular cylinder models that ignore electrolyte friction have been thoroughly studied by a number of investigators. The most comprehensive investigation is that of Garcia de la Torre and coworkers (Tirado and Garcia de la Torre, 1979; Garcia de la Torre and Bloomfield, 1981; Tirado et al., 1984) in which D_i is written

(6)

(7)

where A_i = 2 (i = 1 (parallel to rod axis)) or 4 (i = 2 or 3 (perpendicular to rod axis)), L is the cylinder length, p = L/2R where R is the cylinder radius, and a_i, b_i, c_i are constants tabulated elsewhere (Garcia de la Torre and Bloomfield, 1981; Tirado et al., 1984). In the remainder of the present work, p represents a reduced length (L/2R) rather than the pressure. The orientationally averaged diffusion constant, D, is also expressed in the same form as Eqs. 6-7 above with A = 3, a = 0.312, b = 0.565, c = 0.10 (Garcia de la Torre and Bloomfield, 1981; Tirado et al., 1984). For long rods, it is seen that the D_i (and also D) vary roughly as ln(n)/n.

To include the effects of ion relaxation on diffusion, account must be taken of the external forces on the fluid near the polyion, and this requires solution ot the nonequilibrium Poisson equation

(8)

where (y) is the local dielectric constant and ₁(y) is the local charge distribution. It is also necessary to solve a steady-state ion-transport equation for the local current density of each ion species j(y)

(9)

(10)

where n the local concentration of that species, v the fluid velocity, D the diffusion constant of an  ion, and s the local external force on an  ion. It should be emphasized that Eqs. 1, 2, 8-10 are coupled together.

In the present work, these equations are solved by an iterative Boundary Element procedure described in detail elsewhere (Allison, 1996; Allison and Mazur, 1998; Mazur et al., 2001). Briefly, the polyion is modeled as a discrete rigid array of M plates and representative examples are shown in Fig. 1. These structures are designed to reproduce the translational diffusion constants of n-basepair duplex DNA modeled as right circular cylinders of length 0.34 n (in nm) and 1.0 nm radius (Allison and Mazur, 1998). The interior of the model is represented as a uniform low dielectric medium (_i = 4) containing a line or helical charge distribution appropriate for DNA. The surrounding fluid has uniform viscosity, , and dielectric constant, _o appropriate for water at a particular temperature. The water contains mobile ions that are treated as a continuum. The fluid domain is discretized into 49 shells that increase in thickness moving away from the polyion surface. The outermost shell is located at a distance nearest to the polyion surface of 16/ ( = Debye-Huckel screening parameter). Each shell, in turn, is divided into M discrete volume elements that conform closely to the polyion surface for the innermost shells. The approximation is made that physical quantities are constant on each plate on the polyion surface or each volume element in the fluid. Systematic errors inherent in this discretization approximation can be accounted for by studying a series of models in which M is varied and then extrapolating (electrophoretic mobilities, for example) to the limit 1/M  0. This extrapolated shell procedure (Allison, 1999) as well as more details regarding the iterative BE procedure are described elsewhere (Allison, 1996; Allison and Mazur, 1998; Mazur et al., 2001).

In addition to case 1 transport, which is sufficient for determining friction and diffusion coefficients of model structures, it is also necessary to consider case 2 transport discussed previously in connection with Eq. 4 (stationary polyion in a constant external field, e) when modeling electrophoretic mobility. For a weakly charged polyion in the limit of low salt, the tensor, Z must go to Z_actualI where Z_actual is the actual charge on the polyion (in protonic units) and I is the 3 × 3 identity tensor. In general, however, an external electric field will exert forces on the ion atmosphere of a polyion and this, in turn, generates hydrodynamic forces on the polyion (which is stationary in case 2 transport). Consequently, Z, does not correspond to the total charge of the polyion, in general, and is a tensor quantity. For rodlike models, Z is diagonal in the same reference frame that _t is diagonal in. The diagonal components of Z, Z_i, are determined by placing the model structure in an electric field oriented along the i-direction, computing z (by a numerical BE procedure), and solving Eq. 4 for Z_i. Steady-state electrophoresis can be viewed as a superposition of cases 1 and 2 such that the total force exerted by the polyion on the fluid is zero (O'Brien and White, 1978; Allison and Tran, 1995). Eqs. 3-4 are added together and the sum is set to zero. Eq. 5 is also used to introduce D_i. Define the free solution electrophoretic mobility for the model structure, µ_ie = u for e and u directed along i, then

(11)

The mobility obtained in an experiment is simply the average of the three µ_i provided the applied field is weak enough that no significant orientation of the polyion results as a consequence of the applied field. These conditions are readily achieved in actual experiments (Stellwagen et al., 1997).

Characterizing the Z_i and µ_i length dependencies

As discussed previously, the D_i decrease with length roughly as ln(n)/n or ln(L)/L and deviations from this overall behavior are contained in the correction terms, _i (Eq. 6), which are themselves length dependent. What are convenient representations for Z_i and µ_i? Experimentally, the (orientationally averaged) electrophoretic mobility of duplex DNA exhibits a weak length dependence relative to diffusion and appears to reach a constant plateau value above several hundred basepairs (Stellwagen et al., 1997). In view of Eq. 11 and the known dependence of D_i on length, we can expect the length dependence of Z_i to vary approximately as n/ln(n) (or equivalently n/ln(p)). A simple functional definition which exhibits this general characteristic is

(12)

In the limit of long DNA, only the d_i term remains. The coefficients d_i and e_i (assumed constant in the present work) are determined by fitting the length dependence of the Z, or tether force tensor components.

Once the coefficients for _i and X_i are known, it is straightforward to construct simple equations for the length dependence of parallel and perpendicular mobilities. Combining Eqs. (5, 6, 11, and 12)

(13)

(14)

where  = q/h and h = 0.34 nm is the rise per basepair of duplex DNA. At 20°C in water,  = 1.002 cp and (making use of the conversion factor 1 esu = 300 V cm)  = 1.499 × 10³ cm²/V sec. Some caution should be exercised in applying these equations to very short or very long fragments because the parameters are determined by fitting data from fragments in the size range of 20-100 bp. For this size range, ln(p) varies from 1.224 (20 bp) to 2.833 (100 bp) and Eqs. 13 and 14 should not be used far outside of these ranges.

We are also interested in dimensionless mobilities defined by

(15)

where

(16)

and _0S is the equilibrium electrostatic potential averaged over the polyion surface. For thin double-layer polyions (those whose smallest linear dimensions are large relative to 1/) in the absence of ion relaxation, C_i equals 1 (Smoluchowski, 1921). For thick double-layer spherical polyions of uniform surface potential in the absence of ion relaxation, C_i equals (Huckel, 1924; Henry, 1931). In general, ion relaxation always reduces C_i relative to the no-ion relaxation limit. To characterize the length dependence of the C_i, the following fitting polynomial is used

(17)

Conditions for the Length Studies

A transport property of a particular (highly charged) model particle depends in a complex way on temperature, T, solvent viscosity, , polyion and solvent dielectric constants, (_i and ₀, respectively), salt concentration, c₀ (assumed monovalent in the present work), and the mobility or diffusion constants, D, of the particular small-ion species present. This can be readily understood by considering the various coupled continuum equations that must be solved. One simplifying feature is that mobility appears to be nearly independent of the assumed dielectric constant within the polyion, _i, and this quantity is set to 4 (Mazur et al., 2001). Experimental measurements of the free solution mobility of DNA have been have been carried out under a wide variety of conditions, and, because of that, it will be necessary to carry out length studies under different temperature, ionic strength, and ion-type conditions as well. An additional complicating feature involves specific interactions between small ions and DNA that go beyond the purely electrostatic interactions present in current modeling. There remains little doubt that such interactions occur as the different mobilities of DNA in Tris-acetate and Tris-borate buffers under otherwise identical experimental conditions demonstrate (Stellwagen et al., 1997, 2000). With regard to past modeling, it should be mentioned that the electrophoretic mobility of 20-bp DNA in 0.11 M KCl at 20°C measured by the technique of membrane-confined analytical electrophoresis (Laue et al., 1996) are fairly well explained by continuum modeling. The model mobilities are ~4-6% higher than experiment under these conditions (Allison and Mazur, 1998; Mazur et al., 2001). In contrast, the electrophoretic mobility of 18-bp DNA in 0.04 M Tris-acetate buffer (pH = 8) at 25°C measured by capillary electrophoresis (Stellwagen et al., 1999) has been more difficult to model. Continuum modeling under these conditions yields mobilities that are ~28% higher than experiment (Mazur et al., 2001). Our interpretation is that there is an attractive interaction, possibly hydrogen bonding, between Tris⁺ and DNA, in addition to electrostatic interaction, that is not accounted for in our continuum modeling. In fact, reducing the phosphate charges of DNA by ~50%, or equivalently "binding" half of the anionic phosphates with Tris⁺ cations yields mobilities in good agreement with experiment. There may also be additional interactions between alkali ions such as K⁺ and DNA, but the mobility data mentioned above argues that such interactions would be significantly weaker than for Tris⁺. In the present study, length studies are limited to six cases that cover some of the key experimental conditions used.

The most thorough study of the length dependence of duplex DNA mobilities is the work of Stellwagen et al. (1997). Their experiments were carried out in water at 25°C ( = 0.89 cp, ₀ = 78.3) in a buffer consisting of 0.04 M Tris-acetate plus 1 mM EDTA. The buffer was prepared by titrating Tris base with acetic acid until pH = 8. A buffer prepared in this way contains 0.02 M Tris⁺ and acetate. From limiting molar conductivites of Tris⁺ (Klein and Bates, 1980) and acetate (CRC Handbook of Chemistry and Physics), hydrodynamic radii of 0.3136 and 0.2278 nm can be assigned to the ions (Mazur et al., 2001). As discussed before, a model with the charges of the DNA reduced by 50% gives the best agreement with experiment. Because capped cylinder models yield mobilities in excellent agreement with more realistic surface models for short fragments (Mazur et al., 2001), all models considered in the present work are of the simpler capped-cylinder variety. In addition, the charge distribution within the DNA models are usually approximated with a simple line-charge distribution. However, the effects of replacing the line-charge distribution with a more realistic double-helical one and including curvature in the rod models will be considered briefly. All studies in 0.02 M Tris⁺ acetate at 25°C with DNA phosphates set to half their full charge are called condition I.

All remaining conditions will have the DNA phosphates fully charged and, unless otherwise stated, the polyion charge is represented as a line of charges and the cylinder models are straight. Conditions II-V are in water at 20°C ( = 1.002 cp, ₀ = 80.36). Condition II is in 0.02 M Tris⁺ acetate, condition III in 0.02 M NaCl, condition IV in 0.02 M KCl, and condition V in 0.11 M KCl. The hydrodynamic radii of Na⁺, K⁺, and Cl are set equal to 0.1837, 0.1242, and 0.1242 nm, respectively, on the basis of their limiting molar conductivities (CRC Handbook of Chemistry and Physics). Condition III is chosen to make contact with recent capillary electrophoresis studies of Hoagland et al. (1999) on long duplex DNA. The membrane-confined capillary electrophoresis studies of Laue et al. (1996) on short DNAs were carried out under conditions very similar to condition V. Conditions II-V also provide insight into the effect of small-ion mobility and ionic strength on transport properties of DNA models. Finally, condition VI is carried out in 0.2 M KCl at 1.3°C ( = 1.711 cp, ₀ = 87.49). This reproduces one of the experimental conditions used in a widely cited early study on the mobility of long DNA (Ross and Scruggs, 1964). The key variables used in the model studies for the six conditions are summarized in Table 1.

Electrolyte friction

As discussed previously, ion relaxation of highly charged polyions is expected to reduce the parallel and perpendicular components of the translational diffusion tensor of a straight rod model structure (D and D, respectively) and the orientationally averaged translational diffusion constant, D, relative to that of an uncharged model. Because this electrolyte friction effect is readily obtained in the course of determining electrophoretic mobilities from model structures, results are presented in an abbreviated form. A detailed treatment will be postponed to a later date when experimental data on DNA fragments becomes available. To explore the effects of additional frictional drag due to ion relaxation, define

(18)

where i refers to parallel or perpendicular orientations, and r and nr to ion relaxation included and not included, respectively. Considering Eqs. 3 and 5, it can be seen that _i is proportional to the additional frictional drag ion relaxation generates on a model rodlike polyion in a particular orientation. Figure 2 shows the length dependence of and for fully charged DNA rod models in 0.11 M KCl at 20°C (condition V). In the length studies for diffusion and electrophoresis, six models are examined corresponding to 20, 30, 40, 60, 80, and 100 bp. The number of plates in the structures usually ranges from 112 (20 bp) to 368 (100 bp), but for certain fragment lengths, much more extensive studies are made over a wide range of plate numbers to make the extrapolated shell corrections (Allison, 1999). For the perpendicular orientation, exhibits a nearly linear variation with rod length or number of base pairs, n. Consider a long rod translating perpendicular to its long axis. Ion relaxation results in a perturbed dipolar charge distribution that is essentially constant with position along the rod except at the ends (Stigter, 1978b, Allison et al., 1999). Hence, the additional frictional drag required to maintain that perturbed charge distribution should vary linearly with length. For a long rod translating parallel to its axis, the perturbed charge densities nearly vanish, except at the ends of the rod (Allison, et al., 1999). If we make the reasonable assumption that the perturbed charge densities at the rod ends are independent of rod length, then the frictional drag force required to maintain that charge distribution should be independent of length. Figure 2 is consistent with this simple physical picture.

View larger version (7K): [in this window] [in a new window]   FIGURE 2   i versus n. Diamonds correspond to parallel and squares to perpendicular difference components. The model conditions are condition V (full charge DNA, 0.11 M KCl, 20°C).

Table 2 summarizes the fractional decrease in orientationally averaged translational diffusion constants due to ion relaxation as a function of length (n = 20-100 bp) for the six conditions discussed previously. Under the conditions considered, the electrolyte friction effect is predicted to be approximately 10% or less. Increasing the polyion linear charge density, reducing the length of the polyion, or reducing the salt concentration increases the effect. Also, the effect varies inversely with the mobility of the small ions (comparing conditions II-IV).

View this table: [in this window] [in a new window]   TABLE 2   Electrolyte friction effect versus length under different conditions (100 (1  D(r)/D(nr)))

To put the electrolyte friction effect on a more quantitative footing, the basic mathematical form for the diffusion constants of rodlike polyions, Eqs. 6-7, are retained, but the correction polynomials, , , and , are determined with the inclusion of ion relaxation. As discussed elsewhere (Allison and Mazur, 1998), the rod models themselves are designed to reproduce Eqs. 6-7 when ion relaxation is not included (Garcia de la Torre and Bloomfield, 1981; Tirado et al. 1984), so deviations in the polynomial coefficients of Eq. 7 from the no ion relaxation values reflects the influence of ion relaxation. These polynomial coefficients are summarized in Table 3 for the six conditions discussed previously. Figure 3 shows the behavior of  (diamonds, no ion relaxation; squares, ion relaxation) versus length for condition V, and the solid lines through the data points represent the polynomial fits.

View this table: [in this window] [in a new window]   TABLE 3   Polynomial coefficients for , , and

View larger version (8K): [in this window] [in a new window]   FIGURE 3   versus 1/p. The correction polynomial, , (Eqs. 6 and 7) for translational diffusion in the absence (diamonds) and presence (squares) of ion relaxation. Solid lines represent quadratic fits.

	AVERAGE SURFACE POTENTIAL AND TETHER FORCE TENSOR

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Equilibrium electrostatic potentials of infinitely long (Stigter, 1975) and finite (Allison, 1994) rod models in the framework of the nonlinear Poisson-Boltzmann equation have been available for some time. The reduced surface potential averaged over the polyion surface, y_S (see Eq. 16), is computed at the start of a boundary element (BE) modeling study, and its value for the different lengths under the six conditions studied in this work are summarized in Table 4. At equilibrium, conditions II-IV are equivalent because the ion size enters the model only through their diffusion constants, D, appearing in Eq. 10. These exhibit roughly a linear dependence on 1/L or 1/p (p = L/2R where L is the rod length and R the radius). The infinite rod, or 1/p  0 limit is given in the far right column of Table 4. Although there is considerable variation in y_S, depending on conditions, more uniformity is observed if y_S is divided by the infinite rod limit under the appropriate conditions, y. This is illustrated in Fig. 4, where w is defined as y_S/y. The near linear behavior of y_S or w with 1/p is consistent with a simple two-state model in which the local surface potential is y_end near the ends of the rod and y in the interior. Let  denote the distance from the ends of the rod where the surface potential is best described by y_end. The fraction of the rod surface where the reduced surface potential is y_end is then 2/L, and the fraction of the rod surface where the reduced surface potential is y is 1  2/L. Thus, the overall average reduced surface potential can be written as

(19)

or equivalently,

(20)

The data in Table 4 and Fig. 4 exhibit the length dependence predicted by the simple two-state model described above. At first glance, it might appear unusual that the slopes of w versus 1/p are similar for cases II-VI (full-charge DNA models) despite the wide range of solvent conditions. However, the counterion concentration near the surface of uniform, highly charged polyions is nearly independent of the bulk ion concentrations (Rouzina and Bloomfield, 1996). Thus, we would expect  to be independent of solvent/buffer conditions for a particular charge model and hence the slope of w versus 1/p similar for conditions II-VI.

View this table: [in this window] [in a new window]   TABLE 4   yS Versus length under different conditions

View larger version (8K): [in this window] [in a new window]   FIGURE 4   Variation in average surface potential with length. Diamonds, condition I; filled squares, conditions II-IV; triangles, condition V; and ×, condition VI.

Attention is now turned to the tether force tensor, Z. In a reference frame whose axes correspond to the principal axes of the rod, Z will be diagonal. However, the components parallel, Z, and perpendicular, Z, to the rod axis will be different, in general. As discussed previously, these components cannot be equated to the bare polyion charge, or the electrophoretic charge, which is the bare charge of the polyion itself plus any ions trapped within the hydrodynamic shear surface (Schellman and Stigter, 1977; Stigter, 2000). Also, it is not a thermodynamic charge such as that extracted from the preferential interaction coefficient determined in equilibrium dialysis experiments (Olmstead et al., 1989), because it depends on such nonequilibrium quantities as the solvent viscosity and small-ion diffusion constants or mobilities. The closest direct experimental measurement of Z comes from electrophoretic stretch experiments on high molecular-weight circular plasmid DNAs that are held, or tethered, in place by agarose fibers threaded through their centers (Smith and Bendich, 1990). Theories have been developed to explain electrophoretic stretch (Schurr and Smith, 1990; Stigter, 1991; Stigter and Bustamante, 1998), and the electrophoretic force of Stigter and Bustamante is equivalent to z⁽²⁾ in the present work. Provided that its precise definition is kept in mind, the tether force tensor, Z, should be a useful transport property in the same sense as the translational friction tensor, _t.

Following procedures similar to those described previously for friction/diffusion, we consider fits to X_i defined by Eq. 12 or fits to the orientationally averaged X = 2n/Z, where Z = (Z + 2Z)/3. Figures 5, 6, and 7 show the length dependence and linear fits (see Eq. 12) to X, X, and X, respectively for condition V (full charge DNA, 0.11M KCl at 20°C). Diamonds correspond to the case of no ion relaxation and squares to the case with ion relaxation. Dotted lines represent linear fits to the simple functional form of Eq. 12. One could interpret 1/X as an absolute effective charge per base pair of DNA. (For reasons discussed elsewhere, however, that terminology is discouraged (Stigter, 2000)). From Fig. 7, it is clear that there is a pronounced length dependence of 1/X, and this behavior is seen under all of the conditions studied in this work. The longer the DNA, the lower the absolute effective charge per base pair, and the higher the fraction of associated counterions. This trend is qualitatively consistent with equilibrium effective charges extracted from preferential interaction coefficients (Olmstead et al., 1989). The Case 2 transport coefficients defined by Eq. 12 for all six conditions are summarized in Table 5. With the data contained in Tables 3 and 5, Eqs. 13 and 14 can be used to estimate the electrophoretic mobility for the six conditions considered for rod models of any length. We shall return to this point later.

View larger version (8K): [in this window] [in a new window]   FIGURE 5   Variation in X with length. Triangles correspond to the case without ion relaxation and diamonds to the case with ion relaxation. The model conditions are condition V (full charge DNA, 0.11 M KCl, 20°C).

View larger version (7K): [in this window] [in a new window]   FIGURE 6   Variation in X with length. Triangles correspond to the case without ion relaxation and diamonds to the case with ion relaxation. The model conditions are condition V (full charge DNA, 0.11 M KCl, 20°C).

View larger version (8K): [in this window] [in a new window]   FIGURE 7   Variation in X with length. Triangles correspond to the case without ion relaxation and diamonds to the case with ion relaxation. The model conditions are condition V (full charge DNA, 0.11 M KCl, 20°C).

	REDUCED MOBILITIES

TOP ABSTRACT INTRODUCTION METHODS AVERAGE SURFACE POTENTIAL AND... REDUCED MOBILITIES COMPARISON WITH EXPERIMENT SUMMARY REFERENCES

Rather than fitting the model mobilities directly, we instead fit the reduced mobilities defined by Eq. 15. The length dependence of actual mobilities and a comparison of model and experimental values is given in the next section. Ion relaxation affects the parallel and perpendicular mobilities differently, and these differences depend on length. Consider first the parallel component for condition V (full charge DNA, 0.11 M KCl at 20°C) where C(nr) (upper curve) and C(r) (lower curve) versus length are plotted in Fig. 8. C(nr) varies gradually with 1/p and extrapolates to a value of 0.970 in the limit 1/p  0. This is in fairly good agreement with the Smoluchowski value of 1.00 expected for a polyion of uniform surface potential whose linear dimensions are large compared to the thickness of the ionic double layer (Smoluchowski, 1921). To a good approximation, C(r) approaches the no ion relaxation limit as 1/p  0 as postulated previously (Stigter, 1978b). The solid curves in Fig. 8 represent quadratic fits to the six models studied ranging from 20 to 100 bp. The fact that C(r) and C(nr) do not extrapolate to precisely the same value as 1/p  0 is a consequence of the small number of lengths considered and the functional form of the fitting polynomial. If, for example, the data is fit as a quadratic function of 1/ln(p) rather than 1/p, then C(nr) extrapolates to 1.06 and C(r) extrapolates to 1.17 as 1/ln(p)  0. Additional modeling studies on longer rods would provide more accurate estimates of the long-rod behavior of both C(nr) and C(r), but such studies are not really necessary and would be costly. The equivalence of C(nr) and C(r) in the long-rod limit can be readily understood on physical grounds because ion relaxation for a long-rod polyion undergoing steady-state electrophoresis parallel to the rod axis will be confined to the rod ends (Stigter, 1978b; Allison et al., 1999). Thus, the reduction in parallel mobility due to ion relaxation should go to zero for sufficiently long rods, and that trend is seen at least qualitatively in Fig. 8.

View larger version (11K): [in this window] [in a new window]   FIGURE 8   Variation in C with length. Triangles correspond to the case without ion relaxation and diamonds to the case with ion relaxation. The model conditions are condition IV (full charge DNA, 0.11 M KCl, 20°C).

Very different behavior is seen for the "without relaxation" (top curve) and "with relaxation" (lower curve) perpendicular components in Fig. 9. C(nr) and C(r) do not extrapolate to the same value as 1/p  0. In this case, ion relaxation is not confined to the rod ends, but occurs over the entire length of the rod (Stigter, 1978a). On average, all orientations will be sampled in an experiment and we can conclude that ion relaxation reduces the average mobility of rodlike polyions no matter how long they are, although the effect is expected to be more significant for short rods than for long ones. The polynomial coefficients for the C_i under the six conditions considered in this paper are summarized in Table 6. For the sake of brevity, only the cases with ion relaxation are included.

View larger version (10K): [in this window] [in a new window]   FIGURE 9   Variation in C with length. Triangles correspond to the case without ion relaxation and diamonds to the case with ion relaxation. The model conditions are condition IV (full charge DNA, 0.11 M KCl, 20°C).

	COMPARISON WITH EXPERIMENT

TOP ABSTRACT INTRODUCTION METHODS AVERAGE SURFACE POTENTIAL AND... REDUCED MOBILITIES COMPARISON WITH EXPERIMENT SUMMARY REFERENCES

Measurements of the free solution electrophoretic mobility of 27-bp and longer duplex DNA fragments (Stellwagen et al., 1997) were carried out in 0.04 M Tris-acetate aqueous buffer at 25°C and pH = 8.0. More recently, the same workers reported the mobility of an 18-bp duplex DNA under the same conditions (Stellwagen et al., 1999). Detailed modeling of the 18-bp fragment (Mazur et al., 2001) led us to conclude that a charge model of DNA in which the bare phosphate charges are reduced by 50% reproduced the experimental mobility quite well. Physically, this corresponds to a net charge of about +18 (neutralization of half of the DNA phosphates) from the Tris⁺ cations being bound or entrapped within the hydrodynamic shear surface of the DNA. Thus, we expect the condition I modeling results to be the most appropriate to use in a comparison with the experimental results. The diamonds in Fig. 10 correspond to the condition I model mobilities and the solid line is a fitting curve through these points. The dashed line at µ = 4.24 (in 10⁴ cm²/V s) is an independent model mobility for an infinitely long cylinder using the same polyion radius/linear charge density and solvent conditions used in the BE modeling studies (Stigter, 1978a,b). The underlying methodology used by Stigter and the present BE modeling studies is essentially the same, so the mobilities should agree in the limit of very long rods, and this is confirmed by Fig. 10. The triangles represent experimental mobilities (Stellwagen et al., 1997, 1999). Up to ~60 bp, model and experimental mobilities are in good agreement with each other. However, the model mobilities do not appear to level off with increasing length to the extent seen in experiment. This trend is also seen under the other conditions modeled.

View larger version (9K): [in this window] [in a new window]   FIGURE 10   Length dependence of µ. Comparison of theory and experiment. Diamonds and solid curve, condition I modeling for straight rods; squares, condition I modeling for bent rods; triangles, experiments (Stellwagen et al., 1997). The dashed line is the model mobility of an infinitely long straight rod (Stigter, 1978a,b) under the same conditions as the BE modeling.

A possible explanation could be that bending of DNA is responsible for the leveling off of µ for long fragments. The persistence length, P, of DNA is approximately 50 nm, or 147 bp, though this value exhibits some variation with salt concentration (Rizzo and Schellman, 1981). Thus, for DNA shorter than, say, P/2 (approximately 80 bp), a straight rod model would be expected to be a realistic representation of the fragment. For longer DNAs, a model that accounts for bending would be more realistic. One approach would be to generate an ensemble (100, for example) of different wormlike chain conformations, model the µ of each, and average the results (Hagerman and Zimm, 1981). Unfortunately, this is not feasible due to the long simulation times involved. Computation time varies roughly as the square of the number of plates comprising the structure, and, for a 240-plate straight rod model (typical of 60-bp DNA), a complete calculation of diffusion and electrophoretic mobility with ion relaxation requires approximately two days of single processor time on a Silicon Graphics 4D-380-SX computer. A simpler approach is to generate a single bowed structure that reflects the curvature expected in an actual DNA fragment and compute the mobility for that structure, and that is the procedure followed in the present work. For a wormlike chain, the mean square end-to-end distance, R², is related to contour length, L, and persistence length by the relation (Hagerman and Zimm, 1981, or most texts relevant to polymer theory),

(21)

A single bent structure is generated with sufficient curvature to yield the same R² as predicted by Eq. 21. Two capped cylinder models appropriate for 100-bp DNA are shown in Fig. 11. The upper figure is a straight rod model and the lower a bent rod with a radius of curvature that gives the same R² as a wormlike chain with P = 50 nm and L = 34 nm. Boundary element modeling is then carried out on the bent structure, but it is necessary to consider case 1 and case 2 transport along three orthogonal directions. The filled squares in Fig. 10 represent the extrapolated shell mobilities of bent cylinder models corresponding to P = 50 nm for 60-, 80-, and 100-bp DNAs. These results show that the model mobilities do indeed level off and actually start to decrease. Actual duplex DNAs are not smoothly bending rods such as the bent rod models considered here. Nonetheless, these results do show that rod curvature causes µ to be slightly lower than for the corresponding straight rod. Thus, we conclude that the leveling off of µ for long fragments may be attributed, in part, to bending of the DNA.

View larger version (22K): [in this window] [in a new window]   FIGURE 11   Straight and bent rod capped-cylinder models for 100-bp DNA. The bent structure has the same mean-square end-to-end distance as a wormlike chain of contour length = 34 nm and persistence length = 50 nm.

The effects of global conformation on mobility have also been examined for condition III (0.02 M NaCl at 20°C, full charge model). For straight rods, the orientationally averaged mobilities in the presence of ion relaxation, µ(r) (in 10⁴ cm²/V s), are 4.01, 4.13, 4.22, and 4.32 for 40, 60, 80, and 100 bp, respectively. These are substantially higher than the value of 3.84 inferred from Fig. 11 of Hoagland et al. (1999) for high molecular-weight DNA under the same conditions. However, the bent rod (P = 50 nm) model mobilities for 60, 80, and 100 bp are 4.05, 4.00, and 3.89, respectively. As in the condition I studies considered previously, accounting for DNA bending in the approximate manner of the present work helps to reconcile theory and experiment with regards to the mobilities on long duplex DNA. However, without mobility experiments as a function of length similar to those of Stellwagen et al. (1997), but under different salt/buffer conditions, it is difficult to quantify the effect of bending on length. If we assume the leveling off in µ with length is similar for the six conditions studied in this work, then model mobilities corresponding to 40-60 bp should be more appropriate to compare with experimental mobilities than with model mobilities from very long rods. In 0.02 M NaCl at 20°C and fully charged, straight rod DNA models, the 40- and 60-bp µ(r) are 4.4 and 7.6% higher than the experimental value of 3.84 (Hoagland et al., 1999). In 0.2 M KCl at 1.3°C (condition VI), model mobilities for 40- and 60-bp DNA fragments are 1.96 and 2.00 for 40 and 60 bp. These are 6.5 and 8.7% higher than the experimental mobility of 1.84 (Ross and Scruggs, 1964). Also in KCl, but at a concentration of 0.11 M and at 20°C, a detailed analysis of 20-bp DNA gave a capped-cylinder, model mobility that was larger than experiment by 5.5% (Mazur et al., 2001). In this case, the charge model was a full charge model, but the charges were distributed on a double-helical spiral rather than on a line at the center of the rod axis. A more realistic helical charge distribution reduces the mobility (Allison and Mazur, 1998), but the effect is small and amounts to 1-2%. Also, going to a more realistic surface model reduces mobilities slightly, but again the reduction is in the 1-2% range. The experimental mobilities of long duplex DNA in monovalent alkali salts (Ross and Scruggs, 1964; Hoagland et al., 1999) are reasonably well explained by the model studies once account is taken of the long-range bending of high molecular-weight DNA and the simplifying assumptions of the model studies (line charge, smooth cylinder surface model of polyion).

	SUMMARY