In the Effective Medium, EM, model, the fluid is assumed to obey the Brinkman [18] and solvent incompressibility equations defined by (1) η ∇ 2 v ¯ ( r ¯ ) − ∇ ¯ p ( r ¯ ) = η λ 2 v ¯ ( r ¯ ) (2) ∇ ¯ ⋅ v ¯ ( r ¯ ) = 0 Where η is the solvent viscosity, v̲(r̲) is the local fluid velocity at point r̲, p is the local pressure, and λ (units of 1/length) is the gel screening parameter. This parameter can be related to the gel concentration, M, (in gm dry gel material per gm of solvent) and gel fiber radius, r_f, by the relation [16,25] (3) 1 λ 2 = − 3 ρ g ω s r f 2 20 M [ ln ( M ρ g ω s ) + 0.931 ]

In Equation (3), ρ_g denotes the mass density of dry gel (which equals 1.64 gm/mL for agarose [26]), and ω_s denotes the ratio of dry gel volume to hydrated gel volume (which equals 0.625 for agarose [27]). The term on the right hand side of Equation (1) represents an external force/unit volume due to the viscous drag on the fluid produced by the presence of the gel. In general, other external forces on the fluid may be present as well. This is particularly true in modeling the transport of macroions in external electric fields (electrophoresis) [28]. However, the cases of interest in the present work involve rotational relaxation in the absence of an external electric field. Under these conditions, the additional external force terms can be ignored for a good approximation.

For later reference, it will be useful to consider the local fluid velocity and pressure of a spherical particle of radius “a” rotating about its center with angular velocity ω̲ in an EM that is at rest far from the particle. In this case it is straightforward to solve Equations (1) and (2) and obtain p(r̲) = constant and (4) v ¯ ( r ¯ ) = ( a r ) 3 ( 1 + λ r 1 + λ a ) e − λ ( r − a ) ( ω ¯ x r ¯ )

In Equation (4), “x” denotes the vector cross product. Note that the fluid velocity falls off rapidly moving away from the rotating sphere. The local stress tensor,σ͇(r̲), is related to the velocity and pressure by [29,30] (5) σ ̳ ( r ̲ ) = - p ( r ̲ ) I ̳ + η ( ∇ v ̳ + ∇ v ̳ T )Where I̳ is the 3 by 3 identity tensor (̳I̳)_jk = δ_jk (δ_jk is the Kronecker delta)), ( ∇ v ̳ ) j k = ∇ j v k ( r ¯ ), and ( ∇ v T ̳ ) j k = ∇ k v j ( r ¯ ) For a point r̲= a n̲(r̲) on the surface of the sphere (n̲(r̲) is a local outward (into the fluid) normal to the sphere), the local force/area exerted by the sphere of the fluid, f̲(r̲), is related to the local stress by (6) f ¯ ( r ¯ ) = − σ ̳ ( r ¯ ) ⋅ n ¯ ( r ¯ ) = η γ ( λ a ) ( ω ¯ x n ¯ ( r ¯ ) ) (7) γ ( λ a ) = ( 3 + 3 λ a + λ 2 a 2 ) ( 1 + λ a ) If the sphere were rotating as a rigid body about some point different from its center, then f̲(r̲) would be different. Let r * _ denote the position of the center of the sphere relative to the center of rotation. For a good approximation, we can write (8) f ¯ ( r ¯ ) ≅ F ¯ S + η γ ( λ a ) ( ω ¯ x n ¯ ( r ¯ ) ) In Equation (8) F̲ denotes the net instantaneous force exerted by the bead on the fluid and S = 4πa² is the surface area of the bead. The net force, in turn, equals 6 π η a ( ω _ x r * _ )

In a recent analysis employing the Boundary Element method, a general expression was derived for the local fluid velocity of an array of non overlapping beads translating with uniform velocity through an Effective Medium that obeys Equations (1,2) [16]. It is straightforward to extend that analysis to the rotation of a bead array. As illustrated in Figure 1, the macromolecule is modeled as an array of N non overlapping beads in which their radii, {a_j}, and relative positions are arbitrary. Let r * _ denote the center of rotation of the rigid body bead array, ω̲ the angular velocity of the bead array, and x̲_J and a_J the centroid position vector and radius of bead J. The fluid is assumed to be at rest far from the array. It shall also be assumed that for a point, r̲, on the surface of bead J, we can write (9) f ¯ ( r ¯ ) ≅ F ¯ J S J + η γ ( λ a J ) ( ω ¯ x n ¯ ( r ¯ ) ) Where F̲_J is the net force exerted by bead J on the fluid and S_J is the corresponding surface area. The second term on the right hand side of Equation (9) is similar to the “Volume Correction” approach of Garcia de la Torre and Rodes [31].

It is worthwhile to briefly discuss the physical basis of Equation (9). Hydrodynamic interaction, HI, between two beads arises as a consequence of their relative motion through a viscous fluid. In the absence of a gel, the disturbance of the fluid velocity produced by one bead, J, centered at r̲_J, at the site of a second bead, K, centered at r̲_K separated by distance r̲_JK=|r̲_JK| = |r̲_J−r_K|, falls off as 1/r_JK if their centers are in relative motion [29,30]. On the other hand, if their centers are not in relative motion, but if one or both beads are rotating about their centers, then the disturbance falls off as 1/r_JK ³, which is clear from Equation (4). The presence of a gel modifies these distance dependent interactions, but the important point is that long range HI is determined primarily by the relative translational motion of the bead centers. The dominant long range HI between the beads is contained implicitly in the first term on the right hand side of Equation (9). The second term reflects the fluid stress arising from the rotation of bead J itself relative to the fluid. It should be emphasized that Equation (9) is approximate and ignores shorter range HI interactions.

Following our earlier analysis [16] of a translating bead array, the corresponding results for a rotating bead array can be written (10) ω ¯ x ( r ¯ K − r ¯ ∗ ) = ∑ J = 1 N [ C ̳ K J ⋅ ( r ¯ J − r ∗ ¯ ) + 1 ζ J H ̳ K J ⋅ F ¯ J ] + ∑ J = 1 N ω ¯ x p ¯ K J Where ζ J = 6 π η a J (11) C ̳ K J = { 1 3 λ a K [ ( λ a K − 1 ) + e − 2 λ a K ( λ a K + 1 ) ] I ̳ ( K = J ) 2 9 λ 3 a J 3 i 0 ( λ a K ) [ i 0 ( λ a J ) k 0 ( λ r K J ) I ̳ + 3 i 2 ( λ a J ) k 2 ( λ r K J ) [ 1 r K J 2 r K J r K J − 1 3 I ̳ ] ] − [ 1 3 a J 3 ( 3 v 3 − ( 1 + λ 2 ( a K 2 6 + a J 2 10 ) ) ( 3 w 3 + 3 λ w 2 + λ 2 w 1 ) ) ] [ 1 r K J 2 r K J r K J ̳ − 1 3 I ̳ ] ( K ≠ J ) } (12) H ̳ K J = { i 0 ( λ a K ) e − λ a K I ̳ ( K = J ) λ a J i 0 ( λ a K ) i 0 ( λ a J ) k 0 ( λ r K J ) I ̳ + 3 a J 2 λ 2 [ 3 v 3 − ( 1 + λ 2 ( a J 2 + a K 2 ) 6 ) ( 3 w 3 + 3 λ w 2 + λ 2 w 1 ) ] [ 1 r K J 2 r K J r K J − 1 3 I ̳ ] ( K ≠ J ) } (13) w n = e − λ r K J r K J n (14) v n = 1 r K J n (15) p ¯ K J = i 0 ( λ a K ) ( a J r K J ) 3 e − λ ( r K J − a J ) ( 1 + λ r K J 1 + λ a J ) ( r ¯ K − r ¯ J )

In Equations (11,12), i_n and k_n are modified spherical Bessell functions. Specifically, i₀(z) = sinh(z)/z, i₂(z) = sinh(z)/z − 3cosh(z)/z² + 3sinh(z)/z³, k₀(z) = e^−z/z, k₂(z) = e^−z(1/z+3/z²+3/z³). Also, ( r K J r K J ̳ ) j k = ( r ¯ K − r ¯ J ) j ( r ¯ K − r ¯ J ) k.

The typical procedure followed in a resistance problem [29,30,32-34], as applied to the more general problem in an EM, is to compute the elements of C̳_KJ and H̳_KJ once the geometry of a bead array and conditions of the EM are defined. The bead array is then rotated about three orthogonal axes with unit angular velocity, (16) ω ¯ ( p ) = e ¯ p Where e̲_p is a unit vector along axis p (p = 1, 2, or 3) in some convenient frame of reference. Let F ¯ J ( p ) denote the net force exerted by bead J on the fluid when the array is rotated about axis p. Then Equation (10) can be written (17) ∑ J = 1 N 1 ζ J H ̳ K J ⋅ F ¯ J ( p ) = A ¯ K ( p ) (18) A ¯ K ( p ) = e ¯ p x ( r ¯ K − r ¯ ∗ ) − ∑ J = 1 N [ C ̳ K J ⋅ ( e ¯ p x ( r ¯ J − r ¯ ∗ ) ) + e ¯ p x p ¯ K J ] It is helpful to view F ¯ J ( p ) and A ¯ k ( p ) as 3 by 1 column vectors. From these, we can define 3N by 3 super matrices, F̳ and A̳, by (19) F ̳ = ( F ¯ 1 ( 1 ) F ¯ 2 ( 1 ) ⋮ F ¯ N ( 1 ) F ¯ 1 ( 2 ) F ¯ 2 ( 2 ) ⋮ F ¯ N ( 2 ) F ¯ 1 ( 3 ) F ¯ 2 ( 3 ) ⋮ F ¯ N ( 3 ) ) The super matrix, A̳, has a very similar form to Equation (19) above. Also, the H̳_KJ terms defined by Equation (12) are known 3 by 3 matrices. We can define a 3N by 3N super matrix G̳, by (20) G ̳ = ( H ̳ 11 / ζ 1 H ̳ 21 / ζ 1 ⋮ H ̳ N 1 / ζ 1 H ̳ 12 / ζ 2 H ̳ 21 / ζ 2 ⋮ H ̳ N 2 / ζ 2 ⋯ ⋯ ⋱ ⋯ H ̳ 1 N / ζ N H ̳ 2 N / ζ N ⋮ H ̳ N N / ζ N )

In compact notation, Equation (16) can be written (21) G ̳ ⋅ F ̳ = A ̳

The matrix G̳, is invertible and let C̳⁻¹ denote the inverse. Then (22) F ̳ = G ̳ − 1 ⋅ A ̳

A very similar relation was derived previously for a bead array translating with uniform velocity, u̲.

Before it is possible to compute various resistance tensors from modeling, it is necessary to compute the total force, F̲_tot, and total torque, T̲_tot(r̲*), exerted by the bead array on the fluid if it is translated with velocity, u̲, and rotated with angular velocity, ω̲, about some point, r̲*, in the fluid which is at rest far from the array. In the present work, r̲* is chosen as the center of mass of the bead array. The total force and torque can be written [35] (23) F ¯ tot = Ξ ̳ T . u ¯ + Ξ ̳ C T ( r ¯ ∗ ) . ω ¯ (24) T ¯ tot ( r ¯ ∗ ) = Ξ ̳ C ( r ¯ ∗ ) . u ¯ + Ξ ̳ R ( r ¯ ∗ ) . ω

Where Ξ̳_T, Ξ̳_R (r∗), and Ξ̳_C(ṟ∗) denote translation, rotation, and coupling resistance tensors, respectively. Quantities with argument “r̲*” depend on the choice of r̲*. Also, the “T” superscript on the coupling tensor in Equation (23) denotes transpose. F̲_tot and T̲_tot(r̲*) can be obtained from modeling by the relations (25) F ¯ tot = ∑ J = 1 N ∫ S J d S r f ¯ ( r ¯ ) = ∑ J = 1 N F ¯ J (26) T ¯ tot ( r ¯ ∗ ) = ∑ J = 1 N ∫ S J d S r ( r ¯ − r ¯ ∗ ) x f ¯ ( r ¯ )

For a bead array that is not translating (u̲ = 0̲), but is rotating with angular velocity e̲_p about r̲*, (27) T ¯ ( p ) ( r ¯ ∗ ) = ∑ J = 1 N [ ( r ¯ J − r ¯ ∗ ) x F ¯ J ( p ) + 8 π a j 3 3 η γ ( λ a J ) e ¯ p ]

From Equations (23,24), the net force and net torque give the p-th row of Ξ̳_C(r_∗), and p-th column of Ξ̳_R(r_∗), respectively. The components of Ξ̳_T are obtained by translating the array along three orthogonal axes, computing the net forces, and then using Equation (23).

The connection between the resistance tensors and corresponding mobility or diffusion tensors is well known [29,30,35,36]. The origin dependent translational diffusion tensor, D̳_T(r_∗), and origin independent rotational diffusion tensor, D̳_R, are given by (28) D ̳ T ( r ¯ ∗ ) = k B T [ Ξ ̳ T − Ξ ̳ C T ( r ¯ ∗ ) . Ξ ̳ R − 1 ( r ¯ ∗ ) . Ξ ̳ | C ( r ¯ ∗ ) ] − 1 (29) D ̳ R = k B T Ξ ̳ R − 1 (30) Ξ ̳ R = [ Ξ ̳ R ( r ¯ ∗ ) − Ξ ̳ C ( r ¯ ∗ ) . Ξ ̳ T − 1 . Ξ ̳ C T ( r ¯ ∗ ) ]

In modeling, the origin independent rotational friction tensor, Ξ̳_R can be diagonalized and let Λ_n′ (n = 1, 2, or 3) denote the n-th eigenvalue. We shall define the eigenvalues such that Λ₁′ ≥ Λ₂′ ≥ Λ₃′. Also define the reduced dimensionless eigenvalue for an array of N identical beads of radius a, (31) Λ n = Λ n ′ 8 π η a 3 N

The denominator in Equation (31) represents the rotational friction factor of a sphere of volume equal to that of our bead array.

The modeling results of the present work shall be given in terms of these dimensionless eigenvalues. These, in turn, can be related to the eigenvalues of Ξ̳_R and D̳_R through Equations (29,31). We also want to relate these to the longest lifetime, τ_l, of the “off-field” electric birefringence decay of a dilute solution of macromolecules. We can write [12,37] (32) τ l = 1 6 D R 1 = 4 π η a 3 N 3 k B T Λ 1

A key element in the parameterization of the bead radii of our “coarse grained” models is to find arrays of identical beads that are able to reproduce the rotational friction, Ξ̳_R, or diffusion, D̳_R, tensor of the actual structure to considerable accuracy. From Section 2.1, this is equivalent to matching the reduced eigenvalues, {Λ_n}. Furthermore, since experiments such as electric birefringence [12,37] or dichroism [38,39] are sensitive to particular eigenvalues (Λ₁ for example), then we can focus on matching that particular eigenvalue.

This shall be illustrated for the special case of a right circular cylinder of length L and axial radius, R. For the right circular cylinder in an EM with λ = 0 [40] (33) Λ 1 rod ( L , R ) = L 2 18 R 2 [ ln ( p ) + δ ⊥ ( p ) ] (34) p = L 2 R (35) δ ⊥ ( p ) = − 0.662 + 0.917 / p − 0.050 / p 2

For the corresponding array made up of a linear string of N touching beads of radius a, we shall set (36) L = 2 a ( N − c )

In Equation (36), a and c are left as adjustable parameters but it is assumed that a is proportional to R. For an array made up of N beads, Λ₁ from Equation (31) is independent of η and a, and is computed by the procedure described in Section 2.1. The quantity ∧₁^rod from Equation (33) is independent of η and only depends on the ratio, L/R, or equivalently L/a. For assumed values of a and c, Equation (36) then gives us a direct correspondence between L, N, a, and c. It is straightforward to construct an Excel spreadsheet in which a and c are defined as input parameters. ∧₁^rod and Λ₁ are compared for a range of N values. (In the present work, we are interested in long rods and N is varied from 20 to 100). In the fitting procedure, we start by setting c equal to 0 and then vary a until the sum of the square of the differences, SR, between ∧₁^rod and Λ₁ is minimized. Then, c is incremented by a small amount and the procedure is repeated. This process is continued until that combination of a and c is found that minimizes SR overall. For a good approximation, this is given by c = 0.20 and a/R = 1.20.

We can apply this duplex DNA made up of n_bp base pairs and L (in nm) = 034n_bp. Also, R is 1.0 ± 0.1 [41] and this parameter shall be set equal to 1.0 nm in the present work. This model is very similar to that of Hagerman and Zimm [42] and involves minor corrections in the choice of the a and c parameters. Unless the duplex DNA is very short, however, it is better to model the DNA as a wormlike chain of persistence length P (contour length equal to L and axial radius equal to R) rather than a right circular cylinder. For DNA, P is typically in the 50 nm size range, but this varies with ionic strength [43,44]. As an illustrative example, L = 211.5 nm for 622 bp DNA which is more than 4 persistence lengths long. Also, for a = 1.206 and c = 0.20, Equation (36) requires N be set to 88. For DNA fragments of this size, a rigid rod model is inadequate. Fortunately, it is straightforward to generalize the “linear string” model of the previous paragraph to a “discrete wormlike chain” model that has been widely used in the past [42,45]. Let <cos θ> = <e̲_J. e̲_J+1 > where e̲_J denotes the unit vector along the J-th virtual bond of our discrete wormlike chain model and brackets denote an ensemble or long time average. If the distance between adjacent beads is 2a, then [42,43,45] (37) p = 2 a 1 − < cos θ > Using random number generators, it is straightforward to generate chains that satisfy this condition [42,45]. Since overall conformational features, such as end to end distances, vary greatly from one randomly generated chain to another (when L is significantly larger than P), it is necessary to average transport results over many different chains in order to obtain good statistics.

At this point, we can ask whether or not it is reasonable to equate average transport properties, such as <Λ₁>, derived from model studies of ensembles of chains “frozen” in their starting configuration, to the actual transport properties of flexible particles. The answer to this question depends, in part, on what is actually measured in a particular experiment. Considerable attention to this point has been given to the “off field” electric birefringence or dichroism decay of duplex DNA [12,37,38,42]. In general, the decay is multi-exponential and consists of end over end tumbling as well as more complex “internal” decay processes [46]. However, provided we are interested in the slowest decay process, which is also the decay process of greatest amplitude for comparatively short DNA fragments, equating Λ₁ in Equation (32) to the average, <Λ₁>, obtained from an ensemble of “frozen” chains, is expected to be an accurate approximation [46].

Figure 2 illustrates the equivalence between the “end-over-end” reduced rotational friction coefficient, Λ₁, of a right circular cylinder and a linear string of touching beads in the absence of a gel (λ = 0). The solid line represents the right circular cylinder of length L and axial radius R and is computed from Equation (33) [40]. The filled squares, computed using the procedure described in Section 2.1, are for a linear string of N touching beads of radius a with a/R = 1.207. The length, L is related to a and N by Equation (36) with c = 0.20.

We would next like to consider the effect of the gel on Λ₁. For a sphere of radius a in an EM, it is straightforward to show (using Equations (6,7,24,26)), that (38) Λ 1 Λ 1 n g − 1 = λ 2 a 2 3 ( 1 + λ a )where ∧₁^ng is the reduced friction factor for a rotating sphere in the absence of a gel (which equals 8πηa³). For a linear string of touching beads (radius = a), we computed Λ₁ for N ranging from 20 to 100 and λa ranging from 0 to 0.604 using the procedure described in Section 2.1. To within an accuracy of ±2% over the entire range of N and λa, the data can be fit with the following simple semi-empirical form: (39) Λ 1 Λ 1 n g − 1 = d 1 x 2 + d 2 x 3 (40) x = ln ( 1 + γ ) (41) γ = λ R e = λ a ( N Λ 1 n g ) 1 / 3 (42) d 1 = 0.534 − 0.049 ln ( N ) − 0.004 ( ln ( N ) ) 2 (43) d 2 = − 0.001

Above Λ 1 n g represents the reduced eigenvalue of the bead array in the absence of a gel (λ = 0), and R_e is the radius of a sphere that has the same reduced rotational friction coefficient as the bead array in the absence of a gel.