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In this work, the retarding influence of a gel on the rotational motion of a macromolecule is investigated within the framework of the Effective Medium (EM) model. This is an extension of an earlier study that considered the effect of a gel on the translational motion of a macromolecule [Allison, S.

The subject of biomolecular transport in congested media is of vital interest in such diverse subjects as drug delivery across membranes and the sieving action of a gel in electrophoresis. In particular, diffusion (translational and rotational) has been studied in a wide range of environments including the cytoplasm of cells [

In the EM model, the “fluid” surrounding the particle is treated as a hydrodynamic continuum, and includes both solvent and the “gel” support medium. A special screening term is added to the external force/volume on the fluid in the low Reynolds number Navier-Stokes equation that accounts for the presence of a gel. The resulting equation is what we call the Brinkman equation. Starting from a microscopic model, Felderhof and Deutch were able to derive the Brinkman equation as a mean field approximation [

The principle objective of the present work is to extend our earlier study of the translational diffusion of a macromolecule modeled as an array of non overlapping beads in an EM to the case of rotational diffusion. In Section 2.1, the Brinkman equation is introduced and the rotation of a single sphere in an EM is discussed. This is then extended to the more complex problem of an array of non-overlapping beads. In Section 2.2, we focus first on the parameterization of a linear string of touching beads and later extend that to a wormlike chain model. The wormlike chain model is relevant to modeling the electric birefringence or dichroism decay of duplex DNA [

In the Effective Medium, EM, model, the fluid is assumed to obey the Brinkman [_{f}

In _{g}_{s}

For later reference, it will be useful to consider the local fluid velocity and pressure of a spherical particle of radius “

In _{jk} = _{jk} (_{jk} is the Kronecker delta)),
^{2}

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 _{j}_{J}_{J}_{J}_{J}

It is worthwhile to briefly discuss the physical basis of _{J}_{K}_{JK}_{JK}_{J}_{K}_{JK}_{JK}^{3}, which is clear from

Following our earlier analysis [

In _{n}_{n}_{0}(z)_{2}(z)^{2}^{3}_{0}(z)^{−z}/z_{2}(z)^{−z}(1/z+3/z^{2}+3/z^{3})

The typical procedure followed in a resistance problem [_{KJ} and _{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,
_{p}_{KJ} terms defined by

In compact notation,

The matrix ^{−1} denote the inverse. Then

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

Before it is possible to compute various resistance tensors from modeling, it is necessary to compute the total force, _{tot}_{tot}

Where Ξ̳_{T}, Ξ̳_{R} (_{C}(_{tot}_{tot}

For a bead array that is not translating (_{p}

From _{C}_{R}_{T}

The connection between the resistance tensors and corresponding mobility or diffusion tensors is well known [_{T}_{R}

In modeling, the origin _{R}_{n}′_{1}′ ≥ Λ_{2}′ ≥ Λ_{3}′

The denominator in

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}_{R}_{l}

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}_{R}_{n}_{1}

This shall be illustrated for the special case of a right circular cylinder of length

For the corresponding array made up of a linear string of

In _{1}_{1}^{rod}_{1}^{rod}_{1}_{1}^{rod}_{1}

We can apply this duplex DNA made up of _{bp}_{bp}_{J}_{J+1} > where _{J}

At this point, we can ask whether or not it is reasonable to equate average transport properties, such as <_{1}_{1}_{1}

_{1}

We would next like to consider the effect of the gel on _{1}_{1}^{ng}^{3}_{1}

Above
_{e}

Stellwagen [_{l}_{l}_{1}_{1}

There are two model parameters that need to be determined and they are the persistence length of the DNA, _{f}_{1}_{1}_{g}_{g}_{g} value along with

The EM model that we have used accounts for long range hydrodynamic interactions, HI, but ignores direct interactions. If a characteristic length of our macromolecule is much smaller than the average spacing between gel fibers,

See the discussion following _{g}_{s}_{g}

The objective of the present study is to apply the Effective Medium (EM) model to the rotational motion of a macromolecule modeled as an array of non-overlapping beads and then apply it to several cases including duplex DNA in agarose gels. This is an extension of earlier work which focused on the translational motion of similar model macromolecules in an EM [_{g}

It has been recognized for some time that the behavior of macromolecules in gel electrophoresis fall into well defined “regimes” [

Array of _{j}

_{1}

A 88 Subunit Discrete Wormlike Chain. The persistence length,

Model and Experimental _{1}_{g}