Averaging and Smoothing in the Special Euclidean Group
with Application to Continuum Modelling
of the Structural Properties of DNA


 

Motivation:

The continuum elastic theory of rods has been used with success to determine the equilibrium structures of supercoiled DNA. However, most of the published works assume uniform physical properties (bending and torsional stiffnesses) and intrinsic shape (minimum energy shape). Some numerical works did account for sequence dependent physical properties and intrinsic shape. However, the methods employed to construct continuum physical properties and intrinsic shape from the discrete experimental (X-ray crystal structures) or computational (molecular dynamics) data are ad-hoc and are based on trial and error. With the availability of efficient computational tools for elastic rods the need for a rational method to construct continuum properties from the discrete base-pair structure is now increasing.

In the continuum approach, the DNA double helix is described by a centerline r(s) giving the location of the axis of the double helix at arclength s and a triad (d1(s), d2(s), d3(s)) that gives the orientation of the DNA axis at arclength s with respect to a fixed orthogonal coordinates system. At the discrete level, each base pair of the DNA double helix is described by the location ri of its center and a triad (d1i, d2i, d3i) that gives the orientation of the base pair with respect to a fixed orthogonal coordinates system.

The sequence-dependent discrete stiffnesses and intrinsic shape are obtained from the experimental or computational data by the so-called wedge model which is a nearest neighbor model. The underlying assumptions of this model are that the origin of the triad (d1i, d2i, d3i) is at the center of the base-pair i, the d3i vector points to the next base-pair center, and the d1i axis tracks the rotation of the DNA sugar-phosphate double helix.

The transformation that gives the relation between the triads of two successive base pairs is obtained by the wedge angle set that gives the tilt, roll and twist for the 16 combinations of dimers (AA, AC, AG, AT, CA, CC, CG, CT, GA, GC, GG, GT, TA, TC, TG and TT). Several wedge angle sets have been proposed by researchers in the field of molecular biology.

The DNA intrinsic shape obtained from the wedge angle model contains noise and thus is not suitable to be used in the continuum computations. Therefore, smoothing the discrete intrinsic shape given by the wedge model is required. The smoothing is done by a moving window filter in the group of rotations described below.
 

Smoothing base-pair frames:

The rotation matrix that transforms the frame attached with base pair i-1 of a DNA segment to the frame attached with base pair i is denoted by and is given by a wedge model.

The orientation of each base pair of a DNA sequence is given by a rotation matrix that transforms a fixed frame to the frame attached with base pair i.

Then we have

Let us define the rotation matrices

Now the W-filtered rotation matrix at the base pair i is given by a weighted average of with weights .

By the bi-invariance property of the weighted average we have

where is the weighted average of with weights .


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