Euclidean group.

We can numerically compute moments of this final distribution, or its various marginal distributions, by a Monte Carlo procedure. Idealizing the polymer as a chain of finite elements, each is related to its predecessor by a shift along the latter’s 3-axis, a twist about the same axis, and some random bend and twist. Rather than represent the random part using Euler angles, a more invariant formulation is to draw a $3\times3$ generator matrix from a Gaussian distribution on the Lie algebra so(3), then exponentiate it. The Gaussian distribution is determined by a covariance matrix, which represents the bend and twist elasticity of the DNA, together with bend-twist couplings. We estimated it up to an overall rescaling factor from structural data on DNA, then chose the overall factor to yield a desired value of the persistence length of DNA.

Turning from the idealized problem above to TPM, we see that we must implement steric constraints: One end of the DNA tether is attached to a wall, which the DNA may not penetrate. Moreover, the other end is attached to the sphere, which itself must not penetrate the wall. Nevertheless, each segment of the intervening DNA is otherwise free to 