In this thesis we describe a path integral formalism to evaluate approximations to the probability density function for the location and orientation of one end of a continuum polymer chain at thermodynamic equilibrium with a heat bath. We concentrate on those systems for which the associated energy density is at most quadratic in its variables. Our main motivation is to exploit continuum elastic rod models for the approximate computation of DNA looping probabilities. We first re-derive, for a polymer chain system, an expression for the second order correction term due to quadratic fluctuations about a unique minimal energy configuration. The result, originally stated for a quantum mechanical system by G. Papadopoulos (1975), relies on an elegant algebraic argument that carries over to the real-valued path integrals of interest here. The conclusion is that the appropriate expression can be evaluated in terms of the energy of the minimizer and the inverse square root of the determinant of a matrix satisfying a certain non-linear system of differential equations. We then construct a change of variables, which establishes a mapping between the solutions of the aforementioned non-linear Papadopoulos equations and a matrix satisfying an initial value problem for the classic linear system of Jacobi equations associated with the second variation of the energy functional. This conclusion is trivial if no cross-term is present in the second variation, but ceases to be so otherwise. Cross-terms are always present in the application of rod models to DNA. We therefore can conclude that the second order fluctuation correction term to the probability density function for a chain is always given by the inverse square root of the determinant of a matrix of solutions to the Jacobi equations. We believe this conclusion to be original for the real-valued case when the second-variation involves cross-terms. Similar results are known for quantum mechanical systems, and, in this context, a connection between the so called Van-Hove-Morette determinant, which involves partial derivatives of the classical action with respect to the boundary values of the configuration variable, and the Jacobi determinant have also been established. We next apply the formula described above to the specific context of rods, for which the configuration space is that of framed curves, or curves in R3 × SO(3). An immediate application of our theory is possible if the rod model encompasses bend, twist, stretch and shear. However the constrained case, where the rod is considered to be inextensible and unshearable, is more standard in polymer physics. In this last case, our results are more delicate as the Lagrangian description breaks down, and the Hamiltonian formulation must be invoked. It is known that the unconstrained local minimizers approach constrained minimizers as the coefficients in the shear and extension terms of the energy are sent to infinity. Here we observe that the Hamiltonian form of the unconstrained Jacobi system similarly has a limit, so that the fluctuation correction in the path integral can still be expressed as the square root of the determinant of a matrix solution of a set of Jacobi equations appropriate to the constrained problem. As in reality DNA or biological macromolecules are certainly at least slightly shearable and extensible, the limit of the fluctuation correction is undoubtedly physically appropriate. The above theory provides a computationally highly tractable approach to the estimation of the appropriate probability density functions. For application to sequence-dependent models of DNA the associated systems of equations has non-constant coefficients, which is of little consequence for a numerical treatment, but precludes the possibility of finding closed form expressions. On the other hand the theory also applies to simplified homogeneous models. Accordingly, we conclude by applying our approach in a completely analytic and closed-form way to the computation of the approximate probability density function for a uniform, non-isotropic, intrinsically straight and untwisted rod to form a circular loop.
EPFL2008
