| Kathleen A. Hoffman
Stability theory for constrained calculus of variations problems
DNAProblems arising in applications frequently have a parameter dependent variational structure. In this variational setting, the study of stability is equivalent to determining which equilibria are minima in an appropriate sense. For the unconstrained problem, positivity of the second variation determines which equilibria correspond to minima. In the case of constrained minima, the positivity of the second variation over a subspace of the full domain is the quantity of interest. We define a constrained index to measure the dimension of the appropriate constrained subspace on which the second variation is negative. Thus an equilibrium is stable provided it has constrained index zero.
For functionals with an explicit parameter dependence, Maddocks has presented stability exchange arguments based on distinguished bifurcation diagrams, a specific projection of the bifurcation diagram which encodes stability information in the shape of the branch. Stability exchanges, or changes in constrained index, occur at folds of the branches in the appropriate projection. I've extended Maddocks' work on stability exchange results in two different directions. First, I've extended the concept of distinguished diagram to include constrained systems, where there are both parameters and multipliers. Secondly, I've also included the case when the bifurcation parameter appears in the boundary condition. Both of these extensions are necessary in order to address the stability of elastic rod models of DNA supercoiling.
A twisted elastic rod is an example of a constrained variational problem with the bifurcation parameter in the boundary conditions. A rod is a curve in space with an associated frame of directors, which give the orientation of the cross section of the rod. A rod that is twisted and then closed upon itself to form a stressed loop (see figure below) is widely considered to be a good model for DNA minicircles the application that motivated my analysis. In this setting, the boundary conditions contain the bifurcation parameter, namely, the angle by which one end of the rod is twisted relative to the other.
For this problem, I have shown that the distinguished diagram is twist moment plotted against angle. The shape of the diagram, and thus the stability of the equilibria, depends on the ratio of the twisting stiffness of the rod to the bending stiffness of the rod. For DNA, the generally accepted range of this ratio is between 0.7 and 1.5. Some of the stability results are presented in figure below. For the ratio less than 1.3, the branch shown is unstable whereas for the range between 1.3 and 2.1, there is a change of stability on the branch. For values of the ratio larger than 2.1, the non-trivial branch is stable.
Some elastic rod models of DNA assume that it is inherently straight and untwisted. Manning et al (1996) developed an elastic rod model which incorporates inherent curvatures and twist that were obtained by averaging experimental DNA data, and thus reflect the local DNA geometry. Although the distinguished diagram can predict changes in index for this more general problem, it cannot determine the value of the index. The index can be computed numerically using a generalization of Jacobi's conjugate point test for isoperimetrically constrained calculus of variations problems. Applying these results to the DNA-sequence-dependent rods, we can determine which equilibria are stable and thus experimentally accessible.
The classification of the stability properties of equilibria within elastic rod models of deformed DNA molecules provided the inspiration for an extension of the techniques from finite-dimensional equality constrained optimization theory to the infinite dimensional case of an isoperimetrically constrained one-dimensional calculus of variations problem. In the context of finite-dimensional equality constrained optimization theory, it has recently been observed that the index of a constrained critical point regarded as a function of the basic unknowns, is simply related to an index of the same critical point regarded as an unconstrained solution to an extended variational principle in which the Lagrange multipliers are treated as additional independent variables. An alternative proof of this fact can be constructed from various theories of restricted quadratic forms in finite dimensional spaces applied to the Kuhn-Tucker bordered matrix, which arises as the appropriate Hessian, or second variation, matrix with respect to the extended variables (see for example. A completely analogous theory holds true for a `bordered operator' that arises as the `Hessian' in the infinite dimensional case of an isoperimetrically constrained one-dimensional calculus of variations problem.