The sequence-dependent curvature and flexibility of DNA is critical for its packaging into the cell, recognition by other molecules, and conformational changes during biochemically important processes. However, few experimental methods are available for probing these properties at the basepair level. The objective of this research is to develop a computational method for estimating local, sequence-dependent curvature parameters for DNA from hydrodynamic data on short, relatively stiff fragments. The method consists of minimizing a least-squares functional which quantifies the difference between theoretical and experimental sedimentation speeds for a given collection of fragments, and its numerical implementation requires the repeated solution of an exterior Stokes-type problem around various slender, three-dimensional domains. The research effort is centered on three areas: (1) the study of the theoretical sedimentation speed of a stiff polymer in a Stokes-type fluid with thermal fluctuations in different asymptotic limits, (2) the design and analysis of efficient, high-order boundary element methods for exterior Stokes flows, and (3) the design and analysis of fast, reliable methods for minimizing the least-squares sedimentation functional over a space of sequence-dependent curvature parameters.
An understanding of how material filaments may be optimally packed in confined geometries, and how they may supercoil or wrap around themselves, is of great interest in the study of macromolecules such as DNA and other systems in chemistry and biology. For example, evidence published in the journals Nature and Proteins suggest that structural motifs in double-helical DNA and alpha-helical proteins may be explained by optimal packing rules. Moreover, experimental data on knotted DNA molecules suggest that certain physical properties of DNA knots can be predicted from a corresponding optimally tight shape. The objective of this research is to exploit the novel concept of global curvature to develop concise variational formulations, establish existence and regularity results, derive necessary conditions, and design numerical methods for various different optimal packing problems for curves.
Differential equations in the engineering and physical sciences are often derived from global balance laws pertaining to mass, momentum and energy, and are often formulated in terms of variables that evolve on smooth constraint manifolds. However, balance laws are typically lost under discretization and numerical solutions often drift away from their underlying manifold. The objective of this research is to develop special discretization techniques for preserving balance laws and constraint manifolds for various types of ordinary and partial differential equations, and to establish their stability and approximation properties.