DNA

The replication of DNA requires the assistance of topology changing enzymes called topoisomerases. Efforts to discover the mechanisms by which various enzymes act on DNA has focused attention on questions for which geometrical and topological considerations provide insights as well as raising new questions. In this first lecture, we will explore the biological context and the mathematics that continues to be employed to resolve these questions.

In 1996 Kasianowicz et al. [PNAS vol 93, pp 13770-13773] reported a remarkable experiment. They created a synthetic version of a nanopore using molecular self-assembly on lipid bilayer of a bacterial exotoxin (alpha-hemolysin). The creation of the nanopore was signalled by the passage of an ion current that could be detected with a patch clamp amplifier. When negatively charged DNA is added to the reservoir on one side of the bilayer, the ion current is interrupted momentarily every time a single DNA molecule translocates the pore.

Biological macromolecules are often charged and perform their functions in ionic environments. For this reason the mechanical and electrostatic behavior of macromolecules are connected. In this talk we will shed light on the interplay of mechanics and electrostatics of rod-like macromolecules and cells. In the first part of the talk we will study the deformations of a cell constrained between two plates and subject to a change in potential difference across its membrane.

Newly replicated circular chromosomes are topologically linked. Controlling these topological changes, and returning the chromosomes to an unlinked monomeric state is essential to cell survival. XerCD-dif-FtsK recombination acts in the replication termination region of the Escherichia coli chromosome to remove links introduced during replication. We use topological methods to show definitively that there is a unique shortest pathway of unlinking by XerCD-dif-FtsK that strictly reduces the complexity of the links at every step.

The talk will review the development in the field of DNA-related topological problems, knots and links formed by double-stranded DNA molecules. It will start from purely theoretical problem of calculating the equilibrium probability of knots in a polymer chain. Although solving this problem was an achievement in polymer statistical physics, it did not look useful for anything else at that time. Eventually, however, it helped greatly in the studies of DNA general properties and its topological transformations catalyzed by site-specific recombinases and DNA topoisomerases.

Within only a decade since the first draft of the human genome, we’ve witness astonishing pace of development of technologies for high throughput molecular profiling that probe various aspects of genome biology and its relationship to tumorigenesis and cancer treatment. There have been giant steps towards cataloging massive amounts of data and providing fairly good annotation information.

DNA and RNA are versatile construction materials.
By appropriately designing the sequence of bases in each strand, synthetic nucleic acid
systems can be programmed to self-assemble into complex structures that implement dynamic mechanical
tasks. Motivated by the challenge of encoding arbitrary mechanical function into
nucleic acid sequences, we are developing a suite of computational algorithms for
analyzing the underlying free energy landscapes that control the