Controlling Knots in Simple Polymer Models and its Applications to DNA and Protein Knots

Knotting is a prevailing phenomenon that occurs in everyday objects and polymers. Due to the widespread presence, knotting is of great interest to a broad range of scientists, not only mathematicians, but also polymer physicists studying the effect of knotting on polymer dynamics, biologists investigating the biological functions of protein knots and DNA knots, chemists studying the special catalysis of knotted molecules, and technologists who wish to understand the effects of DNA knots in sequencing technologies. Addressing a wide range of problems about knots and utilizing knots in applications require effective methods to produce stable polymer knots.Despite many studies in recent years, no simple and effective method is available to produce stable swollen polymer knots or compact polymer knots with structural specificity. Swollen polymer knots can be produced by compression, but these knots eventually get untied by knot diffusion or swelling. Compact polymer knots can be produced by worsening the solvent quality, but these compact polymer knots lack structural specificity. Specific knotted structures may be achieved by designing sequences in heteropolymers, but searching sequences for given structures usually requires tremendous computational effort.In this research, we will develop simple yet effective methods to produce stable knots in swollen and compact polymers using novel strategies. For swollen polymers, we will use side chains to lock a polymer knot. For compact polymers, we will use the preferred knotted structures of homopolymers as templates and then change a few monomers to improve the structural specificity. The effectiveness of this approach has been demonstrated by our preliminary results.In addition to producing stable knots, we will reveal mechanisms of polymer knot formation, which should guide the rational control of polymer knots. Furthermore, the results obtained from swollen and compact knots in simple polymer models will be applied to DNA knots and protein knots, respectively. In particular, results about compact polymer knots may help identify the key factor responsible for protein knot formation.This project relies on Monte Carlo simulations and molecular dynamics simulations of simple polymer models and full-atom proteins. These simulations will provide conformational and dynamic properties of polymer and protein knots.The results from this project are expected to reveal mechanisms of polymer knot formations, identify the key factor responsible for protein knot formation, and develop simple ways to produce stable DNA knots and protein knots for practical applications, such as sequencing technologies.