The investigator develops equations of motion for elastic strings with rigid "bouquets" of charges that may swivel around the elastic base. In addition to elastic energy of the bend, arbitrary nonlocal interactions between the charges are considered. The studies are kept completely general and can include electrostatic interactions, the Lennard-Jones interactions preventing self-intersection of the string, or any other nonlocal interactions. The investigator and his colleagues derive an exact geometric theory of the motion of the nonlocally interacting elastic strings and apply it to a variety of problems. This theory is based on the Simo-Marsden-Krishnaprasad theory of exact elastic rods. After the theory is derived, an explicit method for computing exact helical solutions of the equations is found. The investigator also shows that for geometric reasons, an explicit dispersion relation for the propagation and stability of linear waves on helical solutions can be derived. This technique also provides a method for computing linear combinations of elastic constants if the shape of helical molecules and the interactions between the charges are known. The investigator also derives a discrete analogue of the equations allowing a wide class of exact solutions, called n-helices. These are discrete helices with the pattern repeating after n shifts. The bifurcation structure of these discrete n-helices is examined. Finally, it is shown how to use helices as basic building blocks for construction of more complex molecular conformations.
Long chains of organic molecules, like proteins, form the basis of all life on Earth. The shape of these molecules plays a crucial role for their function in a living body, and thus understanding of the inner structure of these molecules is an important task. It has been found that the inner structure of many proteins consists of tightly wound helices, and those helices are packed together in a complex fashion. The helices are held together by a complex interplay of electrostatic and elastic (and other) forces. How can one describe the formation of these complex shapes? Why are helices so ubiquitous in Nature? The investigator and his colleagues approach these questions by deriving a theory of molecular chain evolution that can include many possible interactions between individual atoms. The investigator shows that, because of geometric reasons, helices are stationary states for simple molecules, no matter how complex the interaction between the atoms may be. This theory forms a basis for understanding of the inner structure of biological molecules and their dynamics. Better understanding of the spatial structure of molecules and their formation helps in designing better polymers for the chemical industry, better drugs for medical applications, and materials with novel and improved properties for high-tech engineering of the future.
