Knots in Soft Condensed Matter
- Ivan Coluzza (Center for Cooperative Research in Biomaterials, Spain)
- Mark Miller (Durham University, United Kingdom)
Interest in the mathematical theory of knots started nearly 150 years ago with Lord Kelvin's theory (ultimately abandoned) that atoms are knots in the aether . However, it is only in the last decade that the prevalence and importance of knots in the physical and biological sciences has become truly apparent. Knots are now attracting intense interest in contexts as apparently disparate as organic synthesis , biopolymers , colloidal clusters , liquid crystals  and soap films . The purpose of this workshop is to bring together researchers in the broadly-defined field of soft condensed matter to establish connections on the shared theme of knots and to identify the key challenges for future progress.
The proposed CECAM meeting will span both natural systems and novel artificial systems in which topological properties are fundamental to the behaviour. The importance of topology and of knots in particular has been recognized in the biophysics, colloids and soft matter communities, yet the conventional separation between sub-disciplines means that progress in one area is not always communicated to another. It is therefore timely to bring together scientists from these different fields. Computational research into knots would also benefit from closer interactions with experiment, and the workshop is designed to foster these interactions. A forum for discussing the current understanding of topologically complex systems in its full diversity is of paramount importance for the realization of more controllable self-assembling structures and materials that can be designed to have desired microscopic and macroscopic properties.
Formally, a knot is a state of a closed loop that is defined by its topology the geometrical properties of an object that are not changed by continuous deformations. Once a particular topological state has been established, it may be difficult (depending on the system) to change it. Confinement to a particular topological state can have important consequences. For example, in contrast to ideal linear polymers, ideal ring polymers experience an effective repulsion between the chains purely due to the additional topological constraint of closing each chain into a loop [7, 8]. Important implications of topology have also been identified in biophysical systems ranging from proteins to umbilical cords . In some systems, there are mechanisms for a change in topology, resulting in a sudden change in physical properties .
Molecular knots are found in nature mainly in DNA [9-12] and protein molecules [2, 13-15]. The function of and evolutionary advantage behind natural knots in biological systems are still controversial questions. However, such knots are receiving increasing interest from the scientific community for their peculiar structural and enzymatic properties [16-21]. Proteins and DNA display a rich variety of topological effects, both structural and dynamic, and will form an important part of the workshop. For example, the DNA of bacteria is present in the traditional double-helix form, but the helix is closed into a ring a structural property that is crucial for the way the molecule behaves. The cell has evolved a highly sophisticated system to regulate topology by expressing specific proteins called topoisomerases  that control the supercoiling state of the DNA molecule during replication and interaction with other proteins.
Proteins themselves are a prominent example of a natural system with intricate topological properties. Proteins are the fundamental building blocks of living organisms and viruses, with the unique property that their function is encoded in the same structural elements that composes them. While there are many different proteins, each with its own structure and function, they are all composed of chains of the same 20 fundamental chemical units called amino acids. Although only relatively few sequences produce a well-defined and stable ground state, the chemical heterogeneity of the amino acid alphabet ensures an enormous variety of combinations, each of which folds into complex arrangements of predominantly two types of secondary structure: alpha helices and beta sheets [22-24]. Although proteins are linear, rather than cyclic polymers, it is possible to identify knots in a surprising number of proteins by notionally (or even chemically ) connecting the two ends of the molecule.
The workshop will also give prominence to recent developments in topological properties in synthetic chemistry and materials science. The synthesis of non-trivial topology goes back to the first catenanes (linked cyclic molecules) by Wasserman . Much more recently, knotted molecules have received attention from chemists who aim to produce more complex topologies and interconnected molecules that may assemble into larger intertwined macromolecules [27, 28]. In polymer science the study of knots is of great interest, especially in the synthesis of molecules to be used for drug delivery [29-33], since polymer architecture is an essential parameter to control the diffusion of drugs into the body [15, 34-37]. For example, the cyclization of a protein-based pain killer, normally administered intravenously, resulted in a highly stable variant that could resist the gastrointestinal digestion process .
A key objective of this workshop will be to stimulate interactions between researchers working in materials science and those working on natural systems. The overlap between bio-polymers and materials science has been been made more evident by the recent developments in colloidal matter [4, 39-43] where there is now the possibility of building chains of particles with controlled shape and interactions to the extent that specific topology can be designed into the equilibrium configuration of the chains. The workshop will be an opportunity for scientists from these two communities to exchange ideas and increase the area of overlap even further.
To date, control over the synthesis of knotted molecules is a challenging task, especially if a specific internal structure of the knot is required. Furthermore, topology plays a crucial role in some soft-matter systems where the closed cycle that contains the knot is not molecular in origin. A fascinating example of this are the lines of defects in the director field of a nematic liquid crystal when spherical colloidal particles are suspended in the fluid . Anchoring of the nematogens to the colloids' surface causes a local difference in the direction of the liquid crystalline field that can be traced as a line through the fluid. These lines may become knotted and entangled with each other, resulting in effective forces on the colloidal particles.
In order to expand the discussion, the workshop will also provide a discussion forum on development of topological characterization in the physics and mathematics communities. This field is particularly active in its effort to develop new algorithms that are capable of efficiently detecting and identifying the "delocalized" knots that easily arise in fluctuating chains . Since topology is the fundamental concept that binds all these research communities together, a discussion on the state-of-the-art should stimulate the interdisciplinary interactions that this workshop aims to promote.
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