Kenneth Millett (University of California, Santa Barbara)
Dorothy Buck (Imperial College London)
Mario Eudave-Munoz (University Nacional Autonoma de Mexico)
Eric Rawdon (University of Saint Thomas)
Joanna Sulkowska (University of Warsaw)
The proposed workshop is guided by dual goals: exploring mathematical methods used within the context of new experimental methods and data concerning knotting, linking and entanglement found in proteins or occurring in site-specific recombination in DNA; and secondly, inspiring research into new mathematical and biological questions based on this recent progress. The workshop will provide an ideal environment for talented graduate students, post-docs, and top researchers in a wide variety of fields to establish a new vocabulary for discussing outstanding problems and to develop new multidisciplinary research interactions. Knotting and entanglement in biology constantly surprises scientists. It is commonly accepted that knots do not arise spontaneously in proteins, which is not the case for polymers. Many ask why they arose and are still present despite the expected evolutionary pressure for their avoidance and elimination. Experimental data has revealed that knots in denatured proteins are unexpectedly. Another surprise comes from membrane proteins, whose structures are most difficult to determine experimentally. Current results show that at least 60\% of known membrane protein structures contain slipknots and examples of knotted membrane protein have been found. Moreover, recently, new types of complex topology in proteins have been identified: the lasso motif, the Hopf link, and Solomon’s link. From another perspective, applying topological concepts (linking number and knotting) has led to advances that have profoundly affected nanotechnology, molecular biology, and medical research. Notable examples include the well-recognized insights into the topological dimensions underlying the biological functions of topoisomerases and recombinases, two classes of enzymes essential to life. This understanding has grown out of the mathematical work of Ernst and Sumners on tangle calculus in the 1980s. From this experience, it is also expected that the atypical behavior of knots and other entanglements in proteins will lead to new types of materials, or could be used in the design of new drugs. Many fundamental mechanisms remain poorly understood thereby requiring the development and application of increasingly powerful and complex mathematical structures. Mathematical advances have provided new course grained methods to study the ensemble of conformations, both open and closed, for which an excluded volume has been prescribed. One is, therefore, no longer constrained to "ideal’’ conformations for large-scale structural studies. These algorithms provide a powerful tool that is still evolving in order to make it an effective large-scale research method. Independently, employing methods from symplectic geometry and topology, new mathematically rigorous methods for the study of random chains have been recently developed to illuminate their average character of such configurations. This latter approach provides the mathematical structure required to give rigorous proofs of fundamental features. In another challenging and important direction, one has advances in the course grained simulation of confined structures. From experimental perspective, recent improvements in single molecule techniques and methods to model biomolecules based on different resolution experimental data have provided new or more detailed pictures of complex structures. Progress in understanding those data strongly depends on the development of tools in mathematical sciences. Multidisciplinary efforts engaging the energies of mathematicians and life and physical scientists provide the most promising strategy to solve problems on this frontier of science. An Elaboration of Specific Workshop Foci: (1) The existence of knots and slipknots in crystallized structures of proteins is currently well accepted. Recently a joint mathematical (Millett, Rawdon), physical (Sulkowska, Onuchic) and biological (Stasiak) effort has largely advanced the quantitative characterization of complexity of protein structures by performing comprehensive analysis of protein structures deposited in Protein Data Bank (PDB). This analysis has shown that: 1) at least 2\% of deposited proteins have non-trivial topology, 2) knotted protein chains often contain of subknotted segments, 3) a significant group of protein forms contain slipknot motifs, including a new motif not previously identified. These results show that the topological landscape of proteins is much more complex than expected. Moreover, proteins with non-trivial topology have been conserved in proteins over periods longer than a billion years of evolution. It is well known that the geometry of proteins with trivial topology is most conserved during evolution and is directly correlated with their function. Our new methods have illustrated the presence of conserved linking signatures in both knotted and unknotted proteins. Whether this topology is even more conserved than geometry of protein and how the topology and geometry influence the function of proteins are basic and still unanswered questions. (2) Knotted proteins have been found in many important biological processes e.g. posttranscriptional RNA modification, or methylation (in tRNA or rRNA), or membrane transport (LeuT(Aa)), etc., and in particular they can play an important role in some diseases, such as Parkinson’s disease. Sophisticated experiments have shown that proteins can self-tie and that the process of knotting is probably a rate-limiting step. These experiments also show that it is extremely difficult to untie knotted proteins by thermal or chemical denaturation. Current progress in Small Angle X-ray Scattering (SAXS) confirms this phenomenon for just a few proteins. This unexpected behavior, the retention of knots in denatured conditions (not observed in the case of polymers), creates the biggest limitation for experimental progress. Currently it is expected that physical experiments, such as Nuclear Magnetic Resonance, supported by computer simulations and mathematical methods, can specify those properties of proteins which are responsible for retaining knots on their chains. (3) The unexpected thermal stability of knotted proteins raises an obvious question: how those proteins could be degraded by molecular machines, such as proteasomes. Recent theoretical work (Cieplak, Szymczak, Sulkowska, Dziubiela, Li) shows that knots tie along a chain, moreover not randomly but around its native position. Pulling knotted proteins across a proteasome shows that when the radius of a tightened knot is larger than the size of an entrance (hole) to the proteasome, the tightened knot cannot be degraded. Analysis of different knotted proteins showed that the blocking probability increases with the applied force. However it was also shown that knotted chains could be untied in some particular direction. Current progress in single molecule methods, such as optical tweezers, supported by numerical and mathematical methods, should lead to understanding how those proteins could be degraded. Those examples show that interdisciplinary teams represent the best approach to discover function of knotted proteins. (4) Another new type of complex topology for protein structures has been identified recently (Haglund, Sulkowska). The lasso topology represents a new type of complexity in proteins. Lasso arises in proteins with closed loops (e.g. by disulphide bridges), when at least one termini of a protein backbone pierces through an auxiliary surface of minimal area, spanned on a covalent loop. Currently we know that more than 18\% of all proteins with disulphide bridges in a non-redundant subset of PDB posses lasso topology. Similar topology has been identified in so called miniproteins, where a loop is closed by amide linkage as in the antibacterial peptide microcin, which inhibits bacterial transcription by binding within, and obstructing, the nucleotide uptake channel of bacterial RNA polymerase. Over the years it was shown that those proteins are main target of drug design. Existence of these new topologies raises many questions from a purely mathematical perspective: (i) how to mathematically describe structures which posses at the same time knot and supercoiling motif or how to describe complexity between three open curves e.g. lasso with two tails threaded through the close loop), (ii) from a biophysics perspective, how can such proteins fold, overcoming a topological barrier, and (iii) for biology (what is the functional advantage) and to medicine (miniproteins are the best well known structures used to design drugs).
The main goals of the proposed 2017 workshop are: advance understanding of the relationships between different complex geometries and topologies and the function of associated proteins, the mechanisms of site specifying recombination, and the interplay between mathematical modeling and experimental results. For the proposed workshop we plan to invite theoretical and experimental biologists, biophysicists, biochemists, geometers and topologists as well as applied mathematicians. We expect that this will be the first opportunity for many of the invitees from different disciplines to meet each other. Those new and well-established topics will create stimulating environment to solve existing problems and will give rise to new questions.