07-14 September 2019

Pawel Dabrowski-Tumanski

Home Institution
University of Warsaw, Warsaw, Poland

Host Contact
Dimos Goundaroulis

Host Institution
University of Lausanne, Lausanne, Switzerland

Aim of the mission
The purpose of this STSM was to initiate the methodological analysis of the topology of θ-curves (the embeddings of Greek letter θ, i.e. two three-valent vertices, joined by three arcs) and other spatial graphs, and to study their possible applications to biological processes. In particular, five topics were planned to be discussed during the visit:
1. Invariants of the theta-curves;
2. Sampling and classification of theta-curves and other spatial graphs;
3. The localization of the entanglement in the theta-curve;
4. The concept of ideal theta-curves;
5. The existence of theta-curves in biopolymers, including proteins and replicating DNA chains.
The aim was to plan our future collaboration which is going to be continued remotedly. In particular, we decided on specific task for the projects concerning the topic of θ-curves and spatial graphs, discussed possible problems which may occur and their solutions, and planned the dissemination of the results.

Summary of the Results
The visit was successful, as all the goals previously set were realized. Moreover, the effect of the work, while its finished, will impact various working groups in the Eutopia action, in particular:
1. Discussed the possible ways of representing, generating and classifying the spatial graphs (relevant for WG1);
2. Created a way of representing the entanglement core of the θ-curve (relevant for WG1);
3. Discussed other topological invariants, which may be used in the classification of θ-curves (relevant for WG1);
4. Invented the way of creating the ideal θ-curves (done in cooperation with prof. Andrzej Stasiak – the creator of the notion of “ideal knot”), including the precise mathematical description. The ideal θ-curves may be especially relevant for WG4, as may serve as a template for modelling some biological processes involving the DNA chains.

The aim of this STSM was to initiate the projects, which will be carried on in the future remotedly. Therefore, naturally the cooperation between the University of Warsaw and University of Lausanne will be continued. Moreover, we hope that the biological relevance of the project will be supported by prof. Andrzej Stasiak from University of Lausanne, and the physical aspects (especially concerning the dynamics of polymers) will be supported by prof. Joanna Sulkowska from University of Warsaw. We hope that the outcome of this work will be formalized as a set of a few papers, collecting the methods and the results obtained. In particular, the papers including:
1. The classification of some spatial graphs;
2. The study of the ideal θ-curve;
3. The localization of the θ-curve entanglement core
are planned for the future.
We also do not exclude other collaborations with Eutopia members, or other researchers interested in the topic of θ-curves.