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Dimos Goundaroulis

Publications and source records attributed to Dimos Goundaroulis.

13 recordsLinked to original sources
arXiv2021

A topological selection of folding pathways from native states of knotted proteins

Understanding the biological function of knots in proteins and their folding process is an open and challenging question in biology. Recent studies classify the topology and geometry of knotted proteins by analysing the distribution of a protein's planar projections using topological objects called knotoids. We approach the analysis of proteins with the same topology by introducing a topologically inspired statistical metric between their knotoid distributions. We detect geometric differences between trefoil proteins by characterising their entanglement and we recover a clustering by sequence similarity. By looking directly at the geometry and topology of their native states, we are able to probe different folding pathways for proteins forming open-ended trefoil knots. Interestingly, our pipeline reveals that the folding pathway of shallow knotted Carbonic Anhydrases involves the creation of a double-looped structure, differently from what was previously observed for deeply knotted trefoil proteins. We validate this with Molecular Dynamics simulations.

q-bio.BM↗
arXiv2021

$f$-distance of knotoids and protein structure

Recent studies classify the topology of proteins by analysing the distribution of their projections using knotoids. The approximation of this distribution depends on the number of projection directions that are sampled. Here we investigate the relation between knotoids differing only by small perturbations of the direction of projection. Since such knotoids are connected by at most a single forbidden move, we characterise forbidden moves in terms of equivariant band attachment between strongly invertible knots and of strand passages between $θ$-curves. This allows for the determination of the optimal sample size needed to produce a well approximated knotoid distribution. Based on that and on topological properties of the distribution, we propose a numerical measure for the determination of deeply knotted proteins that does not require the computationally expensive method of subchain analysis.

math.GT↗
arXiv2019

A generalized skein relation for Khovanov homology and a categorification of the $θ$-invariant

The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are able to generalize the Khovanov homology in order to obtain a categorification of the $θ$-invariant, which is itself a generalization of the Jones polynomial.

math.GT↗
arXiv2019

Framization of a Temperley-Lieb algebra of type $\mathtt{B}$

We extend the Framization of the Temperley-Lieb algebra to Coxeter systems of type $\mathtt{B}$. We first define a natural extension of the classical Temperley-Lieb algebra to Coxeter systems of type $\mathtt{B}$ and prove that such an extension supports a unique linear Markov trace function. We then introduce the Framization of the Temperley-Lieb algebra of type $\mathtt{B}$ as a quotient of the Yokonuma-Hecke algebra of type $\mathtt{B}$. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra of type $\mathtt{B}$ to pass to the quotient algebra. Using the main theorem, we construct invariants for framed links and classical links inside the solid torus.

math.RA↗
arXiv2019

$θ$-curves in proteins

Apart from the knots formed by the main-chain, the proteins can form numerous topological structures, when included the covalent and ion-mediated interactions. In this work, we define the protein non-trivial $θ$-curves and identify 7 different topologies in all structures known up to date. We study the correlation of the motif with the function and organism of origin, and pointing the similarity with main-chain knots, we show that some motifs may indeed be functional. We also analyze the folding and bridge-induced stability of an exemplary protein with $θ$-curve motif and provide a catalogue of possible $θ$-curves in proteins.

cond-mat.soft↗
arXiv2018

A new two-variable generalization of the Jones polynomial

We present a new 2-variable generalization of the Jones polynomial that can be defined through the skein relation of the Jones polynomial. The well-definedness of this new generalization is proved both algebraically and diagrammatically as well as via a closed combinatorial formula. This new invariant is able to distinguish more pairs of non-isotopic links than the original Jones polynomial such as the Thistlethwaite link from the unlink with two components.

math.GT↗
arXiv2018

Knoto-ID: a tool to study the entanglement of open protein chains using the concept of knotoids

The backbone of most proteins forms an open curve. To study their entanglement, a common strategy consists in searching for the presence of knots in their backbones using topological invariants. However, this approach requires to close the curve into a loop, which alters the geometry of curve. Knoto-ID allows evaluating the entanglement of open curves without the need to close them, using the recent concept of knotoids which is a generalization of the classical knot theory to open curves. Knoto-ID can analyse the global topology of the full chain as well as the local topology by exhaustively studying all subchains or only determining the knotted core. Knoto-ID permits to localize topologically non-trivial protein folds that are not detected by informatics tools detecting knotted protein folds.

q-bio.BM↗
arXiv2017

Studies of global and local entanglements of individual protein chains using the concept of knotoids

We study here global and local entanglements of open protein chains by implementing the concept of knotoids. Knotoids have been introduced in 2012 by Vladimir Turaev as a generalization of knots in 3-dimensional space. More precisely, knotoids are diagrams representing projections of open curves in 3D space, in contrast to knot diagrams which represent projections of closed curves in 3D space. The intrinsic difference with classical knot theory is that the generalization provided by knotoids admits non-trivial topological entanglement of the open curves provided that their geometry is frozen as it is the case for crystallized proteins. Consequently, our approach doesn't require the closure of chains into loops which implies that the geometry of analysed chains does not need to be changed by closure in order to characterize their topology. Our study revealed that the knotoid approach detects protein regions that were classified earlier as knotted and also new, topologically interesting regions that we classify as pre-knotted.

q-bio.BM↗
arXiv2016

A Survey on Temperley-Lieb-type quotients from the Yokonuma-Hecke algebras

In this survey we collect all results regarding the construction of the Framization of the Temperley-Lieb algebra of type $A$ as a quotient algebra of the Yokonuma-Hecke algebra of type $A$. More precisely, we present all three possible quotient algebras the emerged during this construction and we discuss their dimension, linear bases, representation theory and the necessary and sufficient conditions for the unique Markov trace of the Yokonuma-Hecke algebra to factor through to each one of them. Further, we present the link invariants that are derived from each quotient algebra and we point out which quotient algebra provides the most natural definition for a framization of the Temperley-Lieb algebra. From the Framization of the Temperley-Lieb algebra we obtain new one-variable invariants for oriented classical links that, when compared to the Jones polynomial, they are not topologically equivalent since they distinguish more pair of non isotopic oriented links. Finally, we discuss the generalization of the newly obtained invariants to a new two-variable invariant for oriented classical links that is stronger than the Jones polynomial.

math.GT↗
arXiv2016

Classical link invariants from the framizations of the Iwahori-Hecke algebra and the Temperley-Lieb algebra of type $A$

In this paper we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma-Hecke algebras ${\rm Y}_{d,n}(q)$, which are not topologically equivalent to the Homflypt polynomial. We then present the algebra ${\rm FTL}_{d,n}(q)$ which is the appropriate Temperley-Lieb analogue of ${\rm Y}_{d,n}(q)$, as well as the related 1-variable classical link invariants, which in turn are not topologically equivalent to the Jones polynomial. Finally, we present the algebra of braids and ties which is related to the Yokonuma-Hecke algebra, and also its quotient, the partition Temperley-Lieb algebra ${\rm PTL}_n(q)$ and we prove an isomorphism of this algebra with a subalgebra of ${\rm FTL}_{d,n}(q)$.

math.GT↗
arXiv2015

Framization of the Temperley-Lieb Algebra

We propose a framization of the Temperley-Lieb algebra. The framization is a procedure that can briefly be described as the adding of framing to a known knot algebra in a way that is both algebraically consistent and topologically meaningful. Our framization of the Temperley-Lieb algebra is defined as a quotient of the Yokonuma-Hecke algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra to pass through to the quotient algebra. Using this we construct 1-variable invariants for classical knots and links, which, as we show, are not topologically equivalent to the Jones polynomial.

math.QA↗
arXiv2013

The Yokonuma-Temperley-Lieb Algebra

In this paper we introduce the Yokonuma-Temperley-Lieb algebra as a quotient of the Yokonuma-Hecke algebra over a two-sided ideal generated by an expression analogous to the one of the classical Temperley-Lieb algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra to pass through to the quotient algebra, leading to a sequence of knot invariants which coincide with the Jones polynomial.

math.GT↗