Design of a single-chain polypeptide tetrahedron assembled from coiled-coil segments

Gradišar, Helena; Božič, Sabina; Doles, Tibor; Vengust, Damjan; Hafner-Bratkovič, Iva; Mertelj, Alenka; Webb, Ben; Šali, Andrej; Klavžar, Sandi; Jerala, Roman

doi:10.1038/nchembio.1248

Cited by 293 publications

(322 citation statements)

References 34 publications

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“…[59]) as well as various geometrical objects that are constructed for nano-technology needs from strands of DNA or polypeptides, such as those studied in the work of Jerala et al (of which a representative example is Ref. [60]. )…”

Section: Epiloguementioning

confidence: 99%

A Dual of the Cycle Theorem and its Application to Molecular Complexity

Kirby¹,

Mallion²,

Pollak³

et al. 2017

Croat. Chem. Acta

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Abstract:The duality alluded to in the title is that between the faces and vertices of a graph embedded on a surface. Its recognition in the context of the five Platonic solids is classic. Algebraically, it is present in the equation for Euler's Polyhedron Theorem and in the various extensions thereof. The Cycle Theorem (CT) establishes a formula for the number of spanning trees contained in a graph embedded on a surface. It is based on the mutual incidences of its cycles (circuits which also carry a sense of direction), i.e., of sub-graphs of the Cn type, endorsed with a sense. These appear (though not exclusively) as the boundaries of faces, so that, so to speak, the Cycle Theorem establishes a result which is essentially about vertices via relations between faces. Among several possible duals of the Cycle Theorem there might thus be one that establishes a relation which is essentially about faces via relations between vertices. In order to formulate one, we define, for an embedded graph, a feature concerning faces that is dual to a spanning tree. We call it a ladder. A formula is presented for the number of ladders contained in a graph which, in some cases, introduces the concept of 'artificial vertices'. It is based on the mutual incidences of its vertices. Its form is clearly analogous, or 'dual', to the Cycle Theorem formula for spanning trees, previously proposed (in this journal -2004) by three of the present authors, together with Klein and Sachs. A new index is proposed, which involves ladders. We call it the Patency Index of a graph; its numerical value may be related to molecular complexity. It is effectively the dual of, and is entirely analogous to, the Spanning-Tree Density Index which was earlier (2003) proposed, defined and applied to molecular graphs by one of the present authors and Trinajstić.

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Section: Epiloguementioning

confidence: 99%

A Dual of the Cycle Theorem and its Application to Molecular Complexity

Kirby¹,

Mallion²,

Pollak³

et al. 2017

Croat. Chem. Acta

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“…[7][8][9][10][11][12][13] Furthermore, by connecting two or more peptide using short linkers, fairly complex assembled structures can be formed as result of the well-defined folding pattern of each single peptide folding motif. [14][15][16][17][18] Peptides have also been conjugated to larger synthetic polymeric backbones, such as poly(ethylene glycol) (PEG), in order to create a wider range of supramolecular hybrid materials and nanostructures, [19][20][21] including peptide-polymer hybrid hydrogels, fibrils, spherulities and micells. [21][22][23][24][25][26] PEG is an attractive component in hydrogels and have been extensively investigated for 3D cell culture, and is used in a large number of clinical and pharmaceutical applications.…”

Section: Introductionmentioning

confidence: 99%

Tailoring Supramolecular Peptide–Poly(ethylene glycol) Hydrogels by Coiled Coil Self-Assembly and Self-Sorting

2016

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Physical hydrogels are extensively used in a wide range of biomedical applications. However, different applications require hydrogels with different mechanical and structural properties.Tailoring these properties demands exquisite control over the supramolecular interactions involved. Here we show that it is possible to control the mechanical properties of hydrogels using de novo designed coiled coil peptides with different affinities for dimerization. Four different non-orthogonal peptides, designed to fold into four different coiled coil heterodimers with dissociation constants spanning from µM to pM, were conjugated to star-shaped 4-armpoly(ethylene glycol) (PEG). The different PEG-coiled coil conjugates self-assemble as a result of peptide heterodimerization. Different combinations of PEG-peptide conjugates assemble into PEG-peptide networks and hydrogels with distinctly different thermal stabilities, supramolecular and rheological properties, reflecting the peptide dimer affinities. We also demonstrate that it is possible to rationally modulate the self-assembly process by means of thermodynamic self-sorting by sequential additions of non-pegylated peptides. The specific interactions involved in peptide dimerization thus provides means for programmable and reversible self-assembly of hydrogels with precise control over rheological properties, which 2 can significantly facilitate optimization of their overall performance and adaption to different processing requirements and applications.

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“…Namely, a graph H ∈ Ξ admits retracting-free double circuits traversing every edge twice, but not necessarily in each direction, as is for graphs of Ω . The minimum graph from Ξ \ Ω is that of tetrahedron [9][10][11][12]. The following result is due to Sabidussi [13] and was later independently proven by Eggleton and Skilton ([3; Theorem 9]).…”

Section: Preliminariesmentioning

confidence: 89%

Traversing every edge in each direction once, but not at once: Cubic (polyhedral) graphs

Rosenfeld

2017

ejgta

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A retracting-free bidirectional circuit in a graph G is a closed walk which traverses every edge exactly once in each direction and such that no edge is succeeded by the same edge in the opposite direction. Such a circuit revisits each vertex only in a number of steps. Studying the class Ω of all graphs admitting at least one retracting-free bidirectional circuit was proposed by Ore (1951) and is by now of practical use to nanotechnology. The latter needs in various molecular polyhedra that are constructed from a single chain molecule in the retracting-free way. Some earlier results for simple graphs, obtained by Thomassen and, then, by other authors, are specially refined by us for a cubic graph Q. Most of such refinements depend only on the number n of vertices of Q.Keywords: cubic graph, spanning tree, cotree, retracting-free bidirectional circuit Mathematics Subject Classification : 05C10, 05C30, 05C45, 94C15 DOI:10.5614/ejgta.2017.5.1.13 PreliminariesLet Q = (V, E) be a simple cubic graph with the vertex set V and edge set E |V | = n, |E| = m = 3n/2 . A spanning tree T of Q is a subtree covering all the vertices of Q |V (T )| = n; |E(T )| = n − 1 . Its cotree Q − E(T ) V Q − E(T ) = n; E Q − E(T ) = (n + 2)/2 is a graph Q less all edges belonging to T .

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Design of a single-chain polypeptide tetrahedron assembled from coiled-coil segments

Cited by 293 publications

References 34 publications

A Dual of the Cycle Theorem and its Application to Molecular Complexity

A Dual of the Cycle Theorem and its Application to Molecular Complexity

Tailoring Supramolecular Peptide–Poly(ethylene glycol) Hydrogels by Coiled Coil Self-Assembly and Self-Sorting

Traversing every edge in each direction once, but not at once: Cubic (polyhedral) graphs

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