Natural proteins are characterised by a complex folding pathway defined uniquely for each fold. Designed coiled-coil protein origami (CCPO) cages are distinct from natural compact proteins, since ...their fold is prescribed by discrete long-range interactions between orthogonal pairwise-interacting coiled-coil (CC) modules within a single polypeptide chain. Here, we demonstrate that CCPO proteins fold in a stepwise sequential pathway. Molecular dynamics simulations and stopped-flow Förster resonance energy transfer (FRET) measurements reveal that CCPO folding is dominated by the effective intra-chain distance between CC modules in the primary sequence and subsequent folding intermediates, allowing identical CC modules to be employed for multiple cage edges and thus relaxing CCPO cage design requirements. The number of orthogonal modules required for constructing a CCPO tetrahedron can be reduced from six to as little as three different CC modules. The stepwise modular nature of the folding pathway offers insights into the folding of tandem repeat proteins and can be exploited for the design of modular protein structures based on a given set of orthogonal modules.
Several bibliographic databases offer a free tool that enables one to determine the collaboration distance or co-authorship distance between researchers. This paper addresses a real-life application ...of the collaboration distance. It concerns somewhat unusual clustering; namely clustering in which the average distances in each cluster need to be maximised. We briefly consider a pair of clusterings in which two cluster partitions are uniform and orthogonal in the sense that in each partition all clusters are of the same size and that no pair of elements belongs to the same cluster in both partitions. We consider different objective functions when calculating the score of the pair of orthogonal partitions. In this paper the Wiener index (a graph invariant, known in chemical graph theory) is used. The main application of our work is an algorithm for scheduling a series of parallel talks at a major conference.
The “Grünbaum Incidence Calculus” is the common name of a collection of operations introduced by Branko Grünbaum to produce new
(
n
4
)
configurations from various input configurations. In a previous ...paper, we generalized two of these operations to produce operations on arbitrary
(
n
k
)
configurations, and we showed that for each
k
, there exists an integer
N
k
such that for all
n
≥
N
k
, there exists at least one
(
n
k
)
configuration, with current records
N
5
≤
576
and
N
6
≤
7350
. In this paper, we further extend the Grünbaum Calculus; using these operations, as well as a collection of previously known and novel ad hoc constructions, we refine the bounds for
k
=
5
and
k
=
6
. Namely, we show that
N
5
≤
166
and
N
6
≤
585
.
Polypeptides and polynucleotides are natural programmable biopolymers that can self-assemble into complex tertiary structures. We describe a system analogous to designed DNA nanostructures in which ...protein coiled-coil (CC) dimers serve as building blocks for modular de novo design of polyhedral protein cages that efficiently self-assemble in vitro and in vivo. We produced and characterized >20 single-chain protein cages in three shapes-tetrahedron, four-sided pyramid, and triangular prism-with the largest containing >700 amino-acid residues and measuring 11 nm in diameter. Their stability and folding kinetics were similar to those of natural proteins. Solution small-angle X-ray scattering (SAXS), electron microscopy (EM), and biophysical analysis confirmed agreement of the expressed structures with the designs. We also demonstrated self-assembly of a tetrahedral structure in bacteria, mammalian cells, and mice without evidence of inflammation. A semi-automated computational design platform and a toolbox of CC building modules are provided to enable the design of protein cages in any polyhedral shape.
Solving the Mostar index inverse problem Alizadeh, Yaser; Bašić, Nino; Damnjanović, Ivan ...
Journal of mathematical chemistry,
05/2024, Letnik:
62, Številka:
5
Journal Article
Recenzirano
A nonnegative integer
p
is realizable by a graph-theoretical invariant
I
if there exists a graph
G
such that
I
(
G
)
=
p
. The inverse problem for
I
consists of finding all nonnegative integers
p
...realizable by
I
. In this paper, we consider and solve the inverse problem for the Mostar index, a recently introduced graph-theoretical invariant which attracted a lot of attention in recent years in both the mathematical and the chemical community. We show that a nonnegative integer is realizable by the Mostar index if and only if it is not equal to one. Besides presenting the complete solution to the problem, we also present some empirical observations and outline several open problems and possible directions for further research.
Large networks not only have a large number of vertices but also have a large number of edges. Although such networks are generally sparse they are usually difficult to visualise, even locally. This ...paper considers the case where large weights on edges represent similarity between the corresponding end-vertices. We follow two main ideas in this paper. The first one is network pruning, that is removal of edges that makes the resulting network more manageable while keeping the main characteristic of the original network. The other idea is to partition the network vertex set in such a way that the induced connected components represent groups of network elements that fit together. Furthermore, we assume that the vertices of the network are labeled by types. Here we apply our approach to co-authorship network of researchers in Slovenia in order to identify research groups, finding group leaders and the degree of interdisciplinarity of the group. For the network pruning phase we use a MST-pathfinder network and for vertex partition appropriate linecuts. Each cluster is assigned a distribution of types. In this case, the types correspond to scientific fields, also known as research interests of authors. A measure of interdisciplinarity of research group is derived from such a distribution.
Knots are some of the most remarkable topological features in nature. Self-assembly of knotted polymers without breaking or forming covalent bonds is challenging, as the chain needs to be threaded ...through previously formed loops in an exactly defined order. Here we describe principles to guide the folding of highly knotted single-chain DNA nanostructures as demonstrated on a nano-sized square pyramid. Folding of knots is encoded by the arrangement of modules of different stability based on derived topological and kinetic rules. Among DNA designs composed of the same modules and encoding the same topology, only the one with the folding pathway designed according to the 'free-end' rule folds efficiently into the target structure. Besides high folding yield on slow annealing, this design also folds rapidly on temperature quenching and dilution from chemical denaturant. This strategy could be used to design folding of other knotted programmable polymers such as RNA or proteins.
Vertex and Edge Orbits in Nut Graphs Bašić, Nino; Fowler, Patrick W.; Pisanski, Tomaž
The Electronic journal of combinatorics,
05/2024, Letnik:
31, Številka:
2
Journal Article
Recenzirano
Odprti dostop
A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. If the isolated vertex is excluded as trivial, ...nut graphs have seven or more vertices; they are connected, non-bipartite, and have no leaves. It is shown that a nut graph $G$ always has at least one more edge orbit than it has vertex orbits: $o_e(G) \geq o_v(G) + 1$, with the obvious corollary that edge-transitive nut graphs do not exist. We give infinite familes of vertex-transitive nut graphs with two orbits of edges, and infinite families of nut graphs with two orbits of vertices and three of edges. Several constructions for nut graphs from smaller starting graphs are known: double subdivision of a bridge, four-fold subdivision of an edge, a construction for extrusion of a vertex with preservation of the degree sequence. To these we add multiplier constructions that yield nut graphs from regular (not necessarily nut graph) parents. In general, constructions can have different effects on automorphism group and counts of vertex and edge orbits, but in the case where the automorphism group is ‘preserved’, they can be used in a predictable way to control vertex and edge orbit numbers.
Classification of Cubic Tricirculant Nut Graphs Damnjanović, Ivan; Bašić, Nino; Pisanski, Tomaž ...
The Electronic journal of combinatorics,
05/2024, Letnik:
31, Številka:
2
Journal Article
Recenzirano
Odprti dostop
A nut graph is a simple graph whose adjacency matrix has the eigenvalue zero with multiplicity one such that its corresponding eigenvector has no zero entries. It is known that there exist no cubic ...circulant nut graphs. A bicirculant (resp. tricirculant) graph is defined as a graph that admits a cyclic group of automorphisms having two (resp. three) orbits of vertices of equal size. We show that there exist no cubic bicirculant nut graphs and we provide a full classification of cubic tricirculant nut graphs.
The well-known Petersen graph
G
(
5
,
2
)
admits a semi-regular automorphism
α
acting on the vertex set with two orbits of equal size. This makes it a
bicirculant. It is shown that trivalent ...bicirculants fall into four classes. Some basic properties of trivalent bicirculants are explored and the connection to combinatorial and geometric configurations are studied. Some analogues of the polycirculant conjecture are mentioned.
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