The braid index of complicated DNA polyhedral links

Progressive cancer targeting by programmable aptamer-tethered nanostructures

Scientific research in recent decades has affirmed an increase in cancer incidence as a cause of death globally. Cancer can be considered a plurality of various diseases rather than a single disease, which can be a multifaceted problem. Hence, cancer therapy techniques acquired more accelerated and urgent approvals compared to other therapeutic approaches. Radiotherapy, chemotherapy, immunotherapy, and surgery have been widely adopted as routine cancer treatment strategies to suppress disease progression and metastasis. These therapeutic approaches have lengthened the longevity of countless cancer patients. Nonetheless, some inherent limitations have restricted their application, including insignificant therapeutic efficacy, toxicity, negligible targeting, non-specific distribution, and multidrug resistance. The development of therapeutic oligomer nanoconstructs with the advantages of chemical solid-phase synthesis, programmable design, and precise adjustment is crucial for advancing smart targeted drug nanocarriers. This review focuses on the significance of the different aptamer-assembled nanoconstructs as multifunctional nucleic acid oligomeric nanoskeletons in efficient drug delivery. We discuss recent advancements in the design and utilization of aptamer-tethered nanostructures to enhance the efficacy of cancer treatment. Valuably, this comprehensive review highlights self-assembled aptamers as the exceptionally intelligent nano-biomaterials for targeted drug delivery based on their superior stability, high specificity, excellent recoverability, inherent biocompatibility, and versatile functions.

The braid index of DNA double crossover polyhedral links

In this paper, the authors study the mathematical properties of a class of alternating links called polyhedral links which have been used to model DNA polyhedra. The motivation of such studies is to provide guidance and aid in the research of the properties of certain DNA molecules. For example, such studies can provide characterizations of the structural complexity of DNA molecules. In an earlier work, Cheng and Jin studied the mathematical properties of such polyhedral links and were able to determine the braid index of a double crossover polyhedral link with 4 turn. However, the braid index of a double crossover polyhedral link with 4.5 turn remained an unsolved problem to this date, even though the graphs that admit the double crossover polyhedral links with 4.5 turn have been synthesized. In this paper, we provide a complete formulation of the braid index of a double crossover polyhedral link with an arbitrary turn number. Our approach is more general and it allows us to completely determine the braid indices for a much larger class of links. In the case of the double crossover polyhedral links, our formulation of the braid index is a simple formula based on a simpler graph used as a template to build the double crossover polyhedral links.

The calculation of topological structures of strands-based DNA trigonal bipyramids

With the rapid development of DNA nanotechnology, strands-based DNA polyhedra have been reported and widely applied in chemical biology, drug delivery and materials science. A fundamental problem is to determine which topological structures such a DNA polyhedron will allow. Our goal in this paper is to determine all permissible nontrivial topological structures for DNA trigonal bipyramids with double-helical edges, which have been partially realized by one-step self-assembly of multiple DNA single-strands designed rationally. Here oriented trigonal bipyramid links (OTB links) are constructed as the mathematical models for DNA trigonal bipyramids and are further classified into 451 link types. In this process, an OTBL program is designed in the Python programming language to generate all OTB link diagrams and also to remove the same topological structures produced by the rotational symmetry of trigonal bipyramid. Our result gives a list of candidates for further synthesized DNA trigonal bipyramid prisms with required topological structures.

Intrinsic Dynamics Analysis of a DNA Octahedron by Elastic Network Model

DNA is a fundamental component of living systems where it plays a crucial role at both functional and structural level. The programmable properties of DNA make it an interesting building block for the construction of nanostructures. However, molecular mechanisms for the arrangement of these well-defined DNA assemblies are not fully understood. In this paper, the intrinsic dynamics of a DNA octahedron has been investigated by using two types of Elastic Network Models (ENMs). The application of ENMs to DNA nanocages include the analysis of the intrinsic flexibilities of DNA double-helices and hinge sites through the calculation of the square fluctuations, as well as the intrinsic collective dynamics in terms of cross-collective map calculation coupled with global motions analysis. The dynamics profiles derived from ENMs have then been evaluated and compared with previous classical molecular dynamics simulation trajectories. The results presented here revealed that ENMs can provide useful insights into the intrinsic dynamics of large DNA nanocages and represent a useful tool in the field of structural DNA nanotechnology.

A general method for computing the Homfly polynomial of DNA double crossover 3-regular links

In the last 20 years or so, chemists and molecular biologists have synthesized some novel DNA polyhedra. Polyhedral links were introduced to model DNA polyhedra and study topological properties of DNA polyhedra. As a very powerful invariant of oriented links, the Homfly polynomial of some of such polyhedral links with small number of crossings has been obtained. However, it is a challenge to compute Homfly polynomials of polyhedral links with large number of crossings such as double crossover 3-regular links considered here. In this paper, a general method is given for computing the chain polynomial of the truncated cubic graph with two different labels from the chain polynomial of the original labeled cubic graph by substitutions. As a result, we can obtain the Homfly polynomial of the double crossover 3-regular link which has relatively large number of crossings.

Ear decomposition of 3-regular polyhedral links with applications

In this paper, we introduce a notion of ear decomposition of 3-regular polyhedral links based on the ear decomposition of the 3-regular polyhedral graphs. As a result, we obtain an upper bound for the braid index of 3-regular polyhedral links. Our results may be used to characterize and analyze the structure and complexity of protein polyhedra and entanglement in biopolymers.