Structure of nanoscale truncated octahedral DNA cages: variation of single-stranded linker regions and influence on assembly yields

Comparative Simulative Analysis and Design of Single-Chain Self-Assembled Protein Cages

Coiled-coil protein origami (CCPO) is a modular strategy for the de novo design of polypeptide nanostructures. It represents a type of modular design based on pairwise-interacting coiled-coil (CC) units with a single-chain protein programmed to fold into a polyhedral cage. However, the mechanisms underlying the self-assembly of the protein tetrahedron are still not fully understood. In the present study, 18 CCPO cages with three different topologies were modeled in silico. Then, molecular dynamics simulations and CC parameters were calculated to characterize the dynamic properties of protein tetrahedral cages at both the local and global levels. Furthermore, a deformed CC unit was redesigned, and the stability of the new cage was significantly improved.

Aptamer-based targeted delivery systems for cancer treatment using DNA origami and DNA nanostructures

Due to the limitations of conventional cancer treatment methods, nanomedicine has appeared as a promising alternative, allowing improved drug targeting and decreased drug toxicity. In the development of cancer nanomedicines, among various nanoparticles (NPs), DNA nanostructures are more attractive because of their precisely controllable size, shape, excellent biocompatibility, programmability, biodegradability, and facile functionalization. Aptamers are introduced as single-stranded RNA or DNA molecules with recognize their corresponding targets. So, incorporating aptamers into DNA nanostructures led to influential vehicles for bioimaging and biosensing as well as targeted cancer therapy. In this review, the recent developments in the application of aptamer-based DNA origami and DNA nanostructures in advanced cancer treatment have been highlighted. Some of the main methods of cancer treatment are classified as chemo-, gene-, photodynamic- and combined therapy. Finally, the opportunities and problems for targeted DNA aptamer-based nanocarriers for medicinal applications have also been discussed.

Insights on the molecular mechanisms of cytotoxicity induced by AS1411 linked to folate-functionalized DNA nanocages in cancer cells

Self-assembled multivalent DNA nanocages are an emerging class of molecules useful for biomedicine applications. Here, we investigated the molecular mechanisms of cytotoxicity induced by AS1411 free aptamer, AS1411-linked nanocages (Apt-NCs) and nanocages harboring both folate and AS1411 functionalization (Fol-Apt-NCs) in HeLa and MDA-MB-231 cancer cell lines. The three treatments showed different cytotoxic efficacy and Fol-Apt-NCs resulted the most effective in inhibiting cell proliferation and inducing apoptotic pathways and ROS activation in both HeLa and MDA-MB-231 cells. RNA-seq analysis allowed to identify biological functions and genes altered by the various treatments, depending on the AS1411 route of intracellular entry, highlighting the different behavior of the two cancer cell lines. Notably, Fol-Apt-NCs altered the expression of a subset of genes associated to cancer chemoresistance in MDA-MB-231, but not in HeLa cells, and this may explain the increased chemosensitivity to drugs delivered through DNA nanocages of the triple-negative breast cancer cells.

Folate-Functionalization Enhances Cytotoxicity of Multivalent DNA Nanocages on Triple-Negative Breast Cancer Cells

DNA is an excellent programmable polymer for the generation of self-assembled multivalent nanostructures useful for biomedical applications. Herein, we developed (i) folate-functionalized nanocages (Fol-NC), very efficiently internalized by tumor cells overexpressing the α isoform of the folate receptor; (ii) AS1411-linked nanocages (Apt-NC), internalized through nucleolin, a protein overexpressed in the cell surface of many types of cancers; and (iii) nanostructures that harbor both folate and AS1411 aptamer functionalization (Fol-Apt-NC). We analyzed the specific miRNA silencing activity of all types of nanostructures harboring miRNA sequestering sequences complementary to miR-21 and the cytotoxic effect when loaded with doxorubicin in a drug-resistant triple-negative breast cancer cell line. We demonstrate that the presence of folate as a targeting ligand increases the efficiency in miR-21 silencing compared to nanocages functionalized with AS1411. Double-functionalized nanocages (Fol-Apt-NC), loaded with doxorubicin, resulted in an increase of over 51% of the cytotoxic effect on MDA-MB-231 cells compared to free doxorubicin, demonstrating, besides selectivity, the ability of nanocages to overcome Dox chemoresistance. The higher efficiency of the folate-functionalized nanocages is due to the way of entrance, which induces more than four times higher intracellular stability and indicates that the folate-mediated route of cell entry is more efficient than the nucleolin-mediated one when both folate and AS1411 modifications are present.

Loading of DOX into a tetrahedral DNA nanostructure: the corner does matter

With the rapid development of nanotechnology, various DNA nanostructures have been synthesized and widely used in drug delivery. However, the underlying mechanisms of drug molecule loading into the DNA nanostructure are still elusive. In this work, we systematically investigate the interactions of a tetrahedral DNA nanostructure (TDN) with the anti-cancer drug doxorubicin (DOX) by combining molecular docking and all-atom molecular dynamics simulations. It is found that there are five possible binding modes in the single TDN-DOX interactions, namely the outside-corner mode, the inside-corner mode, the major-groove mode, the minor-groove mode, and the intercalation mode, where the van der Waals (VDW) interaction and the electrostatic (ELE) interaction dominate in the case of unionized DOX and ionized DOX, respectively. Moreover, with the increase of the DOX number, some of the interaction modes may disappear and the inside-corner mode is the most energy-favorable mode. The present study enhances the molecular understanding of the role of TDN as the drug carrier, which may provide a useful guideline for the future design of DNA nanostructures.

DNA origami nano-mechanics

Invention of DNA origami has transformed the fabrication and application of biological nanomaterials. In this review, we discuss DNA origami nanoassemblies according to their four fundamental mechanical properties in response to external forces: elasticity, pliability, plasticity and stability. While elasticity and pliability refer to reversible changes in structures and associated properties, plasticity shows irreversible variation in topologies. The irreversible property is also inherent in the disintegration of DNA nanoassemblies, which is manifested by its mechanical stability. Disparate DNA origami devices in the past decade have exploited the mechanical regimes of pliability, elasticity, and plasticity, among which plasticity has shown its dominating potential in biomechanical and physiochemical applications. On the other hand, the mechanical stability of the DNA origami has been used to understand the mechanics of the assembly and disassembly of DNA nano-devices. At the end of this review, we discuss the challenges and future development of DNA origami nanoassemblies, again, from these fundamental mechanical perspectives.

AS1411 Aptamer Linked to DNA Nanostructures Diverts Its Traffic Inside Cancer Cells and Improves Its Therapeutic Efficacy

The nucleolin-binding G-quadruplex AS1411 aptamer has been widely used for cancer therapy and diagnosis and linked to nanoparticles for its selective targeting activity. We applied a computational and experimental integrated approach to study the effect of engineering AS1411 aptamer on an octahedral truncated DNA nanocage to obtain a nanostructure able to combine selective cancer-targeting and anti-tumor activity. The nanocages functionalized with one aptamer molecule (Apt-NC) displayed high stability in serum, were rapidly and selectively internalized in cancer cells through an AS1411-dependent mechanism, and showed over 200-fold increase in anti-cancer activity when compared with the free aptamer. Comparison of Apt-NCs and free AS1411 intracellular distribution showed that they traffic differently inside cells: Apt-NCs distributed through the endo-lysosomal pathway and were never found in the nuclei, while the free AS1411 was mostly found in the perinuclear region and in nucleoli. Molecular dynamics simulations indicated that the aptamer, when linked to the nanocage, sampled a limited conformational space, more confined than in the free state, which is characterized by a large number of metastable conformations. A different intracellular trafficking of Apt-NCs compared with free aptamer and the confined aptamer conformations induced by the nanocage were likely correlated with the high cytotoxic enhancement, suggesting a structure-function relationship for the AS1411 aptamer activity.

Biological and Medical Applications of Calcium Phosphate Nanoparticles

Calcium phosphate nanoparticles have a high biocompatibility and biodegradability due to their chemical similarity to human hard tissue, for example, bone and teeth. They can be used as efficient carriers for different kinds of biomolecules such as nucleic acids, proteins, peptides, antibodies, or drugs, which alone are not able to enter cells where their biological effect is required. They can be loaded with cargo molecules by incorporating them, unlike solid nanoparticles, and also by surface functionalization. This offers protection, for example, against nucleases, and the possibility for cell targeting. If such nanoparticles are functionalized with fluorescing dyes, they can be applied for imaging in vitro and in vivo. Synthesis, functionalization and cell uptake mechanisms of calcium phosphate nanoparticles are discussed together with applications in transfection, gene silencing, imaging, immunization, and bone substitution. Biodistribution data of calcium phosphate nanoparticles in vivo are reviewed.

Combined and selective miR-21 silencing and doxorubicin delivery in cancer cells using tailored DNA nanostructures

MicroRNAs play an important role in tumorigenesis and, among them, miR-21 is found to be aberrantly up-regulated in various tumors. The tumor-associated antigen, folate receptor alpha is a GPI-membrane protein overexpressed in many malignant tumors of epithelial origin, including ovarian and cervical cancers. Covalently bound octahedral DNA nanocages were functionalized with folate molecules and utilized as scaffolds to engineer four sequestering units with a miR-21 complementary sequence for obtaining biocompatible Fol-miR21-NC non-toxic nanostructures, to be able to selectively recognize folate receptor alpha-overexpressing cancer cells and sequester the oncogenic miR-21. qPCR assays showed that Fol-miR21-NCs reduce the miR-21 expression up to 80% in cancer cells in the first 2 days of treatment. Functional assays demonstrated that miR-21 sequestering leads to up-regulation of miR-21 tumor suppressor targets (i.e., PTEN and Pdcd4), reduction in cancer cell migration, reduction in proliferation, and increase in cell death. Fol-miR21-NCs can be efficiently loaded with the chemotherapeutic agent doxorubicin. Co-delivery of anti-miR-21 and doxorubicin showed additive cytotoxic effects on tumor cells, paving the way for their use as selective nucleic acid drugs.

The allosteric activation mechanism of a phospholipase A2-like toxin from Bothrops jararacussu venom: a dynamic description

The activation process of phospholipase A₂-like (PLA₂-like) toxins is a key step in their molecular mechanism, which involves oligomeric changes leading to the exposure of specific sites. Few studies have focused on the characterization of allosteric activators and the features that distinguish them from inhibitors. Herein, a comprehensive study with the BthTX-I toxin from Bothrops jararacussu venom bound or unbound to α-tocopherol (αT) was carried out. The oligomerization state of BthTX-I bound or unbound to αT in solution was studied and indicated that the toxin is predominantly monomeric but tends to oligomerize when complexed with αT. In silico molecular simulations showed the toxin presents higher conformational changes in the absence of αT, which suggests that it is important to stabilize the structure of the toxin. The transition between the two states (active/inactive) was also studied, showing that only the unbound BthTX-I system could migrate to the inactive state. In contrast, the presence of αT induces the toxin to leave the inactive state, guiding it towards the active state, with more regions exposed to the solvent, particularly its active site. Finally, the structural determinants necessary for a molecule to be an inhibitor or activator were analyzed in light of the obtained results.

In Silico and In Cell Analysis of Openable DNA Nanocages for miRNA Silencing

A computational and experimental integrated approach was applied in order to study the effect of engineering four DNA hairpins into an octahedral truncated DNA nanocage, to obtain a nanostructure able to recognize and bind specific oligonucleotide sequences. Modeling and classical molecular dynamics simulations show that the new H4-DNA nanocage maintains a stable conformation with the closed hairpins and, when bound to complementary oligonucleotides produces an opened conformation that is even more stable due to the larger hydrogen bond number between the hairpins and the oligonucleotides. The internal volume of the open conformation is much larger than the closed one, switching from 370 to 650 nm³, and the predicted larger conformational change is experimentally detectable by gel electrophoresis. H4-DNA nanocages display high stability in serum, can efficiently enter the cells where they are stable and maintain the ability to bind, and sequester an intracellular-specific oligonucleotide. Moreover, H4-DNA nanocages, modified in order to recognize the oncogenic miR21, are able to seize miRNA molecules inside cells in a selective manner.

Engineering a responsive DNA triple helix into an octahedral DNA nanostructure for a reversible opening/closing switching mechanism: a computational and experimental integrated study

We propose an experimental and simulative approach to study the effect of integrating a DNA functional device into a large-sized DNA nanostructure. We selected, as a test bed, a well-known and characterized pH-dependent clamp-switch, based on a parallel DNA triple helix, to be integrated into a truncated octahedral scaffold. We designed, simulated and experimentally characterized two different functionalized DNA nanostructures, with and without the presence of a spacer between the scaffold and the functional elements. The experimental and simulative data agree in validating the need of a spacer for the occurrence of the pH dependent switching mechanism. The system is fully reversible and the switching can be monitored several times without any perturbation, maintaining the same properties of the isolated clamp switch in solution.

Probing the Functional Topology of a pH-Dependent Triple Helix DNA Nanoswitch Family through Gaussian Accelerated MD Simulation

The topology of a pH-dependent triple helix DNA nanoswitch family has been characterized through simulative analysis to evaluate the efficiency of the switching mechanism varying the length of the loop connecting the two strands forming the double helix portion. In detail, the system is formed by a double helix made by two six base complementary sequences, connected by one loop having an increasing number of thymidines, namely 5, 7, or 9. The triplex-forming sequence made by six bases, connected to the double helix through a constant 25 base loop, interacts at pH 5.0 through Hoogsteen hydrogen bonds with one strand of the double helical region. We demonstrate, through molecular dynamics simulation, that the thymidine loop length exerts a fine regulatory role for the stability of the triple helix structure and is critical in modulating the switching mechanism triggered by the pH increase.

Closing the Loop: Constraining TAT Peptide by γPNA Hairpin for Enhanced Cellular Delivery of Biomolecules

Based on the exceptionally high stability of γPNA duplexes, we designed a peptide/γPNA chimera in which a cell-penetrating TAT peptide is flanked by two short complementary γPNA segments. Intramolecular hybridization of the γPNA segments results in a stable hairpin conformation in which the TAT peptide is constrained to form the loop. The TAT/γPNA hairpin (self-cyclized TAT peptide) enters cells at least 10-fold more efficiently than its nonhairpin analog in which the two γPNA segments are noncomplementary. Extending one of the γPNA segments in the hairpin results in an overhang that can be used for binding and delivering a variety of nucleic acid-conjugated molecules into cells via hybridization to the overhang. We demonstrated efficient cellular delivery of a protein (as low as 10 nM) and a DNA tetrahedron by a TAT/γPNA hairpin.

Position Accuracy of Gold Nanoparticles on DNA Origami Structures Studied with Small-Angle X-ray Scattering

DNA origami objects allow for accurate positioning of guest molecules in three dimensions. Validation and understanding of design strategies for particle attachment as well as analysis of specific particle arrangements are desirable. Small-angle X-ray scattering (SAXS) is suited to probe distances of nano-objects with subnanometer resolution at physiologically relevant conditions including pH and salt and at varying temperatures. Here, we show that the pair density distribution function (PDDF) obtained from an indirect Fourier transform of SAXS intensities in a model-free way allows to investigate prototypical DNA origami-mediated gold nanoparticle (AuNP) assemblies. We analyze the structure of three AuNP-dimers on a DNA origami block, an AuNP trimer constituted by those dimers, and a helical arrangement of nine AuNPs on a DNA origami cylinder. For the dimers, we compare the model-free PDDF and explicit modeling of the SAXS intensity data by superposition of scattering intensities of the scattering objects. The PDDF of the trimer is verified to be a superposition of its dimeric contributions, that is, here AuNP-DNA origami assemblies were used as test boards underlining the validity of the PDDF analysis beyond pairs of AuNPs. We obtain information about AuNP distances with an uncertainty margin of 1.2 nm. This readout accuracy in turn can be used for high precision placement of AuNP by careful design of the AuNP attachment sites on the DNA-structure and by fine-tuning of the connector types.

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.

DNA tetrahedron nanostructures for biological applications: biosensors and drug delivery

Herein, we review and summarise the development and biological applications of DNA tetrahedron, including cellular biosensors and drug delivery systems.

DNA hairpins promote temperature controlled cargo encapsulation in a truncated octahedral nanocage structure family

In the present study we investigate the mechanism behind temperature controlled cargo uptake using a truncated octahedral DNA cage scaffold functionalized with one, two, three or four hairpin forming DNA strands inserted in one corner of the structure. This investigation was inspired by our previous demonstration of temperature controlled reversible encapsulation of the cargo enzyme, horseradish peroxidase, in the cage with four hairpin forming strands. However, in this previous study the mechanism of cargo uptake was not directly addressed (Juul, et al., Temperature-Controlled Encapsulation and Release of an Active Enzyme in the Cavity of a Self-Assembled DNA Nanocage, ACS Nano, 2013, 7, 9724-9734). In the present study we use a combination of molecular dynamics simulations and in vitro analyses to unravel the mechanism of cargo uptake in hairpin containing DNA cages. We find that two hairpin forming strands are necessary and sufficient to facilitate efficient cargo uptake, which argues against a full opening-closing of one corner of the structure being responsible for encapsulation. Molecular dynamics simulations were carried out to evaluate the atomistic motions responsible for encapsulation and showed that the two hairpin forming strands facilitated extension of at least one of the face surfaces of the cage scaffold, allowing entrance of the cargo protein into the cavity of the structure. Hence, the presented data demonstrate that cargo uptake does not involve a full opening of the structure. Rather, the uptake mechanism represents a feature of increased flexibility integrated in this nanocage structure upon the addition of at least two hairpin-forming strands.

Shape and Interhelical Spacing of DNA Origami Nanostructures Studied by Small-Angle X-ray Scattering

Scaffolded DNA origami nanostructures enable the self-assembly of arbitrarily shaped objects with unprecedented accuracy. Yet, varying physiological conditions are prone to induce slight structural changes in the nanoscale architecture. Here, we report on high precision measurements of overall shape and interhelical distance of three prototypic DNA origami structures in solution using synchrotron small-angle X-ray scattering. Sheet-, brick-, and cylinder-shaped DNA constructs were assembled and the shape factors determined with angstrom resolution from fits to the scattering profiles. With decreasing MgCl2 concentration electrostatic swelling of both shape cross section and interhelical DNA spacing of the DNA origami structures is observed. The structures tolerate up to 10% interhelical expansion before they disintegrate. In contrast, with increasing temperature, the cylinder-shaped structures show no thermal expansion in a wide temperature window before they abruptly melt above 50 °C. Details on molecular structure of DNA origami can also be obtained using in-house X-ray scattering equipment and, hence, allow for routine folding and stability testing of DNA-based agents that are designed to operate under varying salt conditions.

Receptor-Mediated Entry of Pristine Octahedral DNA Nanocages in Mammalian Cells

DNA offers excellent programming properties for the generation of nanometer-scaled polyhedral structures with a broad variety of potential applications. Translation to biomedical applications requires improving stability in biological fluids, efficient and selective cell binding, and/or internalization of the assembled DNA nanostructures. Here, we report an investigation on the selective mechanism of cellular uptake of pristine DNA nanocages in cells expressing the receptor "oxidized low-density lipoprotein receptor-1" (LOX-1), a scavenger receptor associated with cardiovascular diseases and, more recently, identified as a tumor marker. For this purpose a truncated octahedral DNA nanocage functionalized with a single biotin molecule, which allows DNA cage detection through the biotin-streptavidin assays, was constructed. The results indicate that DNA nanocages are stable in biological fluids, including human serum, and are selectively bound and very efficiently internalized in vesicles only in LOX-1-expressing cells. The amount of internalized cages is 30 times higher in LOX-1-expressing cells than in normal fibroblasts, indicating that the receptor-mediated uptake of pristine DNA nanocages can be pursued for a selective cellular internalization. These results open the route for a therapeutic use of pristine DNA cages targeting LOX-1-overexpressing tumor cells.

Influence of the single-strand linker composition on the structural/dynamical properties of a truncated octahedral DNA nano-cage family

The structural/dynamical properties of three truncated octahedral DNA nano-cages composed by identical double helices but single strand linkers with different composition, namely 7 thymidines, 7 adenines, and 7 alternated thymidines and adenines, have been investigated through classical molecular dynamics simulations. Trajectories have been analyzed to investigate the role of the linkers in defining nano-cages stability and flexibility, including possible influence on the internal cages motions. The data indicate that the cages behavior is almost identical and that the structural/dynamical parameters measured along the trajectories are not particularly affected by the presence of different bases. These results demonstrate that the constraints imposed by the nano-structure geometry are the main factor in modulating these properties

A Simple and Fast Semiautomatic Procedure for the Atomistic Modeling of Complex DNA Polyhedra

A semiautomatic procedure to build complex atomistic covalently linked DNA nanocages has been implemented in a user-friendly, free, and fast program. As a test set, seven different truncated DNA polyhedra, composed by B-DNA double helices connected through short single-stranded linkers, have been generated. The atomistic structures, including a tetrahedron, a cube, an octahedron, a dodecahedron, a triangular prism, a pentagonal prism, and a hexagonal prism, have been probed through classical molecular dynamics and analyzed to evaluate their structural and dynamical properties and to highlight possible building faults. The analysis of the simulated trajectories also allows us to investigate the role of the different geometries in defining nanocages stability and flexibility. The data indicate that the cages are stable and that their structural and dynamical parameters measured along the trajectories are slightly affected by the different geometries. These results demonstrate that the constraints imposed by the covalent links induce an almost identical conformational variability independently of the three-dimensional geometry and that the program presented here is a reliable and valid tool to engineer DNA nanostructures.

The Potential of Nucleic Acid-Based Nanoparticles for Biomedical Application

The recent development of biomedical nanotechnology is providing the appropriate know-how to build novel nanocarriers/nanocapsules to be used in the pharmaceutical industry for delivery of drugs and/or therapeutic payloads to specific cells. DNA is an extremely suitable polymer for the generation of nanocapsules being biocompatible, stable and chemically modifiable. Moreover, the simple four bases mechanism allows the auto-assembly of geometrically defined systems. In this review, we describe the general properties of DNA nanocarriers, how they can be functionalized for different tasks, their interaction with cellular systems and we provide an outlook of their use in a therapeutic perspective.

Comparative simulative analysis of single and double stranded truncated octahedral DNA nanocages

Spacefill view of double (DSL) and single (SSL) stranded linkers DNA cages. The blue atoms represent the shared cages scaffold, while the yellow atoms show the single stranded DNA oligonucleotides shaping the double stranded linkers of the DSL cage.

Temperature-controlled encapsulation and release of an active enzyme in the cavity of a self-assembled DNA nanocage

We demonstrate temperature-controlled encapsulation and release of the enzyme horseradish peroxidase using a preassembled and covalently closed three-dimensional DNA cage structure as a controllable encapsulation device. The utilized cage structure was covalently closed and composed of 12 double-stranded B-DNA helices that constituted the edges of the structure. The double stranded helices were interrupted by short single-stranded thymidine linkers constituting the cage corners except for one, which was composed by four 32 nucleotide long stretches of DNA with a sequence that allowed them to fold into hairpin structures. As demonstrated by gel-electrophoretic and fluorophore-quenching experiments this design imposed a temperature-controlled conformational transition capability to the structure, which allowed entrance or release of an enzyme cargo at 37 °C while ensuring retainment of the cargo in the central cavity of the cage at 4 °C. The entrapped enzyme was catalytically active inside the DNA cage and was able to convert substrate molecules penetrating the apertures in the DNA lattice that surrounded the central cavity of the cage.

The role of defects on the assembly and stability of DNA nanostructures

Defects are known to underlie the mechanical properties of materials, especially so at the nanoscale. Using four compositionally identical DNA triangles, defect density is found to be inversely correlated with assembly efficiency and melting temperature. These findings are supported by a series of experiments with more complex DNA pyramids. Because they are naturally responsive to stresses, defects present an attractive opportunity as design elements for responsive DNA materials.

A new Euler's formula for DNA polyhedra

DNA polyhedra are cage-like architectures based on interlocked and interlinked DNA strands. We propose a formula which unites the basic features of these entangled structures. It is based on the transformation of the DNA polyhedral links into Seifert surfaces, which removes all knots. The numbers of components , of crossings , and of Seifert circles are related by a simple and elegant formula: . This formula connects the topological aspects of the DNA cage to the Euler characteristic of the underlying polyhedron. It implies that Seifert circles can be used as effective topological indices to describe polyhedral links. Our study demonstrates that, the new Euler's formula provides a theoretical framework for the stereo-chemistry of DNA polyhedra, which can characterize enzymatic transformations of DNA and be used to characterize and design novel cages with higher genus.

Nucleic acid-based nanoengineering: novel structures for biomedical applications

Nanoengineering exploits the interactions of materials at the nanometre scale to create functional nanostructures. It relies on the precise organization of nanomaterials to achieve unique functionality. There are no interactions more elegant than those governing nucleic acids via Watson-Crick base-pairing rules. The infinite combinations of DNA/RNA base pairs and their remarkable molecular recognition capability can give rise to interesting nanostructures that are only limited by our imagination. Over the past years, creative assembly of nucleic acids has fashioned a plethora of two-dimensional and three-dimensional nanostructures with precisely controlled size, shape and spatial functionalization. These nanostructures have been precisely patterned with molecules, proteins and gold nanoparticles for the observation of chemical reactions at the single molecule level, activation of enzymatic cascade and novel modality of photonic detection, respectively. Recently, they have also been engineered to encapsulate and release bioactive agents in a stimulus-responsive manner for therapeutic applications. The future of nucleic acid-based nanoengineering is bright and exciting. In this review, we will discuss the strategies to control the assembly of nucleic acids and highlight the recent efforts to build functional nucleic acid nanodevices for nanomedicine.