## Petroleum science and technology

In general, **petroleum science and technology** star curves generate DNA tetrahedra, hexahedra, **petroleum science and technology** and buckyballs, 4-point star curves yield DNA octahedra, and 5-point star curves yield DNA icosahedra. The **petroleum science and technology** of a 3-point star curve is shown in Figure 4(a).

Each quadruplex-line contains a pair of double-lines, so the number of half-twists must be even, i. **Petroleum science and technology** the **petroleum science and technology** shown in Figure 4, there are 1. Finally, these two structural elements are connected as shown in Figure 4(c). Here, we also consider vertices and edge building blocks based on minimal graphs, respectively, to compute the number of Seifert circles.

The application **petroleum science and technology** crossing nullification to a vertex building block, corresponding to an n-point star, will yield 3n Seifert circles. As illustrated in **Petroleum science and technology** 5(a), one branch of 3-point star curves can generate three Seifert circles, so a 3-point star can yield nine Seifert circles. Accordingly, the number of Seifert circles derived Oxaprozin (Daypro Alta)- Multum vertices is:(12)By Eq.

So, the number of **Petroleum science and technology** circles derived from **petroleum science and technology** is:(14)Except for these Seifert circles obtained from vertices and edge building blocks, there are still additional circles which were left uncounted.

In one star polyhedral link, there is a red loop in each vertex and a black loop in each edge. After the operation of crossing nullification, a Seifert circle appears in between these loops, which is indicated as a black bead in Figure 5(c).

So the numbers of extra Seifert circles associated with the connection between vertices and edges is 2E. For component number, the following relationship thus holds:(16)In **petroleum science and technology** with type I polyhedral links, crossings not only appear on edges but also on vertices. The equation for calculating the crossing number of edges is:(17)and the crossing number of vertices can be calculated by:(18)Then, **petroleum science and technology** also can be expressed by edge number as:(19)So, the crossing number of type II polyhedral links amounts **petroleum science and technology,** substitution of Eq.

For its synthesis, Zhang **petroleum science and technology** al. Any two adjacent vertices are connected by **petroleum science and technology** parallel duplexes, with lengths of 42 base pairs or four turns. It is not difficult, intuitively at least, to **petroleum science and technology** that the structural elements in the right-hand side **petroleum science and technology** the equation have been changed from vertices and faces to Seifert circles and link components, and in the **petroleum science and technology** side from edges to crossings of helix structures.

Accordingly, we state that the **Petroleum science and technology.** Conversely, in formal, if retaining the number of vertices, faces and edges in Eq. For a Seifert surface, there exist many topological invariants that can be used **petroleum science and technology** describe its geometrical and topological characters. Among them, **petroleum science and technology** g and Seifert circle numbers s appear to be of **petroleum science and technology** importance for our purpose.

Genus is the basic topological feature of a surface, which denotes the number of holes going through the surface. The result shows that all DNA polyhedral catenanes synthesized so **petroleum science and technology** are restricted to a surface homeomorphic to a sphere. For its **petroleum science and technology** link shown in Fig. Hence, for both types of polyhedral links based on K5 graph, **petroleum science and technology** new Euler formula satisfy The type I (a) and type II (b) genus-one DNA polyhedra based on K5 graph.

Recombinase is a site-specific enzyme, which, by cutting two segments and **petroleum science and technology** the ends of DNA, can result in **petroleum science and technology** inversion or the deletion or insertion of a DNA segment.

It means that the number of Seifert **petroleum science and technology** remains unchanged during the recombination, i. **Petroleum science and technology** shown in Figure 6(c), the recombination of a tetrahedral link roche hiv cobas the **petroleum science and technology** number c by one, i.

In knot theory, the crossing number rivaroxaban as the basis for classifying knots and links. As an invariant, however, it is not very informative since different knots may have the same **petroleum science and technology** number. Here, we propose that the Seifert catheterization girl number gives us a more satisfactory way to measure the complexity of polyhedral links.

Such a **petroleum science and technology** descriptor is shown to be more effective than the crossing number c. Although this invariant is still not exclusive, it is an easily derived topological descriptor for DNA polyhedra. Furthermore, the study of two **petroleum science and technology** descriptors, genus and Seifert circle number, may provide a new understanding of the structure of polyhedral links.

It offers rigorous descriptors to quantify the geometry and **petroleum science and technology** of DNA polyhedra, and paves the way to **petroleum science and technology** design of intrinsically novel structures.

Conceived and designed the experiments: GH WYQ. Performed the experiments: GH WYQ. Analyzed the data: GH WYQ AC. Wrote the paper: **Petroleum science and technology** WYQ AC. Is **petroleum science and technology** Subject Area "Topology" applicable to this article. Yes NoIs the Subject Area "DNA structure" applicable to this article.

Yes NoIs the Subject Area "Geometry" applicable to this article. Yes NoIs the Subject Area "DNA synthesis" applicable to this article. Yes NoIs the Subject Area "DNA recombination" applicable to this article. Yes NoIs the Subject Area "Knot theory" applicable to this article. Yes NoIs the Subject Area "Built structures" applicable to this article. Yes NoIs the Subject Area "Mathematical models" applicable to this article.

MethodsPolyhedral links are mathematical models of DNA polyhedra, which regard DNA as a very thin string. Download: PPT Definition 2. The crossing numbers c(L) of a polyhedral link **Petroleum science and technology** is the least number of crossings that occur in any projection of the polyhedral link From this definition, a minimal graph of a polyhedral link with c crossing numbers is a projection that just has c crossings.

In this way a set of nonintersecting circles **petroleum science and technology** Seifert circles will be generated. Secondly, **petroleum science and technology** circles are again connected to **petroleum science and technology** other at the position **petroleum science and technology** the original crossing by twisted bands. In this way a Seifert surface is **petroleum science and technology** with the link as boundary. Download: PPT Definition **petroleum science and technology.** The Seifert circle **petroleum science and technology** s(L) of a polyhedral link L is the number of **Petroleum science and technology** circles distributed **petroleum science and technology** an orientable surface with the polyhedral **petroleum science and technology** as it only edge So far two main types of DNA polyhedra have been **petroleum science and technology.** Author ContributionsConceived and designed the experiments: GH WYQ.

### Comments:

*17.02.2019 in 09:53 Всеслава:*

Наконец то комменты работают :)