Prediction generation method and system for wireframe DNA polyhedral topological structure

Abstract

The invention provides a prediction generation method and system for a wireframe DNA polyhedral topological structure, and relates to the technical field of DNA polyhedral structure prediction, and the method comprises the steps: obtaining a planar graph and an incidence matrix of the planar graph; based on the incidence matrix of the planar graph, generating all empowerment incidence matrixes of the planar graph according to a set matrix iterative algorithm; extracting an edge set and a vertex set from the incidence matrix of the planar graph, generating all surface rings of the planar graph based on a planar graph algorithm, constructing an ordered set by utilizing all the surface rings to generate a plane graph of the planar graph, traversing the plane graph according to each weighting incidence matrix, and for each edge of the plane graph, carrying out weight determination on each edge of the plane graph; finding out two surface rings containing the edge from the ordered set, respectively extracting two adjacent edges of the edge on the two surface rings, and constructing an ordered array by using the obtained four adjacent edges; and according to the weight of the edge of the plain map in the weighting incidence matrix and the related ordered array, traversing and outputting the anti-parallel plain map chain rings and the chain ring branches and the number of the anti-parallel plain map chain rings.

Description

Technical Field

[0001] The present disclosure relates to the technical field of DNA polyhedron structure prediction, and in particular to a method and system for predicting and generating the topological structure of a wireframe DNA polyhedron. Background Art

[0002] The statements in this section merely provide background information related to the present disclosure and do not necessarily constitute prior art.

[0003] Wireframe DNA polyhedra are topological molecules with a polyhedral geometric framework formed by single-stranded DNA as a topological primitive. Wireframe DNA polyhedra not only include polyhedral knot molecules formed by the folding of a single DNA strand, but also catenane molecules formed by each DNA strand wandering around a face or point of the polyhedron, such as tetrahedrons, cubes, octahedrons, triangular prisms, and so on. Therefore, for any given polyhedron, it is impossible to know how many DNA strands can form its wireframe polyhedron topological molecule. Furthermore, it is also impossible to know whether the topological structure design is unique, or how to calculate the folding path of each DNA strand. These are some of the fundamental unresolved issues in DNA nanoassembly.

[0004] Currently, there are no effective, universal prediction methods or computational software that can theoretically address these issues. The main existing theoretical methods or computational programs can only predict all possible topological structures of four polyhedrons (i.e., tetrahedron, triangular prism, triangular bipyramid, and octahedron) assembled from a single DNA strand. This presupposes that all possible orientations of these four polyhedrons must be determined. Therefore, existing theoretical methods cannot be generalized to polyhedrons in general. Other theoretical methods only involve the construction of knots for a specific type of wireframe polyhedron. They cannot predict or determine all possible knot structures formed by these polyhedrons from a single DNA strand, nor can they determine the probability that any polyhedron can be formed into a topological structure from a single DNA strand. Summary of the Invention

[0005] In order to solve the above problems, the present disclosure proposes a prediction and generation method and system for the topological structure of wireframe DNA polyhedrons, establishes a mathematical model of antiparallel flat graph chains, and predicts all topological structures including knots, catenaries and other chains generated by the assembly of multiple DNA single strands of any polyhedron. By inputting any polyhedron, all edge weighting matrices of the polyhedron can be output, thereby generating all topological structures generated by the assembly of the polyhedron by several DNA single strands, and outputting the chain branch and number of antiparallel flat graph chains corresponding to each weighting matrix, that is, the assembly path and number of DNA single strands of each wireframe DNA polyhedron based on the polyhedron.

[0006] According to some embodiments, the present disclosure adopts the following technical solutions: A method for predicting and generating a wireframe DNA polyhedron topology structure, comprising: Get a planar graph and its incidence matrix; Based on the incidence matrix of the planar graph, all weighted incidence matrices of the planar graph are generated according to a set matrix iteration algorithm; Extract edge sets and vertex sets from the incidence matrix of the planar graph, generate all face cycles of the planar graph based on the planar graph algorithm, and construct an ordered set using all face cycles to generate a flat graph of the planar graph; Traverse the flat graph according to each weighted association matrix. For each edge of the flat graph, find the two face circles containing the edge from the ordered set, and extract the two adjacent edges of the edge on the two face circles respectively. Use the obtained four adjacent edges to construct an ordered array. Four edge pair sets are derived from the edge of the flat graph and its four adjacent edges. According to the weight of the edge of the flat graph in the weighted association matrix and the related ordered array, the four edge pair sets are divided into two binary sets; the two binary sets are traversed and compared with each element in the given set respectively. If there is a common edge with the given element, they are merged into a new set. Otherwise, the binary set will be collected as a new element into the given set. After traversing each edge of the flat graph, the antiparallel flat graph chain ring and its chain ring branches and number are finally output.

[0007] According to some embodiments, the present disclosure adopts the following technical solutions: A prediction and generation system for wireframe DNA polyhedron topology structure, comprising: A data acquisition module, used for acquiring a planar graph and an incidence matrix of the planar graph; The prediction generation module is used to generate all weighted association matrices of the planar graph according to the set matrix iteration algorithm based on the association matrix of the planar graph; extract the edge set and vertex set from the association matrix of the planar graph, generate all the face circles of the planar graph based on the planar graph algorithm, and use all the face circles to construct an ordered set to generate a flat graph of the planar graph; traverse the flat graph according to each weighted association matrix, for each edge of the flat graph, find the two face circles containing the edge from the ordered set, and extract the two adjacent edges of the edge on the two face circles respectively, and use the obtained An ordered array is constructed from the four adjacent edges of the flat graph; four edge pair sets are derived from the edge of the flat graph and the four adjacent edges, and the four edge pair sets are divided into two binary sets according to the weight of the edge of the flat graph in the weighted association matrix and the related ordered array; the two binary sets are traversed and compared with each element in the given set respectively. If there is a common edge with the given element, they are merged into a new set. Otherwise, the binary set will be collected as a new element into the given set. After traversing each edge of the flat graph, the antiparallel flat graph chain ring and its chain ring branches and number are finally output.

[0008] According to some embodiments, the present disclosure adopts the following technical solutions: A computer program product includes a computer program, which implements the method for predicting and generating the topological structure of a wireframe DNA polyhedron when executed by a processor.

[0009] According to some embodiments, the present disclosure adopts the following technical solutions: A non-transitory computer-readable storage medium is used to store computer instructions. When the computer instructions are executed by a processor, the method for predicting and generating a wireframe DNA polyhedron topology structure is implemented.

[0010] According to some embodiments, the present disclosure adopts the following technical solutions: An electronic device comprises: a processor, a memory, and a computer program; wherein the processor is connected to the memory, the computer program is stored in the memory, and when the electronic device is running, the processor executes the computer program stored in the memory to enable the electronic device to implement a method for predicting and generating wireframe DNA polyhedron topology structures.

[0011] Compared with the prior art, the present invention has the following beneficial effects: The present invention discloses a method for predicting and generating the topological structure of wireframe DNA polyhedrons. The method establishes a universal mathematical model and generation method to describe and predict all topological structures, including knots, catenanes, and other chain links, generated by assembling any polyhedron from multiple DNA single strands. This method fills the current gap in universal calculation methods for the topological structure of wireframe DNA polyhedrons. That is, without being restricted by a particular polyhedron or a specific single knot structure, any polyhedron can be input and the following outputs can be obtained: 1) all edge weight matrices of the polyhedron, which can generate all topological structures generated by assembling the polyhedron from a certain number of DNA single strands; and 2) the chain branching and number of antiparallel planar chain links corresponding to each weighting matrix, which is the assembly path and number of DNA single strands of each wireframe DNA polyhedron based on the polyhedron.

[0012] This paper presents a method for predicting and generating the topology of wireframe DNA polyhedra, providing a general computational software package for the precise design of new topological structures for wireframe DNA polyhedra. The software, packaged as a software package, directly predicts and generates wireframe DNA polyhedra topologies. This software, based on Fortran code, calculates the topology of wireframe DNA polyhedra and their chain branches. Simply inputting a flat graph, the software outputs all antiparallel flat graph chains, their chain branches, and their number. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] The accompanying drawings, which constitute a part of the present disclosure, are used to provide a further understanding of the present disclosure. The exemplary embodiments of the present disclosure and their descriptions are used to explain the present disclosure and do not constitute an improper limitation to the present disclosure.

[0014] Figure 1 Schematic diagram of the modeling of the type I DNA polyhedron and its local structure according to an embodiment of the present disclosure; in, Figure 1 (a) in the figure shows four type I DNA polyhedra; Figure 1 (b) in the figure is the vertex area and its node model; Figure 1 (c) is the spiral edge and its 2-winding model; Figure 2 Schematic diagram of the modeling of the Class II DNA polyhedron and its local structure according to the embodiment of the present disclosure; wherein, Figure 2 (a) in the figure shows four type II DNA polyhedra; Figure 2 (b) in the figure is the vertex area and its node model; Figure 2 (c) is the spiral edge and its 2-winding model; Figure 3 A relationship diagram between a chain link graph D and a flat graph G of an antiparallel flat graph chain link L according to an embodiment of the present disclosure; Figure 4 These are four operations based on polyhedrons in an embodiment of the present disclosure: a represents thinning edges, b represents emphasizing edges, c represents differentiation points, and d represents truncation.

[0015] Figure 5 is a transformation diagram of an embodiment of the present disclosure; in, Figure 5 (a) Nodes -O3 and -O4 are transformed into nodes V3 and V4 through node transformation; Figure 5 The antiparallel graph link D(G) in (b) can be transformed into a special antiparallel graph link D through two nodes. S (G).

[0016] Figure 6 This is a flowchart for calculating all antiparallel flat graph links based on any flat graph according to an embodiment of the present disclosure.

[0017] Figure 7 The four pairs of edges [e,e] formed by the edge e and its four adjacent edges in the embodiment of the present disclosure are: s ]、[e,e a ]、[e,e o ] and [e,e d ], which can represent the corresponding four arc segments of D(G).

[0018] Figure 8 This is a flow chart for calculating the chain link branches and their number of antiparallel planar chain links in an embodiment of the present disclosure. DETAILED DESCRIPTION

[0019] The present disclosure will be further described below with reference to the accompanying drawings and embodiments.

[0020] It should be noted that the following detailed descriptions are illustrative and intended to provide further explanation of the present disclosure. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art to which the present disclosure belongs.

[0021] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present disclosure. As used herein, unless the context clearly indicates otherwise, the singular form is intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, they indicate the presence of features, steps, operations, devices, components and / or combinations thereof.

[0022] Example 1 In one embodiment of the present disclosure, a method for predicting and generating a wireframe DNA polyhedron topology structure is provided, the method comprising the following steps: Step 1: Obtain the planar graph and the incidence matrix of the planar graph; Step 2: Based on the incidence matrix of the planar graph, generate all weighted incidence matrices of the planar graph according to the set matrix iterative algorithm; Step 3: Extract the edge set and vertex set from the incidence matrix of the planar graph, generate all face cycles of the planar graph based on the planar graph algorithm, and use all face cycles to construct an ordered set to generate a flat graph of the planar graph; Step 4: Traverse the flat graph according to each weighted association matrix. For each edge of the flat graph, find the two face circles containing the edge from the ordered set, and extract the two adjacent edges of the edge on the two face circles respectively. Use the obtained four adjacent edges to construct an ordered array. Step 5: Derive four edge pair sets from the edge of the flat graph and the four adjacent edges. Divide the four edge pair sets into two binary sets according to the weight of the edge of the flat graph in the weighted association matrix and the related ordered array; traverse and compare the two binary sets with each element in the given set respectively. If there is a common edge with the given element, they will be merged into a new set. Otherwise, the binary set will be collected as a new element in the given set. After traversing each edge of the flat graph, the antiparallel flat graph chain ring and its chain ring branches and number are finally output.

[0023] As an embodiment, the present disclosure provides a prediction and generation method for the topological structure of a wireframe DNA polyhedron. It establishes a universal mathematical model and generation method for describing and predicting all topological structures, including knots, catenanes, and other chain rings, generated by the assembly of multiple DNA single strands of any polyhedron. It includes establishing a universal mathematical model "antiparallel flat graph chain ring" for the topological structure of a wireframe DNA polyhedron, establishing a universal method for generating all antiparallel flat graph chain rings based on any (no-cut) flat graph, and thereby establishing a universal method and software algorithm for generating chain ring branches and their number of antiparallel flat graph chain rings. By inputting any polyhedron, it can output: 1) all edge weight matrices of the polyhedron, which can generate all topological structures generated by the assembly of the polyhedron by several DNA single strands; 2) the chain ring branches and number of antiparallel flat graph chain rings corresponding to each weighting matrix, which are the assembly path and number of DNA single strands of each wireframe DNA polyhedron based on the face. The specific implementation process is as follows: Step 1: Establish a universal mathematical model of the wireframe DNA polyhedron topology structure, the "antiparallel flat graph chain ring"; Specifically, step 11: obtaining a wireframe DNA polyhedron topological molecule, and dividing the wireframe DNA polyhedron topological molecule into two types according to the number of its double helical edges: type I wireframe DNA polyhedron and type II wireframe DNA polyhedron. 1) Type I wireframe DNA polyhedron: A polyhedron molecule whose edges consist of a DNA double helix, and whose vertex regions are either "superimposed single-stranded domains" or "non-superimposed single-stranded domains", such as Figure 1 As shown in (a).

[0024] 2) Type II wireframe DNA polyhedron: A polyhedron molecule with at least one edge consisting of two parallel DNA double helices, whose vertex regions are "overlapping / non-overlapping single-stranded domains" or "non-overlapping double-stranded domains", e.g. Figure 2 As shown in (a).

[0025] Step 12: Decompose the double helical edges and vertex regions of all wireframe DNA polyhedra into some basic building blocks and build corresponding antiparallel 2-winding edge and node models, including: By decomposing the edges and points of all wireframe DNA polyhedrons, the basic building block of the edge, the "DNA double helix", is simplified to an antiparallel 2-winding T. The antiparallel 2-winding T can be divided into positive and negative categories according to the orientation of the arc segment at its lower left corner, and can be divided into odd and even categories according to the parity of the crossing number. Figure 1 As shown in (c), T4 is negative even 2-winding and T3 is positive odd 2-winding.

[0026] The basic building block of the vertex region, the "stacked single-chain domain", is simplified to V m node, and simplify the "non-stacked single-chain domain" to Node and 0 m Node. Take 2-winding T and node V m , 0 m The opposite direction of all the curves in the winding is recorded as their inverse -T, -V m and -0 m .

[0027] Step 13: Establish an antiparallel flat graph chain link based on the correspondence between the chain link graph and the flat graph, including: Antiparallel flat graph links can be defined by establishing a correspondence between directed link graphs and flat graphs, where each antiparallel 2-winding edge of the link graph corresponds to an edge of the flat graph.

[0028] Furthermore, we show that any antiparallel flat graph chain-link graph can be obtained from a flat graph by replacing the edges of the flat graph with antiparallel 2-twists and connecting the endpoints of the two 2-twists along each face. The chain-link of the antiparallel flat graph is obtained by projecting its chain-link graph into space.

[0029] like Figure 3 As shown, according to the chessboard correspondence rule between the chain graph and the flat graph, Theorem 1.1 can be proved: Theorem 1.1: Given any directed link L and its link graph D, there exists a (cut-free) flat graph G such that every edge of G corresponds to a directed 2-winding edge of D. If every 2-winding edge is antiparallel, then the directed link D is defined as an antiparallel flat graph link graph, and the directed link L is defined as an antiparallel flat graph link.

[0030] Conversely, an antiparallel flat graph chain loop can be constructed from a flat graph: given a flat graph G, replace each of its edges with an antiparallel 2-twist and connect the endpoints of the two 2-twists along each face. The resulting directed chain loop is an antiparallel flat graph chain loop graph D(G). Projecting D(G) into three-dimensional space yields an antiparallel flat graph chain loop L(G). Therefore, the antiparallel flat graph chain loop model preserves both the geometric framework and topological structure of the DNA polyhedron.

[0031] Furthermore, we prove Theorem 1.2: There is a correspondence between any planar graph and a class of antiparallel planar graph links, which are a series of antiparallel planar graph links generated by taking the number of twists of the 2-winding edges as a variable.

[0032] According to the construction method of antiparallel planar graph chain ring, an important property of antiparallel planar graph chain ring is obtained: Important Property 1.3: The vertex area of ​​each type of antiparallel graph link is 0 m Node or -0 m node, m is the degree of the corresponding point in the flat graph G.

[0033] Step 14: The polyhedron is a three-connected planar graph. The extended polyhedron is a planar graph obtained by performing a finite number of operations on the polyhedron, such as "truncation", "weighting edges", "differentiation points" and "thinning edges".

[0034] Based on the correspondence between polyhedron or extended polyhedron and type I and type II DNA polyhedron topological structures, it can be proved that antiparallel flat graph chain ring is a universal mathematical model of wireframe DNA polyhedron topological structure.

[0035] Specifically, based on the topological structures of type I and type II DNA polyhedra and the correspondence between polyhedra and extended polyhedra, Theorem 1.4 can be proved: Proof of Theorem 1.4: Type I or II DNA polyhedral topologies can be generated based on the antiparallel flat graph links of the polyhedron / extended polyhedron flat graph.

[0036] The antiparallel planar chain loop covers and predicts all knots, catenanes and other chain loop structures of wireframe DNA polyhedrons, and is a universal mathematical model of the topological structure of wireframe DNA polyhedrons.

[0037] Step 2: Based on any flat graph, the general method for generating all antiparallel flat graph links includes the following steps: Step 21: Prove that the antiparallel graph chain is equivalent to each edge being a positive even winding edge and each vertex being 0 m or V m Furthermore, any wireframe DNA polyhedron can be designed into a special antiparallel polyhedral chain with the same double helix edge length, including: (1) Establish an important isochronous transformation: given an antiparallel flat graph link, it will be transformed with a -0 m Each crossing of a 2-winding edge associated with a node is continuously transformed to -0 m The position of the node, originally -0 m The node becomes V m Node, the continuous transformation is called node transformation, such as Figure 5 As shown in (a).

[0038] The two links before and after the node transformation are identical under the same trace equivalence. Furthermore, if the node transformation is applied to all -0 links of the antiparallel graph, m The resulting flat graph link at the node is called a special antiparallel flat graph link, such as Figure 5 As shown in (b).

[0039] It can be shown that special antiparallel planar graph links have property 1.5: all 2-winding edges of every special antiparallel planar graph link are positive and even, and every vertex is a node 0 m or Vm .

[0040] Prove Theorem 1.6: Every antiparallel planar graph link is equivalent to a graph with positive even winding edges and 0 vertices. m or V m A special antiparallel flat graph link.

[0041] This leads to Corollary 1.7: Any DNA polyhedral topological molecule can be designed into a special antiparallel polyhedral chain ring with the same double helix edge length.

[0042] And we can also get Property 1.8: For any antiparallel planar graph chain, the directed chains generated by taking the reverse orientation of several chain branches are not antiparallel planar graph chains except its reverse chain.

[0043] Furthermore, Lemma 1.9 is established: For any flat graph G, there is a one-to-one correspondence between the vertex subset of its point set and each type of antiparallel flat graph link based on the flat graph G.

[0044] Step 22: Define an edge-weighted graph of any flat graph using a subset of its vertices, and establish a one-to-one correspondence between the edge-weighted flat graph and the antiparallel flat graph link class. Furthermore, the antiparallel flat graph link based on any flat graph can be generated by establishing all the edge-weighted graphs of the flat graph, including: By inputting the adjacency matrix of any flat graph, an array of 0s and 1s with the number of vertices n as the dimension is generated, where each array corresponds to a subset of the flat graph. Then, according to these 2 n arrays to generate the adjacency matrix of all edge-weighted flat graphs, which can uniquely determine all edge-weighted graphs of the flat graph.

[0045] Step 23: By establishing a linear equivalence algorithm for the adjacency matrix of the edge-weighted flat graph, all equivalent edge-weighted flat graphs can be removed, and finally all edge-weighted matrices are output to generate all antiparallel flat graph links.

[0046] As an embodiment, an edge weighted graph is defined by a vertex set of a flat graph, thereby generating all antiparallel flat graph links, as follows: 1) Given any (possibly) flattenable graph G, take any subset S of the point set V(G). If an edge of G is associated with exactly one point in S, then the edge is weighted 2; otherwise, the edge is weighted 1. This generates an edge-weighted graph W s (G).

[0047] 2) In the edge weighted graph W s In (G), for any antiparallel flat graph link determined by the subset S, the edge weight corresponding to the odd 2-winding edge is 2, and the edge weight corresponding to the even 2-winding edge is 1.

[0048] 3) Therefore, we can prove Theorem 1.10: A weighted graph W of a flat graph G s There is a one-to-one correspondence between (G) and each type of antiparallel flat graph link based on G. Therefore, all antiparallel flat graph links can be generated by computing all edge weighted graphs of the (possibly) flat graph.

[0049] As an example, Figure 6 As shown in FIG, the application process of generating all antiparallel flat graph links based on any flat graph includes the following steps: Input: n×m-order incidence matrix M(P) of any (planar) graph P.

[0050] Thought: Generate 2 n n-dimensional arrays, and each array X i The component is 0 or 1 (1≤i≤2 n ).

[0051] For each X i , if X i If the jth component of is 1, then the 1 or 2 in the jth row of M(P) changes to 2 or 1; otherwise, the value of the row remains unchanged. i When all components of P are considered, a weighted correlation matrix M of P can be obtained. i . Collect each weighted association matrix M i (1≤i≤2 n ) into the set M.

[0052] Then, delete any isomorphic weighted graphs. Calculate the number of columns containing 2 in each weighted incidence matrix in M ​​and partition M into a finite number of sets G i (1≤i≤s). Delete G i The equivalent matrix in , and then merge all sets G i And replace all weighted association matrices in M.

[0053] Initialization: i=1, j=1, M =∅.

[0054] Iterative process: (1) For i≤2 n , there are the following iterations: For j≤n, if X i If the jth component of is 1, then the 1 or 2 in the jth row of M changes to 2 or 1; otherwise, the non-zero value of the row remains unchanged. The index j increases to j+1 and the next cycle is carried out until j=n+1, the iteration stops, and the output weighted association matrix M i Collected into set M. Then, index i increases to i+1 for the next cycle until i=2 n +1, the iteration stops, all weighted association matrices are collected into M and labeled.

[0055] (2) For i≤s, there are the following iterations: For j≤|G k |, G k The row vectors of the j-th matrix in G are permuted by n, and the resulting matrix is ​​then combined with G k Compare each matrix with a label greater than j. If there are two matrices with the same column vector, delete the corresponding G i For matrices with labels greater than j, for G i The remaining matrices in are re-labeled. The index j is increased to j+1, and the next cycle is performed until j=|G k |+1, the iteration stops. Then, the index i increases to i+1 and the next cycle continues until i=s+1. Merge all sets G k (1≤k≤s), and replace all elements in set M, and output set M.

[0056] Using the matrix iteration algorithm above, we can obtain all weighted incidence matrices of P. If P is a planar graph, all antiparallel planar graph links can be generated by relying on a planar graph of P. When P is a polyhedron, its planar graph is unique and can therefore be generated directly from P.

[0057] Step 3: Generate the chain link branches and their number of antiparallel flat graph chain links; Specifically, step 31: for each edge of any planar graph, combine its four adjacent edges on the two facets with it respectively to form four edge pairs associated with the edge, each pair of edges describing an arc segment connecting two adjacent 2-tangle edges in the antiparallel planar graph link, including: For any (no-cut) flat graph G, a face cycle of G consists of edges that traverse the face in a counterclockwise / clockwise direction. A flat graph can be determined by its face cycle.

[0058] For a flat graph G, every edge e appears on two facets, so e has four adjacent edges e on the two facets. s 、e a 、e o 、e d , according to their relative positions to e, they are called "starting edge", "adjacent edge", "opposite edge" and "diagonal edge". These four adjacent edges and edge e form four pairs of edges [e,e s ]、[e,e a ]、[e,e o ] and [e,e d ], each pair of edges describes an arc segment connecting two adjacent 2-tangle edges in an antiparallel flat graph link, such as Figure 7 shown.

[0059] Each chain branch can be given by the pair of edges it traverses.

[0060] The branching of the antiparallel planar graph chain depends primarily on the parity of each 2-tangle, independent of its orientation. Therefore, each type of antiparallel planar graph chain based on G has the same branching and number of chains.

[0061] Step 32: For any antiparallel flat graph link based on the flat graph, the four edge pair sets corresponding to the flat graph edges can be divided into two binary sets according to the parity of each 2-winding edge.

[0062] Take any antiparallel graph chain D(G), and its corresponding weighted graph is W s (G), if w(e)=1, then [e,e s ] and [e,e o ] and [e,e a ] and [e,e d ] can form two binary sets respectively. Otherwise, [e,e s ] and [e,e d ] and [e,e o ] and [e,e a ] can form two binary sets respectively. Therefore, according to W s The weight of each edge in (G) can give all binary sets.

[0063] Step 33: Each link branch can be represented as a set of binary sets such that an edge pair set of each binary set has exactly one edge in common with an edge pair set of one of the binary sets. The number of such set classes is exactly the number of link branches in the antiparallel graph link.

[0064] Specifically, all binary sets are divided into k classes, so that an edge pair set in any binary set in the same class contains exactly one same edge as an edge pair set in a binary set. Therefore, each class of sets can give a chain branch of D(G) and the number of chain branches is k, including: Input any planar graph and all its edge weight matrices, and give its planar graph by calculating its face cycles.

[0065] For each edge of the flat graph, according to its face circle set, the four adjacent edges of each edge on the two face circles are collected to generate a four-dimensional ordered array.

[0066] Four corresponding edge pairs can be generated from each edge and its four-dimensional array, and the four edge pairs can be divided into two binary sets according to the weights of the corresponding edges in each edge weight matrix.

[0067] Collect all binary sets and classify them so that an edge pair set of any binary set in the same class of sets contains exactly one same edge as an edge pair set of one of the binary sets. The number of sets of each class and its class of sets generated is the chain branch and its number of chain links of an antiparallel flat graph.

[0068] According to the above method, the chain branches of an antiparallel flat graph chain can be calculated through the corresponding weighted association matrix. As an example, Figure 8 As shown in FIG, a general method for generating chain link branches and their number of antiparallel planar chain links includes the following steps: Input: a planar graph P and a set M = {collect each weighted incidence matrix M of P k , 1≤k≤|M|}.

[0069] Idea: The edge set E(P) and the vertex set V(P) are given by the incidence matrix M(P), and then all the face cycles of P are given according to the planar graph algorithm. Each face cycle is collected into F as an ordered set, thus giving a flat graph G of P.

[0070] Then, according to each weighted association matrix M in M k , give all the chain branches of each antiparallel flat graph chain (graph) based on G. For each edge e of E(G) i , find the number of cells in F that contains e i Two doughnuts i1 and f i2 . Collect e i In f i1 The adjacent edge e ia and e id , and in f i2 The adjacent edge e is and e io , thus generating an ordered array (e is ,e ia ,e id ,e io ), where edge e is and e ia Share a vertex.

[0071] Then, by edge e i And the four adjacent edges can derive four edge pairs, according to e i In M k The weight w(e i ) and the related ordered array, divide the four edge pairs into two binary sets O i1 and O i2 .

[0072] Furthermore, O i1and O i2 Compare with each element O in set O respectively. If set O i1 or O i2 If there are common edges with O, they are merged into one set; otherwise, set O i1 or O i2 Will be collected into O as a new element.

[0073] Finally, when every edge of E(G) is taken, M k Each chain branch of the determined antiparallel flat graph chain will be output as each element in O.

[0074] Initialization: F = ∅, O = ∅, S = ∅.

[0075] Initialization: F = ∅, O = ∅, S = ∅.

[0076] Iterative process: For k≤|M|, we have the following iterations: For i≤m, iterate the edge set E(P). For each edge e of E(P) i , find the number of cells in F that contains e i Two dough circles f i1 and f i2 , collect e i In f i1 The adjacent edge e ia and e id , and in f i2 The adjacent edge e is and e io , thus generating an ordered array (e is ,e ia ,e id ,e io ), where edge e is and e ia Share a vertex.

[0077] If in M k Middle w(e i )=1, the edge pair set generated by the ordered array [e i ,e is ] and [e i ,e io ] and [e i ,e ia ] and [e i , e id ] are collected separately i1 and O i2 Otherwise, [e i ,e is ] and [e i ,e id] and [e i ,e io ] and [e i ,e ia ] were collected separately into O i1 and O i2 middle.

[0078] Then, the set O i1 and O i2 Compare with each element O in O, if the set O i1 or O i2 If there are common edges with O, they are merged into one set; otherwise, set O i1 or O i2 will be collected into O as new elements.

[0079] The index i increases to i+1, and the next cycle is carried out until i=m+1, and the iteration stops, and M k Each link branch of the determined antiparallel flat graph link will be given as each element in O, and O is copied into the set S and initialized.

[0080] The index k increases to k+1, and the next loop is performed until k=|M|+1. The iteration stops and each element in the set S and the cardinality of the element as a set are output, that is, all the chain link branches and the number of chain links in an antiparallel flat graph.

[0081] In the algorithm above, if P is a polyhedron, its face cycles can be uniquely determined. Otherwise, the flat graph G is randomly generated from P's incidence matrix. If we need to specify a specific flat graph of P, we can directly specify each of its faces in the input file.

[0082] As an embodiment, the above method process can be used to implement a general software for calculating the topological structure of a wireframe DNA polyhedron and its chain branches based on Fortran code. When encapsulating the application: only a flat graph needs to be input through the software to output all the antiparallel flat graph chains and their chain branches and their number.

[0083] Example 2 In one embodiment of the present disclosure, a system for predicting and generating a wireframe DNA polyhedron topology structure is provided, comprising: A data acquisition module, used for acquiring a planar graph and an incidence matrix of the planar graph; The prediction generation module is used to generate all weighted association matrices of the planar graph according to the set matrix iteration algorithm based on the association matrix of the planar graph; extract the edge set and vertex set from the association matrix of the planar graph, generate all the face circles of the planar graph based on the planar graph algorithm, and use all the face circles to construct an ordered set to generate a flat graph of the planar graph; traverse the flat graph according to each weighted association matrix, for each edge of the flat graph, find the two face circles containing the edge from the ordered set, and extract the two adjacent edges of the edge on the two face circles respectively, and use the obtained An ordered array is constructed from the four adjacent edges of the flat graph; four edge pair sets are derived from the edge of the flat graph and the four adjacent edges, and the four edge pair sets are divided into two binary sets according to the weight of the edge of the flat graph in the weighted association matrix and the related ordered array; the two binary sets are traversed and compared with each element in the given set respectively. If there is a common edge with the given element, they are merged into a new set. Otherwise, the binary set will be collected as a new element into the given set. After traversing each edge of the flat graph, the antiparallel flat graph chain ring and its chain ring branches and number are finally output.

[0084] Example 3 In one embodiment of the present disclosure, a computer program product is provided, including a computer program. When the computer program is executed by a processor, the computer program implements the method for predicting and generating a wireframe DNA polyhedron topology structure.

[0085] Example 4 In one embodiment of the present disclosure, a non-transitory computer-readable storage medium is provided, wherein the non-transitory computer-readable storage medium is used to store computer instructions. When the computer instructions are executed by a processor, the method for predicting and generating a wireframe DNA polyhedron topology structure is implemented.

[0086] Example 5 In one embodiment of the present disclosure, an electronic device is provided, comprising: a processor, a memory, and a computer program; wherein the processor is connected to the memory, and the computer program is stored in the memory. When the electronic device is running, the processor executes the computer program stored in the memory, so that the electronic device executes the method for predicting and generating a wireframe DNA polyhedron topological structure.

[0087] The present disclosure is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present disclosure. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 a process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0088] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operational steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing the instructions executed on the computer or other programmable device for implementing the process. Figure 1 a process or multiple processes and / or boxes Figure 1 The steps for the function specified in one or more boxes.

[0089] Although the above describes the specific implementation methods of the present disclosure in conjunction with the accompanying drawings, it is not intended to limit the scope of protection of the present disclosure. Those skilled in the art should understand that on the basis of the technical solution of the present disclosure, various modifications or variations that can be made by those skilled in the art without creative work are still within the scope of protection of the present disclosure.

Claims

1. A method for predicting and generating the topological structure of a wireframe DNA polyhedron, characterized in that: include: Get a planar graph and its incidence matrix; Based on the incidence matrix of the planar graph, all weighted incidence matrices of the planar graph are generated according to a set matrix iteration algorithm; Extract edge sets and vertex sets from the incidence matrix of the planar graph, generate all face cycles of the planar graph based on the planar graph algorithm, and construct an ordered set using all face cycles to generate a flat graph of the planar graph; Traverse the flat graph according to each weighted association matrix. For each edge of the flat graph, find the two face circles containing the edge from the ordered set, and extract the two adjacent edges of the edge on the two face circles respectively. Use the obtained four adjacent edges to construct an ordered array. Four edge pair sets are derived from the edge of the flat graph and its four adjacent edges. According to the weight of the edge of the flat graph in the weighted association matrix and the related ordered array, the four edge pair sets are divided into two binary sets; the two binary sets are traversed and compared with each element in the given set respectively. If there is a common edge with the given element, they are merged into a new set. Otherwise, the binary set will be collected as a new element into the given set. After traversing each edge of the flat graph, the antiparallel flat graph chain ring and its chain ring branches and number are finally output.

2. A method for predicting and generating a wireframe DNA polyhedron topology structure according to claim 1, characterized in that: A polyhedron is a three-connected planar graph. The graph produced by performing corner truncation, edge weighting, point splitting, and edge thinning operations is called an extended polyhedron, but it is still a planar graph.

3. The method for predicting and generating the topological structure of a wireframe DNA polyhedron according to claim 1, wherein: According to the correspondence between the Class I and Class II DNA polyhedron topological structures and the polyhedron and the extended polyhedron, the Class I and Class II DNA polyhedron topological structures can be generated based on the antiparallel flat graph links of the polyhedron or the extended polyhedron flat graph. The antiparallel flat graph links cover and predict all knots, catenaries and other link structures of the wireframe DNA polyhedron. It is a universal mathematical model of the wireframe DNA polyhedron topological structure. Each of its link branches can be represented as a set composed of a class of binary sets. The binary sets satisfy that an edge pair set of each binary set contains exactly the same edge as an edge pair set of one of the binary sets. The number of set classes is the number of link branches of the antiparallel flat graph links.

4. The method for predicting and generating the topological structure of a wireframe DNA polyhedron according to claim 3, wherein: Wireframe DNA polyhedron topological molecules are divided into Class I and Class II DNA polyhedron topological structures according to the number of their double helical edges. The double helical edges and vertex regions of all wireframe DNA polyhedrons are decomposed to establish corresponding antiparallel 2-winding edge and node models. Antiparallel flat graph links are defined by establishing a correspondence between directed link graphs and flat graphs, and each antiparallel 2-winding edge of the link graph must correspond to an edge of a flat graph.

5. The method for predicting and generating the topological structure of a wireframe DNA polyhedron according to claim 3, wherein: Any antiparallel flat graph chain-link graph can be obtained from a flat graph by replacing the edges of the flat graph with antiparallel 2-windings and connecting the two endpoints of the 2-windings along each face. The antiparallel flat graph chain-link graph is obtained by the spatial projection of its chain-link graph.

6. The method for predicting and generating the topological structure of a wireframe DNA polyhedron according to claim 1, wherein: For each edge of any flat graph, the four adjacent edges of the edge on the two face circles are combined with it respectively to form four edge pairs related to the edge. Each pair of edges describes an arc segment connecting two adjacent 2-entangled edges in the antiparallel flat graph chain. For any antiparallel flat graph chain based on the flat graph, according to the parity of each 2-entangled edge, the four edge pair sets corresponding to the flat graph edge are divided into two binary sets. Each chain branch is represented as a set composed of a class of binary sets. The set satisfies that an edge pair set of each binary set contains exactly the same edge as an edge pair set of one of the binary sets. The number of these sets is exactly the number of chain branches of the antiparallel flat graph chain.

7. A prediction and generation system for the topological structure of wireframe DNA polyhedrons, characterized by: include: A data acquisition module, used for acquiring a planar graph and an incidence matrix of the planar graph; A prediction generation module, configured to generate all weighted association matrices of a planar graph according to a set matrix iteration algorithm based on the association matrix of the planar graph; Extract edge sets and vertex sets from the incidence matrix of the planar graph, generate all face cycles of the planar graph based on the planar graph algorithm, and construct an ordered set using all face cycles to generate a planar graph of the planar graph; traverse the planar graph according to each weighted incidence matrix, and for each edge of the planar graph, find the two face cycles containing the edge from the ordered set, and extract the two adjacent edges of the edge on the two face cycles respectively, and construct an ordered array using the obtained four adjacent edges; derive four edge pair sets from the edge of the planar graph and the four adjacent edges, and divide the four edge pair sets into two binary sets according to the weight of the edge of the planar graph in the weighted incidence matrix and the related ordered array; traverse and compare the two binary sets with each element in the given set respectively, and merge them into a new set if there is a common edge with the given element, otherwise, the binary set will be collected as a new element in the given set, and finally output the antiparallel planar graph chain ring and its chain ring branches and their number.

8. A computer program product comprising a computer program, characterized in that When the computer program is executed by a processor, the method for predicting and generating the topological structure of a wireframe DNA polyhedron according to any one of claims 1 to 6 is implemented.

9. A non-transitory computer-readable storage medium, characterized in that The non-transitory computer-readable storage medium is used to store computer instructions, and when the computer instructions are executed by the processor, the method for predicting and generating the topological structure of a wireframe DNA polyhedron according to any one of claims 1 to 6 is implemented.

10. An electronic device, characterized in that: include: A processor, a memory, and a computer program; wherein the processor is connected to the memory, the computer program is stored in the memory, and when the electronic device is running, the processor executes the computer program stored in the memory to enable the electronic device to implement a method for predicting and generating a wireframe DNA polyhedron topology structure as described in any one of claims 1 to 6.

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