A method and system for predicting the generation of wireframe DNA polyhedral topology

Abstract

The present disclosure provides a prediction generation method and system for wireframe DNA polyhedral topology, relating to the technical field of DNA polyhedral structure prediction, comprising: obtaining a planar graph and an associated matrix of the planar graph; generating all weighted associated matrices of the planar graph according to a set matrix iteration algorithm based on the associated matrix of the planar graph; extracting an edge set and a vertex set in the associated matrix of the planar graph, generating all face rings of the planar graph based on a planar graph algorithm, constructing an ordered set using all the face rings, and generating a planar graph of the planar graph; traversing the planar graph according to each weighted associated matrix, for each edge of the planar graph, finding two face rings containing the edge from the ordered set, and extracting two adjacent edges of the edge on the two face rings respectively, and constructing an ordered array using the obtained four adjacent edges; and traversing and outputting an anti-parallel planar graph chain ring and the number of chain ring branches according to the weight of the edge of the planar graph in the weighted associated matrix and the related ordered array.

Description

TECHNICAL FIELD

[0001] The present disclosure relates to the technical field of DNA polyhedral structure prediction, in particular to a prediction generation method and system for wireframe DNA polyhedral topology. BACKGROUND

[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 polyhedron is a topological molecule with a polyhedral geometric framework formed by single-stranded DNA as a topological unit. Wireframe DNA polyhedron not only includes polyhedral knot molecules formed by folding a single strand of DNA, but also includes catena molecules formed by one face or point of each single-stranded DNA walking polyhedron, such as tetrahedron, cube, octahedron, triangular prism, etc. Therefore, for any polyhedron, it is impossible to know how many DNA single strands can form a wireframe polyhedral topological molecule, and further, it is impossible to know whether the design of the topological structure is unique and how to calculate the folding path of each DNA single strand. The above problems are some basic problems that have not been solved in DNA nanometer assembly.

[0004] At present, there is no effective and general prediction method and calculation software to theoretically solve the above problems. The existing main theoretical methods or calculation programs can only predict all possible topological structures of four polyhedrons (i.e. tetrahedron, triangular prism, triangular bipyramid and octahedron) assembled by DNA single strands, with the prerequisite of determining all possible orientations of the four polyhedral topological structures, so the existing theoretical methods cannot be applied to general polyhedrons. For other theoretical methods, only one type of special wireframe polyhedral knot is involved, which cannot predict or give all knot structures of these polyhedrons formed by a DNA single strand, nor can it determine the possibility of forming a topological structure by folding a DNA single strand. SUMMARY

[0005] In order to solve the above problems, the present disclosure proposes a prediction generation method and system for wireframe DNA polyhedral topology, establishes an anti-parallel planar graph loop mathematical model, predicts all topological structures of knots, catenanes and other loops generated by assembling a plurality of DNA single strands into any polyhedron, and outputs all edge weighted matrices of the polyhedron, so as to generate all topological structures of the polyhedron assembled by a plurality of DNA single strands, and output the loop branches and the number of anti-parallel planar graph loops corresponding to each weighted matrix, i.e. the assembly path and the 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:

[0007] A method for predicting the generation of a wireframe DNA polyhedral topology, comprising:

[0008] Obtaining a planar graph and an incidence matrix of the planar graph;

[0009] Based on the incidence matrix of the planar graph, generating all weighted incidence matrices of the planar graph according to a set matrix iterative algorithm;

[0010] Extracting an edge set and a vertex set from the incidence matrix of the planar graph, generating all face cycles of the planar graph based on a planar graph algorithm, constructing an ordered set using all face cycles, and generating a planar graph of the planar graph;

[0011] According to each weighted incidence matrix, traversing the planar graph, for each edge of the planar graph, finding two face cycles containing the edge from the ordered set, extracting two adjacent edges of the edge in the two face cycles respectively, and constructing an ordered array using the four adjacent edges obtained;

[0012] Deriving four edge pair sets from the edge and the four adjacent edges of the planar graph, dividing the four edge pair sets into two binary sets according to the weight of the edge in the weighted incidence matrix and the related ordered array, comparing each element in the two binary sets with each element in a given set, merging them into a new set if there is a common edge, otherwise, the binary set will be collected as a new element into the given set, and finally outputting the anti-parallel planar graph chain ring and the number of chain ring branches.

[0013] According to some embodiments, the present disclosure adopts the following technical solutions:

[0014] A system for predicting the generation of a wireframe DNA polyhedral topology, comprising:

[0015] A data acquisition module for acquiring a planar graph and an incidence matrix of the planar graph;

[0016] The prediction generation module is configured to generate all weighted incidence matrices of the planar graph according to a set matrix iterative algorithm based on an incidence matrix of the planar graph; extract an edge set and a vertex set from the incidence matrix of the planar graph, generate all face cycles of the planar graph based on a planar graph algorithm, construct an ordered set by using all the face cycles, and generate a planar graph of the planar graph; traverse the planar graph according to each weighted incidence matrix, for each edge of the planar graph, find two face cycles containing the edge from the ordered set, extract two adjacent edges of the edge in the two face cycles respectively, and construct an ordered array by using the obtained four adjacent edges; derive four edge pair sets from the edge and the four adjacent edges of the planar graph, divide the four edge pair sets into two binary sets according to the weight of the edge in the weighted incidence matrix and the related ordered array, compare each element in a given set with the two binary sets respectively, if there is a common edge, merge them into a new set, otherwise, the binary set will be collected as a new element into the given set, and finally output the anti-parallel planar graph chain ring and the chain ring branches and the number thereof after traversing each edge of the planar graph.

[0017] According to some embodiments, the present disclosure adopts the technical scheme as follows:

[0018] A computer program product comprising a computer program which, when executed by a processor, implements the prediction generation method for a wireframe DNA polyhedral topology.

[0019] According to some embodiments, the present disclosure adopts the technical scheme as follows:

[0020] A non-transitory computer-readable storage medium for storing computer instructions, which, when executed by a processor, implement the prediction generation method for a wireframe DNA polyhedral topology.

[0021] According to some embodiments, the present disclosure adopts the technical scheme as follows:

[0022] An electronic device comprising a processor, a memory, and a computer program; wherein the processor is connected with 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 implements the prediction generation method for a wireframe DNA polyhedral topology.

[0023] Compared with the prior art, the present disclosure has the beneficial effects that:

[0024] The prediction generation method for the wireframe DNA polyhedron topology of the present disclosure fills the gap in the general calculation method of the wireframe DNA polyhedron topology, that is, any polyhedron can be inputted, and then the following can be outputted: 1) the edge weighting matrix of the polyhedron, that is, all the topological structures of the polyhedron generated by the assembly of several DNA single strands; 2) the loop branch and the number of the loop branch of the anti-parallel planar graph loop corresponding to each weighting matrix, that is, the assembly path and the number of DNA single strands of each wireframe DNA polyhedron based on the polyhedron.

[0025] The prediction generation method for the wireframe DNA polyhedron topology of the present disclosure provides a general calculation software for the precise design of the new topology of the wireframe DNA polyhedron, and encapsulates the direct prediction generation of the wireframe DNA polyhedron topology into software. A general software for calculating the wireframe DNA polyhedron topology and the loop branch thereof is realized based on the Fortran code, and all the anti-parallel planar graph loops and the loop branch and the number thereof can be outputted by inputting a planar graph into the software. BRIEF DESCRIPTION OF DRAWINGS

[0026] The accompanying drawings, which form a part of the present disclosure, are used to provide further understanding of the present disclosure, and the schematic embodiments of the present disclosure and the description thereof are used to explain the present disclosure, and do not constitute an improper limitation on the present disclosure.

[0027] Figure 1 The schematic diagram of the I-type DNA polyhedron and the local structure modelization thereof according to the embodiments of the present disclosure;

[0028] wherein, Figure 1 (a) in FIG. 1 is four I-type DNA polyhedrons; Figure 1 (b) in FIG. 1 is the vertex region and the node model thereof; Figure 1 (c) in FIG. 1 is the helical edge and the 2-winding model thereof;

[0029] Figure 2 The schematic diagram of the II-type DNA polyhedron and the local structure modelization thereof according to the embodiments of the present disclosure; wherein, Figure 2 (a) in FIG. 2 is four II-type DNA polyhedrons; Figure 2 (b) in FIG. 2 is the vertex region and the node model thereof; Figure 2 (c) in FIG. 2 is the helical edge and the 2-winding model thereof;

[0030] Figure 3 The relationship diagram of the loop graph D and the planar graph G of the anti-parallel planar graph loop L according to the embodiments of the present disclosure;

[0031] Figure 4 The four operations based on polyhedron for the embodiments of the present disclosure: a represents refining edges, b represents strengthening edges, c represents differentiating points, and d represents truncating corners.

[0032] Figure 5 The transformation graph for the embodiments of the present disclosure;

[0033] wherein, Figure 5 The (a) nodes -O3 and -O4 in D(G) are transformed into nodes V3 and V4 by node transformation; Figure 5 The (b) anti-parallel planar graph chain ring D(G) in D(G) can be transformed into a special anti-parallel planar graph chain ring D S (G) by two node transformations.

[0034] Figure 6 The flow chart for calculating all anti-parallel planar graph chain rings based on any planar graph for the embodiments of the present disclosure.

[0035] Figure 7 The four pairs of edges [e,e s ], [e,e a ], [e,e o ] and [e,e d ] formed by the edge e and four adjacent edges for the embodiments of the present disclosure, which can represent the corresponding four arc segments of D(G).

[0036] Figure 8 The flow chart for calculating the chain ring branches of anti-parallel planar graph chain rings and the number of the branches for the embodiments of the present disclosure. DETAILED DESCRIPTION

[0037] The present disclosure will be further described below in conjunction with the accompanying drawings and embodiments.

[0038] It should be noted that the following detailed description is exemplary and is 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 one of ordinary skill in the art to which the present disclosure belongs.

[0039] It should be noted that the terms used herein are only for the purpose of describing specific embodiments and are not intended to limit the exemplary embodiments according to the present disclosure. As used herein, the singular form is intended to include the plural form unless the context clearly indicates otherwise, and it should also be understood that when the terms “comprise” and / or “include” are used in the specification, there is a presence of a feature, step, operation, device, component and / or a combination thereof.

[0040] Embodiment 1

[0041] In an embodiment of the present disclosure, a method for predicting the generation of a wireframe DNA polyhedral topological structure is provided, and the method steps include:

[0042] Step 1: Obtain a planar graph and an associated matrix of the planar graph;

[0043] Step 2: Based on the associated matrix of the planar graph, generate all weighted associated matrices of the planar graph according to a set matrix iterative algorithm;

[0044] Step 3: Extract the edge set and vertex set in the associated matrix of the planar graph, generate all face rings of the planar graph based on a planar graph algorithm, construct an ordered set using all face rings, and generate a planar graph of the planar graph;

[0045] Step 4: Traverse the planar graph according to each weighted associated matrix, for each edge of the planar graph, find two face rings containing the edge from the ordered set, and extract two adjacent edges of the edge in the two face rings, respectively, and construct an ordered array using the four adjacent edges obtained;

[0046] Step 5: Derive four edge pair sets from the edge and the four adjacent edges of the planar graph, divide the four edge pair sets into two binary sets according to the weight of the edge in the weighted associated matrix and the related ordered array, and traverse and compare each element in the two binary sets with each element in a given set, if there is a common edge, merge them into a new set, otherwise, the binary set will be collected as a new element into the given set, and finally output the anti-parallel planar graph chain ring and the number of chain ring branches.

[0047] As an embodiment, the method for predicting the generation of a wireframe DNA polyhedral topological structure of the present disclosure establishes a general 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, including establishing a general mathematical model "anti-parallel planar graph chain ring" of a wireframe DNA polyhedral topological structure, establishing a general method for generating all anti-parallel planar graph chain rings based on any (non-cut point) planar graph, and establishing a general method and software algorithm for generating the chain ring branches and the number of chain ring branches of the anti-parallel planar graph chain ring. That is, by inputting any polyhedron, the following outputs can be obtained: 1) all edge weighted matrices of the polyhedron, which can generate all topological structures generated by the assembly of multiple DNA single strands of the polyhedron; 2) the chain ring branches and the number of chain ring branches of the anti-parallel planar graph chain ring corresponding to each weighted matrix, which is the assembly path and the number of DNA single strands based on each wireframe DNA polyhedron of the polyhedron. The specific implementation process is as follows:

[0048] Step 1: Establish a general mathematical model "anti-parallel planar graph chain ring" of a wireframe DNA polyhedral topological structure;

[0049] Specifically, step 11: obtaining wireframe DNA polyhedral topology molecules, and dividing the wireframe DNA polyhedral topology molecules into two types of I-type wireframe DNA polyhedron and II-type wireframe DNA polyhedron according to the number of double helix edges of the wireframe DNA polyhedral topology molecules:

[0050] 1) I-type wireframe DNA polyhedron: a polyhedral molecule composed of one DNA double helix on each edge, and the vertex region is "stacked single-stranded domain" or "non-stacked single-stranded domain", as shown in (a) of Figure 1 .

[0051] 2) II-type wireframe DNA polyhedron: a polyhedral molecule composed of two parallel DNA double helix on at least one edge, and the vertex region is "stacked / non-stacked single-stranded domain" or "non-stacked double-stranded domain", as shown in (a) of Figure 2 .

[0052] Step 12: disassembling the double helix edges and vertex regions of all wireframe DNA polyhedrons into some basic building blocks, and establishing corresponding anti-parallel 2-wound edges and node models, including:

[0053] Disassembling the edges and points of all wireframe DNA polyhedrons, and simplifying the basic building block "DNA double helix" constituting the edge into an anti-parallel 2-wound T, wherein the anti-parallel 2-wound T can be divided into positive and negative types according to the orientation of the lower left corner arc segment, and can be divided into odd and even types according to the parity of the crossing number. As shown in (c) of Figure 1 , T4 is negative even 2-wound, and T3 is positive odd 2-wound.

[0054] Simplifying the basic building block "stacked single-stranded domain" constituting the vertex region into V m node, and simplifying "non-stacked single-stranded domain" into node and 0 m node. Taking the reverse direction of all curves in 2-wound T and nodes V m , 0 m , the winding obtained is denoted as their inverse-T, -V m and -0 m .

[0055] Step 13: establishing anti-parallel planar graph chain according to the corresponding relationship between chain graph and planar graph, including:

[0056] The anti-parallel planar graph chain can be defined by establishing the corresponding relationship between the directed chain graph and the planar graph, and it needs to satisfy that each anti-parallel 2-wound edge of the chain graph corresponds to an edge of the planar graph.

[0057] Further, it is proved that any anti-parallel planar chain-link graph can be obtained from a planar graph: replace each edge of the planar graph with an anti-parallel 2-twist, and connect the two end points of each 2-twist along each face. Anti-parallel planar chain-link is obtained by the spatial projection of its chain-link graph.

[0058] As shown in Fig. 1.1, according to the chessboard correspondence rule between chain-link graph and planar graph, it is proved that Theorem 1.1: Figure 3

[0059] Theorem 1.1 Given a directed chain-link L and its chain-link graph D, there exists a (vertex-cut-free) planar graph G such that each edge of G corresponds to a directed 2-twist edge of D. If each 2-twist edge is anti-parallel, then the directed chain-link D is defined as an anti-parallel planar chain-link graph, and the directed chain-link L is defined as an anti-parallel planar chain-link.

[0060] Conversely, an anti-parallel planar chain-link graph can be constructed from a planar graph: given a planar graph G, replace each edge of G with an anti-parallel 2-twist, and connect the two end points of each 2-twist along each face. The resulting directed chain-link graph is an anti-parallel planar chain-link graph D(G). Projecting D(G) into three-dimensional space yields an anti-parallel planar chain-link L(G). Therefore, the anti-parallel planar chain-link model preserves both the geometric framework and the topological structure of DNA polyhedra.

[0061] Further, it is proved that Theorem 1.2: any planar graph corresponds to a class of anti-parallel planar chain-links, which is a series of anti-parallel planar chain-links generated by varying the twist number of 2-twist edges.

[0062] According to the construction method of anti-parallel planar chain-link, an important property of anti-parallel planar chain-link is obtained:

[0063] Important Property 1.3: The vertex region of each class of anti-parallel planar chain-link is 0 m Vertex or -0 m Vertex, m is the degree of the corresponding vertex in the planar graph G.

[0064] Step 14: The polyhedron is a three-connected planar graph, and the expanded polyhedron is a planar graph obtained by performing a finite number of "corner cutting", "edge thickening", "point differentiation", and "edge refinement" operations on the polyhedron.

[0065] According to the correspondence between polyhedra or expanded polyhedra and the topological structures of Class I and Class II DNA polyhedra, it is proved that anti-parallel planar chain-link is a general mathematical model of wireframe DNA polyhedra topological structure.

[0066] Specifically, according to the correspondence between Class I and Class II DNA polyhedra topological structures and polyhedra and expanded polyhedra, it is proved that Theorem 1.4:​

[0067] Proof of Theorem 1.4: Both Class I and Class II DNA polyhedral topologies can be generated based on the anti-parallel planar chain of a polyhedral / expanded polyhedral planar graph.

[0068] The anti-parallel planar chain encompasses, predicts all the knotted, catenated and other chain structures of a wireframe DNA polyhedron, and is a universal mathematical model of wireframe DNA polyhedral topology.

[0069] Step 2: The universal method for generating all anti-parallel planar chains based on any planar graph includes the following steps:

[0070] Step 21: It is proved that an anti-parallel planar chain is homotopically equivalent to a special anti-parallel planar chain in which every edge is a positive even-wound edge and every vertex is either a 0 m or V m vertex. Further, any wireframe DNA polyhedron can be designed to have a special anti-parallel polyhedral chain with the same double helix edge length, including:

[0071] (1) An important homotopy transformation is established: Given an anti-parallel planar chain, every crossing of a 2-wound edge associated with a -0 m vertex is continuously transformed to the position of a -0 m vertex, and the original -0 m vertex becomes a V m vertex, which is called a vertex transformation, as shown in (a) of Figure 5 .

[0072] The two chains before and after the vertex transformation are the same under homotopic equivalence. Further, if the vertex transformation is applied to all -0 m vertices of an anti-parallel planar chain, the resulting planar chain is called a special anti-parallel planar chain, as shown in (b) of Figure 5 .

[0073] It can be proved that a special anti-parallel planar chain has the property 1.5: all 2-wound edges of every special anti-parallel planar chain are positive and even, and every vertex is either a 0 m or V m vertex.

[0074] It can be proved that Theorem 1.6: every anti-parallel planar chain is homotopically equivalent to a special anti-parallel planar chain in which every edge is a positive even-wound edge and every vertex is either a 0 m or V m vertex.

[0075] From this, we get Corollary 1.7: any DNA polyhedral topological molecule can be designed to have a special anti-parallel polyhedral chain with the same double helix edge length.

[0076] and also obtain property 1.8: for any anti-parallel planar chain, the anti-orientation of the chain produces a directed chain, which is not an anti-parallel planar chain except for its inverse chain.

[0077] Further, establish lemma 1.9: for any planar graph G, the vertex subset of the point set is in one-to-one correspondence with each class of anti-parallel planar chain based on the planar graph G.

[0078] Step 22: define an edge-weighted graph of any planar graph by a vertex subset of the planar graph, and establish a one-to-one correspondence between the edge-weighted planar graph and the class of anti-parallel planar chains. Further, based on the anti-parallel planar chain of any planar graph, all edge-weighted graphs of the planar graph can be generated, including:

[0079] By inputting the adjacency matrix of any planar graph, a 0, 1 array with vertex number n as the dimension is generated, where each array corresponds to a subset of the planar graph. Then, according to the two n arrays, the adjacency matrix of all edge-weighted planar graphs can be generated, which can uniquely determine all edge-weighted graphs of the planar graph.

[0080] Step 23: by establishing the linear equivalence algorithm of the adjacency matrix of the edge-weighted planar graph, all equivalent edge-weighted planar graphs can be removed, and finally all edge-weighted matrices can be output, that is, all anti-parallel planar chains can be generated.

[0081] As an embodiment, all anti-parallel planar chains are generated by defining edge-weighted graphs by vertex sets of planar graphs, as follows:

[0082] 1) Given any (planar) graph G, take a subset S of the point set V(G), if a edge of G is associated with a point in S, then the edge is weighted as 2; otherwise, the edge is weighted as 1. Thus, an edge-weighted graph W s (G) can be generated.

[0083] 2) In the edge-weighted graph W s (G), any anti-parallel planar chain determined by the subset S, the odd 2-winding edge corresponds to the edge weight of 2, and the even 2-winding edge corresponds to the edge weight of 1.

[0084] 3) Thus, prove theorem 1.10: the weighted graph W s (G) of the planar graph G is in one-to-one correspondence with each class of anti-parallel planar chain based on G. Thus, all anti-parallel planar chains can be generated by calculating all edge-weighted graphs of (planar) graphs.

[0085] As an embodiment, as Figure 6 shown, the application process based on all anti-parallel planar chains of any planar graph includes the following operation steps:

[0086] Input: n×m-order incidence matrix M(P) of any (planar) graph P.

[0087] Thought: Generate 2 n n-dimensional arrays, and each array X i The component is 0 or 1 (1≤i≤2 n ).

[0088] 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.

[0089] 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.

[0090] Initialization: i=1, j=1, M =∅.

[0091] Iterative process:

[0092] (1) For i≤2 n , there are the following iterations:

[0093] 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.

[0094] (2) For i≤s, there are the following iterations:

[0095] 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 kCompare each matrix with index greater than j. If there exist two matrices whose column vectors are identical, delete the corresponding G i Compare matrices with index greater than j. If there exist two matrices whose column vectors are identical, delete the corresponding G i Re-index the remaining matrices. Index j is increased to j+1 and the next loop is performed until j=|G k |+1, the iteration stops. Then, index i is increased to i+1 and the next loop is performed until i=s+1. Merge all sets G k (1≤k≤s) and replace all elements in set M, output set M.

[0096] According to the above setting matrix iteration algorithm, all the weighted incidence matrices of P can be obtained. If P is a planar graph, all the anti-parallel planar chain loops can be generated depending on one planar graph of P. When P is a polyhedron, its planar graph is unique, thus can be directly generated from P.

[0097] Step 3: generating the chain loop branches of anti-parallel planar chain loops and their numbers;

[0098] Specifically, step 31: for each edge of any planar graph, combine its four adjacent edges on two face cycles with it respectively to form four edge pairs related to the edge, each pair of edges describes an arc segment connecting two adjacent 2-knot edges in an anti-parallel planar chain loop, including:

[0099] For any (cut-vertex-free) planar graph G, a face cycle of G is composed of edges traversed in reverse / clockwise along the face, and a planar graph can be determined by its face cycles.

[0100] For a planar graph G, each edge e appears on two face cycles, so e has four adjacent edges e s , e a , e o , e d on two face cycles according to their relative positions with e. They are called “starting edge”, “adjacent edge”, “opposite edge” and “diagonal edge” respectively. The four adjacent edges form four pairs of edges [e,e s ], [e,e a ], [e,e o ] and [e,e d ] with edge e respectively, each pair of edges describes an arc segment connecting two adjacent 2-knot edges in an anti-parallel planar chain loop, as shown in Figure 7 .

[0101] Each chain loop branch can be given by the edge pairs traversed.

[0102] The chain loop branches of anti-parallel planar chain loops mainly depend on the parity of each 2-knot, which is independent of its orientation. Therefore, each type of anti-parallel planar chain loop based on G has the same chain loop branches and numbers.

[0103] 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.

[0104] 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.

[0105] 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.

[0106] 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:

[0107] Input any planar graph and all its edge weight matrices, and give its planar graph by calculating its face cycles.

[0108] 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.

[0109] 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.

[0110] 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.

[0111] According to the above method, the chain branches of an antiparallel flat graph chain can be calculated through the corresponding weighted association matrix.

[0112] 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:

[0113] Input: a planar graph P and a set M = {collect each weighted incidence matrix M of P k , 1≤k≤|M|}.

[0114] 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.

[0115] 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.

[0116] 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 .

[0117] Furthermore, O i1 and 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 Oi1 or O i2 Will be collected into O as a new element.

[0118] 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.

[0119] Initialization: F = ∅, O = ∅, S = ∅.

[0120] Initialization: F = ∅, O = ∅, S = ∅.

[0121] Iterative process: For k≤|M|, we have the following iterations:

[0122] 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.

[0123] 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 Oi1 and O i2 in O.

[0124] Then, set O i1 and O i2 is compared with each element O in O, if set O i1 or O i2 has common edges with O, it is merged into one set; otherwise, set O i1 or O i2 will be collected as a new element into O.

[0125] The index i is increased to i+1, and the next loop is performed until i=m+1, and the iteration stops. Each chain branch of the anti-parallel planar graph chain ring determined by M k is given as each element in O, and O is copied into set S and initialized.

[0126] The index k is increased to k+1, and the next loop is performed until k=|M|+1, and the iteration stops. Each element in set S and the cardinality of the element, i.e. all chain branches of an anti-parallel planar graph chain ring and the number of chain branches, are output.

[0127] In the above algorithm design, if P is a polyhedron, its face ring can be uniquely determined. Otherwise, the planar graph G is randomly generated from the incidence matrix of P. If we need to specify a certain planar graph of P, we can directly give each face of P in the input file.

[0128] As an embodiment, the above method process can be implemented into a general software for calculating the topology of a wireframe DNA polyhedron and its chain branches based on Fortran code, and when the software is packaged and applied: only by inputting a planar graph into the software, all anti-parallel planar graph chain rings and their chain branches and the number of chain branches can be output.

[0129] Embodiment 2

[0130] In an embodiment of the present disclosure, a prediction generation system for the topology of a wireframe DNA polyhedron is provided, comprising:

[0131] A data acquisition module for acquiring a planar graph and an incidence matrix of the planar graph;

[0132] The prediction generation module is configured to generate all weighted incidence matrices of the planar graph according to a set matrix iterative algorithm based on an incidence matrix of the planar graph; extract an edge set and a vertex set from the incidence matrix of the planar graph, generate all face cycles of the planar graph based on a planar graph algorithm, construct an ordered set by using all the face cycles, and generate a planar graph of the planar graph; traverse the planar graph according to each weighted incidence matrix, for each edge of the planar graph, find two face cycles containing the edge from the ordered set, extract two adjacent edges of the edge in the two face cycles respectively, and construct an ordered array by using the obtained four adjacent edges; derive four edge pair sets from the edge and the four adjacent edges of the planar graph, divide the four edge pair sets into two binary sets according to the weight of the edge in the weighted incidence matrix and the related ordered array, compare each element in the given set with the two binary sets respectively, if there is a common edge, merge them into a new set, otherwise, the binary set will be collected as a new element into the given set, and finally output the anti-parallel planar graph chain ring and the chain ring branches and the number thereof after traversing each edge of the planar graph.

[0133] Embodiment 3

[0134] In an embodiment of the present disclosure, a computer program product is provided, comprising a computer program which, when executed by a processor, implements the prediction generation method for wireframe DNA polyhedral topology.

[0135] Embodiment 4

[0136] In an embodiment of the present disclosure, a non-transitory computer-readable storage medium is provided for storing computer instructions, which, when executed by a processor, implement the prediction generation method for wireframe DNA polyhedral topology.

[0137] Embodiment 5

[0138] In an embodiment of the present disclosure, an electronic device is provided, comprising a processor, a memory and a computer program; wherein the processor is connected with 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 implements the prediction generation method for wireframe DNA polyhedral topology.

[0139] The computer program instructions can also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer-implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart block or blocks. Figure 1 one or more flow or blocks Figure 1 means for functionally implementing the steps in one or more flow or blocks

[0140] The computer program instructions can also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer-implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart block or blocks. Figure 1 one or more flow or blocks Figure 1 steps for functionally implementing the steps in one or more flow or blocks

[0141] Although the present disclosure has been described with reference to specific implementations, it will be apparent to those skilled in the art that various modifications and changes can be made thereto without departing from the scope 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; Wireframe DNA polyhedron topological molecules are obtained and divided into two types according to the number of double helical edges: type I wireframe DNA polyhedron and type II wireframe DNA polyhedron. Type I wireframe DNA polyhedron: A polyhedron molecule in which each edge consists of a DNA double helix, and its vertex regions are either "overlapping single-stranded domains" or "non-overlapping single-stranded domains"; Type II wireframe DNA polyhedron: A polyhedron with at least one edge consisting of two parallel DNA double helices, whose vertex regions are either "overlapping / non-overlapping single-stranded domains" or "non-overlapping double-stranded domains"; The double helix edges and vertex regions of all wireframe DNA polyhedra were decomposed into basic building blocks. The basic building block of the edge, the "DNA double helix," was simplified to an antiparallel 2-winding T. The antiparallel 2-winding T was divided into positive and negative categories based on the orientation of its lower left corner arc segment, and into odd and even categories based on the parity of the crossing number. The corresponding antiparallel 2-winding edge and node models were established. The antiparallel flat graph chain is defined by establishing a correspondence between a directed chain graph and a flat graph, and it must satisfy that each antiparallel 2-winding edge of the chain graph corresponds to an edge of the flat graph; Based on the incidence matrix of the planar graph, all weighted incidence matrices of the planar graph are generated according to a set matrix iterative algorithm; specifically, the method includes: 1) Given any flat graph, take any subset of the point set. If an edge of the flat graph is associated with exactly one point in the subset, then the edge is assigned a weight of 2; otherwise, the edge is assigned a weight of 1, generating an edge-weighted graph. 2) In the edge-weighted graph, for any antiparallel flat graph link determined by the subset, the edge weight corresponding to its odd 2-entangled edge is 2, and the edge weight corresponding to its even 2-entangled edge is 1; Input: n×m-order incidence matrix M of any flattenable graph P; Thought: Generate 2 n n-dimensional arrays, and each array X i The component is 0 or 1, 1≤i≤2 n ; For each X i , if X i If the jth component of is 1, then the 1 or 2 in the jth row of the incidence matrix changes to 2 or 1; otherwise, the value of the row remains unchanged; when j takes X i When all components of the flat graph are obtained, a weighted correlation matrix M is obtained. i ; Collect each weighted association matrix M i to the set M; Then, delete any isomorphic weighted graphs, calculate the number of columns containing 2 in each weighted incidence matrix in M, and divide 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; Initialization: i=1, j=1, M = ; (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, and the index j increases to j+1 for the next cycle 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 correlation matrices are collected into M and labeled; (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 the matrix are re-numbered, 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 for the next cycle until i=s+1, merging all sets G k , 1≤k≤s, and replace all elements in the set M, output set M; according to the above set matrix iteration algorithm, obtain all weighted association matrices of P; 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 1, 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.

5. 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.

6. A system for predicting and generating the topological structure of a wireframe DNA polyhedron, specifically implementing the method for predicting and generating the topological structure of a wireframe DNA polyhedron according to any one of claims 1 to 4, characterized in that: 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.

7. 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 4 is implemented.

8. 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 a processor, a method for predicting and generating a wireframe DNA polyhedron topology structure according to any one of claims 1 to 4 is implemented.

9. 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 the topological structure of a wireframe DNA polyhedron as described in any one of claims 1 to 4.

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