CN113113090A - Construction method of DNA polyhedron topological structure - Google Patents
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Abstract
The invention belongs to the field of DNA polyhedron design and synthesis, and particularly relates to a method for constructing a DNA polyhedron topological structure, which comprises the following steps: step 1, acquiring a target polyhedron, and marking nonadjacent three-degree vertexes in the target polyhedron; step 2, replacing edges connected with all nonadjacent three-degree vertexes in the target polyhedron by odd-numbered twisted edges, and replacing the rest edges in the target polyhedron by even-numbered twisted edges to obtain an optimized DNA polyhedron topological structure; step 3, judging whether nonadjacent three-degree vertexes exist in the optimized DNA polyhedron topological structure, if so, repeating the steps 1 to 2 until an optimal DNA polyhedron topological structure is obtained; on the contrary, the optimized DNA polyhedron topological structure is the optimal DNA polyhedron topological structure, the method can rapidly, simply and qualitatively analyze the reasonable scheme for constructing the DNA polyhedron topological structure by the minimum number of DNA single chains, and the result is accurate and reliable.
Description
Technical Field
The invention belongs to the field of DNA polyhedron design and synthesis, and particularly relates to a construction method of a DNA polyhedron topological structure.
Background
The DNA nanotechnology is a method of designing and constructing a complex structure and precisely controlling its nanoperformance using DNA as a synthetic material, and aims to synthesize a three-dimensional or two-dimensional nanostructure using a DNA strand as a raw material. DNA is suitable for creating nanoscale structures because the binding between nucleic acid strands follows the well-known rules of simple base-complementary pairing, forming unique double-helical nanostructures. By utilizing this property, the assembly of the structure can be easily controlled by the design of the nucleic acid strand.
Due to the special base complementary pairing property of the DNA, the DNA nano structure can show good addressability.
Traditional DNA nanostructures have been synthesized by means of multiple DNA single strands, and in order to construct DNA nanostructures consisting of a minimum number of DNA single strands, it is necessary to provide synthetic templates for them through theoretical studies.
The DNA nano structure consisting of the minimum number of DNA single chains is designed, in order to meet the requirement, a structure meeting the requirement can be designed only by means of parameter adjustment of experienced technicians on software in the design process of the DNA nano structure, the whole process is long in time consumption and complex in operation, and the technical barrier to designers is high. Therefore, research and development of an applicable rapid design method are carried out, a topological structure of a DNA polyhedron formed by the minimum number of single strands and the number of the required DNA single strands are given, and the method has great significance for synthesis and application of the DNA nanostructure.
In recent years, the graph theory and the topological approach have been more and more extensive in solving physical and biological problems, and thus it is a very effective approach to attempt to solve a DNA polyhedron composed of a minimum number of DNA single strands from a topological viewpoint.
Disclosure of Invention
Aiming at the problems in the prior art, the invention aims to provide a method for constructing a DNA polyhedron topological structure, which can quickly, simply and qualitatively analyze a reasonable scheme for constructing the DNA polyhedron topological structure by a minimum number of single chains, can quantitatively give the quantity and distribution conditions of odd number of twisting edges required in the scheme, and has accurate and reliable result and easy operation.
In order to achieve the purpose, the invention adopts the following technical scheme:
the invention provides a method for constructing a DNA polyhedron topological structure, which comprises the following steps:
step 1, obtaining a target polyhedron, and marking non-adjacent three-degree vertexes in the target polyhedron, wherein the three-degree vertexes refer to vertexes connected with three edges simultaneously;
step 3, judging whether nonadjacent three-degree vertexes exist in the optimized DNA polyhedron topological structure, if so, repeating the steps 1 to 2 until an optimal DNA polyhedron topological structure is obtained; and if not, the optimized DNA polyhedron topological structure obtained in the step 2 is an optimal DNA polyhedron topological structure.
Optionally, in step 1, the target polyhedron includes V three-degree vertexes, E edges and F faces, where V is greater than or equal to 4, E is greater than or equal to 6, F is greater than or equal to 4, and V, E and F are integers respectively.
Optionally, in step 1, a specific process of marking the non-adjacent three-degree vertex is as follows: and selecting a three-degree vertex in the target polyhedron, marking the three-degree vertex, and marking all three-degree vertices which are not adjacent to the marked three-degree vertex.
Optionally, in step 1, the number of nonadjacent three-degree vertices in the target polyhedron is M, and M is an integer part of F/3.
Optionally, marking nonadjacent three-degree vertexes in the target polyhedron in step 1 specifically means that all nonadjacent three-degree vertexes are numbered sequentially, and each nonadjacent three-degree vertex corresponds to one number.
Optionally, in step 2, the odd-numbered twisted sides refer to sides of a half-helix containing an odd number of DNAs; the even-numbered twisted sides refer to sides of a half-helix containing an even number of DNAs; the optimized DNA polyhedron topological structure and the optimized DNA polyhedron topological structure respectively comprise N odd-numbered twisting edges, wherein N is 3M and E-3M even-numbered twisting edges.
Optionally, each three-degree vertex in the DNA polyhedron topological structure is connected to three edges, and the three edges are located on different faces, so that three faces corresponding to each three-degree vertex are respectively composed of three DNA single strands; in step 2, the odd-numbered twisted edges are used for replacing three edges connected with each non-adjacent three-degree vertex in the target polyhedron, and the odd-numbered twisted edges are replaced from any one non-adjacent three-degree vertex, so that three DNA single chains forming three surfaces of the topological structure of the DNA polyhedron are reduced into one DNA single chain.
Optionally, in step 3, the step of judging whether non-adjacent three-degree vertexes exist in the optimized DNA polyhedron topological structure specifically includes: whether the optimized DNA polyhedron topological structure consists of two DNA single chains or not.
Optionally, according to the base complementary pairing principle, the DNA double helix line on each edge of the optimized DNA polyhedron topological structure and the DNA double helix line on each edge of the optimized DNA polyhedron topological structure are respectively antiparallel.
Optionally, the target polyhedron is a polyhedron whose vertex is a vertex of three degrees or a convex polyhedron whose vertex is a vertex of three degrees.
Compared with the prior art, the invention has the beneficial effects that:
(1) the invention provides a method for constructing a DNA polyhedron topological structure, which is characterized in that three edges connected with a selected three-degree vertex are all replaced by odd-number twisted edges by selecting non-adjacent three-degree vertices, so that the topological structure of a DNA polyhedron formed by a minimum number of single strands and the number of required DNA single strands can be quickly constructed on the premise of ensuring that the base complementary pairing principle is met, and a great deal of attempts in designing the DNA polyhedron topological structure formed by the minimum number of DNA single strands are avoided.
(2) The method for constructing the topological structure of the DNA polyhedron provided by the invention can rapidly, simply and qualitatively analyze a reasonable scheme for constructing the topological structure of the DNA polyhedron by using the minimum number of single chains, can quantitatively give the quantity and the distribution condition of odd-number twisted edges required in the scheme, can qualitatively and quantitatively give a synthetic template for constructing the topological structure of the DNA polyhedron by using the minimum number of single chains without requiring a large amount of time and energy of experienced technicians, has accurate and reliable result and easy operation, is suitable for both a polyhedron with a vertex of three degrees and a convex polyhedron with a vertex of three degrees, and is suitable for wide popularization.
Additional aspects and advantages of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.
Drawings
FIG. 1 is a schematic structural diagram of an odd-numbered twisted side and an even-numbered twisted side provided by an embodiment of the present invention;
FIG. 2 is a schematic diagram of the labeling of the three-degree vertices of a DNA tetrahedron and the topology of the DNA tetrahedron consisting of 2 DNA single strands according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of the three-degree vertex labeling of DNA hexahedron and the topological structure of DNA hexahedron composed of 2 DNA single strands according to the embodiment of the present invention;
FIG. 4 is a schematic diagram of the labeling of three-degree vertices in a DNA dodecahedron and the topological structure of the DNA dodecahedron composed of 2 DNA single strands according to the embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. Test methods in which specific conditions are not specified in the following examples are generally carried out under conventional conditions or under conditions recommended by the respective manufacturers.
The odd-numbered twisted sides in the present invention can be represented as a DNA double helix containing 2n +1 crossovers at the leftmost side in fig. 1(a), and can be further simplified as a DNA double helix containing only one crossovers at the rightmost side in fig. 1 (a); the even-numbered twisted edges may be represented as a DNA double helix containing 2n crossovers at the leftmost side in fig. 1(b), and may be further simplified as a DNA double helix containing only zero crossovers at the rightmost side in fig. 1 (b).
The method comprises the steps of firstly obtaining a target polyhedron, wherein the target polyhedron is a randomly selected polyhedron with a vertex of three degrees or a convex polyhedron with a vertex of three degrees, and marking all nonadjacent three-degree vertexes in the target polyhedron, wherein the target polyhedron comprises V three-degree vertexes, E edges and F faces, V is more than or equal to 4, E is more than or equal to 6, F is more than or equal to 4, V, E and F are respectively integers, the number of nonadjacent three-degree vertexes in the target polyhedron is M, M is an integer part of [ F/3], and the specific process of marking all nonadjacent three-degree vertexes in the target polyhedron is as follows: the method comprises the steps of firstly selecting a three-degree vertex in a target polyhedron, then finding out all three-degree vertexes which form nonadjacent three-degree vertexes with the three-degree vertex, and then sequentially numbering all the nonadjacent three-degree vertexes, wherein each nonadjacent three-degree vertex corresponds to a number, the number of each nonadjacent three-degree vertex is used as a mark for identifying each nonadjacent three-degree vertex, and the number of each nonadjacent three-degree vertex can be a number, a letter or a figure.
Replacing edges connected with all nonadjacent three-degree vertexes in the target polyhedron by using odd-numbered twisting edges in the graph 1(a), and replacing the rest edges in the target polyhedron by using even-numbered twisting edges in the graph 1(b) to obtain an optimized DNA polyhedron topological structure, wherein the number of the odd-numbered twisting edges in the optimized DNA polyhedron topological structure is 3M, and the number of the even-numbered twisting edges is E-3M;
if the optimized DNA polyhedron topological structure consists of two DNA single chains, the optimized DNA polyhedron topological structure is the optimal DNA polyhedron topological structure; otherwise, finding out all nonadjacent three-degree vertexes in the optimized DNA polyhedron topological structure, marking the nonadjacent three-degree vertexes, replacing the edges connected with all nonadjacent three-degree vertexes in the optimized DNA polyhedron topological structure by utilizing the odd-numbered twisted edges in the graph 1(a), replacing the rest edges in the optimized DNA polyhedron topological structure by utilizing the even-numbered twisted edges in the graph 1(b), and repeating the steps until the optimal DNA polyhedron topological structure is obtained;
specifically, each three-degree vertex in the DNA polyhedron topological structure is connected with three edges, the three edges are positioned on different faces, and then three faces corresponding to each three-degree vertex are respectively composed of three DNA single chains; and replacing three edges connected with each nonadjacent three-degree vertex in the target polyhedron by using the odd-numbered twisted edges, wherein the odd-numbered twisted edges are replaced from any nonadjacent three-degree vertex, so that three DNA single chains forming three surfaces of the DNA polyhedron topological structure are reduced into one DNA single chain.
According to the base complementary pairing principle, the DNA double helix line on each edge of the optimized DNA polyhedron topological structure and the DNA double helix line on each edge of the optimized DNA polyhedron topological structure are respectively in antiparallel.
Example 1
As shown in fig. 2, a tetrahedron has 4 faces, wherein the number of nonadjacent three-degree vertices in the tetrahedron is 1, the nonadjacent three-degree vertices in the tetrahedron are labeled, three edges connected to the nonadjacent three-degree vertices in the tetrahedron are replaced by odd-numbered twisted edges in fig. 1(a), and the remaining three edges in the tetrahedron are replaced by even-numbered twisted edges in fig. 1(b), so as to finally obtain an optimized DNA tetrahedron topology structure composed of two DNA single strands, and the nonadjacent three-degree vertices in the optimized DNA tetrahedron topology structure no longer exist, that is, the optimized DNA tetrahedron topology structure is the optimal DNA tetrahedron topology structure.
Example 2
As shown in fig. 3(a), there are 6 faces in the hexahedron, wherein, one three-degree vertex in the hexahedron is marked by letter a, and the remaining three-degree vertices which are not adjacent to the three-degree vertex a are the following two cases:
in the first case: the nonadjacent three-degree vertexes distributed in a face-to-face angle mode with the three-degree vertexes A are marked as B, the number of the nonadjacent three-degree vertexes B is 3, the three nonadjacent three-degree vertexes B are equivalent, and the three nonadjacent three-degree vertexes B are respectively distributed in a face-to-face angle mode with the three-degree vertexes A;
in the second case: the number of the nonadjacent three-degree vertexes which are distributed in a body diagonal manner with the three-degree vertex A is C.
As shown in fig. 3(B), three sides connected to the three-degree vertex a are replaced by odd-numbered twisted sides in three fig. 1(a), three sides connected to the non-adjacent three-degree vertex B are replaced by odd-numbered twisted sides in 3 fig. 1(a), and the rest sides in the hexahedron are replaced by even-numbered twisted sides in fig. 1(B), wherein the number of DNA single strands synthesizing the optimized DNA hexahedral topology is reduced to two, and at this time, the number of DNA single strands synthesizing the optimized DNA hexahedral topology cannot be further reduced, that is, the optimized DNA hexahedral topology obtained in fig. 3(B) is the optimized DNA hexahedral topology;
as shown in fig. 3(C), the odd-numbered twisted edges in 3 pieces of fig. 1(a) are used to replace three edges connected to the three-degree vertex a, the odd-numbered twisted edges in 3 pieces of fig. 1(a) are used to replace three edges connected to the three-degree vertex C, and the remaining 6 edges in the hexahedron are replaced with the even-numbered twisted edges in fig. 1(b), so as to reduce the number of DNA single strands synthesizing the optimized DNA hexahedral topology to two, at this time, the number of DNA single strands synthesizing the optimized DNA hexahedral topology cannot be further reduced, that is, the optimized DNA topology obtained in fig. 3(C) is the optimal DNA hexahedral topology;
wherein the double helix of DNA on each side of the optimal hexahedral topology of DNA after replacement by an odd or even number of twisted sides is antiparallel according to the base complementary pairing rules, as indicated by the arrows in FIGS. 3(b) and 3 (c).
As can be seen from the optimal DNA hexahedral topology structure in fig. 3(b) and the optimal DNA hexahedral topology structure in fig. 3(c), the selection manner of the triangle vertices that are not adjacent to the triangle vertex a is different, and the optimal DNA hexahedral topology structures obtained finally are different, which indicates that the optimal DNA polyhedron topology structure obtained by the method for constructing a DNA polyhedron topology structure provided by the present invention has uniqueness.
Example 3
As shown in fig. 4(a), the dodecahedron has 12 faces, wherein the number of nonadjacent three-degree vertexes is 4, the four nonadjacent three-degree vertexes in the dodecahedron are sequentially numbered A, B, C and D, specifically, when one three-degree vertex is selected and numbered as a, the marks of the remaining three-degree vertexes which are nonadjacent to a are limited by the three-degree vertex a, and the dodecahedron has only the unique distribution configuration of the four nonadjacent three-degree vertexes marked in fig. 4 (a);
each of the three-degree vertexes is connected with three sides, there are four nonadjacent three-degree vertexes in the dodecahedron in total, which are respectively a three-degree vertex a, a three-degree vertex B, a three-degree vertex C and a three-degree vertex D, the three sides connected with each nonadjacent three-degree vertex in the DNA dodecahedron are replaced by odd-numbered times of twisted sides in fig. 1(a), 12 odd-numbered times of twisted sides in fig. 1(a) are required in total, the rest sides in the DNA dodecahedron are replaced by even-numbered times of twisted sides in fig. 1(B), and finally the optimized DNA dodecahedron topology shown in fig. 4(B) is obtained, which is composed of 4 DNA single-strands, the optimized DNA dodecahedron topology shown in fig. 4(B) is composed of 4 DNA single-strands, that is illustrated that the number F of faces of the DNA dodecahedron topology shown in fig. 4(B) is 4, and the number V of nonadjacent three-degree vertexes is an integer part of [ F/3], the number V of the three-degree vertexes is 1;
further, there are three equivalent three-degree vertices in the DNA dodecahedral topology shown in FIG. 4(b) consisting of 4 DNA single strands, which are represented by circles, and any one of the three equivalent three-degree vertexes in FIG. 4(B) is not adjacent to the four non-adjacent three-degree vertexes A, B, C and D in FIG. 4(a), and 3 sides connected to any one of the three-degree vertexes represented by circles in FIG. 4(B) are replaced with odd-numbered twisted sides in FIG. 1(a) to obtain the optimized DNA dodecahedron topology composed of 2 DNA single strands in FIG. 4(C), the number of DNA single strands synthesizing the optimized DNA dodecahedral topology cannot be further reduced, that is, the optimized DNA dodecahedral topology obtained in FIG. 4(c) is the optimal DNA dodecahedral topology;
wherein the double helix of DNA on each side of the optimized and optimized DNA dodecahedral topologies replaced by either an odd or even number of twisted sides is antiparallel according to the base complementary pairing rules, as indicated by the arrows in FIGS. 4(b) and 4 (c).
In addition, the convex polyhedron means that any one surface of the polyhedron is stretched into a plane, and if all other surfaces are on the same side of the plane, the regular dodecahedron is also the convex polyhedron, so that the method for quickly constructing the target polyhedron consisting of the minimum number of single chains disclosed by the invention is also suitable for other convex polyhedrons only containing three-degree vertexes.
In conclusion, the invention provides a method for constructing a DNA polyhedron topological structure, particularly, three edges connected with a selected three-degree vertex are replaced by odd-numbered twisted edges by selecting the non-adjacent three-degree vertex, the topological structure of the DNA polyhedron formed by the minimum number of single strands and the number of the required DNA single strands can be quickly constructed under the condition of ensuring that the base complementary pairing principle is met, and a great deal of attempts in the process of designing the DNA polyhedron topological structure formed by the minimum number of DNA single strands are avoided; in addition, the method can rapidly, simply and qualitatively analyze a reasonable scheme of DNA polyhedron topological structure formed by the minimum number of single chains, can quantitatively give the quantity and distribution condition of odd number of twisting edges required in the scheme, does not need to require a great deal of time and energy of experienced technicians, can qualitatively and quantitatively give a synthetic template of DNA polyhedron topological structure formed by the minimum number of single chains, has accurate and reliable result and easy operation, is suitable for both polyhedrons with vertexes of three degrees and convex polyhedrons with vertexes of three degrees, and is suitable for wide popularization.
It will be apparent to those skilled in the art that various changes and modifications may be made in the present invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include such modifications and variations.
Claims (10)
1. A method for constructing a DNA polyhedron topology, comprising the steps of:
step 1, obtaining a target polyhedron, and marking non-adjacent three-degree vertexes in the target polyhedron, wherein the three-degree vertexes refer to vertexes connected with three edges simultaneously;
step 2, replacing edges connected with all the nonadjacent three-degree vertexes in the target polyhedron by odd-numbered twisting edges, and replacing the rest edges in the target polyhedron by even-numbered twisting edges to obtain an optimized DNA polyhedron topological structure;
step 3, judging whether nonadjacent three-degree vertexes exist in the optimized DNA polyhedron topological structure, if so, repeating the steps 1 to 2 until an optimal DNA polyhedron topological structure is obtained; and if not, the optimized DNA polyhedron topological structure obtained in the step 2 is an optimal DNA polyhedron topological structure.
2. The method for constructing a topological structure of DNA polyhedrons according to claim 1, wherein in step 1, said target polyhedron comprises V three-degree vertices, E edges and F faces, wherein V is not less than 4, E is not less than 6, F is not less than 4, and V, E and F are integers respectively.
3. The method for constructing a topological structure of a DNA polyhedron of claim 1, wherein the specific process of labeling the non-adjacent three-degree vertices in step 1 is: and selecting a three-degree vertex in the target polyhedron, marking the three-degree vertex, and marking all three-degree vertices which are not adjacent to the marked three-degree vertex.
4. The method of constructing a topology of DNA polyhedrons according to claim 2, wherein in step 1, the number of non-adjacent three-degree vertices in the target polyhedron is M, and M is an integer of F/3.
5. The method for constructing a topological structure of a DNA polyhedron of claim 1, wherein the step 1 of labeling nonadjacent three-degree vertices in the target polyhedron is to sequentially number all the nonadjacent three-degree vertices, and each nonadjacent three-degree vertex corresponds to one number.
6. The method for constructing a DNA polyhedron topology according to claim 4, wherein in step 2, the odd-numbered twisted sides are sides of a half-helix containing an odd number of DNAs;
the even-numbered twisted sides refer to sides of a half-helix containing an even number of DNAs;
the optimized DNA polyhedron topological structure and the optimized DNA polyhedron topological structure respectively comprise N odd-numbered twisting edges, wherein N is 3M and E-3M even-numbered twisting edges.
7. The method for constructing a topological structure of DNA polyhedron of claim 1, wherein each vertex of three degrees in the topological structure of DNA polyhedron is connected to three sides, and the three sides are located on different faces, so that the three faces corresponding to each vertex of three degrees are respectively composed of three DNA single strands;
in step 2, the odd-numbered twisted edges are used for replacing three edges connected with each non-adjacent three-degree vertex in the target polyhedron, and the odd-numbered twisted edges are replaced from any one non-adjacent three-degree vertex, so that three DNA single chains forming three surfaces of the topological structure of the DNA polyhedron are reduced into one DNA single chain.
8. The method for constructing a DNA polyhedron topology structure according to claim 1, wherein the step 3 of determining whether there are non-adjacent three-degree vertices in the optimized DNA polyhedron topology structure specifically means: whether the optimized DNA polyhedron topological structure consists of two DNA single chains or not.
9. The method for constructing a DNA polyhedron topology according to claim 1, wherein the DNA double helix on each side of the optimized DNA polyhedron topology and the DNA double helix on each side of the optimized DNA polyhedron topology are respectively antiparallel according to the base complementary pairing principle.
10. The method of constructing a DNA polyhedron topology according to claim 1, wherein the target polyhedron is a polyhedron whose vertex is a three-degree vertex or a convex polyhedron whose vertex is a three-degree vertex.
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