Universal graph problem solving method and system based on DNA strand displacement

Abstract

The invention belongs to the technical field of biological computing, and discloses a general graph problem solving method and system based on DNA strand displacement, and the method comprises the steps: S1, carrying out graph structure modeling, and carrying out graph representation module based on DNA strand displacement; s2, constructing a detection module, and realizing parallel biochemical identification of constraint conditions by using a detection gate D (u) containing a fluorescence label; and S3, carrying out parallel detection on candidate solutions, supporting a unified solution framework of problems such as a minimum control set and a maximum independent set, and realizing rapid screening of the solutions through candidate solution enumeration and a fluorescence signal criterion. The method and the system provided by the invention adopt a modular cascade structure, realize direct solution of a complex problem on the premise of not depending on electronic calculation conversion, compress a solution space in combination with a graph feature value preprocessing technology, have the technical advantages of high neighborhood recognition accuracy, strong leakage reaction inhibition, strong experimental process universality and the like, and have a wide application prospect. And the application efficiency and stability of DNA calculation in the field of combinatorial optimization are remarkably improved.

Description

Technical Field

[0001] This invention relates to molecular computing technology, and more particularly to a general graph problem-solving method and system based on DNA strand substitution. It proposes a graph structure modeling method based on the DNA strand substitution mechanism and uses it to construct a general DNA computing system for solving NP-complete graph theory problems, belonging to the interdisciplinary field of biological computing and graph theory optimization. Background Technology

[0002] Combinatorial optimization problems are widespread in modern information systems and engineering practices, especially in security-sensitive areas such as access control, key management, and authentication. Essentially, they involve efficiently searching a complex solution space to select the optimal solution under specific constraints. However, these problems are often classified as NP-complete, possessing exponential computational complexity, and to date, no polynomial-time algorithm has been able to solve them completely.

[0003] Faced with the "combinatorial explosion" property of NP-complete problems, traditional electronic computing methods, limited by serial architecture, storage density, and processing power, are inadequate for solving large-scale instances. Therefore, developing novel computational models has become a key path to address complex combinatorial problems.

[0004] DNA computing, as a novel computational paradigm based on molecular encoding and reaction rules, has gradually demonstrated its unique advantages in parallelism, information density, and distributed operations since Adleman first experimentally solved the Hamiltonian path problem in directed graphs in 1994. In DNA computing, graph structure, paths, and logical relationships can all be encoded through base sequences and driven by strand substitution reactions, thereby simulating the generation of solution space, constraint judgment, and result selection for combinatorial optimization problems. While this characteristic provides new possibilities for solving NP-complete problems, it also presents numerous technical challenges.

[0005] However, existing technologies still face many challenges when using DNA computing to solve combinatorial optimization problems. First, the problems are highly specific, requiring reconstruction of the computational model for different NP problems. Current DNA computing models are mostly designed for single types of NP-complete problems (such as 0-1 programming, graph coloring, Hamiltonian path, etc.), lacking versatility. Second, the problem of unified modeling for multiple types of graph problems has not been solved. The lack of a unified architecture leads to low reuse of experimental resources, requiring redesign of the computational structure for different problems, resulting in complex R&D processes and high resource consumption, which seriously restricts the practicality and scalability of DNA computing. Third, the computation process relies on electronic conversion, leading to increased system complexity. In addition, although theoretically NP-complete problems can be converted to each other through reduction, in practical applications such conversions usually rely on electronic computing environments, resulting in fragmented operation processes, increased costs, and additional errors and instabilities in hybrid computing environments.

[0006] For example, CN111598242A suffers from system delays due to electronic signal conversion, approximately >30 minutes per cycle, and is unable to construct a fully biochemical closed-loop system; CN107437003A uses a step-by-step reaction, which leads to error accumulation, with an average leakage rate ≥8%, lacks a solution space compression mechanism, and consumes a lot of resources.

[0007] In view of the above problems, there is an urgent need for a general DNA computational model that can handle multiple types of NP-complete graph theory problems under a unified structural framework, so as to achieve the reuse of computational structures and seamless integration of problem instances, thereby improving the efficiency, accuracy and system integration of combinatorial optimization problem solving. Summary of the Invention

[0008] To address the limitations of existing technologies, this invention proposes a general graph problem-solving method and system based on DNA strand substitution. It constructs a general DNA computational model by combining a graph representation (GR) module with a detection module. This model supports the processing of multiple types of NP-complete graph theory problems within a unified structural framework. Through a module cascading approach, it unifies the encoding of graph structures, execution of strand substitution reactions, and output of fluorescence signals. The problem-solving process is completed without relying on electron conversion, enabling the reuse of computational structures and seamless integration of problem instances. This improves the efficiency, accuracy, and system integration of combinatorial optimization problem solving.

[0009] To achieve the above objectives, the present invention provides the following technical solution: A general graph problem-solving method based on DNA strand substitution, characterized by the following steps: S1. Graph structure modeling: In graph G=(V,E), vertex u∈V is encoded as DNA single strand γ(u), which is composed of Toehold domain a(u), first branch migration domain b(u), and second branch migration domain c(u); construct computation gate Γ(u), and form closed neighborhood representation unit by hybridizing main strand β(u) with γ(u) and its adjacent vertex chains; S2. Detection module construction: Design a detection gate D(u) for each vertex u, including an incumbent chain with a fluorescent group and a complementary target chain with a quenching group. The fluorescent signal is released through chain substitution reaction as the basis for determining the constraint condition. S3. Parallel Detection of Candidate Solutions: Configure computational gate combinations according to the target problem type, add the input chain set Input(S) corresponding to the candidate solutions, and detect the output signal of each detection gate D(u) by real-time PCR. If the following conditions are met, it is determined to be a valid solution: For the minimum control set problem, the detection gates corresponding to all vertices trigger fluorescence signals; For the maximal independent set problem, no fluorescence signal is output from any of the detection gates; This allows for the efficient solving of NP-complete graph theory problems, including minimum control sets and maximum independent sets.

[0010] A DNA computing system for implementing the method, characterized in that it comprises: The graph representation module contains multiple DNA computation gates Γ(u) assembled from the main strand β(u) and γ strand, with each computation gate corresponding to the closed neighborhood relation expression of a vertex in the graph; Detection module: A parallel detection array consisting of multiple detection gates D(u), where each detection gate reflects the satisfaction state of the corresponding vertex neighborhood constraints through the intensity of the fluorescence signal; Control module: Used to configure computational gate combination schemes for different problem types, including closed neighborhood detection mode Γ(u) and open neighborhood detection mode Γ'(u), where Γ'(u) removes the c(u) field from the original chain.

[0011] Compared with the prior art, the method and system provided by the present invention have at least the following advantages: 1. This invention overcomes three major bottlenecks in the field of DNA computing through a synergistic approach of dynamic neighborhood modeling, mathematical pre-optimization, and signal threshold collaboration: It constructs the first biomolecular architecture adaptable to multiple types of NP problems and achieves the first fusion optimization of DNA computing and graph theory mathematical models, providing an efficient, stable, and scalable solution for DNA-based combinatorial optimization. The method and system provided by this invention adopt a modular cascade structure, enabling direct solutions to complex problems without relying on electronic computation conversion. Combined with graph eigenvalue preprocessing technology to compress the solution space, it possesses technical advantages such as high neighborhood recognition accuracy, strong leakage reaction suppression, and strong experimental procedure versatility, significantly improving the efficiency and stability of DNA computing in the field of combinatorial optimization.

[0012] 2. The GR module proposed in this invention is based on a single-stranded complex configuration, mapping each graph vertex and its adjacency relationship to a molecular computational gate structure. The input strand activates the computational gate to release the target strand, and a detection module reads the reaction results to determine structural satisfaction or constraint conflict. Compared with traditional DNA computational models, this invention, for the first time, achieves the simultaneous solution of multiple NP-complete problems such as minimum control set and maximum independent set within a unified model framework, avoiding model reconstruction and logical transformation, reducing development and experimental costs, and improving system robustness.

[0013] 4. This invention incorporates graph structure properties and eigenvalue theory for solution space pruning in the experimental design, combined with theorem preprocessing methods, effectively reducing the experimental scale and improving candidate screening efficiency. Simultaneously, optimization of fluorescence signal reading and leakage reaction ensures the stability of the calculation process and the accuracy of detection.

[0014] 5. This invention provides an efficient, universal, and scalable general molecular computation solution for combinatorial optimization problems, with good engineering adaptability and theoretical innovation value, and has broad application prospects in multiple fields such as data security, graph analysis, and bioinformatics. Attached Figure Description

[0015] Figure 1 is a schematic diagram of the vertex modeling structure based on DNA computing in an embodiment of the present invention; Figure 2 is a schematic diagram of the vertex relationship modeling method based on DNA computing in an embodiment of the present invention; Figure 3 is a schematic diagram of the computational gate structure for generating closed vertex neighborhoods in an embodiment of the present invention; Figure 4 is a schematic diagram of the chain substitution reaction for vertex closed neighborhood detection through a detection gate in an embodiment of the present invention; Figure 5 A schematic diagram of a computational gate constructed for an embodiment of the present invention; Figure 6 shows the fluorescence detection experimental results of the first group of candidate solutions in solving the minimum control set problem using the method of this embodiment of the invention; Figure 7 shows the fluorescence detection experimental results of the 12th group of candidate solutions in solving the minimum control set problem using the method of this embodiment of the invention; Figure 8 is a schematic diagram of the vertex neighborhood generation structure for the maximum independent set problem in an embodiment of the present invention; Figure 9 is a schematic diagram of the chain substitution reaction for vertex neighborhood detection through a detection gate in an embodiment of the present invention; Figure 10 shows the fluorescence detection experimental results of the first group of candidate solutions in solving the maximum independent set problem using the method of this embodiment of the invention; Figure 11 shows the fluorescence detection experimental results of the 49th candidate solution in solving the maximum independent set problem using the method of this embodiment of the invention. Detailed Implementation

[0016] See appendix Figures 1 to 11 The feasibility of the proposed general computational system based on DNA graph representation structure in solving NP-complete graph theory problems is verified through several specific implementation examples. The experiments use Petersen graphs as the target graph structure, solving their minimum control set and maximum independent set problems respectively, to evaluate the performance of the invention in terms of universality, computational efficiency, and molecular recognition accuracy.

[0017] Basic Implementation The general graph problem-solving method based on DNA strand substitution provided in this invention includes the following steps: S1. Graph structure modeling: In graph G=(V,E), vertex u∈V is encoded as DNA single strand γ(u), which is composed of Toehold domain a(u), first branch migration domain b(u), and second branch migration domain c(u); construct computation gate Γ(u), and form closed neighborhood representation unit by hybridizing main strand β(u) with γ(u) and its adjacent vertex chains; The computational gate Γ(u) is constructed by annealing and hybridizing the main chain β(u) with the γ(u) chain of vertex u itself and the γ(v) chains of all its adjacent vertices v∈N(u) to form an activatable complex structure that expresses the closed neighborhood N[u] of vertex u, specifically including: S1-1. Represent the problem to be solved as a graph G=(V,E), where each vertex u∈V is encoded as a single strand of DNA γ(u), consisting of a Toehold domain a(u) and two branch migration domains b(u) and c(u); S1-2, Construct the computational gate Γ(u), which is formed by the DNA strand γ(u) of the vertex itself and its neighboring vertices. i The hybridization of β(u) with the main chain is used to realize the representation and release of closed neighborhoods; S1-3. For scenarios requiring open neighborhood detection, a simplified version of the DNA strand γ′(u) is further constructed, which removes the c(u) domain and constructs the corresponding computational gate Γ′(u) to realize the neighborhood detection function.

[0018] S2. Detection module construction: Design a detection gate D(u) for each vertex u, including an incumbent chain with a fluorescent group and a complementary target chain with a quenching group. The fluorescent signal is released through chain substitution reaction as the basis for determining the constraint condition. Wherein, the sequence of the incumbent chain contains regions complementary to b(u) and c(u), and the target chain contains regions complementary to a(u). b(u) In the complementary region of c(u), when the input chain γ(u) binds to the detection gate D(u) through the a(u) domain, it triggers branch migration and releases the fluorescently labeled output chain; specifically including: S2-1. Construct a detection gate D(u) for each vertex u. The gate consists of an incumbent chain with fluorescent label and its complementary target chain. The former is composed of b(u) and c(u), and the latter is composed of a(u)*, b(u)*, and c(u)*. The release of fluorescent signal depends on the chain substitution mechanism. S2-2, When the target chain γ(u) pairs with the Toehold of D(u) and initiates a substitution reaction, the Output chain is released and a detectable fluorescent signal is generated; S3. Parallel Detection of Candidate Solutions: Configure computational gate combinations according to the target problem type, add the input chain set Input(S) corresponding to the candidate solutions, and detect the output signal of each detection gate D(u) by real-time PCR. If the following conditions are met, it is determined to be a valid solution: For the minimum control set problem, the detection gates corresponding to all vertices trigger fluorescence signals; For the maximal independent set problem, no fluorescence signal is output from any of the detection gates; This allows for the efficient solving of NP-complete graph theory problems, including minimum control sets and maximum independent sets, specifically: S3-1. For the minimum control set problem, design an experimental scheme to enumerate candidate sets and construct corresponding computational gate combinations to detect whether the closed neighborhood of all vertices is contained; if all detection gates produce fluorescence signals, then the set is the control set. S3-1. For the maximum independent set problem, construct a simplified computational gate Γ′(u) and enumerate candidate sets; check if any vertex's neighborhood produces a fluorescence signal; if there is no fluorescence output, then the set is an independent set. S3-3. In the experiment, each candidate solution was configured with a corresponding test tube combination, and calculation gates and detection gates were added one by one. The reaction results were detected by a real-time PCR instrument to determine the validity of the solution.

[0019] Step S3-3 further includes a pre-optimization process: For the eigenvalue parameters of the target graph G, the Lovász eigenvalue theorem or the lower bound theorem of the control set are used to impose cardinality constraints on the candidate solution set, thereby compressing the size of the candidate solution, which can usually be compressed to 20%-30% of the original solution space.

[0020] A DNA computing system for implementing the method, comprising: The graph representation module contains multiple DNA computation gates Γ(u) assembled from the main strand β(u) and γ strand, with each computation gate corresponding to the closed neighborhood relation expression of a vertex in the graph; Detection module: A parallel detection array consisting of multiple detection gates D(u), where each detection gate reflects the satisfaction state of the corresponding vertex neighborhood constraints through the intensity of the fluorescence signal; Control module: Used to configure computational gate combination schemes for different problem types, including closed neighborhood detection mode Γ(u) and open neighborhood detection mode Γ'(u), where Γ'(u) removes the c(u) field from the original chain.

[0021] The general graph problem-solving method based on DNA strand substitution provided in this invention mainly includes the following steps: Step 1: Construct a DNA structure representation of the graph, with vertices being single-stranded γ and adjacency relationships calculated using gates Γ or Γ′; Step 2: Design a detection gate D(u) to achieve full coverage of the closed neighborhood of the control set or neighborhood detection of independent sets; Step 3: Enumerate the candidate solution set and configure the experiment. Construct test tube groups according to the solution set dimension, and configure the target chain, computation gate and detection gate. Step 4: Detect the fluorescence intensity of the chain displacement reaction, screen out all feasible solutions, locate the optimal solution (such as the minimum control set or the maximum independent set), and stop the experiment.

[0022] More specifically, the method includes: Step 1: Define the problem to be solved. The target graph is a Peterson graph P=(V,E) in classical graph theory, with a vertex set size of 10 and an edge set size of 15. It has high symmetry and is suitable as a standard test case for graph structure analysis. This experiment constructs the following two problem scenarios: Scenario 1: Find the minimum control set, that is, the smallest cardinality subset S of the set whose closed neighborhood N[S] of vertices covers the entire set V; Scenario 2: Find the maximum independent set, that is, the vertex set I⊆V, where no two vertices are connected by an edge and the cardinality is the largest.

[0023] Step 2: Construct the graph representation structure (GR) and computation gates. For each vertex v i ∈V, design DNA single-stranded structures γ(v) respectively i ), containing the Toehold field a(v i ), branch migration domain b(v i ),c(v i Each computational gate Γ(v) i ) represents the closed neighborhood structure of this vertex, forming a complex structure, consisting of the main chain β(v i ) and hybridization with all γ chains constitute. In the maximum independent set experiment, the structure that removes c(v) is constructed. i Simplified computational gate Γ′(v) in the domain i This forms a non-closed neighborhood detection mode.

[0024] Step 3: Design the detection module and configure the experimental reaction system. Each vertex corresponds to a detection gate D(v). i It consists of an incumbent chain (containing a fluorescent group) and a target chain (containing a quenching group), and is used to determine the presence of a target chain and initiate a chain substitution reaction.

[0025] Step 4: Introduce graph structure theorems to mathematically reduce the candidate solution space: For the minimum control set problem, referring to the lower bound theorem of control sets (such as the proposition proposed by Bruce in 1996), and combining the fact that the degree of the closed neighborhood of each vertex in graph P is 4, we deduce that the lower bound of the control number is 3, thus compressing the original solution space to contain only combinations of triples; for the maximum independent set problem, using the Lovász eigenvalue bound theorem and the eigenvalues ​​of the adjacency matrix of graph P (maximum 3, minimum -2), we calculate that the upper bound of the independence number is 4. Further combining the lower bound property of the coloring number, we eliminate all candidate solutions with more than 4 tuples. Finally, in the minimum control set problem, only 120 triples are retained from the original solution space, each containing 10 test tubes (corresponding to all vertices); in the maximum independent set problem, 210 quadruples are selected from the original as experimental objects, each containing 1 test tube.

[0026] Step 5: Chain Substitution and Fluorescence Signal Detection. Prepare each set of reagents into reaction tubes, including the target input chain, computation gate, and detection gate, and place them in a quantitative real-time PCR instrument for reaction reading. Determine whether candidate solutions satisfy graph theory constraints by comparing fluorescence signal intensities. If all 10 tubes in the minimum control set experiment produce fluorescence, it is considered a control set; if no fluorescence is observed in any tube in the maximum independent set experiment, it is considered an independent set.

[0027] The results are as follows Figures 6-11 As shown, the normalized output of the fluorescence signal is used to accurately determine the independence of the solution set. In the minimum control set experiment, it was found that the 12th combination satisfies the control set constraint, that is, the solution is the 12th combination {v1,v3,v7}, which verifies the complete solution space coverage capability of the model of the present invention. In the maximum independent set experiment, the 49th combination did not produce fluorescence, meaning the solution is the 49th combination {v1, v3, v9, v...}. 10 This verifies the independent set detection function.

[0028] Example 1: Based on the basic embodiment, this embodiment further uses the Peterson diagram as an example to solve for the minimum control set, and its specific steps include the following: 1) Construct 10 computation gates Γ(u), each containing a γ chain of a central vertex and 3 adjacent vertices ( Figure 5 ); 2) Pre-select 120 candidate solutions for triplets according to the lower bound theorem of control number; 3) Input(S) trigger chain permutation is added group by group. Fluorescence detection shows that the 12th solution (v1, v3, v7) satisfies full vertex coverage. Figure 7 ).

[0029] Example 2 Based on the basic embodiment and Embodiment 1, this embodiment further uses the Peterson graph as an example to solve for the maximum independent set, and specifically includes the following steps: 1) Employ the Γ'(u) computation gate that removes the c(u) domain; 2) Screening candidate solutions for quadruples based on eigenvalue bounds; 3) Experiments show that the 49th solution group (v1, v3, v9, v10) has no neighborhood collision signals. Figure 11 According to qPCR testing, the system can achieve 100% accuracy in neighborhood detection.

[0030] In the above embodiments, the present invention employs a general DNA computing system composed of a graph representation structure (GR) based on a DNA strand substitution mechanism and its detection module. This system can efficiently solve multiple types of NP-complete graph problems, including minimum control sets and maximum independent sets, without relying on problem transformation. The embodiments of the present invention use graph structures as the computational basis, leveraging the high parallelism, programmability, and complementary base properties of DNA molecules to encode vertex and edge information in the graph into single-stranded and complex DNA structures. Input strand activation gates trigger strand substitution reactions to release the target strand, and fluorescently labeled detection gates achieve biochemical identification of feasible solutions.

[0031] In this system, each vertex u∈V corresponds to a single-stranded DNA γ(u), consisting of a Toehold domain a(u), a branching migration domain b(u), and a branching migration domain c(u). The computational gate Γ(u) forms a graph-structured encoding device that responds to the input strand Input(u) by compounding with the strands of all adjacent vertices. The strand substitution reaction releases the γ strand, including itself and all its neighbors, and forms a stable double-stranded byproduct.

[0032] To meet the detection requirements of different graph problems, this embodiment of the invention further designs a detection gate D(u), which consists of an incumbent chain (carrying a fluorescent label) and a target chain (with a quenching group). If the chain substitution reaction is successful, the output chain triggers fluorescence release, which serves as a criterion for whether the constraints are met.

[0033] For the minimum control set problem, this embodiment of the invention constructs a closed neighborhood detection structure to determine whether the closed neighborhood of the candidate vertex set covers the entire set V. If all corresponding detection gates emit fluorescence signals, then the set is a control set; otherwise, it is not.

[0034] For the maximal independent set problem, this invention introduces a simplified computational gate Γ′(u), which removes the c(u) domain from the original chain structure, preventing neighboring nodes from triggering the detection gate through chain permutation, thereby enabling the determination of neighborhood conflicts. If any detection gate emits a fluorescent signal, it indicates the existence of an adjacency conflict and that the set is not independent.

[0035] To verify the feasibility of the method, each candidate solution was evaluated through simulation and biochemical experiments. A quantitative real-time PCR instrument was used to detect the fluorescence output of the chain displacement reaction in each test tube as the criterion for judgment.

[0036] To improve experimental efficiency, this invention combines the control number and eigenvalue theorem of graphs to perform mathematical preprocessing on the solution space. For example, the maximum and minimum eigenvalues ​​of the graph constrain the upper limit of the cardinality of independent sets, and the neighborhood coverage property of the control set is used to prune candidate sets, thereby reducing the number of experimental rounds and reaction time.

[0037] Compared to CN111598242A, the above embodiments of this invention provide a single-objective problem-solving model based on DNA hairpin structures, which only supports linear solutions to vertex covering problems. It uses fixed hairpin probes to mark vertices and relies on electronic chips for result reading, requiring reconstruction of the reaction system for each type of problem. In contrast, the programmable computational gate Γ(u) (dynamically adaptable closed / open neighborhood detection) provided by this invention employs a parallel criterion based on fluorescent chemical signals (fully biochemical autonomous computation) and proposes a Lovász pre-optimization algorithm to achieve solution space compression (overcoming biological hardware limitations). This represents a breakthrough in both the detection module (functional configurability) and the computational architecture (modular stacking). Compared to CN107437003A, which provides a maximum flow algorithm based on DNA walking, it relies on the gradual extension of linear DNA strands (step-by-step reaction), requires real-time capture of intermediate states during electrophoresis (high operational complexity), and cannot solve combinatorial optimization problems such as independent sets (limited application scope). This invention achieves simultaneous detection of multiple vertices through branch migration domain self-assembly, and completes the validity verification of the solution in one step using qPCR quantitative signals. It supports mapping of multiple problems such as control sets and independent sets, and has made breakthroughs in neighborhood relation expression and solution screening mechanisms.

[0038] The embodiments of the present invention described above, through the cascaded design of graph representation structures and detection modules, achieve an executable and automated biochemical solution path for complex graph structure problems by relying on strand substitution reactions and fluorescence signal recognition without electronic computing assistance. The modular DNA computing architecture provided by the embodiments of the present invention employs a unified graph representation model and dynamically expresses vertex adjacency relationships through a programmable DNA computing gate structure Γ(u). Figure 1 Dual-mode detection mechanism: supports parallel detection of closed neighborhood (minimum control set) and open neighborhood (maximum independent set). Figure 3 , Figure 8Eigenvalue pre-optimization: Mathematical pruning of the solution space is performed using the spectral properties of graphs, reducing experimental rounds by more than 70%. It is the first to propose a DNA computational model with a cascaded design of graph representation and detection modules, supporting the solution of various graph problems under a unified architecture; it eliminates the need for problem type reduction and electronic computation conversion, improving system versatility and flexibility; it utilizes highly parallel chain substitution reactions and fluorescence signal discrimination to enhance the automation and recognition accuracy of experimental operations; and it combines graph structure theory for mathematical preprocessing to achieve solution space pruning, improving experimental efficiency. It is applicable to graph theory optimization, data security modeling, biological network analysis, and other fields, possessing strong promotional value and engineering application prospects.

[0039] The general DNA computation method and system provided in the above embodiments of the present invention are applicable to efficiently solving various types of NP-complete graph theory problems, and have broad application prospects, especially in combinatorial optimization scenarios in fields such as data security. This method is based on the DNA strand permutation principle, constructing a unified graph representation module (GR) for modeling graph structural features. It achieves rapid detection of vertex neighborhoods through parallel strand permutation, and combines the detection module to screen target subsets that satisfy specific graph theory constraints. The system adopts a general encoding strategy, eliminating the need for separate computational frameworks for different NP problems. It can directly adapt to typical graph optimization tasks, including minimum control sets and maximum independent sets, greatly simplifying the modeling process and experimental configuration. Through simulation analysis and biochemical experiments, the designed graph representation module achieves 100% neighborhood recognition accuracy and effectively suppresses DNA leakage reactions, ensuring system stability and detection accuracy. Without relying on electronic computation conversion, it achieves direct solution of complex problems through module cascading, providing an efficient, stable, and scalable solution for DNA-based combinatorial optimization. Experimental verification shows that the method and system have good versatility, stability and reaction accuracy, and can be widely used as a solution to NP-complete graph problems in fields such as data security, communication networks, and biological system modeling.

[0040] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A general graph problem-solving method based on DNA strand substitution, characterized in that, Includes the following steps: S1. Graph structure modeling: In graph G=(V,E), vertex u∈V is encoded as DNA single strand γ(u), which is composed of Toehold domain a(u), first branch migration domain b(u), and second branch migration domain c(u); construct computation gate Γ(u), and form closed neighborhood representation unit by hybridizing main strand β(u) with γ(u) and its adjacent vertex chains; S2. Detection module construction: Design a detection gate D(u) for each vertex u, including an incumbent chain with a fluorescent group and a complementary target chain with a quenching group. The fluorescent signal is released through chain substitution reaction as the basis for determining the constraint condition. S3. Parallel Detection of Candidate Solutions: Configure computational gate combinations according to the target problem type, add the input chain set Input(S) corresponding to the candidate solutions, and detect the output signal of each detection gate D(u) by real-time PCR. If the following conditions are met, it is determined to be a valid solution: For the minimum control set problem, the detection gates corresponding to all vertices trigger fluorescence signals; For the maximal independent set problem, no fluorescence signal is output from any of the detection gates; This allows for the efficient solving of NP-complete graph theory problems, including minimum control sets and maximum independent sets.

2. The general graph problem solving method according to claim 1, characterized in that, In step S1, the computation gate Γ(u) is constructed by annealing and hybridizing the main chain β(u) with the γ(u) chain of vertex u itself and the γ(v) chains of all its adjacent vertices v∈N(u) to form an activatable complex structure that expresses the closed neighborhood N[u] of vertex u, specifically including: S1-1. Represent the problem to be solved as a graph G=(V,E), where each vertex u∈V is encoded as a single strand of DNA γ(u), consisting of a Toehold domain a(u) and two branch migration domains b(u) and c(u); S1-2, Construct the computational gate Γ(u), which is formed by the DNA strand γ(u) of the vertex itself and its neighboring vertices. i The hybridization of β(u) with the main chain is used to realize the representation and release of closed neighborhoods; S1-3. For scenarios requiring open neighborhood detection, a simplified version of the DNA strand γ′(u) is further constructed, which removes the c(u) domain and constructs the corresponding computational gate Γ′(u) to realize the neighborhood detection function.

3. The method according to claim 1, characterized in that, In step S2, the sequence of the incumbent chain contains regions complementary to b(u) and c(u), and the target chain contains regions complementary to a(u), b(u), and c(u)*. When the input chain γ(u) binds to the detection gate D(u) through the a(u) domain, it triggers branch migration and releases the fluorescently labeled output chain; specifically including: S2-1. Construct a detection gate D(u) for each vertex u. The gate consists of an incumbent chain with fluorescent label and its complementary target chain. The former is composed of b(u) and c(u), and the latter is composed of a(u)*, b(u)*, and c(u)*. The release of fluorescent signal depends on the chain substitution mechanism. S2-2, When the target chain γ(u) pairs with the Toehold of D(u) and initiates a substitution reaction, the Output chain is released and a detectable fluorescent signal is generated.

4. The method according to claim 1, characterized in that, Step S3 includes: S3-1. For the minimum control set problem, design an experimental scheme to enumerate candidate sets and construct corresponding computational gate combinations to detect whether the closed neighborhood of all vertices is contained; if all detection gates produce fluorescence signals, then the set is the control set. S3-1. For the maximum independent set problem, construct a simplified computational gate Γ′(u) and enumerate candidate sets; check if any vertex's neighborhood produces a fluorescence signal; if there is no fluorescence output, then the set is an independent set. S3-3. In the experiment, each candidate solution was configured with a corresponding test tube combination, and calculation gates and detection gates were added one by one. The reaction results were detected by a real-time PCR instrument to determine the validity of the solution.

5. The method according to claim 4, characterized in that, Step S3 further includes a pre-optimization process: For the eigenvalue parameters of the target graph G, the Lovász eigenvalue theorem or the lower bound theorem of the control set are used to impose cardinality constraints on the candidate solution set, thereby compressing the size of the candidate solution.

6. A DNA computing system implementing the method of any one of claims 1-5, characterized in that, It includes: The graph representation module contains multiple DNA computation gates Γ(u) assembled from the main strand β(u) and γ strand, with each computation gate corresponding to the closed neighborhood relation expression of a vertex in the graph; Detection module: A parallel detection array consisting of multiple detection gates D(u), where each detection gate reflects the satisfaction state of the corresponding vertex neighborhood constraints through the intensity of the fluorescence signal; Control module: Used to configure computational gate combination schemes for different problem types, including closed neighborhood detection mode Γ(u) and open neighborhood detection mode Γ'(u), where Γ'(u) removes the c(u) field from the original chain.

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