阅读说明：本技术 一种基于dna链置换的三维混沌振荡系统的实现方法 (Implementation method of three-dimensional chaotic oscillation system based on DNA (deoxyribonucleic acid) strand displacement ) 是由 孙军伟 李智 刘磊 杨宇理 杨秦飞 梁恩豪 凌丹 王延峰 王英聪 黄春 方洁 于 2019-11-05 设计创作，主要内容包括：本发明提出了一种基于DNA链置换的三维混沌振荡系统的实现方法,其步骤为：分别确定各个逻辑门的小支点结构及反应过程中辅助物、反应物的DNA链结构,并确定各个DNA链中结构域的碱基序列；基于DNA链置换的反应机制分别构建倍加逻辑门、置换逻辑门、催化逻辑门、调整逻辑门和两种降解逻辑门；使用Visual DSD软件对逻辑门电路进行验证；根据逻辑关系分别确定数学表达式,将数学表达式转化为对应的微分表达式并合并,得到设计的三阶混沌振荡系统；利用Matlab软件验证三阶混沌振荡系统的动力学行为。本发明构建的混沌振荡系统具有合理性和有效性,可以作为今后研究混沌同步的动力学行为的基础。(The invention provides a method for realizing a three-dimensional chaotic oscillation system based on DNA strand displacement, which comprises the following steps: respectively determining the small branch point structure of each logic gate, the DNA chain structures of the auxiliary substance and the reactant in the reaction process, and determining the base sequence of the structural domain in each DNA chain; respectively constructing a multiplying logic gate, a replacing logic gate, a catalyzing logic gate, an adjusting logic gate and two degrading logic gates based on a reaction mechanism of DNA strand replacement; verifying the logic gate circuit by using Visual DSD software; respectively determining mathematical expressions according to the logical relations, converting the mathematical expressions into corresponding differential expressions and combining the differential expressions to obtain a designed three-order chaotic oscillation system; and (3) verifying the dynamic behavior of the three-order chaotic oscillation system by using Matlab software. The chaotic oscillating system constructed by the invention has rationality and effectiveness, and can be used as the basis for researching the dynamic behavior of chaotic synchronization in the future.)

1. A method for realizing a three-dimensional chaotic oscillation system based on DNA strand displacement is characterized by comprising the following steps:

the method comprises the following steps: respectively determining the small branch point structure of each logic gate and the DNA chain structures of an auxiliary substance and a reactant in the reaction process according to the logical relations of the double-adding logic gate, the replacing logic gate, the catalyzing logic gate, the adjusting logic gate and the two degrading logic gates, and determining the base sequence of the structural domain in each DNA chain;

step two: constructing a double-adding logic gate, a replacement logic gate, a catalysis logic gate, an adjustment logic gate and two degradation logic gates based on a reaction mechanism of DNA strand replacement; visual DSD software is used for verifying circuits of the designed multiplication logic gate, the designed replacement logic gate, the designed catalysis logic gate, the designed adjustment logic gate and the designed two degradation logic gates, so that five logic operation functions of multiplication, replacement, catalysis, adjustment and degradation are realized respectively;

step three: respectively determining mathematical expressions according to the logical relations of the double-adding logic gate, the permutation logic gate, the catalysis logic gate, the adjustment logic gate and the two degradation logic gates, converting the mathematical expressions into corresponding differential expressions and combining the differential expressions to obtain a designed three-order chaotic oscillation system; and (3) verifying the dynamic behavior of the three-order chaotic oscillation system by using Matlab software.

2. The method for implementing the three-dimensional chaotic oscillating system based on DNA strand displacement according to claim 1, wherein the reaction mechanism based on DNA strand displacement in the second step is that a DNA input strand with a small branch domain and a DNA double strand undergo a DNA strand displacement reaction to displace a DNA output strand; the DNA strand displacement reaction is a spontaneous, dynamic, cascadable, reversible reaction process.

3. The method for realizing the three-dimensional chaotic oscillating system based on the DNA strand displacement according to claim 1 or 2, wherein the addition logic gate is an addition logic gate X → 2X, the displacement logic gate is a displacement logic gate X + Y → 2Y, the catalytic logic gate is a catalytic logic gate 2X → 3X, the adjustment logic gate is an adjustment logic gate 2X + Z → 2Z, one degradation logic gate is a degradation logic gate Y → Φ 1, and the other degradation logic gate is a degradation logic gate Z → Φ 2, wherein X, Y, Z represents a reactant of the logic operation, and Φ 1 and Φ 2 represent generated products which cannot participate in the displacement reaction.

4. The method for realizing the three-dimensional chaotic oscillating system based on the DNA strand displacement according to claim 3, wherein the mathematical expression of the multiplication and addition logic gate reaction is as follows:

wherein A1 and C1 are different auxiliary substances, sp4(1), sp5(1) and wait (1) all represent products, and products sp4(1) and sp5(1) can still participate in the displacement reaction when encountering complementary paired small-fulcrum structures due to the exposed single-chain small-fulcrum structures; the single-chain small-pivot structure which is not exposed in the waste (1) can not participate in the replacement reaction; x represents a reactant of a logical operation; q. q.s_i、q_mRepresents the rate of the chemical reaction;

the equations (1) and (2) of the reaction add to give:

A1+X+C1→sp4(1)+waste(1)+2X (3)

removing the auxiliary A1, the auxiliary C1 and the resultant sp4(1), and reducing waste (1) to obtain a double logic gate: x → 2X;

the mathematical expression of the reaction of the permutation logic gate is:

wherein, A2, B2, D2, F2 and G2 are auxiliary substances in the reaction process, and sp8(2), sp9(2), sp10(2), sp11(2), sp12(2) and sp13(2) all represent products; x and Y represent reactants of a logical operation; q. q.s_i、q_m、q_sRespectively represent different chemical reaction rates;

adding 2 times of the reaction formula (4) and the reaction formula (5) to obtain:

A2+B2+2D2+2X→2sp8(2)+2sp10(2)+2Y (9)

the removal of the auxiliary A2, the auxiliary B2, the auxiliary D2 and the products sp8(2) and sp10(2) is simplified to obtain:

X→Y (10)

the equations (7) and (8) of the reaction add to give:

F2+G2+Y→sp11(2)+sp13(2)+Y (11)

the removal of the auxiliary substance F2, the auxiliary substance G2 and the products sp11(2) and sp13(2) can be simplified as follows:

Y→Y (12)

the addition of equations (10) and (12) of the reaction reduces to a replacement logic gate:

X+Y→2Y (13)

the mathematical expression of the reaction of the catalytic logic gate is:

wherein, A1, C1 and D3 are auxiliary substances in the reaction process, sp5(3), sp6(3) and wait (1) all represent products, products sp5(3) and sp6(3) are DNA chains with exposed single-chain small pivot structures, and wait (1) is a DNA chain without exposed single-chain small pivot structures, so that the products cannot participate in the displacement reaction;

the multiplication of equation (14) by 2 adds equations (15) and (16) to give:

2A1+2X+C1+D3→2sp5(3)+2waste(1)+3X (17)

removing the auxiliary substance A1, the auxiliary substance C1, the auxiliary substance D3 and the product sp5(3), and reducing waste (1) to obtain a catalytic logic gate: 2X → 3X;

the mathematical expression of the reaction of the trim logic gate is:

wherein, a4, B2, D2, F4 and G4 are auxiliary substances in the reaction process, and sp8(4), sp9(2), sp10(2), sp11(2), sp12(2) and sp13(2) represent products; z represents a reactant of a logical operation;

the reaction of formulae (18), (19) plus formula (20) gives:

A4+D2+2X→2sp8(4)+sp10(2)+Z (23)

the removal of the auxiliary A4, the auxiliary D2 and the products sp8(4) and sp10(2) can be simplified as follows:

2X→Z (24)

the equations (21), (22) and (24) of the reaction are added to give:

2X+F4+G4+Z→sp11(2)+sp13(2)+2Z (25)

the removal of the auxiliary substance F4, the auxiliary substance G4, the auxiliary substance Z4 and the products sp11(2) and sp13(2) simplifies the adjustment logic gate: 2X + Z → 2Z;

the mathematical expression of the reaction of the one degradation logic gate is:

wherein Na is an auxiliary substance, Y is a reactant, and sp3(5), sp4(5), and sp5(5) represent products, respectively; y represents a reactant of a logical operation;

the equations (26) and (27) of the reaction add to give:

the removal of auxiliary substances Na and products sp4(5) and sp5(5) simplifies the degradation logic gate: y → Φ 1

The mathematical expression for the reaction of another degradation logic gate is:

wherein Z is a reactant, and sp3(6), sp4(6), and sp5(6) represent products, respectively

The sum of equations (29) and (30) of the reaction yields:

the removal of auxiliary substances Na and products sp4(6) and sp5(6) simplifies the degradation logic gate: z → Φ 2.

5. The method for realizing the three-dimensional chaotic oscillating system based on DNA strand displacement according to claim 4, wherein in the adding logic gate X → 2X, a1, a3 and p1 are domains; the auxiliary A1 is a DNA double strand { a1^ a } [ a2a3^ a ] < X1a1^ X1a1^ >, the auxiliary C1 is a DNA double strand { a3^ a } < p 1> [ X1a1^ a2a3^ > < p1^ > [ X1a1^ a2a3 a, and the reactant X is a DNA single strand < p1^ X1a1^ a2a3^ >; the DNA replacement reaction is as follows: the double-stranded DNA of the auxiliary substance A { a2a ^ a } [ X a ^ a ] < X a ^ a > reacts with the double-stranded DNA of the reactant X < p ^ X a a ^ > to generate strand sp (1), namely the double-stranded DNA < p ^ X > [ a a a ^ a ] and strand sp (1), namely the single-stranded DNA < a ^ X a ^ a >, the double-stranded DNA of the auxiliary substance C { a ^ a } < p ^ X [ X a > < a a ^ a > < p > < X a > < p ^ X a > < X a > reacts with the single-stranded DNA of the strand sp (1), and generates strand waste (1), namely the double-stranded DNA < a > [ a ^ X a ^ X a > and the concentration of the single-stranded DNA < a2a ^ X a ^ X a > of the reactant X < a2a ^ a >; in the catalytic logic gate 2X → 3X, a1, a3 and p1 are all structural domains; the auxiliary substance A1 is a DNA double strand { a1^ a } [ a2a3^ a ] < X1a1^ X1a1^ a, the auxiliary substance C1 is a DNA double strand { a3^ a } < p 1> [ X1a1^ a2a3^ p1^ > [ X1a1^ a2a3 a ^ and the auxiliary substance D3 is a DNA double strand { a3^ p1^ X1a1^ a2a 3> { X1a1^ and the reactant X is a DNA single strand < p 1X 1a 1a 3 a2a 6338 ^ a; the DNA replacement reaction is as follows: the double-stranded DNA of the auxiliary A1 { a1^ a } [ a2a3^ a ] < X1a1^ X1a1^ reacts with the single-stranded DNA of the reactant X < p1^ X1a 1a 2a3^ to generate outstranded sp5(3), namely the double-stranded DNA < p1^ X1> [ a 1a 2a3^ and strand sp6, namely the single-stranded DNA < a2a3^ X1a1^ X1a1^ through displacement; the DNA double strand of the auxiliary C1 { a3^ p1^ > [ X1a1^ a2a3^ p1^ X1a1^ a2a3^ and the strand sp6(3), namely the DNA single strand < a2a3^ X1a1^ X1a1^ react to generate the strand waste (1), namely the DNA double strand < a2> [ a3^ X1a1^ X1a1^ and the strand 2X which is 2 times of the initial reactant concentration, namely 2< p 1X 1a 1a 2a3^ which are displaced; the DNA strand { a3^ a } < p1^ > [ X1a1^ > < a2a3^ > { X1a1^ and the single strand sp6(3) < a2a3^ X1a1^ > react to generate a strand waste (1), i.e., a double-stranded DNA strand, [ a1^ X1a1^ and a single-stranded DNA strand of reactant X < p 1X 1a1 < >, wherein "[ ]" represents an un-complementary paired upper strand structure, "[ ]" represents a complementary double-stranded structure, "{ }" un-paired lower strand structure, "" represents a small lower strand complementary structure, "> represents a complementary lower strand, and" < "" > represents a complementary small strand complementary lower strand, respectively connected small strand, and a complementary strand 1 represents a complementary strand.

6. The method for realizing the three-dimensional chaotic oscillating system based on DNA strand displacement according to claim 4, wherein in the displacement logic gate X + Y → 2Y, a1, a3, a4, a5, p1, p2 and p3 are all domains; an auxiliary substance A2 is a DNA double-chain { a1^ a } [ a2a3^ a ] < a2a3^ X1a 1a, an auxiliary substance B1 is a DNA double-chain { a 1a ^ a } < a 1a1^ a 1a, an auxiliary substance D1 is a DNA double-chain { a1^ X } < a 1a ^ > [ a 1a, an auxiliary substance G1 is a DNA double-chain { a1^ a ^ X } < p 1X 1a 36x 1a 363672 a 1a 363672 a 36363672 a 1a 363672 a 1a 363672 a 3636x 1a 363672 a 3636x 1a 363672 a 1a 3636363672 a 363636x 1a 1; the DNA replacement reaction is as follows: the DAN double strand of the auxiliary substance A { a ^ a } [ a ^ a ] < a ^ X a ^ a > reacts with the DNA single strand of the reactant X < p ^ X a a ^ > to generate a double strand sp (2) of the DNA double strand < p ^ X > [ a ^ a a ] and a strand sp (2) of the DNA single strand < a ^ a X a >, the DNA double strand of the auxiliary substance B { a ^ a } < a > [ a a ^ X a > reacts with the DNA single strand of the reactant X < a ^ X a > ", and the strand sp (2) of the DNA double strand < p ^ X a > < X a > reacts with the DNA single strand of the reactant X to generate a double strand sp (2) of the DNA double strand < a ^ a > and a single strand of the reactant X < a ^ X > a, the auxiliary substance A < a < X > a < a > and a < X > a < a > a < X > a ^ X > a < X > of the reactant X, the displacement generates a chain sp (2), namely a double-chain DNA (a > [ a a ^ a ] < x a ^ a > and a single-chain DNA (p ^ x a ^ a >) which are the products Y, the double-chain DNA (a ^ x a > < a ^ x a >) of the auxiliary substance F and the single-chain DNA (p ^ x a a ^ a) of the products Y react, the displacement generates the chain sp (2), namely the double-chain DNA (p ^ x > [ a a ^ a ] and the chain sp (2), namely the single-chain DNA (p ^ x a a ^) and the double-chain DNA (a ^ a >) of the auxiliary substance G, the double-chain DNA (a ^ x a >) of the auxiliary substance G and the double-chain DNA (p ^ x a ^ a >) and the single-chain X a ^ a > generate the double-chain DNA (p ^ x a ^ a) and the double-chain (x a ^ x a) of the auxiliary substance Y; wherein "< >" represents an upper strand structure not complementarily paired, "[ ]" represents a double-stranded structure complementarily paired, "{ }" a lower strand structure not complementarily paired, "[ lambda ] represents a small branch point structure of an upper strand," [ lambda ] represents a small branch point structure of a lower strand, and a2 represents a base sequence.

7. The method for implementing the three-dimensional chaotic oscillating system based on DNA strand displacement according to claim 4, wherein in the adjustment logic gate 2X + Z → 2Z, a1, a3, a4, a5, p1, p2 and p3 are all domains; the auxiliary substance A is a double-stranded DNA { a ^ a } [ a ^ a ]; the DNA strand displacement reaction is: the double-stranded DNA of the auxiliary substance A { a ^ a } [ a ^ a ]; the double strand of DNA of the auxiliary D2 { a3^ p3^ x3 a5^ a2a3 ] and the strand sp9(2), namely the single strand of DNA < a2a3^ a2a3^ x1a1^ react to generate the strand sp10(2), namely the double strand of DNA < a2> [ a3 a2a3^ x1a1^ and the single strand of DNA of the reactant Z < p3^ x3 a5^ a2a3^ through displacement; the double strand of DNA of the auxiliary F4 { a5^ p } < p2^ x2a4^ a2a3^ reacts with the single strand of DNA of the reactant Z < p3^ x3 a5 a2a3^ to generate strand sp11(2), namely the double strand of DNA < p3^ x3> [ a5^ a2a3^ and the strand sp12, namely the single strand of DNA < p2^ x2a4^ a2a3^ through displacement; the double-stranded DNA of the auxiliary G4 { a4^ x } < p3^ x3 a5^ a2a3^ reacts with the single-stranded DNA of the chain sp12(2) < p2^ x2a4^ a2a3^ to generate the outstranded chain sp13(2) by displacement, namely the double-stranded DNA < p2^ x2> [ a4^ a2a3^ and the single-stranded DNA of the reactant Z < p3^ x3 a5^ a2a3 >; wherein "< >" represents an upper strand structure not complementarily paired, "[ ]" represents a double-stranded structure complementarily paired, "{ }" a lower strand structure not complementarily paired, "^ represents a small branch point structure of an upper strand," ^ represents a small branch point structure of a lower strand, ": indicates that a single-stranded structure not complementarily paired in an upper strand and a double-stranded structure complementarily paired are connected, and a2, x1 and x2 represent different base sequences, respectively.

8. The method for realizing the three-dimensional chaotic oscillating system based on DNA strand displacement according to claim 4, wherein in the degradation logic gate Y → Φ 1, a1, a3, p1 and t1 are all domains; the auxiliary material Na is a DNA double chain { a1^ a } [ a2 t1^ and the reactant Y is a DNA single chain < p1^ x1a1^ a2a3 ^; the DNA strand displacement reaction is: the double-stranded DNA of the auxiliary Na { a1^ a } [ a2 t1^ reacts with the single-stranded DNA of the reactant Y < p1^ x1a1^ a2a3^ > to be linked into a strand sp3(5), namely the double-stranded DNA < p1^ x1> [ a1^ a2] < a3^ a2> [ t1^ ]; the DNA double chain < p1^ x1> [ a1^ a2] < a3^ a2> [ t1^ is decomposed into a chain sp4(5), namely the DNA double chain < p1^ x1> [ a1^ a2] < a3^ t1^ and a chain sp5(5), namely the DNA single chain < a2 t1^ >; in the degradation logic gate Z → phi 2, a1, a3, p1 and t1 are all structural domains; the auxiliary material Na is a double-stranded DNA { a1^ a } [ a2 t1^ a ], the reactant Z is a single-stranded DNA < p1^ x1a1^ a2a3^ a, the chain sp3(6) is a double-stranded DNA < x1> [ a1^ a2] < a 3> < a2> [ t1^ a ], the chain sp4(6) is a double-stranded DNA < x1> [ a1^ a2] < a3^ t1^ a ], and the chain sp5(6) is a single-stranded DNA < a2 t 1> DNA strand displacement reaction: the double-stranded DNA of the auxiliary Na { a1^ a } [ a1 t1^ a ] and the single-stranded DNA of the reactant Z < p1^ x1a1^ react to be linked into a strand sp1 (6), namely the double-stranded DNA < p1^ x1> [ a 1a1 ] < a1^ a 1> < a 1> [ t1^ a, the double-stranded DNA < p1^ x1> [ a 1a1 ] < a1^ a 1> < t1^ a 1> is decomposed into the strand sp1 (6), namely the double-stranded DNA < p1^ x1> { t1^ t 1> [ a 1a 1> < a 1> and the strand 1 a; wherein "< >" represents an upper strand structure not complementarily paired, "[ ]" represents a double-stranded structure complementarily paired, "{ }" a lower strand structure not complementarily paired, "[ lambda ] represents a small branch point structure of an upper strand," [ lambda ] represents a small branch point structure of a lower strand, and a2 and x1 represent different base sequences, respectively.

9. The method for realizing the three-dimensional chaotic oscillating system based on DNA strand displacement according to any one of claims 5 to 8, wherein the base sequence of the domain a1 is CATCG, the base sequence of the domain a2 is CCCTTTACATTACATAACAA, the base sequence of the domain a3 is GTCA, the base sequence of the domain a4 is ACTACAC, the base sequence of the domain a5 is GCTA, the base sequence of the domain x1 is CCCTTTTCTAAACTAAACAA, the base sequence of the domain x2 is CCCTTATCATATCAATACAA, the base sequence of the domain p1 is TATCAG, the base sequence of the domain p2 TCC is CTCAG, the base sequence of the domain p3 is TACCAA, and the base sequence of the small branch point structure t1 is GTCA;

the doubler logic gate X → 2X medium auxiliary A1 and auxiliaryThe concentration of C1 was 10⁴nM, concentration of reactant X5 nM, reaction rate q_i＝10^-7nMs^-1Reaction rate q_m＝10^-3nMs^-1(ii) a The concentrations of the auxiliary substances A2, B2, D2, F2 and G2 in the replacement logic gate X + Y → 2Y are all 10³nM, concentration of reactant X1 nM, concentration of reactant Y1 nM, reaction rate q_i＝10^-6nMs^-1Reaction rate q_m＝10^-3nMs^-1Reaction rate q_s＝7nMs^-1(ii) a The concentrations of the auxiliary substances A1, C1 and D3 in the catalytic logic gate 2X → 3X are all 10³nM, concentration of reactant X2 nM, reaction rate q_-i＝10^-6nMs^-1Reaction rate q_m＝10^{- 3}nMs^-1(ii) a The concentration of the aids A4, B2, D2, F4 and G4 of the adjusting logic gate 2X + Z → 2Z are all 10³nM, concentration of reactant X2 nM, concentration of reactant Z1 nM, reaction rate q_-i＝10^-6nMs^-1Reaction rate q_m＝10^-3nMs^-1Reaction rate q_s＝7nMs^-1. The concentration of the auxiliary substance Na in the degradation logic gate Y → phi 1 is Na-10²nM, concentration of reactant Y5 nM, reaction rate q_i＝3×10^-4nMs^-1Reaction rate q_-m＝1.1126×10^-1nMs^-1(ii) a The concentration of the auxiliary substance Na in the degradation logic gate Z → phi 2 is 10²nM, concentration of the reactant Z of 5nM, reaction rate q_i＝3×10^-4nMs^-1Reaction rate q_m＝1.1126×10^-1nMs^-1。

10. The method for realizing the three-dimensional chaotic oscillating system based on DNA strand displacement according to claim 9, wherein the method for obtaining the third-order chaotic oscillating system in the third step is as follows: based on the DNA strand displacement technology, the mathematical expression of the chaotic oscillation system is given as follows:

wherein r is₁-r₆X, Y and Z are three different species, representing different chemical reaction rates, respectively, and for each species, a differential expression is given for the corresponding equation as follows:

dX(32)＝r₁X，

dX(33)＝-r₂XY，

dX(34)＝r₃X²，

dX(35)＝-r₄X²Z，

dY(33)＝r₂XY，

dY(36)＝-r₅Y，

dZ(35)＝r₄X²Z，

dZ(37)＝-r₆Z；

wherein dX (32), dX (33), dX (34), and dX (35) represent differential expressions with respect to the logical operation reactant X in the chemical reaction formulae (32), (33), (34), and (35), respectively; dY (33) and dY (36) represent differential expressions with respect to the logical operation reactant Y in chemical reaction formulas (33) and (36), respectively; dZ (35) and dZ (37) represent differential expressions with respect to the logical operation reactant Z in chemical reaction formula (35) and formula (37), respectively;

the corresponding three categories X, Y, Z of differential expressions are merged:

dX＝dX(32)+dX(33)+dX(34)+dX(35)

＝r₁X-r₂XY+r₃X²-r₄X²Z，

dY＝dY(33)+dY(36)

＝r₂XY-r₅Y，

dZ＝dZ(35)+dZ(37)

＝r₄X²Z-r₆Z，

let dX ═ dX, dY ═ dY, dZ ═ dZ, r₁＝a，r₂＝m＝n，r₃＝b，r₄＝c＝h，r₅＝g，r₆D, the chaotic oscillating system is expressed by a third order equation:

wherein a, b, c, d, m, n, g and h are parameters of the system, and x, y and z are state variables of the system;

when a is 1, b is 2, c is 2.9851, d is 3, m is 1, n is 1, g is 1, and h is 2.9851, the dynamic behavior of the chaotic oscillating system shows a chaotic state.

Technical Field

The invention relates to the technical field of DNA strand displacement, in particular to a method for realizing a three-dimensional chaotic oscillation system based on DNA strand displacement.

Background

The modern age is known as the information age, and computers are the most popular carriers for processing information and need continuous innovation. DNA computing, which combines computational and molecular biology, is a new area of research. In view of the advantages of good operability and nanoscale space size of biomolecules, biomolecules are found to provide a greater possibility for assembling integrated biological computers using biomolecule elements, and an important development direction of biological computing is DNA computing which is an important direction to be studied and searched in detail. Adleman, an American scientist, in 1994, subtly solved the traveler NP problem in a single tube of specified DNA, only taking 7 days, but at the computer level of that time required at least about two years more, and therefore the solution of this problem opened a new era of DNA computers. DNA calculation also derives the techniques of DNA self-assembly, DNA origamy, DNA single-strand self-assembly, DNA strand displacement, etc. The invention mainly utilizes the related principle and technology of DNA strand displacement.

DNA computing has dealt with a number of molecular manipulations, such as self-assembly, fluorescent labeling, strand displacement, and probe machines. The DNA strand displacement technology is developed on the basis of the DNA self-assembly technology, so the DNA self-assembly technology and the DNA strand displacement technology are two important support technologies for the research of the DNA nanotechnology. The DNA strand displacement reaction can realize not only digital calculation but also analog calculation, and the constructed DNA molecular circuit not only can realize the analog arithmetic operation of addition, subtraction, multiplication and division, but also can solve the problem of solving the root of the equation. The DNA nano structure constructed by combining DNA strand displacement and DNA paper folding can be used as a carrier of an intelligent medicament, can be specifically combined with target cells to release the medicament to achieve the aim of targeted treatment, and avoids damaging normal cells. The DNA biosensor can be used for detecting toxic biomolecules such as oncogenic aflatoxin molecules and HIV molecules, and has the characteristics of quick detection, convenience and practicability. In addition, the DNA strand displacement technology has been intensively studied in the fields of molecular computing, nanomachining, diagnosis and disease treatment by virtue of high-capacity information accumulation, high-performance parallel computing, programming and simulation. DNA strand displacement technology also has great research significance in solving mathematical problems, managing nanomachines, and discussing life histories. The chaotic system has various structures and complex motion forms, constructs a novel chaotic system, analyzes the dynamic characteristics of the system, not only promotes the development of the chaotic system in the theoretical aspect, but also makes the chaotic system make great progress in the practical application aspect.

Disclosure of Invention

Aiming at the technical problems that the existing DNA chain displacement technology cannot be applied to a chaotic oscillation system and the chaotic oscillation system is slow in calculation speed, the invention provides a method for realizing a three-dimensional chaotic oscillation system based on DNA chain displacement.

In order to achieve the purpose, the technical scheme of the invention is realized as follows: a method for realizing a three-dimensional chaotic oscillation system based on DNA strand displacement comprises the following steps:

the method comprises the following steps: respectively determining the small branch point structure of each logic gate and the DNA chain structures of an auxiliary substance and a reactant in the reaction process according to the logical relations of the double-adding logic gate, the replacing logic gate, the catalyzing logic gate, the adjusting logic gate and the two degrading logic gates, and determining the base sequence of the structural domain in each DNA chain;

step two: constructing a double-adding logic gate, a replacement logic gate, a catalysis logic gate, an adjustment logic gate and two degradation logic gates based on a reaction mechanism of DNA strand replacement; visual DSD software is used for verifying circuits of the designed multiplication logic gate, the designed replacement logic gate, the designed catalysis logic gate, the designed adjustment logic gate and the designed two degradation logic gates, so that five logic operation functions of multiplication, replacement, catalysis, adjustment and degradation are realized respectively;

step three: respectively determining mathematical expressions according to the logical relations of the double-adding logic gate, the permutation logic gate, the catalysis logic gate, the adjustment logic gate and the two degradation logic gates, converting the mathematical expressions into corresponding differential expressions and combining the differential expressions to obtain a designed three-order chaotic oscillation system; and (3) verifying the dynamic behavior of the three-order chaotic oscillation system by using Matlab software.

The reaction mechanism based on DNA strand displacement in the second step is that a DNA input strand with a small branch point domain and a DNA double strand are subjected to DNA strand displacement reaction to displace a DNA output strand; the DNA strand displacement reaction is a spontaneous, dynamic, cascadable, reversible reaction process.

The double-adding logic gate is a double-adding logic gate X → 2X, the replacing logic gate is a replacing logic gate X + Y → 2Y, the catalytic logic gate is a catalytic logic gate 2X → 3X, the adjusting logic gate is an adjusting logic gate 2X + Z → 2Z, one degrading logic gate is a degrading logic gate Y → phi 1, the other degrading logic gate is a degrading logic gate Z → phi 2, wherein X, Y, Z represents a reactant of logic operation, and phi 1 and phi 2 represent generated products which cannot participate in the replacing reaction.

The mathematical expression of the multiplication logic gate reaction is as follows:

wherein A1 and C1 are different auxiliary substances, sp4(1), sp5(1) and wait (1) all represent products, and products sp4(1) and sp5(1) can still participate in the displacement reaction when encountering complementary paired small-fulcrum structures due to the exposed single-chain small-fulcrum structures; the single-chain small-pivot structure which is not exposed in the waste (1) can not participate in the replacement reaction; x represents a reactant of a logical operation; q. q.s_i、q_mRepresents the rate of the chemical reaction;

the equations (1) and (2) of the reaction add to give:

A1+X+C1→sp4(1)+waste(1)+2X (3)

removing the auxiliary A1, the auxiliary C1 and the resultant sp4(1), and reducing waste (1) to obtain a double logic gate: x → 2X;

the mathematical expression of the reaction of the permutation logic gate is:

wherein, A2, B2, D2, F2 and G2 are auxiliary substances in the reaction process, and sp8(2), sp9(2), sp10(2), sp11(2), sp12(2) and sp13(2) all represent products; x and Y represent reactants of a logical operation; q. q.s_i、q_mQs respectively represent different chemical reaction rates;

adding 2 times of the reaction formula (4) and the reaction formula (5) to obtain:

A2+B2+2D2+2X→2sp8(2)+2sp10(2)+2Y (9)

the removal of the auxiliary A2, the auxiliary B2, the auxiliary D2 and the products sp8(2) and sp10(2) is simplified to obtain:

X→Y (10)

the equations (7) and (8) of the reaction add to give:

F2+G2+Y→sp11(2)+sp13(2)+Y (11)

the removal of the auxiliary substance F2, the auxiliary substance G2 and the products sp11(2) and sp13(2) can be simplified as follows:

Y→Y (12)

the addition of equations (10) and (12) of the reaction reduces to a replacement logic gate:

X+Y→2Y (13)

the mathematical expression of the reaction of the catalytic logic gate is:

wherein, A1, C1 and D3 are auxiliary substances in the reaction process, sp5(3), sp6(3) and wait (1) all represent products, products sp5(3) and sp6(3) are DNA chains with exposed single-chain small pivot structures, and wait (1) is a DNA chain without exposed single-chain small pivot structures, so that the products cannot participate in the displacement reaction;

the multiplication of equation (14) by 2 adds equations (15) and (16) to give:

2A1+2X+C1+D3→2sp5(3)+2waste(1)+3X (17)

removing the auxiliary substance A1, the auxiliary substance C1, the auxiliary substance D3 and the product sp5(3), and reducing waste (1) to obtain a catalytic logic gate: 2X → 3X;

the mathematical expression of the reaction of the trim logic gate is:

wherein, a4, B2, D2, F4 and G4 are auxiliary substances in the reaction process, and sp8(4), sp9(2), sp10(2), sp11(2), sp12(2) and sp13(2) represent products; z represents a reactant of a logical operation;

the reaction of formulae (18), (19) plus formula (20) gives:

A4+D2+2X→2sp8(4)+sp10(2)+Z (23)

the removal of the auxiliary A4, the auxiliary D2 and the products sp8(4) and sp10(2) can be simplified as follows:

2X→Z (24)

the equations (21), (22) and (24) of the reaction are added to give:

2X+F4+G4+Z→sp11(2)+sp13(2)+2Z (25)

the removal of the auxiliary substance F4, the auxiliary substance G4, the auxiliary substance Z4 and the products sp11(2) and sp13(2) simplifies the adjustment logic gate: 2X + Z → 2Z;

the mathematical expression of the reaction of the one degradation logic gate is:

wherein Na is an auxiliary substance, Y is a reactant, and sp3(5), sp4(5), and sp5(5) represent products, respectively; y represents a reactant of a logical operation;

the equations (26) and (27) of the reaction add to give:

the removal of auxiliary substances Na and products sp4(5) and sp5(5) simplifies the degradation logic gate: y → Φ 1

The mathematical expression for the reaction of another degradation logic gate is:

wherein Z is a reactant, and sp3(6), sp4(6), and sp5(6) represent products, respectively

The sum of equations (29) and (30) of the reaction yields:

the removal of auxiliary substances Na and products sp4(6) and sp5(6) simplifies the degradation logic gate: z → Φ 2.

In the addition logic gate X → 2X, a1, a3, p1 are domains; an auxiliary substance A1 is a DNA double strand { a1^ a } [ a2A3^ a ] < X1a1^ X1a1^ a ], an auxiliary substance C1 is a DNA double strand { A3^ a } < p1^ X1a1^ a2A3^ p1^ X1a1^ a2A3^ and a reactant X is a DNA single strand < p1^ X1a1^ a2A3 ^; the DNA replacement reaction is as follows: the double DNA chains { a ^ a } [ a ^ a ] < X a ^ X a > of the auxiliary substance A and the double DNA chains < p ^ X a a > of the reactant X react to generate strand sp (1), namely the double DNA chains < p ^ X > [ a ^ a ] and strand sp (1), namely the single DNA chains < a ^ X a ^ X a >, the double DNA chains { a ^ a } < p ^ X [ X a > ] of the auxiliary substance C react to generate strand wa (1), namely the double DNA chains < a > [ a ^ X a ^ X a > and the single DNA chains < a ^ X a > of the strand (1), and the double DNA chains < a ^ X a ^ X a > of the auxiliary substance C react to generate strand wa (1), namely the double DNA chains < a ^ X a ^ X a ^ X a > and the concentration of the reactant X < a2a ^ X a > X a ^ X a ^ X2; in the catalytic logic gate 2X → 3X, a1, a3 and p1 are all structural domains; the auxiliary substance A1 is a DNA double strand { a1^ a } [ a2a3^ a ] < X1a1^ X1a1^ a, the auxiliary substance C1 is a DNA double strand { a3^ a } < p 1> [ X1a1^ a2a3^ p1^ > [ X1a1^ a2a3 a ^ and the auxiliary substance D3 is a DNA double strand { a3^ p1^ X1a1^ a2a 3> { X1a1^ and the reactant X is a DNA single strand < p 1X 1a 1a 3 a2a 6338 ^ a; the DNA replacement reaction is as follows: the double DNA chain { a1^ a } [ a2A3^ a ] < X1a1^ X1a1^ of the auxiliary A1 and the single DNA chain < p1^ X1a 1a 2A3 > of the reactant X react to generate the strand sp5(3), namely the double DNA chain < p1^ X1> [ a1^ a2A3^ and the strand sp6, namely the single DNA chain < a2A3^ X1a1^ X1a 1); the DNA double strand of the auxiliary C1 { a3^ p1^ X1a1^ a2a3^ p1^ X1a1^ a2a3^ and the strand sp6(3), namely the DNA single strand < a2a3^ X1a1^ X1a1^ react to generate the strand waste (1), namely the DNA double strand < a2> [ a3^ X1a1^ X1a1^ ] and the strand 2X which is 2 times of the initial reactant concentration, namely 2< p1^ X1a 1a 2a3^ a; the DNA strand { a3^ a } < p1^ > [ X1a1^ > < a2a3^ > { X1a1^ and the single strand sp6(3) < a2a3^ X1a1^ > react to generate a strand waste (1), i.e., a double-stranded DNA strand, [ a1^ X1a1^ and a single-stranded DNA strand of reactant X < p 1X 1a1 < >, wherein "[ ]" represents an un-complementary paired upper strand structure, "[ ]" represents a complementary double-stranded structure, "{ }" un-paired lower strand structure, "" represents a small lower strand complementary structure, "> represents a complementary lower strand, and" < "" > represents a complementary small strand complementary lower strand, respectively connected small strand, and a complementary strand 1 represents a complementary strand.

In the replacement logic gate X + Y → 2Y, a1, a3, a4, a5, p1, p2 and p3 are all domains; an auxiliary substance A2 is a DNA double-chain { a1^ a } [ a2a3^ a ] < a2a3^ X1a 1a, an auxiliary substance B1 is a DNA double-chain { a 1a ^ a } < a 1a1^ a 1a, an auxiliary substance D1 is a DNA double-chain { a1^ X } < a 1a ^ > [ a 1a, an auxiliary substance G1 is a DNA double-chain { a1^ a ^ X } < p 1X 1a 36x 1a 363672 a 1a 363672 a 36363672 a 1a 363672 a 1a 363672 a 3636x 1a 363672 a 3636x 1a 363672 a 1a 3636363672 a 363636x 1a 1; the DNA replacement reaction is as follows: the DAN double strand of the auxiliary substance A { a ^ a } [ a ^ a ] < a ^ X a ^ a > reacts with the DNA single strand of the reactant X < p ^ X a a ^ > to generate a strand sp (2), namely the DNA double strand < p ^ X > [ a ^ a a ] and a strand sp (2), namely the DNA single strand < a ^ a X1a >, the DNA double strand of the auxiliary substance B { a ^ a } < a > [ a a ^ X a > reacts with the DNA single strand of the reactant X < p ^ X1a > ", and the strand sp (2), namely the DNA double strand < p ^ a > < a > a ^ a ] and the DNA single strand of the reactant X < a ^ X > a > < a > react to generate a double strand sp (2), namely the DNA double strand < a ^ a > < a ^ a > and the DNA single strand < a ^ X > a ^ X < X > a </X > a > < a > and the auxiliary substance X < a > a < X > a </a > a </X > < a </X >,x > and a </X > a </X >,x > a </, the displacement generates a chain sp (2), namely a double-chain DNA (a > [ a a ^ a ] < x a ^ a > and a single-chain DNA (p ^ x a ^ a >) which are the products Y, the double-chain DNA (a ^ x a > < a ^ x a >) of the auxiliary substance F and the single-chain DNA (p ^ x a a ^ a) of the products Y react, the displacement generates the chain sp (2), namely the double-chain DNA (p ^ x > [ a a ^ a ] and the chain sp (2), namely the single-chain DNA (p ^ x a a ^) and the double-chain DNA (a ^ a >) of the auxiliary substance G, the double-chain DNA (a ^ x a >) of the auxiliary substance G and the double-chain DNA (p ^ x a ^ a >) and the single-chain X a ^ a > generate the double-chain DNA (p ^ x a ^ a) and the double-chain (x a ^ x a) of the auxiliary substance Y; wherein "< >" represents an upper strand structure not complementarily paired, "[ ]" represents a double-stranded structure complementarily paired, "{ }" a lower strand structure not complementarily paired, "[ lambda ] represents a small branch point structure of an upper strand," [ lambda ] represents a small branch point structure of a lower strand, and a2 represents a base sequence.

In the adjustment logic gate 2X + Z → 2Z, a1, a3, a4, a5, p1, p2 and p3 are all structural domains; the auxiliary substance A4 is a DNA double-chain { a1^ a } [ a2a3^ a ]; the DNA strand displacement reaction is: the double DNA chains of the auxiliary substance A { a ^ a } [ a ^ a ]; the double strand of DNA of the auxiliary D2 { a3^ p3^ x3 a5^ a2a3 ] and the strand sp9(2), namely the single strand of DNA < a2a3^ a2a3^ x1a1^ react to generate the strand sp10(2), namely the double strand of DNA < a2> [ a3 a2a3^ x1a1^ and the single strand of DNA of the reactant Z < p3^ x3 a5^ a2a3^ through displacement; the double strand of DNA of the auxiliary substance F4 { a5^ p2^ x2a4^ a2a3^ reacts with the single strand of DNA of the reactant Z < p 3x 3 a5 a2a3^ to generate strand sp11(2), namely the double strand of DNA < p3^ x3> [ a5 a2a3 ] and the strand sp12(2), namely the single strand of DNA < p2^ x2a4^ a2a3^ through displacement; the double-stranded DNA of the auxiliary G4 { a4^ x } < p3^ x3 a5^ a2a3^ reacts with the single-stranded DNA of the chain sp12(2) < p2^ x2a4^ a2a3^ to generate the outstranded chain sp13(2) by displacement, namely the double-stranded DNA < p2^ x2> [ a4^ a2a3^ and the single-stranded DNA of the reactant Z < p3^ x3 a5^ a2a3 >; wherein "< >" represents an upper strand structure not complementarily paired, "[ ]" represents a double-stranded structure complementarily paired, "{ }" a lower strand structure not complementarily paired, "^ represents a small branch point structure of an upper strand," ^ represents a small branch point structure of a lower strand, ": indicates that a single-stranded structure not complementarily paired in an upper strand and a double-stranded structure complementarily paired are connected, and a2, x1 and x2 represent different base sequences, respectively.

In the degradation logic gate Y → phi 1, a1, a3, p1 and t1 are all structural domains; the auxiliary material Na is a DNA double chain { a1^ a } [ a2 t1^ and the reactant Y is a DNA single chain < p1^ x1a1^ a2a3 ^; the DNA strand displacement reaction is: the double-stranded DNA of the auxiliary Na { a1^ a } [ a2 t1^ reacts with the single-stranded DNA of the reactant Y < p1^ x1a1^ a2a3^ > to be linked into a strand sp3(5), namely the double-stranded DNA < p1^ x1> [ a1^ a2] < a3^ a2> [ t1^ ]; the DNA double chain < p1^ x1> [ a1^ a2] < a3^ a2> [ t1^ is decomposed into a chain sp4(5), namely the DNA double chain < p1^ x1> [ a1^ a2] < a3^ t1^ and a chain sp5(5), namely the DNA single chain < a2 t1^ >; in the degradation logic gate Z → phi 2, a1, a3, p1 and t1 are all structural domains; the auxiliary material Na is a double-stranded DNA { a1^ a } [ a2 t1^ a ], the reactant Z is a single-stranded DNA < p1^ x1a1^ a2a3^ a, the chain sp3(6) is a double-stranded DNA < x1> [ a1^ a2] < a 3> < a2> [ t1^ a ], the chain sp4(6) is a double-stranded DNA < x1> [ a1^ a2] < a3^ t1^ a ], and the chain sp5(6) is a single-stranded DNA < a2 t 1> DNA strand displacement reaction: the double-stranded DNA of the auxiliary Na { a1^ a } [ a1 t1^ a ] and the single-stranded DNA of the reactant Z < p1^ x1a1^ react to be linked into a strand sp1 (6), namely the double-stranded DNA < p1^ x1> [ a 1a1 ] < a1^ a 1> < a 1> [ t1^ a, the double-stranded DNA < p1^ x1> [ a 1a1 ] < a1^ a 1> < t1^ a 1> is decomposed into the strand sp1 (6), namely the double-stranded DNA < p1^ x1> { t1^ t 1> [ a 1a 1> < a 1> and the strand 1 a; wherein "< >" represents an upper strand structure not complementarily paired, "[ ]" represents a double-stranded structure complementarily paired, "{ }" a lower strand structure not complementarily paired, "[ lambda ] represents a small branch point structure of an upper strand," [ lambda ] represents a small branch point structure of a lower strand, and a2 and x1 represent different base sequences, respectively.

The base sequence of the domain a1 is CATCG, the base sequence of the domain a2 is CCCTTTACATTACATAACAA, the base sequence of the domain a3 is GTCA, the base sequence of the domain a4 is ACTACAC, the base sequence of the domain a5 is GCTA, the base sequence of the domain x1 is CCCTTTTCTAAACTAAACAA, the base sequence of the domain x2 is CCCTTATCATATCAATACAA, the base sequence of the domain p1 is TATTCC, the base sequence of the domain p2 is CTCAG, the base sequence of the domain p3 is TACCAA, and the base sequence of the small branch point structure t1 is GTCA;

the concentrations of the auxiliary A1 and the auxiliary C1 in the doubling logic gate X → 2X are both 10⁴nM, concentration of reactant X5 nM, reaction rate q_i＝10^-7nMs^-1Reaction rate q_m＝10^-3nMs^-1(ii) a The concentrations of the auxiliary substances A2, B2, D2, F2 and G2 in the replacement logic gate X + Y → 2Y are all 10³nM, concentration of reactant X1 nM, concentration of reactant Y1 nM, reaction rate q_i＝10^-6nMs^-1Reaction rate q_m＝10^-3nMs^-1Reaction rate q_s＝7nMs^-1(ii) a The concentrations of the auxiliary substances A1, C1 and D3 in the catalytic logic gate 2X → 3X are all 10³nM, concentration of reactant X2 nM, reaction rate q_i＝1 0^-6nMs^-1Reaction rate q_m＝10^-3nMs^-1(ii) a The concentration of the aids A4, B2, D2, F4 and G4 of the adjusting logic gate 2X + Z → 2Z are all 10³nM, concentration of reactant X2 nM, concentration of reactant Z1 nM, reaction rate q_i＝10^-6nMs^-1Reaction rate q_m＝10^-3nMs^-1Reaction rate q_s＝7nMs^-1. The concentration of the auxiliary substance Na in the degradation logic gate Y → phi 1 is Na-10²nM, concentration of reactant Y5 nM, reaction rate q_i＝3×10^-4nMs^-1Reaction rate q_m＝1.11 26×10^-1nMs^-1(ii) a The concentration of the auxiliary substance Na in the degradation logic gate Z → phi 2 is 10²nM, concentration of reactant Z5 n M, reaction rate q_i＝3×10^-4nMs^-1Reaction rate q_m＝1.1126×10^-1nMs^-1。

The method for obtaining the third-order chaotic oscillation system in the third step comprises the following steps: based on the DNA strand displacement technology, the mathematical expression of the chaotic oscillation system is given as follows:

wherein r is₁-r₆X, Y and Z are three different species, representing different chemical reaction rates, respectively, and for each species, a differential expression is given for the corresponding equation as follows:

dX(32)＝r₁X，

dX(33)＝-r₂XY，

dX(34)＝r₃X²，

dX(35)＝-r₄X²Z，

dY(33)＝r₂XY，

dY(36)＝-r₅Y，

dZ(35)＝r₄X²Z，

dZ(37)＝-r₆Z；

wherein dX (32), dX (33), dX (34), and dX (35) represent differential expressions with respect to the logical operation reactant X in the chemical reaction formulae (32), (33), (34), and (35), respectively; dY (33) and dY (36) represent differential expressions with respect to the logical operation reactant Y in chemical reaction formulas (33) and (36), respectively; dZ (35) and dZ (37) represent differential expressions with respect to the logical operation reactant Z in chemical reaction formula (35) and formula (37), respectively;

the corresponding three categories X, Y, Z of differential expressions are merged:

dX＝dX(32)+dX(33)+dX(34)+dX(35)

＝r₁X-r₂XY+r₃X²-r₄X²Z，

dY＝dY(33)+dY(36)

＝r₂XY-r₅Y，

dZ＝dZ(35)+dZ(37)

＝r₄X²Z-r₆Z，

let dX be dX, dY be dY, dZ be dZ, r1 be a, r2 be m be n, r3 be b, r4 be c be h, r5 be g, r6 be d, the chaotic oscillating system is represented by the third-order equation:

wherein a, b, c, d, m, n, g and h are parameters of the system, and x, y and z are state variables of the system;

when a is 1, b is 2, c is 2.9851, d is 3, m is 1, n is 1, g is 1, and h is 2.9851, the dynamic behavior of the chaotic oscillating system shows a chaotic state.

The invention has the beneficial effects that: based on a reaction mechanism of DNA chain replacement, an adding logic gate, a replacing logic gate, a catalyzing logic gate, an adjusting logic gate and a degrading logic gate are constructed and realized, the effectiveness of each logic gate is verified through Visual DSD simulation software, a novel chaotic oscillation system is designed by utilizing the 6 logic gates, the chaotic dynamics behavior of the vibration system is verified by Matlab, and the rationality and the effectiveness of the system are proved by simulation results. The chaotic oscillating system designed by the invention can be used as the basis for researching the dynamic behavior of chaotic synchronization in the future.

Drawings

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.

FIG. 1 is a flow chart of the present invention.

FIG. 2 is a simulation diagram of 6 logic gate circuits of the present invention, wherein (a) is a multiplication logic gate X → 2X, (b) is a replacement logic gate X + Y → 2Y, (c) is a catalysis logic gate 2X → 3X, (d) is an adjustment logic gate 2X + Z → 2Z, (e) is a degradation logic gate Y → φ 1, and (f) is a degradation logic gate Z → φ 2.

Fig. 3 is a Matlab simulation diagram of the chaotic oscillating system constructed by the invention, wherein (a) is a Lyapunov exponent diagram, and (b) is a three-dimensional chaotic attractor.

Detailed Description

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be obtained by a person skilled in the art without inventive effort based on the embodiments of the present invention, are within the scope of the present invention.

As shown in fig. 1, a method for implementing a three-dimensional chaotic oscillating system based on DNA strand displacement includes the following steps:

the method comprises the following steps: and respectively determining the small branch point structure of each logic gate and the DNA chain structures of the auxiliary substances and the reactants in the reaction process according to the logical relations of the double-adding logic gate, the replacing logic gate, the catalytic logic gate, the adjusting logic gate and the two degradation logic gates, and determining the base sequence of the structural domain in each DNA chain.

In the doubler logic gate X → 2X, a1, a3 and p1 are small branch point structures of the structural domain, and are used for improving the chemical reaction rate. Adjuvant A1 was designed in Visual DSD software as "{ a1^ } [ a2a3^]<x1 a1^x1 a1^>", the auxiliary C1 was designed as" { a3^ X }<p1^>[x1 a1^]<a2 a3^>:<p1^>[x1 a1^]<a2 a3^>", the reactant X is designed as"<p1^x1a1^a2 a3^>", chain sp4(1) was designed as"<p1^x1>[a1^a2 a3^]", chain sp5(1) was designed as"<a2 a3^x1 a 1^x1 a1^>", chain waste (1) is designed as"<a2>[a3^x1 a1^x1 a1^]", auxiliary A1 { a1^ } [ a2a3^]<x1a1^x1 a1^>Of small branch domains "a 1 ^" and of reactant X<p1^x1 a1^a2 a3^>The small branch domain of 'a 1 ^' reacts, and the substitution generates double-chain sp4(1) "<p1^x1>[a1^a2 a3^]"with single strand sp5 (1)"<a2 a3^x1 a1^ x1 a1^>"{ a3^ of" } of auxiliary C1<p1^>[x1 a1^]<a2 a3^>:<p1^>[x1 a1^]<a2 a3^>"Small branch domain" a3^ "and single-stranded sp5 (1)"<a2 a3^x1 a1^x1 a1^>The small branch domain of "a 3 ^" reacts to displace to generate double-stranded waste (1) "<a2>[a3^x1 a1^x1 a1^]"chain 2 with 2 times the concentration of the original reactant X<p1^x1 a1 ^a2 a3^>". The base sequence of the domain a1 is "CATCG", the base sequence of a2 is "CCCTTTACATTACAT AACAA", the base sequence of a3 is "GTCA", the base sequence of x1 is "CCCTTTTCTAAACTAAACAA", and the base sequence of p1 is "TATTCC". The concentrations of the adjuvant a1 and the adjuvant C1 were equal, and a1 ═ C1 ═ 10⁴nM, concentration of reactant X5 nM, reaction rate q_i＝10^{- 7}nMs^-1，q_m＝10^-3nMs^-1. nM is the concentration unit, n is expressed in nanometers, and M is "mol/L", i.e., moles per liter, so nM is the concentration unit nanomolar per liter.

In the replacement logic gate X + Y → 2Y, the domains a1, a3, a4, a5, p1, p2 and p3 are all small branch point structures, and the purpose is to improve the chemical reaction rate. The base sequence of a1 is "CATCG", the base sequence of a2 is "CCCTTTACATTACATAACAA", the base sequence of a3 is "GTCA", the base sequence of a4 is "ACTACAC", "the base sequence of a5 is" GCTA ", the base sequence of x1 is" CCCTTTTCTAAACTAAACAA ", the base sequence of x2 is" CCCTTATCATATCAATACAA ", the base sequence of p1 is" TATTCC ", the base sequence of p2 is" CTCAG ", and the base sequence of p3 is" TACCAA ". Adjuvant A2 was designed in software Visual DSD as "{ a1^ } [ a2a3^]<a2 a3^x1 a1^>", auxiliary B2 was designed as" { a1^ X }<a2 a3^>[a2 a3^]<x1 a1^>", the auxiliary D2 is designed as" { a3^ a }<p3^x3a5^>[a2 a3^]", the auxiliary F2 was designed as" { a5^ X }<p2^x2 a4^>[a2 a3^]", the auxiliary G2 is designed as" { a4^ X }<p3^x3 a5^>[a2 a3^]", the reactant X is designed as"<p1^x1 a1^a2 a3^>", the reactant Y is designed as"<p3^x3 a5^a2 a3^>", chain sp8(2) was designed as"<p1^x1>[a1^a2 a3^]", chain sp9(2) was designed as"<a2 a3^a2 a3^x1 a1^>", chain sp10(2) was designed as"<a2>[a3^a2 a3^]<x1 a1^>", chain sp11(2) was designed as"<p3^x3>[a5^a2 a3^]", chain sp12(2) was designed as"<p2^x2 a4^a2 a3^>", chain sp13(2) was designed as"<p2^x2>[a4^a2a3^]", the DNA strand of the auxiliary A2" { a1^ } [ a2a3^]<a2 a3^x1 a1^>"Small branch region" a1^ "and DNA strand of reactant X"<p1^x1 a1^a2 a3^>The small branch domain of 'a 1 ^' reacts, and double chains sp8(2) are generated by replacement "<p1^x1>[a1^a2 a3^]"and Single chain sp9 (2)"<a2 a3^a2 a3^x1 a1^>", the DNA strand of the adjuvant B2" { a1^ X }<a2 a3^>[a2a3^]<x1 a1^>"Small branch region" a1^ "and DNA strand of reactant X"<p1^x1 a1^a2 a3^>The small branch domain of 'a 1 ^' reacts, and double chains sp8(2) are generated by replacement "<p1^x 1>[a1^a2 a3^]"and Single chain sp9 (2)"<a2 a3^a2 a3^x1a1^>", the DNA strand of the adjuvant D2" { a3^ X }<p3^x3 a5^>[a2 a3^]"Small branch domain" a3^ "and single-chain sp9 (2)"<a2 a3^a2 a3^x1 a1^>The small branch domain of 'a 1 ^' reacts, and double chains sp10(2) are generated by replacement "<a2>[a3^a2 a3^]<x1 a1^>"and DN A chain of product Y"<p3^x3 a5^a2 a3^>", the DNA strand of the adjuvant F2" { a5^ X }<p2^x2a4^>[a2 a3^]"Small branch region" a5^ "and product Y DNA strand"<p3^x3 a5^a2 a3^>The small branch domain of 'a 5 ^' reacts, and double chains sp11(2) are generated by replacement "<p3^x3>[a5^a2 a3^]"and Single chain sp12 (2)"<p2^x2 a4^a2 a3^>", the DNA strand of the adjuvant G2" { a4^ X }<p3^x3 a5^>[a2 a3^]"Small branch domain" a4^ "and single-chain sp12 (2)"<p2^ x2 a4^a2 a3^>The small branch domain of 'a 4 ^' reacts, and double chains sp13(2) are generated by replacement "<p2^x2>[a4^a2 a3^]"and product Y“<p3^x3 a5^a2 a3^>". The concentrations of the auxiliary substances A2, B2, D2, F2 and G2 are equal, and A2 ═ B2 ═ D2 ═ F2 ═ G2 ═ 10³nM, concentration of reactant X, X ═ 1nM, concentration of reactant Y, Y ═ 1nM, reaction rate q_i＝10^-6nMs^-1，q_m＝10^-3nMs^-1，q_s＝7nMs^-1。

In the catalytic logic gate 2X → 3X, a1, a3 and p1 are all small branch point structures of the structural domain, and the purpose is to improve the chemical reaction rate. The base sequence of a1 is "CATCG", the base sequence of a2 is "CCCTTTACATTACATAACAA", the base sequence of a3 is "GTCA", the base sequence of x1 is "CCCTTTTCTAAACTAAACAA", and the base sequence of p1 is "TATTCC". Adjuvant A1 was designed in software Visual DSD as "{ a1^ } [ a2a3^]<x1 a1^x1 a1 ^>", the auxiliary C1 was designed as" { a3^ X }<p1^>[x1 a1^]<a2 a3^>:<p1^>[x1 a1^]<a2 a3^>", the auxiliary D3 is designed as" { a3^ X }<p1^>[x1 a1^]<a2 a3^>{ X1a1^ with reactants X ^ with'<p1^x1 a1^a2 a3^>", chain waste (1) is designed as"<a2>[a3^x1 a1^x1 a1^]". The DNA chain of the auxiliary A1 { a1^ } [ a2a3^]<x1 a1 ^x1 a1^>The small branch domain "a 1 ^" and the DNA strand of the reactant X<p1^x1 a1^a2 a3^>The small branch domain of 'a 1 ^' reacts, and double-chain sp5(3) is generated by replacement<p1^x1>[a1^a2 a3^]And single chain sp6<a2 a3^x1 a1^x1 a1^>(ii) a DNA strand of the auxiliary C1 { a3^ X }<p1^>[x1a1^]<a2 a3^>:<p1^>[x1 a1^]<a2 a3^>The small branch domain "a 3^ and the single-chain sp6(3)<a2 a3^x1 a1^x1a1^>The small branch domain of 'a 3 ^' reacts, and double-chain waste (1) is generated by displacement<a2>[a3^x1 a1^x1 a1^]And 2 times the initial reactant concentration of chain 2X 2<p1^x1 a1^a2 a3^>(ii) a DNA strand of the auxiliary D3 { a3^ X }<p1^>[x1 a1^]<a2a3^>{ x1a1^ small branch domains "a 3^ a1^ and single-chain sp6(3)<a2 a3^x1 a1^x1 a1^>The small branch domains 'a 3 ^' and 'a 1 ^' react to generate double-chain waste (1) through displacement<a2>[a3^ x1 a1^x1 a1^]And DNA strand of reactant X<p1^x1 a1^a2 a3^>. The concentrations of the auxiliary substances A1, C1 and D3 are equal, and A1-C1-D3-10³nM, concentration of reactant X, X ═ 2nM, reaction rate q_i＝10^-6nMs^-1，q_m＝10^-3nMs^-1。“<>"denotes an un-complementary paired cochain structure," is defined as]"represents a complementary paired double-stranded structure," { } "a non-complementary paired lower-stranded structure," ^ "represents a small branch point structure of the upper strand," ^ "represents a small branch point structure of the lower strand,": indicates that a non-complementary paired single-stranded structure and a complementary paired double-stranded structure in the lower strand are connected.

The a1, a3, a4, a5, p1, p2 and p3 in the adjustment logic gate 2X + Z → 2Z are all small branch point structures of the structural domain, and the purpose is to improve the chemical reaction rate. The base sequence of a1 is "CATCG", the base sequence of a2 is "CCCTTTACATTACATAACAA", the base sequence of a3 is "GTCA", the base sequence of a4 is "ACTACAC", the base sequence of a5 is "GCTA", the base sequence of x1 is "CCCTTTTCTAAACTAAACAA", the base sequence of x2 is "CCCT TATCATATCAATACAA", the base sequence of p1 is "TATTCC", the base sequence of p2 is "CTCAG", and the base sequence of p3 is "TACCAA". The auxiliary A4 is designed in software DSD as { a1^ } [ a2a3^]::{a1^*}[a2 a3^]<x 1a1^>The auxiliary object B2 is designed as { a1^ X }<a2a3^>[a2 a3^]<x1 a1^>The auxiliary object D2 is designed as { a3^ X }<p3^ x3 a5^>[a2 a3^]The auxiliary object F4 is designed as { a5^ X }<p2^x2 a4^>[a2 a3^]The auxiliary object G4 is designed as { a4^ X }<p3^x3 a5^>[a2 a3^]The reactant X is designed as<p1^x1 a1^a2 a3^>The reactant Z is designed as<p3^x3 a5^ a2 a3^>The chain sp8(4) is designed as<p1^x1>[a1^a2 a3^]The chain sp9(2) is designed as<a2 a3^a2 a3^x1 a1^>The chain sp10(2) is designed as<a2>[a3^a2 a3^]<x1 a1^>The chain sp11(2) is designed as<p3^x3>[a5^a2 a3^]The chain sp12(2) is designed as<p2^x2 a4^a2 a3^>The chain sp13(2) is designed as<p2^x2>[a4^a2 a3^]. DN A chain { a1^ of [ a2a3^ of auxiliary A4]::{a1^*}[a2a3^]<x1 a1^>The small branch domain "a 1 ^" and the DNA strand of the reactant X<p1^ x1 a1^a2 a3^>The small branch domain of 'a 1 ^' reacts, and double-chain sp8(4) is generated by replacement<p1^x1>[a1^a2 a3^]And the DNA strand of the auxiliary B2 { a1^ X }<a2 a3^>[a2a3^]<x1 a1^>The DNA strand of the auxiliary B2 { a1^ X }<a2 a3^>[a 2a3^]<x1 a1^>"Small branch region" a1^ "and DNA strand of reactant X<p1^x1 a1^a2 a3^>The small branch domain of 'a 1 ^' reacts, and double-chain sp8(4) is generated by replacement<p1^x1>[a1^a2 a3^]And a single chain sp9(2)<a2 a3^a2 a3^x1 a1^>(ii) a DNA strand of the auxiliary D2 { a3^ X }<p3^x3 a5^>[a2 a3^]The small branch domain "a 3^ and the single-chain sp9(2)<a2 a3^a2 a3^x1 a1^>The small branch domain of 'a 3 ^' reacts, generates double-chain sp10(2) by replacement<a2>[a3^a2 a3^]<x1 a1^>And the DNA strand of the reactant Z<p3^x3 a5^a2 a3^>(ii) a DNA strand of the auxiliary F4 { a5^ X }<p2^x2 a4^>[a2 a3^]The small branch domain "a 5 ^" and the DNA strand of the reactant Z<p3^x3 a5^a2 a3^>The small branch domain of (a 5 ^) reacts, and double-chain sp11(2) is generated by displacement<p3^x3>[a5^a2 a3^]And a single chain sp12(2)<p2^x2 a4^a2 a3^>(ii) a DNA strand of the adjuvant G4 { a4^ X }<p3^x3 a5^>[a2 a3^]The small branch domain "a 4^ and the single-chain sp12(2)<p2^x 2a4^a2 a3^>The small branch domain of 'a 4 ^' reacts, generates double-chain sp13(2) by replacement<p2^x2>[a4^a2 a3^]And chain Z<p3^x3 a5^a2 a3^>. The concentrations of the auxiliary substances A4, B2, D2, F4 and G4 are equal, and A4 ═ B2 ═ D2 ═ F4 ═ G4 ═ 10³nM, concentration of reactant X, X ═ 2nM, concentration of reactant Z, Z ═ 1nM, reaction rate q_i＝10^-6nMs^-1，q_m＝10^-3nMs^-1，q_s＝7nMs^-1. ": indicates that the non-complementary paired single-stranded structure and the complementary paired double-stranded structure in the upper strand are connected, and a2, x1, x2 and p3 respectively indicate different base sequences.

In the degradation logic gate Y → phi 1, a1, a3, p1 and t1 are all small branch point structures of the structural domain, and the purpose is to improve the chemical reaction rate. The base sequence of the small branch point structure a1 is CATCG; the base sequence of the small branch point structure a2 is CCCTTTACATTACATAACAA; the base sequence of the small branch point structure a3 is GTCA; the base sequence of the small branch structure x1 is CCCTTTTCTAAACTAAACAA; the base sequence of the small branch point structure p1 is TATTCC; the nucleotide sequence of the small branch point structure t1 is GTCA. The auxiliary Na is designed as { a1^ in software Visual DSD [ a2 t1^ in](ii) a The reactant Y is designed as<p1^x1 a1^a2a3^>(ii) a The chain sp3(5) is designed as<x1>[a1^a2]<a3^>:<a2>[t1^](ii) a The chain sp4(5) is designed as<x1>[a1^a2]<a3^>{ t1^ t }; the chain sp5(5) is designed as<a2 t1^>. DNA chain of auxiliary Na { a1^ } [ a2 t1^ A]The small branch domain "a 1 ^" and the DNA strand of the reaction substance Y<p1^x1 a1^a2 a3^>Reacts to the small branch field 'a 1 ^', links into a double-chain sp3(5)<p1^x1>[a1^a2]<a3^>:<a2>[t1^](ii) a Double-stranded sp3(5)<p1^x1>[a1^a2]<a3^>:<a2>[t1^]Decomposition into double strands sp4(5)<p1^x1>[a1^a2]<a3^>{ t1^ and single-chain sp5(5)<a2 t1^>. Concentration of auxiliary Na (10 ═ 10)²nM, concentration of reactant Y, Y ═ 5nM, reaction rate q_i＝3×10^-4nMs^-1，q_m＝1.1126×10^-1nMs^-1。

In the degradation logic gate Z → phi 2, the domains a1, a3, p1 and t1 are small branch point structures of the domains, and are used for improving the chemical reaction rate. The base sequence of the small branch point structure a1 is CATCG, the base sequence of the domain a2 is CCCTTTACATTACATAACAA, the base sequence of the small branch point structure a3 is GTCA, the base sequence of the domain x1 is CCCTTTTCTAAACTAAACAA, the base sequence of the small branch point structure p1 is TATTCC, and the base sequence of the small branch point structure t1 is GTCA. The auxiliary Na is designed as { a1^ in software Visual DSD [ a2 t1^ in]The reactant Z is designed as<p1^x1 a1^a2 a3^>The chain sp3(6) is designed as<x1>[a1^a2]<a3^>:<a2>[t1^]The chain sp4(6) is designed as<x1>[a1^a2]<a 3^>{ t1^ chain sp5(5) designed<a2 t1^>. DNA chain of auxiliary Na { a1^ } [ a2 t1^ A]The small branch domain "a 1 ^" and the DNA strand of the reactant Z<p1^x1 a1^a2 a3^>"Small branch field" a1^ "reacts, links into a double chain sp3(6)<p1 ^x1>[a1^a2]<a3^>:<a2>[t1^]Double strand sp3(6)<p1^x1>[a1^a2]<a3^>:<a2>[t1^]Decomposition into double strands sp4(6)<p1^x1>[a1^a2]<a3^>{ t1^ and single-chain sp5(5)<a2 t1^>. Concentration of auxiliary Na (Na is 10)²nM, concentration of the reactant Z, Z ═ 5nM, reaction rate q_i＝3×10^-4nMs^-1，q_m＝1.1126×10^-1nMs^-1。

Step two: a multiplying logic gate, a replacing logic gate, a catalyzing logic gate, an adjusting logic gate and two degrading logic gates are constructed based on a reaction mechanism and a logic reaction of DNA strand replacement. And verifying the circuits of the double logic gate, the replacement logic gate, the catalysis logic gate, the adjustment logic gate and the two degradation logic gates designed in the step two by using Visual DSD software, and respectively realizing five logic operation functions of double, replacement, catalysis, adjustment and degradation.

The reaction mechanism of DNA strand displacement is that the reaction power of DNA strand displacement comes from the molecular acting force between base complementary pairing, and the DNA strand displacement reaction can be realized spontaneously and in cascade at normal temperature without enzyme or transcription mechanism. The DNA input strand with the small branch region reacts with the DNA double strand to displace the DNA output strand. Single strand<T S₁>Is the upstream input chain, { T } [ S }₁T]Is an upstream complex double strand, single strand<S₁T>Is the upstream output chain. Upstream output chain<S₁T>Can be used as input chain of downstream reaction and complex double chain [ T ] with downstream][S₂S₁]{ T } reaction, liberating the downstream export chain<T S₂S₁>. Wherein T is the minor branch domain and T is the Watson Crick base complementary pairing domain of the minor branch domain. Once the original DNA species are mixed together, the system begins to spontaneously perform a DNA strand displacement reaction, which is known to be a spontaneous, dynamic, cascadable, reversible reaction process.

For a double-add logic gate, the mathematical expression for the reaction is:

wherein, A1 and C1 are different auxiliary substances, X is a reactant of logic operation, sp4(1), sp5(1) and wait (1) all represent products, but products sp4(1) and sp5(1) can still participate in the displacement reaction when encountering the complementary pair small branch point structure because of the exposed single-chain small branch point structure, and wait (1) can not participate in the displacement reaction because of the unexposed single-chain small branch point structure; q. q.s_i、q_mIndicating the rate of the chemical reaction.

The equations (1) and (2) of the reaction add to give:

A1+X+C1→sp4(1)+waste(1)+2X (3)

removal of the auxiliary A1, the auxiliary C1 and the substitution of the generated chain sp4(1), waste (1) are simplified to obtain:

X→2X (4)

this results in the construction of the multiply-add logic gate X → 2X.

For permuting logic gates, the mathematical expression for the reaction is:

wherein, A2, B2, D2, F2 and G2 are auxiliary substances in the reaction process, and sp8(2), sp9(2), sp10(2), sp11(2), sp12(2) and sp13(2) all represent products; x, Y denotes the reactants of the logical operation; q. q.s_i、q_mQs denotes the chemical reaction rate.

Adding 2 times of the reaction formula (5) and the reaction formula (6) to obtain:

A2+B2+2D2+2X→2sp8(2)+2sp10(2)+2Y (10)

removal of auxiliary A2, auxiliary B2, auxiliary D2 and the resulting displaced chains sp8(2) and sp10(2) reduces the process to:

X→Y (11)

the equations (8) and (9) of the reaction add to give:

F2+G2+Y→sp11(2)+sp13(2)+Y (12)

the removal of the auxiliary substance F2, the auxiliary substance G2 and the chains sp11(2) and sp13(2) generated by the replacement are simplified to obtain:

Y→Y (13)

the addition of the formula (11) and the formula (13) of the reaction is simplified to obtain:

X+Y→2Y (14)

this results in the construction of X + Y → 2Y for the replacement logic gate.

For a catalytic logic gate, the mathematical expression for the reaction is:

a1, C1 and D3 are auxiliary substances in the reaction process, and sp5(3), sp6(3) and wait (1) all represent products, but products sp5(3) and sp6(3) can still participate in the displacement reaction when encountering complementary paired small-fulcrum structures because of exposed single-chain small-fulcrum structures, and wait (1) cannot participate in the displacement reaction because of unexposed single-chain small-fulcrum structures; x represents a reactant of a logical operation; q. q.s_i、q_mIndicating the rate of the chemical reaction.

The product of equation (15) of the reaction multiplied by 2 is added to equations (16) and (17) of the reaction:

2A1+2X+C1+D3→2sp5(3)+2waste(1)+3X (18)

removal of auxiliary A1, auxiliary C1, auxiliary D3 and the resulting displaced chain sp5(3), waste (1) are simplified:

2X→3X (19)

this resulted in the construction of catalytic logic gate 2X → 3X.

For the trim logic gate, the mathematical expression for the reaction is:

wherein, a4, B2, D2, F4 and G4 are auxiliary substances in the reaction process, and sp8(4), sp9(2), sp10(2), sp11(2), sp12(2) and sp13(2) represent products; z represents a reactant of a logical operation; q. q.s_i、q_m、q_sIndicating the rate of the chemical reaction. The reaction of formulae (20), (21) plus formula (22) gives:

A4+D2+2X→2sp8(4)+sp10(2)+Z (25)

the removal of the auxiliary A4, the auxiliary D2 and the replacement generated chains sp8(4) and sp10(2) is simplified to obtain:

2X→Z (26)

the sum of equations (23), (24) and (26) of the reaction yields:

2X+F4+G4+Z→sp11(2)+sp13(2)+2Z (27)

the removal of the auxiliary substances F4, G4, Z4 and the substitution of the resulting chains sp11(2) and sp13(2) is simplified:

2X+Z→2Z (28)

this results in the construction of the adjustment logic gate 2X + Z → 2Z.

For a degradation logic gate, the mathematical expression for the reaction is:

wherein Na is an auxiliary substance, Y is a reactant, and sp3(5), sp4(5), and sp5(5) represent products; y represents a reactant of a logical operation; q. q.s_i、q_mIndicating the rate of the chemical reaction.Representing a reversible reaction, the sum of formula (29) and formula (30) of the reaction gives:

removal of the auxiliary Na and substitution of the resulting chains sp4(5) and sp5(5) are simplified:

Y→Φ1 (32)

this results in the construction of the degradation logic gate Y → φ 1.Φ 1 indicates that a product incapable of participating in a substitution reaction was produced.

For a degradation logic gate, the mathematical expression for the reaction is:

wherein Z is a logical operation reactant, and sp3(6), sp4(6), and sp5(6) represent products respectively

The equations (33) and (34) of the reaction add to give:

removal of the auxiliary Na and substitution of the resulting chains sp4(6) and sp5(6) are simplified:

Z→Φ2 (36)

this results in the construction of the degradation logic gate Z → φ 2.Φ 2 indicates that a product incapable of participating in the substitution reaction was produced.

The simulation condition in FIG. 2(a) is that the concentrations of both adjunct A1 and adjunct C1 in the doubler logic gate X → 2X are 10⁴nM, concentration of reactant X5 nM, reaction rate q_i＝10^-7nMs^-1Reaction rate q_m＝10^-3nMs^-1nM is the concentration unit, n is expressed in nanometers, and M is "mol/L", i.e., moles per liter, so nM is the concentration unit nanomolar per liter. The simulation condition of FIG. 2(b) is that the concentrations of adjunct A1 and adjunct C1 in the replacement logic gate X + Y → 2Y are both 10⁴nM, concentration of reactant X5 nM, reaction rate q_i＝10^-7nMs^-1Reaction rate q_m＝10^-3nMs^-1. The simulation condition of FIG. 2(C) is that the concentrations of the auxiliary substances A1, C1 and D3 in the catalytic logic gate 2X → 3X are all 10³nM, concentration of reactant X2 nM, reaction rate q_i＝10^-6nMs^-1Reaction rate q_m＝10^-3nMs^-1(ii) a The simulation condition of FIG. 2(D) is that the concentrations of the assistants A4, B2, D2, F4, and G4 of the adjustment logic gate 2X + Z → 2Z are all 10³nM, concentration of reactant X2 nM, concentration of reactant Z1 nM, reaction rate q_i＝10^-6nMs^-1Reaction rate q_m＝10^-3nMs^-1Reaction rate q_s＝7nMs^-1. The simulation condition of FIG. 2(e) is that the concentration of the auxiliary Na in the degradation logic gate Y → φ 1 is Na ═ 10²nM, concentration of reactant Y5 nM, reaction rate q_i＝3×10^-4nMs^-1Reaction rate q_m＝1.1126×10^-1nMs^-1(ii) a The simulation condition of FIG. 2(f) is that the concentration of the auxiliary Na in the degradation logic gate Z → φ 2 is 10²nM, concentration of the reactant Z of 5nM, reaction rate q_i＝3× 10^-4nMs^-1Reaction rate q_m＝1.1126×10^-1nMs^-1。

The simulation structure is shown in fig. 2, and it can be seen from fig. 2 that (a), (b), (c), (d), (e), and (f) meet the requirements of multiplication, replacement, catalysis, adjustment, and degradation in the trend of the curve, and meet the design of the coefficients of logical operation expressions of multiplication, replacement, catalysis, adjustment, and degradation in concentration.

Step three: respectively determining mathematical expressions according to the logical relations of the double-adding logic gate, the permutation logic gate, the catalysis logic gate, the adjustment logic gate and the two degradation logic gates, converting the mathematical expressions into corresponding differential expressions and combining the differential expressions to obtain a designed three-order chaotic oscillation system; and (3) verifying the dynamic behavior of the three-order chaotic oscillation system by using Matlab software.

Based on a DNA strand displacement technology, a chaotic oscillation system is designed, and a mathematical expression is given as follows:

wherein r is₁-r₆X, Y and Z are three different species, respectively representing different chemical reaction rates, for each of which the differential expression for the corresponding equation is given as follows:

dX(37)＝r₁X

dX(38)＝-r₂XY

dX(39)＝r₃X²

dX(40)＝-r₄X²Z

dY(38)＝r₂XY

dY(41)＝-r₅Y

dZ(40)＝r₄X²Z

dZ(42)＝-r₆Z

wherein dX (37), dX (38), dX (39), and dX (40) represent differential expressions with respect to the logical operation reactant X in chemical reaction formula (37), formula (38), formula (39), and formula (40), respectively; dY (38), dY (41) represent differential expressions with respect to the logical operation reactant Y in chemical reaction formula (38), formula (41), respectively; dZ (40) and dZ (42) represent differential expressions with respect to the logical operation reactant Z in chemical reaction formulas (40) and (42), respectively.

The corresponding three categories X, Y, Z of differential expressions are merged:

dX＝dX(37)+dX(38)+dX(39)+dX(40)

＝r₁X-r₂XY+r₃X²-r₄X²Z

dY＝dY(38)+dY(41)

＝r₂XY-r₅Y

dZ＝dZ(40)+dZ(42)

＝r₄X²Z-r₆Z

let dX ═ dX, dY ═ dY, dZ ═ dZ, r₁＝a，r₂＝m＝n，r₃＝b，r₄＝c＝h，r₅＝g，r₆D, the designed chaotic oscillating system is expressed by the following third order equation:

wherein a, b, c, d, m, n, g and h are parameters of the system, and x, y and z are state variables of the system; when a is 1, b is 2, c is 2.9851, d is 3, m is 1, n is 1, g is 1, and h is 2.9851, the dynamic behavior of the chaotic oscillating system shows a chaotic state.

When the initial condition x(0) When y (0) is 1.2 and z (0) is 1.2, the Lyapunov exponent of the system is shown in fig. 3(a), and it can be seen that the Lyapunov exponent is λ₁>0，λ₂＝0，λ₃<0, indicating that the system is in a chaotic state at this time. Fig. 3(b) shows how a three-dimensional chaotic attractor is obtained, and it can be seen that when an initial value x (0) is 1.2, y (0) is 1.2, and z (0) is 1.2, the system is in a chaotic state.

The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.