Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Chengye Zou^{a},
Xiaopeng Wei^{a} and
Qiang Zhang*^{ab}
^{a}School of Computer Science and Engineering, Dalian University of Technology, Dalian 116024, China. E-mail: zhangq@dlut.edu.cn
^{b}Key Laboratory of Advanced and Intelligent Computing, Dalian University, Ministry of Education, Dalian 116622, China

Received
13th February 2018
, Accepted 21st May 2018

First published on 7th June 2018

DNA strand displacement as a theoretic foundation is helpful in constructing reaction networks and DNA circuits. Research on chemical kinetics is significant to exploit the inherent potential property of biomolecular systems. In this study, we investigated two typical reactions and designed DNA strands with a fluorophore and dark quencher for reaction networks using a 3-variable Lotka–Volterra oscillator system as an example to show the convenience of and superiority for observation of dynamic trajectory using our design, and took advantage of the synchronization reaction module to synchronize two 3-variable Lotka–Volterra oscillators. The classical theory of chemical reaction networks can be used to represent biological processes for mathematical modeling. We used this method to simulate the nonlinear kinetics of a 3-variable Lotka–Volterra oscillator system.

With the aid of DNA strand displacement, massive DNA circuits, including digital and analog circuits, have been designed and validated by experiments.^{4–13} The outputs of the DNA circuit are represented by the concentrations of the DNA strands, but the detection of concentration is difficult or even impossible in some cases. For example, the structure of a DNA polymer is massive, complex, and unstable, such that detection of the concentration of the DNA polymer is impossible.^{14} If the DNA strand is designed with a fluorophore and dark quencher, observation of the output is convenient and fast. Luminous intensity is enhanced with an increase in the concentration of DNA strands.^{15} Thus, it is helpful to measure the concentration of DNA strands by fluorescence intensity. This property is particularly useful for digital circuits because these circuits are encoded with binary bits, where 2 possible values of each bit are “0” and “1”. If the luminous intensity is strong enough, the value of the output is “1”, otherwise the value of the output is “0”. For an analog circuit, the output is represented qualitatively by luminous intensity; for polymerization of a hybridization chain reaction,^{14} characterization of the hyperbranched chain reaction can be recognized by fluorescence intensity.

Synchronization is an important dynamical behavior and a general and ubiquitous phenomenon that has been investigated in a variety of fields such as complex networks, chemical reaction networks,^{16–18} secure communication, and genetic regulatory networks. Recently, it was found that synchronization is beneficial to DNA nanotechnology; for example, synchronization is applied in self-assembly,^{19} gene expression^{20} and DNA replication.^{21} Synchronized systems can enjoy common dynamical behavior and isochronous signal transition or transmission. Thus, synchronization has extensive application potential in digital and analog DNA circuits.

As compared to previous studies, we have proposed catalysis, degradation and synchronization reaction modules with a fluorophore and dark quencher, and identified different DNA strands with different colors of fluorophores, where the synchronization reaction module is a new reaction module. Then, we compiled these reaction modules into the 3-variable Lotka–Volterra oscillator and realized synchronization of two 3-variable Lotka–Volterra oscillator systems. From the simulations of these reaction modules, it was demonstrated that the DNA implementation of CRN can imitate the dynamical behavior of an ideal formal reaction, and there is a relationship between the reaction rates of DNA CRNs, intermediate CRNs and ideal formal CRNs.

We used visual DSD software to design the DNA strand and test the feasibility of the chemical reaction with DNA strand displacement, and utilized chemical reaction network (CRN) software to numerically simulate the dynamic behavior of the 3-variable Lotka–Volterra oscillator.

The fluorophore and quencher exist at the end of the strand. When the fluorophore and quencher appear in pairs, the fluorophore strand does not fluoresce, but when the fluorophore strand releases from the dark quencher strand, it can be detected by fluorescence intensity.

In the catalysis reaction module (1), as shown in Fig. 3(b), is produced by X_{M} in (5), and the increase in will catalyze (6), while the producer X_{M} can catalyze the production of . According to (1), d[X]_{t}/dt = −q_{i}C_{m}[X]_{t} and d[X_{M}]_{t}/dt = −θ_{1}[X_{M}]_{t} by (5); if DSD (1) and (2) can approximate to the intermediate (5), q_{i} = θ_{1}/C_{m} is obtained. According to (3), d[X′]_{t}/dt = −q_{m}C_{m}[X′]_{t}, while based on (6). If DSD (3) and (4) can approximate to the intermediate (6), q_{m} = θ_{m}/C_{m}. d[X]_{t}/dt = −θ_{2}[X]_{t} and d[X_{M}]_{t}/dt = −q_{j}C_{m} [X_{M}]_{t}, according to (4) and (6); if (4) is equivalent to (6), q_{j} = θ_{2}/C_{m}. Similarly, we can obtain k_{1} = θ_{1} based on (5), (6) and (7). Initial concentrations of the auxiliary species and are set to C_{m}, where [X]_{0} ≪ C_{m}, and [X]_{0}, [Y]_{0} are the initial concentrations of X and Y. As shown in Fig. 3(c), the evolutions of X in DNA CRNs, intermediate CRNs and ideal CRNs are approximately coincident. This shows that DNA CRNs, intermediate CRNs and ideal formal CRNs in the catalysis reaction module (1) are equivalent, but there are nuances between DNA CRNs, intermediate CRNs and ideal formal CRNs.

Fig. 3 Catalysis reaction module (1) X → 2X. DNA reactions are listed in (a); (b) exhibits the DNA CRNs and appropriate ideal CRNs, where intermediate CRNs is the transition between DNAs and ideal formal CRNs; (c) shows the result of the variable X in DNA CRNs, media CRNs and ideal formal CRNs respectively, where C_{m} = 10^{4} nM and [X]_{0} = [X_{M}]_{0} = [X_{I}]_{0} = 5 nM; q_{i} = 10^{−8} nM s^{−1}, q_{j} = 10^{−7} nM s^{−1} and q_{m} = 10^{−5} nM s^{−1}. DNA implementation of formal species X′ is gained by species X. Species X reacts with to displace in (1), releases two X′ in (2). In (3), species X′ reacts with hairpin to produce , and and releases X in (4) on reaction with . This reaction replaces the ssDNA strand without the fluorophore to the ssDNA strand with the fluorophore, as adapted from.^{4} q_{m} represents a maximum strand displacement rate constant under the assumption of equal binding strength for all full toeholds; when q_{i} and k_{i} are satisfied q_{i},q_{j} ≪ q_{m}, k_{i} ≪ q_{m}C_{m}; [X]_{0} and C_{m} are satisfied [X]_{0} ≪ C_{m}, as adapted from.^{1} |

In catalysis reaction module (2), as shown in Fig. 4(b), is produced by X_{M} and Y_{M} in (13), and the increase in will catalyze (14), while the producer Y_{M} can catalyze the production of and consumption of X_{M}. According to ref. 1, we have. q_{i} = θ_{3}. Similar to catalysis reaction module (1), q_{j} = θ_{4}/C_{m}, q_{m} = θ_{m}/C_{m} and k_{2} = θ_{3}. Initial concentrations of the auxiliary species A_{i}, B_{i}, H_{i}, P_{j}, Sp_{j}, and Sq_{j} are set to C_{m}, where [X]_{0}, [Y]_{0} ≪ C_{m}. As shown in Fig. 4(c), the evolutions of Y in DNA CRNs, intermediate CRNs and ideal CRNs are approximately coincident. This indicates that DNA CRNs, intermediate CRNs and ideal CRNs in catalysis reaction module (2) are equivalent, but there are nuances between DNA CRNs, intermediate CRNs and ideal CRNs.

Fig. 4 Catalysis reaction module (2) X + Y → 2Y. DNA reactions are listed in (a) and (b) exhibits the DNA CRNs and corresponding ideal formal CRNs, where intermediate CRNs is the transition between DNAs and ideal CRNs; (c) shows the result of the variable Y in DNA CRNs, media CRNs and ideal formal CRNs, respectively, where C_{m} = 10^{4} nM, [X]_{0} = [X_{M}]_{0} = [X_{I}]_{0} = 5 nM, and [Y]_{0} = [Y_{M}]_{0} = [Y_{I}]_{0} = 1 nM; q_{i} = 10^{−5} nM s^{−1}, q_{j} = 10^{−6} nM s^{−1} and q_{m} = 10^{−3} nM s^{−1}. DNA implementation of formal species is gained by species X_{M} and Y_{M} in intermediate CRN (13). Species X reacts with A_{i} to displace T_{i}. In (9), C_{i} is produced when Y irreversibly displaces T_{i}, and H_{i} releases 2 Y′ in reaction (10), as adapted from.^{1} In (11), species Y′ reacts with hairpin P_{j} to produce S_{j}, and S_{j} releases X in (12) on reacting with Sp_{j}, as adapted from.^{4} |

In the synchronization reaction module, it is assumed that X and Ma_{i} attain instant equilibrium through (18), with [X]_{t}/[Ma_{i}]_{t} = q_{m}/q_{i}; when X_{M} and Ma_{i} attain instant equilibrium through (24), we obtain [X_{M}]_{t}/[Ma_{i}]_{t} = θ_{5}/θ_{m}. According to (19) and (25), we have d[Ma_{i}]_{t}/dt = −C_{m}qs_{i}[Ma_{i}]_{t} = −θ_{6}[Ma_{i}]_{t}; as a result, qs_{i} = θ_{6}/C_{m}. Taking (25) and (28) into account, we obtain d[Y_{M}]_{t}/dt = θ_{6}[Ma_{i}]_{t} and d[Y_{I}]_{t}/dt = k_{4}[X_{I}]_{t}, respectively; if intermediate reaction (25) can be approximated to the ideal formal reaction (28), θ_{6}[Ma_{i}]_{t} = k_{4}[X_{I}]_{t}. Because [X]_{t}/[Ma_{i}]_{t} = q_{m}/q_{i} when X and Ma_{i} attain instant equilibrium and [X_{I}]_{t} = [X]_{t}+[Ma_{i}]_{t}, we can obtain k_{4} = q_{i}θ_{6}/(q_{m} + q_{i}) = qs_{i}q_{i}C_{m}/(q_{m} + q_{i}). The initial concentrations of the auxiliary species Na_{i}, , Nb_{i}, Nc_{i}, Nd_{i}, , Ne_{i}, and Nf_{i} are set to C_{m}, where [X]_{0},[Y]_{0} ≪ C_{m}. Fig. 6(c) shows that X and Y can share the same dynamic behavior in DNA CRNs, intermediate CRNs and ideal formal CRNs, but their synchronous values are different. The reasons for this phenomenon are as follows:

I. In DNA CRNs

[X]_{0} + [Y]_{0} = [X]_{t} + [Y]_{t} + [Ma_{i}]_{t} + [Mb_{i}]_{t} + [Mc_{i}]_{t} + [Md_{i}]_{t}
| (30) |

When DNA CRNs approach reaction equilibrium, the following can be obtained:

(31) |

[Mb_{i}]_{t} = [Md_{i}]_{t} ≈ 0
| (32) |

Considering (30) and (31), we can obtain the synchronous value:

(33) |

II. In intermediate CRNs

[X_{M}]_{0} + [Y_{M}]_{0} = [X_{M}]_{t} + [Y_{M}]_{t} + [Ma_{i}]_{t} + [Mc_{i}]_{t}
| (34) |

When intermediate CRNs approach reaction equilibrium, we can obtain

(35) |

III. In ideal CRNs

[X_{I}]_{0} + [Y_{I}]_{0} = [X_{I}]_{t} + [Y_{I}]_{t}
| (36) |

Eqn (36) is satisfied when X_{1} and Y_{1} approach instant equilibrium through (27) and (28)

[X_{I}]_{t} = [Y_{I}]_{t}
| (37) |

Therefore, the synchronous value of X_{1} and Y_{1} are obtained as follows:

(38) |

(39a) |

(39b) |

(39c) |

The product of X_{1} will increase the product X_{2}, and X_{2} can catalyze reaction (39b). When the concentration of X_{1} increases to a certain degree, an increase in X_{2} will lead to consumption of X_{1} and the decrease in X_{1} will slow down reaction (39b), which can lead to a decrease in X_{2}. Therefore, in the cycle, the concentrations of X_{1} and X_{2} will fluctuate over time.

According to the basic Lotka–Volterra oscillator, a 3-variable Lotka–Volterra oscillator model can be described by ideal CRNs as follows:

(40a) |

(40b) |

(40c) |

(40d) |

(40e) |

CRNs (40) can be represented by the following set of ODEs:

Ẋ_{1} = r_{1}X_{1} − r_{2}X_{1}X_{2} − r_{3}X_{1}X_{3} = F_{1}(X_{1})
| (41) |

Ẋ_{2} = r_{2}X_{1}X_{2} − r_{4}X_{2} = F_{2}(X_{2})
| (42) |

Ẋ_{3} = r_{1}X_{1}X_{3} − r_{5}X_{3} = F_{3}(X_{3})
| (43) |

Fig. 8(a) shows the evolution of Lyapunov exponents λ_{i} (i = 1,2,3) of ideal formal CRNs (40) with the chemical reaction rate parameter, r_{1}. It is known that the kinetic characteristics of the system can be characterized by the Lyapunov exponent, if exist in the range in which one Lyapunov exponent is positive and another Lyapunov exponent is negative, indicating that the system is at a chaotic state. Furthermore λ_{1} < 0, λ_{2} > 0 and λ_{3} > 0 when the parameter is taken as r_{1} = 5.5/300 s^{−1}, and the phase map of the state variables X_{1}, X_{2} and X_{3} is shown in Fig. 8(b).

Fig. 7 Evolution of concentrations of x_{1}, x_{2}, x_{3}, y_{1}, y_{2} and y_{3} of two 3-variable Lotka–Volterra oscillator systems based on ideal CRNs and DNA intermediate CRNs, respectively. |

(44a) |

(44b) |

(44c) |

(44d) |

(44e) |

In order to realize internal synchronization between two chemical reaction networks, we added coupling terms as follows:

(45a) |

(45b) |

(45c) |

(45d) |

(45e) |

(45f) |

(45g) |

(45h) |

(46a) |

(46b) |

(46c) |

(46d) |

(46e) |

(46f) |

(46g) |

(46h) |

ODEs of CRNs (45) and (46) can be described as follows:

Ẋ_{1} = r_{1}X_{1} − r_{2}X_{1}X_{2} − r_{3}X_{1}X_{3} + r_{6}(Y_{1} − X_{1}) = F_{1}(X_{1}) + r_{6}(Y_{1} − X_{1})
| (47) |

Ẋ_{2} = r_{2}X_{1}X_{2} + r_{6}Y_{2} − r_{4}X_{2} −r_{6}X_{2} = F_{2}(X_{2}) + r_{6}(Y_{2} − X_{2})
| (48) |

Ẋ_{3} = r_{3}X_{1}X_{3} + r_{6}Y_{3} − r_{5}X_{3} − r_{6}X_{3} = F_{3}(X_{3}) + r_{6}(Y_{3} − X_{3})
| (49) |

Ẏ_{1} = r_{1}Y_{1} − r_{2}Y_{1}Y_{2} − r_{3}Y_{1}Y_{3} + r_{6}(X_{1} − Y_{1}) = G_{1}(Y_{1}) + r_{6}(X_{1} − Y_{1})
| (50) |

Ẏ_{2} = r_{2}Y_{1}Y_{2} + r_{6}X_{2} − r_{4}Y_{2} − r_{6}Y_{2} = G_{2}(Y_{2}) + r_{6}(X_{2} − Y_{2})
| (51) |

Ẏ_{3} = r_{3}Y_{1}Y_{3} + r_{6}X_{3} − r_{5}Y_{3} − r_{6}Y_{3} = G_{3}(Y_{3}) + r_{6}(X_{3} − Y_{3})
| (52) |

The error state was defined as e_{i} = y_{i} − x_{i} (i = 1, 2, 3), and the dynamics of error can be described as follows:

(53) |

In simulations, the two 3-variable Lotka–Volterra oscillator systems with different initial values, and the evolution of the maximum Lyapunov exponent of eqn (53) with the chemical reaction rate r_{6} is given in Fig. 9. When the maximal Lyapunov exponent λ_{m} is negative, the synchronization of the two 3-variable Lotka–Volterra oscillator systems can be obtained.^{30}

CRNs (45) and (46) can be described by the abovementioned DNA reaction modules as follows:

(54a) |

(54b) |

(54c) |

(54d) |

(55a) |

(55b) |

(55c) |

(55d) |

(55e) |

(56a) |

(56b) |

(56c) |

(56d) |

(56e) |

(57) |

(58) |

(59a) |

(59b) |

(59c) |

(59d) |

(60a) |

(60b) |

(60c) |

(60d) |

(60e) |

(61a) |

(61b) |

(61c) |

(61d) |

(61e) |

(62) |

(63) |

(64a) |

(64b) |

(64c) |

(65a) |

(65b) |

(65c) |

(66a) |

(66b) |

(66c) |

Visual DSD software can be used to realize the approximations of the CRNs (45) and (46).^{22} The DSD implementation of the catalysis reactions (54) and (59) are shown in Fig. 3. The DSD implementation of the catalysis reactions (55), (56), (60) and (61) are shown in Fig. 4. The reactions (57), (58), (62) and (63) are degradation reactions. The DSD implementation of reactions (57), (58), (62) and (63) are shown in Fig. 5. Reactions (64)–(66) are used as the synchronization module. The DSD implementation of reactions (64)–(66) are shown in Fig. 6.

Values | Values | Values | |||
---|---|---|---|---|---|

r_{1} |
5.5/300 s^{−1} |
q_{1} |
r_{1}C^{−1}_{m} |
C_{m} |
2 × 10^{−5} M |

r_{2} |
10^{6} M s^{−1} |
q_{2} |
10^{5} M s^{−1} |
q_{m} |
10^{6} M s^{−1} |

r_{3} |
1.008 × 10^{6} M s^{−1} |
q_{3} |
r_{2} |
||

r_{4} |
4/300 s^{−1} |
q_{4} |
10^{5} M s^{−1} |
||

r_{5} |
4/300 s^{−1} |
q_{5} |
r_{3} |
||

r_{6} |
0.8/300 s^{−1} |
q_{6} |
10^{5} M s^{−1} |
||

q_{7} |
r_{4}C^{−1}_{m} |
||||

q_{8} |
r_{5}C^{−1}_{m} |
||||

q_{9} |
2 × 10^{5} M s^{−1} |
||||

qs | r_{6}(q_{m} + q_{9})/(q_{9}C_{m}) |

The evolution of the 3-variable Lotka–Volterra oscillator based on ideal CRNs and the 3-variable Lotka–Volterra oscillator based on DNA CRNs are shown in Fig. 7 and 8. When the three coupling terms are added to the system, the evolution of the 3-variable Lotka–Volterra oscillator based on DNA CRNs and the 3-variable Lotka–Volterra oscillator based on ideal CRNs are given in Fig. 9. According to Fig. 10, it can be found that the error between Y_{i} and X_{i} (i = 1, 2, 3) approaches zero, which indicates that synchronization among X_{i} and Y_{i} with different initial concentrations is realized. It is clear that errors between X_{i} and Y_{i} in the ideal CRNs approach zero, which is similar to errors in DNA CRNs, as shown in Fig. 10 and 11.

Fig. 10 Process of synchronization of two 3-variable Lotka–Volterra oscillator systems based on DNA and ideal CRNs. |

Fig. 11 Evolution of errors between two 3-variable Lotka–Volterra oscillator systems based on DNA and ideal CRNs. |

In CRNs, luminous intensity relates to the concentration of DNA strands. We can observe the process of synchronization of the two 3-variable Lotka–Volterra oscillator systems through luminous intensity. The luminous intensity of X_{i} and Y_{i} are different without adding coupling terms to the systems. The luminous intensity of X_{i} and Y_{i} reach unanimity when there is synchronization among X_{i} and Y_{i}.

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