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Materials Express
2158-5849/2021/11/1700/007
Copyright © 2021 by American Scientific Publishers
All rights reserved. doi:10.1166/mex.2021.2081
Printed in the United States of America www.aspbs.com/mex
Visual solution to minimum spanning tree problem
based on DNA origami
Jing Yang1 , Zhixiang Yin2, ∗ , Zhen Tang1 , Xue Pang1 , Jianzhong Cui1 , and Congcong Liu1
1
School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, 232001, PR China
2
School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai, 201620, PR China
ABSTRACT
DNA origami is a highly precise nanometer material based on DNA molecular. In the current study, we present
a visual computing model of minimum spanning tree that combines advantages of DNA origami, hybridization
chain reaction and nano-gold particles. Nano-gold particles were used to represent vertices and molecular
Article
beacons with fluorescent labels were used as anchor strands, which were fixed on origami substrate with
staple strands according to theIP: shape
192.168.39.151
in graph. On:
We Sat,
then27 Nov 2021
induced 00:03:42 chain reaction using initiator
hybridization
Copyright: American Scientific Publishers
strands and fuel strands. Lastly the problem was detected using fluorescence. The model provides a visualized
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calculation model of minimum spanning tree by using hybridization chain reaction and fluorescence labeling on
origami bases. This model utilizes their advantages and demonstrates effectiveness of the model through case
simulation. It also reduces computational complexity of the problem and improve the way of solution reading.
Keywords: Minimum Spanning Tree, DNA Origami, Hybridization Chain Reaction, Visual Computing Model.
1. INTRODUCTION The resulting pattern is called DNA origami base, and it is
Deoxyribonucleic acid (DNA) is a kind of indispensable a brand-new idea for DNA nanoscale self-assembly using
biological macromolecules in biomass. It is a macromolec- DNA molecules as materials proposed by Rothemund [7].
ular polymer composed of deoxyribonucleic acid. DNA DNA origami is widely used for intracellular drug deliv-
molecular has four kinds of bases: adenine (A), guanine ery [8] and DNA computing. Many scholars took it as a
(G), thymine (T) and cytosine (C). It has widespread appli- nanomaterial to participate in the calculation of graph and
cation in the fields of genetic engineering, forensic iden- combinatorial optimization problems [9–12].
tification and so on. With the development of molecular Hybridization chain reaction (HCR) is a self-assembly
biology technology, DNA has become one of the main reaction that does not need participation of biological
molecules to construct nanometer material. DNA origami enzymes. It simulates molecular chain-growth polymeriza-
is a highly precise nanometer material based on DNA tion that uses the initiator strand to induce two different
molecular for various applications. The nanostructure com- types of molecular beacons to alternately hybridize and
plexity and inter-strands connectivity makes its distinct open the hairpin structures alternately, then form double-
mechanical properties, simple design, high assembly effi- stranded DNA polymer. Therefore, HCR need only three
ciency, and clear nano-addressability. It folds a single- distinct DNA strands (initiating strand and two alternating
strand DNA (M13mp18) of phage containing more than monomer stands). Its mild reaction conditions and simple
7000 bases into arbitrary nanoscale geometric [1–6] with a operation can be applied to signal amplification of other
series of designed short DNA fragments as staple strands. sensors. HCR has been widespread application in nucleic
acid, protein detection, biosensors and other fields [13–18],
and can also be combined with DNA origami to participate
∗
Author to whom correspondence should be addressed. in the calculation of NP problems [9, 10].
1700 Mater. Express, Vol. 11, No. 10, 2021Visual solution to minimum spanning tree problem based on DNA origami
Yang et al.
Materials Express
Nano-gold is a kind of nanomaterial that has been using paste and delete models. In 2012, Kang [25] com-
studied earlier and is widely used. The diameter of the bined DNA calculation to gradually generate the minimum
gold nanoparticles is generally 1–100 nm [19]. Due to its spanning tree using the method for solving the minimum
unique nanosized structure, the nano-gold particles have edge of the cut set. In 2014, Yang and Yin [26] used the
four basic effects, namely quantum size effect, specific sur-triple helix DNA strands combined with magnetic particles
face effect, volume effect and macroscopic quantum tun- separation technology to give a minimum spanning tree
neling effect. Therefore, DNA can be combined with gold calculation model. In 2015, Wang et al. [27] constructed a
particles after SH modification to achieve DNA fixation. In three-dimensional (3D) DNA self-assembly model to solve
addition, the unique optical and electronic characteristics the minimum spanning tree problem. This algorithm can
unique to nano-gold can enhance and amplify the signal in effectively solve the minimum spanning tree problem with
electrochemical detection. The surface of the nano-gold is high efficiency. This study uses the visual model based
bound to a single-stranded DNA, which generally requires on origami to realize the biological estimation process of
higher content of A, C, and G in the single-stranded DNA. Kruskal algorithm. This method is also effective when the
To improve the affinity between gold nanoparticles and edge weights have the same value.
DNA molecules, we should consider the content of A, C, Given a weighted undirected connected graph G =
and G to be higher than the content of T when designing V , E, W , V = v1 , v2 , , vn is the vertex set of G,
the single chain linked to the gold nanoparticles [20]. and E = e1 , e2 , , em is the edge set of G. W =
1 , 2 , , m is the weight set corresponding to the
2. EXPERIMENTAL AND METHODS edge set. The spanning tree T of graph G is a connected
In the study, in order to fully demonstrate the advantages sub-graph. It contains all n vertices of graph G, but it has
of these nanometer materials in DNA computing, we com- and only has n − 1 edges to form a tree. That is a spanning
bined the advantages of DNA origami, HCR and gold tree with n vertices that has only n − 1 edges. If another
nanoparticles to build a computing model of minimum edge is added to the spanning tree, it must form a loop.
In all spanning trees, if the weight sum of all edges is
Article
spanning tree.
minimum, such it is called the minimum spanning tree of
2.1. Experimental Subjects
IP: 192.168.39.151 On: Sat,graph
27 Nov there is W T ≤ W T for the spanning
2021is,00:03:42
G. That
Copyright: American Scientific Publishers
history bytree
The minimum spanning tree problem is a longDelivered T .
Ingenta
problem in combinatorial optimization. It has been widely Regarding the DNA computing model for the minimum
used in network optimization and other problems (such spanning tree, most of them need to sequence the resulting
as the network topology design problems of communi- product to read the solution of problem. In this study, we
cation network design, channel laying, VISI, and multi- use DNA origami and HCR to give an implementation of
cast backbone network). Commonly used algorithms are the Kruskal algorithm. This model also takes advantage of
Prim algorithm and Kruskal algorithm. Each step of the the fluorescent labeling of molecular beacons and the bind-
Prim algorithm adds an edge to a growing tree. Initially, ing advantages of gold nanoparticles and DNA strands.
the tree has only one vertex, and then adds n − 1 edges The computing model designed by us can visualize the
according to the algorithm. The time complexity for the solution of the problem without sequencing.
Prim algorithm is Om + n log n. The Kruskal algorithm
is added one by one edge according to the weight order of 2.2. Experimental Material
the edges (from small to large). But if the edge is added The specific design process of experiment is as follows.
to generate a loop, the edge is discarded until there are (1) Construct DNA origami substrates, nano-gold parti-
n − 1 edges in the spanning tree, and its time complex- cles representing vertices, DNA strands representing their
ity is Om log m. In 1995, Karger et al. [21] proposed a degrees, anchor DNA strands representing edges and edge
Boruvka algorithm and flip-deletion algorithm for a spe- weight, fuel strands and initiator strands. (2) The nano-
cial model where the weights of edges can be compared gold particles representing vertices are combined with
in pairs. It can obtain the minimum spanning tree in lin- DNA strands representing their degrees. Then they are
ear time. Moreover, Chazelle [22, 23] proposed the fastest anchored on the origami substrate with anchoring DNA
non-random comparison algorithm in 2000, which relied strands according to the shape of graph G. (3) Putting fuel
on a data structure similar to a priority queue with soft strands and initiator strands corresponding to the edges
heap. Its running time is Omm, n, where is the with smallest weight into the test tube, and after fully
classical functional inverse of Ackermann’s function and reflecting, the test tube is divided into dvi equal parts,
n (respectively, m) is the number of vertices (respectively, dvi representing the degree of vertex vi . (4) Putting
edges). The complexity of the algorithm can be regarded fuel strands and initiator strands of the edge with small-
as linear. In 2011, Min and Jian [24] gave a DNA algo- est weight among the remaining sides into one of the test
rithm for solving the least-weight spanning tree problem tubes, and observing whether there is a loop formation
Mater. Express, Vol. 11, pp. 1700–1706, 2021 1701Materials Express Visual solution to minimum spanning tree problem based on DNA origami
Yang et al.
with an electron microscope. (5) If there is no loop for- the difference of the initiator strand will cause the anchor
mation, this step 4 is repeated, and if there is a loop for- strand on the gold nanoparticle to indicate a certain edge
mation, we continue to select the smallest remaining edge to respond.
weights and place it in another test tube, repeating the
above steps. If there is a loop formation, and there is the
3. RESULTS AND DISCUSSION
same value of the edge weight according to the different
We will use the example below to verify the experiment
DNA strands linked on the gold nanoparticles, we add fuel
and discuss the results. The given graph G = V , E, W
strands and initiator strands representing another edge with
is a simple weighted undirected connected graph. Herein,
the same weight. Then we continue to observe and repeat
V = v1 , v2 , v3 , v4 , v5 , v6 is the vertex set of graph G,
the above steps. (6) Until n − 1 edges are added and no
E = v1 v2 , v1 v3 , v1 v4 , v2 v3 , v2 v5 , v3 v4 , v3 v5 , v3 v6 , v4 v6 , v5 v6
loop is generated, we can obtain the minimum spanning
is the edge set of G and W = 6, 1, 5, 5, 3, 5, 6, 4, 2, 6 is the
tree of graph G.
weight set corresponding to the edge set. The minimum
The model is mainly composed of the following five
spanning tree of graph G is shown in Figure 1.
components.
(1) Two-dimensional (2D) DNA origami. Which uses a 3.1. Biological Operation
short staple strand to fold the single-stranded circular For the above graph G with 6 vertices and 10 edges, we
phage (M13mp18) into a two-dimensional rectangle as the give its biological algorithm. Figure 2 shows structures
origami substrate. The technology is well-established now. of six vertices of the nano-gold particles and its initiator
(2) Vertex structure. The composition of vertices consists strands (Here, the illustration omits the short strand linked
of two components: the gold nanoparticles that represent to the staple strand in the origami substrate.). The structure
the vertices, and the short DNA strands that connect the of the anchor DNA strand, the fuel strand is shown in
vertices. There are two types of DNA short strands. One Figure 3.
type representing the degree of the vertex, and the other Step 1: Origami construct. Here, the single-stranded
is used to link with the staple strand on the origami sub- cyclic phage (M13mp18) containing more than 7000 bases
Article
strate. If the degree of the vertex is n, the short strands is still selected as the scaffold strand, and the short strand
anchored on the gold nanoparticles are n + 1. The short
IP: 192.168.39.151 On: Sat,is27
usedNovas 2021 00:03:42
the staple strand to fold it into a two-dimensional
strand used for anchoring only needs to Copyright: American Scientific
be complementary rectanglePublishers
as the origami substrate.
Delivered
to the staple strand to be fixed on origami substrate, and by Ingenta
Vertex construct. Different DNA short strands are linked
it is easy to fix the gold nanoparticles in the correspond- to the surface of the gold nanoparticles representing the
ing position. The length of the short DNA strand is half vertices, and the number of short strands is equal to the
the length of the initiator strand, that is, complementary degree of the vertices. For example, the degree dv1 = 3
to half of the initiator strand. of vertex v1 has v2 , v3 , and v4 associated with it. Therefore,
(3) Anchor DNA strands. The anchor strand is a molecu- adding the four short DNA strands on the surface of the
lar beacon with a fluorescent label. It consists of the fol- gold nanoparticles representing v1 represent v1 v2 , v1 v3 , and
lowing three components: a loop region, complementary v1 v4 . Construction of initiator strands and staple strand. In
stem region, and stretched sticky end. It can be combined this calculation model, the number of initiator strands is
with a staple strand with sticky end and anchored on the 2m (m is the number of edges in graph G). One of them is
DNA origami substrate according to the shape of graph G. vi vj → xi . vi vj is complementary to the single strand vi vj
The number of bases in the loop is equal to the number of linked to the gold nanoparticles representing the vertex vi .
bases in the stem. The length of the anchor strand is equal xi is complementary to a and b1 of the anchor strand. The
to the length of the fuel strand that is added to the sticky other is xj → vj vi . vj vi is complementary to the single
end, and it is complementary to the misalignment of the strand vj vi linked to the gold nanoparticles representing
fuel strand. Fluorescence can then be detected when the
reaction occurs.
(4) Fuel strands. The fuel strand is also a hairpin structure 5
without fluorescent label. These fuel strands are abundant 6
1
in test tubes and do not need to be anchored on the DNA
5 5 1
origami substrate. The number of bases in the loop region 5
is equal to the number of bases in the stem of the fuel
6 4 2
strand. Its structure representing edge weight is the same, 3
4 2
but the initiator strand of every edge is different. 6 3
(5) Initiator strands. The initiator strand is a single DNA
strand that acts to initiate HCR. One part is complemen-
tary to the single strand linked to the vertex, and the other
is complementary to half of the anchor strand. Therefore, Fig. 1. Graph G and its minimum spanning tree T.
1702 Mater. Express, Vol. 11, pp. 1700–1706, 2021Visual solution to minimum spanning tree problem based on DNA origami
Yang et al.
Materials Express
+
Fig. 2. Substrate structure of vertex of nano-gold particles and initia-
tor strands.
+
the vertex vj . xj is complementary to a and b2 of the
anchor strand.
The anchor strand and fuel strand construct. The anchor
strand in the model that uses a molecular beacon struc-
ture. The number of anchor strands is used to express the
edge weight, and stem end of the anchor strand is marked
with a fluorescent group and quenching group, and one
end of it links a short DNA strand as a sticky end. Its
structure is presented as; Tori → ā → b2 → b1 → a, where +
a is complementary to ā, and Tori is a sticky end. The fuel
strand is also hairpin structure, and its role is to continue
the HCR. These fuel strands are abundant in test tubes Fig. 4. Schematic diagram of reaction between vi and vj after adding
and do not need to be anchored on the DNA origami sub- initiator strands and fuel strands.
strate. Its structure is represented as; c → d1 → d2 → c̄,
where c is complementary to c̄. The fuel strand and anchor weight can be determined according to the position of flu-
strand are complementary in misalignment, as shown in orescence and amount of fluorescence, and presence or
Figure 4 (When adding initiator strand vi vj → xi to a test
Article
absence of a loop can also be detected. Which edge of
tube, vi vj and vi vj are complementary, then hybrid reac-
IP: 192.168.39.151 On: Sat,the 27HCRNov occurs is determined by the initiator strand. This
2021 00:03:42
tion occurs. The xi of initiator strand is complementary
Copyright: to Scientific
American design can Publishers
greatly reduce the workload of coding. We fix
b1 → a of the anchor strand, and hybridizationDelivered
reaction bythem
Ingentaon the origami substrate using staple strand accord-
continues. Therefore, the hairpin structure of the anchor ing to the shape of graph G. The anchor strand and the
strand is opened, releasing fluorescence. Since there are a gold nanoparticles representing the vertices are fixed on
large number of fuel strands in the test tube, the ā → b2 of the origami substrate according to the shape of graph G
the anchor strands will be complementary to the c̄ → d2 (Fig. 5).
of the fuel strand, and then a hybridization reaction will Step 2: Sort according to the weight of each side
occur, and the fuel strand hairpin structure will open. Until of graph G, v1 v3 < v4 v6 < v2 v5 < v3 v6 <
the anchor strand on this edge participates in the reac- v1 v4 = v2 v3 = v3 v4 < v1 v2 < v3 v5 <
tion, the initiator strand xj → vj vi is complementary to
the anchor strand and the vj vi strand on vertex, and reac-
tion ends.). c̄ → d2 of the fuel strand is complementary
to b2 → ā of the anchor strand. d1 → c of the fuel strand
is complementary to a → b1 of the anchor strand. When 5
the fuel strand is added, the hairpin structure opens. When 6 1
emitting fluorescence, the selection of edges and size of
5
5
6 4 2
3
anchor fuel initiator
strand strand strand
6
Fig. 3. Structure of anchoring DNA strand, fuel strand and initiator
strand. Fig. 5. Graph G fixed on origami substrate.
Mater. Express, Vol. 11, pp. 1700–1706, 2021 1703Materials Express Visual solution to minimum spanning tree problem based on DNA origami
Yang et al.
v5 v6 .v1 v3 = 1, therefore, we firstly choose the ini-
tiator strands Iv1 v3 , Iv3 v1 and fuel strands to add to the test
tube T0 . The HCR starts and product is shown in Figure 6.
In the test tube T0 , the hairpin structure is opened, releas-
1
ing fluorescence. We divide the test tube T0 into two test
tubes T11 and T12 after the full reaction.
Step 3: We choose the test tube T11 and Add initiator
strands Iv4 v6 , Iv6 v4 and fuel strands to the test tube T11 . 5
After full reflection, we use a transmission electron micro-
scope to observe the test tube for no loop formation. So
we divide test tube T11 into two test tubes T21 and T22 , and
4 2
then we add initiator strands Iv2 v5 , Iv5 v2 and fuel strands to
the test tube T21 . We also repeat the above operation. 3
Step 4: If there are same weight edges, we only choose
the corresponding initiator strands and their fuel strands
to add to the test tube. Then we observe the reaction like
step3. If no loop is generated, we repeat step3, and if there
has a loop generated in this test tube, we will choose the Fig. 7. Minimum spanning tree is detected.
other edge and add it to the other test tube, and then we
repeat step4. involved in the reaction, concentration and setting of reac-
Step 5: Until n − 1 = 5 edges are added and no loop tion time. The setting area sets the syntax mode and sim-
is generated, the experiment ends. The minimum spanning ulation mode. The display area shows the curve for DNA
tree is obtained (Fig. 7) and can be observed with electron sequence as reaction progresses, structure of the product,
microscope. etc.
Article
In the simulation, the concentration of initiator strand
3.2. Simulation Analysis IP: 192.168.39.151 On: Sat,(input),
27 Novorigami substrate (origami), and fuel strand (fuel)
2021 00:03:42
In order to verify the feasibility of the Copyright: American
model, we choose to Scientific
are set toPublishers
20 nM, 20 nM and 40 nM, respectively, and
use simulation software for simulation verification.Delivered
Visual bythe
Ingenta
reaction time is set to 7000 s. With addition of the
DSD is currently the universal DNA self-assembly simu- initiator strand (input), the induced hybridization reaction
lation software. Its interface is mainly divided into three begins. In Figure 8, the concentration of initiator strand
components: coding area, setting area and display area. (input) and origami is the same, so their curves coincide.
The coding area shows the composition of DNA sequence The initiator strand (input) first reacts with single strand
on the nanoparticles, and then the fuel strand (fuel) also
participates in the reaction with anchor strand on origami
substrate in turn. Therefore, the concentration of the fuel
strand (fuel) as the fuel strand is consumed in the reaction,
5 then shows a downward trend. The reaction continues and
6 1
concentration of the intermediate products increases first.
After reaching the maximum peak, the concentration of
intermediate products gradually decrease, and eventually
tend to 0. These intermediate products correspond to prod-
5 ucts generated by the intermediate reaction from vertex v1
5
to vertex v3 , respectively. When the reaction is complete,
the concentration of intermediate products tends to 0. The
final products (sp10) tend to be stable as the reaction time
6 4 2 goes on. Figure 8 also shows the structure of generated
3 strand from v1 to v3 , and the other reactions are similar.
This model gives a minimum spanning tree calculation
algorithm for a graph with 5 vertices and 10 edges. The
model can also be generalized to solve the minimum span-
ning tree of graphs with n vertices and m edges.
Fig. 6. Result of the edge HCR with smallest weight. (The addition
of initiator strands Iv1 v3 , Iv3 v1 and fuel strands makes reaction to occur 3.3. Discussion
on path between vertices v1 and v3 , which is also the embodiment of From the solving process of example, we will con-
addressability of HCR.). duct a complexity analysis of the general situation. In a
1704 Mater. Express, Vol. 11, pp. 1700–1706, 2021Visual solution to minimum spanning tree problem based on DNA origami
Yang et al.
Materials Express
Article
IP: 192.168.39.151 On: Sat, 27 Nov 2021 00:03:42
Copyright: American Scientific Publishers
Delivered by Ingenta
Fig. 8. Reaction simulation diagram of test tube T0 and strand structure.
positive integer weight undirected graph G = V , E, W , The structure of DNA origami is stable, and it is
V = v1 , v2 , , vn is vertex sets of graph G, E = nano-addressable. It combines the fluorescent labeling
e1 , e2 , , em is the edge sets for graph G, and W = characteristics of molecular beacons. During the reaction,
1 , 2 , , m is the weight corresponding to the edge. the hairpin structure opens and emits fluorescence, and
The implementation process for this algorithm uses 2m their reaction path can be observed through the release
kinds of initiator strands coding, 1 kind of fuel strand of fluorescence. That is, the solution to the problem can
coding, and 1 kind of anchor strand coding. 2m kinds of be detected by a fluorescence detector. So, the calcula-
short strands are needed on the gold nanoparticles. There- tion process for the model can be visualized. This model
fore, the encoding type required by the entire model is is a biological estimation implementation of the Kruskal
2m + 2m + 2. The model requires a limited number of test
algorithm for minimum spanning tree problem, and fully
tubes (recyclable). The total number of operations does
demonstrates that the DNA computing model combined
not exceed 2n − 1 − 2 + 1 + 1 = 2n − 4 times, (m is
with nanometer material has a natural advantage in solving
the number of edges and n is the number of vertices in
graph G). some NP complete problems in graph and combinatorial
optimization.
4. CONCLUSION
Ethical Compliance
This study synthesizes the advantages of gold nanoparti-
There are no researches conducted on animals or humans.
cles, origami and HCR to give a DNA computing model
for minimum spanning tree problem. Gold nanoparticles
have better DNA fixation and better optical and elec- Conflicts of Interest
trical effects, which can amplify photoelectric signals. There are no conflicts to declare.
Mater. Express, Vol. 11, pp. 1700–1706, 2021 1705Materials Express Visual solution to minimum spanning tree problem based on DNA origami
Yang et al.
Acknowledgments: The project is supported by 13. Peng, D.M. and Yuan, R., 2014. Study of aptamer sensor based on
National Natural Science Foundation of China (Nos. hybrid chain reaction amplification signal. Chemical Sensors, 34(4),
pp.14–19.
61702008, 62072296), Natural Science Foundation of
14. Zhang, B., Liu, B., Tang, D., Niessner, R. and Knopp, D., 2012.
Anhui Province (No. 1808085MF193), Foreign Visit and DNA-Based hybridization chain reaction for amplified bioelectronic
Study Project of Outstanding Young Talents in Col- signal and ultrasensitive detection of proteins. Analytical Chemistry,
leges and Universities (No. gxgwfx2019015) and Anhui 84(12), pp.5392–5398.
Province postdoctoral fund (No. 2019B331). 15. Chen, Y., Xu, J., Su, J., Xiang, Y., Yuan, R. and Chai, Y., 2012.
In situ hybridization chain reaction amplification for universal and
highly sensitive electrochemiluminescent detection of DNA. Analyt-
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Received: 28 March 2021. Accepted: 16 August 2021.
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