Graph Rewriting System (GRS) for Chemical Reactions on DNA ...
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Graph Rewriting System (GRS) for Chemical Reactions on DNA Nanostructures • PhD Candidate (6 th year) • Department of Computer Science • Duke University
Transcript of Graph Rewriting System (GRS) for Chemical Reactions on DNA ...
Graph Rewriting System (GRS)for Chemical Reactions on DNA Nanostructures
• PhD Candidate (6th year)
• Department of Computer Science
• Duke University
Source: http://www.coriell.org/assets/images/personalized-medicine/dna-genes-snps-enlarged.jpgSource: Wikipedia.com
Source: Youtube.com, BIOMOD2011
Modeling DNA NanodevicesUsing Graph Rewriting Systems
Reem Mokhtar, Sudhanshu Garg, Harish Chandran, HieuBui, Tianqi Song, and John Reif
Modeling DNA Nanodevices Using Graph Rewriting Systems Invited Chapter, Advances in Unconventional Computing, Springer (2015).
M odel ing D N A N anodev ices U sing G r aph R ew r i t e Sy st em s 3
on DNA nanostructures with relatively smaller scales of complexity (where steric
hindrance does not greatly influence reactions).
2.2 Prior and Related Work
(a)
(b)
Fig. 1: (a) A rendering of
a double-stranded DNA duplex
(nanoengineer). (b) A cartoon
rendering of the duplex.
The visual representation of DNA nanostruc-
tures often termed DNA cartoons is widely
used (such as in Figure 1b as a representa-
t ion of Figure 1a). They provide a visual rep-
resentation of the secondary structure of DNA
nanostructures, including 5’ to 3’ directional-
ity and hybridizations between single strands
of DNA. Their goal is to abstract away individ-
ual base-pairings and helical twists to provide
a domain-based representation. Though there
are many variations on this, most cartoon ren-
derings represent a strand with a line that ter-
minates in an arrow at the 30-end. When two
complementary antiparallel strands hybridize
to each other through base pairing, a series
of lines between the strands represent the
hybridization bonds between each base pair.
DNA cartoons are frequently used in practice
for describing DNA nanostructures and may also be used by visual GUI software,
e.g., for graphical specification and rendering of DNA nanostructures. Alterna-
tives to string-based methods, such as explicit graph structures, are needed in
order to model certain DNA nanostructures, especially those with branched or
pseudo-knotted regions. Many earlier examples of such graph-based approaches
that provide abstractions of DNA nanostructures and their reactions have been
developed. Jonoska made early use of graphs for representing DNA-based com-
putations Jonoska et al. (1998). Reif developed a graph model for representing
DNA nanostructures with representation of 50 to 30 directionality and hybridiza-
t ions between single strands of DNA Reif (1999). In addit ion, Birac et al. have
made use of graph-based data structures for the design and modeling of DNA
nanostructures Birac et al. (2006). Other more recent graph models for DNA
nanostructures include those of Kawamata et al. Ibuki et al. (2011); Kawamata
et al. (2012). None of these model base-stacking which can be essential for some
DNA reactions Woo and Rothemund (2011). There are various prior works on
coarse-grained modeling of dynamic DNA nanodevices that transit ion between
distinct DNA nanostructures. Cardelli (2011); Phillips and Cardelli (2009); Lakin
et al. (2012a, 2011) developed an algebra for representing a restricted set of
DNA-based reactions, such as see-saw gates Qian and Winfree (2009). Yin et al.
made use of a graphical representation of biomolecular self-assembly pathways, in
Our Graph Rewriting System (GRS)for DNA Nanostructures:
GRS Notation:
An Example: A DNA Nanostructure Graph Model
Rule 1: Toehold Binding
Rule 2: Strand Displacement
Rules forBase Stacking
Rules for Remote Toehold mediated Strand Displacement
using a connectivity predicate check
M odel ing D N A N anodev ices U sing G r aph R ew r i t e Sy st em s 25
Table 1: One sequence of rule-applications representing one possible reaction
pathway
Cartoon Rendering Intermediat e Graph GiM atching
Rule
d1
d2
d4d3
d5
d̄3
d̄2
d3
d4
d1
d2
d̄4
d̄1
d5 d̄3 d̄2
Init ial input
species (G0 )
has matching
subgraph
consist ing of
vert ices with
domains d3 and
d̄3 , which
matches the
LHS of Rule
# 1.
d1
d2
d4
d3d5
d̄3
d̄2
d5d̄3
d̄2
d3
d4
d1
d2
d̄4
d̄1
Rule # 1
d1
d2
d4
d3d5
d̄3
d̄2
+
d3d̄5 d2
d5d̄3
d̄2d1
d4
d3
d2
d̄4
d̄1Rule # 1
Continued on next page
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Table 1 – Continued from previous page
Cartoon Rendering Intermediate Graph GiM atching
Rule
d1
d2
d4 d3 d5
d̄3
d̄2
d̄5d3
d2
d5d̄3
d̄2
d2d3
d̄5
d1
d4
d3
d2
d̄4
d̄1
Rule # 2
d1
d2
d4
d3
d5
d̄3
d̄2
d̄5
d3
d2 d5d̄3
d̄2
d2
d3 d̄5
d1
d4
d3
d2
d̄4
d̄1
Rule # 2
d1
d2
d4d3
d5
d̄3
d̄2
d̄5
d3
d2
d̄5d3d2
d̄2 d̄3 d5d1
d4
d3
d2
d̄4
d̄1
7 Brief Descript ion of our Prototype Software System
We implemented a prototype that generates all the states possible depending
on a subset of the basic rules defined in Section 4. Choosing a subset of the
states, we display one specific sequence of reactions in Table 1. The DNA Graph
Rewrit ing System (DAGRS) is a simple prototype implemented in Python 2.7
Python Software Foundation (2014) that uses the graph-tool library Peixoto
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We start with a set of 4 DNA complexes, 3 hairpins A, B , C and an init iator
I . Init iator I and hairpin A hybridize via a toehold a, followed by subsequent
strand displacement to give complex A + I . On strand displacement of hairpin
A , the stem loop is opened, exposing domain b̄, and allowing it to react with
hairpin B . Again, this is followed by stem loop opening of hairpin B , forming
complex A + B + I . Domain c̄ is now exposed, which hybridizes with hairpin C
and subsequently opens the stem loop.
Note that a region of hairpin C, namely b̄̄a is complementary to the arm
ab of the original hairpin A. By a process of remote-toehold mediated strand
displacement Genot et al. (2011), the domain b̄̄a displaces the attached init iator,
creating the final 3-arm junction A + B + C, and releasing init iator I , which is
free to catalyze the formation of another structure A + B + C.
Fig. 6: Catalytic hairpin-based trigger branched junction
M odel ing D N A N anodev ices U sing G r aph R ew r i t e Sy st em s 29
Fig. 7: The figure shows a sequence of graph rewrite rules that can be applied
in succession. We obtain the 3-arm junction in the above design by Yin et al.
(2008). Note that the application of rule # 5 here includes the variat ion of the
original (see Subsection 4.1).
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Fig. 9: The set of states generated by the graph rewrit ing rules given the input species. There are 26 states. Those circled in red (state 0) and blue mark
the chosen subset of states in Figure 8
M odel ing D N A N anodev ices U sing G r aph R ew r i t e Sy st em s 35
8.3 Hybridizat ion Chain React ion
Figure 13 shows how to apply the rewrite rules to basic HCR system designed by
Dirks and Pierce (2004). We use two graph rewrite rules: toehold binding and
strand displacement.
Fig. 12: Depict ion of HCR system reaction
A description
of basic HCR sys-
tem as shown in
Figure 12: there
are two types of
hairpins (H 1 and
H 2) and one type
of single strand
(I ) in this sys-
tem. The reac-
tion starts with
hybridization be-
tween a domain
of I and a do-
main of H 1 (rule
# 1). Hairpin H 1
is opened by ini-
tiator I after toe-
hold binding (rule
# 1). c domain of
I ⇤H 1 hybridizes
with c domain of
hairpin H 2 (rule
# 1). Hairpin H 2
is opened by I ⇤H 1 after toehold
binding (rule# 2).
The ba domain of
I ⇤H 1⇤H 2 will
be the init iator in
the next cycle.
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Fig. 13: The figure shows how to apply our graph rewrit ing systems to the basic
HCR system designed by Dirks and Pierce (2004)
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Fig. 15: States generated by DAGRS given the subset of rules supplied. Those circled in red (state 0) and blue mark the chosen
subset of states in Figure 14
Qian, L., & Winfree, E. (n.d.). A simple DNA gate motif for synthesizing large-scale circuits. rsif.royalsocietypublishing.org
Example: See-Saw Gates
Example 1: GRS Graph Transformations Modeling See-Saw Gates
Example Path of GRS Graph Transformations
Example Path of GRS Graph Transformations (Cont)
Example Path of GRS Graph Transformations (Cont)
State 0
State 2
State 1
State 3
(a)
State 3 State 4
State 5State 6
State 7
(b) .
[1] Dirks, R., Pierce, N., 2004. Triggered amplification by hybridization chain reaction. Proceedings of the National Academy of Sciences of
the United States of America 101 (43), 15275–15278.
[2] Peixoto, Tiago P, 2014. graph-tool: Efficient network analysis (version 2.2.31) [software].
URL ht t p: / / gr aph- t ool . skewed. de/
[3] Python Software Foundation, 2001–2014. PythonTM.
[4] Qian, L., Winfree, E., 2011. Scaling up Digital Circuit Computation with DNA Strand Displacement Cascades. Science 332 (6034),
1196–1201.
[5] Riverbank Computing Limited, 2014. Pyqt5 (version 5.3.2) [software].
URL ht t p: / / www. r i ver bankcomput i ng. com/ sof t war e/ pyqt / downl oad5
[6] Ullmann, J. R., 1976. An Algorithm for Subgraph Isomorphism. Journal of the ACM 23 (1), 31–42.
[7] Yin, P., Choi, H., Calvert, C., Pierce, N., 2008. Programming Biomolecular Self-assembly Pathways. Nature 451 (7176), 318–322.
[8] Yurke, B., Turberfield, A., Mills, A., Simmel, F., Neumann, J., 2000. A DNA-fuelled Molecular Machine Made of DNA. Nature 406 (6796),
605–608.
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Enzymatic DNA Graph Rewriting Rules
