1
o.:ȋͬͭͮͯͰͱͲͳʹ͵Ȍ
Scientific Reports
| (2023) 13:11295
|
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Molecular system
for an exponentially fast growing
programmable synthetic polymer
Nadine Dabby
1
, Alan Barr
1
& Ho‑Lin
Chen
2
*
In this paper, we demonstrate a molecular system for the first
active
self
‑assembly
linear
DNA polymer
that exhibits programmable molecular exponential growth in real time, also the first to implement
“internal” parallel insertion that does not rely on adding successive layers to “external” edges for
growth. Approaches like this can produce enhanced exponential growth behavior that is less limited
by volume and external surface interference, for an early step toward efficiently building two and
three dimensional shapes in logarithmic time. We experimentally demonstrate the division of these
polymers via the addition of a single DNA complex that competes with the insertion mechanism and
results in the exponential growth of a population of polymers per unit time. In the
supplementary
material, we note that an “extension” beyond conventional Turing machine theory is needed to
theoretically analyze exponential growth itself in programmable physical systems. Sequential physical
Turing Machines that run a roughly constant number of Turing steps per unit time cannot achieve an
exponential growth of structure per time. In contrast, the “active” self
‑assembly model in this paper,
computationally equivalent to a Push‑Down Automaton, is exponentially fast when implemented in
molecules, but is
taxonomically less powerful
than a Turing machine. In this sense, a physical Push‑
Down Automaton can be
more powerful
than a sequential physical Turing Machine, even though the
Turing Machine can compute any computable function. A need for an “extended” computational/
physical theory arises, described in the supplementary material section
S1.
Molecular programming, nanotechnology and synthetic biology raise the prospect of bottom-up fabrication, the
manufacture of complex devices that assemble themselves from simpler components. Biological systems fabri
-
cate structures with enormous scale and complex behaviors, defined at atomic-scale resolution, which can grow
quickly with small programs relative to their object size and algorithmic
complexity
1
. A goal in the molecular
synthesis field is to build biophysical systems with great complexity and power, with applications to medicine,
the environment, and green manufacturing.
In natural biological systems, periods of programmed exponential growth per unit time is common, and
perhaps is almost ubiquitous. Understanding and controlling exponential growth will become key, to obtain
acceptable reaction yield and performance for practical applications of bottom-up self-fabrication.
Over the past several years, new directions in research have been translating computational algorithms into
and out of molecular systems using DNA and other molecular substrates. DNA has been used to build autono
-
mous
walkers
2
–
10
, logic and catalytic
circuits
8
,
11
–
13
, and triggered assembly of
linear
14
,
15
and dendritic
structures
8
.
The primary task in this paper is to build an exponentially quickly growing molecular assembly in the physical
world. We present a programmable molecular model and a molecular implementation of the first
active
synthetic
linear polymer
system that can grow exponentially quickly in Real Time (see Fig.
2
). Our molecular system is
not
,
however, the first exponentially fast growing structure ever synthesized. Yin et al. constructed a
binary molecular
tree
out of
DNA
8
. Our system is implemented with DNA and is also capable of a second behavior—splitting or
division of polymers. By encoding the order of the nucleotides in the DNA sequence, we can control the interac-
tion of DNA strands, which is how the system is programmed. Our molecular construction (Fig.
1
) is inspired by
the Hybridization Chain Reaction (HCR) system developed by Dirks and
Pierce
14
. The molecular DNA system
is computationally equivalent to a Pushdown
Automaton
16
,
17
.
The molecular insertion system we designed, computationally equivalent to a Pushdown Automaton, when
implemented in molecules as described in Fig.
1
C, could perform exponential growth tasks that some Turing-
Complete molecular systems could
not
perform, such as the DNA Tile Assembly
Model
18
, even though Turing
Complete systems can in principle, implement any computable function. The Turing-Complete DNA Tile system
OPEN
1
California Institute of Technology, Pasadena, USA.
2
National Taiwan University, Taipei City, Taiwan.
*
email:
holinchen@ntu.edu.tw
2
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cannot achieve “exponentially quick” growth in real physical time, while the Pushdown Automaton in Fig.
1
C
can, even when both are implemented molecularly. Yet the Turing-Complete molecular system can, in theory,
compute “anything.”
This was surprising to us at first, since a Turing-Complete system is taxonomically more powerful, in a purely
computational sense, than a Push-Down Automaton, which is
not
Turing complete.
The observation led us to have a discussion in the supplementary material section S1 to identify a need for a
new type of combined physical/computational theory for “Extended Physical Computation.” This would aid the
understanding of
programmable physical systems
, especially in the context of exponential growth. It also relates to
the the body of work initiated in the 1980’s by Carver Mead, John Hopfield and Richard Feynman, on the Phys-
ics of
Computation
19
,
20
. A new type of theoretical and conceptual framework could be useful for understanding
how to build and analyze exponentially growing, complex, programmable physical systems for the technology
of bottom-up self-fabrication.
The remainder of this paper is organized as follows: In section "
Schematic and computational model for
“active” self-assembly
" we describe the formal Push-Down Automaton computational model we use for our active
insertion method for self-assembly. In section "
Molecular implementation
", we present experimental methods
that describe our molecular implementation. In section "
Exponential growth results
", we present the exponential
growth results, as well as the exponential growth mechanism controls and the kinetics of parallel insertion, and
time lapse experiments. In section "
Methods to generate other behaviors
", we describe methods to generate other
Figure 1.
Schematic of our insertional polymer implementation using DNA. The insertional polymer
implementation shows the first two rounds of growth. (
A
) Legend shows (i) schematics of the Initiator complex,
Hairpin 1, Hairpin 2, Hairpin 3, and Divider complex with sequences color-coded by domain below. Each
oligonucleotide is shown with color-coded motifs that correspond to the DNA subsequences below. (ii) The
Initiator-ROX complex is a modified Initiator complex with a single fluorophore tag for gel electrophoresis
experiments. Hairpin 2RQ is a modified Hairpin 2 molecule with a quencher and fluorophore pair on
opposite ends of the molecule, used in the spectrofluorimetry experiments. (iii) Hairpin 2L and Hairpin 3L
are inactivated versions of Hairpins 2 and 3, in which the loops are replaced with an inactive poly-T sequence.
The color of the boxes around each oligonucleotide in (i) correspond to the insertion arrows in (iv) as follows:
a blue arrow indicates an insertion site for Hairpin 1, a pink arrow indicates an insertion site for Hairpin 2,
Hairpin 2RQ or Hairpin 2L, a purple arrow indicates an insertion site for Hairpin 3 or Hairpin 3L, a green arrow
indicates an insertion site for the Divide complex. (
B
) The abstract model of our system and (
C
) the molecular
implementation of our polymer display exponential growth occurs as follows: (0) The Initiator has one insertion
site for Hairpin 1 (blue arrow). Insertion of Hairpin 1 is driven forward by the hybridization of 6 new base
pairs. (1) After Hairpin 1 inserts into the Initiator, two new insertion sites are generated: one for Hairpin 2 (pink
arrow) and one for Hairpin 3 (purple arrow). Hairpin 2 and Hairpin 3 are sequentially inserted (in solution
insertion occurs asynchronously), each one generates a new insertion site for Hairpin 1 (blue arrows). After the
first round of insertion, two insertion sites for Hairpin 1 are generated from what was initially (in round (0)) one
site. (2) A second round of insertion is illustrated. The 4-way branch migration mechanism used in the insertion
process is demonstrated in Fig. S2 in the supplementary material.
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behaviors, such as division and treadmilling. We end the paper in section "
Conclusions
" with conclusions, and
then present a Supplementary Section containing additional materials for the experimental methods.
Schematic and computational model for “active” self‑assembly
In this section we define the first molecularly implementable active self-assembly model (shown in Fig.
1
). We
introduce a theoretical framework for provably knowing what actions, behaviors, and life-like qualities can
emerge from a given set of simple modular units. We will use some of the theoretical approaches that computer
science has for determining the complexity and difficulty of solving computational problems. Our approach
arises out of the fact that molecules do certain things well and other things badly, and digital computers do other
types of things well and badly.
As a starting point we note that the abstract Tile Assembly Model (aTAM)
18
is a “passive” self-assembly system
that formally couples computation with shape construction. It is a computational model that can be directly
implemented in DNA
molecules
21
–
23
. Implements aTAM systems, which assemble self-similar fractals, counters,
and digital circuits using DNA molecules. Winfree showed that the tiles are capable of universal
computation
18
.
Such a system is said to be “Turing-complete”. Because the Tile Assembly Model is Turing-complete, it is capable
of computing anything that another computer can compute with at most a polynomial time slowdown, but it
cannot compute any arbitrary task. There are many behaviors that are not “computations” in a classical sense.
Examples include exponential growth, and molecular motion relative to a surface. The tiles cannot implement
these behaviors because (a) there is no instruction for moving or rotating a tile relative to a surface and (b) pas-
sive self-assembly is exponentially slower than active self assembly. Several extensions of the aTAM have been
Figure 2.
Gel time-lapse studies of linear and exponential polymer growth. The bottom edges of the thick
horizontal red lines indicate 1000 base pairs. Top left: Gel time-lapse studies of linear polymer growth. Note
that it takes 480 min to reach nearly 1000 base pairs. Super Fine Resolution Agarose non-denaturing gels of the
product of a polymerization reaction with 80 nM ROX-labeled Initiator, 1.5
μ
M Hairpin 1, and 1
μ
M of Hairpin
2 and Hairpin 3L. ROX fluorescence was imaged prior to staining with SYBR Gold. (The SYBR Gold stained
gel can be found in Fig. S12). A more complete analysis of this gel was precluded due to the interference of the
fluorescent loading dye bromophenol blue as discussed in section "
Time Lapse experiments
". Top right: Average
length of the polymers versus time in the linear system. The size of the purple dots represent the total molecular
weight of molecules with that size. Bottom left: Gel time-lapse studies of exponential polymer growth. It takes
less than 60 min to reach 1000 base pairs. Super Fine Resolution Agarose non-denaturing gels of the product
of a polymerization reaction with 80 nM ROX-labeled Initiator, 1.5
μ
M Hairpin 1, and 1
μ
M of Hairpin 2 and
Hairpin 3. ROX fluorescence was imaged prior to staining with SYBR Gold. (The SYBR Gold stained gel can
be found in Fig. S13). Three additional experimental runs of this experiment can be found in Figs. S10, S11
and S14. A more complete analysis of this gel was precluded due to the interference of the fluorescent loading
dye bromophenol blue as discussed in section "
Time Lapse experiments
". Bottom right: Average length of the
polymers versus time in the exponential system. Note the rapid increase on the growth rate after the 30-min
point which indicates that the growth is nonlinear on the bottom right. Also note from the time labels that the
time scale of the columns of the gel and graphs on the left and right do not match, which makes the thick red
1000-base-pair line appear to be longer in the graph on the bottom left and shorter on the bottom right, even
though each red line in the bottom two graphs is still around 60 min long. The 1000-base-pair time for the
bottom “exponential” graphs, 60 min, is much shorter than the time for the 1000-base-pair red line in the top
two “linear” graphs, which is around 480 min long.
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proposed, such as staged (two-handed, hierarchical)
assembly
24
–
27
and signal passing
tiles
28
have been proposed.
These systems are more powerful than the aTAM (signal passing tiles are more fuel-efficient in simulating Turing
machines
28
; staged assembly system can perform superlinear
growth
26
. However, these extensions are still passive
and cannot achieve the abovementioned behaviors. The theoretical nubot
model
29
is not molecularly imple-
mentable today, but it could achieve exponential growth in theory. The behaviors that we catalogued are different
from typical computations because they demonstrate a notion of physicality that is not captured by traditional
computational theory. These behaviors are in a class that we call “programmable behaviors”. In order to achieve
these behaviors, a tile system requires the logic for how it will grow and move, and the speed conferred by allow-
ing individual molecules to change the existing structure, but the system does not need to be Turing-complete.
As we will demonstrate, a sufficiently expressive implementation of an “active” molecular self-assembly
approach can achieve these behaviors. We derive a new type of “active” self-assembly system that can be formally
defined and easily implemented in molecules.
Molecular biology needs a better theoretical framework for understanding the complexity of subsets of mol-
ecules that interact with each other to generate behaviors. Computer Science has such a framework but it deals
with computational complexity—thus we can say how “hard” a particular mathematical problem is by analyzing
how much time and space a computer requires to solve it. On the other hand, in other parts of Biology, we can’t
say how computationally “hard” it is to generate behaviors like metamorphosis (the changing of one shape into
another) or treadmilling (the growth of a linear polymer in one direction while it shrinks in the other direction).
The absence of this ability to distinguish how hard it is to produce a desired behavior in a molecular system is a
limitation in the fields of synthetic chemistry and biology.
In the absence of biological measures of complexity we map our system onto a computational framework, by
proving theorems regarding the “expressive power” of the model we define. The expressive power of our system
is equivalent to context-free
grammars
16
,
17
. This system is capable of implementing exponential growth and can
construct fixed-length linear polymers in poly-logarithmic
time
16
,
17
,
30
.
Formal molecular model description.
In our model, each construction begins with an initiator, and
grows via the insertion of simple units that we call monomers (Fig.
1
). We assume that each type of monomer
in the system is present in infinite amounts. Monomers can be inserted into the middle of the structure and
increase the length of the structure.
The detailed description of initiators, monomers, and the insertion rules follows:
1.
We have two finite sets of symbols
Ŵ
={
a
1
,
a
2
,
a
3
,
a
4
,
...
}
and
Ŵ
∗
={
a
∗
1
,
a
∗
2
,
a
∗
3
,
a
∗
4
,
...
}
. Each pair
a
i
and
a
∗
i
are called
complementary
to each other.
2. There are
k
monomers, each is described by a quadruple of symbols (
a
,
b
,
c
,
d
) and either a plus sign or a
minus sign. The plus and minus sign indicate the directionality of the molecules and are used in mapping
the model onto a direct DNA implementation, which requires both
5
′
and
3
′
sequences. (For example,
(
a
4
,
a
7
,
a
∗
6
,
a
1
)
+
or
(
a
5
,
a
7
,
a
∗
2
,
a
∗
3
)
−
.) Each monomer has a concentration
c
. We assume that the total con-
centration is at most 1. For example, in the exponential growing system described in Fig.
1
, Hairpin 1 can
be described as
(
b
∗
,
e
∗
,
f
∗
,
c
∗
)
+
and Hairpin 2 can be described as
(
c
,
a
∗
,
e
,
b
)
−
.
Notice that all hairpin sequences are represented in a clockwise ordering, but the ones marked with a “+”
and the ones marked with a “−” begin at different locations on the hairpin. The hairpins marked with a “+”
begin at one of the toeholds and the hairpins marked with a “−” begin in the middle of the hairpin loop. The
insertion of a “+” hairpin generates insertion sites where only “−” hairpins can insert, and vice versa, but
the choice of “+” and “−” can be arbitrary.
3.
The initial state can be described by two pairs of symbols (
a
,
b
), (
c
,
d
). Either
a
and
d
are complementary to
each other or
b
and
c
are complementary to each other. Each of these pairs is considered a monomer.
4. An
insertion site
can only exist between two consecutive monomers: e.g., in the initial state (
a
,
b
) and (
c
,
d
)
belong to two different monomers. For example, in Fig.
1
, the initial state (Initiator) can be described as
(
a
,
b
)
,
(
c
,
a
∗
)
forming an insertion site.
5.
Only the following insertion rules are possible:
(a)
If there are two consecutive monomers connected in the structure such that the first one ends with
the pair
(
e
,
a
∗
)
and the second one starts with the pair
(
d
∗
,
f
)
, where
e
and
f
are complementary with
each other, then any monomer of the form
(
a
,
b
,
c
,
d
)
+
can insert between those two groups, and add
a group of symbols (
a
,
b
,
c
,
d
) in the middle.
(
e
,
a
∗
)
,
(
d
∗
,
f
)
is called an
insertion site
.
(b)
If there are two consecutive monomers connected in the structure such that the first one ends with
(
d
∗
,
e
)
and the second one starts with
(
f
,
a
∗
)
, where
e
and
f
are complementary with each other, then
any monomer of the form
(
a
,
b
,
c
,
d
)
−
can insert between these two groups and add a group of sym
-
bols (
c
,
d
,
a
,
b
) in the middle.
(
d
∗
,
e
)
,
(
f
,
a
∗
)
is called an
insertion site
.
6. If a particular insertion is applicable, it occurs at time
x
, where
x
is an exponential random variable with rate
c
, where
c
is the concentration of the monomer inserted.
7. A
polymer
is a sequence of tuples of symbols reachable from the initial state, where the first and last tuples
are pairs of symbols and the middle tuples are monomers (as defined in rule 2). A
terminal polymer
is a
polymer such that no monomers exist in the system that can be inserted at any of the insertion sites avail
-
able on that polymer. The
length
of the polymer is defined as the number of monomers that it contains. For
example, in Fig.
1
, Hairpin 1,
(
b
∗
,
e
∗
,
f
∗
,
c
∗
)
+
, can insert into the Initiator, which implements the initial state
(
a
,
b
)
,
(
c
,
a
∗
)
to form a new polymer
(
a
,
b
)
,
(
b
∗
,
e
∗
,
f
∗
,
c
∗
)
,
(
c
,
a
∗
)
with two new insertion sites
(
a
,
b
)
,
(
b
∗
,
e
∗
)
and
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(
f
∗
,
c
∗
)
,
(
c
,
a
∗
)
. Notice that the sequences
a
and
a
∗
were bound together before the insertion and are separated
after the insertion. After that, Hairpin 2,
(
c
,
a
∗
,
e
,
b
)
−
can insert into the first new insertion site
(
a
,
b
)
,
(
b
∗
,
e
∗
)
to form the structure
(
a
,
b
)
,
(
c
,
a
∗
,
e
,
b
)
,
(
b
∗
,
e
∗
)
, generating another new insertion site
(
a
,
b
)
,
(
c
,
a
∗
)
. Hairpin 3
can also insert into the second insertion site
(
f
∗
,
c
∗
)
,
(
c
,
a
∗
)
and generate the same insertion site
(
a
,
b
)
,
(
c
,
a
∗
)
.
How do we tell if a polymer is terminal? The only way is to check for the condition, at each site, whether there
exists a monomer that can insert according to the insertion rules listed in 5(a) and 5(b). If no monomers can
insert, then the polymer is terminal.
Molecular implementation
Given any system described above, there is a direct implementation of monomers into a set of DNA molecules.
By encoding the order of the nucleotides in a DNA sequence, we can control the interaction of DNA strands.
Subsequences of these strands are called domains and it is their binding (hybridization) and unbinding (disasso-
ciation) from complementary domains that determines what a system can do. In DNA nanotechnology, dynamic
systems of DNA molecules can be controlled by toeholds, the short sequences of DNA that are complementary
to single stranded domains in a target
molecule
31
,
32
. Toeholds serve as the inputs to dynamic DNA systems and
initiate branch migration processes, the random walk process of bond breaking and formation that results in the
exchange of one strand in the duplex for another single strand with the same sequence.
Any system described in our model can be implemented by designing DNA hairpins and an initiator complex
as follows:
For every monomer
(
a
,
b
,
c
,
d
)
−
, we add a hairpin with domains
(
a
,
x
,
b
,
c
,
x
∗
,
d
)
, where
x
(composed of
18 bases) is the long stem of the hairpin. For every monomer
(
a
,
b
,
c
,
d
)
+
, we add a hairpin with domains
(
a
,
x
∗
,
b
,
c
,
x
,
d
)
. The initiator is
(
a
,
x
∗
,
b
)
binding with (
c
,
x
,
d
). The insertion rules defined in the model corre-
spond to all possible reactions that can happen in the corresponding molecular system.
In addition to the monomer
(
a
,
b
,
c
,
d
)
+
(or minus), we can also have a new type of monomer
(
a
,
b
)(
c
,
d
)
+
that we call a divider monomer. The reaction available for
(
a
,
b
)(
c
,
d
)
+
is exactly the same as that for
(
a
,
b
,
c
,
d
)
+
,
except that after
(
a
,
b
)(
c
,
d
)
+
inserts, the polymer will be cut between (
a
,
b
) and (
c
,
d
) and divided into two parts
, as will be described in section "
Division
".
Figure
1
C shows the molecular implementation of our exponential growth system. Hairpin 1 (H1) and the
Initator (I) react first, this results in two new insertion sites: one that is complementary to Hairpin 2 (H2), and
another that is complementary to Hairpin 3 (H3). Upon insertion of H2 and H3 into the growing polymer, two
new insertion sites that are complementary to H1 are generated. Thus for every initial H1 insertion site, each
round of insertions creates two new H1 insertion sites.
The initial reaction (insertion of H1 into the Initiator complex) is driven by the hybridization of six new
base pairs. After that, each new hairpin that is inserted adds nine base pairs to the system. Some of these steps
become reversible as the system approaches equilibrium. The free energy and reversibility of toehold-medi-
ated four-way branch migration is explored in depth
in
33
. Other design lengths and sequences were explored
(Tables S1, S2, S3, S4), but these resulted in a larger system leak (an undesired molecular interaction) than the
sequences presented here (see Figs. S3, S24, S25, S26 and the discussion in section "
Treadmilling
").
We created a new molecular system that grows linearly, which we compared to the exponential growth sys
-
tem. To achieve this, we used the same monomers as in the exponential growth system, except that we replaced
Hairpin 3 with an inactive version called Hairpin 3L. Hairpin 3L contains a poly-T sequence instead of the loops
that normally create a new insertion site for Hairpin 1 after Hairpin 3 inserts. As a result, when Hairpins 1, 2,
and 3 are inserted, they generate only one new insertion site for Hairpin 1 to bind to, instead of the two insertion
sites created by the exponential growth system. This limits the system’s growth to be linear instead of exponential.
In addition to the insertional monomers that grow the polymer, we introduce a new type of monomer, which
we call a Divide complex, that upon insertion splits the polymer into two pieces. This tool can be used to generate
a population of polymers in exponential time (Section "
Division
").
Figure
1
A is a legend for the set of DNA molecules used in this section. Each oligonucleotide complex (Initia-
tor, Hairpin 1, Hairpin 2, Hairpin 3, and Divide) is shown with color-coded motifs (purple, green, blue, brown,
pink, and black) that correspond to the colored DNA subsequences (Fig.
1
A; see Table S1 for all sequences).
The Initiator-ROX complex is a modified Initiator complex with a single fluorophore tag for gel electrophoresis
experiments. Hairpin 2RQ (H2RQ) is a modified Hairpin 2 molecule with a quencher and fluorophore pair
on opposite ends of the molecule, used in the spectrofluorimetry experiments. Hairpin 2L (H2L) and Hairpin
3L (H3L) are inactivated versions of Hairpins 2 and 3, in which the loops are replaced with a poly-T sequence.
The boxes around each oligonucleotide correspond to the insertion arrows as follows: a blue arrow indicates an
insertion site for Hairpin 1, a pink arrow indicates an insertion site for Hairpin 2, Hairpin 2RQ or Hairpin 2L,
a purple arrow indicates an insertion site for Hairpin 3 or Hairpin 3L, and a green arrow indicates an insertion
site for the Divide complex (Fig.
4
A).
In each diagram, we utilize a domain abstraction for referring to stretches of consecutive nucleotides that act
as a unit in binding to complementary stretches of nucleotides. Domains are represented by Latin letters (Fig.
1
).
Letters followed by an asterisk denote complementary domains, e.g.:
x
is complementary to
x*
. Single-stranded
molecules of DNA (henceforth strands) are comprised of concatenated domains. DNA complexes are composed
of two or more noncovalently-bound strands. There are two types of toeholds in our system: long toeholds that
indicate a stronger desired interaction (six bases in length) and short toeholds that indicate a weaker desired
interaction (three bases in length).
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| (2023) 13:11295 |
https://doi.org/10.1038/s41598-023-35720-5
www.nature.com/scientificreports/
Exponential growth results
We confirm exponential growth by measuring the conversion of monomers into a product. We then qualitatively
measure the size of products over time. Finally, we verify the predicted structure using Atomic Force Microscopy.
Exponential growth mechanism controls.
We tested each insertion step in the exponential growth
mechanism by using the inactivated versions of of Hairpins 2 and 3 in which the binding loop is replaced by an
inactive sequence of nucleotides (Fig. S3). Hairpin 2L and Hairpin 3L were added to the Initiator and Hairpin 1
both individually (this results in exactly one insertion event) and together with the normal version of the other
hairpin (i.e.: Hairpin 2L and Hairpin 3), which results in linear growth. We note that there is more product in
lanes 14 (I, H1, H3L) and 15 (I, H1, H3) than there is in lanes 12 (I, H1, H2L) and 13 (I, H1, H2).
The reactants in lanes 12 and 14 can only proceed through two steps of the polymerization reaction due to
the inactivated strands. At equilibrium (after 6 hours) there is more dimerization between the Initiator-Hairpin
1 complex and Hairpin 3L than there is between the Initiator-Hairpin 1 complex and Hairpin 2L. Thus Hairpin
3 appears to have a greater affinity to the Initiator-Hairpin 1 complex than Hairpin 2. This observation implies
that the two reactions have different rate constants, Hairpin 2 is either slower to react with its insertion site or
faster to dissociate from its insertion site than Hairpin 3 (or both).
The reader may observe the presence of faint extra bands in the lanes that contain only individual hairpins.
These are dimerized hairpins that form in small amounts from individual hairpins when the strands are annealed.
We minimize their presence by snap cooling. Snap cooling the hairpins results in the same amount of dimerized
monomers as gel purification (data not shown). All hairpins except for the Initiator were snap cooled prior to
experiments. The Initiator is a gel-purified duplex composed of two molecules of DNA.
In order to ensure that polymers were not randomly joining each other over time, we observed the linear
systems (I, H1, H2) and (I, H1, H3) and the exponential system (I, H1, H2, H3) when all molecules were added
at the same concentration. If the polymers were randomly joining at the ends, we would see a shift in the length
of the polymers over time. Figure S4 shows that there is minimal joining as the polymer bands do not shift
upward over time.
The kinetics of parallel insertion.
We examined the kinetics of the conversion of monomers into the
polymer by adding a fluorophore and quencher pair to the opposite ends of Hairpin 2. Before reaction, the fluo-
rophore is quenched. Upon incorporation of the hairpin into the DNA polymer, the quencher and fluorophore
pair are separated, and the fluorescence of the solution increases (Fig. S5).
We probed both the linear and exponential polymerization over eight different Initiator concentration val
-
ues. The time course of fluorescence intensity confirmed linear conversion of hairpins in the system with one
inactivated strand (Fig.
3
A), and exponential conversion of hairpins in the full system (Fig.
3
B).
In order to derive both the linear and exponential growth equations, we made the approximation that the
hairpin concentrations remain constant until 10% of the monomers are consumed as
in
8
. For more details see
Supplementary Section S3.
In a linear growth system, the total mass of polymer product,
P
, grows as a function of initial Initiator con
-
centration,
I
0
, and time,
t
, as follows:
The time at which
10%
of monomers are consumed,
t
10%
, is
Thus, in a linear growth system, the time to
10%
completion of polymer growth (
10%
conversion of hairpins)
is inversely proportional to initial Initiator concentration. When plotted on a logarithmic concentration scale,
the time to
10%
conversion exponentially decays as a function of increasing initial Initiator concentration. This
model fits our linear growth system data (Fig.
3
A).
In an exponential growth system, the total mass of polymer product,
P
, grows as a function of initial Initiator
concentration,
I
0
, and time,
t
, as follows:
The time at which
10%
of monomers are consumed,
t
10%
, is
Thus, in an exponential growth system, the time to
10%
completion of polymer growth (
10%
conversion of hair
-
pins) is a linear function of the logarithm of the initial Initiator concentration. When plotted on a logarithmic
concentration scale, the time to
10%
conversion linearly decreases with increasing initial Initiator concentration.
This is what we observe in our exponential growth system data (Fig.
3
B).
We quantify the leak via spectrofluorimetry experiments in Fig.
3
8
: we adjust the Initiator concentration [
I
]
by an additional term
[
I
]
leak
to obtain an effective Initiator concentration
[
I
]
effective
=[
I
]+[
I
]
leak
. We then fit
the
[
I
]
leak
parameter to our data and find that in the exponential system
[
I
]
leak
=
0.04
×
and in the linear system
[
I
]
leak
=
0.01
×
. Reactions were started with the addition of Hairpin 1 in order to avoid the leak. (The baseline
in Fig.
3
contains all hairpins except for Hairpin1).
(1)
P
=
k
(
I
0
+
I
leak
)
t
.
(2)
t
10%
=
P
10%
k
(
I
0
+
I
leak
)
.
(3)
P
=
(
I
0
+
I
leak
)
e
(
kt
)
.
(4)
t
10%
=
1
k
(
ln
(
P
10%
)
−
ln
(
I
0
+
I
leak
))
.