Journal of Nanomaterials
Finally, the output production can be acquired through the
concentration of the reporter molecular. In principle, enough
outputcanbeobtainedinagivenconditionwithenoughfuel
and gate:output complex.
e chemical reactions described in Figure can be
equivalently translated into some kinetic equations. e main
equations are shown below from () to ().Intheseequa-
tions,
2,5
and others indicate reacting molecular species,
𝑠
indicates the slow strand displacement rate of seesawing
and reporting reactions, and
𝑓
indicates the fast strand
displacement rate of thresholding reactions. Other side-
reactions are ignored, which are also discussed in detail in
[].
Seesawing reactions:
2,5
+
5:5,6
𝑘
𝑠
⇐⇒
𝑘
𝑠
2:5,5
+
5,6
,
2,5
+
5:5,7
𝑘
𝑠
⇐⇒
𝑘
𝑠
2:5,5
+
5,7
.
()
resholding reactions:
2,5
+
2,5:5
𝑘
𝑓
→waste.
()
Reporting reactions:
5,6
+Rep
6
𝑘
𝑠
→Flour
6
.
()
ese chemical reactions modelling the toehold exchange
steps and threshold absorption steps can be written uni-
formly. For all ,,∈{1,2,...,}, where the variables refer
to the molecular species,
𝑗𝑖
+
𝑖:𝑖𝑘
𝑘
0
⇐⇒
𝑘
0
𝑗𝑖:𝑖
+
𝑖𝑘
,
𝑖𝑗
+th
𝑖𝑗:𝑗
𝑘
1
→waste
𝑘
1
←th
𝑖:𝑖𝑗
+
𝑖𝑗
.
()
Using standard mass action chemical kinetics, it gives rise
to a system of ordinary dierential equations (ODEs) for the
dynamics. In the following,
𝑖𝑗
and similar terms refer to
the concentration of the respective species, rather than to the
species themselves:
𝑤
𝑖𝑗
𝑡
=
0
𝑁
𝑛=1
𝑛𝑖
⋅
𝑖:𝑖𝑗
+
𝑗𝑛
⋅
𝑖𝑗:𝑗
−
𝑖𝑗
⋅
𝑛𝑖:𝑖
−
𝑖𝑗
⋅
𝑗:𝑗𝑛
,
𝑔
𝑖:𝑖𝑗
𝑡
=
0
𝑁
𝑛=1
𝑖𝑗
⋅
𝑛𝑖:𝑖
−
𝑛𝑖
⋅
𝑖:𝑖𝑗
,
𝑔
𝑖𝑗:𝑗
𝑡
=
0
𝑁
𝑛=1
𝑖𝑗
⋅
𝑗:𝑗𝑛
−
𝑗𝑛
⋅
𝑖𝑗:𝑗
,
th
𝑖:𝑖𝑗
𝑡
=−
1
⋅
𝑖𝑗
⋅th
𝑖:𝑖𝑗
,
th
𝑖𝑗:𝑗
𝑡
=−
1
⋅
𝑖𝑗
⋅th
𝑖𝑗:𝑗
.
()
ese dynamics have conserved quantities for each gate
node and for each signal wire :
𝑁
𝑛=1
𝑛𝑖:𝑖
+
𝑖:𝑖𝑛
≡
𝑖
,
𝑖:𝑖𝑗
−th
𝑖:𝑖𝑗
+
𝑖𝑗
+
𝑖𝑗:𝑗
−th
𝑖𝑗:𝑗
≡
𝑖𝑗
,
𝑐
𝑖
𝑡
=
𝑐
𝑖𝑗
𝑡
=0.
()
With these dynamic equations above, simulations have
been performed masterly by Mathematic in a PC platform
(Windows OS, I processor, G RAM).
3. Molecular Adders Based on the Concept of
Seesaw Gate
In electronics, an adder is a digital circuit that performs
addition of numbers. In many computers and other types of
processors, adders are used not only in arithmetic logic units,
but also in other parts of the processor, where they are used
to calculate addresses, table indices, and similar operations.
In this section, we propose full adder and serial binary adder,
with the concept described above, which is based on toehold-
mediated DNA strand displacement.
3.1. Molecular Full Adder. A full adder adds binary numbers
and accounts for values carried in and out. A single-bit full
adder adds three single-bit numbers, oen written as , ,
and
0
; and are the operands, and
0
is a bit carried in
from the next less signicant stage. e full adder is usually a
component in a cascade of adders which add -, -, and -
bit binary numbers, and so on. e circuit produces a two-
bit output, namely, output carry and sum, which are typically
represented by the signals
1
and . e logic expression of
the full adder is shown below:
=
0
,
1
=⋅+
0
⋅
.
()
Add a le containing a digital circuit netlist, which can
translate into an equivalent dual-rail circuit, in which each
input is replaced by a pair of inputs, representing logic ON
andOFFseparately.eequivalentseesawcircuitofthefull
adder is shown in Figure , in which numbers ahead of or at
the top of nodes indicate identities of nodes (or interfaces to