ChemPhysChem, in print (2017)
DOI link http://dx.doi.org/10.1002/cphc.201700018
Design of High Quality Chemical XOR Gates with Noise Reduction
Mackenna L. Wood, Sergii Domanskyi, Vladimir Privman*
Department of Physics, Clarkson University, Potsdam, NY 13676
Corresponding author’s contact information:
*[email protected], http://www.clarkson.edu/Privman, +1-315-268-3891
ABTRACT
We describe a chemical XOR gate design that realizes gate-response function with
filtering properties. Such gate-response function is flat (has small gradients) at and in the vicinity
of all the four binary-input logic points, resulting in analog noise suppression. The gate
functioning involves cross-reaction of the inputs represented by pairs of chemicals to produce a
practically zero output when both are present and nearly equal. This cross-reaction processing
step is also designed to result in filtering at low output intensities by canceling out the inputs if
one of the latter has low intensity as compared to the other. The remaining inputs that were not
reacted away, are processed to produce the output XOR signal, by chemical steps that result in
filtering at large output signal intensities. We analyze the tradeoff due to filtering, which
involves loss of signal intensity. We also discuss practical aspects of realizations of such XOR
gates.
KEYWORDS
chemical computing, XOR, binary gate, digital, noise reduction
–1–
INTRODUCTION
Chemical and especially biomolecular computing,1–13 as well as other systems that
involve (bio)chemical analytes constituting input signals and/or output,13–24 the latter of interest
in biosensing, forensics, and other applications, have become important fields of research as
subfields of signal and information processing and unconventional computing.23,25,26 Extensive
recent theoretical and experimental developments have laid out the foundations for design of
novel systems and devices that involve conversion of biological and/or (bio)chemical signals into
digital binary-type YES/NO response. In chemical computing many types of digital gates,
including AND, OR, NAND, NOR, CNOT, XOR, INHIBIT, etc.10–13 have been realized.
However, there has been a challenge to design high-quality digital gates that do not
amplify and preferably suppress noise in the inputs27–38 as it is reflected in the output. Highquality gates with “sigmoid” (flat-slope near the logic-point values) response functions have
been developed recently.27–38 However, these studies primarily focused on AND gates, which
have generally been the most studied binary gates27,31,32,39,40 in chemical and biochemical
computing, because AND is the most natural chemically realized function: Two irreversibly
reacting chemicals, A and B, when both of them are present will produce the third, S, which is
then defined as the output signal,
→ ,
(1)
otherwise there will be no output.
The simplest AND gate, Eq. 1, is not high quality, because it amplifies analog noise in
the inputs.19,27,41 Therefore, additional pre-processing of the inputs or post-processing of the
output is required. Such “filtering” by added chemical or biocatalytic reactions can yield highquality AND gates by modifying typical convex chemical-reaction, Eq. 1, response.27,28,30,42–47
Other, more complicated approaches to get high-quality AND gates by using properties of
biocatalytic enzymatic reactions have also been explored.48,49 We note that another useful
–2–
property of enzyme-catalyzed processes is their high selectivity and specificity towards specific
analytes as substrates.
In the present work we consider the XOR gate, which was much less studied35,36,50 in the
chemical/biochemical computing context. Like AND gate, it can also be potentially realized in a
very simple chemical setting. For example, the reaction in Eq. 1 can be viewed as XOR if A and
B are initially supplied in equal concentrations, and each of them is considered an “output” or is
converted into the output signal, S, in additional chemical processes,
→ ⋯,
(2)
⋯→ ,
(3)
⋯→ ,
(4)
where the reactions in Eqs. 3-4 should be much slower than the “cancellation” process, Eq. 2,
which produces a non-signal chemical.
However, we will demonstrate that, similar to the simplest AND gate realization with
Eq. 1, this XOR realization here will not yield a high-quality gate for analog noise handling.
Furthermore, it might not be easy to find a reasonably large selection of chemicals to realize the
required reactions, Eqs. 2-4. As with AND gates, additional chemical or biochemical steps might
be needed. We will argue that a convenient approach for XOR is to have each input a
combination of two chemicals, one of which “cancels out” a counterpart from the other input
pair, whereas the other is converted to the output. In practice, realization of such gates can be
enabled by some or all of the chemicals being substrates for enzymes that catalyze the relevant
processes. We will show that, with proper selection of the process parameters high quality XOR
gates can be designed.
–3–
THEORETICAL SECTION
In order to characterize (bio)chemical logic gates, it is useful to introduce scaled, binaryrange dimensionless variables , , . Specifically, we seek to represent the dependence of the
;
output signal as a function of the two inputs,
0 ,
0 , in terms of variables that assume
value 0 at zero signal and 1 at the maximum value of that signal. Note that here the inputs are the
concentrations at time t = 0, A(0) and B(0), whereas the output is at a later, “gate time”
. Thus,
we define
0 ⁄
,
0 ⁄
,
(5)
⁄
,
,
0 ∈ 0,
where the inputs vary in the ranges
and
0 ∈ 0,
, and we define the
largest value of the output as
;
,
,
for AND,
(6)
; 0,
;
,0 ,
Hence, all three variables are in the range
for XOR.
, , ∈ 0,1 ; the maximum value of
is 1
corresponding to the logic value 1. The zero-level output signal corresponds to logic 0.
Typical chemical realizations1,2,27,31,32,39,40,51,52 of AND gates result in convex-shape
response of the type shown in Fig. 1(a). The signal gradient is large near small values of
which causes noise amplification near small logic values of the inputs
and
and ,
. Conversion of
such a response to the sigmoid shaped surface would aid in noise reduction at the low inputs.24
Chemical and biochemical AND gates have been extensively studied,27,31,32,39,40,51,52 and their
–4–
modifications were designed to yield “sigmoid” response of the type shown in Fig. 1. Generally,
besides the nature of the actual processing (chemical or other), the shape of the response curve or
surface also depends on the choice of the variables considered input(s) or output(s),3 a notable
example being, for instance, the use of H+ concentration vs. pH, e.g., in protonation reaction
equilibria.
Figure 1. Schematics of gate-response functions. (a) A typical convex-shaped AND
response, represented in terms of logic variables , , , where the latter is the verticalaxis variable here and in all the other three-dimensional plots. (b) A desired sigmoidshaped AND response, which can be achieved by the means of additional “filtering”
processes. The inset shows the AND gate “truth table.”
–5–
Figure 2. Schematics of gate-response functions. (a) XOR response of the type realized
by processes shown Eqs. 2-4, represented in terms of logic variables
,
, . Unless
various conditions described in the text are satisfied, this realization is approximate; here
the slight inaccuracy due to the large but finite rate in Eq. 2 is illustrated to result in the
output at logic 1,1 being not exactly zero. (b) A desired double-sigmoid-shaped XOR
response, which is the main objective of the present study, with the output at logic 1,1
pushed very close to zero, in addition to small gradients at and near all the logic points.
The inset shows the XOR gate “truth table.”
–6–
Optimization of the response surface of XOR gates in the chemical-computing context
has not been investigated in details for the few previously considered35,36,50 XOR realizations.
Before addressing it, let us consider the quality of XOR at the logic-point input values 0 and 1.
The realization of XOR with chemical processes shown in Eqs. 2-4 will yield a good
approximation to the XOR function at the logic values provided the reaction in Eq. 2 is fast as
compared to those in Eqs. 3-4. Otherwise, the logic 1,1 inputs will not yield the precise output 0
(for logic 0 of the output); see Fig. 2. Another inaccuracy, not considered here, can result if
processes in Eq. 3-4 produce the output in different quantities by the time
condition
; 0,
;
. This will cause the
, 0 , see Eq. 6, to be only approximately satisfied. Note
that the condition that A and B are “initially supplied at equal concentrations,” mentioned in
connection with Eqs. 2-4 for XOR, actually applies at the logic 1 values only, i.e.,
.
For high-quality XOR gate-response function of the type shown in Fig. 2(b) to replace
the response shown in Fig. 2(a), the value z(1,1) should be decreased as much as possible. In
addition, “filtering” is needed to make the absolute value of the gradient of the function z(x,y) at
and near all the logic points small. This is more complicated than “filtering” required to have
high-quality AND gate, because for AND we only need to filter the output signal at low values
of z (the saturation of chemical reactions at large output values ensures small gradient at z = 1).
As a result, for high-quality XOR gates more chemical/biocatalytic processes need to be added,
than for AND gates.
Here we consider one such option: The XOR realization schematically shown in Fig. 3.
We note that some chemicals are not inputs or output “signals” but are introduced into the
system as “gate machinery.” The latter can be the required reactants, such as C, D, F (Fig. 3) or
additional biocatalysts. Indeed, the complexity of the processes in Fig. 3 suggests that, to avoid
“cross-talk” and also to allow “clocking” of the processes,53–58 specifically, have the crossreactions proceed to completion before the other process steps, enzyme-catalyzed processes
should be utilized, with potentially flow-systems59–66 used to spatially separate (for “clocking”)
various reaction steps.
–7–
Figure 3. A diagram of chemical processes that yield a high-quality XOR gate. Inputs are
“duplicated” (details are given in the text) to start with ,
and ,
, which react cross-
wise to produce some inactive chemicals. The non-reacted parts of both inputs are
converted into chemical
in much slower processes, or by adding the required reactants,
here C and D, at a later time. The product P is processed in another, filtering step to
produce the output signal .
Let us describe the system shown in Fig. 3 in detail. The original input A (and similarly
for input B) is “duplicated” in a process not shown in the figure. Generally, signal splitting is a
non-digital function, called FANOUT, chemical realizations64,67–69 of which have been
considered to some extent in the literature, with the main challenge being not to lose signal
intensity. One can always process part of the original amount of A by an added chemical process
to convert it into A'. In our case, for high-quality XOR we actually require a somewhat more
sophisticated “pre-processing.” Let us denote
0 ,
(7)
0 .
We expect that the provided chemical of the type A is pre-processed to be equally split, but also
that a certain amount of the chemical A' is present in the system at time t = 0 as the “gate
–8–
machinery” chemical, in the amount ∆ , and similarly for the preparation of the initial chemicals
concentrations for B and B',
0
∆ ,
(8)
′ 0
∆ .
The first two processes shown in Fig. 3 should be fast (or some of the reactants and/or
biocatalysts for other processes should be added later) to have the reactions
→ …,
(9)
→ …,
proceed to near completion for most possible pairs of values of
0,
∈ 0,
and
∈
, which is required for high-quality XOR. Furthermore, to have equal outputs at logic
0,1 and 1,0 inputs, as well as output practically 0 at logic 1,1 inputs, we have to impose the
conditions
,
(10)
∆
∆ .
To have the gate-response function “flat” at small |
|, i.e., near the lower-plane diagonal,
see Fig. 2(b), we also impose the requirement that the ratio
∆ /
is
(11)
< 1 and is in fact rather small, as will be explained shortly.
–9–
The remaining (slower or carried out later) processes, shown in Fig. 3, correspond to
reactions
→ ,
→ ,
(12)
→ ,
If the processes in Eq. 9 are carried to completion, the amount of unreacted A left is
|
|
∆
⁄2, and similarly for B. Then, if processes in Eq. 12 are
∆
carried to completion in the next step, with plentiful supply of the chemicals C and D , sufficient
to fully convert what was left of A and B into P, the output signal S will be
min |
where
|
∆
|
|
∆
2∆ , 2
⁄2,
(13)
is the “initial” concentration of the reactant F at the time at which the processes in
Eq. 12 were initiated (which, due to “clocking,” might be later than t = 0).
, such that the processes in Eq. 12 are practically
At a late enough gate time,
completed, we then get the following expression for the gate-response function in terms of the
variables defined in Eq. 5,
|
,
| ,
where the initial value
≡
| |
1
,
,
(14)
should be selected such that
,
(15)
– 10 –
which yields “filtering” at large outputs (at z = 1).
Figure 4. “Ideal” XOR gate response function, Eq. 14, illustrated here for a = 0.25 and
f = 0.3964 (from Eq. 16). Note that here, unlike in Fig. 2(b), the two top triangles are flat
(z = 1), as is the bottom (z = 0) yellow-shaded area around the main diagonal.
The function in Eq. 14 is illustrated in Fig. 4. This XOR gate-response function is an
idealization because in practice not all the reactions will carried to completion as specified to get
such a response: first, the cross-wise reactions shown in Eq. 9; then, the reactions in Eq. 12. In
reality a high-quality gate function will more realistically look like that illustrated in Fig. 2(b).
We will revisit this matter in the next section. In the rest of this section we discuss the selection
of the parameters a and f for the “ideal” gate (Fig. 4). The reason for designing gate-functions of
the type shown in Fig. 2(b), with built-in filtering properties, has been that chemical and
biochemical data are rather noisy, with noise levels typically in the 3% up to 10% range.19,27
Therefore, binary gates will magnify such noise to unmanageable in just few steps and they
cannot be concatenated to form even small circuits, unless the gate-functions have slopes
(gradients) much less than 1 in magnitude, at and near all the logic points.19,20 Thus, analog noise
can be “corrected” by using gates with built in filtering. Note that for very large circuits one also
– 11 –
has to incorporate digital error correction accomplished by circuit design with built in
redundancy, but this is not relevant at the level of single gates design.
Our ideal gate function has zero slope in regions that extend distance a away from the
logic points 0,0 and 1,1, along the x and y axes, see Fig. 4. Therefore, a should be selected
comparable to the level of noise in the inputs, typically up to 10%. To get zero-slope regions
near logic points 0,1, and 1,0, as in Fig. 4, we have to impose the restriction on f as shown in
Eq. 15. These regions extend distance (1 – a) – f along the x and y axes away from these points.
However, the shortest distance, along the diagonal, is 1– – ⁄√2. Thus, for efficient error
suppression we should select the latter expression approximately equal a, i.e., the value of f
should be
≃1
√2
1 ,
(16)
which in turn suggest that a should be kept less than 1/ √2
1 ≃ 0.4.
We note that, as usual with this type of “intensity filtering,” the tradeoff involved in
improving noise reduction is the loss of the overall signal intensity. Here, the intensity is reduced
by the factor f, so the percentage loss is 1 – f. Therefore, we should always keep the value of f as
large as possible, which implies a as small as possible. For example, if the level of noise in the
data is 5%, then according to the present criteria for an “ideal” XOR of the form considered here,
we would select
≃ 0.05, and thus
≃ 0.88 according to Eq. 16, which will smooth out the
noise in the inputs, but at the expense of the loss of 12% of the output signal intensity as
compared to the “unfiltered” gate. The values selected for Fig. 4 were exaggerated to better show
the features of the response surface.
RESULTS AND DISCUSSION
Multiple chemicals are needed for realizing the proposed XOR gate. The input signal is
represented by , , ′ and ′ (for the latter, by their excess over the baseline quantities ∆ and
– 12 –
∆
, whereas the output chemical is , and an intermediate product is . The other chemicals, ,
and , as well as ∆ and ∆ , are the “gate machinery” reactants, which should be selected to
achieve an optimal performance for our XOR gate, as discussed in the preceding section and
below. However, before optimizing the gate functioning, it is important to consider that, this is a
rather complicated system. Therefore, a direct chemical-reaction realization of it might be
difficult, especially if we consider the fact that such gates should be modular, i.e., incorporable
as part of even larger systems for “chemical computing.” The problem will be prevention of
crosstalk between various XOR-gate and other “networked” chemical processes. The desired
selectivity, as well as clocking (timing of starting some of the reactions) can be enabled by
utilizing enzyme-catalyzed processes, which offer selectivity and specificity.
Thus, it is likely that most or all the required chemical reactions, Eqs. 9 and 12, will each
actually be a few-step biocatalytic processes. Such systems have been devised70 and are presently
being experimentally explored. For modeling of these processes for gate-design applications, we
use a simplified single-pathway approximation,28 similar to the standard Michaelis-Menten
description71–73 of enzyme catalysis. Let us consider the first process in Eq. 12,
assuming that this reaction is catalyzed by enzyme E of the initial concentration
→ ,
. One of the
substrates, e.g., , will form an intermediate complex, I, with . This complex will react with the
second substrate, here , to produce the product, , and restore the enzyme,
,
(17)
.
Typically, the first reaction is reversible, but here the concentration of substrate
large, and the process of back-conversion of
into
and
is typically
can be ignored. We note that using
large initial concentrations of the gate-machinery chemicals C and D enables control of the
process rates,
and
, see Eq. 12, in the way that affects the gate-response function, and
also prevents loss of signal intensity. Indeed, in the steady state, when the enzyme is bound in the
intermediate complex, the rate
(Eq. 12) can be approximately estimated28 as
– 13 –
.
(18)
Note that in terms of the “logic variables,” all the constants cancel out. Therefore, the gate
response, to the extent that it is affected by this reaction can be only controlled by the initial
concentration of C (in the denominator).
For realization of the “ideal” XOR response, such as illustrated in Fig. 4, the reaction
is
rates have to be very fast, and all the processes have to be carried out to completion, i.e.,
large. Another way to approach our “ideal” XOR gate is to let the processes in Eq. 9 run for a
long enough time, then add chemicals
and
, or the corresponding enzymes, which would
initiate the processes in Eq. 12. The latter can be done in a fluidic system, with enzymes
immobilized at different parts of the flow. In reality the chemical reaction rates are finite and
depend on the chemical and physical conditions, etc. Therefore, after a finite time the reactions
in Eq. 9 will still be ongoing. However, at the time when the “second step” chemicals are added,
the former processes can be ignored, considering only conversion of the unreacted
and
into
the signal, , Eq. 12. In this case, with some additional assumptions, it is possible to obtain an
approximate analytical solution for the output signal, . This is presented in the Appendix.
Rate equations corresponding to Eq. 9 and 12, are
,
,
,
,
(19)
,
,
,
,
– 14 –
.
The system of Eq. 19 can be integrated numerically using finite difference methods. An example
of such a solution is shown in Fig. 5. The parameters for this calculation were a representative
set, summarized in Table 1, selected based on various expectations described earlier. Figure 5
illustrates the XOR gate response which is a rounded version of the “ideal” response, Fig. 4, and
also resembles the schematic drawn in Fig. 2(b).
The realization of high quality XOR gate described above is not unique. One possible
simplification could be to use the chemicals
and
to perform filtering at large signal
intensities, avoiding the last reaction in Eq. 12, i.e., using P instead of S as the output. However,
as mentioned earlier, for enzymatic realizations it is preferable to reserve C and D for controlling
the gate response and making these two gate-machinery chemicals have large initial
concentrations, whereas for filtering their initial concentrations would have to be smaller than .
Figure 5. XOR gate response function obtained by numerically solving Eq. 19, illustrated
for a and f parameter values chosen the same as in Fig. 4. These and the other
parameters are listed in Table 1.
– 15 –
Table 1. List of the representative model parameters used for the
numerical calculation yielding Fig. 5, and their description.
Parameter
Value
Description
Maximum value of the input, chosen typical
1mM
for
such
systems
when
realized
27,29,35,38
experimentally.
The fraction of the extra amount Δ
≡Δ ⁄
in the
counterpart of the input needed for the other
0.25
input cancellation at low input values.
⁄
≡
The fraction of the filtering chemical F as
0.3964
compared to the maximum input value.
Rate constants of the input cancellation, larger
,
0.167mM s
compared to the other rates discussed here.
The initial values of C and D. Only the
,
10mM
products
long as
,
,
≫
, shown below, matter as
.
Rate constants of conversion of the input
,
0.0017 s
chemicals into an intermediate chemical, P.
Rate constant of the conversion of the
0.033mM s
chemical P into the output signal S.
Gate time chosen arbitrarily, however this
60 min
choice is typical for existing realizations of
enzymatic XOR and AND gates.27,29,35,38
– 16 –
Figure 6. Plots along the diagonal in the (x,y) plane that connects the points (1,0) and (0,1). (a)
Output signal intensity for varying parameter a (and f), as described in the text. (b) Absolute
values of the gradient, Eq. 20.
The issue of noise amplification and suppression, discussed in the preceding section, for
our high-quality XOR gate realization is related to the overall signal intensity loss, as illustrated
in Fig. 6, where we graphically demonstrate this effect by varying the parameter . Note that the
choice of a for Fig. 5 (same as for Fig. 4), see Table 1, is exaggerated to better show the gateresponse function features: Such a large value of a yields small f (significant loss of signal
– 17 –
intensity). All the other parameters were kept fixed for Fig. 6 calculations, except f, recalculated
according to Eq. 16. Signal intensity loss due to filtering is quantified in Fig. 6(a), along the
diagonal in the (x,y) plane that connects the points (1,0) and (0,1), for convenience drawn as a
function of x – y instead of the actual distance from the center, which is (x – y)/√2. The quantity
shown is the “filtered” signal, S, at
normalized per the maximum possible signal,
, that
would be obtained if we used the idealized “fast-process” gate without any filtering, with
response of the type sketched in Fig. 2(a). The quality of analog noise suppression is measured
by the smallness of the values of the gradient,
,
,
(20)
at and in the vicinity of logic points. Figure 6(b) shows this quantity along the same diagonal.
Actually, one has to make the gradient small in the vicinity of all the logic points. Figure 6 only
illustrates that there is a trade-off of losing signal intensity for suppressing the gradient along the
selected diagonal near logic 0,1 and 1,0.
Generally, the more asymmetry is introduced in the gate response functions beyond that
imposed by the need to pass at the logic point values, the more gradients are built up somewhere,
and therefore there is more noise amplification. As a result, parameter values are usually selected
to have as symmetrical a response as possible. Specifically, see Table 1, we took equal rates
, cf. the discussion in connection with Eq. 18. In our case, the surface shown in Fig. 5
primarily varies as a function of
, and is practically constant as a function of
have such a symmetrical response, we set the parameters (rate constants)
Increase in the value of
(and
and
. To
.
) leads to both smaller gradient and also z-values closer to
0 in the central region around the diagonal
, see Fig. 7(a), which not only improves
noise handling but also decreases the imprecision in the gate function at logic 1,1. The value of
parameter
controls the quality of filtering near logical 0,1 and 1,0, see Fig. 7(b).
Modifications of the parameter
(and
) would provide changes similar to those in Fig. 7,
affecting both of the aforementioned features in addition to the slope of the steep part of the
curves in Fig. 7, though the latter is primarily affected by the choice of the parameter .
– 18 –
Figure 7. Plots along the same diagonal in the (x,y) plane as in Fig. 6. The multiplicative
factors show the changes in the parameter values as compared to those used for Fig. 5
(listed in Table 1). (a) Increasing the parameters
and
leads to decrease in the signal
for near-equal input values. (b) Increasing the value of the parameter
leads to smaller
gradients near high inputs, which improves analog noise filtering. Note that the black
curve for doubled
value is obscured by the red curve for the original
near the corners of the curves at the top of panel (b).
– 19 –
value except
ACKNOWLEDGEMENTS
We thank Dr. B. E. Fratto and Prof. E. Katz for useful discussions and collaboration70 in
developing the ideas reported here.
APPENDIX: ANALYTICAL APPROXIMATION
Here we outline an analytical solution of Eq. 19, for the reactions in Eq. 9 and 12, that are
“clocked” and separated into several stages. Let us consider a process
→
where
⋯,
and
(A1)
react, with rate constant
species. The initial, at time
,
, to produce chemical
and various inactive
, concentrations of these three chemicals are
,
,
,
and
, respectively. The rate equations describing the process in Eq. A1 are
,
(A2)
.
(A3)
The analytical solution of Eq. A2 can be found in closed form
,
,
,
,
,
,
,
,
(A4)
,
,
,
,
,
,
,
,
– 20 –
from which the product chemical
,
can be calculated from the relation
(A5)
,
or more symmetrical expressions. For the case when
,
,
the solution reduces to
.
(A6)
,
We will now assume that the processes depicted in Fig. 3 are either separately clocked or
their rates are such that we can approximately consider them proceeding nearly to completion in
stages as follows. Stage 1 corresponds to the processes color-coded with black arrows in Fig. 3,
ongoing from time 0 until
, with all other processes disabled during this time interval. Stage 2
then corresponds to the time interval from
to
with the processes color-coded with red
arrows ongoing. Finally, Stage 3 is the filtering step, corresponding to the purple-colored arrows,
ongoing during the time interval from
to the “gate time”
.
For Stage 1 there will be two sets of reactions of the type of Eq. A2,
,
,
with
0
,
′ 0
∆ ;
(A7)
,
,
with
0
,
0
Their solution according to Eq. A4 will provide the initial conditions for
∆ .
and
for the
two Stage 2 reactions,
,
;
(A8)
,
.
– 21 –
These produce the same product,
,
0.
with
(A9)
By adding up the two reaction outputs, each obtained similarly to Eq. A5, we can calculate the
initial condition for
for the Stage 3 reaction,
,
,
The output signal at time
,
with
.
(A10)
is then calculated according to
with
0,
(A11)
and is obtained as in Eq. A5.
We note that while writing an explicit expression for the overall result of all these steps is
possible, it is unilluminating and redundant, because the three steps can be programmed
separately, for the product(s) of each to be calculated in sequence by using the expressions
suggested by Eq. A5, until the final product
is obtained. This can simplify the
computational effort for initial modeling estimates and data fitting, before carrying out numerical
solutions of the full set of Eq. 19.
REFERENCES
[1]
A. P. de Silva, Nature, 2008, 454, 417–418.
[2]
A. P. de Silva and S. Uchiyama, Nat Nano, 2007, 2, 399–410.
[3]
A. P. de Silva, Molecular Logic-based Computation, Royal Society of Chemistry,
Cambridge, UK, 2012.
[4]
Y. Jia, R. Duan, F. Hong, B. Wang, N. Liu and F. Xia, Soft Matter, 2013, 9, 6571–6577.
– 22 –
[5]
E. Katz, Isr. J. Chem., 2011, 51, 132–140.
[6]
J. Wang and E. Katz, Anal. Bioanal. Chem., 2010, 398, 1591–1603.
[7]
G. Strack, M. Ornatska, M. Pita and E. Katz, J. Am. Chem. Soc., 2008, 130, 4234–4235.
[8]
M. L. Simpson, G. S. Sayler, J. T. Fleming and B. Applegate, Trends Biotechnol., 2001, 19,
317–323.
[9]
M. Krämer, M. Pita, J. Zhou, M. Ornatska, A. Poghossian, M. J. Schöning and E. Katz, J.
Phys. Chem. C, 2009, 113, 2573–2579.
[10] Y. Benenson, Nat. Rev. Genet., 2012, 13, 455–468.
[11] M. N. Stojanovic, D. Stefanovic and S. Rudchenko, Acc. Chem. Res., 2014, 47, 1845–1852.
[12] M. N. Stojanovic and D. Stefanovic, J Comput Theor Nanosci, 2011, 8, 434–440.
[13] E. Katz and V. Privman, Chem. Soc. Rev., 2010, 39, 1835–1857.
[14] U. Pischel, J. Andréasson, D. Gust and V. F. Pais, ChemPhysChem, 2013, 14, 28–46.
[15] Y. Liu, E. Kim, I. M. White, W. E. Bentley and G. F. Payne, Bioelectrochemistry, 2014,
98, 94–102.
[16] Hua Jiang, M. D. Riedel and K. K. Parhi, Des. Test Comput. IEEE, 2012, 29, 21–31.
[17] P. Hillenbrand, G. Fritz and U. Gerland, PLoS ONE, 2013, 8, e68345.
[18] C. G. Bowsher, J. R. Soc. Interface, 2011, 8, 186–200.
[19] V. Privman, Isr. J. Chem., 2011, 51, 118–131.
[20] V. Privman, in Molecular and Supramolecular Information Processing, Wiley-VCH
Verlag GmbH & Co. KGaA, Weinheim, Germany, 2012, pp. 281–303.
[21] J. Halámek, V. Bocharova, S. Chinnapareddy, J. R. Windmiller, G. Strack, M.-C. Chuang,
J. Zhou, P. Santhosh, G. V. Ramirez, M. A. Arugula, J. Wang and E. Katz, Mol. Biosyst.,
2010, 6, 2554–2560.
[22] E. Katz, Biomolecular Information Processing. From Logic Systems to Smart Sensors and
Actuators, Wiley-VCH, Weinheim, Germany, 2012.
[23] E. Katz, Molecular and Supramolecular Information Processing: From Molecular
Switches to Logic Systems, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany,
2012.
[24] S. Domanskyi and V. Privman, J. Phys. Chem. B, 2012, 116, 13690–13695.
– 23 –
[25] C. S. Calude, J. F. Costa, N. Dershowitz, E. Freire and G. Rozenberg, Unconventional
Computation. Lecture Notes in Computer Science, Springer, Berlin, Germany, 2009, vol.
5715.
[26] A. Adamatzky, B. De Lacy Costello, L. Bull, S. Stepney and C. Teuscher, Unconventional
Computing, Luniver Press, UK, 2007.
[27] V. Privman, M. A. Arugula, J. Halámek, M. Pita and E. Katz, J. Phys. Chem. B, 2009, 113,
5301–5310.
[28] V. Privman, O. Zavalov, L. Halámková, F. Moseley, J. Halámek and E. Katz, J. Phys.
Chem. B, 2013, 117, 14928–14939.
[29] O. Zavalov, V. Bocharova, V. Privman and E. Katz, J. Phys. Chem. B, 2012, 116, 9683–
9689.
[30] O. Zavalov, V. Bocharova, J. Halámek, L. Halámková, S. Korkmaz, M. A. Arugula, S.
Chinnapareddy, E. Katz and V. Privman, Int J Unconv Comput, 2012, 8, 347–365.
[31] S. Bakshi, O. Zavalov, J. Halámek, V. Privman and E. Katz, J. Phys. Chem. B, 2013, 117,
9857–9865.
[32] V. Privman, B. E. Fratto, O. Zavalov, J. Halámek and E. Katz, J. Phys. Chem. B, 2013,
117, 7559–7568.
[33] A. Vallée-Bélisle, F. Ricci and K. W. Plaxco, J. Am. Chem. Soc., 2012, 134, 2876–2879.
[34] V. Privman, J. Halámek, M. A. Arugula, D. Melnikov, V. Bocharova and E. Katz, J. Phys.
Chem. B, 2010, 114, 14103–14109.
[35] J. Halámek, V. Bocharova, M. A. Arugula, G. Strack, V. Privman and E. Katz, J. Phys.
Chem. B, 2011, 115, 9838–9845.
[36] V. Privman, J. Zhou, J. Halámek and E. Katz, J. Phys. Chem. B, 2010, 114, 13601–13608.
[37] V. Bocharova, O. Zavalov, K. MacVittie, M. A. Arugula, N. V. Guz, M. E. Dokukin, J.
Halámek, I. Sokolov, V. Privman and E. Katz, J. Mater. Chem., 2012, 22, 19709–19717.
[38] D. Melnikov, G. Strack, J. Zhou, J. R. Windmiller, J. Halámek, V. Bocharova, M.-C.
Chuang, P. Santhosh, V. Privman, J. Wang and E. Katz, J. Phys. Chem. B, 2010, 114,
12166–12174.
[39] J. Halámek, O. Zavalov, L. Halámková, S. Korkmaz, V. Privman and E. Katz, J. Phys.
Chem. B, 2012, 116, 4457–4464.
– 24 –
[40] Z. Li, M. A. Rosenbaum, A. Venkataraman, T. K. Tam, E. Katz and L. T. Angenent, Chem.
Commun., 2011, 47, 3060–3062.
[41] V. Privman, J Comput Theor Nanosci, 2011, 8, 490–502.
[42] K. MacVittie, J. Halámek, V. Privman and E. Katz, Chem. Commun., 2013, 49, 6962–
6964.
[43] X. Chen, J. Zhu, R. Tian and C. Yao, Sens. Actuators B Chem., 2012, 163, 272–280.
[44] F. Tian and G. Zhu, Anal. Chim. Acta, 2002, 451, 251–258.
[45] L. Zhu, R. Yang, J. Zhai and C. Tian, Biosens. Bioelectron., 2007, 23, 528–535.
[46] S. Mailloux, O. Zavalov, N. Guz, E. Katz and V. Bocharova, Biomater. Sci., 2014, 2, 184–
191.
[47] S. P. Rafael, A. Vallée-Bélisle, E. Fabregas, K. Plaxco, G. Palleschi and F. Ricci, Anal.
Chem., 2012, 84, 1076–1082.
[48] M. Pita, V. Privman, M. A. Arugula, D. Melnikov, V. Bocharova and E. Katz, Phys. Chem.
Chem. Phys., 2011, 13, 4507–4513.
[49] V. Privman, S. Domanskyi, S. Mailloux, Y. Holade and E. Katz, J. Phys. Chem. B, 2014,
118, 12435–12443.
[50] A. Credi, V. Balzani, S. J. Langford and J. F. Stoddart, J. Am. Chem. Soc., 1997, 119,
2679–2681.
[51] D. C. Magri, G. J. Brown, G. D. McClean and A. P. de Silva, J. Am. Chem. Soc., 2006,
128, 4950–4951.
[52] D. C. Magri and A. P. de Silva, New J. Chem., 2010, 34, 476–481.
[53] R. Pei, E. Matamoros, M. Liu, D. Stefanovic and M. N. Stojanovic, Nat. Nanotechnol.,
2010, 5, 773–777.
[54] V. Privman, Nat. Nanotechnol., 2010, 5, 767–768.
[55] S. Mailloux, Y. V. Gerasimova, N. Guz, D. M. Kolpashchikov and E. Katz, Angew. Chem.
Int. Ed., 2015, 54, 6562–6566.
[56] N. Guz, T. A. Fedotova, B. E. Fratto, O. Schlesinger, L. Alfonta, D. M. Kolpashchikov and
E. Katz, ChemPhysChem, 2016, 17, 2247–2255.
[57] E. Katz, J. Wang, M. Privman and J. Halámek, Anal. Chem., 2012, 84, 5463–5469.
[58] J. Zhou, J. Halámek, V. Bocharova, J. Wang and E. Katz, Talanta, 2011, 83, 955–959.
[59] M. W. Toepke, V. V. Abhyankar and D. J. Beebe, Lab Chip, 2007, 7, 1449–1453.
– 25 –
[60] B. E. Fratto and E. Katz, ChemPhysChem, 2015, 16, 1405–1415.
[61] B. E. Fratto, L. J. Roby, N. Guz and E. Katz, Chem. Commun., 2014, 50, 12043–12046.
[62] A. Verma, B. E. Fratto, V. Privman and E. Katz, Sensors, 2016, 16, 1042.
[63] B. E. Fratto, J. M. Lewer and E. Katz, ChemPhysChem, 2016, 17, 2210–2217.
[64] B. E. Fratto and E. Katz, ChemPhysChem, 2016, 17, 1046–1053.
[65] B. E. Fratto, N. Guz and E. Katz, Parallel Process. Lett., 2015, 25, 1540001.
[66] F. Moseley, J. Halámek, F. Kramer, A. Poghossian, M. J. Schöning and E. Katz, Analyst,
2014, 139, 1839–1842.
[67] L. Fedichkin, E. Katz and V. Privman, J Comp Theor Nanosci, 2008, 5, 36–43.
[68] M. Privman, T. K. Tam, M. Pita and E. Katz, J. Am. Chem. Soc., 2009, 131, 1314–1321.
[69] M. Pita, M. Privman and E. Katz, Top. Catal., 2012, 55, 1201–1216.
[70] B. E. Fratto, M. Wood, V. Privman and E. Katz, unpublished, 2016.
[71] D. L. Nelson and M. M. Cox, Lehninger Principles of Biochemistry, W.H. Freeman, New
York, USA, 6th edition., 2012.
[72] D. L. Purich, in Enzyme Kinetics: Catalysis & Control, Elsevier, Boston, USA, 2010, pp.
287–334.
[73] A. R. Schulz, Enzyme Kinetics: From Diastase to Multi-enzyme Systems, Cambridge
University Press, Cambridge, New York, USA, 1st edition, 1995.
– 26 –
TABLE OF CONTENTS GRAPHICAL ABSTRACT AND CAPTION
Chemical XOR gate design that realizes gateresponse function with filtering properties is
described. The gate-response function is flat
(has small gradients) at and in the vicinity of
all the four binary-input logic points, resulting
in
analog
noise
suppression.
The
gate
functioning involves cross-reaction of the
inputs represented by pairs of chemicals to
produce a practically zero output when both
are present and nearly equal. The remaining
inputs that were not reacted away, are
processed to produce the output XOR signal.
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