Compiler-aided systematic construction of large-scale DNA strand displacement circuits using unpurified components

ARTICLE

Received 7 Aug 2016

Accepted 21 Dec 2016

Published 23 Feb 2017

Compiler-aided systematic construction

of large-scale DNA strand displacement circuits

using unpurified components

Anupama J. Thubagere

, Chris Thachuk

, Joseph Berleant

, Robert F. Johnson

, Diana A. Ardelean

Kevin M. Cherry

& Lulu Qian

1,2

Biochemical circuits made of rationally designed DNA molecules are proofs of concept for

embedding control within complex molecular environments. They hold promise for trans-

forming the current technologies in chemistry, biology, medicine and material science by

introducing programmable and responsive behaviour to diverse molecular systems. As the

transformative power of a technology depends on its accessibility, two main challenges are an

automated design process and simple experimental procedures. Here we demonstrate the

use of circuit design software, combined with the use of unpurified strands and simplified

experimental procedures, for creating a complex DNA strand displacement circuit that

consists of 78 distinct species. We develop a systematic procedure for overcoming the

challenges involved in using unpurified DNA strands. We also develop a model that takes

synthesis errors into consideration and semi-quantitatively reproduces the experimental data.

Our methods now enable even novice researchers to successfully design and construct

complex DNA strand displacement circuits.

DOI: 10.1038/ncomms14373

OPEN

Bioengineering, California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125, USA.

Computer Science, California Institute of

Technology, 1200 East California Boulevard, Pasadena, California 91125, USA.

Applied and Computational Mathematics, California Institute of Technology, 1200

East California Boulevard, Pasadena, California 91125, USA. Correspondence and requests for materials should be addressed to L.Q. (email: luluqia

n@caltech.edu).

NATURE COMMUNICATIONS

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1

T

he success of computer engineering has inspired attempts

to use hierarchical and systematic approaches for devel-

oping molecular devices with increasing complexity. To

enable the design and construction of a wide range of functional

molecular systems, we need software tools such as a compiler that

can automatically translate high-level functions to low-level

molecular implementations and provide models and simulations

for predicting and debugging the behaviours of designed

molecular systems. The mechanism of DNA strand displacement

has been used to create a variety of synthetic molecular systems

including circuits, motors and triggered assembly of structures

Software tools have been developed for designing and analysing

DNA strand displacement systems, capable of generating nucleic

acid sequences from well-defined structures and molecular

interactions

2,3

, calculating the thermodynamic

2,4,5

and kinetic

properties of designed molecules, and evaluating if the behaviours

of the molecular systems agree with the higher-level designs

3,7–11

There also exist a few molecular compilers that can translate

abstract functions such as a logic function to DNA strand

displacement implementations without requiring an under-

standing of the molecular level details

12,13

. However, there

has been little independent experimental validation of these

compilers, most of which were developed in parallel with or after

experimental findings

12,14

In addition to software tools that facilitate automated design

and analysis of DNA strand displacement circuits, we also need to

simplify the experimental procedures for creating these circuits

in vitro

, so that it is possible for researchers with diverse

backgrounds to build their own circuits and explore potential

applications. A great inspiration is DNA origami

, a technique

that folds DNA into sophisticated structures. In just 10 years

since its birth, DNA origami has become one of the most

significant successes in the field of DNA nanotechnology.

Over 170 research groups have contributed to advancing this

technique or developing it for applications in a variety of

research areas

16–19

. A fundamental reason why DNA origami

was able to quickly spread around the world is that the

experimental procedure is extremely simple and makes use of

cheap, unpurified nucleic-acid strands. In contrast, other

than a few very simple circuits with just one or two double-

stranded components

, most DNA strand displacement circuits

were constructed using strands that were purchased either

purified or unpurified, but all followed by in-house

polyacrylamide gel electrophoresis (PAGE) purification to

reduce undesired products due to synthesis errors and

stoichiometry errors

12,14,21

. Purified strands are approximately

ten times more expensive than unpurified strands, which

significantly increases the cost for building large-scale DNA

circuits. In-house PAGE purification is both time consuming and

labour intensive.

In this work, we show that one can successfully build

a complex DNA strand displacement circuit, using DNA

sequences automatically generated from a molecular compiler.

We also show that one can even do so using cheap, unpurified

DNA strands, following simple and systematic experimental

procedures.

Results

Circuit design

. A simple DNA strand displacement motif called

the seesaw gate was developed to scale up the complexity of

DNA circuits

and was used to demonstrate digital logic

computation

and neural network computation

. The Seesaw

Compiler

12,24

was developed to automatically translate an

arbitrary feed forward digital logic circuit into its equivalent

seesaw DNA circuit (Fig. 1). The compiler takes an input file that

describes a logic circuit with a list of input and output terminals,

and a list of AND, OR, NOT, NAND and NOR gates with the

connectivity of their terminals specified. First, a technique called

dual-rail logic is applied to translate the original logic circuit into

an equivalent circuit that contains AND and OR gates only

This is because the NOT gate cannot be directly implemented in

multi-layer use-once DNA circuits, if the OFF and ON state of a

signal is represented by low and high concentration of a single

DNA strand, respectively. If a NOT gate were implemented this

way, then output molecules of the gate could be immediately

produced in the absence of input. However, once this reaction

reaches equilibrium it cannot be reversed, even if input mole-

cules are added at a later point. With dual-rail logic, each terminal

in the original circuit is replaced by two terminals, representing

the OFF and ON states of a signal separately (for example,

each input signal

is replaced by

and

). Thus, no reaction

will take place until signal molecules on one of the two wires

have arrived. With this representation, the NOT gate

can be implemented by exchanging the two wires of an

input and output signal. Each AND, OR, NAND and NOR

gate in the original circuit is replaced by a pair of AND and OR

gates.

Next, the compiler translates the dual-rail logic circuit into an

equivalent seesaw DNA circuit. In a seesaw DNA circuit, each

signal is defined as a wire

connecting seesaw nodes

and

and implemented using a single-stranded DNA molecule.

Each AND and OR gate in the dual-rail circuit is replaced by a

seesaw AND and OR gate, respectively, which is defined as a

pair of integrating and amplifying seesaw nodes connected with a

set of input and output wires

. The seesaw nodes are

composed of double-stranded threshold and gate:output

molecules and single-stranded fuel molecules (Fig. 1, bottom

right). We will explain how the seesaw logic gates work in the

next section. Input fan-out gates are introduced to take an input

signal that is used for multiple logic gates and produce the

corresponding number of output signals. Reporters are

introduced to take each output signal and generate a distinct

fluorescence signal for readout.

Finally, the compiler generates Visual DSD

3,26

code and

Mathematica code for simulating and analysing the seesaw DNA

circuit and a file that contains DNA sequences for all molecular

species in the circuit. The Visual DSD code can be used to

automatically produce diagrams of species, reactions and network

graphs with domain-level representation of DNA and to simulate

the circuit behaviour based on the network of chemical reactions.

The Mathematica code provides more customized and efficient

simulations of seesaw circuits. The simulation uses the

CRNSimulator package

and models a specific set of side

reactions in addition to the designed reactions in a seesaw

network

As a demonstration of using the Seesaw Compiler, we

designed a single DNA strand displacement circuit that

implements two distinct elementary cellular automata

transition functions. An elementary cellular automaton (CA) is

one of the simplest models of computation

. It consists of a

one-dimensional grid of cells, collectively called a generation,

where each cell has a binary state of 0 or 1. In each subsequent

generation, the state for a cell

is determined by its current

state and those of its left neighbour

and right neighbour

. A state transition rule maps each of the 2

8 possible

combinations of states for

and

to either 0 or 1.

Thus, a length 8 binary string uniquely identifies one of the 2

possible transition functions that specify how an elementary

CA will evolve between generations. The rule 110 elementary CA

(binary number 01101110 written in decimal) is famously known

to be Turing universal

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Another rule that is equally powerful is rule 124 (binary

number 01111100 written in decimal), generated by applying the

following mirror transformation: the new state of the centre cell

for

LCR

zyx

in rule 124 is the same as the new state for

LCR

xyz

in rule 110. Our circuit was designed to compute a

combined logic function of the two transition rules (Fig. 2a). It

consists of five logic gates in two layers, including a three-input

two-output NAND gate. It is noteworthy that we designed the

circuit to demonstrate an interesting logic function associated

with cellular automata and not to implement the actual cellular

automata model. The circuit operates in a well-mixed test tube

environment that does not involve spatial dynamics (that is, no

geometry of cells).

The DNA circuit generated by the Seesaw Compiler consisted

of 6 layers and a total of 78 distinct initial DNA species (Fig. 2b

and Supplementary Fig. 1). Mathematica simulations of the DNA

INPUT(1)

# x1

INPUT(2)

# x2

INPUT(3)

# x3

INPUT(4)

# x4

OUTPUT(11)

# y1

OUTPUT(8)

# y2

5 = NOR(1, 2)

6 = NOT(4)

7 = AND(3, 6)

8 = OR(3, 4)

9 = NAND(5, 3, 4)

10 = OR(5, 7)

11 = NAND(9, 10)

Feedforward logic circuit

INPUT(2)

# x1^0

INPUT(3)

# x1^1

OUTPUT(22)

# y1^0

OUTPUT(23)

# y1^1

10 = OR(3, 5)

11 = AND(2, 4)

14 = OR(6, 9)

15 = AND(7, 8)

16 = AND(6, 8)

17 = OR(7, 9)

18 = AND(11, 7, 9)

19 = OR(10, 6, 8)

INPUT(2) = w[5,22]

# x1^0

INPUT(3) = w[7,20]

# x1^1

OUTPUT(22) = Fluor[52] # y1^0

OUTPUT(23) = Fluor[54] # y1^1

inputfanout[13,12,{28,32,38}]

inputfanout[15,14,{30,34,36}]

seesawOR[20,21,{7,11},{38,40}]

seesawAND[22,23,{5,9},{36,42}]

seesawOR[28,29,{13,19},{40}]

seesawAND[30,31,{15,17},{42}]

Reporter[52,45]

Reporter[54,47]

Seesaw DNA circuit

directive plot <_ _ _ Fluor52> (* y1^0 *)

directive plot <_ _ _ Fluor54> (* y1^1 *)

def normal = 0.0003 (* normal toehold binding rate constant nM^-1 s^-1*)

def slow = 0.000015 (* slow toehold binding rate constant nM^-1 s^-1*)

(* a seesaw signal *)

def signal(N,iL,i,iR,jL,j,jR) = ( N * <iL^ iiR^ T^ jL^ j jR^>

)

(* 2-input 2-output seesaw OR gate *)

def seesawOR2I2O(i1L,i1,i1R,i2L,i2,i2R,k1L,k1,k1R,k2L,k2,k2R)

(gateL(20*N,i1L,i1,i1R,i2L,i2,i2R)

| thresholdL(6*N,i1R,i2L,i2,i2R)

| gateL(10*N,i2L,i2,i2R,k1L,k1,k1R)

| gateL(10*N,i2L,i2,i2R,k2L,k2,k2R)

| signal(40*N,i2L,i2,i2R,fL,f,fR))

( signal(ON,S5L,S5,S5R,S22L,S22,S22R) (* x1^0 *)

| signal(OFF,S7L,S7,S7R,S20L,S20,S20R) (* x1^1 *)

| seesawOR2I2O(S20L,S20,S20R,S21L,S21,S21R,S38L,S38,S38R,S40L,S40,S40R)

| seesawAND2I2O(S22L,S22,S22R,S23L,S23,S23R,S36L,S36,S36R,S42L,S42,S42R)

Visual DSD code

(* Rate constants: *)

kf = 2*10^6; (* fast strand displacement rate, unit: M^ –1 s^ –1 *)

ks = 5*10^4; (* slow strand displacement rate, unit: M^ –1 s^ –1 *)

(* Translates a seesaw gate into a list of reactions: *)

seesaw[x_,l_List,r_List]:={

(* Toehold exchange reactions *)

Outer[revrxn[w[#1,x]+g[x,w[x,#2]],g[w[#1,x],x]+w[x,#2],ks,ks]&,l,r],

(* Translates logic OR operation into a list of seesaw gates *)

seesawOR[x1_,x2_,l_List,r_List]:=Module[{f},

{seesaw[x1,l,{x2}],

(* Simulation *)

SIMcircuit=Table[gatesys={

seesawOR[20,21,{7,11},{38,40}],

seesawAND[22,23,{5,9},{36,42}],

(* Plot *)

Plot[Evaluate[SIMcircuit],{t,0,time},

Mathematica code

x1^0: w5,22 = S22 T S5

x1^1: w7,20 = S20 T S7

Th12,13:13-t = S13

Th12,13:13-b = s12* T* S13*

w13,28

= S28 T S13

w13,32

= S32 T S13

G13-b

= T* S13* T*

w13,f

= Sf T S13

Rep48-t

= RQ S48

Rep48-b

= T* S48* ATTO590

DNA sequences

Visual DSD simulation

Mathematica simulation

Gate:Output

(

5:5,6

)

Fuel (

5,f

)

Reporter (

Rep

)

Input (

53,5

)

Threshold (

53,5:5

)

S41L

S41

S41R

S41L*

S41*

S41R*

S40R*

S41L

S41

S41R

S40R

S40

S40L

↓

S41L

S41

S41R

S41L

S41

S41R

S40R

S40

S40L

S41L*

S41*

S41R*

S40R*

S41L

S41

S41R

S42L

S42 S42R

S41L*

S41*

S41R*

S41L

S41

S41R

S40R

S40

S40L

S41L

S41

S41R

S42L

S42

S42R

S41L*

S41*

S41R*

S41L

S41

S41R

S40R

S40

S40L

Dual-rail logic circuit

1.0

= 1001

0.8

0.6

0.4

0.2

0.0

Time (hours)

Output

s53

S53

–0.5

–1.5

S5 =

S7 =

S9 =

S5* =

S7* =

S9* =

Figure 1 | Automated circuit design steps using the Seesaw Compiler.

A feedforward digital logic circuit is first translated into an equivalent dual-rail logic

circuit and then translated into an equivalent seesaw DNA circuit. Visual DSD code and Mathematica code are generated for analysing and simulating th

seesaw DNA circuit, and DNA sequences are generated for constructing the circuit. Bottom right diagram introduces the notations of seesaw circuits:

black

numbers indicate identities of nodes. The locations and values of red numbers indicate the identities of distinct DNA species and their relative init

ial

concentrations, respectively.

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circuit predicted correct computation for all 8 possible input

combinations under ideal experimental conditions (Fig. 2c).

The next step was to construct the DNA circuit using

strands that were purchased unpurified and with no additional

in-house purification. We expected that the main challenges

would be to understand how synthesis errors and stoichiometry

errors affect the behaviours of DNA circuits and to

explore solutions that restore the desired circuit behaviour.

We took a bottom-up approach and began building the DNA

circuit from the simplest functi

onal component—digital signal

restoration.

Calibrating effective concentrations

. Digital signal restoration is

a process that pushes the intrinsically analog signal towards either

the ideal ON or OFF state, therefore cleaning up the noise and

compensating for the signal decay that occurs during circuit

execution. In seesaw circuits, digital signal restoration is a

component of every logic gate, and is implemented by an

amplifying seesaw node with the following idealized input-output

function:

At the molecular level, the digital signal restoration process

consists of two basic reactions: catalysis and thresholding.

Catalysis is implemented with two toehold exchange pathways

that release free output strands

from double-stranded gate

molecules

, using the input strands

as a catalyst (Supple-

mentary Fig. 2a):

;

:

;

LCR

= 100

LCR

= 001

LCR

= 101

LCR

= 110

LCR

= 111

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

LCR

= 000

Rule 110

Rule 124

LCR

= 010

LCR

= 011

L C R

124

110

124

110

1.0

0.8

0.6

0.4

0.2

0.0

0246810

Time (h)

Output

1.0

0.8

0.6

0.4

0.2

0.0

0246810

Time (h)

Output

1.0

0.8

0.6

0.4

0.2

0.0

0246810

Time (h)

Output

1.0

0.8

0.6

0.4

0.2

0.0

0246810

Time (h)

Output

1.0

0.8

0.6

0.4

0.2

0.0

0246810

Time (h)

Output

1.0

0.8

0.6

0.4

0.2

0.0

0246810

Time (h)

Output

1.0

0.8

0.6

0.4

0.2

0.0

0246810

Time (h)

Output

1.0

0.8

0.6

0.4

0.2

0.0

0246810

Time (h)

Output

124

110

Figure 2 | Design of a rule 110–124 circuit using the Seesaw Compiler.

(

) Gate diagram and truth table of a digital logic circuit that computes the

transition rules 110 and 124 of elementary cellular automata. (

) Seesaw gate diagram of the equivalent DNA strand displacement circuit. Each seesaw

node connected to a dual-rail input implements input fan-out. Each pair of seesaw nodes labelled

and

implements a dual-rail AND and OR gate,

respectively. Each pair of dual-rail AND and OR gates implements an AND, OR or NAND gate in the original logic circuit. Each dual-rail output is convert

to a fluorescence signal through a reporter, indicated as a half node with a zigzag arrow. Each circle and dot inside a seesaw node indicates a double-

stranded threshold and gate molecule, respectively. Each dot on a wire indicates a single-stranded fuel molecule. (

) Simulations of the DNA strand

displacement circuit using the previously developed model for purified seesaw circuits. Trajectories and their corresponding outputs have matchin

g colours.

Overlapping trajectories were shifted to be visible. Dotted and solid lines indicate dual-rail outputs that represent logic OFF and ON, respectivel

y. For

example, when input

LCR

001, meaning

and

were introduced at a high concentration and

and

at a low concentration, two output

trajectories

124

and

110

reached an ON state and the other two output trajectories

124

and

110

remained in an OFF state, indicating that the output

was computed to be 0 and 1 for rule 124 and 110, respectively. Simulations were performed at 1

50 nM—the compiler recommended standard

concentration for large-scale purified seesaw circuits.

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Catalysis can be used for signal amplification, since a small

amount of input can trigger the release of a much larger amount

of output.

Thresholding is implemented with double-stranded threshold

molecules

consuming the input at a much faster rate

(

) than the input acting as a catalyst (Supplementary

Fig. 2b):

;

:

+

Asshowninsimulationsge

nerated using the Seesaw

Compiler (Fig. 3a), when the concentration of the

threshold molecule is 0.5

(where 1

is a standard

concentration of 100 nM), we expect that input less

than the threshold (for example, 0.3

)shouldbecleaned

up to an ideal OFF state via reaction 3 and input greater

than the threshold (for example, 0.7

) should be amplified

to an ideal ON state via reaction 2. However, the

observed circuit behaviour

was different: when input

0.7

the output signal was higher than an ideal OFF state, but

did not reach an ideal ON state (

Fig. 3b). This experimental

result suggested that the input did not sufficiently exceed

the threshold, which was an indication that the effective

concentration of an unpurified

threshold species, compared

with that of an unpurified signal species, was higher than

expected.

The nominal concentration of a DNA species can be

measured using ultraviolet absorbance, but it can be higher

than the effective concentration, which is the concentration of

the DNA species actually perform

ing the desired reactions. If

the sequences of the DNA strands are properly designed, the

difference between nominal concentration and effective con-

centration is typically caused by synthesis errors including

nucleotide insertion, deletion and mismatch. To calibrate the

effective concentrations of unpurified DNA molecules, we

defined the following ratio between effective (eff) and nominal

(nom) concentrations of an arbit

rary signal, threshold and gate

species:

;

eff

;

nom

;

:

eff

;

:

nom

;

:

;

eff

:

;

nom

The effective to nominal concentration of a DNA species

cannot be measured in isolation. More importantly, the absolute

values of

and

should only affect the speed but not the

correctness of computation, if the values remain comparable to

each other. Thus, we chose to estimate the ratio between

and

for a threshold consuming a signal, by comparing simulation

with experimental result of a signal restoration circuit. For

example, manipulating the threshold value in simulation (sim)

identified that

;

:

sim

agreed with the experimental

data (Fig. 3c), which means the effective concentration of the

threshold was similar to that of the signal for

;

:

nom

and

;

nom

. Thus, the threshold to signal ratio can be

calculated as:

;

:

eff

;

:

nom

;

nom

;

eff

;

nom

;

:

nom

;

:

eff

;

eff

A possible explanation for an unpurified threshold having a

higher effective concentration than an unpurified signal, when the

nominal concentrations are the same, is the following: the

synthesis errors of an unpurified strand depend on the length of

the strand, because in the process of chemical synthesis each

nucleotide is attached to a growing chain of oligonucleotide

one at a time and the coupling efficiency of each step is less than

Input

53,5

–1.5

AND

Input

0.3

0.7

Input

0.3

0.7

Input

0.3

0.7

Output

1.0

0.8

0.6

0.4

0.2

0.0

Output

1.0

0.8

0.6

0.4

0.2

0.0

Output

1.0

0.8

0.6

0.4

0.2

0.0

Output

1.0

0.8

0.6

0.4

0.2

0.0

Output

1.0

0.8

0.6

0.4

0.2

0.0

Output

1.0

0.8

0.6

0.4

0.2

0.0

Time (h)

0246810

Time (h)

0246810

Time (h)

0246810

Time (h)

0246810

Time (h)

0246810

Time (h)

02468

[

53,5:5

]

nom

= 0.85 ×

[

10,1:1

]

nom

= 0.35 ×

–0.6

–1.5

–1.2

–1.5

[

1:1,23

]

tri

= 0.8 ×

[

53,5:5

]

sim

= 0.5 ×

[

53,5:5

]

nom

= 0.5 ×

[

53,5:5

]

sim

= 0.7 ×

[

53,5:5

]

nom

= 0.5 ×

–0.5

–1.5

Figure 3 | Calibrating effective concentrations.

(

) Simulations and (

) experimental data of digital signal restoration. (

) Estimating effective threshold

concentration by fitting simulations to the data obtained. (

) OR and AND logic gates constructed using adjusted nominal threshold concentrations.

(

) Estimating effective gate concentration. Data show steady-state fluorescence level. 1

100 nM. Here and in later figures, all output signals in the

data were normalized using the minimum fluorescence signal (the first data point) of an OFF trajectory as 0 and the maximum fluorescence signal

(the average of the last five data points) of an ON trajectory as 1.

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100% (ref. 30). Threshold mol

ecules are composed of shorter

strands (15 and 25 nucleotides) than signal molecules

(33 nucleotides) and thus may contain fewer synthesis errors.

Additional signal restoration experiments suggested that the

threshold to signal ratio

1.4 was consistent for different

threshold and signal molecules (Supplementary Fig. 3). Thus,

using this ratio, we can then calculate how to adjust the nominal

thresholds for correctly computing logic AND and OR.

Each seesaw logic gate has an integrating node upstream of an

amplifying node. Ideally, an integrating node outputs the sum of

all inputs:

A two-input logic function can be computed as:

Assuming that an ideal OFF state is [0, 0.2] and an ideal ON state

is [0.8, 1],

0.6 will compute logic OR and

1.2 will

compute logic AND, if the effective concentrations of the

threshold and input signals are comparable to each other (that

is,

1).

1 for unpurified threshold and signal molecules, we

can take this ratio into consideration while calculating the lower

and upper bounds of the nominal threshold for an

-input logic

gate:

nom

ðÞþ

AND

nom

Using

1.4, we chose a nominal threshold of 0.35

and

0.85

for two-input OR and AND gate, respectively, and

0.4

and 1.6

for three-input OR and AND gate. Experiments

of the logic gates showed desired behaviours (Fig. 3d and

Supplementary Fig. 4).

An alternative approach for adjusting the nominal threshold is

to use the following equations:

nom

AND

nom

ðÞþ

Compared with choosing a nominal threshold based on

the lower and upper bounds, this approach is less flexible but

simpler.

Next, we can estimate the ratio between

and

for a gate

releasing a signal, using an ex

periment that compares the

fully triggered (tri) concentration of the gate with the signal

when their nominal concentrat

ions are the same. For example,

thedatainFig.3eshowedthat

:

;

tri

when

:

;

nom

;

nom

.Thus,thegatetosignalratiocan

be calculated as:

;

:

;

eff

:

;

nom

;

nom

;

eff

:

;

eff

;

eff

:

;

nom

;

nom

Additional gate calibration experiments suggested that the ratio

0.8 was consistent for different gate and signal molecules

(Supplementary Fig. 5). We suspect that due to synthesis errors in

gate molecules, not all gates can successfully release a signal,

which is why an unpurified gate has a lower effective concen-

tration compared to a signal.

As signal restoration was built in within every logic gate to

accept an ON state of [0.8, 1], we decided not to make any

adjustment for nominal gate concentrations if

0.8. Other-

wise, nominal concentration of an amplifying gate and an

-input

integrating gate can be adjusted as:

AMP

nom

INT

nom

Importantly, the values of

and

should depend on the

strand quality and thus could vary with different DNA synthesis

providers, procedures and even batches. It is necessary to

recalculate the ratios

and

, if these conditions change.

Identifying outliers

. With calibrated logic gates, we investigated

how well they compose together in larger circuits. We constructed

a two-layer logic circuit that is part of the rule 124 sub-circuit and

is composed of an AND gate and two upstream OR gates

(Fig. 4a). The expected circuit behaviour is that the output should

remain OFF when only one of the upstream OR gates is ON.

However, the observed circuit behaviour showed that the output

was reasonably OFF when one upstream OR gate was ON, but

was half ON when the other upstream OR gate was ON. This

experimental result suggested that the ON signals pushed onto

the two input wires of the downstream AND gate (that is, the

output wires of the two upstream OR gates) were significantly

different from each other, which was an indication that the

effective concentrations of the two unpurified gate species that

released the output signals were different—one of the gates must

be an outlier with

0.8.

Indeed, with a gate calibration experiment shown in Fig. 4b,

we measured that

18,53

0.8

for one gate and

22,53

0.44

for another. A possible explanation is that

the synthesis errors of unpurified strands somewhat depend on

DNA sequences

and variations of effective concentrations may

occur between different gate or threshold species. We suspect it

was not a coincidence that the outlier gate had a lower effective

concentration compared with other unpurified gates, because a

particular DNA strand having much worse quality than average is

probably more likely than it having much better quality.

Once an outlier is identified, either a threshold or a gate, the

nominal concentration can be adjusted using its own threshold to

signal ratio (that is,

) or gate to signal ratio (that is,

), the

common nominal concentration described in the previous

section, and the common ratio for other thresholds and gates:

;

:

nom

;

:

nom

;

:

;

nom

:

;

nom

;

We constructed the two-layer logic circuit using the adjusted

nominal gate

:

;

nom

(Fig. 4c). The

trajectories that compute logic ON reached an ideal high

fluorescence state faster than the previous experiments shown

in Fig. 4a and the trajectories that compute logic OFF remained at

a lower fluorescence state that were roughly identical for all three

input combinations, regardless of which upstream OR gate was

ON. However, after identifying and adjusting the outlier gate, we

still had a problem: the OFF trajectories were not at an ideal low

ARTICLE

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14373

6

NATURE COMMUNICATIONS

| 8:14373 | DOI: 10.1038/ncomms14373 | www.nature.com/naturecommunications

fluorescence state. This led to the next tuning step that is

necessary for unpurified seesaw circuits.

Tuning circuit output

. Comparing the behaviour of the AND

gate when it was in isolation (Fig. 3d) and that when it was

connected with two upstream OR gates (Fig. 4c), the ON/OFF

separation was significantly decreased in the latter. These

experimental results suggest that, compared with purified seesaw

DNA circuits in which the ON/OFF separations were roughly

identical from a single logic gate to four-layer logic circuits

unpurified circuits are much noisier and the behaviour becomes

less robust with more than one layer. We suspect this is caused by

the stoichiometry errors in unpurified gate species. The double-

stranded gate molecules were annealed with the same amount of

top and bottom strands, because both strands have combinations

of toehold and branch migration domains that can cause

undesired interactions with other circuit components and thus

neither should be in excess. However, due to variations in the

pipetting volume and in the accuracy of concentrations, the equal

stoichiometry cannot be guaranteed. Without purification, a

small excess of one strand or another in the gate species cannot be

removed. Therefore, the excess of strands would result in

undesired release of output signals in logic gates, even without

input signals, and introduce extra noise to downstream logic

gates.

Fortunately, thanks to the thresholding function in every logic

gate, we can tune the circuit output by increasing a threshold.

A simple method for estimating how much threshold adjustment

is needed is based on the ON/OFF separation of the circuit

output. Using experimental data of a logic circuit with different

inputs, we can choose a trajectory that should compute logic ON

and OFF, respectively, and calculate the difference (

) between

the observed OFF value and an ideal OFF value, when the ON

trajectory reaches an ideal ON value. Considering 0.7 and 0.3 as

the lower bound, and 0.9 and 0.1 as the upper bound for an ideal

ON/OFF separation, the range of

can be determined as:

OFF

The nominal threshold in the logic gate that produces the circuit

output can then be adjusted accordingly:

;

:

nom

;

:

nom

Using the data of the two-layer logic circuit shown in Fig. 4c,

we chose the trajectory with input

01010 and 11100 as the

reference ON and OFF trajectory, respectively, and calculated

0.08

0.41. We then increased the threshold in the down-

stream AND gate to

;

:

nom

and

repeated the experiment. The circuit behaviour was improved

with a much better ON/OFF separation (Fig. 5a).

With the same method, we constructed another two-layer logic

circuit that is composed of an OR gate and two upstream AND

gates (Fig. 5b). In this case, using input

00011 and 01110 as the

reference ON and OFF trajectories, we obtained a similar range of

and decided to apply the same amount of increase to the

threshold in the downstream OR gate.

It is noteworthy that a rule of thumb is to choose the slowest

ON trajectory and the fastest OFF trajectory as the references for

threshold adjustment, but different choices can be made if one

has the knowledge of which data set is experimentally more

reliable. Also note that increasing the threshold not only

suppresses the OFF trajectories but also slows down the ON

trajectories and thus this method of tuning the circuit output is

only applicable if all ON trajectories are significantly faster than

all OFF trajectories (which should be true if the thresholds and

gates are properly calibrated).

Combining the two logic circuits shown in Fig. 5 and adding

fan-out gates for input signals that are used in multiple logic

gates, we successfully demonstrated the rule 124 sub-circuit

consisting of 54 distinct DNA species (Supplementary Fig. 6).

OFF

–1.5

–0.6

Output

1.0

0.8

0.6

0.4

0.2

0.0

Time (h)

0246810

Output

1.0

0.8

0.6

0.4

0.2

0.0

Output

1.0

0.8

0.6

0.4

0.2

0.0

Time (h)

0123

0246810

S53

22:22,53

18:18,53

[

22:22,53

]

tri

= 0.44 ×

[

22:22,53

]

′

nom

= 1.8 ×

[

18:18,53

]

tri

= 0.8 ×

S53

S18

S22

S53

S22

S53

S18

[

53,5:5

]

nom

= 0.85 ×

[

34,18:18

]

nom

= 0.4 ×

[

39,22:22

]

nom

= 0.35 ×

–1.2

–1.5

–0.6

Figure 4 | Identifying an outlier gate.

(

) Logic circuit diagram, seesaw circuit diagram and experimental data of a two-layer logic circuit. (

) Measuring

the effective concentrations of the gate species. Three independent circuits were used to measure the effective concentrations of two gates fully tr

iggered

and

, respectively, comparing with the effective concentration of

(using signal strand

18,53

). (

) Experimental data of the two-layer logic circuit

using adjusted nominal gate concentration. 1

100 nM.

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14373

ARTICLE

NATURE COMMUNICATIONS

| 8:14373 | DOI: 10.1038/ncomms14373 | www.nature.com/naturecommunications

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