Design and Analysis of Compact DNA Strand Displacement Circuits

Subscriber access provided by Uppsala universitetsbibliotek

Article

Design and Analysis of Compact DNA Strand Displacement Circuits for Analog Computation Using Autocatalytic Amplifiers Tianqi Song, Sudhanshu Garg, Reem Mokhtar, Hieu Bui, and John Reif ACS Synth. Biol., Just Accepted Manuscript • DOI: 10.1021/acssynbio.6b00390 • Publication Date (Web): 04 Dec 2017 Downloaded from http://pubs.acs.org on December 5, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

ACS Synthetic Biology is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Synthetic Biology

Design and Analysis of Compact DNA Strand Displacement Circuits for Analog Computation Using Autocatalytic Amplifiers Tianqi Song, Sudhanshu Garg, Reem Mokhtar, Hieu Bui, and John Reif∗ Department of Computer Science, Duke University, Durham, North Carolina 27708, United States E-mail: [email protected] Abstract A main goal in DNA computing is to build DNA circuits to compute designated functions using a minimal number of DNA strands. Here, we propose a novel architecture to build compact DNA strand displacement circuits to compute a broad scope of functions in an analog fashion. A circuit by this architecture is composed of three autocatalytic amplifiers, and the amplifiers interact to perform computation. We show DNA circuits to compute functions sqrt(x), ln(x) and exp(x) for x in tunable ranges with simulation results. A key innovation in our architecture, inspired by Napier’s use of logarithm transforms to compute square roots on a slide rule, is to make use of autocatalytic amplifiers to do logarithmic and exponential transforms in concentration and time. In particular, we convert from the input that is encoded by the initial concentration of the input DNA strand, to time, and then back again to the output encoded by the concentration of the output DNA strand at equilibrium. This combined use of strandconcentration and time encoding of computational values may have impact on other forms of molecular computation.

1

ACS Paragon Plus Environment

ACS Synthetic Biology 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Keywords: DNA circuits; analog computation; DNA self-assembly

1

Introduction

1.1

Capability of DNA Circuits

Deoxyribonucleic acid (DNA) is a highly programmable material because of its WatsonCrick base pairing property(1 ). This programmability provides the potential to use DNA to build complex systems for designated tasks. To date, DNA circuits have been developed for multiple purposes such as detection(2 –5 ), implementing chemical reaction networks (such as controllers, oscillators, etc)(6 –8 ), programming molecular reaction pathways(9 ), and computation(10 –12 ).

1.2

Digital and Analog DNA Circuits

Previous efforts to engineer DNA-based systems to perform computation have focused primarily on digital logic circuits. Digital circuits have the advantage that they are more scalable and error-resilient compared to analog circuits. Seelig et al. first demonstrated large-scale DNA circuits for digital computation(10 ). Qian et al. developed a motif called “seesaw gate” which can be used to construct logic gates(13 ). This architecture based on “seesaw gate” is more scalable than previous ones and able to build circuits for complex tasks like computing square root function. Using the same motif, Qian et al. also created DNA circuits that perform neural network computation(12 ). Recently, Groves et al. reported DNA logic gates in mammalian cells(14 ). Analog circuits also have advantages over digital circuits (15 ). The main advantage is that, compared to digital circuits, analog circuits may require fewer gates to compute functions(15 ), because analog circuits have a different encoding strategy from digital circuits. For example, an input or output in analog circuits is directly represented by a physical quantity (e.g. voltage, the concentration of a DNA strand). However, in digital circuits, an input 2


Page 2 of 22

Page 3 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


or output needs to be encoded into several binary bits. In biological computing, remarkable work has been done on analog genetic circuits(16 , 17 ). Research on analog DNA circuits has mainly been done on making dynamic systems in time domain such as controllers(7 , 18 ), oscillators(8 ), timer(19 ), etc. A fundamental work was done by Soloveichik et al. to show that any chemical reaction network can be implemented by DNA(6 ). Cardelli showed how to program dynamic DNA systems by two-domain motifs(20 ). These works can be used on DNA-based analog computation by translating chemical reaction networks for computing functions(21 , 22 ) to DNA reactions. Song et al. proposed an architecture based on three analog gates, to construct DNA circuits for analog polynomial functions(23 ). This architecture does not give compact DNA circuits for nonpolynomial functions, and the complexity of the circuits depends on the strategy to approximate these functions by polynomials.

1.3

Our Approach and Contribution

In this paper, we propose a novel architecture to design compact DNA strand displacement circuits to compute functions such as sqrt(x), ln(x) and exp(x) for x in tunable ranges, providing a tunable trade-off between the range of values and precision. To motivate our strategy to compute sqrt(x) and other functions, we first briefly discuss the way sqrt(x) is computed using a slide rule(24 ). To compute sqrt(x), a slide rule uses a process of conversion to logarithms, multiplication by 0.5, and then exponentiation. Given an input x, the conversion to logarithm gives ln(x); the multiplication by 0.5 gives 0.5×ln(x) = ln(x0.5 ); and the final exponentiation gives exp(ln(x0.5 )) = x0.5 = sqrt(x). Whereas a slide rule has a physical system for conversion to logarithms (a rod ruled out in logarithm lengths) and its inverse conversion, we will make use of autocatalytic amplifiers for conversion to logarithms and its inverse conversion. As a consequence, the basic components in our architecture are tunable autocatalytic amplifiers. Our method does not require a particular type of autocatalytic amplifier: The amplifier just needs (i) to provide (within small error) an exponential (with respect to time) concentration of the output strand 3



within a time domain, and (ii) to be adjustable to provide a tunable range of constants in the exponent. We can use any known DNA-hybridization based autocatalytic amplifier or enzyme-based autocatalytic amplifier(9 , 25 –31 ). The autocatalytic amplifier used for our examples in this paper is based on seesaw gates(13 ), which will be described in detail later.

2 2.1

Results and Discussion A High-level Chemical Reaction Network (CRN) Description of Our Architecture

Here we give a high-level chemical reaction network (CRN) description of our architecture. The strategy that we design our architecture and its expected performance in the ideal case are described as following.

2.1.1

Overview

There are three modules in a circuit based on our architecture: an input module, a stopper module, and an output module. The basic strategy is that: (i) The input module and output module start to work at the beginning, but the stopper module does not; (ii) The input module computes a time point according to the input (of the circuit), to trigger the stopper module to stop the output module from producing the output (of the circuit) at the computed time point; (iii) The input module and output module are designed such that the output produced totally is the expected value (a function of the input).

2.1.2

The Chemical Reaction Network (CRN)

Specifically, as shown in the chemical reaction network, there are three modules: the input module Gi (reactions (1a) and (1b)), the stopper module Gs (reaction (1c), leak-free), and the output module Go (reactions (1d) and (1e)). Reaction (1f) is for the stopper module

4


Page 4 of 22

Page 5 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


to stop the output module, and reaction (1g) is for the input module to compute the time point to trigger the stopper module.

k

Ii + Gi −−1→ Ii + Is k

Gi −−2→ Ii + Is k

Is + Gs −−3→ Is + S k

Io + Go −−4→ Io + O k

Go −−5→ Io + O k

S + Go −−6→ ∅ k

I + Is −−→ ∅

(1a) (1b) (1c) (1d) (1e) (1f) (1g)

Each module is an autocatalytic amplifier: using the input module as an example, there are two reactions (1a) and (1b), where Gi is the amplifier, Ii is the initiator, and Is is a byproduct. Reaction (1b) is the leak reaction where Ii and Is are produced without the initiator Ii . The input of the circuit is encoded by [I]0 , the initial concentration of the input species I. The output of the circuit is encoded by [O]∞ , the concentration of the output species (a byproduct of output module) at equilibrium. Initially, only input I, and three amplifiers Gi , Gs and Go present in the reaction solution. 2.1.3

The Input Module Computes a Time Point to Trigger the Stopper Module

The input module starts to produce Is by reaction (1a) and (1b). Although there does not exist Ii at the beginning, the leak reaction (1b) can initiate the amplification. Is consumes input species I by reaction (1g). Note that Is may also go to reaction (1c) to trigger the stopper module, but we prevent this by setting k k3 , which means that reaction (1g) is much more preferred by Is than reaction (1c). In the ideal case, no Is goes to reaction (1c)

5



until all input species I has been consumed and reaction (1g) has to stop. Let the input module produce Is by [Is ]= f (t), and let [I]0 = x. It takes f −1 (x) time for the input module to produce enough Is to consume all input species I, and another ∆t1 time is needed for reaction (1g) to complete. Therefore, at time f −1 (x) + ∆t1 , reaction (1c) starts to happen, which means that the stopper module is triggered.

2.1.4

The Stopper Module Stops the Output Module at the Designated Time

For the stopper module, we assume that it is designed to be free of leak. Therefore, the stopper module only starts to produce S by reaction (1c) at time f −1 (x) + ∆t1 , when Is produced by the input module starts to go to reaction (1c). The purpose of stopper module is to produce enough S to consume all Go by reaction (1f) in a short time and stop the output module. To make this stopping process fast, we Let [Gs ]0 [Gi ]0 , [Go ]0 and k6 k4 . Let the time needed to produce enough S and stop the output module be ∆t2 . Therefore, at time f −1 (x) + ∆t1 + ∆t2 , the output module is stopped. 2.1.5

The Output Module Produces the Expected Output

The output module starts to produce O by reaction (1d) and (1e) at the beginning. Although there does not exist Io at the beginning, the leak reaction (1e) can initiate the amplification. Let the output module produce O by [O]= g(t). The output module works for totally f −1 (x) + ∆t1 + ∆t2 time, and produces g(f −1 (x) + ∆t1 + ∆t2 ) concentration of O at equilibrium. We expect to tune the chemical reaction network such that the output [O]∞ =[O]f −1 (x)+∆t1 +∆t2 =g(f −1 (x) + ∆t1 + ∆t2 ) ≈ λ1 K(x) + λ2 , where K(x) is the function that we want to compute, λ1 and λ2 are functions of ∆t1 and ∆t2 but can be treated as constants without causing major errors. The result λ1 K(x) + λ2 can be further calibrated by multiplying

1 λ1

and subtracting λ2 to get K(x) using analog gates proposed in our previous

work(23 ). For better understanding of our architecture, we did a simulation under specific parameters as shown in Figure 1. The simulation code and parameters used are included in

6


Page 6 of 22

Page 7 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Supporting Information. 100 I O S

80

2.5

I O S

2

p2

60 1.5 40

1

0.5

20

0

p1

0 −0.5 −20

0

50

100

150

200

250

300

350

400

450

500

212

214

216

Time (s)

218

220

222

224

Time (s)

(a)

(b)

Figure 1: (a) The input module starts to consume input I since the beginning. When all input is consumed, the stopper module is triggered to produce S. S stops the output module from producing output O in a short time. (b) A zoom-in view: Input I is consumed up when t ≈ 215 s (point p1), and after about three seconds, output O stops being produced (point p2).

2.1.6

Examples of Computing Functions

To design a circuit for a specific function, we need to specify f (t) and g(t). Here, we give examples of computing sqrt(x), ln(x) and exp(x) : • To compute K(x) = sqrt(x), we need f (t) = βI eαI t , g(t) = βO eαO t , and g(f −1 (x) + ∆t1 + ∆t2 ) = βO e

αO ( α1 ln βx +(∆t1 +∆t2 )) I

αO

= βO e αI = βO e

I

ln βx +αO (∆t1 +∆t2 ) I

0.5ln βx +αO (∆t1 +∆t2 ) I

βO = ( √ eαO (∆t1 +∆t2 ) )sqrt(x) βI

7


αO αI

= 0.5 then:

(2) (3) (4) (5)


Page 8 of 22

• To compute K(x) = ln(x), we need f (t) = βI eαI t , g(t) = αO t + βO , and

αO αI

= 1 then:

x 1 ln + (∆t1 + ∆t2 )) + βO αI βI αO αO = ln(x) − ln(βI ) + αO (∆t1 + ∆t2 ) + βO αI αI

g(f −1 (x) + ∆t1 + ∆t2 ) = αO (

= ln(x) + (αO (∆t1 + ∆t2 ) + βO − ln(βI ))

• To compute K(x) = exp(x), we need f (t) = αI t + βI , g(t) = βO eαO t , and

αO (

g(f −1 (x) + ∆t1 + ∆t2 ) = βO e

αO

= βO e αI

x−βI αI

+(∆t1 +∆t2 ))

(6) (7) (8)

αO αI

= 1 then:

(9)

α

x− αO βI +αO (∆t1 +∆t2 ) I

(10)

= βO ex−βI +αO (∆t1 +∆t2 )

(11)

= (βO eαO (∆t1 +∆t2 )−βI )exp(x)

(12)

To perform computing, the amplifiers should be configured such that f (t) and g(t) are as needed. In practice, we may only be able to make f (t) and g(t) be good approximation of needed functions in a time domain [t1 , t2 ], which also means x should be in [f (t1 ), f (t2 )]. Examples of abstract CRNs computing sqrt(x), ln(x) and exp(x) with simulation results are included in Supporting Information. In this paper, we will mainly show DNA-based CRNs that approximately compute sqrt(x), ln(x) and exp(x) for x in certain ranges. Note that this architecture does not have to rely on leak reactions, and theoretically the rate constants of the leak reactions can be zero and the input and output modules can be initiated by external initiators. We use leak reactions here because the DNA implementation of the autocatalytic amplifier usually has leak. In summary, this novel architecture in DNA computing is motivated the 400-year old set

8


Page 9 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


of operations used by a slide rule to compute sqrt(x) (conversion to logarithms, multiplication by 0.5, and then inverse logarithm conversion for the output), except multiplication by 0.5 is not needed since we can set the exponential constants of the autocatalytic amplifiers.

2.2

Examples Implemented by DNA Strand Displacement Circuits

In this section, we present DNA strand displacement circuits to compute sqrt(x), ln(x) and exp(x) for x in certain ranges. We first show the design of a DNA-based autocatalytic amplifier, and then show the circuits based on this amplifier.

2.2.1

A DNA-based Autocatalytic Amplifier

The autocatalytic amplifier is based on seesaw gates(13 , 32 ) as shown in Figure 2. It has three parts: two seesaw gates SG1, SG2 and a fanout gate FG. O1 is both reactant and product, which makes the amplifier autocatalytic. The designed DNA reactions in the amplifier are shown in Figures 3. The leak reactions caused by fuel strands F1 , F2 , and the helper strand H are dismissed in the figure for brevity, and details about these common leak reactions can be found in previous work(8 , 13 ). To map this amplifier to a high-level CRN description, e.g., reactions (1a) and (1b): O1 is Ii , SG1, SG2, and FG together are Gi , and Of is Is . To make a specific circuit by three such amplifiers, as described in Section 2.1, we need: (1) Tune the input and output modules such that f (t) and g(t) are as needed; (2) Make the stopper module leakless or be of low leak; (3) Make reactions (1f) and (1g) much faster than their competitors (k6 k4 , k k3 ). Next, we explain how to fulfill these requirements. 2.2.2

Building Analog Circuits by the DNA-based Autocatalytic Amplifier

Tunable Property of the Amplifier: This amplifier has a property that the function f (t) = [Of ] can be well approximated by an exponential function e(t) = βe eαe t or a linear 9



Page 10 of 22

Figure 2: The autocatalytic amplifier consists of two seesaw gates SG1, SG2, and a fanout gate FG. The reaction pathway of the amplifier is shown in Figure 3. The DNA figures in this paper are drawn by Visual DSD(33 ).

function l(t) = αl t + βl in certain time domains. For example, as shown in Figure 4: In time domain [t1 , t2 ] (seconds), where t1 = 3 × 105 and t2 = 3.5 × 105 , f (t) can be well −6 )t

approximated by an exponential function e(t) = 0.48 × e(8.51×10

or a linear function

l(t) = 6.53 × 10−5 t − 13.51. The parameters αe in e(t) and αl in l(t) can be fine tuned by modifying the initial concentrations of fuel strands F1 and F2 . For example, according to our simulation, if we make the initial concentrations be [F1 ]0 = [F2 ]0 = 1858 nM, the αe in e(t) will be tuned to 4.255 × 10−6 which is half of the case of [F1 ]0 = [F2 ]0 = 200 nM. And if we make the initial concentrations be [F1 ]0 = [F2 ]0 = 29.75 nM, the αl in l(t) will be tuned to 8.51 × 10−6 . We will again see these two examples later when we construct circuits to compute sqrt(x), ln(x) and exp(x). Reducing the Leak in Stopper Module: The stopper module should be leakless in the ideal case. In practice, we can only suppress the leak to some extent. In the simulation of this paper, we assume that the leak rate constants in the stopper module are 5 orders of

10


Page 11 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 3: The brief reaction pathway of the DNA-based autocatalytic amplifier. A detailed version is included in Supporting Information.

magnitude lower than in other modules (see Section 2.4), because the architecture cannot tolerate larger leak rate constants in the stopper module according to our simulation. We concede that achieving such low leak rate constants is a challenge in practice, but it might be doable given that there are a variety of techniques in practice to reduce leaks, such as (1) preparation of extremely pure DNA strands via culturing cells to reduce the leak caused by imperfect strands(4 ); (2) using clamp domains or mismatches to reduce unwanted strand displacement reactions(8 , 9 , 13 , 34 –36 ); (3) using the ”leakless strand displacement” technique to systematically design a low-leak circuit(34 ). Make Reactions (1f ) and (1g) Fast: Reaction (1f) is for stopping the output module. In our DNA design, reaction (1f) is implemented by the irreversible long-domain DNA hy11



80

9.5

70

9

f(t) e(t) l(t)

60 8.5 50

f(t) (nM)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 22

8 40 7.5 30 7 20 6.5

10

0

0

5

10

Time (s)

15

6 2.9

3

5

x 10

(a)

3.1

3.2

Time (s)

3.3

3.4

3.5 5

x 10

(b)

Figure 4: (a) How f (t) =[Of ] grows over time. In the simulation, we let the initial concentrations be [G1 ]0 =[G2 ]0 = 100 nM, [F1 ]0 =[F2 ]0 = 200 nM, [H]0 =[F]0 = 100 nM. The domain lengths are |S1 | = |S2 | = |S3 | = |S4 | = |S5 | = |S6 | = 17, |T | = |r| = 3. The rate constants are described in Section 2.4. (b) In time domain [t1 , t2 ] (seconds), where t1 = 3 × 105 and t2 = 3.5 × 105 , f (t) can be −6 well approximated by an exponential function e(t) = 0.48 × e(8.51×10 )t with mean squared error 1 t2 2 −4 n Σt=t1 (e(t) − f (t)) = 5.73 × 10 , and f (t) can also be well approximated by a linear function 2 l(t) = 6.53 × 10−5 t − 13.51 with mean squared error n1 Σtt=t (l(t) − f (t))2 = 7.92 × 10−4 , where t is 1 integer when calculating mean squared error, and n = t1 − t2 + 1.

bridization reaction (which is fast) between the Of strand produced by the stopper module and the helper strand H in FG of the output module, which consumes up the helper strand and then stops the output module. Reaction (1g) is for consuming the input DNA strand. Reaction (1g) is implemented by the irreversible long-domain DNA hybridization reaction between the input DNA strand and the Of strand produced by the input module. Each reaction is a long-domain DNA hybridization reaction between Of of a module and a single DNA strand, and it works as shown in Figure 5: the Of strand should be modified if needed by adding a domain W which is inactive before Of is released from DNA complex F. A single DNA strand I can hybridize to Of when Of is released and domain W becomes active, where domain s∗ is complementary to part of S6 domain. Domain W should be short enough such that it is inactive as a bulge loop, and domain s∗ should be short enough such that it does not much interact with the helper strand H which also has domain S6. However, strand I should be long enough for an irreversible fast DNA hybridization reaction.

12


Page 13 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 5: Long-domain DNA hybridization reaction between Of and a single DNA strand I: the Of strand is modified by adding a domain W , and W is inactive before Of is released from DNA complex F. When Of is released and domain W becomes active, the single DNA strand I can hybridize to Of , where domain s∗ is complementary to part of S6 domain.

2.2.3

Analog DNA Circuits to Compute sqrt(x), ln(x), and exp(x)

Here we show how to construct DNA circuits by our architecture to compute sqrt(x), ln(x), and exp(x) for x in certain ranges using the DNA-based autocatalytic amplifier. The input is encoded by the initial concentration of the input DNA strand. The output is encoded by the concentration of the output DNA strand (Of of output module) at equilibrium. Analogous to the description in Section 1, the input and output modules should be tuned such that f (t) and g(t) are as desired in a time domain. We set up the initial concentrations as in table 1. Other requirements needed to make the circuits work are fulfilled as described in Section 2.2.2. Table 1: The initial concentration setup of circuits to compute sqrt(x) for x ∈ [6.15, 9.42] (nM), ln(x) for x ∈ [6.15, 9.42] (nM), and exp(x) for x ∈ [0.79, 1.22] (nM).

Module Input

Output

Stopper

sqrt(x) [G1 ]0 =[G2 ]0 = 100 nM [F1 ]0 =[F2 ]0 = 200 nM [H]0 =[F]0 = 100 nM [G1 ]0 =[G2 ]0 = 100 nM [F1 ]0 =[F2 ]0 = 1858 nM [H]0 =[F]0 = 100 nM [G1 ]0 =[G2 ]0 = 2000 nM [F1 ]0 =[F2 ]0 = 4000 nM [H]0 =[F]0 = 2000 nM

ln(x) [G1 ]0 =[G2 ]0 = 100 nM [F1 ]0 =[F2 ]0 = 200 nM [H]0 =[F]0 = 100 nM [G1 ]0 =[G2 ]0 = 100 nM [F1 ]0 =[F2 ]0 = 29.75 nM [H]0 =[F]0 = 100 nM [G1 ]0 =[G2 ]0 = 2000 nM [F1 ]0 =[F2 ]0 = 4000 nM [H]0 =[F]0 = 2000 nM 13


exp(x) [G1 ]0 =[G2 ]0 = 100 nM [F1 ]0 =[F2 ]0 = 29.75 nM [H]0 =[F]0 = 100 nM [G1 ]0 =[G2 ]0 = 100 nM [F1 ]0 =[F2 ]0 = 200 nM [H]0 =[F]0 = 100 nM [G1 ]0 =[G2 ]0 = 1300 nM [F1 ]0 =[F2 ]0 = 2600 nM [H]0 =[F]0 = 1300 nM


For the circuit to compute sqrt(x), by the setup, we have that, in time domain [t1 , t2 ] (seconds) (where t1 = 3×105 , t2 = 3.5×105 ), f (t) ≈ βI eαI t where βI = 0.48, αI = 8.51×10−6 , and g(t) ≈ βO eαO t where αO = 4.255 × 10−6 = 0.5αI , βO = 7.51. Therefore, as described in Section 1, the output g(f −1 (x) + ∆t1 + ∆t2 ) ≈ λ1 sqrt(x) for x ∈ [f (t1 ), f (t2 )] = [6.15, 9.42] (nM), where λ1 =

βO αO (∆t1 +∆t2 ) √ e . βI

The stopper module is at much higher concentration

than other modules to stop the output module in time as required. This is the same for other examples. As shown in Figure 6 (a), the circuit gives out results of less than 1% error for all simulated inputs. The simulation model is described in Section 2.4. For the circuit to compute ln(x), by the setup, we have that, in time domain [t1 , t2 ] (seconds) (where t1 = 3 × 105 , t2 = 3.5 × 105 ), f (t) ≈ βI eαI t where αI = 8.51 × 10−6 , βI = 0.48, and g(t) ≈ αO t + βO where αO = 8.51 × 10−6 = αI , βO = −1.77. Therefore, as described in Section 1, the output g(f −1 (x) + ∆t1 + ∆t2 ) ≈ ln(x) − λ2 for x ∈ [f (t1 ), f (t2 )] = [6.15, 9.42] (nM), where λ2 = ln(βI ) − αO (∆t1 + ∆t2 ) − βO . As shown in Figure 6 (b), the circuit gives out results of less than 2% error for all simulated inputs. For the circuit to compute exp(x), by the setup, we have that, in time domain [t1 , t2 ] (seconds) (where t1 = 3 × 105 ,t2 = 3.5 × 105 ), f (t) ≈ αI t + βI where αI = 8.51 × 10−6 , βI = −1.77, and g(t) ≈ βO eαO t , where αO = 8.51 × 10−6 = αI and βO = 0.48. Therefore, as described in Section 1, the output g(f −1 (x) + ∆t1 + ∆t2 ) ≈ λ1 exp(x) for x ∈ [f (t1 ), f (t2 )] = [0.79, 1.22] (nM), where λ1 = βO eαO (∆t1 +∆t2 )−βI . As shown in Figure 6 (c), the circuit gives out results of less than 3% error for all simulated inputs. Note that the computing results here are calibrated simply by multiplying or subtracting a constant, not by simulating a DNA gate for calibration (which can be done by analog multiplication or subtraction gate proposed in our previous work(23 )). For the errors, there two major sources: (i) The imperfect approximation of f (t) and g(t) to target exponential or linear functions; (ii) The calibration factors λ1 and λ2 are simply treated as constants when we calculate the errors, but they are actually variables with small variances.

14


Page 14 of 22

Page 15 of 22

Percentage Error

0.008 0.006 0.004 0.002 0 6

7

8 x

9

10

(a) sqrt(x)

0.025 Percentage Error

0.015

0.01 Percentage Error

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.01

0.005

0 6

7

8 x

9

10

(b) ln(x)

0.02 0.015 0.01 0.005 0

0.8

1 x

1.2

1.4

(c) exp(x)

Figure 6: (a) The simulated performance of the circuit to compute sqrt(x), x ∈ [6.15, 9.42] (nM): the simulation is conducted for samples {6.15, 6.25, ..., 9.35}. The percentage error is computed by formula (1/λ1 )sqrtc (x)−sqrt(x) , where sqrtc (x) is the result from simulating the circuit, (1/λ1 ) is a calsqrt(x)

ibration factor and sqrt(x) is the theoretical result. (b) The simulated performance of the circuit to compute ln(x), x ∈ [6.15, 9.42] (nM): the simulation is conducted for samples {6.15, 6.25, ..., 9.35}. The percentage error is computed by formula ((lnc (x)−λ2 )−ln(x) , where lnc (x) is the result from simln(x)

ulating the circuit, λ2 is a calibration factor and ln(x) is the theoretical result. (c) The simulated performance of the circuit to compute exp(x), x ∈ [0.79, 1.22] (nM): the simulation is conducted c (x)−exp(x) for samples {0.79, 0.80, ..., 1.22}. The percentage error is computed by formula (1/λ1 )exp , exp(x) where expc (x) is the result from simulating the circuit, (1/λ1 ) is a calibration factor and exp(x) is the theoretical result. The simulation model is in Section 2.4

2.3

Discussion

In this paper, we proposed an architecture of compact DNA circuits to compute functions such as sqrt(x), ln(x) and exp(x) for x in tunable ranges. This architecture can be used to design circuits for a broad scope of functions. For example, by carefully tuning the input and output modules such that

αO αI

= p, a circuit to compute f (x) = xp can be achieved. The

circuit to compute sqrt(x) is just an example where we let p = 0.5. This architecture has a good property that we may not need larger circuits for functions that look more complex to implement than the three examples: we just simply tune the modules by changing the concentrations of fuel strands such that the circuits compute designated functions. There are several potential improvements for this architecture. First, an activation mechanism (e.g., optical or enzymatic activation) can be added to start all the modules simultaneously to make the circuits autonomous in applications. Secondly, the computation precision depends on the accuracy of the parameters such as leak rate constants, concentrations of the fuel strands, the range of input x. In practice, the leak rate constants may vary in modules. A 15



small change in a leak rate constant can make a major difference in the parameters of f (t) or g(t) in a certain time domain. This issue can be addressed by experimentally measuring the leak rate constants of different modules, and then performing tuning. For the concentrations of fuel strands, we observe that they influence f (t) and g(t) in a moderate way, which means that a small change in the concentration of a fuel strand only makes a small difference in the parameters of f (t) or g(t) in a certain time domain. For input x, the computation error will grow gradually when the value of x leaves the designated range. It would be useful to improve the architecture such that the computation precision is less influenced by parameter errors. Thirdly, the computing speed currently is slow according to the simulation and we need to improve the architecture to get higher computing speed.

2.4

Methods

The rate constants in the simulation are: toehold binding (3 nucleotides): (2 × 10−3 ) × 10−2 = 2 × 10−5 nM−1 s−1 (13 , 37 ); toehold unbinding (3 nucleotides): 6000 s−1 (37 ); branch migration:

8000 b2

s−1 , where b is the length of the branch migration domain(37 ); leak in seesaw

gates: 1.4 × 10−9 nM−1 s−1 (8 , 37 ); long domain hybridization: 2 × 10−3 nM−1 s−1 (13 ); leak in fanout gates: 5 × 10−9 nM−1 s−1 (8 ). We assume that the leak rate constant in the seesaw gates of the stopper module is reduced from 1.4 × 10−9 nM−1 s−1 to 1.4 ∗ 10−14 nM−1 s−1 , and also the leak rate constants in all fanout gates are reduced for 5 orders of magnitude. The simulation is done by LBS in Visual GEC(38 ) and Matlab (MathWorks).

Competing Interests The authors do not have competing interests.

16


Page 16 of 22

Page 17 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Authors’ Contributions T.S. designed the gates and circuits, performed simulation and wrote the paper. J.R. conceived and supervised the study and wrote the paper. All authors participated in discussing the design and revising the paper and gave final approval to publish this paper.

Funding This work is supported by NSF Grants CCF-1320360, CCF- 1217457 and CCF-1617791.

Supporting Information The Supporting Information is available free of charge on the ACS Publications website including: code.txt (simulation code) and supplementary.pdf (detailed DNA reactions).

References 1. Watson, J. D., and Crick, F. H. (1953) Molecular structure of nucleic acids. Nature 171, 737–738. 2. Dirks, R. M., and Pierce, N. A. (2004) Triggered amplification by hybridization chain reaction. Proc. Natl. Acad. Sci. U.S.A. 101, 15275–15278. 3. Li, B., Ellington, A. D., and Chen, X. (2011) Rational, modular adaptation of enzymefree DNA circuits to multiple detection methods. Nucleic Acids Res. 39, e110–e110. 4. Chen, X., Briggs, N., McLain, J. R., and Ellington, A. D. (2013) Stacking nonenzymatic circuits for high signal gain. Proc. Natl. Acad. Sci. U. S. A. 110, 5386–5391. 5. Li, B., Chen, X., and Ellington, A. D. (2012) Adapting enzyme-free DNA circuits to the

17



detection of loop-mediated isothermal amplification reactions. Anal. Chem. 84, 8371– 8377. 6. Soloveichik, D., Seelig, G., and Winfree, E. (2010) DNA as a universal substrate for chemical kinetics. Proc. Natl. Acad. Sci. U. S. A. 107, 5393–5398. 7. Chen, Y.-J., Dalchau, N., Srinivas, N., Phillips, A., Cardelli, L., Soloveichik, D., and Seelig, G. (2013) Programmable chemical controllers made from DNA. Nat. Nanotechnol. 8, 755–762. 8. Srinivas, N. Programming chemical kinetics: engineering dynamic reaction networks with DNA strand displacement. Ph.D. thesis, California Institute of Technology, 2015. 9. Yin, P., Choi, H. M., Calvert, C. R., and Pierce, N. A. (2008) Programming biomolecular self-assembly pathways. Nature 451, 318–322. 10. Seelig, G., Soloveichik, D., Zhang, D. Y., and Winfree, E. (2006) Enzyme-free nucleic acid logic circuits. Science 314, 1585–1588. 11. Qian, L., and Winfree, E. (2011) A simple DNA gate motif for synthesizing large-scale circuits. J. R. Soc. Interface 8, 1281–1297. 12. Qian, L., Winfree, E., and Bruck, J. (2011) Neural network computation with DNA strand displacement cascades. Nature 475, 368–372. 13. Qian, L., and Winfree, E. (2011) Scaling up digital circuit computation with DNA strand displacement cascades. Science 332, 1196–1201. 14. Groves, B., Chen, Y.-J., Zurla, C., Pochekailov, S., Kirschman, J. L., Santangelo, P. J., and Seelig, G. (2015) Computing in mammalian cells with nucleic acid strand exchange. Nat. Nanotechnol. 15. Sarpeshkar, R. (1998) Analog versus digital: extrapolating from electronics to neurobiology. Neural Comput. 10, 1601–1638. 18


Page 18 of 22

Page 19 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


16. Daniel, R., Rubens, J. R., Sarpeshkar, R., and Lu, T. K. (2013) Synthetic analog computation in living cells. Nature 497, 619–623. 17. Purcell, O., and Lu, T. K. (2014) Synthetic analog and digital circuits for cellular computation and memory. Curr. Opin. Biotechnol. 29, 146–155. 18. Yordanov, B., Kim, J., Petersen, R. L., Shudy, A., Kulkarni, V. V., and Phillips, A. (2014) Computational design of nucleic acid feedback control circuits. ACS Synth. Biol. 3, 600–616. 19. Fern, J., Scalise, D., Cangialosi, A., Howie, D., Potters, L., and Schulman, R. (2016) DNA Strand-Displacement Timer Circuits. ACS Synth. Biol. 20. Cardelli, L. (2013) Two-domain DNA strand displacement. Mathematical Structures in Computer Science 23, 247–271. 21. Chen, H.-L., Doty, D., and Soloveichik, D. (2014) Deterministic function computation with chemical reaction networks. Nat. Comput. 13, 517–534. 22. Salehi, A., Parhi, K., and Riedel, M. D. (2016) Chemical Reaction Networks for Computing Polynomials. ACS Synth. Biol. 23. Song, T., Garg, S., Mokhtar, R., Bui, H., and Reif, J. (2016) Analog Computation by DNA Strand Displacement Circuits. ACS Synth. Biol. 5, 898–912. 24. Hobson, E. W. John Napier and the Invention of Logarithms, 1614: A Lecture by EW Hobson; Cambridge University Press, 2012. 25. Zhang, D. Y., Turberfield, A. J., Yurke, B., and Winfree, E. (2007) Engineering entropydriven reactions and networks catalyzed by DNA. Science 318, 1121–1125. 26. Levy, M., and Ellington, A. D. (2003) Exponential growth by cross-catalytic cleavage of deoxyribozymogens. Proc. Natl. Acad. Sci. U. S. A. 100, 6416–6421. 19



27. Lincoln, T. A., and Joyce, G. F. (2009) Self-sustained replication of an RNA enzyme. Science 323, 1229–1232. 28. Lam, B. J., and Joyce, G. F. (2009) Autocatalytic aptazymes enable ligand-dependent exponential amplification of RNA. Nat. Biotechnol. 27, 288–292. 29. van Roekel, H. W., Meijer, L. H., Masroor, S., F´elix Garza, Z. C., Est´evez-Torres, A., Rondelez, Y., Zagaris, A., Peletier, M. A., Hilbers, P. A., and de Greef, T. F. (2014) Automated design of programmable enzyme-driven DNA circuits. ACS Synth. Biol. 4, 735–745. 30. Aubert, N., Mosca, C., Fujii, T., Hagiya, M., and Rondelez, Y. (2014) Computer-assisted design for scaling up systems based on DNA reaction networks. Journal of The Royal Society Interface 11, 20131167. 31. Baccouche, A., Montagne, K., Padirac, A., Fujii, T., and Rondelez, Y. (2014) Dynamic DNA-toolbox reaction circuits: a walkthrough. Methods 67, 234–249. 32. Gopalkrishnan, N. (2012) Engineering Exquisite Nanoscale Behavior with DNA. 33. Lakin, M. R., Youssef, S., Polo, F., Emmott, S., and Phillips, A. (2011) Visual DSD: a design and analysis tool for DNA strand displacement systems. Bioinformatics 27, 3211–3213. 34. Thachuk, C., Winfree, E., and Soloveichik, D. DNA Computing and Molecular Programming; Springer, 2015; pp 133–153. 35. Jiang, Y. S., Bhadra, S., Li, B., and Ellington, A. D. (2014) Mismatches Improve the Performance of Strand-Displacement Nucleic Acid Circuits. Angew. Chem., Int. Ed. 53, 1845–1848. 36. Olson, X., Kotani, S., Padilla, J. E., Hallstrom, N., Goltry, S., Lee, J., Yurke, B.,

20


Page 20 of 22

Page 21 of 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Hughes, W. L., and Graugnard, E. Availability: A Metric for Nucleic Acid Strand Displacement Systems. ACS Synth. Biol. 37. Zhang, D. Y., and Winfree, E. (2009) Control of DNA strand displacement kinetics using toehold exchange. J. Am. Chem. Soc. 131, 17303–17314. 38. Pedersen, M., and Phillips, A. (2009) Towards programming languages for genetic engineering of living cells. J. R. Soc. Interface 6, S437–S450.

21



Graphical TOC Entry

22


Page 22 of 22

Design and Analysis of Compact DNA Strand Displacement Circuits

Recommend Documents