Leak-resilient enzyme-free nucleic acid dynamical systems through shadow cancellation

Abstract

DNA strand displacement (DSD) emerged as a prominent reaction motif for engineering nucleic acid-based computational devices with programmable behaviours. However, strand displacement circuits are susceptible to background noise, known as leaks, which disrupt their intended function. The ill effects of leaks are particularly severe in circuits with complex dynamics, as leaks in them amplify nonlinearly, resulting in rapid circuit degradation. Shadow cancellation is a dynamic leak-elimination strategy originally proposed to control the leak growth in such circuits. However, the kinetic restrictions of the method incur a significant design overhead, making it less accessible. In this work, we use domain-level DSD simulations to examine the method’s capabilities, the inner workings of its components and, most importantly, its robustness to the practical deviations in its design requirements. First, we show that the method could stabilize the dynamics of several catalytic and autocatalytic dynamical systems heavily affected by leaks. Then, through several probing experiments, we show that its design restrictions could be significantly relaxed without impacting the circuit function by simply adjusting the circuit parameters. Finally, we discuss several ideas to tackle the practical challenges in applying the method to arbitrary DSD circuits, paving the way for future experimental work.

1. Introduction

Intricate networks of biochemical interactions drive the biological processes responsible for the survival and functioning of an organism. Examples include the biochemical oscillators that underlie circadian rhythms [1,2], autocatalytic reaction networks in metabolism [3], consensus protocols in bacterial quorum sensing [4,5] and feedback control systems that regulate the molecular concentrations inside a cell. It is, therefore, desirable to obtain robust, durable and biocompatible implementations of such systems, as they have the potential to engender novel applications in fields such as molecular diagnostics [6], biomedicine [7,8] and biosensing [9,10].

Molecular computing methods that use DNA as the prime substrate (also referred to as DNA computing) are at the forefront of these efforts. DNA is an excellent substrate for simulating chemical behaviour because the specificity inherent in its base pair interactions leads to programmable reaction pathways. In a prototypical construction of a DNA-based circuit, the desired chemical behaviour, first specified as an abstract reaction system known as a chemical reaction network (CRN) [11,12], is systematically translated into a DNA–DNA reaction system, sometimes involving enzyme action [13] (in this work, however, we are only concerned with enzyme-free DNA circuits [14–19]).

Toehold-mediated DNA strand displacement (alternatively simply referred to as strand displacement; DSD) [19–21] is one such enzyme-free reaction motif that has grown in prominence recently owing to its design simplicity, modularity, and the flexibility in tuning its reaction rates. In its standard form, a strand displacement reaction is akin to transduction, where a single-stranded (ssDNA) invader strand displaces an ssDNA incumbent previously bound in a partially double-stranded (dsDNA) gate complex. The reaction begins with the invader latching onto the single-stranded overhang of the gate complex, known as the toehold and proceeds with the reversible exchange of the bound base pairs between the invader and the incumbent until one of them detaches from the gate complex. The reaction is thermodynamically driven forward (i.e. towards the release of the incumbent) by the formation of additional base pairs at the toehold by the invader strand.

The simple yet effective nature of the strand displacement reaction motif has resulted in its use in the construction of a variety of molecular-scale computational devices, including logic gates [22–24], analogue computation [25], dynamical systems [26–32] and neural network computation [33,34]. Despite this versatility, strand displacement circuits are susceptible to systemic background noise, known as leaks—spurious strand displacement events that occur owing to the noisy interactions among the substrates—that significantly limit their sensitivity and scale [35,36]. Leaks are detrimental to circuit function as the leaked strands can interact with the circuit downstream, causing undesired behaviour. This cascading effect of leaks is particularly severe in circuits with complex dynamics (e.g. feedback loops), as the leaked strands undergo nonlinear amplification, causing the circuit to degrade rapidly. We provide a detailed overview of various sources of leak in strand displacement circuits in electronic supplementary material S3.

Several leak-mitigation techniques have been proposed in the context of strand displacement reactions. Many of these techniques introduce specific modifications into the circuit to either reduce the probability of a leak pathway or prevent it from going to completion, i.e. releasing the incumbent. Examples include the use of stronger G–C base pairs at the helix ends (at both nicks and blunt ends) or short ‘clamp’ domains at the blunt ends, to reduce the rate of fraying [28,37–40], introducing redundant domains so that multiple low-probability events precede a leak pathway [41] and the use of long domains [42]. Other strategies include the use of mismatches at the helix ends [43–45] and spatially separating the leak-causing substrates [46–48]. These techniques are, however, static in nature and, therefore, cannot control the leak growth once it materializes in the circuit. Furthermore, they also require a complete knowledge of all the leak pathways in the circuit. As a result, they alone cannot handle the leaks in circuits with complex dynamics, as even a minuscule amount of leaks in these circuits would be subjected to rapid amplification, resulting in uncontrolled signal growth.

Tackling these issues, Song et al. [27] proposed a non-invasive and dynamic leak-elimination method known as shadow cancellation. This method works by installing a subsidiary shadow circuit alongside the existing primary circuit. The shadow circuit is designed so the leaks from the two circuits cancel each other quickly via a cancellation mechanism, thereby preserving the circuit’s function. Thus, for the method to be effective, both the primary and shadow circuits should be well balanced in terms of the rate at which the leak is generated and the rate at which the leak grows. To satisfy these requirements, Song et al. [27] suggested that the primary and shadow circuits should be designed with a ‘near-identical’ rate profile and similar leak characteristics in conjunction with fast cancellation reactions. However, these requirements, along with the orthogonality constraints between the primary and shadow circuits, impose a significant design overhead for applying the method to arbitrary strand displacement circuits. Further, prior work on shadow cancellation [27] has left several questions unanswered regarding the method’s capabilities, the inner workings of its components and its robustness to practical deviations, such as (i) Can we extend the method to more complex autocatalytic circuits or circuits that involve complex reaction cascades, in which the leak grows much faster than in the cross-catalytic amplifier case? (ii) What is the role of its individual components in achieving effective leak cancellation? (iii) How much can its design requirements be relaxed while preserving the circuit behaviour? (iv) Is it possible to reduce the size of the shadow circuit?

In this work, we use domain-level DSD simulations to answer these questions. First, we show that shadow cancellation can effectively restore the dynamics of several catalytic and autocatalytic circuits of practical importance that are known to be significantly affected by leaks [28]. Then, through several probing experiments on three representative dynamical systems, we argue that the aforementioned design restrictions could be significantly relaxed while preserving the circuit behaviour. Finally, we discuss several practical challenges associated with shadow cancellation, such as the doubling of the circuit size (due to the shadow circuit) and hint at possible ways to tackle these issues.

2. Background

2.1. CRN-to-DSD translation and kinetic model

We use the React–Produce framework proposed by Srinivas et al. [28] to construct the DSD circuits of our target dynamical systems. The framework models each reaction using a two-phase pipeline: (i) the react phase and (ii) the produce phase. Briefly, the reactants are consumed during the React phase and the products are released during the Produce phase. We use the React–Produce circuits of general unimolecular (UNIMOL) and bimolecular (BIMOL) reactions (see electronic supplementary material S1) as building blocks for the construction of our target dynamical systems. Further, to ensure that our simulations are as close as possible to the experiment, we use the rate constants of the individual strand displacement reactions reported by Srinivas et al. [28] in all our circuit constructions. Detailed descriptions of the circuit parameters for all the dynamical systems are provided in electronic supplementary material S9.

2.2. Leak modelling

Three different toeless strand displacement (toeless-SD) leak pathways were identified in the React–Produce framework [28]: (i) React-second input leak, (ii) React–Produce leak, and (iii) Produce–Helper leak. Following the modelling procedure described in Srinivas et al. [28], we only include the Produce–Helper leak pathway in our DSD circuits, as its magnitude is significantly higher than the rest, owing to it involving two high-concentration fuel species (Produce gate complex and auxiliary Helper strand). We refer the reader to electronic supplementary material S3 for an extended description of our leak modelling framework.

In choosing the rate constant for the Produce–Helper leak pathway, we considered the following experimentally observed values. Srinivas et al. [28] reported a Produce–Helper leak rate constant of ${\displaystyle 10{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}$ in their most optimized circuit design. Reynaldo et al. [35] reported toeless-SD leak rate constants ranging from ${\displaystyle 3.6{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}$ at ${\displaystyle {30}^{\circ }\mathrm{C}}$ to a maximum value of ${\displaystyle 79{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}$ at ${\displaystyle {55}^{\circ }\mathrm{C}}$ . Assuming that our simulations are performed closer to room temperature and that the circuits are equipped with so-called ‘leakless’ design principles [39,41,49], we use a rate constant of ${\displaystyle 20{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}$ consistently across all our simulations. Simulations that use a higher leak rate constant are shown in electronic supplementary material S5.

2.3. The components of shadow cancellation

The distinguishing feature of the shadow cancellation methodology is the presence of a subsidiary shadow circuit alongside the primary circuit. It is identical in structure to the primary circuit but runs independently from it. Its kinetics are assumed to be programmed so that its leak activity (signal activity resulting from the leaks) matches closely with the leak activity in the primary circuit. Then, by quickly and conjunctively sequestering the leak activities in the two circuits through a cancellation mechanism, shadow cancellation prevents the leak from amplifying and interfering with the circuit’s function. The cancellation mechanism is enabled through a special class of DNA gate complexes known as cooperative hybridization complexes [50], henceforth referred to as the cancellation complexes. These complexes contain toeholds at both helix ends, facilitating the trimolecular annihilation reactions between matching signal strands of the primary and shadow circuits (see electronic supplementary material, figure S23). The rate constants for the cancellation reactions are adapted from the experimental data provided by Zhang et al. [50].

Intuitively, for the method to be effective, the leak activity in the two circuits must grow at an identical rate, and the cancellation mechanism should be faster than the fastest strand displacement reaction in the overall circuit. Faster leak growth in the primary circuit amplifies the primary signal uncontrollably, leading to the degradation of circuit function, whereas lower leak activity in the primary circuit leads to excessive suppression of its signal by the shadow circuit, leading to underexpression and, in some extreme cases, total suppression of the signal activity (e.g. ‘LeakyCancel’ setting of RPS in electronic supplementary material, figure S26d). Furthermore, fast cancellation reactions ensure that the leak is sequestered in a timely manner, thereby preventing unwanted amplification.

3. Leak-resilient dynamical systems through shadow cancellation

In this section, we present the results of our domain-level simulations, showing that shadow cancellation can restore the dynamics of the leak-affected circuits of several catalytic and autocatalytic systems. Many strand displacement simulation software have been developed, such as Multistrand [51], VisualDSD [52] and Peppercorn [53]. In this work, we use Peppercorn owing to its simplicity, flexibility, and accessibility. Peppercorn takes in as input the domain lengths and their positional arrangement in the substrates and enumerates a set of all possible strand displacement reactions along with their coarsely approximated rate constants. However, we do not use the rate constants generated by Peppercorn in our simulations. Instead, we use the experimental data reported by Srinivas et al. [28] in the context of a rock–paper–scissors (RPS) oscillator and adapt these rate constants to all our circuits.

We simulate the following dynamical systems: (i) a unimolecular autocatalytic amplifier, (ii) a bimolecular autocatalytic amplifier, (iii) a RPS oscillator, (iv) a two-species consensus protocol, and (v) a proportional–integral (PI) feedback controller. Each dynamical system is simulated under four different settings, collectively referred to as the simulation settings, allowing us to incrementally analyse the influence of individual components in a shadow cancellation-enabled circuit: (i) Vanilla, (ii) Leaky, (iii) VanillaOccluded, and (iv) LeakyCancel. Vanilla setting represents the dynamics of a DSD circuit devoid of leaks. Its dynamics differ from the ideal CRN dynamics solely owing to the finiteness of the fuel species. This is the circuit enumerated by Peppercorn. Leaky setting represents the dynamics of the DSD circuit in the presence of leaks. Since Peppercorn does not have native support for enumerating the leak pathways, we add them retroactively to the circuit (note that we only include Produce–Helper leak; see §2.1). LeakyCancel setting describes the dynamics of a shadow cancellation-enabled DSD circuit. It thus combines the ‘Leaky’ primary circuit, the ‘Leaky’ shadow circuit and the cancellation mechanism. However, the presence of cancellation complexes in the ‘LeakyCancel’ setting and their absence in the ‘Vanilla’ setting precludes a fair comparison between the two as these complexes affect the circuit dynamics through toehold occlusion—the temporary sequestration of a signal strand in a gate complex via its toehold; we will discuss more about this effect in later sections. Therefore, to provide a fairer baseline for comparision with the ‘LeakyCancel’ setting, we devised the VanillaOccluded setting, which incorporates the effect of toehold occlusion into the ‘Vanilla’ setting. The circuit for this setting is constructed by simply including the cancellation complexes into the ‘Vanilla’ DSD circuit. Since the shadow circuit is absent, the signals interact with them solely through ‘occlusion’ reactions.

3.1. A leak-resilient. unimolecular autocatalytic amplifier

The unimolecular autocatalytic amplifier (UNIAMP) exponentially amplifies a signal until all the fuel species are exhausted. The formal CRN of the amplifier consists of a single unimolecular reaction ${\displaystyle {\displaystyle \mathrm{C}\stackrel{\hspace{0.17em}}{\to }\mathrm{C}+\mathrm{C}}}$ , with ${\displaystyle \mathrm{C}}$ being the amplified signal. We rewrite the reaction using DNA signal strands as follows: ${\displaystyle {\displaystyle \mathrm{C}\mathrm{j}\stackrel{\hspace{0.17em}}{\to }\mathrm{C}\mathrm{j}+\mathrm{C}\mathrm{k}}}$ , where the strands ${\displaystyle \mathrm{C}\mathrm{j}}$ and ${\displaystyle \mathrm{C}\mathrm{k}}$ are instances of the same formal species ${\displaystyle \mathrm{C}}$ , differing only in their history domains (see electronic supplementary material S1). The UNIAMP DSD circuit is constructed by modifying the UNIMOL DSD circuit such that ${\displaystyle \mathrm{C}\mathrm{j}}$ is consumed during the React phase, and ${\displaystyle \mathrm{C}\mathrm{j}}$ and ${\displaystyle \mathrm{C}\mathrm{k}}$ (in that order) are released during the Produce phase (see electronic supplementary material S2).

Figure 1a juxtaposes the dynamics of the UNIAMP DSD circuit in all four simulation settings. We also include the ideal UNIAMP dynamics (labelled as ‘Ideal’ in figure 1) for reference. In the ‘Vanilla’ setting, the signal ${\displaystyle \mathrm{C}}$ , representing the collective concentration of ${\displaystyle \mathrm{C}\mathrm{j}}$ and ${\displaystyle \mathrm{C}\mathrm{k}}$ , traces an S-shaped curve with an initial exponential amplification region, followed by an inflection region, and culminating in a saturation region, where it reaches a steady-state concentration close to the initial concentration of the fuel species. The inflection region starts when the signal concentration becomes comparable to the fuel concentration, whereas the saturation region is reached once the fuel species are exhausted. In the ‘Leaky’ setting, the excess ${\displaystyle \mathrm{C}\mathrm{k}}$ strands released owing to the Produce–Helper leak undergo amplification, accelerating the fuel consumption, and resulting in steeper amplification and faster saturation. In the ‘VanillaOccluded’ setting, toehold occlusion by the cancellation complexes temporarily sequesters the amplifying signal strands ( ${\displaystyle {\displaystyle \mathrm{C}\mathrm{j}}}$ and ${\displaystyle \mathrm{C}\mathrm{k}}$ ), leading to a reduction in the circuit activity compared with the ‘Vanilla’ setting. Finally, the inclusion of the shadow circuit and the cancellation reactions in the ‘LeakyCancel’ setting restores the circuit dynamics to normality, i.e. brings them closer to the ‘VanillaOccluded’ dynamics. The close-to-zero activity in the shadow circuit in this setting provides further evidence indicating that the leak activity is effectively subdued in the primary circuit.

Figure 1.

(a) Dynamics of the UNIAMP DSD circuit ( ${\displaystyle \mathrm{C}\mathrm{j}\to \mathrm{C}\mathrm{j}+\mathrm{C}\mathrm{k}}$ ) in all four simulation settings juxtaposed with its ideal dynamics (‘Ideal’). Initial concentrations: ${\displaystyle {\displaystyle [\mathrm{C}\mathrm{j}]=1\mathrm{n}\mathrm{M},[\mathrm{C}\mathrm{k}]=0\mathrm{n}\mathrm{M},[\mathrm{F}\mathrm{u}\mathrm{e}\mathrm{l}]=80\mathrm{n}\mathrm{M}}}$ . The signal ${\displaystyle \mathrm{C}}$ ( ${\displaystyle [\mathrm{C}\mathrm{j}]+[\mathrm{C}\mathrm{k}]}$ ) traces an S-shaped curve with an initial exponential region, followed by an inflection region, culminating in a saturation region. In the ‘Vanilla’ setting, the signal reaches a final concentration close to the initial fuel concentration. Leaks accelerate fuel consumption in the ‘Leaky’ setting, leading to a steeper amplification and faster saturation. Including shadow cancellation into the circuit in the ‘LeakyCancel’ setting restores its dynamics to normality, i.e. closer to the ‘VanillaOccluded’ setting. The signal activity is lower in the ‘VanillaOccluded’ setting than in the ‘Vanilla’ setting owing to toehold occlusion by the cancellation complexes. (b) Dynamics of the bimolecular autocatalytic amplifier (BIAMP) circuit ( ${\displaystyle \mathrm{C}\mathrm{j}+\mathrm{B}\mathrm{r}\to \mathrm{C}\mathrm{j}+\mathrm{C}\mathrm{k}}$ ) in all four simulation settings juxtaposed with its ideal dynamics. Initial concentrations: ${\displaystyle [\mathrm{C}\mathrm{j}]=10\mathrm{n}\mathrm{M},[\mathrm{C}\mathrm{k}]=0\mathrm{n}\mathrm{M},[\mathrm{B}\mathrm{r}]=10\mathrm{n}\mathrm{M},[\mathrm{F}\mathrm{u}\mathrm{e}\mathrm{l}]=200\mathrm{n}\mathrm{M}}$ . At the steady state, the amplifying signal ${\displaystyle \mathrm{C}}$ ( ${\displaystyle [\mathrm{C}\mathrm{j}]+[\mathrm{C}\mathrm{k}]}$ ) reaches a concentration equal to the sum of the initial concentrations of the two signals, whereas the concentration of the ‘fuel’ signal B ( ${\displaystyle [\mathrm{B}\mathrm{r}]}$ ) goes to zero. The signal activity in the ‘Vanilla’ setting is slightly lower than in the ideal dynamics owing to toehold occlusion by the fuel complexes. The signal activity in the ‘VanillaOccluded’ setting is lower than in the ‘Vanilla’ setting owing to toehold occlusion by the cancellation complexes. In the ‘Leaky’ setting, the Produce–Helper leak releases excess ${\displaystyle \mathrm{C}\mathrm{k}}$ strands, causing the amplifying signal to grow beyond the expected maximum. The introduction of shadow cancellation in the ‘LeakyCancel’ setting restores the circuit dynamics to normality.

3.2. A. Leak-resilient bimolecular autocatalytic amplifier

The bimolecular autocatalytic amplifier (BIAMP), given by the single reaction CRN: ${\displaystyle \mathrm{C}+\mathrm{B}\stackrel{\mathrm{k}}{\to }\mathrm{C}+\mathrm{C}}$ amplifies a signal ( ${\displaystyle \mathrm{C}}$ ) at the expense of another ‘fuel’ signal ( ${\displaystyle \mathrm{B}}$ ). We rewrite this CRN using DNA signal strands as ${\displaystyle \mathrm{C}\mathrm{j}+\mathrm{B}\mathrm{r}\stackrel{\mathrm{k}}{\to }\mathrm{C}\mathrm{j}+\mathrm{C}\mathrm{k}}$ , where the signal ${\displaystyle \mathrm{C}}$ is represented by the collective concentration of the strands ${\displaystyle \mathrm{C}\mathrm{j}}$ and ${\displaystyle \mathrm{C}\mathrm{k}}$ and the signal ${\displaystyle \mathrm{B}}$ is represented by the concentration of ${\displaystyle \mathrm{B}\mathrm{r}}$ . Consequently, the BIAMP DSD circuit is constructed by modifying the BIMOL DSD circuit such that the strands ${\displaystyle \mathrm{C}\mathrm{j}}$ and ${\displaystyle \mathrm{B}\mathrm{r}}$ (in that order) are consumed during the React phase, whereas the strands ${\displaystyle \mathrm{C}\mathrm{j}}$ and ${\displaystyle \mathrm{C}\mathrm{k}}$ (in that order) are released during the Produce phase.

Figure 1b juxtaposes the dynamics of the BIAMP DSD circuit in all four simulation settings. We also include the ideal BIAMP dynamics for reference. In the ‘Vanilla’ setting, the signal ${\displaystyle \mathrm{C}}$ grows at the expense of ${\displaystyle \mathrm{B}}$ , reaching a steady-state concentration approximately equal to the sum of the initial concentrations of the two species, whereas the concentration of ${\displaystyle \mathrm{B}}$ depletes to zero. In the ‘Leaky’ setting, the excess ${\displaystyle \mathrm{C}\mathrm{k}}$ strands released owing to the Produce–Helper leak cause the amplifying signal to grow beyond the expected maximum concentration ( ${\displaystyle \mathrm{C}}$ continues to grow despite the exhaustion of ${\displaystyle \mathrm{B}}$ ). In the ‘VanillaOccluded’ setting, the recorded signal activity of ${\displaystyle \mathrm{C}}$ is lower than in the ‘Vanilla’ setting owing to toehold occlusion by the cancellation complexes. Enabling shadow cancellation in the ‘LeakyCancel’ setting subdues the leak activity close to zero, restoring the circuit dynamics to normality, i.e. closer to the ‘VanillaOccluded’ dynamics.

3.3. A leak-resilient rock–paper–scissors oscillator

The RPS oscillator is an autocatalytic system in which three competing species engage in a cyclic dominance pattern, tracing sinusoidal-like paths (figure 2a ), with the dominant species alternating periodically. The oscillator derives its name from the non-zero-sum game rock–paper–scissors, whose precedence rules follow a similar cyclic pattern (figure 3b ; arrows indicate the precedence rules). The formal CRN for this system consists of three BIAMP reactions (figure 3a ), each encoding a precedence rule where the amplifying signal dominates the fuel signal (e.g. the BIAMP reaction, ${\displaystyle {\displaystyle \mathrm{A}+\mathrm{C}\stackrel{\hspace{0.17em}}{\to }\mathrm{A}+\mathrm{A}}}$ , encodes the precedence rule ${\displaystyle \mathrm{A}}$ :rock beats ${\displaystyle \mathrm{C}}$ :paper). Cyclic patterns of this kind are found in the population dynamics of various biological systems [55] and in ecological predator–prey models [56].

Figure 2.

Domain-level DSD circuit dynamics of the RPS oscillator in the ‘buffered’ setup, where additional fuel species are introduced into the circuit at a constant rate. Initial concentrations of the signals ${\displaystyle {\displaystyle \mathrm{A},\mathrm{B}}}$ and ${\displaystyle \mathrm{C}}$ are set to ${\displaystyle 11\mathrm{n}\mathrm{M},10\mathrm{n}\mathrm{M}}$ and ${\displaystyle 3\mathrm{n}\mathrm{M}}$ , respectively. (a) ‘Vanilla’ circuit dynamics in the buffered setting. The circuit sustains oscillations as long as the fuel species are replenished. (b) ‘Leaky’ dynamics in the buffered setting. The leaks grow uncontrollably, quickly exhausting the fuel species (despite their buffering), causing the oscillations to die out. (c) ‘VanillaOccluded’ dynamics in the buffered setting. Toehold occlusion owing to cancellation complexes reduces signal activity, although buffering ensures that the oscillations are sustained as long as the fuel species and cancellation complexes are replenished. (d) ‘LeakyCancel’ dynamics in the buffered setting. Shadow cancellation, in conjunction with buffering, stabilizes the oscillations in the circuit.

Figure 3.

(a) CRN of the RPS oscillator consisting of three BIAMP reactions. (b) Schematic of the RPS oscillator depicting the cyclic dominance pattern. (c) CRN of the two-species consensus protocol consisting of two BIAMP reactions and one BIMOL reaction. (d) Schematic of the two-species consensus protocol. At steady state, the protocol elects a leader by converting all the ‘minority’ species into the ‘majority’ species. (e) CRN of the PI feedback controller with the Controller constructed solely using catalysis, annihilation and degradation reactions. Reactions in the plant are on the right. The circuit design is adapted from [54]. (f) Schematic of the PI feedback controller adapted from [54].

Several prior works have demonstrated oscillatory behaviour both in enzymatic [57] and enzyme-free [19,28] DNA circuits. To our knowledge, the most recent enzyme-free implementation of the RPS oscillator is by Srinivas et al. [28], where the circuit was able to sustain oscillations for only a few cycles owing to excessive leaks (this is not a criticism of their work, as their goal was to realize oscillatory behaviour in enzyme-free nucleic acid circuits).

Since the RPS oscillator’s CRN is a linear combination of three BIAMP reactions, its DSD circuit is constructed by linearly combining the BIAMP DSD circuits of the three reactions. However, since the RPS oscillator is a non-equilibrium dynamical system, its fuel species should be continuously replenished, i.e. buffered, to keep the system functional. Indeed, applying shadow cancellation without buffering in this circuit leads to counterintuitive results (electronic supplementary material, figure S26). An example buffering scheme in the context of strand displacement circuits was proposed by Lakin & Stefanovic [34], where the fuel species are allowed to be present in the circuit in an inactive form and could be dynamically activated through special ‘activator’ strands. In this work, we implement a much simpler buffering scheme, where the fuel species of the primary circuit are programmed to diffuse into the circuit at a constant rate (chosen by trial and error).

Figure 2 depicts the dynamics of the RPS oscillator in the buffered setup. The ‘Vanilla’ (figure 2a ) and ‘VanillaOccluded’ (figure 2c ) settings sustain oscillations as long as the fuel and cancellation complexes are replenished. In the ‘Leaky’ setting (figure 2b ), leaks cause all three signals to amplify exponentially, speeding up the fuel consumption and causing the oscillations to die out. Figure 2d shows the dynamics of the buffered ‘LeakyCancel’ setting. Here, we observe that shadow cancellation and buffering can successfully stabilize the oscillations. We hypothesize that this restoration is possible because of buffering as it (i) maintains the primary fuel concentrations to be on par with shadow fuel concentrations so that the shadow circuit’s activity cannot overcome the activity in the primary circut; and (ii) maintains the concentrations of the cancellation complexes so that the cancellation mechanism is fast enough to sequester all of the leak activity in the primary circuit.

3.4. A leak-resilient two-species consensus protocol

Consensus protocols enable agents in a multi-agent environment to agree upon a decision under uncertainty. Developing robust biochemical consensus protocols could enable novel applications that require collective decision-making at the molecular scale [58–61]. Our target dynamical system is a consensus protocol (figure 3d ), where a leader (green) is elected among two competing species by transforming the minority species (red; species in the initial concentration minority) into majority species (green; species in the initial concentration majority) until only the ‘majority’ species remain. The CRN of this system consists of three bimolecular reactions (two BIAMP reactions and one BIMOL reaction; figure 3c ) with ${\displaystyle \mathrm{A}}$ and ${\displaystyle \mathrm{B}}$ being the competing species and ${\displaystyle \mathrm{Y}}$ being the moderator species. The ideal dynamics of this consensus protocol are shown in figure 4a (dashed), where at steady state, the ‘majority’ species reach a concentration equal to the sum of initial concentrations of all three species, whereas the concentrations of the ‘minority’ and the ‘moderator’ species go to zero.

Figure 4.

Dynamics of the two-species consensus protocol in all four simulation settings. The initial concentrations of ${\displaystyle {\displaystyle \mathrm{A},\mathrm{B}}}$ and ${\displaystyle \mathrm{Y}}$ are set to ${\displaystyle 3\text{nM},8\text{nM}}$ and ${\displaystyle 5\mathrm{n}\mathrm{M}}$ , respectively, making ${\displaystyle \mathrm{B}}$ the ‘majority’ species and ${\displaystyle \mathrm{A}}$ the ‘minority’ species. (a) Juxtaposes the ‘Vanilla’ dynamics with the ideal dynamics. The circuit reaches a consensus, albeit with a slightly lower steady-state concentration than the ideal setting owing to toehold occlusion by the fuel complexes. (b) Juxtaposes the ‘Leaky’ dynamics with the ‘Vanilla’ dynamics. Leaks cause the concentration of ${\displaystyle \mathrm{B}}$ to grow beyond the expected maximum, preventing the system from reaching a steady state. (c) Juxtaposes the ‘VanillaOccluded’ dynamics with ‘Vanilla’ dynamics. Toehold occlusion by the cancellation complexes lowers the steady-state concentration reached by ${\displaystyle \mathrm{B}}$ . (d) Juxtaposes the ‘LeakyCancel’ dynamics with the ‘VanillaOccluded’ dynamics. Including shadow cancellation restores the circuit dynamics to normality, i.e. close to the ‘VanillaOccluded’ dynamics.

The most recent DSD implementation of this consensus protocol was owing to Chen et al. [32], where the circuit was constructed using the Join-Fork framework [18]. Here, we reimplement this circuit using the React–Produce framework to maintain homogeneity among our implementations. Figure 4 illustrates the circuit’s behaviour under all four simulation settings. In all the settings, the initial concentrations of the species ${\displaystyle \mathrm{A}}$ , ${\displaystyle \mathrm{B}}$ and ${\displaystyle \mathrm{Y}}$ are set to ${\displaystyle 3\mathrm{n}\mathrm{M},8\mathrm{n}\mathrm{M}}$ and ${\displaystyle 5\mathrm{n}\mathrm{M}}$ , respectively. Here, ${\displaystyle \mathrm{B}}$ is the ‘majority’ species, ${\displaystyle \mathrm{A}}$ is the ‘minority’ species and ${\displaystyle \mathrm{Y}}$ is the ‘moderator’ species. Figure 4a shows the circuit dynamics in the ‘Vanilla’ setting, where ${\displaystyle \mathrm{B}}$ reaches a steady-state concentration of ${\displaystyle 16\mathrm{n}\mathrm{M}}$ , whereas the concentrations of ${\displaystyle \mathrm{A}\text{ and }\mathrm{Y}}$ go to zero. In the ‘Leaky’ setting (figure 4b ), the release of excess signal owing to the Produce–Helper leak causes the dynamics of ${\displaystyle \mathrm{B}}$ to diverge, i.e. grow beyond the expected maximum concentration, never reaching a steady state. Figure 4c shows the dynamics in the ‘VanillaOccluded’ setting, where we observe an initial dip in the signal concentration and a subsequent reduction in the maximum concentration reached at the steady state owing to toehold occlusion by the cancellation complexes. In the ‘LeakyCancel’ setting (figure 4d ), shadow cancellation restores the circuit dynamics to normality, where the concentration of ${\displaystyle \mathrm{B}}$ saturates after reaching a steady-state concentration comparable to the concentration reached in the ‘VanillaOccluded’ setting.

3.5. A leak-resilient proportional–integral feedback controller

A PI feedback controller is a catalytic I/O system that retroactively corrects the error in the output signal of a noise-affected system. The design of our target dynamical system (figure 3f ) is adapted from Yordanov et al. [54], which in turn was based on a result by Oishi & Klavins [29] that a feedback controller can be constructed solely using catalysis, annihilation and degradation reactions. The system consists of two interacting components—Controller and Plant. The Plant is inherently faulty and produces an output signal ${\displaystyle y}$ instead of the expected ‘reference’ output ${\displaystyle r}$ . The Controller reads this output and retroactively generates corrections to the Plant’s input ( ${\displaystyle v}$ ) such that the subsequent calculations of ${\displaystyle y}$ converge to ${\displaystyle r}$ . Feedback control [62–64] is an essential mechanism in the cellular feedback machinery that regulates the molecular concentrations inside a cell [65–67]. For this reason, synthetic biochemical controllers have the potential to lead to novel biomedical and therapeutic applications [31,32,54,68].

Figure 3e shows the CRNs of the Controller and the Plant (adapted from [54]). The signal species of the CRNs ( ${\displaystyle \mathrm{E},\mathrm{X},\mathrm{V},\mathrm{Y},\mathrm{R},\mathrm{L}}$ ) directly map to their signal counterparts ( ${\displaystyle e,x,v,y,r,load}$ ) (see figure 3f ). The ${\displaystyle \pm }$ in the superscript of the signal species indicates that the CRNs are specified in the dual-rail format, where a signal’s value is modelled as the difference in concentrations of two complementary species with opposing parity (e.g. the value of ${\displaystyle x}$ is modelled as ${\displaystyle [{\mathrm{X}}^{+}]-[{\mathrm{X}}^{-}]}$ ). The Controller CRN is constructed using catalysis, annihilation and degradation reactions [29], whereas the Plant’s CRN is constructed using produce, consume and load reactions. We refer the reader to the earlier works [29,32,54] for more details regarding the Controller architecture. We will now briefly discuss the construction of the DSD circuits for its component submodules. In doing so, we will interchangeably use the abstract signal species, their dual-rail notations and the corresponding signal strands to ease the notation, disambiguating them when necessary. A detailed description of the circuit parameters is provided in electronic supplementary material, figure S20.

Catalysis reactions of the Controller are unimolecular reactions of the form ${\displaystyle \mathrm{E}\stackrel{{\mathrm{k}}_{\mathrm{c}\mathrm{a}\mathrm{t}}}{\to }\mathrm{E}+\mathrm{X}}$ . Their DSD circuits are constructed by modifying the UNIMOL DSD circuits such that ${\displaystyle \mathrm{E}}$ is consumed during the React phase and ${\displaystyle \mathrm{E}}$ and ${\displaystyle \mathrm{X}}$ (in that order) are released during the Produce phase. Figure S27a of the electronic supplementary material juxtaposes the dynamics of the ideal catalysis reaction (dashed) with that of our React–Produce implementation (solid).

Annihilation reactions of the Controller are bimolecular reactions of the form ${\displaystyle {\mathrm{X}}^{+}+{\mathrm{X}}^{-}\stackrel{{\mathrm{k}}_{\mathrm{a}\mathrm{n}\mathrm{n}}}{\to }\mathrm{\Phi }}$ . They represent the annihilative reactions among the complementary dual-rail species of the same signal (e.g. ${\displaystyle {\mathrm{X}}^{+}}$ and ${\displaystyle {\mathrm{X}}^{-}}$ are the dual-rail species of signal ${\displaystyle \mathrm{X}}$ ). Since the reaction consumes two reactants and releases no products, its DSD circuit only includes the React phase of the BIMOL circuit. The reactants ${\displaystyle {\mathrm{X}}^{+}}$ and ${\displaystyle {\mathrm{X}}^{-}}$ are consumed during this phase, and the Flux strand released at the end is considered to be ${\displaystyle \mathrm{\Phi }}$ . Furthermore, since the second reactant is consumed slower than the first in the BIMOL DSD circuit, the DSD circuit for the annihilation reactions is constructed as a combination of the DSD circuits of two half-reactions: ${\displaystyle {\displaystyle {\mathrm{X}}^{+}+{\mathrm{X}}^{-}\stackrel{\frac{{\mathrm{k}}_{\mathrm{a}\mathrm{n}\mathrm{n}}}{2}}{\to }\mathrm{\Phi }}}$ and ${\displaystyle {\mathrm{X}}^{-}+{\mathrm{X}}^{+}\stackrel{\frac{{\mathrm{k}}_{\mathrm{a}\mathrm{n}\mathrm{n}}}{2}}{\to }\mathrm{\Phi }}$ , i.e. the reactant order is reversed in the second DSD circuit to ensure equal consumption of the two reactants. Figure S27b of the electronic supplementary material juxtaposes the dynamics of the ideal annihilation reaction (dashed) with that of our React–Produce implementation (solid).

Degradation reactions of the Controller are unimolecular reactions of the form ${\displaystyle \mathrm{V}\stackrel{{\mathrm{k}}_{\mathrm{d}\mathrm{e}\mathrm{g}}}{\to }\mathrm{\Phi }}$ . Their DSD circuit is implemented by modifying the UNIMOL circuit to consume V during the React phase. We forgo the Produce phase by considering the Flux strand released at the end of the React phase to be the ${\displaystyle {\displaystyle \mathrm{\Phi }}}$ species. Figure S27c of the electronic supplementary material juxtaposes the dynamics of the ideal degradation reaction (dashed) with that of our React–Produce implementation (solid).

Produce reactions in the Plant are catalytic unimolecular reactions of the form ${\displaystyle {\displaystyle \mathrm{V}\stackrel{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}}{\to }\mathrm{V}+\mathrm{Y}}}$ . However, their rate constant ( ${\displaystyle produce}$ = ${\displaystyle {\displaystyle 0.2{\mathrm{s}}^{-1}}}$ ) is too high for a strand displacement implementation for an extended time period (as required by the Controller). To preserve the reaction dynamics, we ‘buffer’ the fuel species into the reaction mix at a constant rate, similar to the buffering scheme in the RPS DSD circuit. Figure S27d of the electronic supplementary material juxtaposes the ideal dynamics of the Produce reaction (dashed) with our buffered React–Produce implementation (solid).

Consume reactions in the Plant are unimolecular reactions of the form ${\displaystyle {\displaystyle \mathrm{Y}\stackrel{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{u}\mathrm{m}\mathrm{e}}{\to }\mathrm{\Phi }}}$ . While the reaction looks similar to a degradation reaction, its rate constant ( ${\displaystyle consume}$ = ${\displaystyle {\displaystyle 0.1{\mathrm{s}}^{-1}}}$ ) is too high for a strand displacement implementation. Once again, we use buffering to maintain the reaction dynamics for extended periods. Figure S27e of the electronic supplementary material juxtaposes the dynamics of the ideal Consume reaction with that of our React–Produce implementation.

Load reactions in the Plant are bimolecular reactions of the form ${\displaystyle {\displaystyle \mathrm{Y}+\mathrm{L}\stackrel{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}{\to }\mathrm{L}}}$ , where the second reactant ( ${\displaystyle \mathrm{L}}$ ) catalytically consumes the first ( ${\displaystyle \mathrm{Y}}$ ). Unlike the Produce and Consume reactions in the Plant, Load reactions do not require buffering. Their DSD circuits are constructed by modifying the BIMOL DSD circuit such that the reactants ${\displaystyle \mathrm{Y}}$ and ${\displaystyle \mathrm{L}}$ (in that order) are consumed during the React phase, and only ${\displaystyle \mathrm{L}}$ is released during the Produce phase (not to be confused with the Produce reaction in the Plant). Figure S27f of the electronic supplementary material juxtaposes the dynamics of the ideal Load reaction (dashed) with that of our React–Produce implementation (solid).

The DSD circuit of the PI Controller differs from the other dynamical systems discussed in this work in the following aspects: (i) the complexity of the system arises from its dynamically interacting components rather than from autocatalytic elements and (ii) the system imposes hard constraints on the rate constants of the individual reactions, which requires a more elaborate tuning of the circuit parameters. In this work, we implement a simplistic PI Controller with a constant reference state and a constant load (more complex settings use a variable reference state, a variable load, or both [54]). The ‘Vanilla’ PI Controller circuit is constructed by linearly combining the DSD circuits of its component reactions. Figure 5b depicts the dynamics of this setting with the ideal Controller dynamics (figure 5a ; dashed) added for reference. Here, the Plant’s output ( ${\displaystyle \mathrm{Y}}$ ) oscillates around the reference ( ${\displaystyle \mathrm{R}}$ ) with damping amplitude before finally collapsing onto it. In the ‘Leaky’ setting, we only include the Produce–Helper reactions from the Catalysis DSD reactions of the Controller. While we do not assume that the annihilation and degradation circuits are leak-free, their leak magnitudes are significantly lower in comparison (as they cannot induce Produce–Helper leak) and are, therefore, ignored. Furthermore, the leaks from the Plant are ignored because the Controller is the dynamical system of interest to us. Owing to this, we do not consider this circuit to be a buffered implementation, as the Controller components do not require buffering. Figure 5c depicts the dynamics of the Controller in the ‘Leaky’ setting. Here, we observe that ${\displaystyle \mathrm{Y}}$ diverges from ${\displaystyle \mathrm{R}}$ (instead of converging onto it as expected), rendering the circuit unusable. Figure 5d depicts the circuit dynamics in the ‘LeakyCancel’ setting, where shadow cancellation restores the circuit dynamics to normality, i.e. similar to the ‘Vanilla’ setting. We ignore the ‘VanillaOccluded’ setting for this system as the initial concentration of the output ( ${\displaystyle \mathrm{Y}}$ ) is zero, and, therefore, the effect of toehold occlusion on the circuit dynamics is minimal (in qualitative terms).

Figure 5.

Domain-level DSD circuit dynamics of a PI Controller in different simulation settings. Initial concentrations are set as follows: ${\displaystyle [\mathrm{R}]=4\mathrm{n}\mathrm{M},[\mathrm{Y}]=0\mathrm{n}\mathrm{M},[\mathrm{X}]=2\mathrm{n}\mathrm{M},[\mathrm{L}]=2\mathrm{n}\mathrm{M},[\mathrm{V}]=0\mathrm{n}\mathrm{M}}$ and ${\displaystyle [\mathrm{E}]=0\mathrm{n}\mathrm{M}}$ . Both the reference state ( ${\displaystyle \mathrm{R}}$ ) and the load ( ${\displaystyle \mathrm{L}}$ ) are kept constant. (a) Ideal PI Controller dynamics. (b) Dynamics of the system in the ‘Vanilla’ setting. The Controller adjusts the Plant’s input retroactively so that the Plant’s output oscillates around the reference state with damping amplitude, finally collapsing onto it. (c) Dynamics in the ‘Leaky’ setting. The presence of leaks causes the output signal to diverge from the reference state. (d) Dynamics in the ‘LeakyCancel’ setting. Including shadow cancellation restores the circuit dynamics to normality, i.e. qualitatively closer to the ‘Vanilla’ dynamics.

4. Revisiting the design bottlenecks of shadow cancellation

The central idea behind shadow cancellation is that the leak activity should be the same in both the primary and shadow circuits so that the cancellation mechanism can sequester this activity quickly before it can amplify. To ensure that the leak activity grows alike in both the circuits, Song et al. [27] suggested that there should be a close enough match (ideally an exact match) between their rate profiles and leak characteristics. Additionally, the circuit dynamics should be slowed down to ensure that the cancellation reactions are faster than the regular strand displacement reactions in the circuit. However, since there are no existing techniques to predict the kinetic rates given the nucleotide sequences or to reverse-engineer nucleotide sequences that adhere to a given kinetic profile, constructing an effective primary–shadow circuit pair often incurs a significant design overhead.

In this section, we evaluate both qualitatively and quantitatively the extent to which the requirements prescribed by Song et al. [27] could be relaxed while preserving the circuit dynamics. For this, we used three dynamical systems chosen for their qualitative diversity: (i) UNIAMP (a pure autocatalytic amplifier), (ii) BIAMP (a steady-state system), and (iii) RPS oscillator (a non-equilibrium system). We found that in the presence of fast cancellation reactions (within the capabilities of current DNA nanotechnology), these requirements could be significantly relaxed through simple adjustments to the shadow fuel concentrations. In the rest of the article, we use the following phrases frequently and so define them here for clarity: (i) signal activity—signal strands generated through legitimate strand displacement pathways; (ii) leak activity—signal strands generated through leak pathways and their subsequent interactions with the circuit; and (iii) overall activity—the sum of the signal and leak activities, i.e. observed signals.

4.1. Shadow circuit has a different rate profile from the primary circuit

The first design requirement of shadow cancellation stipulates that the rate kinetics of the shadow circuit match closely with that of the primary circuit. A faster primary circuit leads to excess leak, causing the circuit dynamics to explode, whereas a faster shadow circuit prunes legitimate signal activity in the primary circuit, causing underexpression and, in some extreme cases, extinction of the primary circuit’s activity (see ‘LeakyCancel’ dynamics of the unbuffered RPS oscillator in electronic supplementary material, figure S26d). However, the limited oligonucleotide design space owing to the possibility of secondary structures [69], the orthogonality constraints between the primary and shadow circuits, and the unavailability of methods to reliably map the nucleotide sequences to their rate kinetics, make it highly probable for the designed primary and shadow circuits to differ in their rate profiles. We refer to this disparity as the shadow circuit being out-of-phase with the primary circuit.

In simulating the out-of-phase dynamics, we assume that the shadow circuit has a faster rate profile than the primary circuit. This can be enforced easily, as the two circuits are structurally similar and, therefore, interchangeable. The approximate rate laws derived for the general unimolecular and bimolecular reactions (see electronic supplementary material S2) suggest that a ${\displaystyle p\times }$ increase in the strand displacement rate constants raises the overall circuit speed by ${\displaystyle p\times }$ (see electronic supplementary material S7, for simulation proof). Thus, the out-of-phase behaviour can be simulated by freezing the primary circuit’s rate constants (collectively referred to as ${\displaystyle k_{primary}}$ ) and uniformly scaling up the shadow circuit’s rate constants ( ${\displaystyle k_{shadow}}$ ).

We examine our chosen dynamical systems under two different settings of ${\displaystyle k_{primary}}$ : (i) ${\displaystyle {\displaystyle k_{primary}\approx 1\times {10}^{-5}\mathrm{n}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}}$ ( ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-5}}$ ) and (ii) ${\displaystyle {\displaystyle k_{primary}\approx 1\times {10}^{-4}\mathrm{n}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}}$ ( ${\displaystyle {\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-4}}}$ ). To enforce physical constraints in our simulations, we set the maximum possible value of the DSD rate constant to ${\displaystyle {\displaystyle {10}^{-3}\mathrm{n}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}}$ (equivalently ${\displaystyle {\displaystyle {10}^6{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}}$ ) [19]. Consequently, a scale-up of ${\displaystyle {\displaystyle 100\times }}$ is possible in the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-5}}$ setting and a scale-up of ${\displaystyle {\displaystyle 10\times }}$ is possible in the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-4}}$ setting. Here, the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-5}}$ setting provides insight into the effectiveness of shadow cancellation when there is a high degree of disparity between the primary and shadow rate profiles, whereas the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-4}}$ setting clarifies the method’s effectiveness in conditions where the leak grows at a faster rate. Finally, to isolate the effect of dissimilar rate profiles, we preserve the leak rate constants between the two circuits ( ${\displaystyle k_{leak}^{primary}=k_{leak}^{shadow}=2\times {10}^{-8}{\mathrm{n}\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}$ ).

We choose four representative speed-ups of the shadow circuit in each ${\displaystyle k_{primary}}$ setting, henceforth referred to as the rate-perturbed configurations: (i) ${\displaystyle 1\times ,30\times ,70\times \text{ and}100\times }$ in the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-5}}$ setting (we only show ${\displaystyle 100\times }$ for RPS) and (ii) ${\displaystyle 1\times ,2\times ,5\times \text{ and}10\times }$ in the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-4}}$ setting. Since the shadow circuit has a faster rate profile than the primary circuit in these configurations, the leak in the shadow circuit grows faster than in the primary circuit, damaging the primary circuit dynamics. To counterbalance this disparity in leak growth, we scale down the shadow circuit’s fuel concentrations proportionally in each configuration using the formula

$${\displaystyle [\mathrm{F}\mathrm{u}\mathrm{e}\mathrm{l}{]}_{\mathrm{n}\mathrm{e}\mathrm{w}}={\displaystyle \frac{[\mathrm{F}\mathrm{u}\mathrm{e}\mathrm{l}{]}_{\mathrm{o}\mathrm{l}\mathrm{d}}}{p^{\mathrm{S}\mathrm{C}\mathrm{A}\mathrm{L}\mathrm{E}}}},}$$

where ${\displaystyle p}$ represents the perturbation factor given by ${\displaystyle p={\displaystyle \frac{k_{shadow}}{k_{primary}}}}$ , and ${\displaystyle \mathrm{S}\mathrm{C}\mathrm{A}\mathrm{L}\mathrm{E}}$ represents the scale factor, determined by trial-and-error and depends on the independent rate profiles of the primary and shadow circuits (a complete list of the scale factors used in our simulations is provided in electronic supplementary material S7). Note that, however, this scale-down does not imply that the kinetic profile of the shadow circuit is coerced to match with the primary circuit. For example, we show in electronic supplementary material, figure S21a, that even after scaling down, the shadow circuit runs at a different rate from the primary circuit (note that the inputs of the shadow circuit are not zeroed out in this comparison).

Figure 6 illustrates the results of the out-of-phase simulations in the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-5}}$ setting. We show the results for the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-4}}$ setting in electronic supplementary material, figures S11 and S12. Surprisingly, we found that the signal activity remains relatively unperturbed in all three dynamical systems under different rate-perturbed configurations (UNIAMP in figure 6a ; BIAMP in figure 6b ; and RPS in figure 6c ), despite the relative difference in the rate profiles of the primary and shadow circuits. For example, the RPS oscillator in figure 6c sustained oscillations even when the rate constants of the shadow circuit are two magnitudes ( ${\displaystyle 100\times }$ ) higher than the primary circuit.

Figure 6.

Dynamics of the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-5}}$ rate-perturbed configurations of the UNIAMP (a,d), BIAMP (b,e), and RPS (c,f) circuits. The top row shows the signal dynamics, and the bottom row shows the leak dynamics in the presence (with cancellation) and absence of shadow cancellation (without cancellation). In all the figure labels, ${\displaystyle {\displaystyle p\times }}$ indicates that the rate constants of the shadow circuit are a factor of ${\displaystyle p\times }$ higher than the rate constants of the primary circuit ( ${\displaystyle p={\displaystyle \frac{k_{shadow}}{k_{primary}}}}$ ). Initial concentrations: UNIAMP ( ${\displaystyle [\mathrm{S}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{l}]=1\mathrm{n}\mathrm{M}}$ ); BIAMP ( ${\displaystyle [\mathrm{C}]=10\mathrm{n}\mathrm{M},[\mathrm{B}]=10\mathrm{n}\mathrm{M}}$ ) and RPS ( ${\displaystyle [\mathrm{A}]=11\mathrm{n}\mathrm{M},[\mathrm{B}]=10\mathrm{n}\mathrm{M},[\mathrm{C}]=3\mathrm{n}\mathrm{M}}$ ). In each dynamical system, the signal dynamics remain relatively stable, i.e. comparable to the ${\displaystyle 1\times }$ dynamics, as shown in (a–c). This indicates that the shadow cancellation effectively sequesters the leak before it can amplify. This observation is further bolstered by the fact that the leak activity is subdued close to zero in the presence of shadow cancellation, whereas it amplifies exponentially in its absence, as seen in (d–f).

Note that this is possible only when the instantaneous leak activities in the two circuits are similar to each other, and the cancellation mechanism effectively sequesters this leak activity quickly from the circuit. To test this hypothesis, we measured the leak activity in the circuit under different rate-perturbed configurations. However, since we cannot measure the leak activity in the primary circuit, as it is indistinguishable from the signal activity, we use the residual activity (overall activity) in the shadow circuit as a substitute for the residual leak activity (remaining leak activity after cancellation) in the primary circuit. This is an appropriate substitution as (i) the overall activity in the shadow circuit is solely owing to leaks and (ii) since the shadow circuit is structurally similar to the primary circuit but runs faster, its instantaneous leak activity acts as an upper bound on the leak activity of the primary circuit. To isolate the effect of the cancellation mechanism in preserving the dynamics, we measure this leak activity in both the presence and absence of shadow cancellation (UNIAMP in figure 6d ; BIAMP in figure 6e ; and RPS in figure 6f ). As expected, we observed that the leak activity amplifies exponentially without shadow cancellation (solid; w/o cancellation; note that the curves do not vary linearly with ${\displaystyle p}$ owing to the nonlinear scale-down of concentrations in the shadow circuit), whereas, with shadow cancellation (dashed), it is subdued close to zero.

The close-to-zero leak activity, despite the relative rate difference between the primary and shadow circuits, indicates that the cancellation mechanism sequestered the leak before it had a chance to interact with its parent circuit and amplify. As a result, the task of sequestering the leak activity now diminishes to the task of sequestering the leak signal generated by the leak pathways as soon as it appears in the circuit. It only remains to ensure that the rate of leak generation should remain adequately close in the primary and shadow circuits. This could be easily achieved by simply adjusting the concentrations of the fuel species in the shadow circuit proportionally. Furthermore, the fuel dissipated owing to the leak pathways would not be significant enough to affect the circuit dynamics as the leak pathways are considerably slower than the normal strand displacement reactions (about six orders of magnitude slower [19,35,70]). Thus, assuming that the cancellation reactions are fast and the leak generation rates of the primary and shadow circuits are maintained adequately close (with shadow leak rate slightly higher), shadow cancellation can effectively preserve the circuit dynamics, with the circuit being functional even at ${\displaystyle 100\times }$ rate perturbation.

4.2. Shadow circuit has a different leak-profile from the primary circuit

The second design requirement of shadow cancellation stipulates that the shadow circuit should have similar leak characteristics to the primary circuit. Recall that in the out-of-phase simulations in §4.1, we assumed that the leak pathways in the two circuits have identical rate constants. However, since base pair fraying at the helix ends is the rate-determining step in a toeless-SD leak pathway [35,70], a difference in the nucleotide sequence at the frayed ends could lead to dissimilar leak characteristics between the two circuits. Once again, we assume that the leak rate constants of the shadow circuit ( ${\displaystyle k_{leak}^{shadow}}$ ) are higher than the leak rate constants of the primary circuit ( ${\displaystyle k_{leak}^{primary}}$ ), as the leak activity in the primary circuit would grow unhindered otherwise. Here, we only alter the leak rate constants between the two circuits to isolate the effect of dissimilar leak characteristics while keeping their normal strand displacement rate constants the same ( ${\displaystyle k_{primary}=k_{shadow}}$ ). This difference in the leak characteristics is simulated as follows. We freeze the value of ${\displaystyle k_{leak}^{primary}}$ at ${\displaystyle 1\times {10}^{-8}\mathrm{n}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}$ and vary the value of ${\displaystyle k_{leak}^{shadow}}$ from ${\displaystyle 1\times {10}^{-8}}$ to ${\displaystyle 8\times {10}^{-8}}$ in increments of ${\displaystyle {10}^{-8}}$ (the bounds are adapted from [35]), collectively referred to as the leak-perturbed configurations. Note that since ${\displaystyle {\displaystyle k_{leak}^{shadow}>k_{leak}^{primary}}}$ , we once again scale down the shadow fuel concentrations proportionally according to the formula

$${\displaystyle {\displaystyle [\mathrm{F}\mathrm{u}\mathrm{e}\mathrm{l}{]}_{\mathrm{n}\mathrm{e}\mathrm{w}}={\displaystyle \frac{[\mathrm{F}\mathrm{u}\mathrm{e}\mathrm{l}{]}_{\mathrm{o}\mathrm{l}\mathrm{d}}}{p^{\mathrm{S}\mathrm{C}\mathrm{A}\mathrm{L}\mathrm{E}}}}p={\displaystyle \frac{k_{leak}^{shadow}}{k_{leak}^{primary}}},}}$$

where ${\displaystyle p}$ represents the perturbation factor and ${\displaystyle \mathrm{S}\mathrm{C}\mathrm{A}\mathrm{L}\mathrm{E}}$ represents the scale factor determined by trial-and-error (see electronic supplementary material S8 for the detailed list of the scale factors used in our simulations). In practice, the scale factors can be calculated by comparing the independently measured kinetics of the primary and shadow circuits. Once again, we highlight that these concentration adjustments are not meant to equalize the leak activity in the two circuits but to reduce the disparity in the leak generation rates between them. We show in electronic supplementary material, figure S21b, that despite the scale-down, the leak-perturbed configuration of the BIAMP circuit still leaks at a higher rate than the primary circuit.

Figure 7 illustrates the dynamics of our chosen dynamical systems under different leak-perturbed configurations. We chose the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-4}}$ setting for these simulations as its leak growth is relatively faster than the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-5}}$ setting. We provide the results for the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-5}}$ setting in electronic supplementary material, figure S16. In all the leak-perturbed configurations of all three dynamical systems, we observe that the circuit dynamics remain stable even when the values of ${\displaystyle k_{leak}^{primary}}$ and ${\displaystyle k_{leak}^{shadow}}$ are at either extremum of the possible range, i.e. ${\displaystyle k_{leak}^{primary}=1\times {10}^{-8}\mathrm{n}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}$ and ${\displaystyle k_{leak}^{shadow}=8\times {10}^{-8}\mathrm{n}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}$ . We once again hypothesize that this counterintuitive behaviour is owing to the presence of fast cancellation reactions that sequester the generated leak before it can interact with its parent circuit.

Figure 7.

Dynamics of the leak-perturbed configurations of the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-4}}$ setting in the UNIAMP (a,d,g), BIAMP (b,e,h), and RPS (c,f) circuits. ${\displaystyle {\displaystyle \mathrm{\Delta }Leak=\frac{k_{leak}^{shadow}-k_{leak}^{primary}}{k_{leak}^{primary}}\times 100\mathrm{\%}}}$ , where ${\displaystyle k_{leak}^{primary}}$ and ${\displaystyle k_{leak}^{shadow}}$ are the leak rate constants in the primary and shadow circuits, respectively. In all the settings, ${\displaystyle k_{leak}^{primary}}$ is frozen at ${\displaystyle 1\times {10}^{-8}\mathrm{n}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}$ and ${\displaystyle k_{leak}^{shadow}}$ is increased gradually from ${\displaystyle 1\times {10}^{-8}\mathrm{n}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}$ to ${\displaystyle 8\times {10}^{-8}\mathrm{n}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}}$ in increments of ${\displaystyle {10}^{-8}}$ (only ${\displaystyle k_{leak}^{shadow}}$ = ${\displaystyle {\displaystyle \{1\times {10}^{-8}}}$ , ${\displaystyle 3\times {10}^{-8}}$ , ${\displaystyle 6\times {10}^{-8}}$ , ${\displaystyle {\displaystyle 8\times {10}^{-8}\}}}$ corresponding to ${\displaystyle \mathrm{\Delta }Leak=\{0\mathrm{\%},200\mathrm{\%},500\mathrm{\%},700\mathrm{\%}\}}$ are shown here). Initial concentrations: UNIAMP ( ${\displaystyle {\displaystyle [\mathrm{S}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{l}]=1\mathrm{n}\mathrm{M}}}$ ); BIAMP ( ${\displaystyle [\mathrm{C}]=10\mathrm{n}\mathrm{M},[\mathrm{B}]=10\mathrm{n}\mathrm{M}}$ ); RPS ( ${\displaystyle [\mathrm{A}]=11\mathrm{n}\mathrm{M},[\mathrm{B}]=10\mathrm{n}\mathrm{M},[\mathrm{C}]=3\mathrm{n}\mathrm{M}}$ ). In all the simulations, the concentrations of the fuel species in the shadow circuit are scaled down to ensure similar leak activity between the primary and shadow circuits (details in electronic supplementary materials, S8). (a–c) Signal activity in the primary circuit in the presence of shadow cancellation. The circuit dynamics in all three systems remain relatively stable (close to the ${\displaystyle \mathrm{\Delta }Leak=0\mathrm{\%}}$ case) despite the dissimilarity between ${\displaystyle k_{leak}^{primary}}$ and ${\displaystyle k_{leak}^{shadow}}$ , being stable even when the values are at either extremum of the possible range. (d–f) Residual leak in the primary circuit, estimated using the overall activity in the shadow circuit as a proxy. In all the settings, the residual leak is subdued close to zero, indicating that shadow cancellation has successfully sequestered all the leaks. (g,h) The cumulative amount of leak consumed (from the difference between initial and current concentrations) and the cancelled leak (from the slope) are estimated using the concentration dynamics of the cancellation complexes. The dynamics of the cancellation complexes remain similar under different leak-perturbed configurations, indicating that the leak is sequestered before it can amplify (note that the fuel concentrations were adjusted to keep the leak generation rate similar in each configuration).

We adopt the following methodology to explain this behaviour. We assume that the instantaneous leak activity consists of two components: (i) residual leak—the leak activity that remained in the primary circuit after cancellation and (ii) cancelled leak—the leak activity sequestered by the cancellation mechanism. Similar to the previous section (§4.1), the residual leak in the primary circuit is measured using the residual signal activity in the shadow circuit, and the cancelled leak is measured using the concentration dynamics of the cancellation complexes. Here, the cancellation complex dynamics serve two purposes: (i) the absolute value of the concentration of the cancellation complexes consumed could be used to estimate the cumulative amount of leak sequestered from the circuit (calculated as the difference between initial and current concentrations) and (ii) the slope of the curve at a given instant represents the amount of cancelled leak.

We observe that the amount of residual leak is subdued close to zero in all the dynamical systems in all leak-perturbed configurations (UNIAMP in figure 7d ; BIAMP in figure 7e ; and RPS in figure 7f ), thus validating our hypothesis. Furthermore, we observe that the concentration dynamics of the cancellation complexes in each dynamical system follow similar curves under different leak-perturbed configurations (UNIAMP in figure 7g ; BIAMP in figure 7h ; not provided for RPS as the cancellation complexes are buffered) despite the disparity in their leak rate constants. This indicates that the cumulative amount of leak generated and the leak cancelled at a given instant remain similar across the leak-perturbed configurations. Note that both these observations are possible only when the cancellation mechanism precludes the leak from amplifying. This shows that in the presence of fast cancellation reactions, shadow cancellation is robust to the difference in the leak characteristics between the primary and shadow circuits, provided that the necessary concentration adjustments are made.

4.3. Trade-off between fast cancellation reactions and toehold occlusion

The third requirement stipulates that the cancellation reactions should be significantly faster than the strand displacement reactions in the circuit. Recall that fast cancellation reactions were key to the effectiveness of shadow cancellation in both rate-perturbed and leak-perturbed simulations as it would allow for the sequestration of leak as and when it appears in the circuit, preventing its amplification. Currently, the DSD circuits are restricted to the low-concentration regime (maximum concentration possible is ≈ 10⁴ nM [19]) as the leaks grow uncontrollably at higher concentrations. However, by enabling increasingly faster cancellation reactions, it could, in theory, be possible to stem the leak prematurely and achieve the much desired concentration scale-up.

A straightforward way to achieve this speed-up is by increasing the concentrations of the cancellation complexes. This, however, has a retroactive effect on the circuit dynamics as the circuit signals engage in nonspecific interactions with the cancellation complexes known as toehold occlusion [28,49], where the cancellation complexes temporarily sequester the signal strands by their toeholds (in the absence of their matching signal). This manifests as an initial dip in the signal concentrations and a subsequent reduction in the circuit activity (see ‘VanillaOccluded’ and ‘LeakyCancel’ settings of the BIAMP circuit in figure 1b ). Consequently, this scale-up causes an increasingly higher amount of the signal to be sequestered in the cancellation complexes, which results in the suppression of the circuit activity and the slowing down of the circuit dynamics. Figure 8 illustrates this effect in the RPS oscillator circuit. As the concentrations of the cancellation complexes are scaled up, the initial dip in the signal concentrations increases, oscillation amplitude decreases—indicating a drop in the circuit activity, and the oscillation frequency decreases—indicating the slowing down of the circuit dynamics, with the circuit being of no practical use beyond a certain threshold (figure 8c,d ).

Figure 8.

RPS circuit dynamics in the ‘LeakyCancel’ setting under increasing concentrations of the cancellation complexes. Demonstrates the design bottleneck imposed by toehold occlusion for accelerating cancellation reactions by increasing the concentrations of the cancellation complexes. As the concentrations of the cancellation complexes are increased from ${\displaystyle 60\mathrm{n}\mathrm{M}}$ in (a) to ${\displaystyle 600\mathrm{n}\mathrm{M}}$ in (d), more toehold occlusion is observed, as seen from reduced amplitude and slowed-down oscillations, rendering the circuit unusable at higher concentrations, as seen in (b, c,d).

Srinivas et al. [28] studied the issue of toehold occlusion in the React–Produce framework, wherein the React complexes temporarily sequester the signal strands. To combat this sequestering, the authors repurposed the so-called Backward strand in the framework into a fuel species to keep most of the signal in its ssDNA form (electronic supplementary material S1). They also suggested weakening the toeholds of the React complexes to prevent excessive sequestration. However, both these measures have a negative effect of slowing down the circuit dynamics. We provide an example simulation of reducing the toehold binding strength of the cancellation complexes in the RPS oscillator at the ‘primary circuit end’ in electronic supplementary material S6. Circuit designs that could reduce or eschew toehold occlusion without affecting the circuit dynamics could significantly improve the effectiveness of the overall shadow cancellation methodology and provide a plausible way for scaling up strand displacement circuits. In this work, we tune the the initial concentrations of the cancellation complexes, trading-off between maximizing leak suppression and minimizing toehold occlusion.

5. Discussion

One of the primary impediments to applying shadow cancellation in arbitrary strand displacement circuits is the increased circuit complexity caused by the doubling of the circuit size owing to the presence of the shadow circuit. Here, we measure the circuit size by the number of active reaction pathways. Owing to the limited oligonucleotide design space [69], this increased circuit size could lead to cross-talk between the primary and shadow circuits, which violates the orthogonality constraints, and disrupts the circuit dynamics. However, in the prior rate-perturbed and leak-perturbed simulations, we showed that when the concentrations are appropriately adjusted through a proportional scale-down, the leak activity in the circuit could be restricted solely to the signal generated by the leak pathways. This implies that the reaction pathways involving the shadow fuel species would not be triggered (apart from the leak pathways). For example, in the context of the React–Produce implementation of the BIAMP DSD circuit (see electronic supplementary material S1), this means that the reaction where the signal strand interacts with the React complex will not be triggered or will be triggered very minimally, suggesting that the normal strand displacement pathways in the shadow circuit are passive participants in the shadow cancellation process. To verify this hypothesis, we measured the concentration of the React complex consumed in the BIAMP shadow circuit ( ${\displaystyle \mathrm{s}\mathrm{h}\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{C}\mathrm{B}\mathrm{C}\mathrm{j}}$ ) under different rate-perturbed configurations (figure 9a ). Here, we observe that the concentration of this complex remained relatively constant in all of them for the entirety of the simulation, indicating that the normal strand displacement pathways in the shadow circuit are only minimally triggered (note that the React complex is the first substrate to be consumed when the DSD circuit for a reaction is triggered). Therefore, it could be possible, in theory, to intelligently cut down the size of the shadow circuit from ${\displaystyle \mathcal{O}(N)}$ to ${\displaystyle \mathcal{O}(1)}$ (in terms of the number of active reaction pathways), for example, by removing the toeholds of the shadow fuel species. Such a size reduction could further relieve the orthogonality and design restrictions. We believe that this is an exciting avenue for future experimental work.

Figure 9.

(a) Concentration of the React complex in the shadow circuit ( ${\displaystyle \mathrm{s}\mathrm{h}\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{C}\mathrm{B}\mathrm{C}\mathrm{j}}$ ) under different rate-perturbed configurations of the ${\displaystyle {\mathrm{K}}_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{y}}^{1\mathrm{e}-5}}$ setting. We observe that the concentrations stay relatively constant for the entirety of the experiment, indicating that the normal strand displacement pathways in the shadow circuit are not being triggered. (b) Results of scaling up all the circuit concentrations by ${\displaystyle 5\times }$ in the ‘LeakyCancel’ setting of the BIAMP and RPS circuits (indicated in the picture) showing that shadow cancellation cannot restore their dynamics under such a scale-up.

In light of the possibility of such a reduced design, we suggest the following straightforward yet effective automated design strategy for constructing the shadow circuit. Given only a sequence-level description of the primary circuit (note that we only present the strategy for constructing the full circuit with the assumption that the above discussed reductions are applied post facto):

1. The shadow circuit is designed to have a one-to-one correspondence with the primary circuit in terms of domains, substrates and reactions.

2. The nucleotide sequence of a shadow circuit domain is constructed by simply reversing the sequence of its matching primary circuit domain (electronic supplementary material, figure S22a).

3. A shadow circuit strand is constructed by reversing the domain order of its corresponding primary circuit strand and substituting the domains with their matching shadow domains (electronic supplementary material, figure S22b). Note that the combined action of reversing the domain sequences and reversing the domain order in a strand is equivalent to simply reversing the sequence of that strand.

4. The nucleotide sequences of all the domains are additionally modified to ensure no cross-talk between the primary and shadow circuits.

Intuitively, this strategy preserves the domain sequences, especially of the toeholds, which are shown to be strong indicators of corresponding strand displacement kinetics [70]. Furthermore, by assuming that a duplex is stable only when its length exceeds that of a toehold, we could, in theory, ensure orthogonality among the primary and shadow circuits by imposing the constraint that the domain sequences are nonpalindromic and the strand sequences are not self-reverse complements. A palindromic domain preserves the domain sequence between the two circuits and leads to cross-talk, whereas a primary circuit strand that is a reverse complement of itself will bind with its shadow circuit counterpart, also resulting in cross-talk.

Another significant disadvantage of shadow cancellation is that it requires slowing down the kinetics of the primary circuit so that the cancellation reactions are significantly faster than the regular strand displacement reactions in the circuit. One way to circumvent this requirement is by scaling up the concentrations of the cancellation complexes and the circuit’s fuel concentrations. However, since the leak activity grows combinatorially as the circuit is scaled up [36] it becomes increasingly hard for the cancellation mechanism to control the leak growth. Furthermore, the restriction on the concentrations of the cancellation complexes owing to toehold occlusion hinders the scaling-up of cancellation complexes, precluding fast cancellation reactions. We show the results of such a scale-up in the BIAMP and RPS circuits with negative results (figure 9b ). We did not, however, explore this avenue further in this work. Scaling up DNA circuits is a significant challenge for the field and we believe that with more focused studies, shadow cancellation will play an important role in the solution to this problem.

6. Conclusion

Shadow cancellation is a dynamic leak-elimination strategy previously proposed by our group for the purpose of error correction in strand displacement circuits. While the method was successfully demonstrated in a cross-catalytic amplifier circuit, its assumed kinetic restrictions imposed several design bottlenecks that hurt its usability.

In this work, we used domain-level DSD simulations of various catalytic and autocatalytic dynamical systems that are known to leak profusely, to reexamine the method’s capabilities, the inner workings of its components, and its perceived design restrictions. First, we showed that the method can mitigate leaks in DSD circuits of several dynamical systems of practical importance. To ensure that the simulation results are as close as possible to the experimental results, we constructed the circuits using experimentally feasible parameters. In all the dynamical systems, we observed that in the absence of shadow cancellation, leaks swiftly degraded the circuit function, whereas the introduction of shadow cancellation restored the circuit dynamics to normality.

We then performed several probing simulations to revisit the method’s design requirements, i.e. near-identical rate profiles, similar leak characteristics, and fast cancellation reactions, and tested the method’s sensitivity to deviations in each of these requirements. Through these simulations, we showed that these requirements could be significantly relaxed with simple concentration adjustments to the shadow circuit while preserving the circuit dynamics. First, we showed that shadow cancellation is effective even when the rate constants are ${\displaystyle 100\times }$ higher in the shadow circuit than in the primary circuit. Then, we showed that the method can stabilize the dynamics even when the leak rates of the primary and shadow circuits are at either extremum of the possible range of leak rate constants. Finally, we showed that toehold occlusion due to cancellation complexes causes a significant hindrance to enabling fast cancellation reactions and address this issue partially by trading off between maximizing leak suppression and minimizing toehold occlusion.

We then discussed several challenges associated with the method, such as doubling of the circuit size owing to the shadow circuit, the orthogonality constraints, and the issues with scaling up the concentrations for enabling fast strand displacement reactions, and provided rational suggestions for tackling each of the challenges, setting up future experimental work. Our work is thus a positive step towards constructing robust and durable enzyme-free strand displacement systems with complex dynamical behaviours.

Ethics

This work did not require ethical approval from a human subject or animal welfare committee.

Data accessibility

Code is available from Dryad [71].

Supplementary material is available online at [72].

Declaration of AI use

We have not used AI-assisted technologies in creating this article.

Authors’ contributions

R.T.N.: conceptualization, formal analysis, investigation, methodology, resources, software, validation, visualization, writing —original draft, writing—review and editing; J.H.R.: conceptualization, funding acquisition, project administration, resources, supervision, writing—review and editing.

Both authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Competing interests

The authors have no competing interests.

Funding

This work was funded by the National Science Foundation under grant nos. 1909848 and 2113941 to J.H.R.