Social DNA Nanorobots

Abstract

We describe social DNA nanorobots, which are autonomous mobile DNA devices that execute a series of pair-wise interactions between simple individual DNA nanorobots, causing a desired overall outcome behavior for the group of nanorobots which can be relatively complex. We present various designs for social DNA nanorobots that walk over a 2D nanotrack and collectively exhibit various programmed behaviors. These employ only hybridization and strand-displacement reactions, without use of enzymes. The novel behaviors of social DNA nanorobots designed here include: (i) Self-avoiding random walking, where a group of DNA nanorobots randomly walk on a 2D nanotrack and avoid the locations visited by themselves or any other DNA nanorobots. (ii) Flocking, where a group of DNA nanorobots follow the movements of a designated leader DNA nanorobot, and (iii) Voting by assassination, a process where there are originally two unequal size groups of DNA nanorobots; when pairs of DNA nanorobots from distinct groups collide, one or the other will be assassinated (by getting detached from the 2D nanotrack and diffusing into the solution away from the 2D nanotrack); eventually all members of the smaller groups of DNA nanorobots are assassinated with high likelihood. To simulate our social DNA nanorobots, we used a surface-based CRN simulator.

1 Introduction

1.1 Motivation

Due to its simple base-pairing and predictable secondary structure, DNA is an ideal material to construct useful molecular-scale objects and devices. DNA nanorobots are molecular-scale synthetic devices composed primarily of DNA that can execute a variety of operations. Autonomous DNA nanorobots operate without outside mediation. DNA walkers are a class of mobile DNA nanorobots which can move over a nanotrack composed of DNA stepping stones. The nanotrack may be 1D or 2D and may be either a self-assembled DNA nanostructure or a set of DNA strands affixed to a surface. Autonomous DNA nanorobots have been demonstrated to perform a small number of moderately complex tasks including walking over nanostructures, maze traversal, and cargo delivery activities.

A major remaining challenge in the field of DNA nanoscience is to increase the complexity and diversity of the activities DNA nanorobots can perform in spite of practical limitations on the complexity of each individual DNA nanorobot. How can one design molecular-scale systems with multiple mobile autonomous nanorobots which exhibit a group behavior that is significantly more complex than the behavior of individual nanorobots? We take inspiration from the field of Sociobiology pioneered by Wilson, which demonstrated how social insects such as ants and honeybees perform a wide variety of relatively complex organized group behaviors, even though the individual insects have quite limited brains.

1.2 Summary of Our Results

We describe social DNA nanorobots, which are autonomous mobile DNA devices that execute a series of pair-wise interactions between simple individual DNA nanorobots, causing a desired overall outcome behavior for the group of nanorobots which can be relatively complex. We present various designs for social DNA nanorobots that walk over a 2D nanotrack and collectively exhibit various programmed behaviors. In our designs, we employed only hybridization and strand-displacement reactions, without use of enzymes; strand-displacement reactions are used for communication between pairs of nearby DNA nanorobots (where a finite amount of information is transferred between a pair of nearby DNA nanorobots). The novel behaviors of social DNA nanorobots designed here include:

(i) Self-avoiding random walking, where a group of DNA nanorobots randomly walk on a 2D nanotrack and avoid the locations visited by themselves or any other DNA nanorobots.

(ii) Flocking, where a group of DNA nanorobots follow the movements of a designated leader DNA nanorobot, and

(iii) Voting by assassination, a process where there are originally two unequal size groups of DNA nanorobots; when pairs of DNA nanorobots from distinct groups collide, one or the other will be assassinated (by getting detached from the 2D nanotrack and diffusing into the solution away from the 2D nanotrack); eventually all members of the smaller groups of DNA nanorobots are assassinated with high likelihood.

We simulated our social DNA nanorobots using the Surface CRN Simulator of Clamons[1, 2].

1.3 Organization

• In Sect. 2, we overview the field of Sociobiology and in particular the behavior of social insects.

• In Sect. 3, we review prior DNA nanorobots, including prior DNA walkers, and programmable DNA nanomachines.

• In Sect. 4, we present our designs and simulations of various social DNA nanorobots.

• In Sect. 5, we conclude with a discussion of future work and challenges, including: further development of simulation software for social nanorobots, experimental demonstrations of social DNA nanorobots, design of further social DNA nanorobot behaviors, and possible use of the diffusion of pheromone-like DNA molecules for communication between social DNA nanorobots.

2 Sociobiology

The concept of Sociobiology was defined by Wilson [3] in 1975. The field of Sociobiology aims to investigate and explain the evolved social behaviors of social animals. Sociobiology studies have been made for example of mating patterns, aggression, nurturance, pack hunting, and the hive society of social insects. High-level social organizations can be found in social insects that have the following three characteristics: cooperative brood care, overlapping generations, and a division of labor into reproductive and non-reproductive groups[4,5,6,7,8]. Social insects include ants and termites, and some social bees and wasps. Social insects gain several advantages by living together. They work together to gain resources, share their findings with each other, defend their home when under attack, and attack other insects for territory and food. Social insect communities are divided into castes by their function and behavior; these include a reproductive caste (e.g., the queen) and the sterile caste (soldiers and various types of workers) [9]. The reproductive caste carries out the basic function of reproduction, and the sterile castes do various types of tasks including taking care of the reproductive members. There are generally multiple subcastes of insects within the sterile castes, which do specialized tasks. For example, the soldiers defend the colony against predators and the workers are responsible for foraging, construction and repair of the nest and feeding the larvae and brood care, all tasks that are typically done by specialized sterile subcastes (which may depend on the insects age or development).

The communication signals between the social insects required for this control can be mechanical, optical, or chemical. Mechanical communication between social insects includes direct physical contact between members of the insect colony. Optical communication between social insects includes visual displays and stylized movements, sometimes termed dances, that communicate discoveries of food and their locations. Olfactory communication between social insects includes chemical tracks that specify a path to important locations, as well as diffused chemical factors known as pheromones. Pheromones can trigger a social response in members of the same species, also often playing an important role in the development and maintenance of insect society [10]. Also, ants and termites are specialized to perform specific functions (e.g., attacking & foraging) and can communicate and lay down chemical tracks to specify a path to important locations such as food sources. (See Sect. 5 for a discussion of possible use of diffusion of pheromone-like molecules for communication between nanorobots.)

Quite surprisingly, the overall control of these insect communities is not done by the reproductive caste, but instead the control is done distributively within the sterile subcastes via group social interactions. Particularly, complex collective behavior is found in honeybee colonies, where individual honeybees can be specialized for foraging and harvesting. For example, after traveling outside the colony’s nest to find possible locations of flowers with pollen, successful honeybee foragers can share information about the direction and distance to a food source [11] by use of a flying dance known as a waggle dance (a particular figure-eight dance) [11,12,13]. These dances communicate (i) the existence of pollen, (ii) its quantity, and also (iii) its general direction. Also, the honeybees of the species Apis mellifera perform tremble dances, which recruit receiver bees to collect nectar from returning foragers [14]. Seeley [15] demonstrated that honeybees also use a form of democratic voting (executed without input from the queen bee) to make important decisions, such as the best source of flowering plants providing pollen or the best location for a new hive home for the honeybee colony.

Our paper is not concerned with evolution per se, but we propose designs (rather than evolution) of nanorobots which are derived from prior Sociobiology studies of these behaviors of social insects. We are particularly inspired by the social behaviors of social insects such as ants and bees, which exhibit complex collective behavior, even though the individual insects have quite limited brains, a property shared also by DNA nanobots. Notable diverse activities of social insects that we aim to mimic using DNA nanobots include:

• Random walking: where insects of the colony make random walks.

• Flocking: a group of insects of the colony follow a selected leader individual insect.

• Guarding: a group of insects of the colony follow, and guard from attack by another group, a particular individual insect of the colony.

• Attacking: a group of insects of the colony attack another group.

• Communication: between insects of the colony.

• Democratic group decision making (voting): among groups of insects of the colony.

• Foraging: a select foraging group of insects of the colony leave the colony and attempt to discover new sources of food, and then report back to the colony their discoveries.

• Harvesting: a harvesting group of insects of the colony travel (navigating by either (a) following successful foragers or (b) following their chemical trail, or (c) via instruction from successful foragers) to the new sources of food and harvest it.

3 Prior DNA Nanorobots

We use the term nanorobot for a molecular-scale device that can execute a variety of operations. A DNA (or RNA) nanorobot is a nanorobot composed of nucleic acids. Our overall aim is to take inspiration from the field of Sociobiology to design novel nanorobotic systems, but our designs will employ in part various known nanorobotic design techniques, which we briefly overview in this section.

In 2000, Yurke and Turberfield [16] demonstrated the first DNA device, a DNA tweezer, that used DNA hybridization to power its movements. Their DNA tweezer was nonautonomous; its movements were controlled by adding ssDNA strands. The field of DNA nanorobotics has rapidly evolved from nonautonomous molecular devices that each successive movement needed to be controlled externally, to subsequent autonomous molecular devices that operate without control from external environment. Autonomous DNA nanorobot devices that executed in-place motions were demonstrated by [17,18,19,20].

3.1 Prior DNA Walkers

A DNA nanorobot is autonomous if it operates without outside intermediate control, and otherwise is non-autonomous. In the past decades, researchers designed and experimentally realized DNA devices that can autonomously conduct complex tasks such as autonomous walking, maze traversal, and cargo delivery activities.

A mobile DNA nanorobot is a DNA nanorobot that locomotes in some way. A DNA walker (also termed a Mobile DNA nanorobot) is a mobile DNA nanorobot which moves over a nanotrack composed of ssDNA pads. The nanotrack may be 1D or 2D and may be either self-assembled DNA nanostructure or a set of DNA strands affixed to a surface. An autonomous DNA walker is a mobile DNA nanorobot that locomotes autonomously.

The concept of the DNA walker was first defined and named by Reif [21] in 2002, who gave two autonomous designs for bidirectional movement. Sherman and Seeman [22] and Shin and Pierce [23] then experimentally realized the first DNA walkers, which were bipedal walkers that moved along a linear track. But they were nonautonomous walkers that required external control for each step. Yin et al. [24] and Tian and Mao [19] also demonstrated biped non-autonomous walkers that walked foot-over-foot along a linear track.

In 2004, the first autonomous DNA walker was experimentally demonstrated [25, 26] by Reif’s group (in collaboration with Turberfield). This autonomous DNA walker and many subsequent DNA walkers [20, 27,28,29,30,31] made use of a series of enzymic reactions to power the locomotion; for example, Sahu [29] demonstrated a DNA nanotransport device which is powered by strand-displacing polymerase $\phi$29. Other autonomous DNA walkers employed DNAzymes, for example, Pei et al. [32] demonstrated a multipedal DNA walker that moves on a 2D substrate in a biased random walk and Lund [33] demonstrated DNA walkers that traversed paths on a 2D nanostructure guided by landmark molecules affixed to the 2D nanostructure. There are also many known DNA walkers powered by hybridization reactions. Also, autonomous DNA walkers have been demonstrated that navigate networks: Wickham et al. demonstrated a DNA-based molecular motor that can be routed through a network of tracks [34]. Chao et al. [35] demonstrated a DNA walker that conducted single-molecule parallel depth-first search on a 2D DNA origami surface. Reviews of DNA-based walkers are given in [36,37,38].

3.2 Prior Programmable DNA Nanorobots

Various schemes for programmable autonomous DNA nanorobots that do computations as they walk have been described; those of Yin and Reif et al. [39] use enzymes, and those of Reif and Sahu [40] use DNAzymes. One application of the DNAzyme programmable autonomous DNA nanorobot of [40] was a DNAzyme router for programmable routing of nanostructures on a 2D DNA addressable lattice, where the 2D DNA addressable lattice is embedded with a network of DNAzymes and where the routed path for the input nanostructure could be programmed by modifying its sequence. The input was encoded as a set of hairpins on the walker. The transport of the walker across the surface simulated a finite state machine that switched states based on input, where the state of the automaton was indicated by the DNAzyme that currently binds the walker. The various DNAzymes embedded on the 2D surface consumed the input as the walker moved.

The Seeman group [41] demonstrated a non-autonomous programmable DNA nanorobot that transported a series of molecules to form a molecular-scale assembly line. Their nanorobot picked up cargo in a programmable manner when it walked on a DNA track. This process was non-autonomous since it required addition of appropriate fuel strands at specific time instants.

3.3 Prior Autonomous DNA Walkers that Do Molecular Cargo-Sorting on a 2D Nanostructure

The prior work most similar to that reported here is the work of Thubagere [42], which demonstrated an ingenious molecular-scale system with a group of autonomous DNA nanorobots executing a molecular cargo-sorting task on a 2D nanostructure. The 2D nanostructure initially had various kinds of molecular cargo that needed to be transported to different targeted locations on a DNA nanostructure (a DNA origami surface). Each DNA nanorobot traversed a random walk over the 2D nanostructures and when encountering a molecular cargo, they loaded and transported the cargo to the targeted location (the goal). They used a simple protocol to perform a complex cargo-sorting task. When the robot randomly walked on the surface, if it moved local to a cargo molecule, the nanorobot picked the cargo up and continued walking randomly. If it moved local to a goal molecule at the targeted location of the cargo molecule, the robot dropped the cargo off. The nanorobot repeated the above process until all cargo molecules were sorted. The picking up and dropping off process used strand displacement reactions.

4 Design and Simulation of Social DNA Nanorobots

Our approach is to adapt collective behavior strategies of social insects into molecular-scale nanorobots, which will be specialized to perform specific functions. We describe social DNA nanorobots, which are autonomous mobile DNA devices that execute a series of pair-wise interactions (only between pairs of nearby nanorobots) on a 2D DNA nanotrack that determine an overall desired outcome behavior as a group. We give detailed designs that program social behaviors via interactions between individual DNA molecules.

4.1 Social DNA Nanorobot Behaviors Designed and Simulated

A basic behavior of social DNA nanorobots is Random Walking, where a group of DNA nanorobots make random traversals of a 2D nanotrack. For random walking, we will make use of a known design of [42].

The novel behaviors of social DNA nanorobots described here include:

$\bullet$ Self-avoiding random walking, where a group of DNA nanorobots walk on a 2D nanotrack and avoid the locations visited by themselves or any other DNA nanorobots.

$\bullet$ Flocking, where a group of DNA nanorobots follow the movements of a designated leader DNA nanorobot.

$\bullet$ Voting by assassination, a process where there are originally two unequal size groups of DNA nanorobots; when pairs of DNA nanorobots from distinct groups collide, one or the other can be assassinated (by getting detached from the 2D nanotrack and diffusing into the solution away from the 2D nanotrack); eventually all members of the smaller groups of DNA nanorobots are assassinated with high likelihood.

4.2 Software for Stochastic Simulations of the Social DNA Nanorobots Behaviors

We made stochastic simulations of the social DNA nanorobots behaviors listed above. Our simulation model is adapted from the Surface CRN Simulator of Clamons [1, 2]. A chemical reaction network (CRN) contains chemical reactions and their rates. In a 2D surface CRN model, each individual nanorobot to be simulated is a molecule attached at a specific position on a 2D surface, so that the nanorobot can only interact with neighbors and their attachment strands. The behaviors of DNA nanorobots moving on a 2D nanotrack can be modeled in this 2D surface CRN model as a set of chemical reactions between DNA walkers and DNA strands affixed to a surface. State transitions modeled chemical reactions (e.g., toehold-mediated strand displacements and dehybridizations) between DNA walkers and DNA strands affixed to a surface. We applied the Surface CRN Simulator specifically for optimized performance assessment of our social DNA nanorobot designs. Assessment criteria are formulated to specify the performance of the designs of the social DNA nanorobots, and redesigns were made to improve performance.

In the next subsections, we give detailed domain level designs of social DNA nanorobots that conduct random walking, self-avoiding random walking, flocking and voting by assassination.

Fig. 1

Known design of a DNA nanorobot that executes a random walk (for simplicity only illustrated in 1D)

4.3 A Prior DNA Nanorobot that Autonomously Walks

There are many prior known designs for DNA nanorobots that make random traversals on a 2D DNA nanotrack. Our design for a DNA nanorobot that autonomously walks uses the design for a random DNA walker of Thubagere [42]. A nanotrack with an array of attached ssDNA pads that are self-assembled on a surface is illustrated in Figure 1 (for simplicity it is only illustrated in 1D). The 2D arrangement of pads on a 2D nanotrack is illustrated in Fig. 2. There are two types of pads: (i) ssDNA $p_0 = B^*A^*$ attached at its $3'$ end and (ii) ssDNA $p_1 = C^*B^*$ attached at its $5'$ end. (These ssDNA sequences, and all sequences given subsequently, are written in $5'$ to $3'$ order.) There is a single type of ssDNA nanorobot, the DNA Walker $W = A B C$ (see Fig. 1), which operates as follows:

• (a) A low concentration of W walker strands are added to the buffer solution containing the 2D nanotrack, and a few of these W strands hybridize to random pads of the nanotrack.

• (b) The buffer solution is replaced, so as to remove the remaining non-hybridized W walker strands from the solution surrounding the nanotrack.

• (c) As illustrated in Fig. 1, at first, the W strand hybridizes with a $p_0 = B^*A^*$ pad, which is the State 0. In State 1, the unpaired domain C of W hybridizes with domain $C^*$ of $p_1.$ Then domain $B^*$ of pad $p_1$ can displace the domain $B^*$ of pad $p_0$, so domain B of W hybridizes with both $p_0$ and $p_1$, which is the intermediate State 1.5, and this process is reversible. If the strand displacement completes, it enters State 2, in which W detaches fully from the $p_0$ pad and hybridizes with a $p_1 = C^*B^*$ pad. From State 0 to State 2, W walks from $p_0$ pad to $p_1$ pad due to hybridization and strand displacement. Similarly, W walks from $p_1$ pad to $p_0$ pad when it proceeds from State 2 to State 1.5 to State 1 to State 0.

• (d) W walks successively from either (a) the $p_0$ to the $p_1$ pad of the nanotrack, or (b) the $p_1$ to the $p_0$ pad of the nanotrack, as described above. As result, W walks randomly over the nanotrack.

Fig. 2

2D locations of the random walker’s pads on a 2D Nanotrack

Fig. 3

Simulation of a DNA nanorobot that executes a random walk (with orthogonal steps) on a 2D nanotrack

Figure 3 gives an example simulation (also see mp4 at [43]) run for the W nanorobot (in green) randomly walking on a 2D nanotrack. Where it randomly walks on the nanotrack, the grid turns to orange to show the trace of the W nanorobot. The W Walker traverses uniformly randomly over the nanotrack.

4.4 Prior Demonstrated Technique for Hybridization Inhibition of Short Sequences Within the Hairpin Loops

Our DNA nanorobots will make use of hybridization inhibition of short ssDNA sequences within the hairpin loops; this seems well established. Extensive prior published works (see survey [44]) have demonstrated (in simulation as well experimentally) localized reactions that make use of DNA hairpins for inhibition of hybridization on short ssDNA sequences within the hairpin loops. For example, the hybridization chain reaction (HCR) [45] makes use of a series of strand-displacements to open a series of DNA hairpins; while HCR was initially demonstrated [45] in well-mixed solutions, Reif’s group has shown that it also can be made to operate reliably in localized reactions. Reif’s group has made extensive probabilistic analysis and simulations (in the studies [46,47,48]) and has also experimentally implemented (in the paper [49]) a localized version of HCR where the hairpins are localized on DNA Origami.

4.5 A Novel DNA Nanorobot that Executes a Self-Avoiding Walk

A self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. SAWs have a number of important applications, e.g., in the modeling of nucleic acids, peptides, and proteins. It is known [50] that a self-avoiding random walk on the 2D square lattice lasts an average of approximately 71 steps before the walker is trapped. (Note: While we could modify our design given the below to decrease the likelihood the walker gets trapped at a $p_0$ pad position, and so increase the average number of steps before the walker is trapped, but then the resulting system would not correspond to the classical self-avoiding random walk on the 2D square lattice.)

Here, we described the design of a DNA nanorobot that makes a random self-avoiding traversal of a 2D DNA nanotrack (which is a 2D square lattice). It is the first to execute a self-avoiding walk without use of enzymes. The are two types of pads: (i) ssDNA $p_0 = B^*A^*$ attached at its $3'$ end and (ii) DNA hairpin $p_1 = C^* B^* E_1^* E_2^* B$ attached at its $5'$ end. Figure 4 illustrates (for simplicity on a 1D nanotrack) the self-assembly of the pads into DNA hairpins. Figure 5 gives the 2D arrangement of these pads on a 2D nanotrack. Note: To improve stability of the hairpin $p_1$, its loop portion $E_1^* E_2$ needs to be relatively short compared to its stem sequences B and $B^*$. The initial buffer solution $S_0$ is a conventional buffer solution with the component strands of a DNA nanotrack, and the buffer solution $S_1$ is $S_0$ plus a high concentration of another DNA hairpin type $h_1 = E_2 E_1 B E_1^*$.

The operation of a self-avoiding walker $W = ABC$ is as follows:

• (a) A low concentration of W walker strands are added to the initial buffer solution $S_0$ containing the nanotrack, and a few of these W strands hybridize to random pads of the nanotrack.

• (b) The buffer solution $S_0$ is replaced with buffer solution $S_1$, so as to remove the remaining non-hybridized W strands from the solution surrounding the nanotrack, and to add the DNA hairpin $h_1$.

• (c) As in Fig. 4, the Walker W first hybridizes with a $p_0 = B^*A^*$ pad, which is State 0. In State 1, the unpaired domain C of W hybridizes with the domain $C^*$ of the $p_1$ pad. Then the domain $B^*$ of $p_1$ can displace the domain $B^*$ of $p_0$, the domain B of W hybridizes with both $p_0$ and $p_1$ and also hairpin $p_1$ is opened, giving intermediate State 1.5. (Note this transition process is reversible.) If and when the strand displacement finishes, it enters State 2, in which W hybridizes with a $p_1$ pad. In summary, transitioning from State 0 to State 2, the W nanorobot walks from $p_0$ pad to $p_1$ pad due to the hybridization and strand displacement.

• (d) (Note our design includes a way to keep W from stepping to the next pad, therefore letting the track hairpin reform. ) The ssDNA strand $h_1$ is in high concentration in buffer solution $S_1$, with domain $E_2$ that can hybridize with the newly released $E_2^*$ domain of $p_1$, so $h_1$ is then opened and displaces the domain B of W. The unpaired domain AB of W can hybridize with the next $p_0$ and moves from $p_1$ to $p_0$, entering State 1. After some time, W detaches from $p_1$ and hybridizes with a $p_0$ pad, taking it back to State 0. Then the previously visited $p_1$ will hybridize with a hairpin $h_1$ from the buffer solution, which hinders the $p_1$’s reformation of the hairpin, so W cannot move back to the visited $p_1$. (Also, note that the small length of the loop of W deters $h_1$ from binding to the loop of W in State 0.)

• (e) The W nanorobot walks from either (a) $p_0$ to a non-visited $p_1$ pad of the nanotrack, or (b) $p_1$ to a $p_0$ pad of the nanotrack, as described in part c, d. As a result, the W nanorobot avoids the locations visited by itself or any other DNA nanorobot when it moves on the nanotrack. (A similar design can also be used for foraging nanorobots.)

Fig. 4

Design of a DNA nanorobot that executes a self-avoiding walk (for simplicity only illustrated in 1D)

Fig. 5

2D locations of the self-avoiding walker’s pads on a 2D nanotrack

Fig. 6

Simulation of a DNA nanorobot that executes a self-avoiding walk

Figure 6 (also see mp4 at [43]) gives an example simulation run for a W nanorobot (in green) that executes a self-avoiding random walk on a 2D nanotrack; it randomly walks on the nanotrack, and the grid turns to orange to show the trace of the W nanorobot. The W walker randomly moves over the 2D nanotrack and then eventually stops when all the pads around it are visited.

4.6 Flocking: Novel DNA Nanorobots that Follow a Leader

This is a DNA nanorobot system with two types of DNA nanorobots (the designated leader and the followers): the leader makes a random traversal of a 2D nanotrack and the other DNA nanorobots follow the movements of the leader. Let $B_2 = B_{2a}B_{2b}.$ There are two types of pads: (i) $h_1 = B_1 B_2^* A_2^* B_{2b} B_1^* A_1^*$ attached at its $3'$ end and (ii) $h_2 = C_1^* B_1^* B_{2a} C_2^* B_2^* B_1$ attached at its $5'$ end. Figure 7 illustrates (for simplicity on a 1D nanotrack) the self-assembly of the pads into DNA hairpins. The 2D arrangement of these pads on a 2D nanotrack is illustrated in Fig. 8 .

Note: Observe that each of $h_1$ and $h_2$ each self-assemble into a type of hairpin with two separate short loops: $h_1$ has two separate short loops, one with $A_2^*$ and another with $B_{2a}^*$, whereas $h_2$ has one short loop with $B_{2b}^*C_2^*$. This use of small hairpin loops is a deliberate design with the goal of inhibiting the hybridization follower $W_2$ with $A_2^*, B_{2a}^*, C_2^*, B_{2b}^*$. This will make it more difficult for a follower $W_2$ to avoid following a leader $W_1$. Note that to ensure stability of each the hairpins, their loop portions need to be relatively short compared to the $B_1$ and $B_1^*$ portions of their stems.

There are two types of ssDNA nanorobots, the leader $W_1 = A_1 B_1 C_1$ and the follower $W_2 = A_2 B_2 C_2$, which operate as follows:

• (a) $W_1$(leader) and $W_2$(follower) strands are added to buffer solution containing the nanotrack.

• (b) The leader $W_1$ hybridizes with $h_1$ on the domain $A_1$ and opens the hairpin of $h_1$ by strand displacement; then a follower $W_2$ can hybridize with the newly opened $h_1$.

• (c) The leader $W_1$ moves from pad $h_1$ to pad $h_2$ by hybridizing with $C_1^*$ of $h_2$ and opening $h_2$ by strand displacement; then follower $W_2$ can hybridize with the newly opened $h_2$.

• (d) After $W_1$ and $W_2$ Walkers leave these pads, a limited number of further $W_2$ strands can walk nearby, and similarly follow leader $W_1$.

• (e) As in Fig. 7 (for simplicity it is only illustrated in 1D), the leader $W_1$ walks successively from either (a) a $h_0$ pad to a $h_1$ pad of the nanotrack, or (b) a $h_1$ pad to a $h_0$ pad of the nanotrack. Whenever a follower $W_2$ stand hybridizes with a $h_1$ pad, the hairpin $h_1$ needs to be open (where the loop of the hairpin was opened by a leader $W_1$ and the loop closes after some time), so $W_2$ is forced to follow the leader $W_1$. Hence, leader $W_1$ walks randomly over the nanotrack and is followed by a group of $W_2$ followers. (The maximum size of group of $W_2$ followers is limited by the duration that $h_1,h_2$ remain open, and this parameter can be set by appropriate DNA strand design.)

Figure 9 (also see mp4 at [43]) gives an example simulation run for a group of DNA nanorobots that follow a leader DNA nanorobot randomly walk on a 2D nanotrack. The yellow and blue cells are the $W_1$(leader) and $W_2$(follower) nanorobots, respectively. The $W_1$ walker can randomly walk on the nanotrack, while the $W_2$ walker can only follow the $W_1$ walker or stay in place. If the $W_2$ walker follows the $W_1$ walker, the grid it visited turns to green to show the trace of the $W_2$ Nanorobot; if the $W_2$ walker does not follow the $W_1$ walker, it will stay in place and keep blue until the $W_1$ walker visited $W_2$’s neighbor again.

Fig. 7

Design of DNA nanorobots that follow a leader DNA nanorobot

Fig. 8

2D locations of the pads on the 2D nanotrack used for DNA nanorobots that follow a leader DNA nanorobot

Fig. 9

Simulation of DNA nanorobots that follow a leader DNA nanorobot

4.7 Novel DNA Nanorobots that Vote by Assassination

Distributed voting is essential to many distributed computing and population protocols [51,52,53], where processors are restricted to pair-wise interactions. For example, distributed voting can be used for leader election which allows a process in a distributed system some special powers in the distributed system, often allowing for simplified protocols, reduced coordination, and improved efficiency. The task of determining approximate majority in distributed computing can be reduced to the case where each processor has a binary value in $\{0,1\}$; assuming an initial margin of disparity $\delta >1$ between those processors with value 0 and value 1, then the problem is for the set of processors to settle on a majority value in $\{0,1\}$. A fast randomized distributed protocol for approximate majority was given by Angluin et al. [54, 55]. (Interestingly, Cardelli [56] observed that this approximate majority protocol was used in certain cell cycle switches.) Let an event with size parameter n be high probability if it has likelihood $\ge 1-\frac{1}{n^{\alpha }}$ for some constant $\alpha \ge 1.$ Angluin et al. [54, 55] proved that with high probability, their randomized distributed protocol n processors reached consensus on a majority value after $O(n \log n)$ pair-wise interactions, assuming that the initial margin of disparity is $\ge \delta =c \sqrt{n} \log n$ for a constant $c \ge 1$. A slightly modified version of their protocol proceeds in $O(n \log n)$ stages, where in each stage a random pair of processors compare their values; if their values are the same, they do nothing, and otherwise, a random processor of the pair drops out from subsequent stages of the protocol. Afterward, with high probability, only processors with the same majority value remain. Then the other processors that previously dropped out are informed of that majority value.

Fig. 10

Separate walks of assassinator nanorobots $W_1$ and $W_2$

Fig. 11

2D locations of the assassination protocol’s pads on a 2D Nanotrack

Condon et al. [57, 58] have given chemical reaction systems (CRN) for approximate majority, which can in principle be implemented by strand-displacement reactions operating in solution. There has been no previous localized reaction protocol for approximate majority; a localized reaction protocol for approximate majority would likely operate far faster than a solution-based protocol which is delayed by diffusion.

We expect distributed voting to be also of central importance to programming complex behavior in social DNA nanorobots. The fast randomized distributed protocol for approximate majority given by Angluin et al. [54, 55] inspired our DNA nanorobots assassination protocol design described here. Our design for distributed voting of DNA nanorobots has the nanorobots exhibit an anti-social behavior to achieve group decision making. The idea is the DNA nanorobots vote by assassination. There are originally two unequal size (with sufficiently large size difference) groups of DNA nanorobots; when pairs of DNA nanorobots from distinct groups collide, one or the other is assassinated (it is detached from the 2D nanotrack and then diffused into the solution away from the 2D nanotrack). Eventually, all members of the smaller of the two groups of DNA nanorobots are assassinated with high likelihood.

Fig. 12

Design of DNA nanorobots that vote by assassination where $n_1 = 10$ and $n_2 = 6$

Let $C=C_{a}C_{b}C{c}$ where $|C_{a}| =|C_{c}|$ are between 3 to 5 bases pairs (sufficient to act as toeholds), and $|C_{b}| \ge 10$. There are three types of pads: (i) $p_0 = (B_2)^*C^* (B_1)^*$ attached at its $5'$ end and with the ssDNA sequence $C_{b}$ hybridized to the complementary subsequence $C_{b}^*$ of $p_0$, (ii) $p_1 = (B_1)^* (A_1)^*$ attached at its $3'$ end, and (iii) $p_2= (A_2)^* (B_2)^*$ attached at its $3'$ end. The 2D arrangement of pads on a 2D nanotrack is illustrated in Fig. 11. There are two types of ssDNA nanorobots: $W_1 =A_1 B_1 C$ and $W_2 = C B_2 A_2$. Figure 10 illustrates the separate walks of assassinator nanorobots $W_1$ and $W_2$ (for simplicity it is only illustrated in 1D).

There is also a short ssDNA sequence $C_{b}$ initially hybridized to the complementary subsequence $C_{b}^*$ of $p_0$, which has purpose of substantially increasing the likelihood that $W_1$ and $W_2$ will simultaneously bind to a pad $p_0$. Once $W_1$ or $W_2$ is partly bound to $p_0$, it still has to engage in a relatively slow strand-displacement reaction to dislodge the $C_{b}$, which was already hybridized to the complementary subsequence $C_{b}^*$ of $p_0$, increasing the likelihood that a second nanorobot also attaches to $p_0$. ($C_{b}$ needs to be in sufficient concentration in the solution, allow it to re-hybridize to $C_{b}^*$ of $p_0$ if strand-displaced.)

Initially, a combination of an unequal concentration of $W_1$ and $W_2$ strands is added to the buffer solution containing the nanotrack; some $W_1$ and $W_2$ strands hybridize to random pads of nanotrack. The buffer solution is replaced, so as to remove the remaining non-hybridized $W_1$ and $W_2$ strands from the solution surrounding the nanotrack. Let $n_1$ and $n_2$ be the (unknown) numbers of $W_1$ and $W_2$ strands initially attached to the nanotrack and let $n=n_1 +n_2$. We assume $n>0$ and the initial margin of disparity $|n_1 -n_2| \ge \delta$ with $\delta =c\sqrt{n} \log n$ and constant $c \ge 1$. Our goal is: to test if $n_1 > n_2$ or $n_1 < n_2$.

Randomized Assassination Protocol: The nanorobots $W_1$ and $W_2$ operate as follows:

• (a) As in Fig. 10, $W_1$ and $W_2$, when separate, walk randomly over the nanotrack, The $W_1$ nanorobot walks only on the $p_0$ and $p_1$ pads of the nanotrack, whereas the $W_2$ Nanorobot walks only on the $p_0$ and $p_2$ pads of the nanotrack.

• (b) As in Fig. 12, whenever both a $W_1$ and a $W_2$ nanorobot collide at a common $p_0$ pad of the nanotrack, a random one of either $W_1$ or $W_2$ nanorobot is partially detached via strand-displacement at domain C, and then is melted off to enter the buffer solution. By this process, pairs of $W_1$, $W_2$ duel and randomly one of the nanorobots assassinate the other (which disassociates from the nanotrack).

Note that often only one of $W_1$ or $W_2$ will arrive at the $p_0$ pad of the nanotrack, in which case there will be no competition for hybridization at domain C, and no assassination. However, for the protocol to operate correctly, there just needs to be a constant finite probability that both a $W_1$ and a $W_2$ nanorobot collide at a common $p_0$ pad of the nanotrack. It is shown below that there is only constant probability of the detached nanorobot $W_i$ reattaching to the 2D nanotrack.

Probabilistic Analysis of Re-attachment Likelihood of a Detached (assassinated) Nanorobot: We assume:

• The protocol is executed over some finite time duration.

• The size of each 2D nanotrack is very small compared to the width of the test tube containing the buffer solution.

• The detached (assassinated) nanorobot $W_i$ takes a random 3D walk within the volume of the test tube after detaching from its 2D nanotrack.

• There is a very small concentration of 2D nanotracks in the buffer solution, so it is very low likelihood $\rho _0 > 0$ that a detached nanorobot $W_i$ later attaches to another 2D nanotrack also in the buffer solution of the test tube during the duration of the protocol.

Consider a sphere S fully containing the 2D nanotrack (and 3 times its diameter) from which a nanorobot $W_i$ is detached from during the assassination protocol. It is easy to see that there is a constant probability $\rho _1 > 0$ that a random 3D walk of $W_i$ takes it outside S. The further random movement of nanorobot $W_i$ can be modeled by a random walk on a 3D grid whose nodes correspond to spheres of the same diameter as S. By Pólya’s recurrence theorem [59, 60], in a random walk on a 3D grid starting at a given start node, the likely of never re-visiting that start node is a constant $\rho _2 > 0$. Hence during the duration of the protocol:

• There is at most a constant total likelihood $\le (1-\rho _0) \rho _1\rho _2$ that a nanorobot $W_i$ that is detached from a nanotrack during the assassination protocol then is reattached to the same nanotrack or any other nanotrack in the test tube, and so

• There is at least a constant likelihood $\ge 1-(1-\rho _0) \rho _1\rho _2$ that the detached nanorobot $W_i$ never re-attaches to any nanotrack.

Probabilistic Analysis of Outcome of Assassinating Nanorobot Protocol: We further assume: whenever both a $W_1$ and a $W_2$ nanorobot collide at a common $p_0$ pad of the nanotrack, it is equally likely that the $W_1$ or $W_2$ nanorobot is detached from the nanotrack. From the above, there is at least a constant likelihood that the detached nanorobot never re-attaches to any nanotrack during the duration of the protocol. Observe that as a consequence, ultimately either:

• one or more $W_1$ remains attached to the nanotrack and all the $W_2$ are detached, or

• one or more $W_2$ remains attached to the nanotrack and all the $W_1$ are detached.

Angluin et al. [54, 55] gave a similar randomized protocol for approximate majority (for distributed computing applications), and their probabilistic analysis implies that if the initial margin of disparity is $|n_1 -n_2| > c{\sqrt{n_1 +n_2}}$ (for some constant $c > 0$), then ultimately, with high probability:

• if at least one $W_1$ remains attached to the nanotrack, then $n_1 > n_2$, and

• if at least one $W_2$ remains attached to the nanotrack, then $n_2 > n_1$.

Fig. 13

Simulation of DNA nanorobots that vote by assassination where $n_1 = 10$ and $n_2 = 6$

Fig. 14

Simulation of DNA nanorobots that vote by assassination where $n_1 = 8$ and $n_2 = 8$

Fig. 15

Results of a collection of simulation runs for DNA Nanorobots voting by assassination on a 2D nanotrack with different initial n, where $n_1\ge n_2$

Simulation of the Assassination Protocol:

• Figure 13 gives an example simulation run for two groups of DNA nanorobots with unequal sizes that vote by assassination on a 2D nanotrack; the number of $W_1$ is larger than the number of $W_2$ initially attached to the nanotrack $(n_1 = 10, n_2 = 6)$, and eventually all members of $W_2$ DNA nanorobots are assassinated.

• Figure 14 gives an example simulation run for two groups of DNA nanorobots with equal sizes that vote by assassination on a 2D nanotrack; the numbers of $W_1$ and $W_2$ strands initially attached to the nanotrack are equal $(n_1 = n_2 = 8)$, so they have the same chance to win the game, and in this simulation, eventually all members of $W_2$ DNA nanorobots are assassinated.

• Figure 15 (also see mp4 at [43]) gives the results of a collection of simulation runs for DNA nanorobots voting by assassination on a 2D nanotrack with different initial n, where the number of $W_1$ is larger or equal to the number of $W_2 (n_1\ge n_2)$ that are initially attached to the nanotrack. The X-axis and Y-axis represent the initial $n_2/n_1$ and final $n_2/n_1$ respectively. Due to $n_1 \ge n_2$ initially, it is more likely that eventually $W_1$ remains attached to the nanotrack and all the $W_2$ are detached, and the final $n_2/n_1$ will convert to 0. With same n, when initial $n_2/n_1$ goes smaller, the final $n_2/n_1$ will approach 0 with higher likelihood. When n goes larger, the final $n_2/n_1$ will approach 0 with higher likelihood.

5 Discussion

We described social DNA nanorobots: these are autonomous mobile DNA nanorobots that execute a series of pair-wise interactions that determine an overall desired outcome behavior for the group of nanorobots. Our goal was to increase the complexity of the various tasks the nanorobots can execute and at the same time preserve a low design complexity for individual nanorobots. We presented detailed designs for social DNA nanorobots that perform novel behaviors of self-avoiding walking, flocking, and voting by assassination, and their behaviors were simulated in the 2D surface CRN model.

5.1 Further Development of Simulation Software for Social Nanorobots

We are developing software extending the Surface CRN Simulator of Clamons [1, 2] specifically for use with social DNA nanorobots. The software will allow high-level specification and visualizations of state transitions (modeled by chemical reactions such as toehold-mediated strand displacements and dehybridizations) between DNA walkers and DNA strands affixed to a 2D surface. The software could provide an editable catalog of DNA nanorobot devices, improved visualization, and allow automatic incrementally optimized performance assessment. This software should significantly improve performance assessments & design optimizations. We are also using the Visual DSD [61] to simulate the DNA hybridization and strand-displacement reactions of the individual DNA nanorobots and between pairs of DNA nanorobots.

5.2 Experimental Demonstrations of Social DNA Nanorobots

We are planning to experimentally demonstrate the social DNA nanorobots behaviors on 1-dimensional DNA nanotracks and 2-dimensional DNA origami. After experimental demonstrations are made of each design and assessed, they may also be redesigned for further optimization.

5.3 Further Social DNA Nanorobot Behaviors

Our novel designs presented here for DNA nanorobots (self-avoiding walking, flocking, and voting by assassinations) can be employed in designs for even more complex behavior. For example, other behaviors of interest for DNA nanorobots include:

$\bullet$ Guarding, where a group of DNA nanorobots follow and guard a particular DNA nanorobot from attack by another group of DNA nanorobots. Here we expect we can employ parts of our flocking design.

$\bullet$ Attacking, where a group of DNA nanorobots attack another group of DNA nanorobots. Here we expect we can employ a simplification of the assassination design.

$\bullet$ Foraging and Harvesting. In foraging, a group of designated foraging DNA nanorobots randomly walk on the 2D nanotrack, and can transform to a “discovery state” when they discover a target molecule (e.g., a group of gold nanoparticles attached to 2D surface) (this makes use of our designs for self-avoiding random walking). In harvesting, a group of harvesting DNA nanorobots which follow the trail of foraging DNA nanorobots in discovery state and pick up the detected target molecules and deliver the target molecules to a designated region of the 2D nanotrack. Designs for foraging nanorobots may employ our designs for self-avoiding random walking, and designs for harvesting nanorobots may employ our design for flocking.

5.4 Communication Between Distant Social Nanorobots

Prior Use of Potential Fields for Generating Autonomous Group Social Activities: Another source of inspiration for collective behavior strategies by groups can be found in biology: for example, flocking of animals such as birds and schooling of amphibious animals. The behavior of these animals has been modeled by mathematical models and computer programs. In 1989, Beni [62, 63] developed one of the first such flocking model, which he called swarm intelligence and made applications to multi-robot motion planning systems. Subsequently the field of swarm intelligence [64, 65] and artificial flocking grew rapidly and found applications to many applied areas in addition to robotics, such as for computer graphics. In 1994, Reif [66, 67] developed a general programmable scheme for multi-robot motion planning, termed Social Potential Fields, which made use of artificially defined potential fields that controlled the individual robots by weighted sum of decreasing functions of the distance and direction of other local robots; he demonstrated various autonomous group social activities, including flocking, attacking, and guarding, using the Social Potential Fields technique. Unfortunately, the potential field models assume far-distance field effects that are not easy to implement using local interactions between co-located DNA nanorobots.

Using Instead Diffusion of Pheromone-like DNA Molecules for Communication Between Social Nanorobots: Recall that the communication signals between social insects include pheromones; these are chemical factors that can trigger a social response in members of the same species [10]. We are exploring the use of diffusion of DNA molecules for communication between social nanorobots in a manner similar to the use of pheromones in social insects. For example, this technique may be employed by foraging and harvesting nanorobots. Suppose the goal is to detect and harvest a particular target molecule on the 2D surface on which the foraging and harvesting nanorobots walk on. The 2D surface would be decorated with additional hairpins that when opened by a foraging nanorobot (to announce the discovery of a nearby target molecule), would act as pheromones for the foraging nanorobots (e.g., the speed of the motion of nearby harvesting nanorobots could temporarily increased when encountering such an opened hairpin).