Counting, Computing, and Pattern Recognition with Self-Assembling Non-Reciprocal DNA Tiles

TL;DR

The paper tackles physical computation in non-equilibrium matter by demonstrating how self-assembling DNA tiles can realize finite-state automata through controlled, fuel-driven transitions between multiple target structures. It introduces non-reciprocal swap dynamics with energy input $\lambda$ and stabilizing inter-target bonds with energy $\eta$, along with a finite non-reciprocity budget $\mathcal{B}$ and discrete time windows to prevent premature transitions. The authors show how tasks such as counting (modulo 4), modulo-3 computation, and pattern recognition can be implemented with high fidelity (≈95% across 21 runs per input) using four or more target states $S_\ell$ and alternating tile libraries $L_i$. This framework points to energy-efficient, programmable computation embedded in materials, with potential applications in DNA-based systems, enzymes, proteins, and colloids for autonomous sensing and information processing.

Abstract

Harnessing the intrinsic dynamics of physical systems for information processing opens new avenues for computation embodied in matter. Using simulations of a model system, we show that assemblies of DNA tiles capable of self-organizing into multiple target structures can perform basic computational tasks analogous to those of finite-state automata when equipped with programmable non-reciprocal interactions that drive controlled dynamical transitions between these structures. By establishing design rules for multifarious self-assembly while budgeting the energy input required to drive these non-equilibrium transitions, we demonstrate that these systems can execute a wide variety of tasks including counting, computing modulo functions, and recognizing specific input patterns. This framework integrates memory, sensing, and actuation within a single physical platform, paving the way toward energy-efficient physical computation embedded in materials ranging from DNA and enzymes to proteins and colloids.

Figures (8)

• Figure 1: (a) Schematic representation of two target structures, labeled $S_0$ and $S_1$, both composed of a library that consists of nine different building block species. Particle colors in these sketches are arbitrary and only intended to distinguish different target structures, particles of the same color represent different particle species. Neighboring particles within the target structures bind with a strength $\varepsilon$ as shown in the insets. (b) Schematic of a few of the intermediate steps during the transition $S_0\rightarrow S_1$ as facilitated by the non-reciprocal interactions $\lambda$ and the reciprocal interactions $\eta$ between neighboring building blocks in the initial and the subsequent structure. Here we show that the particle of species $\beta$ from target structure $S_1$ replaces one of species $\alpha$ from $S_0$, driven by the availability of four times the non-reciprocal energy contribution $\lambda$ in the swap rate $p_{swap}$, as defined in Eq.(\ref{['eq:swaprate']}). The neighboring particles from two consecutive target structures (here $S_0$ and $S_1$) are bound together during the transition with bond strength $\eta$ to stabilize the intermediate structures, as shown in the inset.
• Figure 2: (a) Typical configuration of $576$ different DNA tiles, modeled as unit cubes with a fixed orientation and endowed with specific directional nearest neighbour binding of the four faces in the $xy$-plane. The simulation box is a $128\times128\times8$ cubic lattice. A complete two-dimensional target structure of 18$\times$16=288 DNA tiles has successfully assembled, and one other particle library of 18$\times$16 different DNA tiles is dispersed in the simulation box. Each species of DNA tiles is assigned a unique color, chosen such that the self-assembled target structure resembles Vincent van Gogh's painting Wheatfield with cypresses. (b) Heatmap of the nucleation time $\tau_{nucl}/\tau_0$ as a function of the inter-target interaction strength $\beta \eta$ and non-reciprocity value $\beta\lambda$, along with the lines denoting the parameter regime where the $S_0 \rightarrow S_1$ transitions occur reliably.
• Figure 3: Transition paths with three target structures using (a) three unique particle libraries or (b) two particle libraries, where structures $S_0$ and $S_2$ share most of their particle species. Each particle species has a unique color, chosen such that $S_0$, $S_1$ and $S_2$ resemble the paintings The milkmaid by Johannes Vermeer, Wheatfield with cypresses by Vincent van Gogh, and The threatened swan by Jan Asselijn, respectively. In panel (b) $S_2$ uses the same particle library as $S_0$, resulting in a scrambled version of The milkmaid. In both systems, the transitions $S_0\rightarrow S_1$ and $S_1\rightarrow S_2$ are triggered by the same input label and are therefore simultaneously accessible. (c,d) The fraction of bonded neighbors $f_\ell$ for all structures $S_\ell$ during a single time window plotted as a function of time (solid lines), for various values of non-reciprocity budget $\mathcal{B}$, averaged over 21 individual simulations (transparent lines). Panel (c) corresponds to the system shown in (a), while panel (d) shows the results for the multifarious system illustrated in (b).
• Figure 4: (a) Transition graph of a finite-state automaton for counting the number of "1"-bits in binary inputs 000 through 111, in (a) a conventional abstract representation and in (b) our proposed physical realization using four multifarious structures $S_\ell$ for $\ell\in\{0,1,2,3\}$ with alternating particle libraries $L_i$ for $i\in\{0,1\}$ and directed transitions $S_{\ell}\rightarrow S_{\ell+1}$ triggered by non-reciprocal interactions. After an input of $\ell$ chemical fuel pulses (see text for details), the final structure is (ideally) $S_\ell$ such that this is a physical device that can count up to three. (b) The fidelity of the Brownian automaton of Fig. \ref{['fig:sketch-counting']}(b), showing the fraction of assembled structures $S_\ell$ of all seven inputs 001 through 111 (realized physically as fuel-budget pulses) as obtained from 21 simulations for each input. For all inputs, the correct result, corresponding to the number of "1" bits in the input, is obtained in at least 95% of the simulations.
• Figure 5: Transition graph of a finite-state automaton for calculating the modulo three of a binary input, in (a) a conventional representation and in (b) our equivalent implementation using multifarious structures with non-reciprocal transitions. Here states labeled $L_i$ are congruent with $i \bmod 3$. The target structures are represented by solid rectangles, and colored according to their particle library because the scrambled paintings of libraries $L_0$ and $L_2$ (as shown in Fig. \ref{['fig:budget-sequence']}(a)) are difficult to distinguish by eye. (c) The fraction of bonds of each target structure $f_\ell$ as a function of time. Typical trajectory of the composition at binary input sequence $1001$ (decimal 9). The input and the transitions graph representation of the specific transition that occurs in each time window is shown overlaid with the figure. (d-e) The fraction of simulations which resulted in (d) the 3 different libraries and (e) the 6 possible output structures of the finite-state automaton that calculates modulo 3 of a binary input signal, for all 16 binary input signals between 0000 and 1111.