Increasing Redundancy Exponentially Reduces Error Rates during Algorithmic Self-Assembly

While biology demonstrates that molecules can reliably transfer information and compute, design principles for implementing complex molecular computations <i>in vitro</i> are still being developed. In electronic computers, large-scale computation is made possible by redundancy, which allows errors to be detected and corrected. Increasing the amount of redundancy can exponentially reduce errors. Here, we use algorithmic self-assembly, a generalization of crystal growth in which the self-assembly process executes a program for growing an object, to examine experimentally whether redundancy can analogously reduce the rate at which errors occur during molecular self-assembly. We designed DNA double-crossover molecules to algorithmically self-assemble ribbon crystals that repeatedly copy a short bitstring, and we measured the error rate when each bit is encoded by 1 molecule, or redundantly encoded by 2, 3, or 4 molecules. Under our experimental conditions, each additional level of redundancy decreases the bitwise error rate by a factor of roughly 3, with the 4-redundant encoding yielding an error rate less than 0.1%. While theory and simulation predict that larger improvements in error rates are possible, our results already suggest that by using sufficient redundancy it may be possible to algorithmically self-assemble micrometer-sized objects with programmable, nanometer-scale features

Ensemble Bayesian Analysis of Bistability in a Synthetic Transcriptional Switch

An overarching goal of synthetic and systems biology is to engineer and understand complex biochemical systems by rationally designing and analyzing their basic component interactions. Practically, the extent to which such reductionist approaches can be applied is unclear especially as the complexity of the system increases. Toward gradually increasing the complexity of systematically engineered systems, programmable synthetic circuits operating in cell-free <i>in vitro</i> environments offer a valuable testing ground for principles for the design, characterization, and analysis of complex biochemical systems. Here we illustrate this approach using <i>in vitro</i> transcriptional circuits (“genelets”) while developing an activatable transcriptional switch motif and configuring it as a bistable autoregulatory circuit, using just four synthetic DNA strands and three essential enzymes, bacteriophage T7 RNA polymerase, <i>Escherichia coli</i> ribonuclease H, and ribonuclease R. Fulfilling the promise of predictable system design, the thermodynamic and kinetic constraints prescribed at the sequence level were enough to experimentally demonstrate intended bistable dynamics for the synthetic autoregulatory switch. A simple mathematical model was constructed based on the mechanistic understanding of elementary reactions, and a Monte Carlo Bayesian inference approach was employed to find parameter sets compatible with a training set of experimental results; this ensemble of parameter sets was then used to predict a test set of additional experiments with reasonable agreement and to provide a rigorous basis for confidence in the mechanistic model. Our work demonstrates that programmable <i>in vitro</i> biochemical circuits can serve as a testing ground for evaluating methods for the design and analysis of more complex biochemical systems such as living cells

Molecular Schema

<div><p>(A) Top center: abstract versions of the four DAE-E Sierpinski rule tiles, VE-00, UE-11, RE-01, and SE-10, highlight their differences from the tiles in <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.0020424#pbio-0020424-g001" target="_blank">Figure 1</a>. The arrangement of 3′ and 5′ ends on DAE-E tiles dictates that two distinct pairs of complementary binding domains must be used for each symbol ‘0' or ‘1,' denoted here by making complementary shapes large or small. Pink legends show the mapping of shape to sticky-end sequences. Top left: a molecular diagram for VE-00 shows how each DAE-E tile is comprised of five DNA strands; small arrows point to crossovers. Top right: a diagram for RE-01 shows how hairpins are attached to ‘1' tiles to provide AFM contrast; the exact orientation of these hairpins is unknown. Below: tiles are shown assembling on a nucleating structure. The nucleating strand for the input row (blue) is generated by assembly PCR and frequently reaches lengths of more than 3 μm (200 tiles). The nucleating strand contains subsequences onto which capping strands (orange) and input tile strands assemble to form an input tile outputting ‘0' at random intervals, the nucleating strand contains a subsequence (asterisk) for a different input tile that outputs a ‘1' on one side.</p> <p>(B) Top center: the six DAO-E Sierpinski rule tiles: S-00, R-00, S-11, R-11, S-01, and R-01. Top left and right: molecular diagrams highlight two notable features: (1) R-type tiles output only to S-type tiles, and vice-versa, as dictated by the 3′/5′ polarity of the molecules—again requiring two distinct pairs of binding domains per symbol. (2) The indicated rotational symmetry of the DAO-E molecules allows each molecule to serve in either of two orientations; no explicit S-10 or R-10 tiles are needed. An input tile outputting a single ‘1' sticky end is shown (asterisk). Sequences are given in <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.0020424#sg004" target="_blank">Figures S4–S7</a>.</p></div

The XOR Cellular Automaton and Its Implementation by Tile-Based Self-Assembly

<div><p>(A) Left: three time steps of its execution drawn as a space–time history. Cells update synchronously according to XOR by the equation shown. Cells at even time steps are interleaved with those at odd time steps; arrows show propagation of information. Right: the Sierpinski triangle.</p> <p>(B) Translating the space–time history into a tiling. For each possible input pair, we generate a tile T-<i>xy</i> that bears the inputs represented as shapes on the lower half of each side and the output as shapes duplicated on the top half of each side.</p> <p>(C) The four Sierpinski rule tiles, T-00, T-11, T-01, and T-10, represent the four entries of the truth table for XOR: 0 ⊕ 0 = 0, 1 ⊕ 1 = 0, 0 ⊕ 1 = 1, and 1 ⊕ 0 = 1. Lower binding domains on the sides of tiles match input from the layer below; upper binding domains provide output to both neighbors on the layer above. Semicircular domains represent ‘0' and rectangular domains, ‘1'. Tiles that output ‘0' (T-00 and T-11) are gray, and we refer to them as ‘0' tiles. Tiles that output ‘1' (T-01 and T-10) are white and are referred to as ‘1' tiles. Initial conditions for the computation are provided by a nucleating structure (blue). Red asterisks indicate sites on the nucleating structure that bear a ‘1' binding domain; elsewhere, sites have all ‘0' binding domains. Black arrows indicate associations that would form two bonds; red arrows, associations that would form one bond. </p> <p>(D) Error-free growth results in the Sierpinski pattern.</p> <p>(E) Error-prone growth from a nucleating structure with three ‘1' domains. Red crosses indicate four mismatch errors.</p></div

Direct Atomic Force Microscopy Observation of DNA Tile Crystal Growth at the Single-Molecule Level

While the theoretical implications of models of DNA tile self-assembly have been extensively researched and such models have been used to design DNA tile systems for use in experiments, there has been little research testing the fundamental assumptions of those models. In this paper, we use direct observation of individual tile attachments and detachments of two DNA tile systems on a mica surface imaged with an atomic force microscope (AFM) to compile statistics of tile attachments and detachments. We show that these statistics fit the widely used kinetic Tile Assembly Model and demonstrate AFM movies as a viable technique for directly investigating DNA tile systems during growth rather than after assembly

AFM Images of DAO-E Crystals

<div><p>(A) A large templated crystal in a 5-tile reaction (no R-11). A single ‘1' in the input row (asterisk) initiates a Sierpinski triangle, which subsequently devolves due to errors. Mismatch errors within ‘0' domains initiate isolated Sierpinski patterns terminated by additional errors at their corners.</p> <p>(B) A large untemplated fragment in a 5-tile reaction (no S-11). Large triangles of ‘0's can be seen. Crystals similar to this are also seen in samples lacking the nucleating structure.</p> <p>(C) Several large crystals in a 6-tile reaction, some with more zeros than ones, some with more ones than zeros. It is difficult to determine whether these crystals are templated or not.</p> <p>(D) An average of several scans of the boxed region from (C), containing roughly 1,000 tiles and 45 errors.</p> <p>(E) An average of several scans of a Sierpinski triangle that initiated by a single error in a sea of zeros and terminated by three further errors (a 1% error rate for the 400 tiles here). Red crosses in (D) and (E) indicate tiles that have been identified (by eye) to be incorrect with respect to the two tiles from which they receive their input. Scale bars are 100 nm.</p></div

Direct Atomic Force Microscopy Observation of DNA Tile Crystal Growth at the Single-Molecule Level

While the theoretical implications of models of DNA tile self-assembly have been extensively researched and such models have been used to design DNA tile systems for use in experiments, there has been little research testing the fundamental assumptions of those models. In this paper, we use direct observation of individual tile attachments and detachments of two DNA tile systems on a mica surface imaged with an atomic force microscope (AFM) to compile statistics of tile attachments and detachments. We show that these statistics fit the widely used kinetic Tile Assembly Model and demonstrate AFM movies as a viable technique for directly investigating DNA tile systems during growth rather than after assembly

AFM Images of DAE-E Crystals

<div><p>(A) Several frequent morphologies that appear in most samples, including all-'0' (upper arrow) and ‘011011'-striped crystals (lower arrow). The all-'0' crystal may be a tube that opened upon adsorption to the mica.</p> <p>(B) A templated crystal. The identification of tiles in this crystal is given in <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.0020424#pbio-0020424-g001" target="_blank">Figure 1</a>E. Crosses indicate mismatch errors. Asterisks indicate ‘1's on the nucleating strand.</p> <p>(C) A crystal containing 10 rows of error-free Sierpinski triangle. A red triangle marks a lattice defect in the input row.</p> <p>(D) Another Sierpinski triangle, better resolved.</p> <p>(E) A crystal containing a perfect 19 × 6 subregion. Individual tiles can be clearly seen; three tiles are outlined in the lower left. Unfortunately, this crystal landed atop a thin sliver of DNA (lower arrow), obscuring the central columns of the Sierpinski triangle. The upper arrow indicates a 4-tile wide tube, near the point where it opens. A pentagon marks a lattice dislocation. Scale bars are 100 nm.</p></div

Simulations with Slow Border Growth and T-00 and T-11 at Doubled Concentrations

<div><p>(A) A common error pattern: termination of triangles at corners.</p> <p>(B) An observed mechanism leading to termination or sideways extension of triangles: preferential nucleation of T-00 on facets.</p> <p>(C) Forward and sideways growth is deterministic: at sites presenting two binding domains, there is always a unique tile that can form exactly two bonds. Backward growth is non-deterministic: at sites where both binding domains agree (e.g., green arrows), there are two possible tiles that can make two bonds (either {T-10, T-01}, or {T-00, T-11}). At sites where the available binding domains disagree (e.g., red arrows), there is no tile that can associate to form two bonds. Since only the output type of tiles are shown, it is impossible to tell from these figures which backward growth sites present agreeing or disagreeing binding domains.</p></div

Typical kTAM Simulation Results

<div><p>(A) A roughly 130 × 70 subregion of an error-free templated crystal.</p> <p>(B) A subregion with 10 mismatch errors (0.1%), shown in red (both false ‘0's and false ‘1's). Grown at <i>G_mc</i> = 17, <i>G_se</i> = 8.8. Large all-zero patches near the template row are due to intact Sierpinski pattern; for simulations with these parameters, asymptotically only approximately 1% of T-00 tiles are in all-zero patches containing more than 90 tiles (referred to as large patches).</p> <p>(C) A subregion from a crystal grown with the T-00 and T-11 tiles at doubled concentration, on a slowly growing nucleating row. Mismatch errors (43 of them, i.e., 0.3%, during growth at <i>G_mc</i> = 17, <i>G_se</i> = 8.6) characteristically terminate the Sierpinski pattern at corners. Asymptotically, approximately 18% of T-00 tiles are in large patches.</p> <p>(D) An untemplated crystal with roughly 4000 tiles and no errors. Inset: The largest subregion of a perfect Sierpinski pattern is small.</p> <p>(E) An untemplated crystal with several errors, grown at <i>G_mc</i> = 17, <i>G_se</i> = 10.4. Note that growth in the reverse direction is more error-prone. Only approximately 1% of T-00 tiles are in large patches.</p> <p>(F) An untemplated crystal with few errors, grown at <i>G_mc</i> = 17, <i>G_se</i> = 8.6, with T-00 and T-11 at doubled stoichiometry. Note the large perfect subregion. Simulation was initiated by a preformed seed larger than the critical nucleus size (roughly 100 tiles). For these simulation parameters, approximately 25% of T-00 tiles are in large patches. According to the approximations used in <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.0020424#pbio-0020424-Winfree2" target="_blank">Winfree (1998a)</a>, <i>G_mc</i> = 17 corresponds to 0.8 μM, <i>G_se</i> = 8.5 corresponds to 41.8 °C, and <i>G_se</i> = 10.4 corresponds to 32.7 °C. The black outline around the crystals is for clarity; it does not represent tiles.</p></div