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Error suppression mechanisms for DNA tile self-assembly and their simulation

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Abstract

Algorithmic self-assembly using DNA-based molecular tiles has been demonstrated to implement molecular computation. When several different types of DNA tile self-assemble, they can form large two-dimensional algorithmic patterns. Prior analysis predicted that the error rates of tile assembly can be reduced by optimizing physical parameters such as tile concentrations and temperature. However, in exchange, the growth speed is also very low. To improve the tradeoff between error rate and growth speed, we propose two novel error suppression mechanisms: the Protected Tile Mechanism (PTM) and the Layered Tile Mechanism (LTM). These utilize DNA protecting molecules to form kinetic barriers against spurious assembly. In order to analyze the performance of these two mechanisms, we introduce the hybridization state Tile Assembly Model (hsTAM), which evaluates intra-tile state changes as well as assembly state changes. Simulations using hsTAM suggest that the PTM and LTM improve the optimal tradeoff between error rate \(\epsilon\) and growth speed r, from \(r \approx \beta \epsilon^{2.0}\) (for the conventional mechanism) to \(r \approx \beta \epsilon^{1.4}\) and \(r \approx \beta \epsilon^{0.7}\), respectively.

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Notes

  1. We consider only association and dissociation reactions between monomer tiles and the assembly. Interactions between assemblies are not considered.

  2. The value of k f has been experimentally measured for hybridization between DNA oligomers and is sequence dependent (Wetmur 1991). It has not been experimentally measured for DNA tiles yet.

  3. G se depends on temperature T and sticky end length s, where it is approximated by \(G_{se}=(\frac{4000}{T}-11)s\) (Winfree 1998). We neglect the initiation entropy for hybridization, as in Winfree and Bekbolatov (2004).

  4. In reality, it is difficult to achieve such near perfect assemblies that the simulation suggests. This discrepancy could be due to overly simplified assumptions of the kinetic model:

    1. (1)

      The concentrations of all the monomer tiles are not kept constant through the entire duration of the assembly process, and also are not in perfect stoichiometry with one another.

    2. (2)

      The binding strengths of all the different sticky ends are not identical.

    3. (3)

      The macroscopic interaction between separate assemblies is ignored, when in reality assemblies join and/or hinder each other’s growth. In addition, there are technical problems such as (a) the nucleating scaffold is so floppy that makes many defects in the assembly, and (b) synthesis of DNA oligonucleotide is not perfect.

  5. In the OTM, the length of the sticky ends is 5 nt. If we follow this design, the length of the protection strands must be shorter than 10 nt. However, this length is not enough to prevent the protection strand from spontaneously dissociating from the foundation tile even at room temperature. This indicates that we need longer sticky ends. We also have to consider the geometrical constraints of maintaining the planarity of the assembly made of DNA tiles. Both the intra- and inter-molecular spacing of the crossover points must be an integer number of half-turns of the B-form DNA double helix. The 14-nt sticky ends we have chosen here satisfies this constraint.

  6. In practice, the concentration [P] of released protection strands or tiles increases while assemblies are formed, and it may affect the growth of the assemblies. This can be solved by external control of concentrations, for instance, by using microfluidic devices (Somei et al. 2006).

  7. There are 24 situations of possible neighborhoods. All of such possibilities are simulated. b is therefore the sum of attachment strengths on all four sticky ends. The cases where tiles are attached on the output is discussed later in this section.

  8. Although the branch migration may occur at both ends of the protection strand simultaneously, for simplicity we have assumed that they occur sequentially, which may give unrealistic transition rates.

  9. k s is essentially treated as a constant value. However at large G mc or G se , this results in a dramatic and unnecessary slowdown in the simulation. This is because all rates except k s become extremely small (e.g., \(T_{11} \rightleftharpoons T_{10}\) in Fig. 8b) and therefore the majority of steps are spent redundantly simulating branch migration that has already reached equilibrium. To avoid this, we use the smallest value of k s that ensures rapid equilibration on the timescale of the other reactions:

    $$ k_s= \left\{ \begin{array}{ll} min(10^3, 10 \times max(r_f^{prot}, r_f^{unp}, r_{r,b})) & (\hbox{a tile matching by two inputs})\\ min(10^3, 10 \times max(r_f^{prot}, r_f^{unp}, r_{r,b}, \frac{k_t}{k_f} r_{r,\frac{11}{14}})) & (\hbox{a tile matching by one input}) \end{array} \right. $$

    Since the strand replacement by the branch migration is isoenergetic, this modification is effective in accelerating the simulation with the guarantee that internal substate probabilities achieve the same pseudo-equilibrium probabilities as they would with the faster rates.

  10. Reactions T 10 → E, T 01 → E and T 00 → E are one-directional reactions, and thus they do not comply strictly with the detailed balance. We neglected them here for the simplicity of the analysis.

  11. The actual computer code is optimized to remove redundant calculations.

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Acknowledgements

This work was supported by Grant-in-Aid for Scientific Research on Priority Areas (No. 17059001) from MEXT and Grant-in-Aid for Scientific Research (A) (No. 19200023) from JSPS to SM, JSPS Research Fellowships for Young Scientists (No. 05697) to KF, with additional support from NSF Grant (No. 0523761) to EW, and the Fannie and John Hertz Foundation to DYZ.

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Authors and Affiliations

  1. Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, Midori-ku, Yokohama, 226-8502, Japan

    Kenichi Fujibayashi & Satoshi Murata

  2. Department of Computation and Neural Systems, California Institute of Technology, Pasadena, CA, 91125, USA

    David Yu Zhang & Erik Winfree

  3. Department of Computer Science, California Institute of Technology, Pasadena, CA, 91125, USA

    Erik Winfree

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  1. Kenichi Fujibayashi
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  4. Satoshi Murata
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Correspondence to Satoshi Murata.

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Fujibayashi, K., Zhang, D.Y., Winfree, E. et al. Error suppression mechanisms for DNA tile self-assembly and their simulation. Nat Comput 8, 589–612 (2009). https://doi.org/10.1007/s11047-008-9093-9

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