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Activatable tiles for compact robust programmable molecular assembly and other applications

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Abstract

Algorithmic DNA self-assembly is capable of forming complex patterns and shapes, that have been shown theoretically, and experimentally. Its experimental demonstrations, although improving over recent years, have been limited by significant assembly errors. Since 2003 there have been several designs of error-resilient tile sets but all of these existing error-resilient tile systems assumed directional growth of the tiling assembly. This is a very strong assumption because experiments show that tile self-assembly does not necessarily behave in such a fashion, since they may also grow in the reverse of the intended direction. The assumption of directional growth of the tiling assembly also underlies the growth model in theoretical assembly models such as the TAM. What is needed is a means for enforce this directionality constraint, which will allow us to reduce assembly errors. In this paper we describe a protection/deprotection strategy to strictly enforce the direction of tiling assembly growth so that the assembly process is robust against errors. Initially, we start with (1) a single “activated” tile with output pads that can bind with other tiles, along with (2) a set of “deactivated” tiles, meaning that the tile’s output pads are protected and cannot bind with other tiles. After other tiles bind to a “deactivated” tile’s input pads, the tile transitions to an active state and its output pads are exposed, allowing further growth. When these are activated in a desired order, we can enforce a directional assembly at the same scale as the original one. Such a system can be built with minimal modifications of existing DNA tile nanostructures. We propose a new type of tiles called activatable tiles and its role in compact proofreading. Activatable tiles can be thought of as a particular case of the more recent signal tile assembly model, where signals transmit binding/unbinding instructions across tiles on binding to one or more input sites. We describe abstract and kinetic models of activatable tile assembly and show that the error rate can be decreased significantly with respect to Winfree’s original kinetic tile assembly model without considerable decrease in assembly growth speed. We prove that an activatable tile set is an instance of a compact, error-resilient and self-healing tile-set. We describe a DNA design of activatable tiles and a mechanism of deprotection using DNA polymerization and strand displacement. We also perform detailed stepwise simulations using a DNA Tile simulator Xgrow, and show that the activatable tiles mechanism can reduce error rates in self assembly. We conclude with a brief discussion on some applications of activatable tiles beyond computational tiling, both as (1) a novel system for concentration of molecules, and (2) a catalyst in sequentially triggered chemical reactions .

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Notes

  1. In the TAM for temperature \(\tau =2\), a tile binds strongly either using at least one strong bond or two weak bonds.

  2. Growth error happens when a tile with one weak bond (weakly binding tile) attaches at a location where a tile with two weak bonds could have, and should have, been placed.

  3. A facet nucleation error happens when weakly binding tile attaches to a site where no tiles should attach at the moment.

  4. Spontaneous nucleation errors occur when a large assembly grows in absence of a seed tile.

  5. In the more complex version of activatable tile set, the seed is the initiator tile in the assembly experiments and consequently does not have any protection on any of its sides. Boundary tiles still need to be protected (Fig. 3d-ii, g-ii). The corresponding states are shown in Fig. 3e, h. Deprotection is simpler for boundary tiles. Since the bonds on the east side of the x-axis boundary tiles (\(B_x\)) and that on the south side of the y-axis boundary tiles (\(B_y\)) are strong bonds, matches on those pads can trigger the deprotection of the north and west pads for \(B_x\) as well as \(B_y\) (Fig. 3f, i).

  6. Dissociation from state S8 involves unbinding of 2 toeholds, and state S12 involves unbinding of 1 bound input pad and 1 toehold, since the toeholds have been assumed to be universal.

  7. The design of these tiles are such that the protection is part of the nanostructure until the outputs are deprotected.

  8. For the subsequent discussion, note that the backward rate is denoted by the negation of forward rate. For instance, if \(k_{poly}\) be the forward rate of association of the DNA polymerase enzyme to the primer, \(k_{-poly}\) denotes the rate of dissociation of the polymerase enzyme.

  9. Note that if the concentration of a growing assembly front at any instant is [GS], and the primer concentration is [P], then the reaction rate is \(k_f[GS][P]\). Likewise, the rate of monomer tile attachment is \(k_f[GS][T]\), where [T] is the tile concentration. By keeping the concentration of primer and tiles the same, the reaction rate \(r_f\) can be assumed to be the same for both these reactions. Also, the dehybridization rate constant \(r_{r,1}\) is kept the same by keeping the primer length the same as the sticky ends. Since a primary assumption of the kTAM is that monomer tile concentrations are kept constant, similarly primer concentrations are also kept constant.

  10. The polymerase extension of the primer hybridized to the protection strand, is modeled as an irreversible atomic process for simplicity. If the exonuclease activity during polymerization occurs at a reasonably low rate, then the assumption is quite justified.

  11. The rate constants \(k_{sd1}\) and \(k_{sd3}\) each correspond to the reaction rate \(r_{dp}\), while the rate constants \(k_{-sd1}\) and \(k_{-sd3}\) each correspond to \(r_p\). The rate constants \(k_{poly}\) and \(k_{ext}\) together contribute to the rate \(r_{dp\_out}\). The rate constant \(k_{sd2}\) is involved in the deprotection of the second input pad by the first activated input pad, and has not been modelled in Fig. 5.

  12. A growth site can only be frozen if the output pads of the tile sitting in that growth site are available for binding. Hence the transitions to FC and FI are only from S6, S8 and S12 and not from S5 or S11.

  13. Another way of deprotected tiles forming is if a protected monomer tile and primer bind with a small chance, and the polymerase extends the primer deprotecting the tile. An explanation of this is given in a following note 2.4.1.

  14. This is a loose upper bound because at \(t\rightarrow \infty\), most growth sites are frozen and the tiles in those growth sites cannot leave and contribute to the expected number of tiles leaving S6. The concentration of completely deprotected tiles, however, is maximum at steady state.

  15. Out of \(r_{\textit{eff}}\) tiles reaching state S6 in unit time, only \(r_{r,2}\) leave. Only those tiles which leave the growth site after complete deprotection can come back to any growth site with a rate constant of \(k_f\). Hence if \(r_{\textit{eff}} \gg r_{r,2}\) then \(r_{f}^{\prime}\) should be much less than \(r_{\textit{eff}}\) and we can safely neglect the contribution from \(r_{f}^{\prime}\) in the denominator \(r^{*}+r_{r,2}\) while computing the value of \(r_{f}^{\prime}\).

  16. Controlled growth is defined to be the growth occurring for parameter values in a certain part of the kinetic parameter space, such that (1) growth does occur, (2) errors are rare and (3) growth not seeded by the seed tile is rare (Winfree 1998).

  17. The time taken for single tile attachment is \(O(\frac{1}{r_{\textit{eff}}})\) which is less than \(\frac{1}{r_{insuf}}\).

  18. The stochastic process of tile attachment and detachment in self -assembly has often been modeled as a random walk (Chen and Goel 2004). Further this is similar to the lattice gas model where modeling interactions as random walks is quite well established.

  19. Note that this is different from the assumption in Sect. 2.3, which assumes that the primer concentration is maintained equal to the monomer tile concentration at all times. In simulations with very low concentrations of the monomer tile, if the primer concentration is also very low, then the primer binding reaction becomes the rate limiting step, and the simulation does not proceed. In order to avoid this, the primer was kept constant at a (relatively high) experimentally realizable concentration of 10 \(\upmu\)M. Thus, \(r_{primer}=k_f\) * [P] = 30 \(\text{s}^{-1}\).

  20. The branch migration rate constant \(k_b\) (Zhang and Winfree 2009), is given by \(k_b\) = \(\frac{400}{x^2} s^{-1}\). In our case, \(x = 10\).

Abbreviations

PCR:

Polymerase chain reaction

DNA:

DeoxyriboNucleic acid

ds-DNA:

Double stranded DeoxyriboNucleic acid

TAM:

Tile assembly model

aTAM:

Abstract tile assembly model

kTAM:

Kinetic tile assembly model

aATAM:

Abstract activatable tile assembly model

kATAM:

Kinetic activatable tile assembly model

LTM:

Layered tile mechanism

PTM:

Protected tile mechanism

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Acknowledgments

The authors thank the anonymous referees, whose suggestions have had a noticeable improvement on the quality of the article, in helping us better articulate ideas and by giving attention to detail. The work was supported by NSF Grants CCF-1217457, CCF-1141847, CCF-0829797,  CCF-1320360.

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Authors and Affiliations

  1. Oracle Corporation, Washington, DC, USA

    Urmi Majumder

  2. Department of Computer Science, Duke University, Durham, NC, USA

    Sudhanshu Garg & John H. Reif

  3. Materials Science and Engineering Department at NC State, Raleigh, NC, USA

    Thomas H. LaBean

Authors
  1. Urmi Majumder
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  2. Sudhanshu Garg
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  3. Thomas H. LaBean
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  4. John H. Reif
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Correspondence to John H. Reif.

Additional information

This paper is a revised version of the conference proceedings extended abstract: Urmi Majumder, Thomas H. LaBean, and John H. Reif, Activatable Tiles: Compact, Robust Programmable Assembly and Other Applications, in DNA Computing: DNA13.

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Majumder, U., Garg, S., LaBean, T.H. et al. Activatable tiles for compact robust programmable molecular assembly and other applications. Nat Comput 15, 611–634 (2016). https://doi.org/10.1007/s11047-015-9532-3

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