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Monomer Control for Error Tolerance in DNA Self-Assembly

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Abstract

This paper proposes the control of monomer concentration as a novel improvement of the kinetic Tile Assembly Model (kTAM) to reduce the error rate in DNA self-assembly. Tolerance to errors in this process is very important for manufacturing scaffolds for highly dense ICs; the proposed technique significantly decreases error rates (i.e. it increases error tolerance) by controlling the concentration of the monomers (tiles) for a specific pattern to be assembled. By profiling, this feature is shown to be applicable to different tile sets. A stochastic analysis based on a new state model is presented. The analysis is extended to the cases of single, double and triple bondings. The kinetic trap model is modified to account for the different monomer concentrations. Different scenarios (such as dynamic and adaptive) for monomer control are proposed: in the dynamic (adaptive) control case, the concentration of each tile is assessed based on the current (average) demand during growth as found by profiling the pattern. Significant error rate reductions are found by evaluating the proposed schemes compared to a scheme with constant concentration. One of the significant advantages of the proposed schemes is that it doesn’t entail an overhead such as increase in size and a slow growth, while still achieving a significant reduction in error rate. Simulation results are provided.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Northeastern University, Boston, MA, 02115, USA

    Byunghyun Jang, Yong-Bin Kim & Fabrizio Lombardi

Authors
  1. Byunghyun Jang
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  2. Yong-Bin Kim
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  3. Fabrizio Lombardi
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Correspondence to Byunghyun Jang.

Additional information

Responsible Editor: N. A. Touba

This manuscript is an extended version of a paper presented at the 21th IEEE International Symposium on Defect and Fault Tolerance in VLSI Systems, Oct 2006.

Appendices

Appendices

Single bonding: The single bonding case in monomer concentration control is described by the following rate equations.

$$\mathbf{\stackrel{\boldsymbol{.}}{p}_{\{AA,AB\}}}(t)=\left[\begin{array}{*{20}c}r^{1} &\kern4pt r_{r,1} &\kern4pt r_{r,0} &\kern4pt 0 &\kern4pt 0 \\r^{4} &\kern4pt r^{2} &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 \\5r_{f,y} &\kern4pt 0 &\kern4pt r^{3} &\kern4pt 0 &\kern4pt 0 \\0 &\kern4pt r^{*} &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 \\0 &\kern4pt 0 &\kern4pt r^{*} &\kern4pt 0 &\kern4pt 0\end{array}\right]\left[\begin{array}{*{20}c}p_{E}(t) \\ p_{C}(t) \\ p_{I}(t) \\ p_{FC}(t) \\ p_{FI}(t)\end{array}\right]$$
$${\kern46.5pt} \doteq\mathbf{Mp_{\{AA,AB\}}}(t) $$

where \(r^{1}=-(r_{f,x}+6r_{f,y})\), \(r^{2}=-(r_{r,1}+r^{*}),\ r^{3}=\) \(-(r_{r,0}+r^{*})\), \(r^{4}=r_{f,x}+r_{f,y}\) and

$$\mathbf{\stackrel{\boldsymbol{.}}{p}_{\{BA,BB\}}}(t)=\left[\begin{array}{*{20}c}r^{1} &\kern4pt r_{r,1} &\kern4pt r_{r,0} &\kern4pt 0 &\kern4pt 0 \\2r_{f,y} &\kern4pt r^{2} &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 \\r^{4} &\kern4pt 0 &\kern4pt r^{3} &\kern4pt 0 &\kern4pt 0 \\0 &\kern4pt r^{*} &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 \\0 &\kern4pt 0 &\kern4pt r^{*} &\kern4pt 0 &\kern4pt 0\end{array}\right]\left[\begin{array}{*{20}c}p_{E}(t) \\ p_{C}(t) \\ p_{I}(t) \\ p_{FC}(t) \\ p_{FI}(t)\end{array}\right]$$
$${\kern45pt} \doteq\mathbf{Mp_{\{BA,BB\}}}(t) $$

where \(r^{1}=-(r_{f,x}+6r_{f,y})\), \(r^{2}=-(r_{r,1}+r^{*}),\ r^{3}=\) \(-(r_{r,0}+r^{*})\), \(r^{4}=r_{f,x}+4r_{f,y}\).

Triple bonding: The rate equations for this bonding case are given below.

$$\mathbf{\stackrel{\boldsymbol{.}}{p}_{AA}}(t)=\left[\begin{array}{*{20}c}r^{1} &\kern4pt r_{r,3} &\kern4pt r_{r,2} &\kern4pt r_{r,1} &\kern4pt r_{r,0} &\kern4pt 0 &\kern4pt 0 \\r_{f,x} &\kern4pt r^{2} &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 &\kern4pt 0\\r_{f,y} &\kern4pt 0 &\kern4pt r^{3} &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 &\kern4pt 0\\2r_{f,y} &\kern4pt 0 &\kern4pt 0 &\kern4pt r^{4} &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 \\3r_{f,y} &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 &\kern4pt r^{5} &\kern4pt 0 &\kern4pt 0 \\0 &\kern4pt r^{*} &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 &\kern4pt 0\\0 &\kern4pt 0 &\kern4pt r^{*} &\kern4pt r^{*} &\kern4pt r^{*} &\kern4pt 0 &\kern4pt 0\end{array}\right]\left[\begin{array}{*{20}c}p_{E}(t) \\ p_{CCC}(t) \\ p_{CCI}(t) \\ p_{ICI}(t) \\ p_{III}(t) \\ p_{FC}(t) \\p_{FI}(t)\end{array}\right]$$
$${\kern28.5pt}\doteq\mathbf{Mp_{AA}}(t) $$

where \(r^{1}=-(r_{f,x}+6r_{f,y})\), \(r^{2}=-(r_{r,3}+r^{*}),\ r^{3}=\) \(-(r_{r,2}+r^{*})\), \(r^{4}=-(r_{r,1}+r^{*})\), \(r^{5}=-(r_{r,0}+r^{*})\) and

$$\mathbf{\stackrel{\boldsymbol{.}}{p}_{\{AB,BA\}}}(t)=\left[\begin{array}{*{20}c}r^{1} &\kern4pt r_{r,3} &\kern4pt r_{r,1} &\kern4pt r_{r,0} &\kern4pt 0 &\kern4pt 0 \\r_{f,y} &\kern4pt r^{2} &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 &\kern4pt 0\\r^{5} &\kern4pt 0 &\kern4pt r^{3} &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 \\2r_{f,y} &\kern4pt 0 &\kern4pt 0 &\kern4pt r^{4} &\kern4pt 0 &\kern4pt 0 \\0 &\kern4pt r^{*} &\kern4pt 0 &\kern4pt 0 &\kern4pt 0 &\kern4pt 0\\0 &\kern4pt 0 &\kern4pt r^{*} &\kern4pt r^{*} &\kern4pt 0 &\kern4pt 0\end{array}\right]\left[\begin{array}{*{20}c}p_{E}(t) \\ p_{CCC}(t) \\ p_{ICI}(t) \\ p_{III}(t) \\ p_{FC}(t) \\p_{FI}(t)\end{array}\right]$$
$${\kern45.2pt}\doteq\mathbf{Mp_{\{AB,BA\}}}(t) $$

where \(r^{1}=-(r_{f,x}+6r_{f,y})\), \(r^{2}=-(r_{r,3}+r^{*}),\ r^{3}=\) \(-(r_{r,1}+r^{*})\), \(r^{4}=-(r_{r,0}+r^{*})\), \(r^{5}=r_{f,x}+3r_{f,y}\) and

$$\mathbf{\stackrel{\boldsymbol{.}}{p}_{BB}}(t)=\left[\begin{array}{*{20}c}r^{1} &\kern4pt r_{r,3} & \kern4ptr_{r,1} & \kern4ptr_{r,0} & \kern4pt0 & \kern4pt0 \\r_{f,y} & \kern4ptr^{2} & \kern4pt0 & \kern4pt0 & \kern4pt0 & \kern4pt0\\r^{5} & \kern4pt0 & \kern4ptr^{3} & \kern4pt0 & \kern4pt0 & \kern4pt0 \\3r_{f,y} & \kern4pt0 & \kern4pt0 & \kern4ptr^{4} & \kern4pt0 & \kern4pt0 \\0 &\kern4pt r^{*} & \kern4pt0 & \kern4pt0 & \kern4pt0 & \kern4pt0\\0 & \kern4pt0 & \kern4ptr^{*} & \kern4ptr^{*} & \kern4pt0 & \kern4pt0\end{array}\right]\left[\begin{array}{*{20}c}p_{E}(t) \\ p_{CCC}(t) \\ p_{ICI}(t) \\ p_{III}(t) \\ p_{FC}(t) \\p_{FI}(t)\end{array}\right]$$
$${\kern28.2pt}\doteq\mathbf{Mp_{BB}}(t) $$

where \(r^{1}=-(r_{f,x}+6r_{f,y})\), \(r^{2}=-(r_{r,3}+r^{*}),\ r^{3}=\) \(-(r_{r,1}+r^{*})\), \(r^{4}=-(r_{r,0}+r^{*})\), \(r^{5}=r_{f,x}+2r_{f,y}\).

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Jang, B., Kim, YB. & Lombardi, F. Monomer Control for Error Tolerance in DNA Self-Assembly. J Electron Test 24, 271–284 (2008). https://doi.org/10.1007/s10836-007-5016-4

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