Young Bok Kim, Yong-Bin Kim, and F. Lombardi- Error Tolerant DNA Self-Assembly by Link- Fracturing

8/3/2019 Young Bok Kim, Yong-Bin Kim, and F. Lombardi- Error Tolerant DNA Self-Assembly by Link- Fracturing

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Error Tolerant DNA Self-Assembly by Link-

Fracturing

Young Bok Kim, Yong-Bin Kim, and F. Lombardi

Dept of Electrical and Computer Engineering Northeastern University

Boston, USA

{youngbok,ybk,lombardi}@ece.neu.edu

Abstract —This paper proposes and evaluates link fracturing as

an approach for error tolerance in self-assembly by utilizing a

DNA chain as a link between two blocks of molecules. Through

the use of restriction enzymes, link fracturing breaks the

connecting DNA chain between two blocks if an incorrect

assembly has occurred due to the erroneous growth of tiles. Two

error tolerant techniques are proposed by fracturing of the DNA

chain links, namely 1-link and 2-link. Using the tool Xgrow,

simulations under the Kinetic Tile Assembly Model (KTAM) areperformed. Results show that 2-link fracturing achieves an

improvement in error rate as compared to a normal assembly;

moreover this is accomplished with little overhead in assembly

size and execution complexity. The 1-link method shows 100%

error free growth with moderate overhead as compared to

normal growth and other existing error tolerant methods.

Keywords-component; error tolerance; defects ;

manufacturing; DNA; self-assembly

I. I NTRODUCTION

Bio-molecular computing is an emerging field in whichmolecules such as DNA can be utilized to perform algorithmiccomputation as well as for manufacturing highly densescaffolds [1] [14] [11] [15]. The technique of DNA self-assembly is of particular interest; DNA self-assembly allowsfor programmable construction of complex patterns (inclusiveof circuits) at a molecular scale to develop systems capable of algorithmic computation. Sef-assembly proceeds by utilizingtiles with no control in the aggregation process. Aggregates areformed by placing tiles next to each other with matching labels.A tile will ”stick” to the assembly if a sufficient number of matches occur .

However, self-assemblies on a molecular scale are prone tohigh error rates (in the range of 1 to 10 percent) [8] [7]; thisseverely restricts highly complex or large-scale crystalstructures from being developed using millions of molecules[12] [10] [11]. Many techniques have been proposed in thedevelopment of error tolerance to reduce error rates [18] [17][6] [16] [13]. These techniques utilize tile sets that prevent the

permanent association of erroneous tiles (as basic building block in the self-assembly process); so, massive redundancy iswidely utilized. Growth proceeds in a very controlled fashionsuch that incorrect tiles can drop off by dis-association. Two of such error tolerant techniques are K × K Proofreading [6]andKxK Snake Proofreading tile sets [13]. Both of thesetechniques rely on controlling the growth and construction of

the crystal to reduce the error rate. Effectively each single tilein a normal growth process is replaced by a set of K2 tiles that

provide a level of redundancy to reduce the error rate. Theobjective of this paper is to present and assess a new techniquereferred to as link fracturing for error tolerance in DNA self-assembly. Differently from previous techniques, link fracturing

breaks the connecting DNA chain between two blocks in the presence of an erroneous growth. This process relies on

restriction enzymes to break at least one of the two links present in the DNA. Therefore, link fracturing can be thoughtas equivalent to a repair process as commonly employed inVLSI systems for yield enhancement at manufacturing. The

proposed technique does not utilize tile redundancy for achieving error tolerance, but rather it employs correctiveactions based on endonucleases. The performance of the

proposed techniques are compared versus that of 3× 3Proofreading and 3 × 3 Snake Proofreading. This paper isorganized as follows. Section 2 introduces link fracturing as

proposed technique for error tolerance; the mechanism andsample operation of the 1-link and 2-link fracturingmethodologies are presented. The evaluation and simulation

results are provided in Section 3 in which the proposedmethods are compared versus existing techniques. Section 4concludes the paper.

II. LINK FRACTURING

An alternative to adding block redundancy for error tolerance is checking and correcting the aggregate [18] [17][12]. Rather than reducing the occurrence of an error, link fracturing fixes errors as they occur. This can be thought asequivalent to a repair process (as commonly employed in VLSIfor yield improvement). The basic premise of link fracturing isto detect a mismatch and cut the bonds between the correct andthe erroneous tiles. As shown in Figure 1, the bond connecting

any two tiles is replaced by a DNA wire. Each unique matchhas its own unique DNA wire attaching the blocks (as denoted

by different DNA patterns). Previous work has shown thatthese wires can still remain as active DNA molecules evenwhen coated with nano-particles to act as wires [2]. Thesewires can be cut using a process known as restrictionendonucleases [2]. Currently, over 200 such enzymes have

been discovered [3] and each can cut a unique pattern of DNA[4]. An example is shown in Figure 2 Using this method for detecting and cutting DNA wires, it is possible to achieve error checking and correction for molecular self-assembly.



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Figure 1. DNA wire links

Figure 1. Example of a figure caption. (figure caption)

Each of the tiles can be designed such that a mismatchresults in a DNA wire with a known pattern that can be cutwith a known restriction enzyme. Two methods (namely 1- and2-link fracturing) are detailed in the next sections.

Figure 2. Cutting a DNA chain with restriction enzymes

A. 2-Link Fracturing

In KTAM, growth occurs by links formed on the bottomand right sides of a tile. For this reason, two different link

fracturing models are proposed in this paper. In the first one,called 2-link, both the bottom and right bonds would have to bemismatched for the error block to be fractured. Also includedin 2-link fracturing is the scenario in which a block is attached

by only a single link and that link is a mismatch. The statediagram of Figure 3 shows the transitions for all states..

Figure 3. Tile Insertion State Diagram (Incorrect state means two mismatches,

Almost correct means only one mismatch) [19]

In this diagram ”frozen” refers to the permanent attachmentof a tile in the self-assembly. Under the 2-link method,whenever a tile is in the incorrect state, its DNA wires will becut and it will regress to the empty state. Thus, the probabilityto transition from the incorrect state to the frozen incorrect

state is reduced to zero. The only way for an error to occur under this method would be if an almost correct (error-free) tiletransitions to the frozen incorrect state. The Xgrow simulationof the 2-link fracturing method is presented in Figure 4: anerroneous tile with two mismatches is attached to the growthand is fixed after approximately 100 events.

Figure 4. Xgrow Simulation for 2-link Fracturing: Error detection and Fixing.

B. 1-Link Fracturing

The second method, that is mostly an extension of the 2-link, is the 1-link fracturing error correction. In this method, if

either the bottom or right DNA wires are mismatches, then theentire tile is disassociated. The 2-link method ensured that anytile in the incorrect state does not become permanent. With the1-link method, both the incorrect and almost correct statesreturn to the empty state. Only tile placements that are correctwill enter a permanent frozen state, that gives 100%correctness in assembly. However, there is a concern with thismethod. While it is possible to detect and detach a bad DNAwire with restriction enzymes [3], there is no provenmechanism to simultaneously detach both a bad link and agood link based on proximity. Some mechanisms are currently




in the very early stage of investigation to essentially connectthe bottom and right faces of the tile so that an effect on one of the links affects both [4]. Simulations and results of the 1-link will be shown to prove the possible effectiveness of link fracturing in DNA self-assembly. The following example takenfrom Xgrow shows how these errors are fixed. Figure 5 (left)shows a pair of errors resulting from 1-link mismatches. Tilesare then removed from those positions. However, those slots

are surrounded by other tiles. As the simulation continues, theempty slots are replaced with correct tiles that create further mismatches as shown in Figure 5 (right). Correspondingly,these errors are also removed and fixed. This continued

progression not only detects and fixes mismatches due toerroneous tiles, but it correctly reassembles the aggregate, thusachieving 100% correctness in the self-assembly process.

Figure 5 Xgrow Simulation for 1-link Fracturing, Initial Errors, Propagation

III. EVALUATION AND SIMULATION R ESULTS

The simulation model used in this paper is KTAM in whichaggregation is governed by two dimensionless free energies:Gmc as the entropic energy cost of attaching a tile and Gse asthe free energy cost of breaking a single bond [19]. The Xgrowsoftware [5] is used for simulation. The tile set utilized for thisstudy is the Sierpinski self-assembly growth set [7] ascommonly used benchmark due to its very complex growth

pattern (as available from previous studies). To simulate DNA

fracturing, some modifications were made to Xgrow. In eachsimulation run, the entire growth region is scanned for errors. If a mismatch is detected between the south or east connections,the tile are removed from the growth and that position is set toempty. Each simulation run typically has fifteen to thirty events(an event is defined as an action occurring such as the additionor removal of a tile). Simulations of each error toleranttechnique are run twenty times with each simulation running

until the growth has 15,000 tiles. The metrics for comparisonfrom the simulation are the number of errors and the number of events. As measure of execution complexity, the count of thenumber of events gives an approximation of the time overheadrequired for implementing the error tolerant technique.

TABLE I. Error tolerance of various techniques for Sierpinski growth

method

G mc Normal 2-Link 1-Link 3X3 Snake 3×3

19.7 2.7% 2.9% 0% 0% 0%

19.4 3.55% 3.7% 0% 0.1% 0%

19.1 8.15% 6.9% 0% 0.5% 0.1%

18.7 11.65% 10.05% 0% 2.85% 0.4%

18.4 16.65% 15.2% 0% 7.05% 0.7%

TABLE II Event Overhead for Error Tolerant Techniques

G mc Normal 2-Link 1-Link 3X3 Snake 3×3

19.7 4.9e+6 4.9e+6 5.2e+6 3.2e+7 7.6e+7

19.4 2.8e+6 2.8e+6 3.4e+6 1.8e+7 3.9e+7

19.1 2.2e+6 2.2e+6 3.8e+6 1.5e+7 2.9e+7

18.7 1.7e+6 1.8e+6 3.7e+6 1.3e+7 2.8e+7

18.4 1.6e+6 1.6e+6 3.2e+6 1.6e+7 2.4e+7

The results of the simulations are reported in Table 1 andTable 2. For each error tolerant technique twenty simulationswere run by varying the values of Gmc with the value of Gsefixed to 10.0. In most cases, the results show error rates lower for the error tolerant methods [13] [6] compared to a normalgrowth. The 1-link method performs as expected and ranwithout errors for even low values of Gmc . The performance of the 3 x 3 snake tile set is very close to error-free with less than1 error occurring on average under even non ideal conditions.The most interesting results came from the 2-link method, inwhich for close to ideal values of Gmc (optimally Gmc =2Gse-log2) error rates are higher than for normal growth. The reasonfor this is likely due to more events occurring, thus causing thenumber of errors with only one bad link to increase. For nonideal conditions (and higher speed of aggregation) however,the 2-link method performs better and has a lower error ratethan normal growth. Figure 6 shows the comparison of error rates for the different Gmc values of each of the error toleranttechniques. In addition to comparing the error rate, theexecution overhead for each error tolerant technique was also




measured. As mentioned earlier the overhead was measured asthe number of events that took place for 15,000 tiles toassemble.

Figure 6. Xgrow Simulation for 1-link Fracturing, Initial Errors,Propagation

As shown in Figure 7, the proofreading error tolerant

techniques add a significant overhead. For the 3 x 3 proofreading and snake proofreading, the overhead is an order of magnitude greater than the normal Sierpinski growth.

The link fracturing techniques however, have an almostnegligible overhead. For a non ideal Gmc , the 1-link methodhas an increased overhead due to the high error correction thatoccurs. As Gmc moves away from the ideal value, the number of events decreases due to fewer constraints in the amount of energy it takes to add versus the energy to remove a tile.

IV. CONCLUSION

This paper has proposed link fracturing as an approach for error tolerance in DNA self-assembly. It relies on fracturing aDNA chain as a link between two blocks of molecules; through

the use of restriction enzymes. The connecting DNA chain is broken between two blocks if an incorrect assembly occurs dueto the erroneous growth of tiles. Therefore, link fracturing can

be through as equivalent to a repair process as commonlyemployed in VLSI systems for yield enhancement atmanufacturing. Two error tolerant techniques have been

proposed for fracturing the DNA chain links, namely 1-link and 2-link. Using the tool Xgrow, simulations using the kinetictile assembly model (KTAM) have been performed. Results

have shown that for the Sierpinski triangle growth, 2-link fracturing achieves an improvement in error rate as comparedto a normal assembly with little overhead in executioncomplexity. The 1-link method shows 100% error free growthwith moderate overhead as compared to normal growth andother existing error tolerant methods.

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Young Bok Kim, Yong-Bin Kim, and F. Lombardi- Error Tolerant DNA Self-Assembly by Link- Fracturing