Compact Error-Resilient Computational DNA Tiling Assemblies

1. Compact Error-Resilient Computational DNA Tiling Assemblies John H.Reif, Sudheer Sahu, and Peng Yin Presenter: Seok, Ho-SIK

2. How to decrease assembling error? • First approach • Optimizing the physical environment, improving the design of the tile itself or using novel material They admit that DNA is not suitable for constructing tiles • Second approach • Design of new tile sets that can reduce the total number of errors in the final structure even with the same intrinsic error rate

3. Winfree’s proofreading - 1 • The basic idea • To exploit cooperative binding at the next higher level: to have several tiles that stabilize each other when they bind together • Essentially, each rule tile in the original tile set is replaced by four tiles with related labels • According to the aTAM, assembly from the seed tile proceeds according to the same logic as the original tile set, but scaled up in size by a factor of two

4. *Kinetic trapping • The essential feature of kinetic trapping within tile self-assembly is that once an error has occurred, both sites above the mismatched tile display an (x, y) pair that is perfectly matched by some monomer tile in solution, because tiles implementing all replacement rules are present. Thus, if such a tile arrives before the mismatched tile dissociates, the mismatched tile becomes locked in by multiple bonds and is now unlikely to dissociate (Winfree)

5. Winfree’s proofreading - 2 • If a mismatch occurs, the assembly process stalls, giving time for the initial mismatched tile to fall off and be replaced by a correct tile They want to escape the kinetic trap If there is a tile structure with desired size, then an experimenter can guarantee that the structure is the wanted one

6. Errors in assemblies • The pad mismatch rate ε • Critical issue in 2D tiling assemblies • It determines the size of the error-free assembly • Key challenge in experimentally demonstrating large scale assemblies is to construct error-resilient tiles Winfree’s approach resulted in a final structure that is four times the size of the original one Winfree wanted to inhibit occurrence of an incorrectly matched tile structure

7. Assembly with no error corrections • NXM array • V(i,j) • The value of i-th bit on the j-th row displayed at position (i.j) and communicated to the position (i,j+1) • V(i,j)= U(i-1,j) OP1 V(i,j-1) • U(i,j) • Boolean value communicated to the position (i+1,j) • U(i,j)= U(i-1,j) OP2 V(i,j-1) U(i,j) V(i,j) V(i,j-1) U(i-1,j) • Bottom pad: V(i,j-1) • Right pad: U(i-1, j) • Top pad: V(i,j) computed by V(i,j-1) OP1 U(i-1,j) • Left pad: U(i,j) computed by V(i,j-1) OP2 U(i-1,j)

8. Error-resilient assembly using two-way overlay redundancy • Error-Resilient Assembly I • Proposed architecture • This architecture drops the probability of a tile assembly error rate to 6ε2 • Basic idea • Two way overlay redundancy • Each tile T1(i, j) computes the outputs for its own position (i, j) and also for its right neighbor’s position (i-1, j) • The redundant computation results obtained by T1(i, j) and its right neighbor T1(i-1, j) is compared via an additional error checking portion on T1(i, j)’s right pad • If only one of T1(i, j) and T1(i-1, j) is in error, the kinetics of the assembly may allow for the incorrectly placed tile to be ejected from the assembly

9. Error checking portion U(i-1,j)=U(i-2,j)OP2V(i-1,j-1) But, V(i, j)=U(i-1,j)OP1V(i,j-1) This tile is a sequential machine! Construction • Note • The bottom portion of the right pad represents the value of V(i-1, j) • The value V(i-1, j) is redundantly determined by T1(i-1, j) and hence the bottom portion performs comparison of the two values and is referred to as error checking portion • The value is communicated to tile T1(i, j) from its immediate right neighbor T1(i-1, j) • The determined value is displayed by the tile T1(i, j) using an extruding ssDNA • We emphasize that through a pad has two portions, it should be treated as a whole unit. A value change in one portion of a pad changes the pad to a completely new pad V(i,j), V(i-1,j) U(i-2,j), V(i-1,j) U(i-1,j), V(i,j) V(i,j-1), V(i-1,j-1)

10. Assumptions of the construction • We can glue a pad selectively on a DNA tile after some internal calculations • We or a nice guy could found a method for pushing sequential machine into a DNA tile • We or a genius could invent a method capable of kicking off mismatched sequences after examining content the sequences • There would be a way delivering information without annealing • Or we escape the limit of DNA, so do not harass us with the notion of DNA! We can use some nice material and we no longer use the double helix structure  We are talking about some nice kind of nano machine not damn it DNA machine!!!

11. Error analysis - 1 • Our intention is that the individual tiling assembly error rate is substantially decreased, due to cooperative assembly of neighboring tiles, which redundantly compute the values at their positions and at their right neighbors • We consider only the cases where the pad binding error occurs on either the bottom pad or the right pad of a tile T1(i, j)

12. Information Some kind of linker of fastener Information Error analysis - 2 • Lemma • Suppose that the neighborhood tiles independent of tile T1(i,j) have correctly computed V, U. If there is a single pad mismatch between tile T1(i,j) or to its just below or immediate right, then there is at least one further pad mismatch in the neighborhood of tile T1(i,j). Furthermore, given the location of the initial mismatch, the location of the further pad mismatch can be determined among at most three possible pad locations • Suspicion • They have invented some kind of DNA fastener so there would be no need for annealing between information portion

13. Error checking portion Error analysis – 3 (Proof) • Error occurs on the bottom pad of tile T1(i, j) 1.1 Error is due to the incorrect value of the right portion V(i-1, j-1) of the bottom pad of tile T1(i, j) In this case, T1(i, j) will compute incorrect value for rightV(i-1, j)(V(i-1, j)=U(i-2,j) OP2 V(i-1, j-1)), which is distinct from the correct value of V(i-1, j) determined by its right neighbor tile T1(i-1, j) V(i,j), V(i-1,j) U(i-2,j), V(i-1,j) U(i-1, j) V(i, j) 1.2 Error is due to the incorrect value of the left portion V(i, j-1) of the bottom pad of tile T1(i, j) In this case, T1(i, j) will compute incorrect value for V(i, j). So there must be a further mismatch at tile T1(i+1, j). If there is no mismatch, then there is a mismatch between T1(i+1, j) and T1(i+1, j-1) V(i,j-1), V(i-1,j-1) Their mismatch may mean a mismatch between a pad and a tile If a tile itself is mismatched, then correct value of U(i-2., j) can not be delivered Or we can believe that tile knows correct value of its neighbor tiles before computation

14. Error checking portion Error analysis – 4 (Proof) 2. Error occurs on the right pad of tile T1(i, j) 2.1 The error occurs on the U(i-2, j) portion of the right pad of tile T1(i, j) V(i,j), V(i-1,j) In this case, T1(i, j) will compute incorrect value for top V(i-1, j) So there must be a further pad mismatch at tile T1(i, j+1) (Because, V(i-1, j+1)=u(i-2, j+1) OP2 V(i-1, j)) U(i-2,j), V(i-1,j) U(i-1, j) V(i, j) Given the location of the initial mismatch, the location of the further pad mismatch can be determined among at most three possible pad locations V(i,j-1), V(i-1,j-1)

15. Error analysis - 5 • The probability of a single pad mismatch between adjacent assembling tiles The probability that there is no pad mismatch between tile T1 and another tile just below or to its immediate right is (1- )2 Hence the probability that there is a pad mismatch between tile T1 and another tile just below or to its immediate rightis 1-(1- )2 =2- 2, which is at most 2 • The probability that there is such a further pad mismatch between tiles at most three possible pad locations is at most 1-(1- )3, which is at most 3. • This implies that with probability (2)(3)=62, there is both (i) a pad mismatch between tile T1 and another tile just below or to its immediate right; and (ii) there is also a further pad mismatch between tiles in the immediate neighborhood of tile T1

16. *Suspicion • However in their tile structure, the above and left tiles do depend on their just below and immediately right tile. Therefore, it is impossible for perfectly matched tile to arrive above mismatched tile • Or tiles stick to some frame and only pads are floating around • In this model, there is no need for distinction between independent and dependent tiles • If distinction is meaningful, then mismatch will spread • If distinction is meaningless, then their computation is just kidding

17. Simulation • Using simulation, our version 1(T1) is comparable to winfree’s proofreading tile set constructions, while our version 2 (T2) outperforms both of them

18. Comparison to Winfree • Winfree’s method can inhibit the spreading of incorrect mismatches • With their notion of dependency, they (Reif & his followers) do not show the method for spreading inhibition • Their 62 was computed with the kinetic trapping • In Winfree’s method, structuring is computation • They assume some internal computation • Their model should implement sequential machine, comparator, and a method for communication. Therefore, their claim that their structure is less than Winfree’s one is meaningless